M.W.L.M. Rijnen D&C

Transcription

1 A numerical and experimental study on passive damping of a 3D structure using viscoelastic materials M.W.L.M. Rijnen D&C 24.8 Master s thesis Coaches: F. Pasteuning, MSc dr. ir. R.H.B. Fey 2 dr. ir. G. van Schothorst Supervisor: prof. dr. H. Nijmeijer 2 Committee: prof. dr. H. Nijmeijer 2 dr. ir. R.H.B. Fey 2 dr. ir. L.E. Govaert 3 F. Pasteuning, MSc Philips Innovation Services Mechatronics Technologies 2 Eindhoven University of Technology Department of Mechanical Engineering Dynamics & Control 3 Eindhoven University of Technology Department of Mechanical Engineering Polymer Technology Eindhoven, May, 24

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3 Abstract Structural vibrations often have unwanted consequences. They can result in noise, decreased performance, and control limitations in high precision instruments. One way to reduce the effects of these vibrations is to create a lighter and stiffer structure, which increases the natural frequencies. However, this has its limitations. Another (complementary) approach is to introduce (semi-) active or passive damping to the structure. Since active damping methods require power and are generally more complex and costly, passive approaches often are preferred. Passive damping can be introduced in many ways. This study focuses on passive damping through the application of engineering materials with viscoelastic properties. There are many ways of using viscoelastic materials (VEM) for damping. In this study, first, the effectiveness of several VEM application configurations is compared for a simple beam structure by looking at the influence on the amount of damping, impact on volume requirements, damped eigenfrequencies, and vibration amplitudes. Subsequently, based on the results of this initial study, two basic methods are selected and used to efficiently damp a more complex 3D structure: an open aluminum box representing a structural component of a high precision instrument. Discrete damping elements and three constrained layer configurations are considered. The dynamics of the box structure including VEM components are simulated with a finite element (FE) model that incorporates the VEM s frequency dependent complex Young s modulus. The VEM model is obtained for several materials through a Dynamic Mechanical Thermal Analysis (DMTA). The FE model is used together with the findings of the initial effectiveness study to find a damping solution that optimizes damping of the box, while taking into account design constraints on mass and volume. Damping ratios in the order of five percent and higher are attained between the real part and the absolute value of the eigenvalues. Validation of the simulation results is done by comparison of both modal parameters and transfer functions with results obtained from experiments on the real structure. For the computation of transfer functions from model simulations, a (new) method based on modal superposition is proposed, where the frequency dependency of the VEM properties is incorporated by means of frequency interpolation of modal parameters. The model and experiments show good resemblance, even without a model updating step or fine-tuning. In conclusion, for effective machine performance improvement through the reduction of vibrations, an experimentally validated FE modeling approach is developed for evaluating viscoelastic damping design. i

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5 Acknowledgements The realization of this master project would not have been possible without the support of a great number of people, whose contribution to the project deserves mention. I would like to offer my special thanks to Frank Pasteuning, MSc for his valuable advice, guidance and supervision to this research work. His help with all aspects of this study, from modeling to experiments and from insightful discussions to giving critical comments on the report, is much appreciated. I would also like to thank dr. ir. Rob Fey for his input to the project. Inspiring discussions with him, his knowledge on the subject and his pain-staking effort in proof reading the draft, to a great extend, helped form the result of this study. Advice and supervision given by prof. dr. Henk Nijmeijer and dr. ir. Gert van Schothorst and their constructive recommendations on this project has been a great help and are much appreciated. I am furthermore grateful that prof. dr. Henk Nijmeijer and dr. ir. Rob Fey brought this research opportunity to my attention. My grateful thanks are also extended to the employees at Philips Innovation Services, whose advice, general input and help with practical aspects of the study have been invaluable for the project completion. Finally, I wish to thank my family for their support and encouragement throughout my study. I would like to thank everybody who was important to the realization of this project, as well as expressing my apology that I could not mention them personally one by one. iii

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7 List of Figures 2. Basic configurations for viscoelastic layer damping. (a) Unconstrained, (b) constrained, and (c) part of a magnetic constrained layer treatment Cantilever beam in magnetic field generated by a permanent magnet A piezoelectric laminate structure disturbed by an external force w. The resulting vibration d is suppressed by the presence of a shunt impedance Z(s). A current i is generated due to the obtained voltage v over Z(s). The poling direction of the transducer is indicated by the shaded arrows Isothermal stress-induced martensitic phase transformation Tuned mass damper operating principle. k is the spring stiffness, m the mass of the tuned-mass damper, c is the damping coefficient, and M the modal mass Strain response to a step in applied stress at t and release at t 2 (left) and stress response to a step in applied strain at t and release at t 2 (right) Stress response to a harmonic strain excitation or vice versa Influence of temperature and frequency on dynamic modulus and loss factor Illustration of the shifting process from curves at different temperatures to a master curve Two illustrations of a Wicket plot Nomograph of a commercially available damping polymer Several classical linear viscoelastic material models consisting of springs and dampers Different approaches to dampen axial vibrations of a bar by means of viscoelastic material Different approaches to dampen transversal vibrations of a pinned-pinned beam by means of viscoelastic material Mass-spring-damper system representing a structural mode. A schematic representation (a) and the frequency response function between F and q, the compliance (b) Schematic representation of a space that is partly filled with a metal bar structure (a) and its representation by a D finite element model (b) Schematic representation of a viscoelastic block (a) and its representation by a 2D finite element model (b) Modal damping ratios β (a) and β 2 (b) for a discrete VEM element connected to a truss as a function of mass ratio m r and stiffness ratio k r for η = and x v = L b Modal damping ratio β 3 as a function of mass ratio m r and stiffness ratio k r for low loss factor (i.e. η =.2) and x v = L b illustrating local "TMD" optima when using a discrete damping element Modal damping ratios β (a) and β 2 (b) as a function of relative discrete damper position x v /L b and stiffness ratio k r for m r = and η = Modal damping ratios β (a) and β 2 (b) as a function of m r and k r for η = and x v = L b for a discrete damper loaded in shear Modal damping ratios β (a) and β 2 (b) with FL damping as a function of m r and relative beam length for modes and 2 and η = Modal damping ratio β 3 using free layer configuration as a function of mass ratio m r and relative beam length L/tb for a small mass density ρ v of the VEM Modal damping ratios β (a) and β 2 (b) of modes and 2 using the CL configuration as a function of mass ratio m r and relative beam length L/t b for η = Modal damping ratios β (a) and β 3 (b) using the CL configuration as a function of mass ratio m r and relative layer thickness t c for η = Modal damping ratio β of mode as a function of mass ratio m r and relative constraining layer thickness t c for η =, where a small mass density of the VEM and CL is used in computations.. 27 v

8 List of Figures 3.5 Illustration of Γ as a function of volume fraction χ and dynamic modulus E d for different damping methods and η = Schematic representation of a space that is partly filled with a beam structure (a) and its representation by a D finite element model (b) Modal damping ratios β (a) and β 2 (b) when using discrete dampers in tension as a function of m r and k r for η = and x v =.4L b Modal damping ratios β (a) and β 2 (b) as a function of relative discrete damper position x v /L b and stiffness ratio k r for m r = 5 and η = Change of the first two mode shapes as the VEM stiffness increases for different damper positions x v Modal damping ratios β (a) and β 2 (b) for discrete shear dampers as a function of m r and k r for η = and and x v =.4L b Schematic representation of a part of the beam (undeformed and bended) illustrating the degrees of freedom and connectivity conditions linking the beam to the VEM Modal damping ratios β (a) and β 2 (b) for FL damping as a function of m r and beam length L for η = Modal damping ratios β (a) and β 2 (b) for FL damping as a function of m r and relative beam length L/t b, with a small mass density ρ v =. kg/m 3 for the VEM Modal damping ratios β (a) and β 2 (b) for CL damping as a function of m r and L/t b for η = and t c = Modal damping ratios β (a) and β 2 (b) for CL damping as a function of m r and relative constraining layer thickness t c for η = Illustration of Γ as a function of volume fraction χ and dynamic modulus E d for different damping methods and η = Rectangular elements in iso-parametric coordinates ξ and ζ illustrating a four (a), eight (b) and nine (c) node element including the node numbering Contributions to the in-plane deformation: deformation due to interpolated nodal displacements in the midplane (a) and deformation due to rotations θ z (b) Schematic representation of the open box structure Illustration of an edge element connecting two walls of the box Illustration of the FE modeling of discrete dampers used for damping a 3D structure: dimensions (a), construction of elements (b) and a single element with the degrees of freedom at a node i (c) Illustration of a CL element and base plate element above each other connected by VEM in undeformed (a) and deformed (b) state A selection of the eigenmodes of an aluminum box structure Scaled frequency as a function of mesh size for the first, fifth and tenth (non-rigid body) mode Dynamic modulus and loss factor at 22 C as a function of frequency for Norsorex R from DMTA measurements and a fractional derivative model fit Dynamic modulus Ev(iω) and loss factor η(ω) at 22 C as a function of frequency for 75 Shore A Viton R SCVBR from DMTA measurements and a fractional derivative model fit Iteration scheme for finding the eigenvalues and -vectors of the damped box structure Cantilever beam with viscoelastic damping element connected near the end Driving point frequency response function of the damped beam at the location of the damper computed using the exact method and IMPS method ( db m/n) Interpolation of the first three σ k of the damped cantilever beam Interpolation of the first three R k of the damped cantilever beam Schematic representation of the box with discrete dampers and auxiliary structures seen from the top Eigenmodes 8,, 3, and 6 of the aluminum box structure damped using discrete VEM elements Eigenmode of the aluminum box damped using discrete VEM elements where the auxiliary beams oscillate on the VEM blocks Damping ratio for mode as a function of CL and VEM layer thickness (t c and t v ) for a plate representing wall and for different viscoelastic materials Schematic representation of the box where the hashed areas illustrate the parts to which the constrained layer patches are added Eigenmodes 9 and 4 of the aluminum open box structure damped using a partial coverage CL approach vi

9 List of Figures 5. Measurement setup for the box structure for different VEM damping approaches Fold-out of the box structure, illustrating the sensor placement and hammer excitation points Measurement setup for a plate to asses influence of layer bonding technique Frequency response function (accelerance) comparison for two plates both undamped and damped to investigate the influence of using glue or two-sided tape Frequency response functions for input force at point 52 and out-of-plane output acceleration at point 34 derived from measurements and model (a) and corresponding coherence function (b) Numerical and experimental frequency response functions for input force at point 52 and outof-plane output acceleration at point 34 for the box with discrete dampers (a) and corresponding coherence function (b) Numerical and experimental frequency response functions for input force at point 52 and out-ofplane output acceleration at point 34 for the box with fully covering CL (thick) (a) and corresponding coherence function (b) Numerical and experimental frequency response functions for input force at point 52 and out-ofplane output acceleration at point 34 for the box with fully covering CL (thin) (a) and corresponding coherence function (b) Numerical and experimental frequency response functions for input force at point 52 and out-ofplane output acceleration at point 34 for the box with partially covering CL (a) and corresponding coherence function (b) Illustrations of two sources of difference between model and reality, i.e. reduced bonding of the CL dampers near the edges and welds along the inside edges of the box A. DMTA measurement system A.2 Wicket plot with spline model fit C. Ratio between first natural frequency according to FEM calculations and thin plate analytical values for different plate thicknesses D. The first elastic eigenmodes of an aluminum box structure D.2 The first four elastic experimental eigenmodes of an aluminum box structure E. The first elastic eigenmodes of an aluminum box structure damped using discrete VEM elements.5 E.2 The first four elastic experimental eigenmodes of an aluminum box structure damped using discrete VEM elements E.3 The first elastic eigenmodes of an aluminum box structure damped using fully covering CL dampers E.4 The first four elastic experimental eigenmodes of an aluminum box structure damped using fully covering CL dampers E.5 The first elastic eigenmodes of an aluminum box structure damped using partially covering CL dampers E.6 The first four elastic experimental eigenmodes of an aluminum box structure damped using partially covering CL dampers F. A four degree of freedom viscoelastically damped system F.2 FRF function computed using different methods for a four degree of freedom viscoelastically damped system F.3 Cantilever beam with viscoelastic damping element connected near the end F.4 Driving point FRF for a cantilever beam at the location of the damper computed using different methods G. Fold-out of the box structure, illustrating the sensor placement and hammer excitation points. The blue squares indicate the actuation points for which FRFs are shown below G.2 Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) G.3 Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) G.4 Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b).. 33 vii

10 List of Figures G.5 Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) G.6 Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H. Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.2 Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.3 Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b).. 36 H.4 Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.5 Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.6 Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.7 Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.8 Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b).. 39 H.9 Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H. Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H. Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.2 Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.3 Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b).. 4 H.4 Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.5 Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.6 Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.7 Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.8 Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b).. 44 H.9 Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) H.2 Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b) viii

11 List of Tables 3. Parameter values used in free layer method damping performance analysis Parameter values used in comparison damping method performance for axial deformation Maximal value of Γ, Γ 2 and Γ 3 when χ.25,.5 MPa E d 5 GPa, and for loss factors η = or η = 2 (axial deformation). The color gives an indication of performance, where red corresponds to (relatively) low Γ k, green to (relatively) high Γ k and yellow indicates the intermediate range Parameter values used in comparison of VEM applications for transversal vibrations (used in computations below unless stated otherwise) Maximal value of Γ, Γ 2 and Γ 3 when χ.25,.5 MPa E d 5 GPa, and for loss factors η = or η = 2 (bending deformation). The color gives an indication of performance, where red corresponds to (relatively) low Γ k, green to (relatively) high Γ k and yellow indicates the intermediate range Parameter values used in modeling the box structure and its final design dimensions First (non-zero) undamped eigenfrequencies found with the Matlab FE model and with ANSYS and the relative difference between them Selected materials for dynamic characterization Parameter values and dimensions used in modeling the discrete dampers applied to the box structure First (nonzero) eigenfrequencies and damping ratios found using the derived Matlab model for the box with discrete dampers Added mass in kg per wall (of viscoelastic and constraining layer) for different constrained layer damping configurations First (nonzero) eigenfrequencies and damping ratios found using the derived Matlab model for the box with constrained layer damping Hardware used in dynamic experimental analyses First (nonzero) eigenfrequencies found using the Matlab model and the corresponding eigenfrequencies and damping ratios from experiments for the box without viscoelastic damping MAC values for comparing the mode shapes from the Matlab model and those derived from experiments (using ME scope VES) for the undamped box structure First (nonzero) eigenfrequencies in Hz found using the Matlab model and the corresponding frequencies from experiments for the box with viscoelastic damping First (non-rigid body mode) damping ratios found using the Matlab model and the corresponding damping ratios from experiments for the box with viscoelastic damping MAC values when comparing the mode shapes from the FE model with those identified by experiments (using ME scope VES) for the box structure with discrete damping elements MAC values when comparing the mode shapes from the FE model with those identified by experiments (using ME scope VES) for the box structure with fully covering constrained layer dampers (relatively thick VEM) MAC values when comparing the mode shapes from the FE model with those identified by experiments (using ME scope VES) for the box structure with fully covering constrained layer dampers (relatively thin VEM) MAC values when comparing the mode shapes from the FE model with those identified by experiments (using ME scope VES) for the box structure with partially covering constrained layer dampers A. Some viscoelastic materials and corresponding manufacturers ix

12 List of Tables B. Maximal objective function in parameter variation for different methods, modes and loss factors (axial deformation) for χ. and L b = 2 cm B.2 Maximal objective function in parameter variation for different methods, modes and loss factors (axial deformation) for χ.25 and L b = 2 cm B.3 Maximal objective function in parameter variation for different methods, modes and loss factors (axial deformation) for χ. and L b = 5 cm B.4 Maximal objective function in parameter variation for different methods, modes and loss factors (axial deformation) for χ.25 and L b = 5 cm B.5 Maximal objective function in parameter variation for different methods, modes and loss factors (bending) for χ. and pinned-pinned boundary condition B.6 Maximal objective function in parameter variation for different methods, modes and loss factors (bending) for χ.25 and pinned-pinned boundary condition B.7 Maximal objective function in parameter variation for different methods, modes and loss factors (bending) for χ. and clamped-clamped boundary condition B.8 Maximal objective function in parameter variation for different methods, modes and loss factors (bending) for χ.25 and clamped-clamped boundary condition C. Illustration of the different four node finite element types considered C.2 Illustration of the different eight node (and mixed) finite element types considered C.3 Illustration of the different nine node (and mixed) finite element types considered F. Parameter values used in FRF method accuracy comparisons for a damped cantilever beam x

13 List of Symbols Greek symbols α Temperature-frequency shifting function [ ] αj Coefficient corresponding to the jth summation term in the Golla-Hughes-McTavish [ ] model β Reference damping ratio for inherent damping of the "undamped" structure [ ] β k The damping ratio corresponding to the kth eigenvalue [ ] Γ Gamma function Γ k Damping performance criterion corresponding to the kth eigenvalue [ ] γ Shear strain, where a subscript indicates the direction of the strain and plane normal [ ] γ k Residue corresponding to the kth mode in modal superposition δ Phase difference between stress and strain for harmonic excitation of VEM [ rad ] ε Strain, where a subscript indicates the direction of the strain and plane normal [ ] ε Complex representation of the strain [ ] ε Laplace transform of ε [ ] ε Harmonic strain amplitude [ ] ε Strain vector [ ] ζ Iso-parametric element coordinate [ ] ζ j Iso-parametric coordinate of the jth node [ ] η Loss factor of the viscoelastic material [ ] η e Vector containing experimentally obtained loss factors [ ] η j Coefficient corresponding to the jth summation term in the generalized Maxwell [ s ] model θ Rotational degree of freedom where a subscript number indicates the corresponding [ rad ] node and a letter refers to the corresponding structure part and axis of rotation κ Factor to account for shear deformation variations [ ] κ v Factor to account for shear deformation variations in the viscoelastic material [ ] λ ref,k The k-th reference eigenvalue of the system in computing the objective function [ rad/s ] λ k The kth eigenvalue of the system [ rad/s ] µ Viscosity, coefficient linking strain rate to stress [ Pa s ] ν Poisson s ratio [ ] ν a Poisson s ratio of the auxiliary structure material [ ] ν b Poisson s ratio of the base structure material [ ] ν c Poisson s ratio of the constraining layer material [ ] ν v Poisson s ratio of the viscoelastic material [ ] xi

14 List of Symbols ξ Iso-parametric element coordinate (Chapter 4) and damping ratio (Chapter 3) [ ] ξ j ρ Coefficient corresponding to the jth summation term in the Golla-Hughes-McTavish [ ] model in Chapter 2 and to the iso-parametric coordinate of the jth node in Chapter 4 Mass density [ kg/m 3 ] ρ Mass density at a reference temperature T [ kg/m 3 ] ρ a Mass density of the auxiliary structure material [ kg/m 3 ] ρ b Mass density of the base structure material [ kg/m 3 ] ρ c Mass density of the constraining layer material [ kg/m 3 ] ρ v Mass density of the viscoelastic material [ kg/m 3 ] Σ Diagonal matrix containing frequency dependent eigenvalues, used for IMPS [ rad/s ] σ Stress, where a subscript indicates the direction of the strain and plane normal [ Pa ] σ Complex representation of the stress [ Pa ] σ Laplace transform of σ [ Pa ] σ Stress vector [ Pa ] σ Harmonic stress amplitude [ Pa ] σ af Stress at which all martensite is transformed to austenite [ Pa ] σ as Stress at which martensite to austenite transformation will start upon unloading [ Pa ] σ k The kth eigenvalue of the quadratic eigenvalue problem used for IMPS [ Pa ] σ m Stress at which all austenite is transformed to martensite [ Pa ] σ ms Stress at which austenite to martensite transformation will start upon loading [ Pa ] τ Shear stress [ Pa ] φ j Fractional derivative order of the jth term of the fractional derivative material model χ Fraction of a reference volume V taken up by VEM [ ] ψ Iso-parametric element coordinate [ ] ω Frequency in forced response [ rad/s ] ω Undamped natural frequency [ rad/s ] ω... ω Nf Frequencies at which the eigenvalue problem for IMPS is solved [ rad/s ] ω c Cut-off frequency defining the upper bound of a frequency range [ rad/s ] ω d Damped natural frequency [ rad/s ] ω j Coefficient corresponding to the jth summation term in the Golla-Hughes-McTavish [ ] model ω n Natural frequency [ rad/s ] ω r Reduced frequency [ rad/s ] xii

15 List of Symbols Roman symbols A Cross sectional area [ m 2 ] A v Cross sectional area of a viscoelastic damping element [ m 2 ] a Location along the length of a beam where a force is applied [ m ] a j Coefficient corresponding to the jth time derivative of stress in a linear differential [ ] equation between stress and strain and (in Appendix A) the jth coefficient in the ratio model B Fitting constant for WLF shift factor model [ K ] B Matrix with shape function derivatives b k Coefficient corresponding to the kth time derivative of stain in a linear differential [ ] equation between stress and strain C Fitting constant for WLF shift factor model [ ] c Viscous damping ratio [ Ns/m ] c f Conditioning factor [ ] D(t) Creep function [ Pa ] D(s) D D(s) Laplace transform of D(t) Isotropic constitutive relation matrix Dynamic stiffness matrix D Complex compliance [ Pa ] D Storage compliance (real part of D ) [ Pa ] D Loss compliance (imaginary part of D ) [ Pa ] D R (s) Reduced dynamic stiffness matrix d Vibration amplitude [ m ] E(t) Relaxation function [ Pa ] E Young s modulus [ Pa ] Ê(ω r ) Young s modulus master curve in terms of reduced frequency [ Pa ] E Complex Young s modulus [ Pa ] E Storage modulus (real part of E ) [ Pa ] E Loss modulus (imaginary part of E ) [ Pa ] E(s) Laplace transform of E(t) E Young s modulus at zero frequency [ Pa ] E a Young s modulus of the auxiliary structure material [ Pa ] E b Young s modulus of the base structure material [ Pa ] E c Young s modulus of the constraining layer material [ Pa ] E d Dynamic modulus of the viscoelastic material ( E v ) [ Pa ] E d,e Vector containing experimentally obtained dynamic modulus data [ Pa ] E j Coefficient corresponding to the jth summation term in different material models [ Pa ] E v Complex Young s modulus of the viscoelastic material [ Pa ] E v Storage modulus of the viscoelastic material (real part of E v) [ Pa ] E v Loss modulus of the viscoelastic material (imaginary part of Ev) [ Pa ] e b Finite element length in modeling a beam base structure [ m ] e v Finite element length in modeling a VEM block using D elements [ m ] e x Finite element length in x-direction [ m ] xiii

16 List of Symbols e y Finite element length in y-direction [ m ] F Force [ N ] F Force vector F Force vector, laplace transform of F(t) F R Reduced force vector f Frequency resolution [ Hz ] G(t) Damping kernel function G(s) Laplace transform of G(t) G Shear modulus [ Pa ] G Complex shear modulus [ Pa ] G Storage shear modulus (real part of G ) [ Pa ] G Loss shear modulus (imaginary part of G ) [ Pa ] G Damping matrix in laplace domain for s = G v Complex shear modulus of the viscoelastic material [ Pa ] H(s) Transfer function matrix H(iω) Frequency response function matrix Ĥ(iω) Approximate frequency response function matrix I Second moment of area [ m 4 ] I b Second moment of area of a beam [ m 4 ] i Complex number, i.e. and (in Section 2..3) current K System stiffness matrix K a K e K K b,e K g K g K l K v Stiffness matrix contributions from edge elements Elementary stiffness matrix corresponding to a single finite element Stiffness matrix contributions of the system excluding VEM Elementary stiffness matrix corresponding to a single beam element System stiffness matrix w.r.t. global degrees of freedom including edge element contributions System stiffness matrix w.r.t. global degrees of freedom excluding edge element contributions System stiffness matrix w.r.t. local degrees of freedom Stiffness matrix contributions of the VEM scaled with Young s modulus k Spring stiffness [ N/m ] k bar Static stiffness of a structural bar element [ N/m ] k b Static stiffness of the base material structure [ N/m ] k r Stiffness ratio of base structure and VEM [ ] k shear Static shear stiffness of the viscoelastic material damping element [ ] k v Static stiffness of the viscoelastic material damping element [ N/m ] L Length of a structural element [ m ] L a Length of an auxiliary beam structure [ m ] L b Length of a base structure [ m ] L v Length of a viscoelastic damping element [ m ] L x Length of the box in X-direction [ m ] L y Length of the box in Y -direction [ m ] L z Length of the box in Z-direction [ m ] xiv

17 List of Symbols M Mass MAC Modal assurance criterion (see []) [ ] M M M a M e M b,e M g M g M l System mass matrix Diagonal mass matrix Mass matrix contributions from edge and corner elements Elementary mass matrix corresponding to a single finite element Elementary mass matrix corresponding to a single beam element System mass matrix w.r.t. global degrees of freedom including edge and corner element contributions System mass matrix w.r.t. global degrees of freedom excluding edge and corner element contributions System mass matrix w.r.t. local degrees of freedom MSF Modal scale factor [ ] m Number of elements used in modeling a beam or bar and (in Section 2..6) a TMD [ ] mass (kg) m bar Mass of a structural bar element [ kg ] m a Number of elements used for modeling auxiliary beam structures [ ] [ kg ] m b Mass of the base material structure [ kg ] m k The kth diagonal entry of M m r Mass ratio of base structure and VEM [ ] m v Mass of the viscoelastic material damping element [ kg ] m X Number of elements in global X-direction used for modeling the box [ ] m x Number of elements in local x-direction used for modeling the VEM [ ] m Y Number of elements in global Y -direction used for modeling the box [ ] m y Number of elements in local y-direction used for modeling the VEM [ ] m Z Number of elements in global Z-direction used for modeling the box [ ] N Summation order for different material models [ ] N Matrix with shape functions [ ] N e Number of entries in the vectors η e and E d,e [ ] N f Number of frequencies used for interpolation in the IMPS method [ ] N lin,j The jth bi-linear shape function [ ] N q Number of degrees of freedom [ ] N quad,j The jth quadrilateral shape function [ ] N r Number of rigid body modes [ ] N uθ,j The jth shape function corresponding to the drilling degrees of freedom for u [ ] N u Matrix with shape functions for interpolating displacements in x-direction [ ] N vθ,j The jth shape function corresponding to the drilling degrees of freedom for v [ ] N v Matrix with shape functions for interpolating displacements in y-direction [ ] N w Matrix with shape functions for interpolating displacements in z-direction [ ] n Number of nodes used in modeling a beam or bar [ ] n x Number of nodes in local x-direction used for modeling the VEM [ ] n y Number of nodes in local y-direction used for modeling the VEM [ ] obj Objective function for fitting a material model [ ] p Summation order for constitutive relation between stress and strain for VEM in Chapter 2 and number of eigenvalues in Chapter 4 xv

18 List of Symbols q q q q c q e q g q l q R (iω) q r R R k Summation order for constitutive relation between stress and strain for VEM Column with degrees of freedom, laplace transform of q(t) Column with degrees of freedom Column with degrees of freedom that can be expressed in terms of other DOFs Column with degrees of freedom of a single element Column with degrees of freedom w.r.t. global reference frame Column with degrees of freedom w.r.t. local reference frames Reduced column with degrees of freedom Column with minimal degrees of freedom Rotation matrix Residue matrix corresponding to the kth mode r Truncation order in model superposition [ ] r j Transition frequency of the jth term in the fractional derivative material model [ rad/s ] s T T Laplace variable Temperature Matrix for incorporating constraint equations T e Kinetic energy contribution of a single finite element [ J ] T Reference temperature in shift factor model [ K ] T... T 3 Vectors spanning a projection basis T A Fitting constant for Arrhenius shift factor model [ K ] T c T p Matrix for incorporating constraint equations Projection matrix T n Fitting constant for nth term in modified Arrhenius shift factor model [ K ] t Time t Thickness of the base structure for the reference case [ m ] t b Thickness of the base structure, where an added numeral in the subscript indicates the [ m ] box wall t c Thickness of the constraining layer [ m ] t c Dimensionless constraining layer thickness [ m ] t v Thickness of the viscoelastic layer [ m ] t v,x Dimension is local x-direction of a viscoelastic damper [ m ] t v,y Dimension is local y-direction of a viscoelastic damper [ m ] U e Potential energy contribution of a single finite element [ J ] U k The kth complex right eigenvector of the system u Translational degree of freedom in x-direction where a subscript number indicates the [ m ] corresponding node and a letter refers to the corresponding structure part V Available volume for a structure [ m 3 ] V v Volume used for viscoelastic material [ m 3 ] v Translational degree of freedom in y-direction where a subscript number indicates [ m ] the corresponding node and a letter refers to the corresponding structure part and (in Section 2..3) current (V ) v j Fractional derivative order of the jth term of stress in the constitutive stress - strain relation in Chapter 2 W Width of a structural element [ m ] W a Width of an auxiliary beam structure [ m ] [ K ] [ s ] xvi

19 List of Symbols w k Fractional derivative order of the kth term of strain in the constitutive stress - strain relation in Chapter 2 w Translational degree of freedom in z-direction where a subscript number indicates [ m ] the corresponding node and a letter refers to the corresponding structure part and (in Section 2..3) disturbance force (N) w f Weighting factor [ ] X Global reference coordinate [ m ] x Local reference coordinate [ m ] x j Local x-coordinate at the jth node [ m ] x v Position along a beam at which a viscoelastic damper is attached [ m ] Y Global reference coordinate [ m ] y Local reference coordinate [ m ] y j Local y-coordinate at the jth node [ m ] Z Global reference coordinate [ m ] Z(s) Shunt impedance [ Ω ] Z Matrix with the eigenvectors used for IMPS [ m ] z Local reference coordinate [ m ] z j Local reference coordinate [ m ] z k The kth eigenvector of an approximate system used for IMPS xvii

20

21 Contents Abstract Acknowledgements List of Figures List of Tables List of Symbols i iii v ix xi Introduction. Motivation and objective Outline Passive damping 3 2. Overview of different methods Viscoelastic material damping Magnetic fields and induction Piezoelectric shunt damping Alloys Viscous damping Discrete dampers Coulomb damping Isolation mountings Properties and modeling of viscoelastic materials Deformation behavior Environmental influences Material properties representations Viscoelastic material models VEM application method analysis and comparison 5 3. Description of comparison cases Objective functions Axial vibration damping Axial vibration: Discrete dampers Axial vibration: Continuous layer damping Axial vibration: Damping method comparison Transversal vibration damping Transversal vibration: Discrete dampers Transversal vibration: Continuous layer damping Transversal vibration: Damping method comparison Conclusions xix

22 Contents 4 Case study: Modeling and analysis 4 4. FEM modeling Base structure Viscoelastic material Base structure design Dynamic modeling for viscoelastic damping Viscoelastic material model Modal analysis FRF derivation Damping design Discrete dampers Constrained layer damping Case study: Experiments and comparison with models Experimental setup Undamped structure Frequency response functions Modal parameters Damped structure Frequency response functions Modal parameters Error sources Conclusions and recommendations Conclusions Recommendations A Viscoelastic material characterization and modeling 83 A. DMTA A.2 Shifting A.3 Model fitting A.4 Materials and manufacturers B Additional results principle method comparison 89 C Finite element modeling 99 C. Generic linear structural element C.2 Plate elements C.2. Element types C.2.2 Comparison C.3 Beam elements D Mode shapes undamped box E Mode shapes damped box 5 E. Discrete dampers E.2 Constrained layer dampers E.2. Fully covering CL dampers E.2.2 Partially covering CL dampers F FRF method comparison 25 F. FRF computation methods F.2 FRF comparison F.2. Case : four-mass-spring system F.2.2 Case 2: Cantilever beam G Additional FRFs undamped box 3 xx

23 Contents H Additional FRFs damped box 35 H. Discrete dampers H.2 Constrained layer dampers H.2. Fully covering CL dampers H.2.2 Partially covering CL dampers I Experiment hardware specifications 47 Bibliography 53 xxi

24

25 Chapter Introduction. Motivation and objective In the high precision industry, demands on positioning accuracy are ever increasing. Unwanted structural vibrations hamper these demands and result in noise, decreased performance, and control limitations. Also outside the high-tech industry, the reduction of vibrations is frequently desired. Generally, to reduce cost, high stiffness and low mass are aimed for in designing high performance structures, often resulting in lightly damped systems. To further increase the vibrational performance, incorporation of energy dissipation mechanisms may be required. This can be realized by adding active, semi-active or passive damping, all having their advantages and disadvantages. In (semi-) active damping techniques, controllers are involved, which are not necessary in passive damping techniques. Active damping usually gives better performance, but since these methods require power, are generally more complex, more costly, and less robust, passive approaches are often preferred. There is a wide variety of dissipation mechanisms, which can be employed to introduce damping (in a passive sense) to a structure. Also in this category, some methods are more complex than others. Dissipating energy in an electric shunt, exploiting viscous or Coulomb friction, or using inherent material dissipation characteristics are some examples of passive damping methods. Tuned-mass dampers are also often used in order to improve dynamic performance. They can be used to address specific cumbersome modes. A fairly simple, but less often used, technique is based on the introduction of viscoelastic materials into the structure. When strained, a part of the deformation energy in such materials is dissipated. These materials have high potential in improving dynamic performance and in reducing vibrations by adding passive damping, which is why damping of structures by means of viscoelastic materials (VEM) is the main topic of this study. How and where the viscoelastic material should be added to a structure to efficiently introduce damping is the main question of interest. One way to employ it would be to use the VEM in a discrete damping element. Another way would be to use it in a constrained layer configuration, where the viscoelastic material is sandwiched between two (usually metal) layers. Even though in literature already quite some attention has been given to damping by means of viscoelastic materials, insight in the capabilities of different application methods for a structure of interest (and corresponding design guidelines) is still limited. Furthermore, add-on damping approaches at the end of the design process are commonly considered, whereas incorporating (VEM) damping in the structure in the early design stage would probably result in higher performance. This requires one to be able to model and accurately predict damping performance even for relatively complex structures. The main objectives of this study are: To attain a basic understanding of the capabilities of different VEM applications to reduce vibration amplitudes and increase the damping of eigenmodes of simple structures and a representative 3D structure. To investigate (and if necessary develop) numerical techniques for characterizing the dynamic properties (eigenvalues, eigenmodes, frequency response functions) of a structure with viscoelastic material. To experimentally validate model based predictions of dynamic characteristics of a representative 3D structure with VEM damping. For the 3D structure, the viscoelastic material damping is applied to obtain maximal damping under certain design constraints. However, optimizing performance is not considered in this study, because in engineering practice that will depend on the specific design requirements and objectives for each particular case study.

26 Chapter : Introduction.2 Outline In Chapter 2, first, an overview is given of passive structural damping methods present in literature. Viscoelastic engineering materials are chosen to introduce passive structural damping in the current study, which is why their dynamic characteristics are explained in Section 2.2. How these materials are commonly modeled is explained in that section as well. In Chapter 3, several approaches of applying the viscoelastic materials in order to add damping to a structure are compared by means of some (relatively simple) test cases. Section 3. gives an overview of the different cases and application methods. In Section 3.2, objective functions are introduced to asses the performance of each method. In Sections 3.3 and 3.4, the effect of all application methods is investigated and compared for axial and transversal vibrations of a beam, respectively. In Section 3.5, conclusions are drawn. The conclusions from Chapter 3 are used in adding damping to a more complex (3D) open box structure. The modeling of the structure is described in Chapter 4. In Section 4., the Finite Element modeling of the structure will be discussed In Section 4.2, the undamped base structure is designed on certain objectives and requirements. Subsequently, adding viscoelastic damping to the system is considered. A frequency dependent VEM model based on fractional derivatives is chosen. Numerical methods to calculate the modal parameters and frequency response functions for a structure viscoelastic material are discussed in Section 4.3. In Section 4.4, designs of applying viscoelastic damping in improving the dynamic properties of the 3D open box structure are discussed. In these designs, discrete damping elements and three different constrained layer damping configurations are used. Chapter 5 contains the model validation step. The undamped and four damped structures are realized and their dynamic behavior is characterized experimentally. The measurement approach and experimental setup are described in Section 5.. The measurement results of the undamped structure can be found in Section 5.2. Experimental and numerical results are compared using modal parameters (such as eigenfrequencies and mode shapes) and frequency response functions. Subsequently, experimental and numerical results are compared for the four damped structures. Some possible error sources causing differences between numerical and experimental results are mentioned in Section 5.4. In Chapter 6, the conclusions of this study and recommendations for further research are presented. 2

27 Chapter 2 Passive damping Passive damping includes all methods for dissipating vibration energy without the need for a power source or active control. This chapter describes passive damping by means of structure additions. In Section 2., an overview of different approaches is given. For each method, a short description is given. Damping by means of viscoelastic material is the main focus of this study. The properties of viscoelastic materials are described in Section 2.2 for that reason. The section describes deformation behavior, environmental influences on the material properties, common ways of representing different properties, and elaborates on modeling. 2. Overview of different methods Passive damping can be introduced to a structure in many ways (see [2] for recent advances). Depending on the application and requirements, one method may be more favorable than the other. A short qualitative description of different damping methods is provided below. The methods and systems that are briefly explained are:. Viscoelastic material damping 2. Eddy current damping 3. Electromagnetic shunt damping 4. Piezoelectric shunt damping 5. Damping using shape memory alloys 6. High damping alloys 7. Thin gas film vibration damping 8. Magnetorheological dampers 9. Particle impact dampers. Tuned-mass dampers. Coulomb damping 2. Isolation mountings 2.. Viscoelastic material damping Viscoelastic materials (VEM) are materials that exhibit a combination of elastic and viscous behavior when deformed. Examples are rubber, polymers, some adhesives, urethanes, epoxies, and enamels. During a process of loading and unloading, some of the deformation energy is restored and a relatively large amount (compared to other materials) is dissipated. Due to the latter, it can be used to introduce damping to a structure. Different means of application have been published. Examples are by using it as a sort of discrete damper element or in the form of a layer added to the structure. Viscoelastic materials cannot be used as structural materials on their own because of lack of strength and rigidity [3] and creep, but can be used as add-on passive damping treatments. The material properties usually depend quite a lot on temperature and frequency. One method of using VEM is free layer damping. A surface layer of viscoelastic material is added to a structural element at the outside (see Figure 2.a). As the base layer deforms, the VEM is strained (mostly directly), but this is not in phase. The strain of the VEM lags behind, which is a measure of damping in the material [4]. The layer thickness has a direct impact on the amount of dissipation energy [3]. The method where the viscoelastic layer is sandwiched between two stiff layers, the host structure and a constraining layer [3] (see Figure 2.b), is called constrained layer damping. Vibrations and deflections of the element result in mostly shear deformations of the viscoelastic layer. The constrained layer configuration is more effective in damping structural vibrations than the free layer configuration [5]. The amount of energy dissipated is related to the VEM s strained volume and the imposed shear strain [3]. While a low thickness VEM means a reduced volume to store strain energy, it is responsible for larger shear strains when the outer plates bend [3]. 3

28 Chapter 2: Passive damping Viscoelastic material N S S N (a) (b) (c) Longitudinal deformation Shear deformation N S S F N Figure 2.: Basic configurations for viscoelastic layer damping. (a) Unconstrained, (b) constrained, and (c) part of a magnetic constrained layer treatment Magnetic constrained layer damping is an extension to the constrained layer principle. The constraining layer is magnetic or equipped with permanent magnets at the ends (see Figure 2.c and [6]). The magnets should be arranged in attraction configuration to increase damping. The shear strain in the viscoelastic layer is increased resulting in a larger energy dissipation. The main advantage of this principle (as for other VEM layer methods) is that it is passive and does not require extensive electronics for example. Furthermore, it is relatively easy to apply. A disadvantage is that the magnets cause an increase in flexibility of the structure (when arranged in attraction) and with that a shift of resonance frequencies Magnetic fields and induction Several techniques make use of magnetic fields and induction to passively damp structural vibrations. These fields can be provided by permanent magnets or magneto-elastic materials. Some of the methods using magnetic fields are briefly explained below. Eddy current damping Electromagnetic forces are generated by the movement of a conductor trough a stationary magnetic field or a time varying magnetic field through a stationary conductor [9]. The generated eddy currents circulate in such a way that they induce a magnetic field opposite to the applied field which causes a resistive force. The induced currents will be dissipated into heat due to the electrical resistance of the conductor [9]. This principle can be used to damp vibrations. A possible way to apply it to a flexible cantilever beam with length L, width W, and thickness t is shown in Figure 2.2. As the beam vibrates, eddy currents are generated in the conductive plate and energy is dissipated. A main advantage of this damping technique is that it does not change other properties of the structure (as mass and stiffness). The solution is maintenance free (no contact) and easy to install. Figure 2.2: Cantilever beam in magnetic field generated by a permanent magnet (from [9]). Magneto-elastic (or magnetostrictive) materials couple mechanical behavior with magnetic field. An applied magnetic field induces strains in the material (Joule effect) and conversely a magnetic field is generated when a strain is applied (Villari effect) [7, 8]. 4

29 Section 2.: Overview of different methods Electromagnetic shunt damping Electromagnetic shunt damping requires a magnetic field and a coil connected to an impedance in closed circuit. In this shunted electromagnetic transducer an opposing electromotive force (emf) is induced as the mechanical structure vibrates. Some of the kinetic energy of the structure is dissipated in the impedance thus passively damping the system []. The magnetic field can be produced by a permanent magnet or straining of a magnetoelastic material for example. The magnetic field in the coil changes as the permanent magnet (or coil) moves or the magneto-elastic material is strained, resulting in an induced current in the coil. The electromagnetic damper gives an opposing force that is linearly dependent on velocity as an ideal dashpot []. The device can also be used in an active manner as an actuator []. The main principle is comparable to that of piezoelectric shunt damping (see below) Piezoelectric shunt damping This damping method makes use of piezoelectric elements. Piezoelectric elements have a permanent polarization and mechanical compression or tension changes the dipole moment associated with that element [2], which creates a voltage (known as the direct piezoelectric effect). The reverse is also true, i.e. applying a voltage over the element results in a compression or extension depending on the polarity. A piezoelectric element can thus be used as a sensor and as an actuator. When operating as a sensor, mechanical energy is converted into electrical energy. To extract vibration energy from a structure, piezoelectric transducers are bonded to it. These transducers are shunted by a passive electric circuit that acts as a medium for dissipating the mechanical energy of the base structure [3]. See Figure 2.3 for a schematic of this method. d d w i w i Z(s) v v Z(s) Figure 2.3: A piezoelectric laminate structure disturbed by an external force w. The resulting vibration d is suppressed by the presence of a shunt impedance Z(s). A current i is generated due to the obtained voltage v over Z(s). The poling direction of the transducer is indicated by the shaded arrows (from [2]) Alloys Different alloys have been produced with certain properties that can be employed to dampen structural vibration. They employ different principles to add damping, e.g. internal damping and state transformations. Shape memory alloys (SMA) Shape memory alloys exhibit two unique properties: the shape memory effect and superelasticity [4]. The shape memory effect refers to the phenomenon that these materials return back to their predetermined shapes upon heating. The superelasticity refers to the phenomenon that shape memory alloys can undergo a large amount of inelastic deformations and recover their shapes after unloading [4]. 5

30 Chapter 2: Passive damping Figure 2.4 shows the stress-strain loop of a superelastic SMA specimen. On loading, when the stress reaches σ ms transformation from austenite to martensite will be induced and this transformation will continue until all the austenite has been transformed to martensite at σ m. Upon release of stress during unloading, martensite unloads elastically down to σ as, where it will transform back to austenite from. When this transformation is completed, there is final elastic unloading of austenite phase [5] from σ af. The area enclosed by the hysteresis loop represents the energy dissipated through the cycle [4]. This characteristic can be used to dampen structural vibrations. The properties of these shape memory alloys depend on temperature. The stress-strain loop shown in the figure below is for a temperature above the so called austenite finish temperature [4]. Figure 2.4: Isothermal stress-induced martensitic phase transformation (from [4]). Segments of shape memory alloys can be added to the structure material, or hybrid composites can be used as structural material with layers of SMA and of glass fiber for example [6]. High damping alloys Most metals have high stiffness, but low damping or low stiffness and high damping (lead for example)[7]. High damping alloys have been developed that have the best of both worlds, i.e. high stiffness and damping [7]. Some examples are indium-tin [8], Mn-Cu sintered [9] and magnesium alloys [7]. These alloys can be used for the purpose of dampening vibrations in structures and machinery Viscous damping The methods described in this section make use of viscous friction forces to dissipate kinetic energy of vibrations. They make use of fluids with or without specific non-newtonian properties. The common viscous dashpot is left out in this description as its operating principle is assumed known by the reader. Thin gas film vibration damping A thin gas film is entrapped between two plates of the structure. As the structure deflects, fluid flow and compression of the film occur and energy is dissipated due to viscous effects [2]. The principle is also called squeeze film damping (SFD) as the fluid between the surfaces is stretched and squeezed [2]. The added fluid layer acts as a mass, spring, and damper having a significant effect on the dynamics of the moving plates [2]. This technique can provide solutions to dampen vibrations in harsh environments that do not allow for using materials as high damping rubbers and other viscoelastic materials for example [2]. Magnetorheological dampers Magnetorheological (MR) fluid is known as a non-newtonian fluid, whose properties change in response to an applied magnetic field [22]. This can be employed to change the characteristics of a conventional viscous damper as it has variable damping properties. An example of such a discrete damper is shown in [22]. For that damper, the damping coefficient varies with position and direction of motion of the piston. These characteristics can be optimized to obtain better damping performance than with a regular viscous damper [22]. 6

31 Section 2.: Overview of different methods 2..6 Discrete dampers This section addresses two types of discrete dampers. In the sections above, some methods are described, which can also be realized in the form of discrete damping elements. The methods below can also be categorized in different groups though. Particle impact damper Particle dampers are containers or structural voids partially filled with particles (e.g., ball bearings, tungsten powders, etc.) [23]. As the structure to which the damper is connected vibrates, the particles inside move. Friction forces and impact between the particles dissipates kinetic energy thus dampening the vibration. The main advantages of particle dampers are ruggedness, reliability, and insensitivity to extreme temperatures [23]. These dampers are therefore especially applied to lightly damped structures in harsh environments, where traditional approaches fail. Tuned-mass damper Dynamic vibration absorbers are add-on mass-spring systems with a tuned resonance frequency to match that of a certain mode of the system (see Figure 2.5). Adding this "splits" the problematic mode into two resonance peaks [7]. By also including a damper to this mass-spring system, the two arising resonant peaks are also suppressed. The tuned-mass damper (TMD) causes high damping of the mode of interest (see Figure 2.5). TMD designs are compact and easily incorporated into a structural system [7]. k m M (modal mass) k c Tuned-mass damper Base structure magnitude Undamped Absorber TMD frequency Figure 2.5: Tuned mass damper operating principle (see [7]). k is the spring stiffness, m the mass of the tunedmass damper, c is the damping coefficient, and M the modal mass Coulomb damping Due to the hysteretic behavior of dry friction, it can be employed to dissipate energy. The method requires at least two surfaces that are rubbing against each other due to relative motion caused by structural vibrations. It is naturally present in joints and when parts are bolted together. An example of applying this technique to dampen structural vibrations is by superimposing two beams with friction in the interface [24]. Another way of using this damping strategy is by using it in a mass-spring-damper [25, 26] Isolation mountings With this method certain support structures are meant that suppress vibrations arising from base disturbances. The transmitted force is lowered in a certain frequency range. The main principle is that the resonance frequency of the isolator is tuned quite low such that vibrations above 2ω n are attenuated (where ω n is the isolators natural frequency) [27]. This resonance frequency is low by using low stiffness which can for example be obtained by using negative stiffness characteristics. Note that since this technique does not really dissipate energy it is questionable if it should be called passive damping. Nevertheless, it reduces structural vibrations in a passive sense. In isolation mountings, friction, viscoelastic materials, and viscous forces can be employed to dissipate energy [27]. An example is a friction-pendulum used to dissipate earthquake energy in buildings [27]. 7

32 Chapter 2: Passive damping 2.2 Properties and modeling of viscoelastic materials As passive damping by means of viscoelastic materials is the focus of this study, a more detailed explanation of the material properties and modeling will be given. First, the deformation behavior under loading will be discussed. Subsequently, the dependency of the material properties on certain quantities (i.e. temperature, frequency, strain amplitude, and other environmental influences) will be addressed, which is followed by an illustration of how they are usually represented. Finally, the mathematical modeling of viscoelastic materials is considered Deformation behavior This subsection is partly inspired by [28]. As mentioned above, viscoelastic materials show a combination of elastic and viscous behavior. For materials that exhibit only elastic deformation behavior, the stress strain relation can be described by Hooke s law: σ = Eε. (2.) The stress σ in the material is linearly dependent on the strain ε with a ratio E between them, which is known as the Young s modulus. A similar relation holds for shear deformation γ and shear stress τ, i.e. τ = Gγ. (2.2) When a stress is applied to such a material, it deforms and all deformation energy is stored. Releasing the stress will cause the material to go back to its original state and no energy is lost during the process. The mechanical analog to such a material is a spring. Harmonic excitation will result in in-phase stress and strain. A material that shows viscous behavior can be considered analog to a linear dashpot and the relation between stress and strain for such a material is given by Newton s law: σ = µ ε. (2.3) The stress is in this case linearly dependent on the applied strain rate ε with a factor µ, the viscosity. For this type of materials all energy is dissipated during deformation. For example, when a constant stress is applied, the part will deform at a constant strain rate. When the stress is subsequently released, the material will remain in its current state. It will require energy to force it back to its original state. Harmonic excitation will result in a 9 degrees phase difference between stress and strain for such materials. Many polymers exhibit a combination of elastic and viscous behavior [28]. Such materials are called viscoelastic: part of the energy is stored and a part is dissipated when it is deformed. A viscoelastic material can be characterized by its creep and relaxation behavior, which are described by the creep function D(t) and relaxation function E(t) respectively. Figure 2.6 (left) shows an illustration of a creep response where a stress step is applied at t = t and released at t = t 2. Figure 2.6 (right) depicts a typical relaxation response to a strain step between t and t 2. ε (t) σ (t) t t 2 t t t 2 t Figure 2.6: Strain response to a step in applied stress at t and release at t 2 (left) and stress response to a step in applied strain at t and release at t 2 (right) (modified from [29]). In general (according to linear viscoelasticity theory), the current stress in a viscoelastic material is a function of the strain history. The stress and strain responses are calculated by the Boltzmann integral which is based on 8

33 Section 2.2: Properties and modeling of viscoelastic materials the superposition principle of infinitesimal excitation steps. This leads to the following expressions for stress and strain responses due to strain and stress excitation respectively [29, 3]: σ(t) = ε(t) = t t Taking the Laplace transform of these expressions gives E(t τ) ε(τ)dτ, (2.4) D(t τ) σ(τ)dτ. (2.5) σ(s) = sē(s) ε(s) E (s) ε(s), (2.6) ε(s) = s D(s) σ(s) D (s) σ(s), (2.7) for zero initial condition, where the bar above the symbols indicates that it is the Laplace transform and E (s) and D (s) are the complex Young s modulus and compliance of the material respectively and s is the Laplace variable. Viscoelastic materials can be characterized by their creep D(t) (or D(s)) and relaxation functions E(t) (or Ē(s)) but the experimental determination of these is often difficult. An alternative characterization is realized by considering harmonic excitations (see for example [3] or [32]). This is also very practical as for most engineering purposes it is convenient to work in the frequency domain. The strain-time history and stress-time history are both harmonic but there exists a phase difference δ between the strain and corresponding stress, see Figure 2.7. σ (t) ε (t) σ ε t δ / ω Figure 2.7: Stress response to a harmonic strain excitation or vice versa (inspired by [33]). Taking σ(t) = σ sin(ωt + δ) and corresponding strain excitation ε(t) = ε sin(ωt), it follows that [3] σ(t) = σ sin(ωt + δ) = σ sin(ωt) cos(δ) + σ cos(ωt) sin(δ) = σ ε cos(δ)ε(t) + σ ε ω sin(δ)dε dt. (2.8) The stress is thus composed of a part that is related to the strain and a part related to the strain rate. This can be seen as a combination of a spring and damper. The material properties usually depend on the frequency of the excitation ω. If it is introduced that E d (ω) = σ(ω) ε, η(ω) = tan (δ(ω)), E (ω) = E d (ω) cos (δ(ω)) and E (ω) = E d (ω) sin (δ(ω)), the stress strain relation can be rewritten as σ(t) = E d (ω) cos (δ(ω)) ε(t) + E d(ω) sin (δ(ω)) dε ω dt, (2.9) σ(t) = E (ω)ε(t) + E (ω) dε (2.) ω dt and as ( σ(t) = E (ω) ε(t) + η(ω) ) dε. (2.) ω dt Here, E d is known as the dynamic modulus, η is known as the loss factor (which sign needs to match that of ω for the model to be causal and dissipative[34]), δ is the loss angle, E is referred to as the storage modulus, and E is the loss modulus. These coefficients are related to each other as η = tan(δ) = E E ; E d = E 2 + E 2. (2.2) 9

34 Chapter 2: Passive damping A similar approach can be used to write the strain response to a sinusoidal stress as ε(t) = D (ω)σ(t) D (ω) dσ ω dt (2.3) where D and D are known as the storage compliance respectively the loss compliance. In literature, the material characteristics are often represented using a complex modulus E (ω) or compliance D (ω), which is only valid for harmonic excitations and responses [29]. Introducing the complex exponential (with i = ) and thus (when ε(t) = Re { ε e iωt} Re {ε (t)}) e iωt = cos(ωt) + i sin(ωt) (2.4) it follows that ( σ(t) = Re {σ (t)} = E (ω) dε dt = Re {iωε (t)} (2.5) = E (ω) ) dε dt ε(t) + η(ω) ω ( Re {ε (t)} + η(ω) ω Re {iωε (t)} = Re {E (ω) ( + iη(ω)) ε (t)} (2.6) = Re {(E (ω) + ie (ω)) ε (t)} (2.7) ) and consequently that the modulus E (ω) = σ ε = E (ω) ( + iη(ω)) = E (ω) + ie (ω). Similarly, for the compliance it follows that D (ω) = D (ω) id (ω). The same approach can be used for the relation between shear strain γ and shear stress τ Environmental influences The parameters commonly used to define the material behavior of viscoelastic polymers (as storage modulus, loss factor and loss modulus) depend on different aspects. The most dominant factor is temperature. An example of how the storage modulus and loss factor vary with temperature for many viscoelastic materials is shown in Figure 2.8. To every storage modulus corresponds a unique loss factor [35]. For low temperatures the storage modulus (and dynamic modulus) is high and the loss factor is low. This region is known as the glassy region and is characterized by low mobility of the polymer chains resulting in mainly elastic deformation. As the temperature increases, the storage modulus drops and η increases up to a certain point and then drops again. The increased mobility causes the chains to interfere and dissipate some of the energy. This temperature region is known as the glass transition region as it is the regime between glassy and rubbery state. For higher temperatures the storage modulus decreases to a steady value and the loss factor goes to a low steady value again. The storage modulus usually changes by a couple orders of magnitude ( 3) over the complete temperature range. Storage modulus (log) Scaled magnitude Loss factor Increasing frequency (log) Increasing temperature Figure 2.8: Influence of temperature and frequency on dynamic modulus and loss factor.

35 Section 2.2: Properties and modeling of viscoelastic materials Another aspect that influences the parameters quite a lot (but at a much slower rate than temperature) is time scale or frequency [3]. The dependence on frequency (on logarithmic scale) shows a trend that is similar to the inverse relation with temperature (see Figure 2.8). There is a relation between the influence of these two aspects. The frequency dependence of the material at a specific temperature can be found by shifting the isotherm curve at a different temperature along the frequency axis by a given factor α(t ) (see Figure 2.9 for an illustration of this process). This motivates the use of a reduced frequency ω r = α(t )ω and a description of the complex modulus in the form E (ω, T ) = Ê (α(t )ω) = Ê (ω r). (2.8) This representation is called the frequency temperature superposition principle or frequency temperature equivalence [36]. The curve Ê is known as the master curve of a viscoelastic material and is generally split up in its components E and η. Various expressions for α(t ) have been proposed to model the shift factor. Some are [3, 36, 37] T T Williams-Landel-Ferry (WLF) log [α(t )] = C (B +T T) Arrhenius log [α(t )] = T A T T Modified Arrhenius Tomlinson et al. log [α(t )] = N n= T n ( T T ) n log [α(t )] = N T n (T T ) n Here, C, B, T A, and T n are constants used for fitting and T is a non-arbitrary reference temperature. The reference temperature needs to be found such that the shift expression is best. At this temperature the shift factor is equal to unity and thus E (ω, T ) = Ê(ω). Strictly speaking, the absolute temperature corresponding to each data point also affects the modulus in another way, whereby the actual value of the real modulus should be factored by the ratio ρt/ρ T [35]. In this, ρ is the mass density at absolute temperature T and ρ the density at the reference temperature T. Only shifting the storage modulus horizontally (with frequency) is sufficient for a thermorheologically simple (TRS) material [35]. For materials that do not fit this class, a vertical shift factor might also be necessary. This factor is found by first finding the horizontal shift factor such that the loss factor curve is aligned and subsequently shifting the storage modulus vertically. The shift factors can be found in different ways. One is to take one measurement temperature as a reference and by trial and error shift measurement curves at other temperatures such that they combine into a smooth line. Another method is explained in [35] and [38] and uses the Wicket plot (explained in the next section). See Appendix A for a detailed description of the shifting process used in this study, based on [35, 38]. n= log E log E log ω log ω Figure 2.9: Illustration of the shifting process from curves at different temperatures to a master curve (inspired by [39]). Other factors that can influence the dynamic properties of viscoelastic materials are strain amplitude, exposure to sunlight and contact with foreign substances (such as petroleum products, alkalis, harmful chemicals, etc.) [4], but usually to a lower extent. For relatively large strain amplitudes, the mechanical properties of viscoelastic materials usually are non-linear. The range, that can be considered linear, depends on the composition of the material, but generally strain below.% can be used.

36 Chapter 2: Passive damping Material properties representations As mentioned above, to each storage modulus corresponds a unique loss factor (and loss modulus) for thermorheologically simple materials. This fact is illustrated in Figure 2.a where the dynamic properties of a viscoelastic material as a function of frequency and temperature are projected on the storage/loss modulus plane. This is known as the Wicket plot. Figure 2.b shows an example of a Wicket plot based on measurements (frequency sweeps for different temperatures). The plot is commonly used to investigate the scatter in measurement data and identify erroneous data points. If the data is close to a smooth curve it is not automatically accurate though, it just is consistent. If the points corresponding to an isotherm do form a smooth curve but the different isotherms do not join nicely, the material is likely to be not TRS and vertical shifting can be introduced. The Wicket plot can also be used to identify the temperature shift function as mentioned above (see Appendix A and [35, 38]), which will be employed in this study. (a) Three-dimensional repr. of complex modulus (b) Example of a Wicket plot Figure 2.: Two illustrations of a Wicket plot (from [37] respectively [38]). Figure 2.: Nomograph of a commercially available damping polymer (from [4]). 2

37 Section 2.2: Properties and modeling of viscoelastic materials The dependence of storage modulus (either in tension, E, or shear, G ) and loss factor on frequency and temperature is often depicted in what is called a nomograph (see Figure 2. for an example). It shows both properties as a function of reduced frequency. The way to read it is:. Locate the desired frequency on the right vertical scale. 2. Follow the chosen frequency line to the desired temperature isotherm. 3. From this intersect go vertically down (or up) until crossing both the modulus and loss factor curves. 4. Where you cross these curves read the appropriate modulus and loss factor values from the dual scale on the left vertical side Viscoelastic material models In order to do computations and design of dynamic damping using viscoelastic materials, it is necessary to include the frequency (and in some cases temperature) dependency of the complex modulus E. Therefore, a model is required that describes this relation and thus accurately fits the material properties. Several models have been proposed in literature and used. In the most general form, classical linear viscoelastic material models are represented by a constitutive relation between stress and strain in the form of a linear differential equation, i.e. [29] p j= Transforming this to the Laplace domain gives and thus σ(s) a j d j dt j σ(t) = q k= p a j s j = ε(s) j= E (s) = σ(s) ε(s) = d k b k ε(t). (2.9) dtk q b k s k (2.2) k= q b k s k k=. (2.2) p a j s j In the past, many apparently simple models that fit this description and are based on combinations of elastic and viscous elements have been introduced. See Figure 2.2 for the most common ones. The choice of which model to use depends on the material of interest and the required accuracy (usually increased accuracy comes at the cost of increased complexity). Of these models, the generalized Maxwell model (also known as the Wiechert model or Maxwell-Wiechert model) fits a lot of viscoelastic solids quite well as it can include many coefficients for fitting. The expression for the complex modulus for this model is given by [3] E (s) = E + N j= j= E j η j s E j + η j s, (2.22) where E can be considered the stiffness of the separate spring and E j and η j the stiffness respectively damping constant of the jth spring damper combination (see Figure 2.2e). N is the order of the model and corresponds to the number of spring damper combinations. A well known higher order model is the Golla-Hughes-McTavish model. It is known in different notations and forms and can be seen as a combination of mini-oscillators. One expression for it is [29] N E (s) = E + αj s 2 + 2ξ j ω j s s 2 + 2ξ j ω j s + ωj 2, (2.23) j= with α j, ξ j, and ω j non-dimensional parameters describing the frequency dependent behaviour of the viscoelastic material. A constraint of dissipativity is required on the parameters. The loss modulus should be greater than zero for all (positive) excitation frequencies ω [29]. 3

38 Chapter 2: Passive damping (a) Maxwell model (b) Kelvin-Voigt model (c) standard linear model (d) generalized Kelvin-Voigt model (e) generalized Maxwell model Figure 2.2: Several classical linear viscoelastic material models consisting of springs and dampers. Another way to model linear viscoelastic materials is by using fractional derivatives in a way similar to the classical models as mentioned above. For a fractional derivative material model, the constitutive relationship between stress and strain can be formulated as [29] p j= d q vj a j σ(t) = vj dt k= d w k b k dt w ε(t), (2.24) k where v j and w k are real constants with v j, w k. A fractional time derivative of order v of an arbitrary function f(t) is defined as d v [ f(t) d t ] f(τ) dt v Γ( v) dt (t τ) v dτ (2.25) where < v < and Γ(.) denotes the gamma function [4]. For fitting the complex material modulus, the fractional derivative model can be expressed as E (s) = E + N j= E j (s/r j ) φj, (2.26) φj + (s/r j ) with E a constant modulus at s =, E j and r j the modulus contribution and transition frequency respectively of the jth term and φ j the corresponding derivative order. The fractional derivative model is quite accurate compared to the classical models and requires a lot fewer parameters to fit the material behavior as usually less terms are necessary (smaller N). A downside is that it is more difficult to interpret. This model is used for fitting the Young s modulus characteristic for different materials considered in a case study in Chapter 4. See Appendix A for a description on fitting material models on measurement data. In this chapter, several methods for passively dissipating vibration energy are described. For each method, a short description is given. Special attention is given to damping by means of viscoelastic materials. The properties of viscoelastic materials are described, i.e. deformation behavior, environmental influences on the material properties, common ways of representing different properties, and elaborates on modeling. 4

39 Chapter 3 VEM application method analysis and comparison This study is on introducing passive damping to a structure by adding viscoelastic material (VEM). This material can be added in many ways and forms. Some ways are already mentioned in the previous chapter, i.e. in a free layer or constrained layer form. Another option is to add a small block or patch and use this as a discrete damping element. In this chapter, it is investigated, which VEM application method is most effective using a number of relatively simple cases. A distinction between structures loaded in axial direction and loading in bending is made. An objective function is introduced to be able to do this comparison. First, the different application configurations are investigated separately, where modal damping is used as a criterion. 3. Description of comparison cases Common methods of adding viscoelastic material are free layer damping and constrained layer damping. Because of that and the easy application of these techniques in practice, they will be considered. Furthermore, the effectiveness of using VEM as discrete damping elements is investigated. In free layer damping, the VEM is mostly strained in tension/compression whereas for constrained layer damping the deformation is for the largest part in shear (but not purely shear deformation). To see if the type of deformation influences damping performance, for the discrete dampers, a distinction is made between loading a VEM block in tension or in shear. Structural elements are usually loaded in axial or transverse direction, which is why an axially vibrating bar and a beam with transversal (bending) vibrations are considered. For each of these load cases, four different application methods of the viscoelastic material are investigated, see Figures 3. and 3.2. For the beam, pinned-pinned and clamped-clamped boundary conditions are used and both cases are compared. Furthermore, a distinction is made for the constrained and free layer treatments between adding the material at the top or both at top and bottom of the base material. L v VEM rigid link rigid link L v VEM x v x v (a) discrete damper (tension) (b) discrete damper (shear) VEM constr. layer VEM (c) free layer damping (d) constrained layer damping Figure 3.: Different approaches to dampen axial vibrations of a bar by means of viscoelastic material. 5

40 Chapter 3: VEM application method analysis and comparison VEM VEM rigid link rigid link x v x v (a) discrete damper (tension) (b) discrete damper (shear) VEM VEM constr. layer (c) free layer damping (d) constrained layer damping Figure 3.2: Different approaches to dampen transversal vibrations of a pinned-pinned beam by means of viscoelastic material. The different cases are thus: Axial vibration of a bar Transversal vibration of a beam (clamped and hinged) with for each the following subcases: Discrete damping element loaded in tension/compression Discrete damping element loaded in shear Free layer damping on one side Free layer damping on two sides Constrained layer damping on one side Constrained layer damping on two sides For all these cases, a finite element model (FEM) is made using Matlab [42]. Parameter variations are performed (subjected to design constraints) and the damping methods are studied separately first. After that the maximal value of an objective function (see next section) is computed and used for performance comparison. 3.2 Objective functions In order to compare the different approaches for each load case, a performance criterion is required. One option could be to use the dimensionless modal damping ratios of the system. They are defined here as β k Re {λ k} λ k (3.) for the kth mode where λ k is the kth eigenvalue of the system. This notion is also commonly used in the case of proportional damping (see for example [43 45]). In this study however, in general the damping is not proportional. Some of the characteristics for that type of damping do not apply here, but it can still be used as a measure for the amount of damping related to an eigenmode. It should be noted further that in the case of weak damping, the proportional damping model gives a good approximation. In practice, just using damping as a performance criterion is not very convenient, since not the damping needs to be maximized, but the dynamic deformations should be minimal. The mass and stiffness should also be included in a way. When using (3.) as an objective function it will always be preferred to use only viscoelastic material whereas this, in general, reduces the (dynamic) stiffness. For the dynamic deformations to be small, the structure needs to be light and stiff and with high damping. As a consequence, the damped eigenfrequencies (imaginary part of the eigenvalues) and damping ratios should both be high. The deflection due to a time-varying load scales inversely with the eigenfrequency squared and approximately inversely with damping ratio. To illustrate this, consider the simple system depicted below. It represents a structural mode with modal mass M, modal stiffness k 6

41 Section 3.2: Objective functions and modal damping c. Note that this actually requires proportional damping in order to decouple a general MDOF system into the sum of subsystems like these. For relatively small damping ratios the analogy holds though. q k M c F log H(iω) ωpeak log ω (a) (b) Figure 3.3: Mass-spring-damper system representing a structural mode. A schematic representation (a) and the frequency response function between F and q, the compliance (b). The modal deflection q as a consequence of the force F is given by the differential equation M q + c q + kq = F (t). (3.2) Transforming this to the Laplace domain gives ( Ms 2 + cs + k ) q(s) = F (s). (3.3) Considering harmonic forces and responses (i.e. s = iω), the relation between force and response, the frequency response function (FRF), is given by H(iω) = q(iω) F (iω) = Mω 2 + cωi + k. (3.4) The response due to an applied force is dominated by the resonance. This peak in the FRF should furthermore be reduced to minimize control limitations. The frequency at which the peak in the FRF (see Figure 3.3b) is seen is ω peak = ω 2ξ2 [46] using the definitions c k λ = ξω ± ω ξ2 i ξ = 2 ω = km M. (3.5) Incorporating these expressions in (3.4) gives The magnitude of the FRF at the peak frequency equals H(iω) = M( ω 2 + 2ξω i + ω 2 (3.6) ). H(iω peak ) = M ω 2 + 2ξ2 ω 2 + 2ξω2 2ξ2 i + ω 2 = M 4ξ4 ω 4 + 4ξ2 ω 4( 2ξ2 ) = M 2ω 2ξ ξ 2 ξ 2 = M 2ω 2 d ξ, (3.7) with ω d = ω ξ2, the imaginary part of the eigenvalue. This maximal value is bounded as M 2ω 2 ξ < H(iω peak) < M 7 2ω 2 d ξ, (3.8)

42 Chapter 3: VEM application method analysis and comparison where the terms are approximately the same for small ξ. This shows that the deformation amplitude approximately scales inversely with damping ratio ξ and inversely with eigenfrequency squared. One way to characterize the modal performance would thus be β k Im {λ k } 2, where the above defined damping ratios and eigenvalues of a generally damped system are substituted. To indicate the amount of reduction of deflection, it should be scaled with a certain reference value. Since the influence of introducing viscoelastic material is the main focus, the structure completely made of metal is taken as reference. A small reference damping ratio β is introduced and taken equal to 5 3, assuming this to be a reasonable damping ratio of a metal structure. The found performance criterion for each mode is Γ k (β k + β ) Im {λ k } 2 β Im {λ ref,k } 2 k =, 2,..., N, (3.9) with N the number of eigenmodes. The reference damping value β is added in the numerator since this inherit damping of the base material is not considered in the FE model. Note however, that in principle, the damping of the two different materials cannot just be added. Adding β in the numerator merely gives more insight on the performance, since the addition makes it possible to discern change of eigenfrequency and dynamic deformation amplitude even when no damping is added (β k ). In case the VEM is applied effectively, it is expected, that β k β, and that the performance criterion (3.9) can be used well in a practical sense. Γ k is smaller than unity if the dynamic deformation is increased relative to the reference case and larger than one if the deformation is reduced. In practice it is often the case that, for the structure under consideration, only a limited volume is available. Therefore, one has to divide this volume over the base material and the VEM. The question is if the increase in damping obtained by adding viscoelastic material compensates for a possible reduction in eigenfrequency. Keeping this in mind, it makes sense to choose the eigenfrequencies of the case where the complete available volume is taken up by the metal structure as a reference. The different VEM damping approaches from Section 3. will be compared using the following steps:. The eigenfrequencies of the reference case are calculated, where the complete available volume is filled with the metal structure. 2. The ratio of the volume used by the VEM is varied. The metal part of the beam will be reduced. 3. Other relevant parameters are varied. For example, for the discrete damper cases, the stiffness of the viscoelastic material can be altered by changing the length/cross sectional area ratio for a specific volume. Other relevant parameters include: damper position, storage modulus, and loss factor of the VEM. 4. The objective function (3.9) is computed for each of the parameter combinations and different modes. 5. The maximal objective function value per eigenmode within a practical range of the parameters is found for comparison of the effectiveness. It must be pointed out that in the derivation of the objective function (3.9) the proportional damping concept is considered, whereas the viscoelastic damping in general is not proportional. For non-proportional damping the "performance" will probably not scale linearly with damping and quadratically with damped frequency as indicated by (3.9). However, this deviation is likely to be small for relatively low damping ratios (<.2) and, therefore, it is assumed that the fairly simple objective function (3.9) still can be used. 3.3 Axial vibration damping The first load case considered is where a bar with rectangular cross section oscillates in axial direction. It is fixed at one end and free at the other end (see Figure 3.4a). The structure has a total volume V. A part of this volume (V v ) is used for the viscoelastic material. Consequently, the fraction of the volume taken up by the VEM is χ = V v V. (3.) The bar has length L b, width W, and height t b = ( χ)t, where t is the total height of the bar for χ =. The situation where only the base material (metal) is used (χ = ) is taken as reference case. The properties of the base material are denoted by ρ b, E b, and ν b being the mass density, Young s modulus, and Poisson s ratio respectively. 8

43 Section 3.3: Axial vibration damping The metal structure is modeled using D finite elements with one degree of freedom per node, i.e. the axial displacement (see Figure 3.4b). The metal structure is divided in m elements of equal length e b = L b /m resulting in n = m + nodes. t b _ t e b n-2 n- n L b (a) W (b) Figure 3.4: Schematic representation of a space that is partly filled with a metal bar structure (a) and its representation by a D finite element model (b). The element mass and element stiffness matrix of the metal part of the structure are [ ] /3 /6 M b,e = ρ b W t b e b and K /6 /3 b,e = E [ ] bw t b (3.) e b [ ] ub,k respectively [47] with the corresponding degrees of freedom for the kth element. For this element, a u b,k+ plane stress assumption is used for the deformations in the lateral directions (since deformation is not restricted) and the axial displacements at the nodes are interpolated linearly within each element. The contributions are assembled into a complete mass and stiffness matrix by imposing C continuity of the displacements u b,i (where i =, 2,..., n) and equilibrium of internal forces. Finally, the boundary conditions are applied. The different VEM application methods are investigated in this section to reduce axial vibrations of the bar structure. The discrete VEM damper elements are considered in Section 3.3., the continuous VEM layer damping methods is focussed on in Section and the methods are compared for this load case in Section Axial vibration: Discrete dampers A block of viscoelastic material with length L v, a square cross section and cross sectional area A v is added in parallel to the base structure and the two are connected via a rigid link (see Figures 3.a and 3.b). Only axial deformations of the bar are considered, so bending because of the off axis application of the VEM is not taken into account. To improve accuracy, the VEM-block is modeled using 4-node 2D elements (instead of D elements) with 3 degrees of freedom at each node, i.e. two in plane translations u and v and one rotation θ, see Figure 3.5. The elements consist of the in plane deformation part of the element Q4383 (u, v and θ z ) described in Appendix C and used in Section 4... A complex modulus E v(+i sign(ω)η) is used, where E v is the storage modulus of the VEM. The number of elements in local x-direction (normal to the length) is taken m x and the number of elements in the axial direction as m y. Again, a plane stress assumption is used for the out of plane direction. It was found that employing a plane strain condition gives the same qualitative conclusions (which will be mentioned below), but a higher stiffness of the VEM in the other directions. In practice, the deformation will likely be something in between plane stress and plane strain. Because the influence of the parameter values E v and η is considered, they are taken constant over frequency. If frequency dependence is added, the influence for each mode will change but the overall conclusions of this preliminary study will most likely still hold. Tension/Compression The block of viscoelastic material is oriented such that axial deformation of the base structure will cause the VEM to be loaded in tension/compression (see Figure 3.a). The mass and complex stiffness matrix contributions of the VEM and bar are linked by imposing that the displacement of the bar at x v equals that of the top of the VEM 9

44 Chapter 3: VEM application method analysis and comparison A v rigid link ny v θ u n - y n y-2 L v n - x n x (a) (b) Figure 3.5: Schematic representation of a viscoelastic block (a) and its representation by a 2D finite element model (b). block, obtaining a full mass and stiffness matrix. The equations of motion for this system, in the frequency domain are ( ω 2 M + K ) qe iωt = F e iωt, (3.2) with q a column containing all the degrees of freedom and F the corresponding external forces. The corresponding eigenvalue problem is given by [ Mλ 2 k + K ] U k = (3.3) with λ k and U k the kth complex eigenvalue and -vector respectively and the zero-vector. The damping β k depends on quite some parameters, i.e. the different dimensions and the material properties of both substructures. A minimal number of variables is looked for that capture the system dynamics. It is found that the damping ratios of this damped truss (as defined by Equation (3.)) primarily depend on the ratio of the effective stiffness and mass of VEM and base structure. Next to that, the loss factor of the damping material η of course influences the overall damping characteristics. Furthermore, the position of the discrete damper x v is a key parameter in maximizing the damping. The axial stiffness of a bar with Young s modulus E b, mass density ρ b, cross sectional area A b, and length L b equals [48] and its mass is k bar = E ba b L b (3.4) m bar = ρ b A b L b. (3.5) Using this, the stiffness ratio k r and mass ratio m r are introduced and are defined for this case as k r = k b E b W t b /L b = k v E v( and m r = m b = ρ bw t b L b (3.6) + iη) A v /L v m v ρ v A v L v respectively. If the length of the VEM block becomes small compared to the other dimensions, the effective stiffness increases significantly and the Poisson s ratio ν v starts playing a role. This is captured to some extent by using 2D elements. When representing the block as a spring with stiffness k v, this effect is commonly adjusted for by altering the effective Young s modulus using a so-called shape factor (not used in this study). The damper is attached at the end of the bar, i.e. x v = L b. The stiffness and mass ratios are varied and the influence on damping of the different modes (for some loss factors) is considered. The major difficulty lies here in the fact that Equations (3.) and (3.9) are evaluated while at the same time the mode shapes change with varying parameters. If the VEM is relatively stiff, the mode shape of the axial displacement in the beam resembles that of a single bar that is fixed at both ends (for x v = L b, see Figure 3.a). Furthermore, extra eigenvalues arise in the VEM due to the FE discretization of the VEM block, which (especially when it is relatively flexible) makes 2

45 Section 3.3: Axial vibration damping sorting by ascending damped frequency incorrect. The correct modes need to be compared for different ratios in order to be able to draw conclusions from the results. Since in practice it will often be the case that it is not desirable that the added VEM changes the mass very much, the mass ratio is limited to m r. The stiffness ratio is limited to k r. based on practical considerations. To compare the correct modes, they will be sorted using the Modal Assurance Criterion (MAC) given by [43] MAC[i, j] = U H i U j 2 (U H i U i )(U H j U (3.7) j ), where (.) H denotes the Hermitian transpose of a complex vector. The MAC value quantifies how much two modes resemble each other by a scalar ranging from zero to one and is independent of scaling of the eigenvectors. The modes at a specific mass and stiffness ratio will be sorted such that the MAC value compared to the undamped modes of the bar is maximal. This approach is found to be adequate as smooth transitions arise. Only the degrees of freedom of the base structure are used in computing the MAC values. Figures 3.6a and 3.6b illustrate the modal damping ratio (see Equation (3.)) as a function of m r and k r for the lowest two modes. The Poisson s ratio ν v is taken.49 (common for rubbery materials [49]) and, as stated before, the damper is attached at the end of the bar, i.e. x v = L b. The base structure is modeled using elements and for the VEM m x = 2 and m y = 6. 2 damping ratio mode 2 damping ratio mode k r k r m r 2 m r (a) Mode (b) Mode 2 Figure 3.6: Modal damping ratios β (a) and β 2 (b) for a discrete VEM element connected to a truss as a function of mass ratio m r and stiffness ratio k r for η = and x v = L b. It can be seen that the damping highly depends on the relative stiffness k r and much less on mass ratio m r. A similar trend can be seen for different η, but with different damping levels. A clear optimum is seen for the stiffness ratio k r for each mode. This optimum originates from a trade off between stiffness and strain. For a spring, the potential energy is given by 2 ku2 and thus depends on the stiffness k and deformation u. Increasing the stiffness of the VEM would increase the strain energy in the material and with that damping (complex Young s modulus). When the VEM stiffness gets too high though, the boundary conditions of the bar start to resemble fixed ones and the VEM will hardly deform reducing strain energy and damping. The stiffness ratio k r corresponding to the optimum shifts down the stiffness ratio axis for increasing mode number. This means that the VEM should be stiffer to attain the desired stiffness ratio for a higher order mode. In practice, this might not be very useful as the viscoelastic block will have to become very thin and wide. Even though the stiffness of the VEM has the most effect on damping performance, the mass ratio also plays a role. This is because obviously the mass ratio m r also affects the eigenvalues and eigenmodes (especially the internal VEM modal displacements) and with that the strains in the VEM. When looking at Figure 3.6 another thing can be observed. There seem to be regions along diagonals in the m r, k r plane with a + slope on logarithmic scale where the damping is high relative to the surrounding points. These regions are more clearly discernable for higher modes and low loss factors, see Figure 3.7. In these regions, a resonating mode in the viscoelastic material in-phase with a base material mode occurs implying that it functions like a tuned mass damper (TMD). In these cases, there is an optimal material loss factor. It does not hold that a 2

46 Chapter 3: VEM application method analysis and comparison higher loss factor is automatically better, because of a trade off between the strain energy in the VEM and the fraction of this energy (η) that is dissipated. In practice, it is not useful to design on these conditions as this only works in a narrow range of parameter values and a slight deviation between model and experiments results in significantly less damping. 2 damping ratio mode k r m r.5 Figure 3.7: Modal damping ratio β 3 as a function of mass ratio m r and stiffness ratio k r for low loss factor (i.e. η =.2) and x v = L b illustrating local "TMD" optima when using a discrete damping element. Next to the geometric and mechanical properties of the base material and the VEM block (captured by k r and m r ), the position x v of the discrete VEM damper influences the damping performance of each mode. To investigate this, the mass ratio m r is fixed at and the x v and k r are varied. A loss factor of η = is used and the Poisson s ratio is again set to ν v =.49. The damping ratios β and β 2 as a function of the varied parameters are shown in Figures 3.8a and 3.8b, respectively. 2 damping ratio mode.22 2 damping ratio mode 2.6 k r k r x v / L b x v / L b.4.2 (a) Mode (b) Mode 2 Figure 3.8: Modal damping ratios β (a) and β 2 (b) as a function of relative discrete damper position x v /L b and stiffness ratio k r for m r = and η =. It can be seen that the damping is zero when the damper is connected at the location of a node in the mode shape of the undamped bar. This makes sense of course as the VEM block will not deform. For a high stiffness ratio k r (relatively low VEM stiffness), the damping β k is optimal when it is placed at the location of maximal amplitude of the undamped mode shape. The optimal locations shift for a decreasing ratio k r though. For a low stiffness ratio k r, the local optima seem to converge to fixed locations (for this structure approximately where the next mode has a node). This again has to do with the trade off between stiffness and strain amplitude. The required 22

47 Section 3.3: Axial vibration damping VEM block stiffness to restrict motion at that location depends on position and is lower near a maximal amplitude point. When motion at the location of the damper is restricted, the VEM block will hardly deform resulting in little damping. The optimal locations for the first and second eigenmode are found to be approximately x v =.72L b and x v =.84L b, respectively. Shear A similar approach as for the discrete VEM damper in tension/compression will be used for the discrete VEM damper loaded in shear (see Figure 3.b). To investigate the influence of different parameters on the modal damping ratios β k, a different definition for k r is introduced. The effective stiffness of a generic shear block is [48] k shear = GA L, (3.8) with G = E 2(+ν), the shear modulus. The stiffness ratio is changed to k r = k b E b W t b /L b = k v E v(. (3.9) + iη) A v /2( + ν v )L v The damping ratios β and β 2 as a function of k r and m r for a loss factor η = and x v = L b are shown in Figures 3.9a and 3.9b respectively. k r 2 damping ratio mode k r 2 damping ratio mode m r 2 m r (a) Mode (b) Mode 2 Figure 3.9: Modal damping ratios β (a) and β 2 (b) as a function of m r and k r for η = and x v = L b for a discrete damper loaded in shear. Clearly, loading the VEM block in shear gives approximately the same results as loading it in tension/compression. It appears that the plots have shifted slightly in vertical direction. This is mainly caused by a difference between the static VEM block stiffness according to the expressions for k v used in Equations (3.6) and (3.9) and the stiffness according to the finite element model (near incompressibility). As a consequence, the stiffness ratios for shear and tension/compression do not completely match. The same conclusions can be drawn in varying the damper position as in the analysis of the viscoelastic damper deformed in axial direction. Loading in shear is preferred when the deformation amplitudes become larger as the stiffness is constant over a larger amplitude range for this method Axial vibration: Continuous layer damping For the case of continuous layer damping, a layer of viscoelastic material is added on top of the aluminum base structure. Only full coverage is considered, i.e. the length and width of the viscoelastic layer equal that of the bar (i.e. L = L b = L v ). For the constrained layer configuration, an extra (fully covering) aluminum layer is added on top of this. The bar and constraining layer deform only axially (bending is not considered) and are modeled using the D finite elements as mentioned above. In practice, the deformation will not be purely axial for an asymmetric 23

48 Chapter 3: VEM application method analysis and comparison structure as considered. The viscoelastic material is modeled using the same 2D elements as described in the previous section (using a plane stress assumption). The VEM is linked to the metal structure(s) by imposing that the horizontal displacements at the interface are the same as that of the metal part and that the other degrees of freedom are zero there. Free layer (FL) damping To dampen the vibrations of an axially vibrating beam, a layer of viscoelastic material with thickness t v is added. To investigate the performance of this approach for the load case considered, some parameter variations are done. The main design parameter is the thickness ratio t b /t v. This is incorporated in a mass ratio m r = m b m v = ρ bw t b L b ρ v W t v L v = ρ bw t b L ρ v W t v L = ρ bt b ρ v t v, (3.2) which again has a lower bound set to. Another parameter of interest is the length L of the bar, which is scaled with the thickness t b to make it non-dimensional. These two parameters do not uniquely define the dynamic behavior and the damping performance, but are the design variables of main interest. In computations, t v and L are varied to obtain different m r and non-dimensional beam lengths. See Table 3. for the other parameter values used. Table 3.: Parameter values used in free layer method damping performance analysis. Symbol Description Value W Width of the bar (and VEM layer).2 m t b Thickness of the bar.2 m ρ b Mass density of the base structure (aluminum) 27 kg/m 3 ρ v Mass density of the VEM (typical) 3 kg/m 3 E b Young s modulus of the base structure (aluminum) 69 GPa E v Storage modulus of the VEM (typical) 5 MPa ν v Poisson s ratio of the VEM (typical).49 m x (= m) Number of elements in axial direction 25 m y Number of elements in lateral direction (VEM) 8 2 damping ratio mode.5 2 damping ratio mode L / t b.3.25 L / t b m r 2 m r (a) Mode (b) Mode 2 Figure 3.: Modal damping ratios β (a) and β 2 (b) with FL damping as a function of m r and relative beam length for modes and 2 and η =. The mass ratio m r and scaled length L/t b are varied and the corresponding damping ratios β and β 2 for the first two modes are computed, see Figures 3.a and 3.b respectively. Similar trends are seen for different VEM loss factors η. There seems to be only a very narrow region where the free layer damping is effective. The 24

49 Section 3.3: Axial vibration damping viscoelastic layer acts as a TMD in this region. The damping does not increase with increasing material loss factor η as one may expect. A higher loss factor broadens the high damping region though. This again is because of the trade off between strain energy and the fraction of it that is dissipated. A higher loss factor η means more coupling between the deformations and therefore smaller local maximal amplitudes. There is an optimal loss factor (which cannot be seen from the figures). Figure 3. also shows that, in the TMD regions, the maximum damping values do not change for different modes, which was the case for the discrete dampers considered above. The optimal parameters (m r and L/t b ) are not the same for different modes though, so it is not possible to attain this maximum for all modes at the same time. For the parameter combinations in Figure 3., outside the TMD region, the amount of damping depends on the axial and shear stiffness of the viscoelastic layer. This can be illustrated by using a very small value for ρ v (e.g.. kg/m 3 ) in computations (see Figure 3. for the resulting damping ratio β 3 of mode 3). For that case, the VEM simply acts as a spring-damper. The trend in Figure 3. originates from a transition from mostly direct to mostly shear strain. This corresponds to the fact that shear stiffness decreases with increasing t v whereas axial stiffness increases. For small VEM thickness, the damping increases approximately linear with t v. For a relatively thick VEM layer, the damping does not increase anymore and levels off due to the fact that the VEM part farthest from the base structure hardly deforms. The attained damping levels are very low, because the VEM is much more flexible than the metal and thus stores, in a relative sense, only little strain energy. In practice, the metal will possess some inherent damping, which may already be larger than the added viscoelastic damping from the free layer damper. 2 damping ratio mode 3 x L / t b m r Figure 3.: Modal damping ratio β 3 using free layer configuration as a function of mass ratio m r and relative beam length L/tb for a small mass density ρ v of the VEM. Constrained layer (CL) damping Constrained layer damping is obtained by fixing an aluminum layer on top of the viscoelastic coating. This layer should result in higher (shear) strains in the VEM. The key design variables are the two layer thicknesses, i.e. t v and the thickness of the constraining layer t c. The length L of the truss again is another design parameter of interest. As done for the free layer method, the mass ratio m r of bar and VEM and the dimensionless length L/t b are varied first by varying t v and L respectively. For the aluminum constraining layer, the Young s modulus is set to E c = 69 GPa, the mass density is chosen ρ c = 27 kg/m 3, and an initial thickness of t c = t v is used (t c is varied later on). For the other parameter values, see Table 3.. Again, the damping ratios β and β 2 of the first two modes respectively are determined for combinations of m r and L/t b within certain ranges, for a loss factor η =, see Figure 3.2. The contour lines shown in Figure 3.2 resemble those of Figure 3. for the free layer damping approach. However, the optimal damping regions have shifted and the damping ratios within the range considered have increased. In the high damping region, the constrained layer damper functions as a tuned-mass damper. The constraining layer represents the mass that sways back and forth on the viscoelastic spring-damper combination. Again, designing on this is not practical as the region is quite narrow and a small deviation from the real parameters can cause a significantly lower damping ratio. Adding the constraining layer has increased the performance outside the TMD region as well. For increasing η, the maximal damping ratios decrease, but the high damping region broadens as was also found for the FL damping approach. 25

50 Chapter 3: VEM application method analysis and comparison 2 damping ratio mode.2 2 damping ratio mode L / t b L / t b m r.2 2 m r. (a) Mode (b) Mode 2 Figure 3.2: Modal damping ratios β (a) and β 2 (b) of modes and 2 using the CL configuration as a function of mass ratio m r and relative beam length L/t b for η =. The influence of constraining layer thickness t c will be considered next. It is scaled with the viscoelastic layer thickness t v to obtain the dimensionless quantity t c = t c t v, (3.2) which is varied in the range from. to by varying t c (note that t v is varied to get different values for m r ). The length of the structure is set to L =.2 m. The resulting damping ratios β and β 3 are depicted in Figures 3.3a and 3.3b respectively. damping ratio mode damping ratio mode t c / t v.5. t c / t v m r 2 m r.2 (a) Mode (b) Mode 3 Figure 3.3: Modal damping ratios β (a) and β 3 (b) using the CL configuration as a function of mass ratio m r and relative layer thickness t c for η =. Figure 3.3 shows parameter combinations, for which the constraining layer acts as a tuned mass damper (regions with high damping ratios). For the higher modes, multiple regions can be discerned. The difference between them is the way in which the constraining layer acts as a TMD. For the diagonals seen in Figures 3.3a and 3.3b (high m r and t c ), the CL sways back and forth and can be seen as a rigid mass. For the other optimum (see Figure 3.3b for m r 25 and small t c ), the first axial mode of the constraining layer and the considered mode of the base material match but are in opposite phase. Both vibrate axially with maximum energy resulting in high damping values since the VEM is strained much. The small circles in the contour plot originate from the fact that at a specific combination of variables, the system really is optimal as a TMD. 26

51 Section 3.3: Axial vibration damping damping ratio mode t c / t v m r Figure 3.4: Modal damping ratio β of mode as a function of mass ratio m r and relative constraining layer thickness t c for η =, where a small mass density of the VEM and CL is used in computations. When excluding the TMD effect by setting ρ v and ρ c to. kg/m 3 in computations, the damping ratio β of the first mode for η = becomes as shown in the Figure 3.4. As for the free layer method, when excluding the TMD effect, the damping performance is not nearly as good as when using discrete damping elements (see Figure 3.6a). The damping levels are larger for this method than for the free layer method though (see Figure 3.). Both CL and FL damping are able to introduce similar levels of damping for all modes. The β k s are larger for relatively thick constraining layer thicknesses and for every t c there is an optimal thickness for the VEM Axial vibration: Damping method comparison Now that the different methods of introducing damping to an axially vibrating beam are considered separately (using Equation (3.) as a criterion), the next step is to compare them. To do this, a case study with specific practical constraints is done. As described in Section 3.2, a fixed volume will be divided over VEM and base structure. The question is if the increase in damping obtained by adding viscoelastic material compensates for a possible reduction in eigenfrequency. The approach taken in this comparison is: An available reference volume is defined with a length of L b = 2 cm, width W of 2 cm and height t of 2 cm (completely used for the aluminium bar for the reference case with eigenvalues λ ref,k ). A length L b of 5 cm is considered as well. The height of the aluminum bar is reduced to t b = ( χ)t and the remaining volume V v = χv is used for the damper (see figure 3.4a). The volume fraction χ is varied over two different ranges, i.e. sometimes χ. and sometimes χ.25. The constraining layer takes up part of the available volume V v. A fixed fraction is used in the variations. From the study above, it is found that a relatively thick constraining layer w.r.t. the VEM layer is preferred. Therefore, composing V v of 5% and 5% viscoelastic material (and the rest for the CL) are considered. The storage modulus E v and loss factor η are varied since there are many viscoelastic materials available with different properties, which also vary with temperature and frequency over several orders of magnitude. The dynamic modulus is varied over the range.5 MPa E d = E v( + iη) 5 GPa. For the free layer and constrained layer damping methods, a distinction is made between adding the layers on one side of the structure or on both top and bottom. Three different damper positions are chosen for the discrete damper approaches (both tension/compression and shear), i.e. x v = L b, x v =.72L b, and x v =.84L b. The last two positions correspond to the optimal positions for the first and second mode, respectively. A fixed ratio between the length and cross sectional area of the discrete damping element is taken, i.e. 5 m is used. Note that for every χ this gives different dimension combinations. For each case and mode, the maximal value of the objective function Γ k is looked for in the specified range of χ and E d and used for comparison. 27

52 Chapter 3: VEM application method analysis and comparison The other parameter values are mentioned in Table 3.2. Note that the mass densities ρ v and ρ c are set to arbitrary low values since designing the dampers on the found tuned mass damper effects is not of interest. This might increase the damped eigenfrequencies slightly compared to the real values, which is somewhat compensated for by the fact that the damping will be slightly lower due to fact that inertial effects in the VEM are not present. Table 3.2: Parameter values used in comparison damping method performance for axial deformation. Symbol Description Value ρ b Mass density of the base structure (aluminum) 27 kg/m 3 ρ v Mass density of the VEM. kg/m 3 ρ c Mass density of the constraining layer. kg/m 3 E b Young s modulus of the base structure (aluminum) 69 GPa E v Storage modulus of the VEM 5 MPa E c Young s modulus of the constraining layer (aluminum) 69 GPa ν v Poisson s ratio of the VEM.49 m Number of beam elements in axial direction (for discrete damper) 5 Number of beam elements in axial direction (for FL and CL) 25 m x Number of elements in axial direction (VEM, all cases) 25 m y Number of elements in lateral direction (VEM, all cases) 8 Figure 3.5 shows examples of the objective function Γ for different viscoelastic damping approaches as a function of volume fraction χ and dynamic modulus E d. The maxima are summarized in Table 3.3. Appendix B also contains the found maximal values of Γ Γ 3 and includes results for η =.5, the maximal values for Γ 4, when the volume fraction is restricted to χ., and for a beam length of 5 cm. Table 3.3: Maximal value of Γ, Γ 2 and Γ 3 when χ.25,.5 MPa E d 5 GPa, and for loss factors η = or η = 2 (axial deformation). The color gives an indication of performance, where red corresponds to (relatively) low Γ k, green to (relatively) high Γ k and yellow indicates the intermediate range. η = η = 2 Γ Γ 2 Γ 3 Γ Γ 2 Γ 3 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v = L) Discrete damper, tension (at x v =.72L) Discrete damper, tension (at x v =.84L) Discrete damper, shear (at x v = L) Discrete damper, shear (at x v =.72L) Discrete damper, shear (at x v =.84L)

53 Section 3.3: Axial vibration damping Γ Γ E v *.8 E v * χ χ (a) Free layer damping (one sided) (b) Constrained layer damping (one sided, 5% VEM) Γ 7 Γ E v * E v * χ (c) Discrete damper tension (x v = L b ) χ (d) Discrete damper shear (x v = L b ) Figure 3.5: Illustration of Γ as a function of volume fraction χ and dynamic modulus E d for different damping methods and η =. From Table 3.3 it can be concluded that: The free layer damping method is least effective in axial vibration reduction by means of VEM. The constrained layer damping approach performs reasonably well for high constraining layer thicknesses and reduces the vibrations of the different modes almost equally. Adding the continuous layer dampers on one side or using a two-sided configuration (for the same volume) gives approximately the same maximal values for Γ k. The discrete dampers are clearly most efficient for this load case. They can be used to address a specific mode by tuning their position and stiffness and really reduce the modes contribution to axial vibrations. However, the effect decreases with increasing mode number. Even when the damper is situated at the location optimal for mode two, the maximum of objective function Γ is larger than the maximal Γ 2. There are some differences in the obtained Γ k for loading in tension or shear, caused by the difference in effective modulus. For shear, G v = E v/2( + ν v ) determines the stiffness (for fixed L v /A v ) whereas for tension/compression, the Young s modulus is the critical parameter. To attain the same stiffness a dynamic modulus approximately three times larger is required when loading in shear than in tension. The required value might be outside the considered range. For the other cases specified (see Appendix B), the same conclusions can be drawn. In reducing axial vibrations it can be concluded that discrete dampers should be used to address specific modes, whereas constrained layer damping is preferred to add damping over a wider range of eigenmodes. 29

54 Chapter 3: VEM application method analysis and comparison 3.4 Transversal vibration damping A commonly encountered structural element is a beam. The mode shapes with lowest frequency of this type of structure usually consist of transversal deflections. Therefore, the effectiveness of the different damping methods is investigated next for bending modes of a simply supported beam with rectangular cross section. Clamped boundary conditions are considered as well. As done for the axial vibration analysis, the different methods are investigated separately first using the damping ratio (see Equation (3.)) and are then compared on the basis of the objective function (3.9). Figure 3.6 shows a schematic of the structure. The beam is modeled using m D Euler beam elements with three degrees of freedom per node, i.e. two translations and one rotation, and n = m + nodes in total. In this, a plane stress assumption is employed in the directions perpendicular to the longitudinal direction and the beam is assumed slender. The latter implies that there is negligible shear deformation in the metal structure. See Section C.3 for the derivation of these elements. _ t t b W e b n-2 n- n un v n θ n L b (a) (b) Figure 3.6: Schematic representation of a space that is partly filled with a beam structure (a) and its representation by a D finite element model (b). The elements are assembled to a system mass and stiffness matrix by imposing C continuity of the degrees of freedom u b,i, v b,i, and θ b,i (where i =, 2,..., n). The displacements of the midplane of the beam are suppressed at the ends (pinned-pinned boundary condition). To introduce damping, viscoelastic material is modeled using the finite elements specified in Section 3.3. and connectivity and equilibrium of internal forces and moments are used to connect the VEM to the beam. The eigenvalues are computed to assess damping. Table 3.4 lists the parameter values used for assessing the different VEM applications addressed in the following sections unless stated otherwise. Table 3.4: Parameter values used in comparison of VEM applications for transversal vibrations (used in computations below unless stated otherwise). Symbol Description Value W Width of the beam (and VEM layer).5 m t b Thickness of the beam. m ρ b Mass density of the base structure (aluminum) 27 kg/m 3 ρ v Mass density of the VEM 3 kg/m 3 E b Young s modulus of the base structure (aluminum) 69 GPa E v Storage modulus of the VEM 5 MPa ν v Poisson s ratio of the VEM.49 m Number of beam elements in axial direction (for discrete damper) Number of beam elements in axial direction (for FL and CL) 25 m x Number of elements in axial direction (VEM, for all cases) 25 m y Number of elements in lateral direction (VEM, for all cases) 8 3

55 Section 3.4: Transversal vibration damping 3.4. Transversal vibration: Discrete dampers The approach considered first for dampening the bending modes of a beam is employing a small block of viscoelastic material as a discrete damping element. See Figure 3.5 for a schematic of how the VEM is modeled. A constant complex Young s modulus is used. Note that this corresponds to hysteretic damping, which is non-causal. The loss factor η actually should have equal sign as the frequency (or imaginary part of the eigenvalue), which is not included. This is justified, because only the effect on the magnitude of the real and imaginary parts of the eigenvalues and the eigenvectors is considered and dynamic responses are not really of interest yet. A distinction is made between loading the discrete damping element in tension/compression and in shear to investigate if the deformation type affects the damping performance. Tension/Compression A block of VEM with square cross section is connected at one end to the fixed world and at the other end (via a rigid link) to the beam at a location x v along the length of the base structure (see Figure 3.2a). Similarly as done for the axially loaded bar, a stiffness ratio k r and mass ratio m r are defined and varied. These do not define the dynamic properties of the system completely, but are relevant intuitive quantities. The stiffness of the VEM block is approximated by k v = E v( + iη) A v L v (3.22) as is used in Section Note that this is a static stiffness. A similar definition is required for the beam. When a transversal force F is applied at a location a along a pinned-pinned beam of length L, with second moment of area I and Young s modulus E, the resulting deflection at that location equals [48] This can be translated into a spring stiffness for the structure of interest as v(x = a) = F a2 (L a) 2. (3.23) 3EIL k b = F v = The stiffness ratio for this damping method is now defined as 3E bi b L b x 2 v(l b x v ) 2. (3.24) k r = k b k v = 3E bi b L b /x 2 v(l b x v ) 2 E v( + iη) A v /L v, (3.25) with I b = W t 3 b /2 the second moment of area of the beam. The mass ratio remains the same as defined earlier for the axial case in Section 3.3., i.e. m r = m b = ρ bw t b L b. (3.26) m v ρ v A v L v The mass ratio m r and stiffness ratio k r are varied by taking different combination of A v and L v. For the other parameter values used in the computations, see Table 3.4. For the first analysis, the positioning of the damper is chosen rather arbitrarily at x v =.4L since the lowest modes do not have a zero deflection point at that location. The modes are sorted based on highest MAC value between the current modes and those found for the previous k r, where the stiffness ratio is varied in decreasing order. For the highest k r, the undamped modes are used as a reference for sorting. This approach is taken because the mode shapes change quite a lot when varying k r and the sorting method used in Section 3.3. is found to be erroneous for this case. This problem will be further discussed below. The effects of the parameter variations on β and β 2 are depicted in Figures 3.7a and 3.7b, respectively. The figures illustrate that, to obtain maximal β k, there is an optimal stiffness ratio k r for each mode k that hardly depends on mass ratio m r. An analytic expression for the location of this optimum is not found and could be part of further research. The optimum originates from the compromise between high stiffness and large deformations in the VEM which are both desired for high damping. As the stiffness of the VEM increases, the mode shapes of the beam change and converge to those with a node at the location of the damper, minimizing the deformation in the discrete damping element. The damping increases slightly when the mass ratio m r reduces because inertia effects affect the mode shape in the VEM. Still, a small viscoelastic block can attain similar damping ratios β k for a specific mode as a larger one with the same stiffness. Increasing the loss factor η increases the obtained damping levels β k and broadens the high damping region, but it remains at the same parameter combinations k r. Note that the optimal k r differs quite a lot for the different modes, which makes it difficult to add a lot of damping to many modes at the same time (for a specific configuration). 3

56 Chapter 3: VEM application method analysis and comparison damping ratio mode.25 damping ratio mode k r.2.5 k r m r 2 2 m r.2 (a) Mode (b) Mode 2 Figure 3.7: Modal damping ratios β (a) and β 2 (b) when using discrete dampers in tension as a function of m r and k r for η = and x v =.4L b. As was found for the axial vibration case, there are certain combinations of k r and m r (not visible in Figure 3.7) where the damping is relatively large (especially for small η) since some sort of internal tuned mass damper effect is encountered. This damping methodology for discrete dampers is not investigated here further. Where to connect the damping element to the simply supported beam is considered next. The damper location x v is varied from zero to half the beam length (symmetry). The mass ratio is fixed at m r = 5 since it had hardly any influence and the stiffness ratio k r is varied as well. Sorting of the modes is again done using the Modal Assurance Criterion (Equation (3.7)) using the eigenvectors at the previous k r as a reference and varying k r in decreasing order. The resulting modal damping ratios β and β 2 for η = are shown in Figures 3.8a and 3.8b, respectively. A clear position optimum is seen for each mode, which is not at the location of maximal amplitude as one might expect. For the first eigenmode, this is at x v =.5L b (incidentally at the maximal amplitude point), for the second at about x v =.33L b, and the third mode is optimally damped for x v =.25L b, for example. The graphs also illustrate the optimum in stiffness ratio k r. For high stiffness ratio (and a fixed beam geometry), the optimum is at the location of maximal mode amplitude. Note that the stiffness of a beam is higher near the supports and k r thus varies when moving the damper for fixed beam and VEM properties and dimensions. 2 damping ratio mode.3 2 damping ratio mode k r.5 k r relative position relative position (a) Mode (b) Mode 2 Figure 3.8: Modal damping ratios β (a) and β 2 (b) as a function of relative discrete damper position x v /L b and stiffness ratio k r for m r = 5 and η =. Figures 3.8a and 3.8b appear to contain some discontinuity in the damping ratio for very low k r (relatively stiff VEM dampers) when changing the position of the damper. This has to do with a sudden change of mode shape for small differences in damper position such that it is difficult to see, which mode is which. This effect seems to have similar characteristics as in buckling. To illustrate this, see Figure 3.9 which depicts the mode shapes of 32

57 Section 3.4: Transversal vibration damping the first two modes as the VEM stiffness increases for different x v. The positions x v are chosen far away from such a discontinuity, just left from it, and just right of a transition point to show the difference in convergence for increasing VEM stiffness (decreasing k r ). Real part of modeshape mode increasing stiffness VEM x / L b [ ] (a) x v =.2L b Real part of modeshape increasing stiffness VEM mode x / L b [ ] (b) x v =.5L b Real part of modeshape increasing stiffness VEM mode x / L b [ ] (c) x v =.43L b Real part of modeshape mode 2 increasing stiffness VEM x / L b [ ] (d) x v =.26L b Real part of modeshape increasing stiffness VEM mode x / L b [ ] (e) x v =.46L b Real part of modeshape mode 2 increasing stiffness VEM x / L b [ ] (f) x v =.3L b Figure 3.9: Change of the first two mode shapes as the VEM stiffness increases for different damper positions x v. 33

58 Chapter 3: VEM application method analysis and comparison Shear The block of VEM with square cross section is now oriented such that a deflection of the beam at a location x v along the length of this base structure gives the same deflection of the top of the VEM but in shear (see Figure 3.2b). A stiffness ratio k r and mass ratio m r are defined and varied. The mass ratio m r is the same as defined for tension/compression. The stiffness of the VEM block is approximated, similarly as in Section 3.3., as k v = E v( + iη) A v 2( + ν v )L v. (3.27) Note that this is a static stiffness. The stiffness ratio for this load case and damping method is now defined by k r = k b k v = 3E bi b L b /x 2 v(l b x v ) 2 E v( + iη) A v /2( + ν v )L v. (3.28) The mass ratio m r and stiffness ratio k r are varied again by taking different combinations of A v and L v. See Table 3.4 for the other parameter values. The results of β and β 2 for the parameter variations are depicted in Figures 3.2a and 3.2b, respectively. The figures show results similar to loading the discrete damper in tension/compression. It clearly has little effect on the attained damping levels whether the viscoelastic material is deformed axially or in shear. As mentioned before (for the case of axial vibrations), there are differences in the characteristics of both deformation methods though. In varying the damper position x v and stiffness ratio k r, the shear block also gives almost the same results as the block that is deformed axially. damping ratio mode.25 damping ratio mode k r.2.5 k r m r 2 2 m r.2 (a) mode (b) mode 2 Figure 3.2: Modal damping ratios β (a) and β 2 (b) for discrete shear dampers as a function of m r and k r for η = and and x v =.4L b Transversal vibration: Continuous layer damping Layers of viscoelastic material are commonly applied for dampening transverse vibrations of structures (an approach often considered in literature, see e.g. [3, 6, 5]), which is why the effectiveness of continuous layer damping is considered here as well. The layer of viscoelastic material is modeled using the same elements as described in Section The viscoelastic material is linked to the metal structure(s) by imposing continuity of the displacements and equilibrium of internal forces at the interface. The former means that for the degrees of freedom (DOF) at the interface between VEM and base structure the following must hold u v,i = u b,i θ b,i t b /2 v v,i = v b,i θ v,i = θ b,i i =, 2,..., n(= n x ) (3.29) where u v,ij, v v,ij and θ v,ij are the x-displacement, y-displacement and rotation of the viscoelastic material at the ith node in x-direction and jth node in y-direction. Similar connectivity conditions are used when a constraining layer is added to link this layer to the VEM or when the viscoelastic layer is added to the bottom of the beam. See Figure 3.2 for an illustration of the DOFs and connectivity conditions. The pinned-pinned (or clamped-clamped) boundary conditions are not applied to the VEM layer or constraining layer. 34

59 Section 3.4: Transversal vibration damping u b,i θ b,i+ VEM connected to the beam at this side v b,i t b Figure 3.2: Schematic representation of a part of the beam (undeformed and bended) illustrating the degrees of freedom and connectivity conditions linking the beam to the VEM. Free layer (FL) damping A fully covering layer of viscoelastic material is added on top of the pinned-pinned beam. Its thickness t v is varied to obtain different mass ratios m r as defined by Equation (3.2). Next to that, different values for the beam length L b = L v = L are considered. For each of the parameter combinations, the eigenvalues and -vectors are computed and from those, the damping ratios β k. The resulting modal damping ratios for the first two modes and η = are shown in Figure 3.22 and are overall very small. The attained values are larger in the region where the VEM is acting as TMD (near the lower bounds of m r and L/t b ) through its internal mass and compliance. To eliminate these effects from the results, the density of the viscoelastic material used in computations is set to ρ v =. kg/m 3. The results are shown in Figure damping ratio mode x damping ratio mode L / t b L / t b m r m r.2. (a) mode (b) mode 2 Figure 3.22: Modal damping ratios β (a) and β 2 (b) for FL damping as a function of m r and beam length L for η =. In Figures 3.23a and 3.23b, it is clear that the damping increases with increasing VEM layer thickness (decreasing m r ) and is hardly influenced by the length L of the structure. Actually, the deformations are greater farther from the base structure and a thicker VEM layer is thus preferred. Sometimes, a separation layer is used to increase the distance between viscoelastic material and the beam [32, 5]. This is not considered here. It is also found that a higher Young s modulus E d gives higher damping ratios β k as the stiffness of the layer becomes larger and, therefore, more strain energy is present in the VEM layer. Furthermore, the maximal damping ratios are similar for each mode. 35

60 Chapter 3: VEM application method analysis and comparison 2 damping ratio mode x damping ratio mode 2 x L / t b.2 L / t b m r.2 2 m r.2 (a) mode (b) mode 2 Figure 3.23: Modal damping ratios β (a) and β 2 (b) for FL damping as a function of m r and relative beam length L/t b, with a small mass density ρ v =. kg/m 3 for the VEM. Constrained layer (CL) damping The constrained layer approach is considered next. An aluminum layer is added on top of the viscoelastic layer. It is modeled using the same finite elements as used for the beam, with three degrees of freedom per node. The key design parameters are the viscoelastic layer thickness t v, the length of the structure L and the constraining layer (CL) thickness t c. First, the viscoelastic layer thickness t v (or mass ratio m r ) and length L of the structure are varied. For the constraining layer thickness, a dimensionless quantity t c = t c t b (3.3) is introduced (different from that used for the axial vibration damping case). For CL thicknesses t c larger than that of the beam, the two could as well be swapped to get a similar structure with smaller t c, which is why t b is used for scaling and only t c is considered. Note that this boundary is actually somewhat above unity as the beam is pinned and the constraining layer (and VEM layer) is not. First, t c is set to.25 and the mass ratio m r and beam length L are varied. The results for β and β 2, when varying m r and L/t b (and η = ) are shown in Figures 3.24a and 3.24b, respectively. The constrained layer method appears effective in adding damping to a beam for transversal deflection modes. The trends are similar for different loss factors, but the obtained damping increases with η. 2 damping ratio mode. 2 damping ratio mode L / t b L / t b m r.2. 2 m r.2. (a) Mode (b) Mode 2 Figure 3.24: Modal damping ratios β (a) and β 2 (b) for CL damping as a function of m r and L/t b for η = and t c =

61 Section 3.4: Transversal vibration damping The results show an optimal damping region for each mode. Unlike encountered above, this is not linked to a TMD effect. Especially apparent for low L, an optimal m r is present. The optimum originates from a trade off between maximal strain and total strain energy. The maximal shear strain reduces as t v increases, whereas the deformed volume of the VEM increases. For larger L or higher VEM dynamic modulus E d the optimum shifts towards a thicker VEM layer. Both E d and L affect the shear stiffness of the VEM and the beam stiffness decreases for increasing length L. These two aspects alter the strain energies in the two parts for a given deformation state. Again, the damping optimum depends on the energy stored in the two parallel springs (beam and VEM). In investigating what the constraining layer thickness t c should be to attain maximal damping β k, the mass density of the viscoelastic material and constraining layer are set to ρ v = ρ c =. kg/m 3 to remove tuned mass damper regions. The length of the beam is set to L = 3 cm and for the other parameter values, see Table 3.4. The mass ratio m r and constraining layer thickness t c are varied and the resulting damping ratios are depicted in Figure Similar trends are seen for different loss factors η of the viscoelastic material. damping ratio mode.8 damping ratio mode t c / t b. t c / t b m r.2 2 m r.2 (a) Mode (b) Mode 2 Figure 3.25: Modal damping ratios β (a) and β 2 (b) for CL damping as a function of m r and relative constraining layer thickness t c for η =. It is possible to obtain damping ratios of well above. for a structure like this by means of adding a viscoelastic and constraining layer. This was also achieved by the discrete dampers, but here it is not just for a limited number of modes (in contrast to the discrete damping element approach). For the discrete VEM dampers, it was found difficult to add a lot of damping to different modes at the same time, because of the relatively large difference in optimal stiffness ratio k r for different modes (see Figure 3.7). Figure 3.25 shows that it is best to use a relatively thick constraining layer. The dissipation is highest when the CL and beam match with respect to stiffness (t c t b when both layers are made of the same material). This is not completely shown in the figures above as slight differences occur due to the fact that inertia effects of the CL and VEM are not accounted for (small ρ v and ρ c ). There is an optimal VEM layer thickness t v for each CL thickness t c. The location of this optimum shifts to lower m r for higher VEM layer modulus. The optimum shifts to higher m r for increasing mode number. The optimal design depends on the base structure of interest Transversal vibration: Damping method comparison The approaches described above are compared via a case study with specific practical constraints. A similar approach as described in Section is employed where the volume fraction χ and dynamic modulus E d of the VEM are varied for the different damping approaches and different loss factors η. The approach used in the comparison case study is (see Table 3.2 for the other parameter values used): Again, an available reference volume V = L b W t is defined. A length of L b = 3 cm, width of W = 5 cm and maximal height of t = cm is considered. This volume is completely used to obtain the reference eigenvalues λ ref,k, both for pinned-pinned and clamped-clamped boundary conditions. As described in Section 3.3.3, the height of the aluminium beam is reduced to t b = ( χ)t and the remaining volume V v = χv is used for the damper (see figure 3.6a). The volume fraction χ is varied over two different ranges, i.e. sometimes χ. and sometimes χ

62 Chapter 3: VEM application method analysis and comparison The constraining layer takes up a fixed fraction of the available volume V v. From the study above, it is found that a relatively thick constraining layer w.r.t. the VEM layer is preferred. Therefore, composing V v of 5% and 5% viscoelastic material (and the rest for the CL) are considered. Also for this load case, the storage modulus E v and loss factor η are varied since there are many VEMs available with different properties which also vary with temperature and frequency over several orders of magnitude. The dynamic modulus is varied over the range.5 MPa E d = E v( + iη) 5 GPa. Again, for the free layer and constrained layer damping methods, a distinction is made between adding the layers on one side of the structure or on both top and bottom. Three different damper positions are chosen for the discrete damper approaches (both tension/compression and shear), i.e. x v =.5L b, x v =.33L b, and x v =.25L b, which correspond to the optimal positions for the first three modes, respectively. A fixed ratio between the length and (square) cross sectional area is taken, i.e. 2 m is used, which, together with the VEM volume, defines the dimensions for every χ. In the specified range of χ and E d, the maximal value of the objective function Γ k is found for each case and mode and used for comparison. Γ Γ E v * E v * χ χ (a) Free layer damping (one sided) (b) Constrained layer damping (one sided, 5% VEM) Γ Γ E v * E v * χ (c) Discrete damper tension (x v =.5L b ) χ (d) Discrete damper shear (x v =.5L b ) Figure 3.26: Illustration of Γ as a function of volume fraction χ and dynamic modulus E d for different damping methods and η =. TMD effects are not considered (by choosing low values for ρ v and ρ c ) as they are not of interest. Figure 3.26 shows examples of the objective function for different VEM damping approaches as a function of volume fraction χ and dynamic modulus E d. The maxima are summarized in Table 3.5. Appendix B also contains the found maximal modal objective function values for η =.5, the fourth mode, when the volume fraction is restricted to χ. and for a clamped-clamped boundary condition. 38

63 Section 3.5: Conclusions From Table 3.5 it can be concluded that: The free layer damping is least effective in transversal vibration reduction by means of VEM. Again, CL damping performs better than FL damping and addresses different modes almost equally much. Adding the layers on one side is only a little bit more effective than on two sides for the same volume/mass. Also for the beam bending configuration, the discrete VEM dampers are most efficient in addressing a specific mode. There are no differences in the obtained maximal Γ k for loading in tension or shear. The same conclusions can be drawn for a clamped-clamped configuration. As for axial vibrations, it can be concluded that discrete VEM dampers should be used to address specific modes, whereas constrained layer damping is better for adding damping over a wider range of eigenmodes. Table 3.5: Maximal value of Γ, Γ 2 and Γ 3 when χ.25,.5 MPa E d 5 GPa, and for loss factors η = or η = 2 (bending deformation). The color gives an indication of performance, where red corresponds to (relatively) low Γ k, green to (relatively) high Γ k and yellow indicates the intermediate range. η = η = 2 Γ Γ 2 Γ 3 Γ Γ 2 Γ 3 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v =.5L) Discrete damper, tension (at x v =.33L) Discrete damper, tension (at x v =.25L) Discrete damper, shear (at x v =.5L) Discrete damper, shear (at x v =.33L) Discrete damper, shear (at x v =.25L) Conclusions In this chapter, a set of case studies is done to investigate the effectiveness of different approaches of applying viscoelastic material to a structure to introduce damping and minimize structural vibrations (high Γ k ). The main conclusions are summarized below. For a large part, similar conclusions are drawn for axial and transversal vibrations, but differences are described below as well. Discrete damping elements (in tension/compression or shear) Applying small blocks of VEM as discrete damping elements is most effective to add damping to specific eigenmodes and suppress vibrations (highest objective function values Γ k ). The stiffness of the VEM block should be tuned such that damping is optimal. This optimum is a trade off between high stiffness and strain to attain highest strain energy in the VEM. When the discrete dampers are very stiff compared to the base structure, the mode shapes may drastically change compared to the undamped case. Where to attach the damping blocks depends on stiffnesses of the structure and the VEM block, and mode of interest. It is found that the optimal location is at the maximal mode amplitude for relatively low VEM 39

64 Chapter 3: VEM application method analysis and comparison stiffness and shifts towards different points for higher stiffness of the VEM block. For the relatively simple structures considered, this is at the locations where the following mode has a node, which might not be the case for more complex structures. Highest damping β k is attained for relatively high VEM stiffness. There are certain local optima in damping ratio β k for this technique. For these optimal mechanical properties and dimensions, a resonating mode is found in the VEM. The viscoelastic material acts somewhat as a TMD, for which there is an optimal (relatively low) η originating for the trade off between high stored strain energy in the material and fraction that is dissipated. It is not very practical to design on these local optima as slight differences between model and reality will result in significantly lower damping ratios β k. Loading the discrete damper in tension or shear makes little difference. The fact that the effective modulus in tension is about three times larger than that in shear is not limiting for a bending beam as the required stiffness can easily be obtained, but it might be limiting for reducing axial vibrations. Loading the damping element in shear gives a larger linear deformation range though. The limitations of this damping approach is that it might require auxiliary structures to link the dampers to the fixed world and that it only addresses a limited number of eigenmodes. Free layer damping Free layer damping is not very effective compared to the other methods, both for dampening axially or transversally vibrating structures. The performance is better when the free layer acts as a TMD, but again, designing for this is not very practical as the conditions need to be just right and damping depends quite a lot on the material properties. This method adds similar (low) damping levels to all mode shapes for full coverage. The damping β k increases with VEM Young s modulus and VEM layer thickness (for a fixed beam thickness). For axial deformation, there seems to be a limit to this, because mainly the part of the VEM close to the base structure deforms. This limiting case will not easily be encountered in practical structures. For bending modes, the performance is best when the distance between base structure and viscoelastic material is largest. This can be attained by adding a separation layer (not investigated here) or by using a relatively thicker VEM layer. Applying the free layer on one or both sides with the same mass makes little difference, but on one side is slightly better for the bending load case. An advantage of this method is its simple application. Constrained layer damping Adding a constraining layer on top of the viscoelastic layer improves damping β k significantly, especially for bending modes. The primary deformation changes from direct strain to shear strain. The attained damping levels β k are higher than for free layer damping but not as high as the maximal levels for the discrete damper methods. The maximal objective function values Γ k found vary only little with mode shape for full coverage of the structure. This method is thus quite effective to dissipate a reasonable amount of vibration energy over a broad frequency range. Unlike for the free layer method, there is an optimal thickness t v for the viscoelastic layer. This has to do with the compromise between maximum strain and total strain energy. A larger t v implies smaller strains but in a larger volume and vice versa. Both aspects combine into an optimal thickness, where the total strain energy is maximal. Both for axial and bending modes, it is best to use a relatively thick (or stiff) constraining layer compared to the VEM. For bending it is optimal if the stiffness of the constraining layer matches that of the base structure, because maximal shear strain is reached at the neutral line. For some dimensions, a maximum in damping is obtained through a sort of tuned-mass damper effect of the CL. Two effects are found. First, the constraining layer could in fact be seen as a rigid mass on the viscoelastic spring damper combination. Another possibility is that there is a comparable mode (but of different order) in the constraining layer and base layer but in opposite phase. For the same reason as mentioned for the other methods, it is not advised to design on these TMD conditions. Applying the layers on one or both sides with the same mass makes little difference. The simple application of constrained layer damping is an advantage of this method. 4

65 Chapter 4 Case study: Modeling and analysis To assess application of VEM damping for a more complex structure and to validate the modeling, various viscoelastic damping approaches will be applied to an (almost) undamped structure. The undamped and damped structures are modeled, realized, and the dynamics are characterized experimentally. Damping is obtained by adding viscoelastic material using the conclusions of the previous chapter as a design guideline. An open aluminum box structure is chosen for this case study. The 3D box obviously has a more complex geometry than a beam (used in the previous chapter) and can be regarded as a simplification of some common structural elements, e.g. a machine housing, object carrier, etc. In this chapter, the modeling and analysis of the open box is considered. In Section 4., the general finite element (FE) modeling approach of the structure is discussed. In Section 4.2, the undamped box structure is designed using some practical constraints and objectives. After that, different VEM damping approaches for the open box are considered. In Section 4.3, some modal and frequency response function analysis techniques incorporating the frequency dependency of the material properties are introduced. Finally, in Section 4.4, the damping materials are selected and modeled and the dimensions are optimized to some extent. 4. FEM modeling A finite element model is made using Matlab to model the structure of interest. Matlab is used instead of a commercial FEM package for necessary modeling freedom when introducing viscoelastic damping and avoiding a gray box approach. ANSYS 4. [5] is still used to validate the finite element model of the undamped structure. This section explains how the FE model of the box and viscoelastic material is constructed. 4.. Base structure The open box structure consists of five plates, which are joined at the edges. The plates are relatively long and wide compared to their thickness. That is why they will be modeled using shell elements. The displacement field within the elements is interpolated between the degrees of freedom at a finite number of nodes. For shell elements, these nodes are arranged on a 2D grid and deflection variations through the third dimension (the thickness of the plates) are based on extrapolation. The box is modeled with elements, which are rectangular when expressed in iso-parametric coordinates. Since the walls are all rectangular as well, a rectangular grid is used in the local coordinates too. Different forms of these elements are commonly used, i.e. four, eight and nine node elements (see Figure 4.). Combinations, for different degree of freedom types, are also possible and will be employed in this study. First, the general derivation of the contributions to the mass and stiffness matrix of a single shell element will be explained. The kinetic and potential energy are used for obtaining these matrices, respectively. The elements are assembled to construct the walls of the box. Subsequently, the walls will be assembled using specific edge elements. Modeling of a plate Consider a rectangular plate with x- and y-direction defined along the length and width respectively and the z axis pointing in the direction normal to it (out-of-plane direction). The mass density is given by ρ, E is its Young s modulus, and ν is Poisson s ratio. The plate has a thickness t and is divided into m x elements of length e x in x-direction and into m y elements of length e y in y-direction. Looking at a single element from arbitrary x j to 4

66 Chapter 4: Case study: Modeling and analysis x j+ and y j to y j+, the coordinates in this element can be transformed to iso-parametric coordinates ξ and ζ ranging from to using x = 2 ( ξ)x j + 2 ( + ξ)x j+ = x j + 2 ( + ξ)(x j+ x j ) = x j + 2 ( + ξ)e x (4.) and y = 2 ( ζ)y j + 2 ( + ζ)y j+ = y j + 2 ( + ζ)(y j+ y j ) = y j + 2 ( + ζ)e y. (4.2) 4 ζ 3 4 ζ 3 4 ζ ξ 8 6 ξ ξ (a) (b) (c) Figure 4.: Rectangular elements in iso-parametric coordinates ξ and ζ illustrating a four (a), eight (b) and nine (c) node element including the node numbering. The elements are composed of a specific number of nodes with a certain number of degrees of freedom (DOF) per node which define the displacements or rotations in the material at that location. The element type and degrees of freedom will be chosen below after a comparison of accuracy. First, the kinetic and potential energy will be used to find the generic element mass and stiffness matrix contributions for a shell element. This requires expressions for the displacement field within the material depending on specific degrees of freedom. The degrees of freedom of a single element are captured in the column q e. The displacements within the element are found by interpolating these degrees of freedom using polynomials. This is done by multiplying the degrees of freedom with corresponding shape-functions, the shape of which depends on the order of the polynomial, which in turn is linked to the number of nodes and degrees of freedom. The displacements u, v, and w in x, y, and z-direction respectively within the element can thus be written as u(ξ, ζ, z) v(ξ, ζ, z) = N u(ξ, ζ, z) N v (ξ, ζ, z) q e = N(ξ, ζ, z)q e (4.3) w(ξ, ζ, z) N w (ξ, ζ, z) with N u (ξ, ζ, z), N v (ξ, ζ, z) and N w (ξ, ζ, z) rows containing the shape-functions corresponding to the DOFs that define the displacements u, v, and w respectively. These functions are combined into a matrix N. The interpolation of degrees of freedom is used next to construct a generic expression for the mass matrix contribution corresponding to a single shell element. Any non-constant state of the degrees of freedom implies that the structure has kinetic energy. The kinetic energy T e of a single element with volume V e can be expressed as [47] T e = 2 ρ V e ( u 2 + v 2 + ẇ 2) dv (4.4) = 2 ρ V e q T e N T N q e dv = 2 ρ yj+ = 2 ρ qt e xj+ t/2 y j x j t/2 t/2 t/2 t/2 q T e N T N q e dzdxdy N T N dx dy dξ dζ dzdξdζ q e = 2 ρ qt e t/2 N T N 4 e xe y dzdξdζ q e = 2 qt e M e q e, (4.5) 42

67 Section 4.: FEM modeling with M e the mass matrix corresponding to a single element expressed as t/2 M e = ρ N T N t/2 4 e xe y dzdξdζ. (4.6) Next, the contributions to the element stiffness matrix will be derived based on stresses and strains in the plate linked to the degrees of freedom. Any non-rigid displacement state will give rise to strains and stresses in the material. Potential energy is then stored in the material, which is assumed to be isotropic. The strains ε T = [ ε xx ε yy γ xy γ xz γ yz ] and corresponding stresses σ T = [ σ xx σ yy τ xy τ xz τ yz ] are linked to each other by means of a constitutive relation. The first subscript represent the direction of the strain (or stress) and the second illustrates the direction of the plane normal at the location of that strain (or stress). A plane stress assumption is employed, which means that the stress σ zz = and the direct strain ε zz in that direction is merely due to the Poisson effect. The constitutive relation is given by [52] σ = E + ν ν ν ν ν 2 2 sym 2 ε Dε. (4.7) For these elements, a factor κ that compensates for shear variations through the thickness is used. The diagonal entries in D corresponding to γ xz and γ yz are multiplied by this factor, which is usually taken equal to π 2 /2 or 5/6 [47]. For small strains, the the displacements in the material can be linked to the strains as follows [52] ε xx = u x ε yy = v y γ xy = u y + v x which can be expressed as x y ε = y x z x z y u v w γ xz = u z + w x γ yz = v z + w y, (4.8) x y = y x Nq e Bq e. (4.9) z x z y Using the relations for stress and strain, the potential energy U e due to deformation within the element considered can be written as [47] U e = σ T εdv (4.) 2 V e = ε T D T εdv 2 V e = 2 yj+ xj+ t/2 y j x j t/2 t/2 q T e BT D T Bq e dzdxdy = 2 qt e t/2 B T D T B 4 e xe y dzdξdζ q e = 2 qt e Ke q e, (4.) with K e the stiffness matrix corresponding to a single element. After assembling all elements and using Lagrange s equation, the equations of motion of the finite element structure model can be obtained, i.e. M q + Kq = F, (4.2) with M and K the complete mass and stiffness matrix respectively, q a column with the degrees of freedom and F the force vector. The element matrices depend on how the displacements are interpolated and which degrees of freedom are used. Different options are considered and compared. See Appendix C, Section C.2, for an overview of the considered interpolation methods and shell element types and a comparison of obtained accuracy. It is checked 43

68 Chapter 4: Case study: Modeling and analysis if the eigenfrequencies of a plate modeled using these elements converge to the analytical values from [53] for thin plates. Slightly lower frequencies are found for relatively thick plates (i.e. thickness in the order of one tenth of the other dimensions or larger). A shell element consisting of a combination of four and eight node elements is found to result in the best fit and will be used in this study. It is referred to as the Q4383 quadrilateral here. The four node elements have the degrees of freedom u j, v j, and θ z,j at the jth node which correspond to the in-plane deformations. The DOFs at the points of the eight node elements are w j, θ x,j, and θ y,j (corresponding to the out-of-plane motion). The translations are given by u, v, and w and the rotations by θ where a subscript indicates the axis of rotation. The corner nodes of the eight node elements coincide with the nodes of the four node elements. The degrees of freedom for one element are q T e = [u, v, w, θ x,, θ y,, θ z,,..., u 4, v 4, w 4, θ x,4, θ y,4, θ z,4, w 5, θ x,5, θ y,5,..., w 8, θ x,8, θ y,8 ]. (4.3) The out-of-plane deflections in an element are expressed as w(ξ, ζ) = 8 N quad,j w j (4.4) with the serendipity quadrilateral shape-functions [47] { N quad,j = 4 ( + ξ jξ)( + ζ j ζ)( + ξ j ξ + ζ j ζ) for j =,..., 4 2 ( + ξ jξ)( + ζ j ζ)( ξ j ζ 2 ζ j ξ 2 ) for j = 5,..., 8 j= (4.5) where, ξ j and ζ j are the iso-parametric coordinates ξ and ζ at node j, so ξ j {,, } and ζ j {,, }. The displacements u and v in x and y direction respectively are composed of three contributions: the interpolation of the nodal displacements in the midplane, a variation of displacement through the plate thickness due to rotations θ x and θ y (see Figure 3.2 for an illustration of this), and a displacement contribution due to rotations θ z at the nodes. The latter is sometimes referred to as deflections linked to the drilling degrees of freedom. The first and third contributions are illustrated in Figure 4.2. u 4 u 3 v v θ z4 θ z3 u v v 2 θ z2 u 2 θ z (a) (b) Figure 4.2: Contributions to the in-plane deformation: deformation due to interpolated nodal displacements in the midplane (a) and deformation due to rotations θ z (b). The displacements u and v within one element can thus be written as u(ξ, ζ, z) = v(ξ, ζ, z) = N lin,j u j + N uθ,j θ z,j + z N quad,j θ y,j (4.6) j= 4 N lin,j v j + j= 4 N vθ,j θ z,j z j= j= j= j= 8 N quad,j θ x,j (4.7) with the bi-linear shape-functions [47, 54] N lin,j = 4 ( + ξ jξ)( + ζ j ζ) j =,..., 4, (4.8) 44

69 Section 4.: FEM modeling and the shape-functions corresponding to the in-plane displacements resulting from the rotations at the nodes [55] N uθ,j = e y 6 ( ζ2 )( + ξ j ξ)ζ j j =,..., 4 (4.9) N vθ,j = e x 6 ( ξ2 )( + ζ j ζ)ξ j j =,..., 4. (4.2) Using these shape-functions and degrees of freedom, an element mass and stiffness matrix of the Q4383 shell element can be found using Equation (4.4) and (4.), respectively. The derivation of this is computerized using the symbolic toolbox in Matlab. The final matrix expressions are not included here because they are too elaborate. Assembling the plates The five plates composing the open box structure are first modeled separately by constructing a mass and stiffness matrix with no link between the DOFs of the five plates. In this, the degrees of freedom for each plate are initially w.r.t. a local coordinate system with directions x j, y j, and z j for the jth plate (j {,..., 5}, see Figure 4.3). The finite element meshes are placed on the mid-planes of the walls and have dimensions equal to those of the inside of the box. To be able to couple the plates easily, a fixed number of elements in X, Y, and Z direction (see Figure 4.3) is used for each plate (i.e. m X, m Y, and m Z respectively). Lz y x4 4 Y 4 z 4 Z 3 y z 5 y x z 5 3 x 5 z y x x 2 2 y 2 z 2 Ly O X Lx Figure 4.3: Schematic representation of the open box structure. The five plates need to be connected to form an open box. This is done later in this section by incorporating constraint equations between specific degrees of freedom. First, q l the local degrees of freedom of all five plates expressed in their local coordinate frame are transformed to global DOFs q g w.r.t. the global coordinate frame. For this, a square rotation matrix R is used, i.e. The new mass and stiffness matrix become q l = Rq g. (4.2) M g = R T M l R respectively K g = R T K l R, (4.22) where M l and K l include the mass and stiffness matrices of the five uncoupled plates with respect to local coordinates and M g and K g are the corresponding matrices with respect to global coordinates. Note that this only works for the DOFs at the corner nodes of the elements since at those points all six DOFs are available. At the midpoint nodes, only three degrees of freedom (two rotations and a displacement) are present. Therefore, these DOFs are not altered. The plates will be joined by edge and corner elements, which will be explained below. They are based on the fact that the parts linking the five plates (edge volume) are relatively stiff compared to the plates themselves. The displacements at an edge node of one plate are extrapolated to find the ones at the closest node of the connected plate. To illustrate how the edges are modeled, consider Figure 4.4. It shows an edge element between walls and 4. 45

70 Chapter 4: Case study: Modeling and analysis t b,4 4 2 t b, 3 Z Y X Figure 4.4: Illustration of an edge element connecting two walls of the box. The two hatched surfaces of an edge element are assumed to remain flat and rectangular. These planes undergo rigid motion. This is a simplification of reality, but applicable as the edge is small (t b is small compared to the plates height and length). As an alternative, the meshes are extended until they meet and connected by linking the degrees of freedom. It is found that the latter (standard) approach yields a connection that is more flexible than in reality as the plates are only connected over a single line. The deformation in the edge volume is not approximated correctly. Therefore, the former approach is used. An iso-parametric coordinate ψ for the Z-direction is introduced, with Z = 2 ( ψ)z + 2 ( + ψ)z 2 = Z + 2 ( + ψ)(z 2 Z ) (4.23) and used to approximate the displacements u, v, and w within the edge element in X, Y, and Z-direction respectively as u(y, ψ) = 2 ( ψ) (u (Y Y )θ z, ) + 2 ( + ψ) (u 2 (Y Y 2 )θ z,2 ) (4.24) v(x, ψ) = 2 ( ψ) (v + (X X )θ z, ) + 2 ( + ψ) (v 2 + (X X 2 )θ z,2 ) (4.25) w(x, Y, ψ) = 2 ( ψ) (w + (Y Y )θ x, (X X )θ y, ) ( + ψ) (w 2 + (Y Y 2 )θ x,2 (X X 2 )θ y,2 ) (4.26) In this, X, X 2, Y, and Y 2 are the X and Y coordinates of nodes and 2 respectively (see Figure 4.4) and t b,4 (X X ) = (X X 2 ), t b, /2 (Y Y ) = (Y Y 2 ) t b, /2 and ψ. t b,i is the thickness of the ith plate constructing the open box. Using the same approach as taken for the plate elements, the displacement field and material properties can be used to build a mass and stiffness matrix for an edge element. This is done for all edges and added to the entries of the mass and stiffness matrices corresponding to the correct DOFs. Note that the degrees of freedom at nodes 3 and 4 are not used, because they are fixed for arbitrary values of the DOFs at nodes and 2. Given the displacement field in the edge element, the degrees of freedom at 3 and 4 become and u 3 = u 2 t b,θ z, v 3 = v + 2 t b,4θ z, w 3 = w + 2 t b,θ x, 2 t b,4θ y, (4.27) θ x,3 = θ x, θ y,3 = θ y, θ z,3 = θ z, u 4 = u 2 2 t b,θ z,2 v 4 = v t b,4θ z,2 w 4 = w t b,θ x,2 2 t b,4θ y,2 (4.28) θ x,4 = θ x,2 θ y,4 = θ y,2 θ z,4 = θ z,2 46

71 Section 4.: FEM modeling respectively. Again, this approach can only be taken for the degrees of freedom at the four corner nodes of the Q4383 element. As a constraint for the DOFs at the midpoint nodes at the connecting edges it is imposed that they result from linear interpolation of the corresponding DOFs at the nearest two nodes. For the corners of the box, a similar approach is used, but instead of two rigid planes, now the complete block is assumed rigid. This means that it only adds mass and no stiffness. In a similar fashion, the DOFs at the corners of two (out of three) joining plates can be written as a function of the third one. The added stiffness contributions (by edge elements) and added mass contributions (by edge and corner elements) in terms of the global coordinates q g are collected in K a and M a, respectively, so that the system matrices (still for the uncoupled five plates) become M g = M g + M a K g = K g + K a. (4.29) To incorporate the attained constraint equations (such as (4.27) and (4.28) for an edge element), all degrees of freedom q g are first partitioned in two, i.e. q c and q r. DOFs q c can be written as a linear function of the independent DOFs q r : [ ] [ ] qc T q g = = c q I r = T q r (4.3) q r where T c is a matrix with the linear coefficients that express q c as a function of q r and I is the identity matrix. To enforce the constraint equations on the equations of motion, the expression above is used, which reduces the degrees of freedom to only the independent DOFs q r. The corresponding mass and stiffness matrix are found as follows M = T T M gt respectively K = T T K gt. (4.3) From here on, the reduced column of degrees of freedom will be simply referred to as q (instead of q r ) Viscoelastic material Damping is introduced in the open box by adding viscoelastic material. This material is also modeled using finite elements. Here, different elements are used than in the Chapter 3 to reduce the number of DOFs and, with that, computation times. This section gives a brief explanation of how the different VEM additions are modeled. Discrete VEM dampers A block of VEM is divided into m v elements of equal length e v in its axial direction. At the ends of each element, a node is added at which six degrees of freedom are introduced (i.e. three translations and three rotations). The damper is constructed using n v nodes. The figure below illustrates the modeling steps, dimensions and degrees of freedom. The displacement field within the element is found by interpolating the DOFs. For this, it is assumed that the cross sections remain flat. t v,y t v,x z Lv nv n v- n v θz,j vj θ y,j w j θ x,j u j y x (a) (b) (c) Figure 4.5: Illustration of the FE modeling of discrete dampers used for damping a 3D structure: dimensions (a), construction of elements (b) and a single element with the degrees of freedom at a node i (c). For the axial coordinate within an element, a transformation to the iso-parametric coordinate ξ is done. Equation (4.) is used for this where x is replaced by z. For a generic element between node j and j +, the assumed 47

72 Chapter 4: Case study: Modeling and analysis displacement fields are as follows u(y, ξ) = 2 ( + ξ) (u j yθ z,j ) + 2 ( + ξ) (u j+ yθ z,j+ ) (4.32) v(x, ξ) = 2 ( + ξ) (v j + xθ z,j ) + 2 ( + ξ) (v j+ + xθ z,j+ ) (4.33) w(x, y, ξ) = 2 ( ξ) (w j + yθ x,j xθ y,j ) + 2 ( + ξ) (w j+ + yθ x,j+ xθ y,j+ ), (4.34) (very similar as for the edge elements) using the assumption that the cross sections remain flat and that the deformations are small. Figure 4.5 illustrates the directions of local coordinates x, y, and z and the degrees of freedom u j, v j, w j, θ x,j, θ y,j, and θ z,j. The coordinates x and y are taken zero at the centerline of the block and thus t v,x /2 x t v,x /2, t v,y /2 y t v,y /2, and ξ. The kinetic energy corresponding to the different degrees of freedom is derived using Equation (4.4). This yields, in a similar fashion as done for the shell elements, a mass matrix corresponding to the degrees of freedom of one VEM element. The damping blocks will have relatively small L v compared to t v,x and t v,y and will be glued to the structure, implying that the lateral contraction at the top and bottom is restricted. Therefore, a plane strain assumption for deformation in the lateral directions is used instead of a plane stress one. This means that the strains ε xx and ε yy are zero. The elastic energy in an element is computed using (4.) where the definitions of σ and ε are changed to σ T = [ ] σ zz τ xz τ yz (4.35) and ε T = [ ε zz γ xz γ yz ]. (4.36) In this Ev ν v σ = κ 2 ( + ν v )( 2ν v ) ( 2ν v) ε Dε (4.37) κ sym 2 ( 2ν v) where Ev is the complex, frequency dependent Young s modulus and ν v is Poisson s ratio of the viscoelastic material. Similarly as done for the plate elements, σ and ε are used to form an element (complex) stiffness matrix corresponding to the damper element DOFs. The derived element mass and stiffness matrices are assembled to construct matrices for the complete damper. Note that linear shape-functions are used in this derivation. In bending, this can result in so-called shear locking where the structure stiffness is overestimated due to the fact that there is no non-trivial configuration where pure bending is found. There is always shear strain. See for example [56] for more information on this effect. The shear locking does not form a problem though for this application as the dampers length is small compared to the other dimensions and thus will mostly deform axially and in shear. The dampers are linked to the open box by assembling their mass and stiffness matrices by coupling the correct DOFs using the approach expressed by (4.3) and (4.3). Each node at a VEM damper end is connected to a single node of the open box. This is justified for dampers that are small compared to the box dimensions and modes where the deflection gradient at the location of the block is not too high. When the point of attachment does not fit the mesh of the box, the degrees of freedom at the four corner nodes are interpolated using bi-linear shape-functions. Any auxiliary beam structures that may be required to couple the VEM dampers to the open box are modeled using the same element type as these dampers, but employing a plane stress assumption instead of plane strain (i.e. a Timoshenko beam element [47, 56] extended to include bending in two directions). Constrained layer damping For constrained layer damping, the viscoelastic material is sandwiched between two plates. Again, expressions for the displacement field within the material for a single element are required that linearly depend on a number of degrees of freedom. Using these expressions, the kinetic and potential energy can be derived and from those, an elementary mass and stiffness matrix. The constraining layer (CL) will be modeled using the same elements (a combination of four and eight node elements, Q4383) as used for the aluminum plates forming the open box. This means that new degrees of freedom corresponding to the CL are added to the system. These DOFs need to be linked elastically to those corresponding to the box. The viscoelastic layer is used for that. For the VEM layer, elements are constructed, which basically are an extension of the description of the VEM layer used in the Miles-Reinhall beam element theory used for constrained layer beams [29] to shells. The displacement field within the viscoelastic material is considered as a linear interpolation of displacement fields at the 48

73 Section 4.: FEM modeling top and at the bottom of the VEM layer. At both sides, the CL and base plate are assumed to be rigidly connected to the VEM layer (see Figure 4.6), which might not be realistic if thick layers of glue are applied. Constraining layer Viscoelastic layer Base layer (a) (b) Figure 4.6: Illustration of a CL element and base plate element above each other connected by VEM in undeformed (a) and deformed (b) state. The displacement field within the VEM is expressed as u(ξ, ζ, ψ) = 2 ( ψ)u b(ξ, ζ, z b ) + 2 ( + ψ)u c(ξ, ζ, z c ) (4.38) v(ξ, ζ, ψ) = 2 ( ψ)v b(ξ, ζ, z b ) + 2 ( + ψ)v c(ξ, ζ, z c ) (4.39) w(ξ, ζ, ψ) = 2 ( ψ)w b(ξ, ζ) + 2 ( + ψ)w c(ξ, ζ), (4.4) where u b, v b, and w b are the displacement fields at the surface of the base layer expressed by Equations (4.6), (4.7), and (4.4) respectively and u c, v c, and w c define the displacements at the surface of the constraining layer, which are expressed by the same equations, but with the constraining layer parameters substituted. z b and z c are the surface coordinates of the plate and constraining layer respectively and are plus or minus half the corresponding thicknesses, depending on the side, to which the VEM is attached. Coordinate ψ is again an isoparametric coordinate in the direction of the thickness for the VEM, where ψ = is at the interface between base layer and VEM and ψ = is at the interface between CL and viscoelastic material. Note that this approach requires that the base material elements and CL elements are of equal size and exactly above each other. This may introduce a limitation for partially covering CL dampers as the layer should fit the mesh of the box. Again, the kinetic energy and elastic energy expressions ((4.4) and (4.), respectively) are used to derive a mass and stiffness matrix addition for the VEM element. Adding these contributions to the mass and stiffness matrix in terms of the DOFs of the box and constraining layers couples the CL to the box via the VEM layers. In computing the potential energy, all unique direct and shear strains are required. There is no zero stress or strain assumption in a certain direction. The stresses and strains become σ T = [ ] σ xx σ yy σ zz τ xy τ xz τ yz (4.4) and ε T = [ ε xx ε yy ε zz γ xy γ xz γ yz ] (4.42) respectively, which are related as follows (isotropic material)[52] ν v ν v ν v ν v ν v Ev σ = ν v ( + ν v )( 2ν v ) 2 ( 2ν v) ε Dε. (4.43) κ 2 ( 2ν v) κ sym 2 ( 2ν v) Again, the factor κ is introduced to account for possible shear variations. 49

74 Chapter 4: Case study: Modeling and analysis 4.2 Base structure design The open box dimensions are chosen such that the following design constraints are fulfilled for the structure without VEM: Mass within the range of 25 kg. The outer dimensions should all be between and 4 mm. Approximately elastic modes in the range 5 Hz. At least one clear elastic mode in each of the five plates of the box. The eigenfrequencies in the frequency range of interest should be well separated to simplify experimental modal analysis. A design that fits these criteria is found using a trial and error approach. The final dimensions and parameter values are shown in Table 4.. The box is made of aluminum and has a mass of.43 kg. The first undamped, non-zero eigenfrequencies (modes 7 to 6) can be found in Table 4.2 for m X = m Y = m Z = 4. They are computed using the Matlab function eigs with the assembled M and K matrices. There are 3 elastic modes within the range 5 Hz and 6 rigid body modes that result from the free boundary condition. Figure 4.7 shows a selection of the mode shapes where the color indicates the modal displacement amplitude. The lines do not illustrate the mesh size used in computations. See Appendix D for the first elastic modeshapes (7 to 6) of the undamped box. Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 7 (b) mode.25 Z+w.2. Z+w Y+v..2 X+u Y+v.. X+u (c) mode 2 (d) mode 5 Figure 4.7: A selection of the eigenmodes of an aluminum box with dimensions and parameters as in Table 4.. 5

75 Section 4.2: Base structure design Table 4.: Parameter values used in modeling the box structure and its final design dimensions (see Figure 4.3 for the wall numbers). Symbol Description Value L x Outer dimension of the box in X-direction 4 cm L y Outer dimension of the box in Y -direction 35 cm L z Outer dimension of the box in Z-direction 25 cm t b, Thickness of front wall (wall ) mm t b,2 Thickness of right wall (wall 2) 6 mm t b,3 Thickness of back wall (wall 3) 7 mm t b,4 Thickness of left wall (wall 4) 9 mm t b,5 Thickness of bottom plate (wall 5) 8 mm ρ b Mass density of the base structure (aluminum) 2625 kg/m 3 E b Young s modulus of the base structure (aluminum) 7 GPa ν b Poisson s ratio of the base structure (aluminum).33 κ Factor to account for shear variation of the plates[47] π 2 /2 Table 4.2: First (non-zero) undamped eigenfrequencies found with the Matlab FE model and with ANSYS and the relative difference between them. Mode nr Matlab [Hz] ANSYS [Hz] rel. err. [%] mode 7 mode mode 6 scaled eigenfrequency nr of elements Figure 4.8: Scaled frequency as a function of mesh size for the first, fifth and tenth (non-rigid body) mode. 5

76 Chapter 4: Case study: Modeling and analysis As a first check for the finite element model, the mesh size is varied. The computed eigenfrequencies should converge as the mesh gets finer. An equal number of elements in X, Y and Z-direction is used in this. Figure 4.8 shows the eigenfrequencies scaled with the converged values (for m X = m Y = m Z = 4) for a number of modes. The figure illustrates that the frequencies converge and that even for relatively coarse meshes, the results are reasonably accurate. To be well within one percent deviation from the true values for the first non-rigid body modes, at least 5 elements in each direction should be used. As a second check, the same structure is modeled and its eigenfrequencies and -modes are computed using the commercial FEM package ANSYS 4.. The box is meshed using tetrahedral elements. The resulting eigenfrequencies are listed in Table 4.2 for a mesh consisting of 4298 elements and 8285 nodes. The table also shows the relative difference between the eigenfrequencies found using ANSYS and using Matlab. The Matlab frequencies are higher than the frequencies found using ANSYS due to the relatively stiff edge and corner elements used in this study. The relative differences are small however and mode shapes are similar, supporting the model validity. 4.3 Dynamic modeling for viscoelastic damping Linear equations of motion can be derived for the open box with viscoelastic material using the finite element method (see Section 4.). They depend on the frequency dependent Young s modulus of the viscoelastic material. In Subsection 4.3., a number of commercially available viscoelastic materials are selected based on their properties and modeled. For characterizing the dynamic behavior of systems, modal parameters or frequency response functions are commonly used. How these can be computed for structures with viscoelastic damping will be explained in Subsections and 4.3.3, respectively Viscoelastic material model A commercially available viscoelastic material is looked for with a loss factor that is as high as possible (e.g. close to unity or higher) within the frequency range of interest (5 Hz). A list of high damping viscoelastic materials and a corresponding manufacturers can be found in Appendix A.4. From these products, a selection is made based on loss factor, availability of information of the material properties, and variety of mechanical properties. The six chosen products are characterized dynamically using a Dynamic Mechanical Thermal Analysis (DMTA) test and are listed in Table 4.3. See Appendix A. for more information on DMTA testing. Table 4.3 also lists the so called Shore A hardness of the materials, a measure for how hard the material is. The lower the Shore value, the more flexible the VEM. For the Isoloss R SL-2 sample, the DMTA characterization was not successful, likely due to its high elastic compliance and damping. The measured dynamic modulus E v(iω) and loss factor η(ω) as a function of frequency ω/2π for Norsorex R 4292 and Viton R SCVBR 75 are depicted as blue circles in Figure 4.9 and 4., respectively. These materials are found to give the best results for the two damping approaches that will be described in the next section. The former is used for discrete damping elements, whereas the latter is used in a constrained layer solution. Table 4.3: Selected materials for dynamic characterization. Manufacturer Product name Shore A Country (headquarters) Astrotech advanced elastomer products GmbH Norsorex R Austria Astrotech advanced elastomer products GmbH Norsorex R Austria E-A-R Aearo Technologies LLC Isodamp R C-2 6 USA E-A-R Aearo Technologies LLC Isoloss R SL-2 2 USA Eriks Viton R SCVBR The Netherlands Eriks Viton R SCVBR 9 9 The Netherlands In order to incorporate the dynamical properties of the viscoelastic materials in the finite element model, a model for the complex Young s modulus E v(s) of the VEM is required. Therefore, the measurement data is fitted with the fractional derivative model (2.26) [3], i.e. E v(s) = E + The DMTA measurements are done at Philips Innovation Services - Materials Analysis N j= E j (s/r j ) φj (4.44) + (s/r j ) φj 52

77 Section 4.3: Dynamic modeling for viscoelastic damping 9 measurement data fit E * v (iω) [Pa] measurementdata fit loss factor [ ] Figure 4.9: Dynamic modulus and loss factor at 22 C as a function of frequency for Norsorex R from DMTA measurements and a fractional derivative model fit. loss factor [ ] E * v (iω) [Pa] measurement data fit measurement data fit Figure 4.: Dynamic modulus E v(iω) and loss factor η(ω) at 22 C as a function of frequency for 75 Shore A Viton R SCVBR from DMTA measurements and a fractional derivative model fit. 53

78 Chapter 4: Case study: Modeling and analysis The fractional derivative model is used because it is accurate even for low N. For the fitting process taken (using the Matlab function fmincon), see Appendix A.3. The resulting fits for N = 2 are shown in Figures 4.9 and 4.. The fit order is relatively high, but does influence computation performance of the dynamic analysis for the methods used in this study. When the state-space approach is used, however, the material model order should be reduced (e.g. to N = 5) as the computation times drastically increase with N. Furthermore, the fractional derivative model is difficult to incorporate for that approach Modal analysis The dynamic behavior of a linear system can be characterized by the eigenmodes U k and corresponding eigenvalues λ k. The latter can be split in damped eigenfrequencies Im {λ k } /2π and a measure for the damping β k (see Equation (3.)). The combination of the modes (or modal contributions known as the residues) and eigenvalues are called the modal parameters and are commonly used to asses the dynamic properties of a structure. How they are computed for a system with linear viscoelastic damping will be described below. Consider a structure with general linear damping. The equations of motion for such a structure are given by [57 6] t M q(t) + G(t τ) q(τ)dτ + K q(t) = F (t) (4.45) with M the system mass matrix, K the stiffness matrix, q(t) the vector with degrees of freedom, F (t) the forcing vector and G(t) a matrix with damping kernel functions. The damping is thus a function of the complete deformation history. Transforming this to the Laplace domain gives s 2 M q(s) + sg(s) q(s) + K q(s) = F (s) (4.46) where G(s), q(s), and F (s) are the Laplace transforms of G(t), q(t), and F (t) respectively. Alternatively, this can be written in the form s 2 M q(s) + K(s) q(s) = F (s) (4.47) using K(s) = K + sg(s). In Chapter 2, it is mentioned that for viscoelastic materials, the damping is linked to a phase difference between stress and strain, which is captured in a frequency dependent complex Young s modulus. This implies that for viscoelastic damping it holds that sg(s) = E v(s)k v (4.48) with E v(s) the complex Young s modulus of the VEM as a function of s and K v a constant, scaled stiffness matrix related to the geometry of the VEM. It thus follows that K(s) = K(E v(s)). The eigenvalue problem corresponding to the structure (symmetric system) is given by [ Mλ 2 k + K (E v(λ k )) ] U k = k =,..., p, (4.49) where p is the number of eigenvalues λ k and corresponding vectors U k in the system. For non-viscously damped systems, this number is higher than 2N q (with N q the number of DOFs). The eigenmodes can be divided into two categories: the elastic modes and the so-called non-viscous modes [58]. In the case of relatively weak damping for all elastic modes, the former consist of N q undercritically damped modes of vibration (complex conjugate eigenvalue pairs) of the structure. When there are N r rigid body modes, the number of undercritically damped elastic modes becomes 2N q 2N r. The non-viscous modes will not be found in a viscously damped system and are all supercritically damped modes and not oscillatory in nature [59]. They originate from the non-viscous effect of the damping mechanism. The eigenvalues corresponding to these modes are real. The non-viscous modes are in general very difficult or even impossible to find accurately [62]. A way would be to transform the equations of motion, including a model for E v(s), to a state-space representation where internal states are introduced [29, 59]. This drastically increases the system order. In this study, they are not computed. For finding the elastic modes an iterative method is employed (see Figure 4.). An initial constant complex value E v, is taken for the Young s modulus E v (an average over the frequency range of interest) and is inserted in the eigenvalue problem (4.49). The r 2N q eigenvalues and modes of interest are computed using eigs with the attained complex stiffness matrix. Subsequently, for each eigenvalue λ k individually, an iteration loop is done. E v(s) for s = λ k is computed, this is inserted in the eigenvalue problem and the eigenvalues and -vectors for the resulting complex K(E v) are computed. The mode of interest k is extracted using the MAC sorting method. This process is repeated until the change in the considered eigenvalue λ k is below a certain threshold. The computation efficiency is increased by only computing a couple of eigenvalues and -vectors closest to the eigenvalue at the previous iteration step. 54

79 Section 4.3: Dynamic modeling for viscoelastic damping Initialize the complex Young s modulus Ev, * Solve 2 * (Mλ k+k(e v,))u k= with k =,...,r for k = :r Update Young s modulus as E v * (λ k) Solve 2 * (Mλ k+ K(E v))u k= λ k converged? Y N Output modal parameters λ kand Uk with k =,...,r Figure 4.: Iteration scheme for finding the eigenvalues and -vectors of the damped box structure. Note that when a constant (frequency independent) complex stiffness matrix is used, the equations of motion correspond to a system with hysteretic damping. The actual elastic eigenvalues of the system do come in stable complex conjugate pairs. The eigenvalues of this system are not complex paired but of the form λ k = ±( µ k + iν k ) [43], one of which has a positive real part and is thus not stable. This is because Ev(λ c k ) = E c v (λ k ) (where a superscript c indicates complex conjugate) and one of the eigenvalues of the system with complex stiffness matrix is not a solution to the real system of equations. This is accounted for in the implementation of the iteration algorithm by computing the eigenvalues for Ev(λ k ) and Ev(λ c k ) and eliminating the incorrect solutions FRF derivation Another way to characterize the dynamic properties of a linear system is through a frequency response function (FRF), which is defined by the eigenvalues and eigenmodes of the system. The FRF is the relation between harmonic actuation and corresponding response at a specific angular frequency ω. The transfer function matrix is given by H(s) = D(s), with D(s) = Ms 2 + K(s) (4.5) the dynamic stiffness matrix. The FRF matrix is found by evaluating this equation for s = iω. Computing the frequency response function in this manner is known as the direct method. To compute FRFs between certain excitation forces and response degrees of freedom a matrix inversion is required at each frequency of interest. Although this method is exact, it becomes very inefficient and computationally expensive especially for large scale systems (large N q ). In modeling the box structure the number of degrees of freedom is in the order N q = O(5 4 ). Therefore, this approach is not followed in determining the FRFs in this study. A way to reduce the computation time is to perform a model reduction step. The number of DOFs is reduced by expressing some degrees of freedom in terms of the others. Several reduction techniques are known, such 55

80 Chapter 4: Case study: Modeling and analysis as Guyan- or static reduction, the Craig-Bampton reduction method, the Rubin method and Impedance coupling [43, 63]. The model reduction approach is not used in this report, although it has potential, certainly for application of the discrete dampers. Applying it correctly for constrained layer damping can be a study in itself. Modal superposition employing the residue theorem is commonly used in finding FRFs. For any damped symmetric linear system, the frequency response function matrix can be expressed by summing the contributions of the different modes as follows[, 43, 59] H(iω) = N r k= U k U T k m k ω2 + N q k=n r+ [ γ k U k U T k iω λ k + γc k U c k U H k iω λ c k ] + p k=2n q+ γ k U k U T k iω λ k (4.5) with m k = U T k MU k and γ k = U T k D(s) s. (4.52) U k s=λk H(iω) has three parts: the first part is due to rigid body modes, the second part is due to the elastic modes and the third part is due to the non-viscous modes. Note that the elastic modes are subcritically damped and come in complex conjugate pairs. In general, not all modes are considered in the equation above. The summation is truncated. This is justified because high frequency modes do not much influence the response in the lower frequency range. Depending on the frequency interval of interest, only the r lowest modes are used in the superposition. Furthermore, a truncation is often necessary as the non-viscous modes might be difficult to compute or even unavailable. For viscoelastic damping, it is found that using the truncated modal superposition scheme is not very accurate when not considering the non-viscous modes. This conclusion is drawn based on comparing several computation techniques to the exact method for a cantilever beam with viscoelastic damper connected at the end. See Appendix F for more elaborate results of this comparison. To improve the accuracy of the truncated modal superposition scheme for non-viscously damped systems, several methods are proposed in literature that approximate the contribution of the unavailable modes. Examples are the improved approximation method (IAM) and mode acceleration method (MAM) (both employing a Neumann expansion as proposed in [64]) and a method using approximate contributions as a projection basis to reduce the truncation error as proposed in [62]. Mainly the latter is found to be quite effective and accurate. In applying this method to the box structure, however, some incorrect peaks in the frequency response function are found. Therefore, this method is not used and a different modal superposition approach is proposed. The proposed method is based on interpolation of eigenvalues and residues over frequency. Consider an approximation of the FRF (Equation (4.5) for s = iω) near an arbitrary frequency s = iω Ĥ(iω) = [ Mω 2 + K(iω ) ]. (4.53) This approximation is only exact for ω = ω. The eigenvalue problem corresponding to this approximation is given by [Mσ k + K(iω )] z k = k =,..., N q, (4.54) with σ k and z k the kth eigenvalue and -vector respectively. Note that σ k λ 2 k and z k U k. It follows that or in matrix form K(iω )z k = Mz k σ k (4.55) K(iω )Z = MZΣ, (4.56) with Σ a diagonal matrix with the N q eigenvalues of this approximate system and Z = [z, z 2,..., z Nq ]. Keep in mind that Σ = Σ(ω ) and Z = Z(ω ). Using this spectral decomposition, the FRF matrix can be written as Ĥ(iω) = [ Mω 2 + K(iω ) ] = [ Z T Z T ( Mω 2 + K(iω ) ) ZZ ] = Z [ Z T ( Mω 2 + K(iω ) ) Z ] Z T = Z [ Z T MZ ( ω 2 I + Σ )] Z T (4.57) Using the orthogonality principle for symmetric mass and stiffness matrix, i.e. [43] Z T MZ = M (4.58) 56

81 Section 4.3: Dynamic modeling for viscoelastic damping with M a diagonal matrix, it follows that Ĥ(iω) = Z [ M ( ω 2 I + Σ )] Z T. (4.59) Considering the fact that the term between brackets is a diagonal matrix, the frequency response function matrix approximation can be expressed as with Ĥ(iω) = N q k= z k z T k m k (ω2 + σ k ) = R k = z k zt k m k N q k= R k ω 2 σ k, (4.6) (4.6) the residue matrix corresponding to σ k and m k the kth diagonal entry of M. Note that Equation (4.6) is only accurate for ω = ω. Now suppose that the eigenvalues and eigenmodes depend on ω. The exact FRF matrix as a function of ω can then be found by using Equation (4.6) with Σ = Σ(ω) and Z = Z(ω), i.e. H(iω) = N q k= R k (ω) ω 2 σ k (ω). (4.62) Using this expression would require solving the full eigenvalue problem at every frequency in the frequency range of interest ω [, ω c ], where ω c is the cut-off frequency. This is very inefficient and computationally expensive. To improve performance, first, the modal superposition is truncated to r terms. Subsequently, it is chosen to compute σ k and R k (with k =, 2,..., r) at a limited number N f of frequencies ω = [ ] T ω,... ω Nf and interpolate the eigenvalues and residues using shape preserving cubic splines for frequencies in between. The real and imaginary parts of the eigenvalues σ k and residues R k are interpolated separately as a function of log (ω). These modal parameters are interpolated instead of the FRFs themselves because they change in a much smoother way when the frequency changes. Interpolation of the FRFs themselves leads to inaccurate approximations. The approach will be referred to as the Interpolated Modal Parameter Superposition (IMPS) method. The proposed method is found to be accurate, as will be shown below and is demonstrated in Appendix F. In contrast to other methods, in this approach it is necessary to sort the modes correctly. The modes and eigenvalues found at different frequency points need to be matched correctly in order to be able to perform the interpolation correctly. This is done using the MAC criterion (see Equation (3.7)). As a test case, a block of Norsorex R that has a cross section of 2 2 cm and a thickness of cm is connected near the end of a 5 cm long steel cantilever beam also with cross section of 2 2 cm (see Figure 4.2). The beam is modeled using Euler-Bernoulli elements [] with a Young s modulus of 2 GPa and mass density of 78 kg/m 3. The VEM block is represented by a spring with frequency dependent complex stiffness Ev(s)A v /L v. Beam VEM Figure 4.2: Cantilever beam with viscoelastic damping element connected near the end. The driving point FRF at the location of the damper is computed both using the direct (exact) method and the proposed approach. For the latter, 5 frequency points are used for interpolation with equal separation on a logarithmic scale. The number of included eigenvalues σ k and corresponding modes z k is equal to r =. Figure 4.3 depicts the driving point FRF between lateral displacement and force at the location of the damper for the two computation approaches. As can be seen, the curves almost completely overlap. 57

82 Chapter 4: Case study: Modeling and analysis FRF [db] exact interpolation phase [deg] Figure 4.3: Driving point frequency response function of the damped beam at the location of the damper computed using the exact method and IMPS method ( db m/n). Figures 4.4 and 4.5 illustrate the interpolation of the eigenvalues (in terms of σ k, which is complex) and residues R k (i.e. the entry of R k under consideration) respectively for the first three modes. Note again that these σk differ from the system eigenvalues λ k. Figure 4.4 shows a similar trend as is found for the material dynamic modulus and loss factor (e.g. Figure 4.9). The real part of σ k is linked to the loss factor and the imaginary part to the dynamic modulus. The residues capture a change in mode amplitude and relative phase at the location considered. As the loss factor increases, so does the imaginary part of R j. The latter converges to zero if η. For high frequencies, Re {R k } converges towards zero, because the damper becomes so stiff that the mode has a node at the location of the damper. imag( σ ) [rad/s] k real( σ ) [rad/s] k mode (interp.) mode mode 2 (interp.) mode 2 mode 3 (interp.) mode Figure 4.4: Interpolation of the first three σ k of the damped cantilever beam. 58

83 Section 4.4: Damping design imag( R ) k real( R k ) mode (interp.) mode mode 2 (interp.) mode 2 mode 3 (interp.) mode Figure 4.5: Interpolation of the first three R k of the damped cantilever beam. As an improvement to the method, the frequencies, at which an eigenvalue problem is solved are not taken equally spaced (on log-scale), but chosen such that the increase in dynamic modulus at each frequency step is constant (not yet considered for Figures 4.4 and 4.5). 4.4 Damping design In Section 4.2, the dimensions and material of the undamped open box have been chosen and consequently, the mode shapes and natural frequencies are fixed. The next step is to add viscoelastic material to attain as much damping as possible, taking into account the following realistically chosen restrictions: A maximal mass addition per wall in the order of.5 kg (2.5 kg in total). As a reference, the undamped box has a mass of.43 kg. No or at least a minimal reduction in eigenfrequencies due to the addition of VEM. The viscoelastic material with the desired dimensions and properties should be commercially available and not too expensive. From Chapter 3, it is known that discrete damping elements are most effective in reducing structural vibration. The downside of this method is that it only addresses specific modes. To add damping to a broad set of modes, the constrained layer damping method is found to be favorable. Both techniques will be applied to the box structure separately. For the CL approach, different configurations are chosen and are tested. The design choices and considerations for the discrete dampers and CL applications are addressed in the Subsections 4.4. and 4.4.2, respectively Discrete dampers Small blocks of viscoelastic material are added at specific points on the box walls/plates. Where to position them is considered first. Looking at the mode shapes of the undamped structure (see Figure 4.7 and Appendix D) it can be concluded that for most of the modes there is quite a large deformation amplitude near the top edges of the box. Therefore it is chosen to add dampers near that edge. In Chapter 3, it is mentioned that the optimal position for such a damper, in general, is not at the location of maximal amplitude. Still, the top edge is chosen as most modes have considerable deformation there. Now, looking at a single wall, the deformation mode shapes at the top edge are similar to those of a pinned-pinned beam. In the frequency range of interest, only the lowest two equivalent 59

84 Chapter 4: Case study: Modeling and analysis beam modes are encountered. Using the beam analogy, in this case an optimal damper location to get the highest ) 2 damping for the first two transversal beam modes is found by maximizing the criterion. This criterion is used instead of the objective function Γ k used in Chapter 3, because the latter requires a reference case, which is difficult to define for this optimization. The resulting location varies with VEM block stiffness (and thus frequency) between approximately 27% of the length at low stiffness and 4% of the length at high stiffness. It is chosen to place the damper at 32% of the inner wall length for application to the box. Note that when a specific mode is of interest, this location is not optimal. In Chapter 3, one end of the VEM dampers was connected to the beam structure and the other end to the fixed world. For quite some structures, a connection to the fixed world is not possible as the structure may be required to be able to move. This case is considered for the box. Instead of a fixed world connection, the viscoelastic blocks thus need to be connected to a stiff part of the box. In this case, auxiliary beam structures are used to link one VEM block to another one at the adjoining plate (see Figure 4.6). The idea behind it is that the plates are stiff in the in-plane directions and relatively flexible in the out-of-plane direction. Furthermore, the adjoining walls move in phase for a number of modes maximizing the relative displacements. A total of eight VEM blocks are added to the box, which are connected by four beams. ( 2 k= β k Box Auxiliary beam structure VEM Figure 4.6: Schematic representation of the box with discrete dampers and auxiliary structures seen from the top. Using the material models of the viscoelastic materials from Table 4.3, the dimensions of the VEM blocks and aluminum beams are designed taking a trial and error approach. The aluminum beams are required to be stiff but light, which could be attained by using hollow structures. For this study solid beams are considered since for the design in Figure 4.6, mass is not critical and this reduces cost. For the beams, a square cross section is used of which the dimensions are increased, until the damping and eigenfrequencies of the box do not change anymore. Norsorex R is found to give the most damping for practical dimensions taking into account the available material thicknesses. The final dimensions and properties of the VEM blocks and beams are listed in Table 4.4. For the other parameter values used, see Table 4.. The total added mass is.392 kg (well below the requirement). Table 4.4: Parameter values and dimensions used in modeling the discrete dampers applied to the box structure. Symbol Description Value t v,x = t v,y Cross sectional dimension of discrete damping blocks (square) 5 mm L v Height of discrete damping blocks 3 mm m v Number of elements used to model a discrete damper 2 ρ v Mass density of the viscoelastic material 8 kg/m 3 ν v Poisson s ratio of the viscoelastic material.499 E a Young s modulus of the auxiliary beam structure material (aluminum) 7 GPa ρ a Mass density of the auxiliary beam structure material (aluminum) 27 kg ν a Poisson s ratio of the auxiliary beam structure material (aluminum).33 W a Width of the auxiliary beam structures 5 mm L a Length of the auxiliary beam structures (at centerline) 58.7 mm m a Number of elements used to model the auxiliary beam structures 2 κ v Factor to account for shear variation of the viscoelastic material[47] π 2 /2 6

85 Section 4.4: Damping design The eigenvalues and -modes are computed using the iteration scheme described in the Subsection and the first non-rigid body mode eigenfrequencies and damping ratios are shown in Table 4.5 for m X = m Y = m Z = 3. They are sorted such that the mode shapes correspond to the modes of the undamped structure (using MAC). This means that the eigenfrequencies no longer are in increasing order, as can be seen from modes 5 and 6 for examples (see Table 4.5). The mode shapes of the box with discrete dampers are similar to the undamped box modes. The real part of four modes are depicted in Figure 4.7 (for the remaining modes corresponding to the undamped modes, see Appendix E). The modes are scaled using the vector element with largest magnitude and then multiplied by a factor of.5 for visualization. This scaling step causes the real part of the eigenvectors to be dominant over the imaginary part. A number of new eigenmodes appear that correspond to the auxiliary structures (not included in Table 4.5). The bending modes of the auxiliary beams are at quite high frequency (above Hz). The modes where the beams can be considered rigid masses on the viscoelastic springs are in the range of 6 7 Hz (different from modes 3-6). The vibration is in Z-direction. These modes have a relatively high damping ratio though (O(.35)). An example of such a mode is shown in Figure 4.8. Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 8 (b) mode Z+w Z+w Y+v.. X+u Y+v.. X+u (c) mode 3 (d) mode 6 Figure 4.7: Some of the eigenmodes of an aluminum box damped using discrete VEM elements with dimensions and parameters as in Table

86 Chapter 4: Case study: Modeling and analysis Table 4.5: First (nonzero) eigenfrequencies and damping ratios found using the derived Matlab model for the box with discrete dampers. Mode nr Undamped eigenfreq. [Hz] Damped eigenfreq. [Hz] Damping ratio β k [ ] It can be concluded from Table 4.5 that this approach is effective in adding damping. A number of modes show relatively low damping, because the adjacent walls move out of phase (and the beams mostly just rotate as rigid body, see Figure 4.7a) or because most of the open box deformation is not at the top edge of the box (see Figure 4.7b). Although most of the lower eigenfrequencies increased compared to the undamped structure, the higher eigenfrequencies show decreased eigenfrequencies since the mass of the structures clearly has more effect than the added stiffness. The decrease is small, however, and acceptable (i.e. < 5% for most modes). Z+w Y+v.. X+u Figure 4.8: Eigenmode of the aluminum box damped using discrete VEM elements where the auxiliary beams oscillate on the VEM blocks Constrained layer damping In applying constrained layer dampers the choice can be made to fully or partially cover the walls. Both approaches are considered. Full coverage of the walls will give more damping than partial coverage when there is no constraint on the amount of material that is added. However, when there is a constraint on the mass addition, it is possible that the obtained damping for partial coverage is higher than for full coverage of the box for optimized layer thicknesses. As mentioned above, the mass constraint is initially set to.5 kg per wall, but considering the available material thicknesses from the manufacturer slight exceedings are accepted. First, full coverage of the box walls is considered. The parameters that are free to vary are the thicknesses of the VEM layer (t v ) and constraining layer t c ) for each wall. Furthermore, the materials used for the two layers need to be chosen. Stainless steel is used for the constraining layer since it is available and practical in low 62

87 Section 4.4: Damping design thickness. To determine the thicknesses t v and t c of the different layers for each wall, the damping is computed for a fully covered (separate) plate (considered hinged along the connected edges) with similar dimensions as that specific wall for different t v and t c. The constraint on mass addition is used to limit the range. Dimensions are then chosen based on the result of this short study and commercially available thicknesses. For the first mode of the plate representing wall, the damping ratio β as a function of the two thicknesses is shown in Figure 4.9 for five different materials (at a temperature of 2 C). The trend for the higher modes is similar. Furthermore, similar conclusions can be drawn from the analyse of the other walls. In Figure 4.9, the black line illustrates the mass constraint and the black circle the location of the optimizer. t v [m] t v [m] t v [m] t c [m] t c [m] t c [m]. (a) Norsorex R 4292 (b) Norsorex R (c) Isodamp R C t v [m] t v [m] t c [m] t c [m] (d) Viton R SCVBR75 (e) Viton R SCVBR9 Figure 4.9: Damping ratio for mode as a function of CL and VEM layer thickness (t c and t v ) for a plate representing wall and for different viscoelastic materials. From the graphs above it can be concluded that the Norsorex R 4292 gives the highest damping at the location of the optimizer for wall. However, the required thickness for the VEM is impractically low. Therefore, this material is not selected. The Viton R SCVBR9 material gives least damping, which is why it also is not a suitable candidate. The remaining viscoelastic products attain similar damping levels and even though it does not give the highest optimal value, Viton R SCVBR75 could be selected as it is readily available. However, the product RX R FPM (with comparable properties) is used instead of Viton R SCVBR75. The latter is namely more expensive due to the fact that it has undergone an extra heat treatment, which reduces outgassing and makes it suitable for clean-room conditions. From the optimizers found for the different plates representing the box walls, the nearest commercially available material thicknesses are selected. A VEM thickness t v of mm is used for each wall and for the upright walls a constraining layer thickness t c of.5 mm is chosen. For the bottom plate of the box, a t c of.3 mm is selected. To check if the trend as shown in Figure 4.9d is correct (both using the model and by doing experiments), a viscoelastic layer of half the thickness (i.e..5 mm) is considered as well. The final added masses for the different configurations can be found in Table 4.6 where a mass density of 2243 kg/m 3 is used for the VEM 63

88 Chapter 4: Case study: Modeling and analysis and of 7689 kg/m 3 for the stainless steel (densities are found from measurements). For the approach with the mm thick viscoelastic layer, some of the added masses are somewhat above the proposed maximum of.5 kg. This is because somewhat higher thicknesses are taken than found from the optimization due to availability of sheet thicknesses. Table 4.6: Added mass in kg per wall (of viscoelastic and constraining layer) for different constrained layer damping configurations (see Figure 4.3 for an illustration of the wall numbering). Wall nr Full coverage (thick) Full coverage (thin) Partial coverage Next, the CL damping approach will be considered where only a part of the box is covered. The material should be placed such that the strain energy in the VEM is maximal. This is not at locations where the modes have maximal amplitude, but at points of maximal displacement gradient. Simply placing patches of constrained layer dampers at those locations will not suffice though as that would result in mostly rotations of the full layers instead of deformation. Via a trial and error approach, it is found that a good approach is to place the patches not just at those locations, but also to let the layers cover a point of maximal amplitude. This causes deformation instead of rigid rotations of the constrained layer dampers. Looking at the mode shapes of the undamped box structure, it can be concluded that the upper parts of the upright walls will contribute most to damping. That is why a strip with a width of 3.5 cm (approximately 55% of the box height) is placed along the upper perimeter of the box. To address the bottom plate mode, a strip with a width of 6 cm (approximately 4% of the box length) is placed along the centerline of the box in Y -direction. Figure 4.2 illustrates where the material is added. Note that in the model, the layers need to match with the finite element mesh and thus dimension deviations of in the order of mm arise. For the constrained layer computations, a mesh of m X = 3, m Y = 25, and m Z = 2 is used. The thicknesses of the different layers are varied manually to find highest damping for similar mass as the full coverage configuration, keeping in mind commercial availability of the thickness. For this optimum, the constraining layer thickness is taken mm. The VEM layers of wall 2 and 4 are 2 mm thick and for the other walls a thickness t v of mm is used. The corresponding added masses can be found in Table 4.6 and are all just slightly higher than was aimed for. Y Z X Figure 4.2: Schematic representation of the box where the hashed areas illustrate the parts to which the constrained layer patches are added. Using the designed constrained layer dampers, the first nonzero eigenvalues and corresponding complex eigenvectors are computed (modes 7 to 6). The mode shapes of the damped box and the undamped box are similar and are in the same order. The real part of modes 9 and 4 are depicted in Figure 4.2 (again, scaled such that the vector element with largest amplitude is real) for partial coverage CL damping (for the remaining modes, see Appendix E). The eigenfrequencies and damping ratios for the different CL approaches are summarized in Table 4.7. The table also contains the natural frequencies of the undamped structure for comparison. 64

89 Section 4.4: Damping design Table 4.7: First (nonzero) eigenfrequencies and damping ratios found using the derived Matlab model for the box with constrained layer damping. Mode nr Undamped CL full (thick) CL full (thin) CL partial freq. [Hz] freq. [Hz] β k [ ] freq. [Hz] β k [ ] freq. [Hz] β k [ ] Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 9 (b) mode 4 Figure 4.2: Eigenmodes 9 and 4 of the aluminum open box structure damped using a partial coverage CL approach. From the results shown in the Table 4.7 it can be concluded that for the full coverage CL damping configurations, the damping is in fact higher for the thick VEM layer than for the thin VEM layer. The results also show that it is possible to attain higher damping levels when using partial coverage for similar mass or volume. By adding the constrained layer additions with thicker t v (full coverage), the damped eigenfrequencies decrease slightly compared to those of the undamped structure. A slight increase is seen for the thinner fully covering layers. This difference is caused by the increased shear stiffness of the viscoelastic layers as t v is reduced. The damped eigenfrequencies of the system with partial coverage CL application show a greater decrease. Since the VEM thickness is larger, the added stiffness is lower and does not weigh up against the mass contribution. The decrease in eigenfrequencies is small enough to be acceptable (i.e. < %). In this chapter, the modeling and analysis of an open box structure is described. FE models using thick shell elements for modeling the box, extended Timoshenko beam elements for modeling discrete damping elements, and a plate equivalent for the VEM layer in the Miles-Reinhall CL beam element are composed and described. The undamped box structure is designed considering constraints on mass, dimensions, and eigenfrequencies. Subsequently, different VEM damping approaches for the open box are considered. Modeling and analysis of structures 65

90 Chapter 4: Case study: Modeling and analysis with viscoelastic material requires different numerical techniques than for e.g. viscously damped systems. For that reason, some modal and frequency response function analysis techniques are introduced. Finally, the damping materials are selected and modeled and the dimensions are optimized to some extent. It is found that the discrete damping elements add significant damping to several (but not all) modes in the considered frequency range. Furthermore, it is concluded that the CL damping configuration is effective for adding damping to most modes and that the partial coverage case can give higher damping ratios than the fully covering CL dampers when there is a constraint on mass addition. 66

91 Chapter 5 Case study: Experiments and comparison with models In Chapter 4, the box structure has been designed, modeled, and damped by different VEM damping applications. In this chapter, to analyze whether the model estimates on damped eigenfrequencies, modal damping levels, mode shapes, and FRFs are accurate, the structure is realized and characterized dynamically. This is done for all cases, i.e. the undamped structure, the box with discrete dampers and the box with different CL damping configurations. The experiments are described and comparisons of experimental results with the model predictions are presented. In Section 5., the experimental setup and test plan are discussed. Some practical considerations considering the VEM fastening (i.e. using glue or two-sided tape) and suspension of the structure are elaborated on. Subsequently, Section 5.2 compares measurement and simulation results for the undamped 3D structure. The comparison is carried out for both frequency response functions and modal parameters. In Section 5.3, the experimental results for the structure with viscoelastic damping applications are discussed and these results are again compared to simulations. Finally, in Section 5.4, some possible causes for differences (error sources) between model and experiments are listed. 5. Experimental setup The box structure is manufactured by welding five plates together. Penetration welds are used to minimize the inherent damping of the structure (no free surfaces moving against each other). The welds introduce a radius along the inside edges causing a slight increase in stiffness (see Figure 5.b). In the box model, free boundary conditions are assumed. For this reason, for limiting vibration noise transmitted from the surroundings and to make sure that there are no suspension modes within the frequency range of interest (5 Hz), the 3D structure is suspended on very flexible material. Packaging material consisting of plastic foil with large air pockets is used for this purpose (see Figures 5.a and 5.b). These flexible supports are placed at the four bottom corners. To characterize the dynamic behavior of the structure, a so-called roving hammer test is performed. The structure is excited at different points using a modal hammer and the resulting accelerations at a specific point are measured with a 3D accelerometer. The FRF between the two signals (the accelerance) is computed by a data acquisition system (using the H estimator). At each excitation point, five hammer impacts are performed and the results are averaged. For actuation, a modal hammer with aluminum tip is used to be able to excite all considered box modes in the frequency range of interest. The equipment used in the experiments is listed in Table 5.. More specifications of the sensor and modal hammer can be found in Appendix I. Table 5.: Hardware used in dynamic experimental analyses. Component Manufacturer Model Specifications Modal hammer Meggitt s Endevco Co. Type 232- Range: 22 N Serial No. 378 Sensitivity: 2.34 mv/n 3D accelerometer Meggitt s Endevco Co. Type 65- Range: 5 g Serial No. 25 Sensitivity: mv/g (piezoelectric) Mass: 5 grams Data acquisition system Müller BBM PAK MKII 67

92 Chapter 5: Case study: Experiments and comparison with models (a) box with discrete dampers (b) box with fully covering constrained layer dampers (c) box with partially covering constrained layer dampers (box is upside down) Figure 5.: Measurement setup for the box structure for different VEM damping approaches Wall 5 Wall Wall Wall 2 Wall Sensor position Hammer position Figure 5.2: Fold-out of the box structure, illustrating the sensor placement and hammer excitation points. 68

93 Section 5.: Experimental setup The 73 used actuation points (see Figure 5.2) should be sufficient to capture the mode shapes. Note the numbering of these points in Figure 5.2. The walls are all impacted from the inside of the box. The margins from the side edges and bottom edges (inside of the box) are 5 cm and from the top edge 3 cm. The middle points at the upright walls are placed cm from the top edge. The other points are equally spaced. The sensor is placed on the inside of the box at point 34 on wall 3 (see Figure 5.2). For measuring the undamped structure dynamic behavior, a measurement time of 8 s is taken for each impact, resulting in a frequency resolution of f =.25 Hz. This is the time required for the vibrations to damp out sufficiently. For the damped structures this is reduced to 4 s ( f =.25 Hz). A sample rate of 3.2 khz is used (Nyquist frequency is.6 khz). The dampers can be fixed to the structure using glue, two-sided tape, or by vulcanizing the viscoelastic material on the metal. The viscoelastic material needs to be well secured to the metal plates and the fixation should not reduce the effectiveness of the VEM damper. Since several damping applications need to be characterized, one should be able to remove the applied viscoelastic material without too much difficulty. For that reason, the VEM will not be bonded by means of vulcanization. For fixing the discrete damping element to the box and auxiliary beam structures, instant adhesive Loctite 4 is used. It is relatively stiff and since the VEM patches are small, not difficult to remove. Figure 5.a shows the structure with discrete damping elements. For the CL approaches, the glue surface gets much larger, which complicates application and makes removal almost impossible. Therefore, two-sided tape is considered, which is only acceptable if its shear stiffness is not so low that it drastically influences the damping performance of the VEM. To test if this is the case, both bonding techniques are employed for aluminum plates ( mm) with CL dampers. The performance of a 2 µm thick tape is compared with that of a two component glue (Araldite R 24). For both plates, a roving hammer test is done using a grid of 4 3 points where the outer points are 5 cm from the edges. The plates are suspended on a layer of foam, see Figure 5.3. Figure 5.3: Measurement setup for a plate to asses influence of layer bonding technique. The out of plane driving point accelerance (with impact just right of the sensor, see Figure 5.3) of the two undamped plates are shown in Figure 5.4a. It can be concluded that the dynamic properties of both undamped plates are practically the same. A mm thick layer of viscoelastic material (Viton R SCVBR75) is fixed to each plate and a mm thick stainless steel plate is placed on top. For one structure, glue is used and for the other, the two-sided tape. The driving point FRF for both damped structures is shown in Figure 5.4b. It can be seen that the FRFs for the two different fixation methods differ only slightly. Therefore, the use of the two-sided tape for laminating the viscoelastic material and constraining layer on the box structure seems justified. Two-sided tape will be used for the CL damping approaches. Figures 5.b and 5.c show the fully respectively partially covering CL dampers applied to the box. 69

94 Chapter 5: Case study: Experiments and comparison with models 2 Magnitude [m/s 2 / N] plate plate Magnitude [m/s 2 / N] two sided tape glue Phase [deg] Phase [deg] (a) undamped plates (b) damped plates Figure 5.4: Frequency response function (accelerance) comparison for two plates both undamped and damped to investigate the influence of using glue or two-sided tape. 5.2 Undamped structure The undamped box structure is considered first. For the roving hammer test, excitation points 22 and 37 (see Figure 5.2) are moved downward by 2 cm to prevent so-called double-hits. The measured frequency response functions are exported to ME scope VES[65], which is software used for fitting modal parameters. In this section, the experimental results for the undamped structure are given and compared to the model predictions by means of FRFs and modal parameters Frequency response functions The finite element model is used to compute the frequency response functions (zero damping) for the same points as used to excite the structure experimentally. When a point does not fit the finite element mesh, the displacement (or force) corresponding to a force (or displacement) is linearly interpolated using the four nearest nodes of the FEM mesh. The FRFs are computed using modal superposition [43] using r = modes (last mode near 5 Hz). The results are compared to measurements. Figure 5.5 shows the FRF (accelerance) of an excitation force at point 52 and the resulting out-of-plane acceleration at the sensor at point 34 (see Appendix G for other FRFs). The figure also contains the coherence. The coherence between actuation and response signal is good for the frequency range of interest (the first mode is just above Hz). The coherence below Hz is better for some other FRFs, but is likely poor due to a bad signal to noise ratio. The FRF and coherence of Figure 5.5 are representative for the results at the other excitation points. It can be concluded from Figure 5.5a that the model predictions are quite accurate. The eigenfrequencies are slightly underestimated by the model, as can be seen from the resonance peaks. This probably has to do with the increased stiffness of the box due to the welds along the inside edges. Note furthermore that no model updating step is taken, so it is likely that there still is some difference between some of the parameter values used in the model and the actual values (e.g. Young s modulus and the dimensions). 7

95 Section 5.2: Undamped structure Magnitude [m/s 2 / N] 3 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure 5.5: Frequency response functions for input force at point 52 and out-of-plane output acceleration at point 34 derived from measurements and model (a) and corresponding coherence function (b) Modal parameters The eigenfrequencies and mode shapes of the undamped box from simulations and experiments are compared next. The measured FRFs are imported into the program ME scope VES and a viscously damped model is fitted. Since the accelerometer measures accelerations in all three directions, three reference degrees of freedom are available. The found eigenfrequencies and damping ratios are shown in Table 5.2 together with those found using the FE model. For comparing mode shapes, the modal assurance criterion (MAC) is used, see Equation (3.7). The MAC values are summarized in Table 5.3. The numerical modes are reduced to the same dimension as the experimental modes for computing the MAC values. Illustrations of the first elastic numerical modes and experimental modes 7 to can be found in Appendix D. Table 5.2: First (nonzero) eigenfrequencies found using the Matlab model and the corresponding eigenfrequencies and damping ratios from experiments for the box without viscoelastic damping. Mode nr Freq. model Freq. exp. Damping ratio exp. [Hz] [Hz] [ ]

96 Chapter 5: Case study: Experiments and comparison with models Table 5.3: MAC values for comparing the mode shapes from the Matlab model and those derived from experiments (using ME scope VES) for the undamped box structure. Experimental modes Numerical modes The mode shapes correspond very well since the diagonal entries in Table 5.3 are all close to unity and the off-diagonal terms are near zero. The eigenfrequencies found from experiments and using the finite element model show good resemblance as well (see Table 5.2). Except for modes and 5, the experimentally obtained frequencies are slightly higher than the model predictions. The differences are relatively low though (in the order of only a few percent). The structure is obviously not completely undamped as considered in the model, but the damping ratios are very low (<.), except for mode 2. This can also be seen in Figure 5.5a. For mode 2, most of the deformation is in the bottom plate (first order plate mode). A clear reason for the higher damping in this specific mode is not found. It can be concluded from this comparison that the finite element model is an accurate representation of the undamped structure. In other words, these experiments validate the model without viscoelastic damping material. 5.3 Damped structure The box structure with added viscoelastic damping is considered for the four different damping cases. In this section, the experimental results for the damped structure will be given and compared to the model predictions based on FRFs and modal parameters Frequency response functions The four damped box FE models are used, together with the IMPS method (see Section 4.3.3), to compute the frequency response functions of the four structures. For each structure, out-of-plane force excitation at point 52 and out-of-plane acceleration response at point 43 (Y -direction) is considered. The modal superposition truncation order is set to r = and for the interpolation, N f = 5 frequency points are used (equally spaced over dynamic modulus on log-scale). Figure 5.6 shows the simulated and experimentally estimated FRF and the corresponding coherence function for the box with discrete damping elements. These results are representative for all FRFs of the considered structure. Other frequency response functions can be found in Appendix H.. Figure 5.6 illustrates a good resemblance between experiments and predicted FRF, especially in the low frequency range 4 Hz. There is some difference between the location and amplitude of the peaks for higher frequencies, but also in the high frequency range, the global trend is similar. The influence of the first three elastic modes is predicted accurately. The first mode (torsion mode of the box near 35 Hz) is highly damped and cannot even be discerned in the graph. The coherence is approximately unity in the frequency Hz. The correspondence between model and experiments could be improved by tuning certain model parameters (e.g. Young s modulus of the aluminum, structure dimensions, etc.). Furthermore, it is found that the results are sensitive to the complex Young s modulus dependency on frequency. In addition, the found FRFs are sensitive to slight differences in temperature and errors in the data from the material characterization (DMTA). 72

97 Section 5.3: Damped structure Magnitude [m/s 2 / N] 2 2 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure 5.6: Numerical and experimental frequency response functions for input force at point 52 and out-of-plane output acceleration at point 34 for the box with discrete dampers (a) and corresponding coherence function (b). Next, the three constrained layer damping approaches are considered, starting with the fully covering layers with thick ( mm) VEM layers. The numerical and experimental FRF and the corresponding coherence are depicted in Figure 5.7. The measured and predicted FRFs almost overlap (apart from the frequency region 35 5 Hz). In that region, two modes are located with quite some deformation of the bottom plate. The differences are likely due to the fact that the higher inherent damping of the undamped structure is not included. Another explanation may be that the constraining layer for that wall is relatively thin (i.e..3 mm) and thus more sensitive to differences in dimensions (tolerances on thickness) and faults in application (e.g. air bubbles). Magnitude [m/s 2 / N] 2 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure 5.7: Numerical and experimental frequency response functions for input force at point 52 and out-of-plane output acceleration at point 34 for the box with fully covering CL (thick) (a) and corresponding coherence function (b). 73

98 Chapter 5: Case study: Experiments and comparison with models The results for the box structure with fully covering CL damping and relatively thin VEM layers (.5 mm) can be found in Figure 5.8. As for the CL damping approach with thicker VEM layers, the model predictions are still quite accurate, with larger differences between model and experiments between 35 5 Hz. The overall deviations between model and experiments have increased slightly though. The eigenfrequencies are sligthly overestimated by the model. This may be explained by the fact that, in laminating the different layers, some air bubbles are formed. Then, the constraining layers add less stiffness to the box. The effect is more apparent for the thin VEM layer dampers, because they are stiffer than the thicker VEM layer dampers. Magnitude [m/s 2 / N] 2 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure 5.8: Numerical and experimental frequency response functions for input force at point 52 and out-of-plane output acceleration at point 34 for the box with fully covering CL (thin) (a) and corresponding coherence function (b). Magnitude [m/s 2 / N] 2 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure 5.9: Numerical and experimental frequency response functions for input force at point 52 and out-of-plane output acceleration at point 34 for the box with partially covering CL (a) and corresponding coherence function (b). 74

99 Section 5.3: Damped structure Finally, the box with partially covering constrained layer dampers is considered. The FRF and coherence are shown in Figure 5.9. Again, it can be concluded that the dynamic properties of the structure are estimated well by the finite element model, even without taking a model updating step. The major differences are again seen for the peaks corresponding to modes of the bottom plate. The results presented in Figures 5.6 to 5.9 are representative for those found for different actuation points (other FRFs for the damped box can be found in Appendix H) Modal parameters The FRFs found from roving hammer tests are imported in ME scope VES and fitted with a viscous model. Considering the relatively high damping levels, fitting is difficult, because of several reasons. First, it is possible to obtain a model that fits the FRFs nicely, but it is difficult to find the exact eigenvalues and -modes since resonance peaks are less pronounced and contributions of the modes to the FRF start to overlap. Furthermore, the model used in ME scope VES is based on viscously damped systems, whereas the considered structure is viscoelastic. By importing model FRFs and fitting them, the influence of this on eigenfrequencies and damping ratios is found small, for the lower modes (at least modes 7, 8, and 9). For the higher modes, the influence may be stronger. The first eigenfrequencies respectively damping ratios (not considering the suspension modes) from model and experiments for the 4 different damping approaches are listed in Tables 5.4 and 5.5. To see if the numerical and experimental modal parameters correspond to each other, the mode shapes of the FE model and measurements are compared based on MAC value. The MAC values for the different damping methods are listed in Tables 5.6 to 5.9. Illustrations of the first numerical elastic modes and experimental modes 7 to for the different damping methods can be found in Appendix E. Note that, in general, it is possible that some physical eigenvalues and -modes are not found in the modal decomposition using the finite element model. Furthermore, experimental modes without physical meaning may be found in fitting (computational modes). Table 5.4: First (nonzero) eigenfrequencies in Hz found using the Matlab model and the corresponding frequencies from experiments for the box with viscoelastic damping. Mode nr Discrete dampers CL full (thick) CL full (thin) CL partial Model Exp. Model Exp. Model Exp. Model Exp Table 5.5: First (non-rigid body mode) damping ratios found using the Matlab model and the corresponding damping ratios from experiments for the box with viscoelastic damping. Mode nr Discrete dampers CL full (thick) CL full (thin) CL partial Model Exp. Model Exp. Model Exp. Model Exp

100 Chapter 5: Case study: Experiments and comparison with models Table 5.6: MAC values when comparing the mode shapes from the FE model with those identified by experiments (using ME scope VES) for the box structure with discrete damping elements. Experimental modes Numerical modes Table 5.7: MAC values when comparing the mode shapes from the FE model with those identified by experiments (using ME scope VES) for the box structure with fully covering constrained layer dampers (relatively thick VEM). Experimental modes Numerical modes Table 5.8: MAC values when comparing the mode shapes from the FE model with those identified by experiments (using ME scope VES) for the box structure with fully covering constrained layer dampers (relatively thin VEM). Experimental modes Numerical modes

101 Section 5.3: Damped structure Table 5.9: MAC values when comparing the mode shapes from the FE model with those identified by experiments (using ME scope VES) for the box structure with partially covering constrained layer dampers. Numerical modes Experimental modes It can be concluded from the eigenfrequencies, damping ratios and MAC values that the model predictions are good for modes 7, 8, and 9 for all damped structures. For the box with discrete dampers, the measured eigenvalues for mode and higher differ significantly from those computed with the model. This is mostly caused by the fact that fitting for higher damping levels and higher frequencies is problematic. As the damping ratios become higher, it gets more and more difficult to discern the individual modes from estimated experimental FRFs, see Subsection This can be seen from Table 5.6. The higher numerical modes with relatively high damping seem to be linear combinations of higher experimentally identified modes and vice versa. Modes and 2 of the FE model for example seem to correspond to combinations of modes and 3 found in fitting. Numerical mode seems to consist of experimental modes and 2. For the discrete damping approach, only nine relevant modes are experimentally identified within the frequency range of interest. It can be concluded that the discrete damping elements adds significant damping to the structure. Looking at the eigenfrequencies of the structure with constrained layer damping, in general it can be concluded that the model slightly overestimates the damped eigenfrequencies. The damping ratios of the modes are sometimes slightly underestimated and sometimes slightly overestimated by the model. Similar to the simulation results, the partially covering CL damping adds most damping to the structure for the constrained layer damping approaches. Second best is the fully covering CL approach with relatively thick VEM. The fully covering CL approach with thinner VEM gives the lowest damping ratios. As for the discrete dampers, the higher modes for the three constrained layer applications do not completely match. For the CL damping with full coverage of the walls for example (see Tables 5.7 and 5.8) a computational mode is experimentally identified in fitting (mode 2), which is numerically not found. The subsequent modes have shifted upward one number (e.g. numerical mode 2 model matches with experimental mode 3). This is especially apparent for the constrained layer configuration with relatively thin VEM layer. For the case where the box walls are partially covered by constrained layer dampers (see Table 5.9), the numerical and experimental mode shapes match quite well since most of the diagonal entries are close to unity and the off-diagonal terms are near zero. However, this is not the case for modes 5 and 6. Note that the corresponding eigenfrequencies are very close to each other (see Table 5.4), which makes it difficult to discern the modes identified from measured FRFs (even when using multiple reference DOFs). It can be concluded that the dynamic characterization by model predictions and experiments match reasonably well to well for the four damped structures. In other words, the measurements validate the model of the structure to a large extent. A good resemblance between model and reality can be seen from the frequency response functions, especially in the frequency range 4 Hz, but sometimes also in the higher frequency range. The modal parameters for the damped cases do not match that well for the higher frequency modes, mainly because it is difficult or impossible to discern those damped modes during fitting of modal parameters. This can be seen clearly when looking at the MAC values. During fitting, modes are found, which are combinations of the modes determined with the model. 77

102 Chapter 5: Case study: Experiments and comparison with models 5.4 Error sources In comparing the model predictions and experimentally obtained dynamic properties of the box structure, certain differences are found. Some (at low frequency, i.e. < Hz) are caused by measurement errors, but most can be assigned to differences between reality and model. Some of the possible error sources are: In the model, specific assumptions are made on the displacement field within the material (e.g. cross sections remain flat) and plane stress and plane strain assumptions are used. This is a simplification of how the structure actually deforms. There are differences in parameter values used in the model and the actual properties of the structure (e.g. dimensions, Young s modulus, mass density, Poisson s ratio). Especially differences in the dimensions such as layer and plate thicknesses are found to have quite some impact on the results, in particular for low thicknesses. For the constrained layer damping approaches, the dimensions of the damping and constraining layers used in the model are limited to the mesh used and thus differ slightly from those in practice. The layers span for example not as close to the outside edges of the box in the model as in reality. This difference is found not to have much influence on the dynamic properties of the system (from simulations). Furthermore, the bonding of the layers to the structure is not very good near the edges (see Figure 5.a) reducing the effect of that material segment. The material model used for the complex Young s modulus of the VEM is prone to errors originating from deviations in temperature and errors in DMTA measurement data. The latter aspect is sensitive to how the isotherm measurement data is shifted. Errors in Young s modulus of the VEM are found to have quite some effect on the resulting FRFs and modal parameters. Changing the temperature used in the model from 22 C to 2 C is done to confirm this. Some differences between model and practice originate from limitations in realization of the structure. Examples of this are the welds along the edges at the inside of the box (see Figure 5.b) and difficulties in laminating the layers for CL damping. During lamination, some small air bubbles get entrapped between the different layers. This reduces the contact area and effective stiffness. In identifying experimental eigenvalues and -modes from experiments sometimes computational modes are found or particular modes are very difficult to discern. The identification is problematic due to the relatively high damping of the modes. Furthermore, errors in these fitted eigenvalues and -modes are introduced by the fact that a viscous model (instead of a viscoelastic) is used for fitting the modal parameters. A robust viscoelastic damping identification approach needs to be developed. A method for identifying hysteretic damping from FRFs is already presented in [66] and could perhaps be used in this development. (a) CL lamination close-up (b) close-up of one of the welds Figure 5.: Illustrations of two sources of difference between model and reality, i.e. reduced bonding of the CL dampers near the edges and welds along the inside edges of the box. 78

103 Chapter 6 Conclusions and recommendations 6. Conclusions Reducing vibrations may be a critical issue in high-tech industry. Vibrations may limit performance, cause noise, and result in control limitations. Damping can be used to address these vibrations and improve the dynamic properties of structures. Passive damping may be favorable over active damping because it does not require a power source, is less complex in implementation, and less sensitive to wear. A simple but effective way to introduce damping is by adding engineering materials with viscoelastic properties. Insight on how and where to apply these materials to efficiently dampen structural vibrations is limited. The effectiveness of different configurations and application techniques of viscoelastic materials is considered in this study. An experimentally validated finite element model, accurately estimating the dynamics of a considered 3D structure is composed. First, different VEM configurations such as free layer damping, constrained layer damping, and discrete damping elements have been considered for two different simple load cases (i.e. axial deformation and bending). The structures are modeled using the finite element method and a complex Young s modulus for the VEM is considered. The different methods are compared. In a general comparison study, the damping ratio is used and for a specific case study, a performance criteria linked reflecting vibration amplitude is used. It is concluded that using the viscoelastic material in the form of discrete damping elements is most effective to address specific modes of interest. To obtain relatively high damping levels (but not as high as for discrete dampers) over a wide range of modes, applying the VEM in a constrained layer configuration is found to be favorable. A free layer approach is not very effective compared to the other techniques. Some design guidelines and important VEM damping design considerations are attained. It is found that, for the discrete dampers, the added damping levels depend mostly on the ratio of the (local) stiffness of the structure and the stiffness of the VEM, for which there is an optimum. Furthermore, the positioning of the discrete dampers is important and is found not to be optimal at the location of maximal displacement amplitude, as one might expect. The optimal position results from a trade-off between maximizing strain amplitude and stiffness. For the constrained layer damping approach, there is an optimal VEM layer thickness that depends on constraining layer thickness and the base structure dimensions and properties. The constraining layer thickness is optimal when its stiffness matches that of the base structure for transversal vibrations (at least for a fully covering CL damping configuration). The two promising approaches (discrete and constrained layer dampers) have been applied to a more complex 3D structure (i.e. an open aluminum box) to asses viscoelastic damping performance for a more realistic structure and using commercially available viscoelastic materials. It is found that viscoelastic materials can be used to effectively introduce damping to a structure. Damping ratios (ratio between real part and absolute value of the eigenvalues) in the order of five percent and higher are attained for the considered structure. Discrete damping element are found to be add most damping. When there are mass constraints for constrained layer dampers, partial coverage of the structure at the parts with the largest deformation is most effective. Partially covering CL dampers should be placed at locations of maximal strain. The structure is realized and characterized experimentally. From comparing model predictions and experiments it is concluded that the model is reasonably accurate to accurate both for estimating frequency response functions and modal parameters. For the computation of transfer functions from model simulations, a method based on modal superposition is proposed, where the frequency dependency of the VEM properties is incorporated by means of frequency interpolation of modal parameters. To the best of the authors knowledge, this is a new method, which is not described in literature. For extracting the modal parameters from experiments, fitting is found to be difficult, because of the attained (relatively high) damping levels and because only a viscous model is available for fitting these parameters. 79

104 Chapter 6: Conclusions and recommendations It can be concluded that it is possible to quite accurately predict and model the dynamics of a structure with viscoelastic passive damping using the techniques employed in this study. This is an important feature in improving the dynamic performance of structures, particularly in the high precision industry. Moreover, it encourages the incorporation of viscoelastic damping in an early phase of the design process. 6.2 Recommendations In Chapter 3, different methods of applying VEM to a simple beam structure are compared for both axial and transversal vibrations. It is found that there are optimal conditions for most cases. For the discrete dampers for example, there is an optimal stiffness related to the effective stiffness of the base structure. A clear and generally applicable expression for the parameter combinations corresponding to these optima for the different damping approaches is not found. Further study on this is recommended. For example by making the equations of motion non-dimensional and extracting relevant non-dimensional quantities. In Chapter 4, shell elements are used to model the walls of the 3D structure and stiff elements are considered in coupling the plates at their edges. Fairly simple (low order) elements are employed in modeling the viscoelastic material. It seems likely that better predictions can be made when different (higher order or higher dimension) finite elements are used, especially for the discrete damping elements. This probably comes at the cost of increased computation times. Furthermore, in this study, a method for computing frequency response functions of generally damped systems is proposed. The method referred to as the interpolated modal parameters superposition (IMPS) method, uses interpolation of certain modal parameters over frequency and modal superposition to avoid computing an inverse of a matrix for all frequencies required in the direct (exact) method. The method requires that the correct eigenvalues and -modes are (incrementally) matched at each frequency, which is done using the modal assurance criterion. Sometimes this is problematic. However, it seems possible to attain some improvement in robustness here. Alternatively, a completely different approach to construct a transfer function model using the finite element model may be found. The experimental modal parameters are found by fitting measured FRFs using ME scope VES in this study. In this, a viscous damping model is used by the software, causing inaccuracies. Therefore, the development of a robust fitting technique based on a general viscoelastic frequency dependent model is recommended. A comparison of effectiveness of different passive damping techniques (e.g. using VEM, TMD s and viscous dampers) for a specific case is furthermore recommended for further research. A clear analysis of the potential of each method for particular cases and geometries can provide a guide for passive damping design. An optimization problem may need to be defined and investigated. 8

105 Appendices 8

106

107 Appendix A Viscoelastic material characterization and modeling To incorporate viscoelastic material in a structure with the objective to add passive damping to the system, it is desirable to be able to accurately predict the added damping level. A model that comprises the viscoelastic material properties to a high degree of accuracy is required. In this appendix, an approach to obtain such a model for a material of interest is described. The dynamic properties are first characterized using a so-called Dynamic Mechanical Thermal Analysis (DMTA) test. This will be elaborated on first. The data is processed by means of shifting using the temperature-frequency superposition method. How this can be done is mentioned in Section A.2. The measurement data are in the format as required and a model can be fitted on them, a process which will be explained in Section A.3. This appendix is concluded with a list of some viscoelastic material products suitable for passive damping and the companies that manufacture them. A. DMTA A dynamic mechanical thermal analysis is often used to obtain the Young s modulus of materials as a function of temperature or frequency. A sample of the material is placed in the machine. A displacement is imposed by actuating a force motor, deforming the specimen. From measuring force and displacement (and considering the dimensions and other properties of the sample), the Young s modulus under the conditions of the measurement can be found. Several configurations are possible, i.e. the sample can be loaded in tension, compression, shear, torsion or a three point bending test can be done. A dual or single cantilever configuration for the sample is also possible. The three point bending and cantilever configurations are commonly used to characterize stiff materials. The measurement system used to characterize the materials for this study is shown in Figure A.. The samples are loaded in tension during the measurements. In assessing the properties of polymers, first, a low frequency harmonic strain (or stress) is imposed and the temperature is varied. From this test, the glass transition temperature can be found by looking at the location of a maximum in the loss factor (or loss modulus). This initial measurement is used to choose the ranges in the variations for another test. It also shows some other characteristics of the material, for example β-transitions and if the polymer melts in the temperature range of interest. More detail on the properties of polymers is not mentioned here. As a second measurement, a harmonic strain is applied of which the frequency is slowly varied (e.g. from to 4 Hz). This is done at different temperatures. For all these conditions, a storage modulus and loss factor are found from the amplitude ratio between stress and strain and the phase difference between the two respectively. The temperature-frequency equivalence can be used to construct the frequency dependency of these properties over a much wider range by shifting the isotherm data as is illustrated in Figure 2.9. How to choose the factors with which each data group (same temperature) is shifted along the frequency axis will be explained in the next section. 83

108 Appendix A: Viscoelastic material characterization and modeling Furnace Low Mass, High Stiffness Sample Clamps Air Bearings Rigid Aluminum Casting Drive motor Optical Encoder Figure A.: DMTA measurement system (from [67]). A.2 Shifting The data found using the DMTA measurements can be processed to construct what is called a master function Ê(ω r ), i.e. the complex Young s modulus (or split into storage modulus and loss factor) over a wide frequency range at a specific reference temperature T. In this, a shift factor α(t ) (where E (ω, T ) = Ê (α(t )ω) = Ê (ω r)) needs to be found for every data group (measurements at equal temperature), such that, when shifting the properties along the frequency axis with that factor, the curves line up nicely (see Figure 2.9). From the shift factors for the data sets, a shift function can be extracted. This can be used to find the material properties at a different temperature (or frequency). Common models for this shift function are [3, 36, 37]: T T Williams-Landel-Ferry (WLF) log [α(t )] = C (B +T T) Arrhenius log [α(t )] = T A T T Modified Arrhenius Tomlinson et al. log [α(t )] = N n= T n ( T T ) n log [α(t )] = N T n (T T ) n with, C, B, T A and T n constants used for fitting. The reference temperature T needs to be found such that the shift expression is best. To obtain the shift factors for the different isotherms, several approaches are possible. The most basic method is starting with the data set at a specific temperature and shifting the data corresponding to measurements at decreasing (and increasing) temperature along the frequency axis until the curves line up. This is done visually by means of a trial and error approach. Although the resulting master function will look good, there is no aspect supporting the correctness of the shift factor function. Furthermore, the method is time-consuming. Another method incorporating a constitutive relation between storage modulus and loss factor as a function of frequency can also be used [35, 38]. The approach is based on fitting a material model (i.e. the ratio model) on a spline interpolation of the Wicket plot to construct the dependency of E on frequency (apart from a scaling factor) and shifting the isotherms such that they line up best with this model. The measurement data of the DMTA test is represented in the Wicket plot. See Figure A.2 for an example. The complex modulus data E for the different temperatures and frequencies is split into its magnitude and loss factor. n= 84

109 Section A.2: Shifting These quantities are plotted against each other. If the points are all on a smooth line, the data can be considered consistent. In Figure A.2, the blue crosses represent measurement data [38]. Figure A.2: Wicket plot with spline model fit (from [38]). A function describing the relation between loss factor and magnitude of Young s modulus (i.e. E d (η)) is required for the next step. A spline is fitted on the data and the curves are extrapolated to find a static dynamic modulus at zero frequency and a steady value for ω [38]. See Figure A.2 for an example of this "Analytical Wicket". A constitutive relation is necessary to include frequency dependence. It is assumed the material characteristics are linear. A ratio model is fitted on the obtained spline. The ratio model can be expressed as [38] E (s) = E N j= + a j s/r j + s/a j r j (A.) with E a zero frequency modulus, r j the transition frequency for the jth term and a j a coefficient determining the height of the riser of that contribution. When fitted, this model describes the dependence of Young s modulus on frequency, but this frequency is not physically relevant. It can be scaled as desired. In [38] it is chosen to place the top in loss factor at Hz. For more detail on how the fitting is done, see [38]. For every data point, the closest point on the model curve (for example in the sense of minimal complex difference amplitude) is found, thus defining the shift factor for that measurement. All the found shift factor increments are plotted against temperature and a model is fitted on that. Some possible functions for this are mentioned above. The obtained shift function expression can then be used to find the complex Young s modulus as a function of frequency for a temperature of interest, by shifting the measurement data accordingly and using a factor of one for the data measured at that specific temperature. In this study, a slightly adjusted technique is considered. The step of fitting the ratio model is bypassed. It holds that [38] δ π d log(e d ) 2 d log(ω), (A.2) where η = tan(δ). This is employed to construct an approximation for the frequency dependence. Starting at an arbitrary frequency, a minimal dynamic modulus is used. The corresponding loss factor is found. This factor is used to find the required modulus magnitude at the next frequency in a used frequency vector considering that the loss factor can be used to approximate the slope. The process is repeated and in the end the frequency vector is scaled, again placing the η peak at Hz. The found relation for dynamic modulus and loss factor as a function of frequency are used to find the shift factor function similarly as described above. The approximation used for constructing the frequency dependency is found to be fairly accurate for the viscoelastic materials considered as the logarithmic slope of modulus magnitude is relatively low (maximally in the order of.7). 85

110 Appendix A: Viscoelastic material characterization and modeling A.3 Model fitting Now that the measurement data is shifted to form the modulus across a much wider frequency range, a model can be fitted. Some common model representations are mentioned in Section Of these, the fractional derivative model is found to give the best results even for relatively low model orders (e.g. N = 5). Higher orders are used in this study to obtain higher accuracy of the results. In this section, a fitting procedure will be described for the fractional derivative model, but an equivalent approach can be taken for different material models. The fraction derivative model can be described by Equation (2.26), i.e. N E (s/r j ) φj (s) = E + E j. (A.3) φj + (s/r j ) j= The Matlab optimalisation function fmincon is used for fitting. The variables are the coefficients E, E j, r j and φ j for j ranging from to N. For the terms E, E j and r j, the log is taken to obtain the variables used in computation (improved conditioning). The experiment data is captured in a vector E d,e containing the dynamic modulus for all data points and a vector η e with the loss factors. The shifted frequencies corresponding to these data points are captured in ω e. For the model to fit the data best, the differences between E (iω e ) and the experiment data should be minimal. An objective function obj is used which needs to be minimized, i.e. obj = c f N e ( + w f ) N e j= [ log (E (iω e,j )) log (E d,e,j ) + w f η(ω e,j ) η e,j ], (A.4) with c f a conditioning factor (taken ), w f a weighting factor, N e the number of data points and E d,e,j, η e,j and ω e,j the jth term in the vectors E d,e, η e and ω e respectively. In this study, w f is set to. The fmincon(.) function uses this objective to find a suitable model fit. The variable E is initialized as the lowest dynamic modulus in the measurement data vector. For the different E j an initial value equally spaced on a log scale between minimum and maximum value of measured dynamic modulus is used. The different r j are initialized equally spaced over log frequency where the log (ω) range is divided by the order and a factor 2 to obtain the frequency step. An initial guess of.5 is taken for φ j. When using a GHM model for example, an extra constraint on the dissipativity of the model is required [29]. This can be included in the fmincon function. A.4 Materials and manufacturers A short search on available engineering materials which can be employed to introduce passive damping to a structure is done. For a list of some viscoelastic material damping products and the companies that manufacture them, see Table A.. This is by no means a full survey and not all relevant product of the manufacturers might be listed. Some polymers with suitable viscoelastic properties are [3] Acrylic rubber Polystryrene Butadiene rubber Polyvinyl Chloride (PVC) Butyl rubber Polymethyl Methacrylate (PMMA) Chloroprene Polybutadiene Chlorinated Polyethylenes Polypropylene Ethylene-Propylene-Diene Polyisobutylene (PIB) Fluorosilicone rubber Polyurethane Fluorocarbon rubber Polyvinyl Acetate (PVA) Nitrile rubber Polyisoprene (e.g. Acrylonitrile-butadiene rubber (NBR)) Styrene-Butadiene (SBR) Natural rubber Silicone rubber Polyethylene Urethane rubber 86

111 Section A.4: Materials and manufacturers Table A.: Some viscoelastic materials and corresponding manufacturers. Manufacturer Country (headquarters) Product name(s) 3M USA Viscoelastic damping polymers (2-, 3- and 242 series) E-A-R Aearo Technologies LLC USA Several damping materials available (e.g. Isoloss R and Isodamp R ) Astrotech advanced elastomer products Austria Norsorex R Eriks The Netherlands Viton R sheets Exxonmobil Chemical USA Butyl rubber Bromobutyl rubber Chlorobutyl rubber Heatcote industrial plastics United Kingdom HIP2 (constrained layer dampers) Hodgson & Hodgson group Ltd. United Kingdom Aludamp (constrained layer dampers) Damping sheets - H&H INC industrial noise control USA Noise and vibration damping sheets D35, D36 Nitto Denko Japan Legetolex R Robinson rubber products company, Inc. USA Several damping polymers available (e.g. ACM, BR, CM, CR, IIR, NR) Roush USA Several damping adhesives and rubbers Sorbothane shock and vibration solutions USA Sorbothane R Soundcoat USA GPDS DYAD (constrained layer dampers) Soundamp (constrained layer dampers) Soundown USA Vibration damping sheet and tile EcoDamp sheet (constrained layer dampers) McGill Airsilence LLC USA Soundscreen TM 87

112

113 Appendix B Additional results principle method comparison In this appendix, additional results corresponding to a principle method comparison for the effectiveness of different viscoelastic material (VEM) applications are presented. This is done for two load cases, i.e. axial deformation and bending. The effectiveness is compared on the basis of a proposed objective function (Equation (3.9)). The considered approach is:. The eigenfrequencies of a reference case are found, where the complete available volume is filled with the metal structure. 2. The ratio of the volume used by the VEM is varied. The beam will have to become thinner. 3. Other relevant parameters are varied. For example, for the discrete damper cases, the stiffness of the viscoelastic material can be altered by changing the length/cross sectional area ratio for a specific volume. Other relevant parameters include: damper position, storage modulus and loss factor of the VEM. 4. The objective function (3.9) is computed for each of the parameter combinations and different modes. 5. The maximal objective function value per eigenmode within a practical range of the parameters is found for comparison of the effectiveness. Tables B. and B.2 list the results corresponding to different viscoelastic material applications for an axially vibrating beam with a length of 2 cm for the first four modes and three different η. For Table B., the volume fraction VEM is limited to χ. whereas for the results in Table B.2 a limit of χ.25 is considered. Tables B.3 and B.4 show the results for the same parameter combinations but for a beam length of L b = 5 cm, again for χ. and χ.25 respectively. The maximum found objective function for the parameter combinations and four different modes and three loss factors for a beam bending load case are presented in Tables B.5 to B.8. Pinned boundary conditions are considered in finding the results shown in Tables B.5 and B.6 (for two different volume fraction ranges). For the results in Tables B.7 and B.8, clamped boundary conditions are applied to the beam of interest (again for χ. and χ.25 respectively). Other parameter values used and examples of the dependence of objective function on volume fraction and modulus for the different VEM approaches can be found in Sections and

114 9 Table B.: Maximal objective function in parameter variation for different methods, modes and loss factors (axial deformation) for χ. and L b = 2 cm. η =.5 η = η = 2 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v = L) Discrete damper, tension (at x v =.72L) Discrete damper, tension (at x v =.84L) Discrete damper, shear (at x v = L) Discrete damper, shear (at x v =.72L) Discrete damper, shear (at x v =.84L) Appendix B: Additional results principle method comparison

115 Table B.2: Maximal objective function in parameter variation for different methods, modes and loss factors (axial deformation) for χ.25 and L b = 2 cm. 9 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v = L) Discrete damper, tension (at x v =.72L) Discrete damper, tension (at x v =.84L) Discrete damper, shear (at x v = L) Discrete damper, shear (at x v =.72L) Discrete damper, shear (at x v =.84L) η =.5 η = η = 2 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode

116 92 Table B.3: Maximal objective function in parameter variation for different methods, modes and loss factors (axial deformation) for χ. and L b = 5 cm. η =.5 η = η = 2 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v = L) Discrete damper, tension (at x v =.72L) Discrete damper, tension (at x v =.84L) Discrete damper, shear (at x v = L) Discrete damper, shear (at x v =.72L) Discrete damper, shear (at x v =.84L) Appendix B: Additional results principle method comparison

117 Table B.4: Maximal objective function in parameter variation for different methods, modes and loss factors (axial deformation) for χ.25 and L b = 5 cm. 93 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v = L) Discrete damper, tension (at x v =.72L) Discrete damper, tension (at x v =.84L) Discrete damper, shear (at x v = L) Discrete damper, shear (at x v =.72L) Discrete damper, shear (at x v =.84L) η =.5 η = η = 2 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode

118 94 Table B.5: Maximal objective function in parameter variation for different methods, modes and loss factors (bending) for χ. and pinned-pinned boundary condition. η =.5 η = η = 2 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v = L) Discrete damper, tension (at x v =.72L) Discrete damper, tension (at x v =.84L) Discrete damper, shear (at x v = L) Discrete damper, shear (at x v =.72L) Discrete damper, shear (at x v =.84L) Appendix B: Additional results principle method comparison

119 Table B.6: Maximal objective function in parameter variation for different methods, modes and loss factors (bending) for χ.25 and pinned-pinned boundary condition. 95 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v = L) Discrete damper, tension (at x v =.72L) Discrete damper, tension (at x v =.84L) Discrete damper, shear (at x v = L) Discrete damper, shear (at x v =.72L) Discrete damper, shear (at x v =.84L) η =.5 η = η = 2 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode

120 96 Table B.7: Maximal objective function in parameter variation for different methods, modes and loss factors (bending) for χ. and clamped-clamped boundary condition. η =.5 η = η = 2 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v = L) Discrete damper, tension (at x v =.72L) Discrete damper, tension (at x v =.84L) Discrete damper, shear (at x v = L) Discrete damper, shear (at x v =.72L) Discrete damper, shear (at x v =.84L) Appendix B: Additional results principle method comparison

121 Table B.8: Maximal objective function in parameter variation for different methods, modes and loss factors (bending) for χ.25 and clamped-clamped boundary condition. 97 Free layer (one sided) Free layer (two sided) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Constrained layer (one sided, 5% VEM) Constrained layer (two sided, 5% VEM) Discrete damper, tension (at x v = L) Discrete damper, tension (at x v =.72L) Discrete damper, tension (at x v =.84L) Discrete damper, shear (at x v = L) Discrete damper, shear (at x v =.72L) Discrete damper, shear (at x v =.84L) η =.5 η = η = 2 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode 4 Mode Mode 2 Mode 3 Mode

122

123 Appendix C Finite element modeling The finite element model approach for modeling structural dynamics is explained in this appendix. First the generic derivation of the equation of motion contributions for a single finite element of the structure will be described. This results in a mass and stiffness matrix corresponding to the different degrees of freedom describing the system. In modeling the case structure used in this study (i.e. an open box), plate elements are used as the geometry is composed of different plates, where for each, one dimension is relatively small compared to the others. Different element types are possible. They differ with respect to interpolation order (and with that, number of nodes) and the degrees of freedom used. A selection is made and the elements are explained in Section C.2. They are compared to each other with respect to accuracy and computation cost in order to choose the element used for the final structure modeling. Finally, how to model a beam by means of finite elements will be explained. C. Generic linear structural element The finite element method (FEM) is based on dividing the volume of interest into different elements with corresponding nodes. Continuous quantities such as displacements in the material are found by interpolating the degrees of freedom defining those quantities at the nodes. For a generic structural element with linear dynamic behavior, the method can be used to construct the equations of motion in terms of the nodal degrees of freedom (DOF) and their time derivatives. For an undamped (non-rotating) structure, this results in a symmetric mass and stiffness matrix. Consider part of the volume of interest constituting a single element with volume V e. Within this sub-volume, deflections are defined as u, v and w in directions x, y and z respectively. These directions correspond to an arbitrary cartesian reference frame. The different deflections within the material are interpolated based on a specific number of DOFs using polynomials. The order of the polynomials needs to be chosen as do the different degrees of freedom and relation between them and the displacements in the material. The degrees of freedom of a single element are captured in the column q e. So-called shape functions defining the relation between these DOFs and the deflections for the coordinates within the considered element are employed for interpolation. The displacements u, v and w in x, y and z-direction respectively within the volume can be written as u(x, y, z) N u (x, y, z) v(x, y, z) = N v (x, y, z) q e = N(x, y, z)q e (C.) w(x, y, z) N w (x, y, z) with N u (x, y, z), N v (x, y, z) and N w (x, y, z) rows containing the shape-functions corresponding to the DOFs that define the displacements u, v and w respectively. These functions are combined into a matrix N. It is common to perform a coordinate transformation to iso-parametric coordinates ranging from to for the x, y and z coordinates within an element, for example as x = 2 ( ξ)x j + 2 ( + ξ)x j+ = x j + 2 ( + ξ)(x j+ x j ) = x j + 2 ( + ξ)e x (C.2) and y = 2 ( ζ)y j + 2 ( + ζ)y j+ = y j + 2 ( + ζ)(y j+ y j ) = y j + 2 ( + ζ)e y. z = 2 ( ψ)z j + 2 ( + ψ)z j+ = z j + 2 ( + ψ)(z j+ z j ) = z j + 2 ( + ψ)e z (C.3) (C.4) 99

124 Appendix C: Finite element modeling for a block-shaped element volume with dimensions e x, e y and e z in x, y and z direction respectively where x j x x j+, y j y y j+ and z j z z j+. In this, the subscripts indicates that it is the coordinate at that specific node. The quantities ξ, ζ and ψ are the iso-parametric coordinates in this case. The displacements within the considered volume V e are now defined by the degrees of freedom and any nonconstant state of them implies that the structure has kinetic energy. The kinetic energy T e of a single element can be expressed as [47] with M e the mass matrix corresponding to a single element expressed as M e = ρ N T NdV. V e T e = 2 ρ V e u 2 + v 2 + ẇ 2 dv (C.5) = 2 ρ V e q T e N T N q e dv (C.6) = 2 qt e M e q e, (C.7) Any non-rigid displacement state will furthermore give rise to strains and stresses in the material. Potential energy is stored (not considering gravity influences). Isotropic material is considered. The strains ε T = [ εxx ε yy ε zz γ xy γ xz γ yz ] and corresponding stresses σ T = [ σ xx σ yy σ zz τ xy τ xz τ yz ] in an arbitrary material volume are linked to each other by means of a constitutive relation. The subscripts represent the direction of the strain (or stress) and the direction of the plane normal at the location of that strain (or stress). It might be possible/necessary to employ assumptions as plane stress (e.g. σ zz = ) or plane strain (e.g. ε zz = ), reducing the stress and strain components required in computations. The constitutive relation for an isotropic material is given by [52] σ = E ( + ν)( 2ν) ν ν ν ν ν ν 2 ( 2ν) ε Dε 2 ( 2ν) sym ( 2ν) with E the Young s modulus and ν the Poisson s ratio. The strains in the material can be linked to the displacements as (for small strains) [52] ε xx = u x γ xy = u y + v x which can be expressed as x y ε = z y x z x z y ε yy = v y γ xz = u z + w x u v w κ 2 ε zz = w z γ yz = v z + w y x y = z Nq y x e Bq e. z x z y (C.8) (C.9), (C.) (C.) Using the relations for stress and strain, the potential energy U e due to deformation within the part of the structure considered can be written as [47] U e = σ T εdv (C.2) 2 V e = ε T D T εdv (C.3) 2 V e = q T 2 V e e BT D T Bq e dv (C.4) = 2 qt e Ke q e, (C.5)

125 Section C.2: Plate elements with K e the stiffness matrix corresponding to a single element. Using Lagrange s equation (assumed known) and assembly of the different generic elements gives the equations of motion for the structure, i.e. M q + Kq = F, (C.6) with M and K the complete mass and stiffness matrix respectively, q a column with the degrees of freedom and F the force vector. For many structures, it is necessary to add specific constraints on some of the DOFs. For linear constraint equations, this can be done by a coordinate transformation to minimal coordinates using a transformation matrix T c. As and example, consider an arbitrary system with 4 degrees of freedom q,..., q 4 with as a boundary condition q 4 = q. The mentioned transformation matrix is then defined as q q q q 2 q 3 q 4 = q 2 q 3 = T c Using this approach, the mass and stiffness matrix are transformed to M min and K min respectively as q 2 q 3. (C.7) M min = T T c MT c respectively K min = T T c KT c. (C.8) Other changes of the degrees of freedom, such as rotations, can also be done by pre- and post-multiplying the mass and stiffness matrices by a specific transformation matrix. C.2 Plate elements The main structure considered in this study consists of a combination of plates. Plates are characterized by the fact that one dimension is small w.r.t. the other dimensions. In finite element modeling, they are commonly modeled using shell (or plate) elements. These elements span a 2D mesh, in this study placed at the midplane of the plates. The coordinates x and y are aligned with this mesh and the z direction is normal to this plane. The deflections within the plane spanned by the elements is found using interpolation of the degrees of freedom at the nodes. Extrapolation is considered in estimation the displacements in the material as a function of the coordinate in the perpendicular direction. It is considered that the plate cross sections remain flat during deformation, meaning that rotations along the x or y-axis spanning the element result in in-plane deflections that linearly increase with z. An illustration of this for a beam structure is shown in Figure 3.2. An assumption for the strain (or stress) in the thickness direction is required as the out of plane displacements are not so easily extrapolated. A plane stress assumption is employed here implying that σ zz =. Triangular and quadrilateral elements are commonly considered for shell elements. It is chosen to use quadrilaterals here. They consist of at least four nodes (using bi-linear interpolation functions). Extra midpoint nodes can be added to obtain eight node (serendipity family) or nine node (lagrange family) elements. For these, second order polynomials are used for interpolation. Figure 4. shows the different elements with respect to iso-parametric coordinates and illustrates how the nodes of these elements are numbered. It is possible to use one of these types for all degrees of freedom, but a combination of elements is also possible. The degrees of freedom at the nodes that will be considered are deflections in x, y and z direction called u, v and w respectively, not to be confused with the displacement fields within the element itself. A subscript will be added later referring to the corresponding node. Furthermore, rotations along the different axes referred to as θ x, θ y and θ z (around the x, y and z axis respectively) at the coordinates of the nodes are required to have all six degrees of freedom represented. It is possible to add other degrees of freedom to make it possible to use higher order polynomials for interpolation in finding expressions for the displacement field, such as spatial derivatives at the nodes. Sometimes, more DOFs are added to model shear deformation more accurately. For plate elements, sometimes only five degrees of freedom per node are considered, i.e. the rotation θ z (often referred to as the drilling degree of freedom) is left out. A selection is made of quadrilateral elements differing in the number of nodes, the degrees of freedom per node and assumptions on displacement characteristics. These will be described below. The different elements are implemented in Matlab to model a simply supported plate and compute its eigenfrequencies. The accuracy of the quadrilaterals is compared and one type is chosen in modeling the box walls.

126 Appendix C: Finite element modeling C.2. Element types The different elements considered are illustrated in Tables C., C.2 and C.3. The tables show the element in isoparametric coordinates, the different degrees of freedom corresponding to specific nodes, what the element type will be referred to as here and the total number of DOFs corresponding to a single element. The latter gives an indication for the computational cost, which increases with the model size. A brief explanation of the different element types and the corresponding considerations will be given below. Table C.: Illustration of the different four node finite element types considered. Element DOF per node Name used here DOF per element u, v, w, θ x, θ y, θ z thin plate assumption Q46t 24 u, v, w, θ x, θ y, θ z Q46 24 u, v, w, θ x, θ y, θ z, w x, w y Q48 32 Table C.2: Illustration of the different eight node (and mixed) finite element types considered. Element DOF per node Name used here DOF per element u, v, w, θ x, θ y Q85 4 u, v, θ x, θ y, θ z w Q u, v, θ z Q w, θ x, θ y 2

127 Section C.2: Plate elements Table C.3: Illustration of the different nine node (and mixed) finite element types considered. Element DOF per node Name used here DOF per element u, v, w, θ x, θ y Q95 45 u, v, w, θ x, θ y, θ z Q96 54 u, v, θ x, θ y, θ z w Q u, v, θ z Q w, θ x, θ y Q46t This element uses the four node construction with all six degrees of freedom per node. The assumption of a thin plate is employed corresponding to Kirchhoff plate theory [53]. This implies that bending of the plate is accompanied by zero shear. The cross sections not only remain flat, but furthermore perpendicular to the midplane. It holds that [47] w x = θ y and w y = θ x (C.9) implying that a higher order polynomial can be used for interpolating the out of plane deflection w as there are more boundary conditions (i.e. displacement and slope in both direction are imposed at the locations of the nodes). The displacement field within the plate element is expressed as u(ξ, ζ, z) = 4 j= v(ξ, ζ, z) = 4 j= ( N lin,j u j + N uθ,j θ z,j z ξ ( N lin,j v j + N vθ,j θ z,j z ζ w(ξ, ζ) = 4 [ ] Nthin,w,j w j + N thin,θx,jθ x,j + N thin,θy,jθ y,j j= with corresponding shape functions [47, 54, 55] [ ] ) 2 e Nthin,w,j x w j + N thin,θx,jθ x,j + N thin,θy,j [ ] ) 2 e Nthin,w,j y w j + N thin,θx,jθ x,j + N thin,θy,j (C.2) N lin,j = 4 ( + ξ jξ)( + ζ j ζ) j =,..., 4 (C.2) N uθ,j = e y 6 ( ζ2 )( + ξ j ξ)ζ j j =,..., 4 (C.22) N vθ,j = e x 6 ( ξ2 )( + ζ j ζ)ξ j j =,..., 4 (C.23) N thin,w,j = 8 ( + ξ jξ)( + ζ j ζ)(2 + ξ j ξ + ζ j ζ ξ 2 ζ 2 ) j =,..., 4 (C.24) 3

128 Appendix C: Finite element modeling N thin,θx,j = e y 6 ( + ξ jξ)(ζ j + ζ)(ζ 2 ) j =,..., 4 (C.25) N thin,θy,j = e x 6 (ξ j + ξ)(ξ 2 )( + ζ j ζ) j =,..., 4 (C.26) The last three expressions constitute a higher order expression with C continuity for the out of plane displacement considering the fact that the derivatives w.r.t. x and y are linked to the degrees of freedom θ y and θ x respectively. The expressions for the displacement field can be used to construct the mass and stiffness matrix and find the equations of motion as is explained in Section C.. Q46 The element referred to as Q46 is very similar as the Q46t element but differs in the fact that the thin plate assumption is not considered. The conditions (C.9) no longer hold implying that a lower order polynomial is used for the displacement w than for Q46t. The displacement field for this element can be expressed as u(ξ, ζ, z) = 4 (N lin,j u j + N uθ,j θ z,j + zn lin,j θ y,j ) j= v(ξ, ζ, z) = 4 (N lin,j v j + N vθ,j θ z,j zn lin,j θ x,j ) j= w(ξ, ζ) = 4 N lin,j w j j= (C.27) Again, using these expressions and the approach described in Section C., the equations of motion are derived. This derivation is implemented in Matlab. Exact integration is done using the symbolic toolbox. The final mass and stiffness matrix are not mentioned as the expressions are too extensive due to the size of the matrices. Q48 For the Q46 element, both θ x, θ y and w are linearly interpolated. The shear strains γ xz and γ yz are given by u z + w x = θ y + w x respectively v z + w y = θ x + w x. The derivative of w with respect to x or y is constant in that direction. This means that it is not possible that the angles θ x and θ y are such that all the plate sections for different elements remain normal when the plate is bent. Pure bending is not possible, thus any deflection will result in shear according to the model which adds a lot of stiffness to the structure. This is called shear locking and its influence increases with decreasing plate thickness. This is sometimes solved by using reduced integration with insufficient integration point for exact integration. Another idea to solve this is by increasing the polynomial order used for w and attain C continuity, which can be done by adding extra degrees of freedom (i.e. the spatial derivatives w.r.t. x and y) that add boundary conditions to the interpolation function. This is considered for the Q48 element. The displacement field in the material element is now given by u(ξ, ζ, z) = 4 (N lin,j u j + N uθ,j θ z,j + zn lin,j θ y,j ) j= v(ξ, ζ, z) = 4 (N lin,j v j + N vθ,j θ z,j zn lin,j θ x,j ) j= w(ξ, ζ) = 4 j= ( ) N thin,w,j w j + N w thin,θx,j y N w thin,θy,j j x j (C.28) The element thus consists of four nodes with eight degrees of freedom each. The total number of DOFs per element is 32 as mentioned in Table C.. Q85 The interpolation function order is increased by considering eight node elements next. The Q85 element is an eight node element with five degrees of freedom per node. Only five DOFs are considered and not the full six DOF element because finding an expression for including the drilling degrees of freedom was not successful. For the four node elements this inclusion is based on [55]. See Figure 4.2 for an illustration of this deformation contribution. Expanding to the eight node configuration is difficult because the number of boundary conditions and interpolation function do not match (even though the order is increased). 4

129 Section C.2: Plate elements For the Q85 element, the displacements u, v and w in terms of z and the iso-parametric coordinates ξ and ζ are given by u(ξ, ζ, z) = 8 (N quad,j u j + zn quad,j θ y,j ) j= v(ξ, ζ, z) = 8 (N quad,j v j zn quad,j θ x,j ) j= w(ξ, ζ) = 8 N quad,j w j j= with the serendipity quadratic shape functions [47] { N quad,j = 4 ( + ξ jξ)( + ζ j ζ)( + ξ j ξ + ζ j ζ) for j =,..., 4 2 ( + ξ jξ)( + ζ j ζ)( ξ j ζ 2 ζ j ξ 2 ) for j = 5,..., 8 (C.29) (C.3) The found expressions for the elementary mass and stiffness matrix are not included. The total number of DOFs per element is increased with respect to the previous mentioned ones to 4 implying higher computation cost. Q458 The Q458 element is a combination of a four and eight node element. The corner nodes of the two are arranged such that they overlap. The degrees of freedom u, v, θ x, θ y and θ y are interpolated using the bi-linear shape functions corresponding to the four node element and the w DOF is linked to the eight node element. The higher order element is used for the out of plane deflection w to reduce the shear lock effect. The other DOFs are only considered at four nodes to reduce computation cost by decreasing the total number of variables. The displacements for this element are expressed as u(ξ, ζ, z) = 4 (N lin,j u j + N uθ,j θ z,j + zn lin,j θ y,j ) j= v(ξ, ζ, z) = 4 (N lin,j v j + N vθ,j θ z,j zn lin,j θ x,j ) j= w(ξ, ζ) = 8 N quad,j w j j= (C.3) These expressions can be substituted in Equations (C.5) and (C.2), the result of which is not included here for the sake of conciseness. Q4383 The Q4383 element is similar to the Q458 type, but for it also θ x and θ y and not only the variable w are interpolated using the quadratic eight node shape functions. The idea behind it is again to obtain more freedom for the states to form a zero shear configuration in the case of pure bending, or at least reduce shear locking contributions. The expressions for the displacement field corresponding to a single Q4383 element are u(ξ, ζ, z) = 4 (N lin,j u j + N uθ,j θ z,j ) + z 8 N quad,j θ y,j j= j= v(ξ, ζ, z) = 4 (N lin,j v j + N vθ,j θ z,j ) z 8 N quad,j θ x,j j= w(ξ, ζ) = 8 N quad,j w j j= j= (C.32) 5

130 Appendix C: Finite element modeling Q95 Another category of quadratic elements is the nine node element type. It corresponds to the Lagrange element family. The nine node elements are considered next. For the Q95 element (similarly as for Q85) only five degrees of freedom are considered for each node. The drilling degree of freedom θ z is not included. The resulting deflections within the material are given by u(ξ, ζ, z) = 9 (N L,j u j + zn L,j θ y,j ) j= v(ξ, ζ, z) = 9 (N L,j v j zn L,j θ x,j ) j= w(ξ, ζ) = 9 N L,j w j with the lagrangian quadratic shape functions (inspired by [47]) 4 ( + ξ jξ)( + ζ j ζ)ξ j ξζ j ζ for j =,..., 4 N L,j = 2 ( + ξ jξ)( + ζ j ζ)(ζ j ζ + ξ j ξ)( ξj 2ζ2 ζj 2ξ2 ) for j = 5,..., 8 ( ξ)( + ξ)( ζ)( + ζ) for j = 9 j= (C.33) (C.34) This type has quite many variables per element. It is therefore not very likely that the Q95 will be favored if the accuracy of the other types with respect to estimating the eigenfrequencies of a plate is comparable. Q96 For the nine node elements, expressions are found to include the effect of θ z on the in-plane deflections. The shape functions are given by and N uθl,j = e y ( ζ 2 ( ζ 2 )(ζ + ζ j )ζj 2 4ζ(ζ 2 ) 2 ( ζj 2 ) )... (C.35) 6 ( + ξ j ξ)( ξj 2 + ξ j ξ)(2 ξj 2 2( ξj 2 )ξ 2 ) j =,..., 9 (C.36) N vθl,j = e x ( ξ 2 ( ξ 2 )(ξ + ξ j )ξj 2 4ξ(ξ 2 ) 2 ( ξj 2 ) )... (C.37) 6 ( + ζ j ζ)( ζj 2 + ζ j ζ)(2 ζj 2 2( ζj 2 )ζ 2 ) j =,..., 9 (C.38) and are inspired by [55]. This makes it possible to use all six degrees of freedom for the nine node element resulting in the Q96 type. The deflection interpolation is similar as for the Q46 type, but with quadratic shape functions instead of bi-linear ones. The expressions are u(ξ, ζ, z) = 9 (N L,j u j + N uθl,j θ z,j + zn L,j θ y,j ) j= v(ξ, ζ, z) = 9 (N L,j v j + N vθl,j θ z,j zn L,j θ x,j ) j= w(ξ, ζ) = 9 N L,j w j j= (C.39) The total number of variables per element is 54, which is the most of all elements considered in this comparison. The accuracy must thus really weight up against the computation cost for it to be chosen in application to the box structure. 6

131 Section C.2: Plate elements Q459 As mentioned above, to reduce the number of states and still attain high accuracy, combinations of linear and quadratic elements are considered. In this case, the nine node element is used for the degree of freedom w. The same reason as mentioned in the description of the Q458 element is behind this choice, i.e. try to reduce shear lock effects. The expressions defining the displacement field within a single Q459 element are u(ξ, ζ, z) = 4 (N lin,j u j + N uθ,j θ z,j + zn lin,j θ y,j ) j= v(ξ, ζ, z) = 4 (N lin,j v j + N vθ,j θ z,j zn lin,j θ x,j ) j= w(ξ, ζ) = 9 N L,j w j j= (C.4) Again, using these expressions and the approach described in Section C., the equations of motion are derived. This derivation is implemented in Matlab, where exact integration is done using the symbolic toolbox. The final mass and stiffness matrix are not mentioned as the expressions are too extensive due to the size of the matrices. Q4393 This element is very similar as the Q4383 type, but differing in the fact that an extra node is used for defining the quadratic interpolation of w, θ x and θ y. For the Q4393 element, the displacements u, v and w in terms of z and the iso-parametric coordinates ξ and ζ are given by u(ξ, ζ, z) = 4 (N lin,j u j + N uθ,j θ z,j ) + z 9 N L,j θ y,j j= j= v(ξ, ζ, z) = 4 (N lin,j v j + N vθ,j θ z,j ) z 9 N L,j θ x,j j= w(ξ, ζ) = 9 N L,j w j j= j= (C.4) C.2.2 Comparison To compare the accuracy of the different elements, a simply supported plate is considered for which the eigenfrequencies are computed for the elements and compared to analytical (thin plate) values. In this, the thickness of the plate with respect to its length is varied. As the ratio length/thickness increases the eigenfrequencies should converge to the analytical thin plate results (Kirchhoff theory, [53]). It is expected that the frequencies should be lower for smaller length/thickness ratios (thick plate) as extra flexibility due to shear starts playing a factor. For a x.5 m simply supported aluminum plate with E = 69 GPa, ρ = 27 kg/m 3 and ν =.3 the eigenfrequencies are computed using a 3x3 element mesh for the different element types and different plate thicknesses. They are divided by the analytical results from [53]. The results for the first natural frequency are shown in Figure C.. The Q46t element is close to the required frequency over the whole thickness range (even for low length/thickness ratios). It is therefore not valid for thick plates. A clear reason for the deviation for thick plates is not found though, but it might possibly be caused by the fact that in the model also kinetic energy corresponding to in-plane motion is considered which is not the case for [53]. The other elements all approach the same trend for thicker plates, but some deviate for thin plates where they should converge to Kirchhoff. This is likely caused by shear lock effects. The elements which do converge are Q85, Q4383, Q95, Q96 and Q4393. Of these, the mixed element Q4383 has the least degrees of freedom per element and therefore has the lowest computation cost. Because of this, the Q4383 element is chosen to build a finite element model of an open box for the case study. Note that it might be possible that higher order elements have faster mesh convergence, thus actually requiring less variables in total. This is not further considered in this analysis. 7

132 Appendix C: Finite element modeling Q46 Q46t Q Q85 Q458 Q Q95 Q96 Q459 Q f / f thin f / f thin f / f thin L x / t L x / t L x / t Figure C.: Ratio between first natural frequency according to FEM calculations and thin plate analytical values for different plate thicknesses. C.3 Beam elements For a beam, D elements are used. Only deflections in the 2D plane are considered, i.e. u and v in x respectively y-direction. The x direction is chosen in the axial direction of the beam and the y axis points in the transverse direction. Consider Figure 3.2. A beam is divided in a number of elements, connected by nodes. The axial displacement u at the midplane of the beam within a specific element between node j and j + is interpolated using linear shape functions [47] as u mid (ξ) = 2 ( ξ)u j + 2 ( + ξ)u j+ (C.42) in terms of an iso-parametric coordinate ξ. In this, u j is the degree of freedom corresponding to the axial displacement of node j. For the displacement in lateral direction, a higher order polynomial is used. The degrees of freedom v j and θ j are introduced corresponding to the jth node. They impose the displacement respectively slope at the location of the node. This implies that a third order polynomial can be used for interpolation as four boundary conditions are available. The lateral deflection within the element can be expressed as [29, 47] v(ξ) = 4 (2 + ξ)( ξ)2 v j + e b 8 ( ξ) 2 ( + ξ)θ j (2 ξ)( + ξ)2 v j+ + e b 8 ( ξ)( + ξ) 2 θ j+, (C.43) where e b is the element length. Bending of the beam causes not just lateral deflections, but since the cross sections rotate, also axial stresses are caused. It is assumed that the beam is then and thus that shear is negligible. This means that the cross sections remain normal to the midline of the beam and the off-axis axial deflections become u(ξ, y) = u mid (ξ) y dw dx (C.44) with y = at the midplane. The expressions for the displacements are substituted in Expressions (C.5) and (C.2) to obtain an elementary mass and stiffness matrix. The deflections normal to the xy plane are considered zero and a plane stress assumption is used for the transverse direction, i.e. σ yy =. The found mass and stiffness matrix are [29] 8

133 Section C.3: Beam elements M b,e = ρ bw 42 and 4e b t b 7e b t b 42t 3 b +56e2 b t b e b 22e 2 b t b + 3.5t 3 54e b 2 b t b 42t 3 b e b 3e 2 b t b + 3.5t 3 b 22e 2 b t b + 3.5t 3 b 4e 3 b t b e bt 3 b 3t b e 2 b 3.5t3 b 3t b e 3 b 7 6 t3 b e b 7e b t b 4e b t b 54e 2 b t b 42t 3 b e b 3t b e 2 b 3.5t3 42t b 3 b +56e2 b t b e b 22t b e 2 b 3.5t3 b 3e 2 b t b + 3.5t 3 b 3t b e 3 b 7 6 t3 b e b 22t b e 2 b 3.5t3 b 4e 3 b t b e bt 3 b K b,e = E bw t b e b t 2 b t 2 e 2 b 2e b b t2 b t 2 e 2 b 2e b b t 2 b t 3 b 2e b 2 t2 b t 2 b 2e b 6 t2 b t2 e 2 b t 2e b b 2 b e 2 b t 2 b t 2 b 2e b 6 t2 b t 2 b 2e b 3 t2 b 2e b (C.45). (C.46) respectively. In this, ρ b is the mass density, W is the beam width, t b is its thickness and E is the Young s modulus of the material. See for example [47] for more information on finite element modeling for structural dynamics. 9

134

135 Appendix D Mode shapes undamped box The mode shapes of the undamped open box structure found from simulations are depicted in Figure D.. The lines used for visualization are not indicative for the mesh used in computations. The color indicates modal deflection amplitude. Modes 7 to found from fitting measured frequency response functions are shown in Figure D.2, where the lines do illustrate the measurement mesh. Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 7 (b) mode 8 Z+w Z+w Y+v.. X+u Y+v.. X+u (c) mode 9 (d) mode Figure D.: The first elastic eigenmodes of an aluminum box with dimensions and parameters as in Table 4..

136 Appendix D: Mode shapes undamped box Z+w Z+w Y+v.. X+u Y+v..2 X+u.3.4 (e) mode (f) mode 2 Z+w Z+w Y+v.. X+u Y+v.. X+u (g) mode 3 (h) mode 4 Z+w Z+w Y+v.. X+u Y+v.. X+u (i) mode 5 (j) mode 6 Figure D.: The first elastic eigenmodes of an aluminum box with dimensions and parameters as in Table 4. (continued). 2

137 Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 7 (b) mode 8 Z+w Z+w Y+v.. X+u Y+v...2 X+u.3.4 (c) mode 9 (d) mode Figure D.2: The first four elastic experimental eigenmodes of an aluminum box structure. 3

138

139 Appendix E Mode shapes damped box The modeshapes of the open box structure with viscoelastic damping applications (discrete dampers, fully covering CL dampers, and partially covering CL dampers) are presented in this appendix. Both the numerical and experimental modeshapes are considered. Since the damping is not proportional, the eigenvectors for the different configurations are complex. The numerical modes are scaled such that the displacement degree of freedom with largest magnitude is real. This is obtained by dividing the eigenvector of interest by the entry with largest magnitude. Subsequently, the eigenvector is multiplied by a factor.5 for visualization purposes. The real part of the eigenvectors is dominant over the imaginary part. To be able to visually compare the experimental eigenmodes to the numerical eigenmodes, the experimental modes need to be scaled correctly. The modal scale factor is used for this [] MSF[i, j] = U eh i U n j U eh i U e i, (E.) where superscripts e and n indicate whether the considered mode is from experiments or a numerical mode (reduced to only the same DOFs as in experiments), respectively. The experimental eigenvector U e i is multiplied by a factor MSF[i, i]. Subsequently, an amplification factor (real) is used to make the real part of the eigenvector entry with largest magnitude equal.5 (for visualization purposes). E. Discrete dampers The mode shapes of the open box structure with discrete dampers (real part) are depicted in Figure E.. The lines used for visualization are not indicative for the mesh used in computations. The color indicates modal deflection amplitude. Modes 7 to (real part) found from fitting measured frequency response functions for the open box with discrete dampers are shown in Figure E.2, where the lines do illustrate the measurement mesh. Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 7 (b) mode 8 Figure E.: The first elastic eigenmodes of an aluminum box damped using discrete VEM elements with dimensions and parameters as in Table

140 Appendix E: Mode shapes damped box Z+w Z+w Y+v.. X+u Y+v.. X+u (c) mode 9 (d) mode Z+w Z+w Y+v.. X+u Y+v..2 X+u.3.4 (e) mode (f) mode 2 Z+w Z+w Y+v.. X+u Y+v.. X+u (g) mode 3 (h) mode 4 Figure E.: The first elastic eigenmodes of an aluminum box damped using discrete VEM elements with dimensions and parameters as in Table 4.4 (continued). 6

141 Section E.: Discrete dampers Z+w Z+w Y+v.. X+u Y+v.. X+u (i) mode 5 (j) mode 6 Figure E.: The first elastic eigenmodes of an aluminum box damped using discrete VEM elements with dimensions and parameters as in Table 4.4 (continued). Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 7 (b) mode 8 Z+w Z+w Y+v.. X+u Y+v.. X+u (c) mode 9 (d) mode Figure E.2: The first four elastic experimental eigenmodes of an aluminum box structure damped using discrete VEM elements. 7

142 Appendix E: Mode shapes damped box E.2 Constrained layer dampers E.2. Fully covering CL dampers The mode shapes of the open box structure with fully covering CL dampers (real part) are depicted in Figure E.3. The mode shapes are similar for the CL dampers with thin or thick viscoelastic layer. The lines used for visualization are not indicative for the mesh used in computations. The color indicates modal deflection amplitude. Modes 7 to (real part) found from fitting measured frequency response functions for the open box with fully covering CL dampers are shown in Figure E.4, where the lines do illustrate the measurement mesh. Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 7 (b) mode 8 Z+w Z+w Y+v.. X+u Y+v.. X+u (c) mode 9 (d) mode Figure E.3: The first elastic eigenmodes of an aluminum box structure damped using fully covering CL dampers. 8

143 Section E.2: Constrained layer dampers Z+w Z+w Y+v.. X+u Y+v..2 X+u.3.4 (e) mode (f) mode 2 Z+w Z+w Y+v.. X+u Y+v.. X+u (g) mode 3 (h) mode 4 Z+w Z+w Y+v.. X+u Y+v.. X+u (i) mode 5 (j) mode 6 Figure E.3: The first elastic eigenmodes of an aluminum box structure damped using fully covering CL dampers (continued). 9

144 Appendix E: Mode shapes damped box Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 7 (b) mode 8 Z+w Z+w Y+v.. X+u Y+v.. X+u (c) mode 9 (d) mode Figure E.4: The first four elastic experimental eigenmodes of an aluminum box structure damped using fully covering CL dampers. E.2.2 Partially covering CL dampers The mode shapes of the open box structure with partially covering CL dampers (real part) are depicted in Figure E.5. The lines used for visualization are not indicative for the mesh used in computations. The color indicates modal deflection amplitude. Modes 7 to (real part) found from fitting measured frequency response functions for the open box with partially covering CL dampers are shown in Figure E.6, where the lines do illustrate the measurement mesh. 2

145 Section E.2: Constrained layer dampers Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 7 (b) mode 8 Z+w Z+w Y+v.. X+u Y+v.. X+u (c) mode 9 (d) mode Z+w Z+w Y+v.. X+u Y+v..2 X+u.3.4 (e) mode (f) mode 2 Figure E.5: The first elastic eigenmodes of an aluminum box structure damped using partially covering CL dampers. 2

146 Appendix E: Mode shapes damped box Z+w Z+w Y+v.. X+u Y+v.. X+u (g) mode 3 (h) mode 4 Z+w Z+w Y+v.. X+u Y+v.. X+u (i) mode 5 (j) mode 6 Figure E.5: The first elastic eigenmodes of an aluminum box structure damped using partially covering CL dampers (continued). 22

147 Section E.2: Constrained layer dampers Z+w Z+w Y+v.. X+u Y+v.. X+u (a) mode 7 (b) mode 8 Z+w Z+w Y+v.. X+u Y+v.. X+u (c) mode 9 (d) mode Figure E.6: The first four elastic experimental eigenmodes of an aluminum box structure damped using partially covering CL dampers. 23

148

149 Appendix F FRF method comparison A quantity that is often used in describing the dynamic properties of a system is the frequency response function (FRF). It gives the relation between harmonic excitation and response. Several techniques to compute and approximate the FRF are possible. In this appendix, some will be compared for a system with damping by means of viscoelastic material. F. FRF computation methods The equations of motion (in the laplace domain) for a structure with viscoelastic damping can be expressed as ( Ms 2 + K(s) ) q(s) = D(s) q(s) = F (s) (F.) or alternatively as ( Ms 2 + G(s)s + K ) q(s) = D(s) q(s) = F (s). (F.2) In this, M is the mass matrix, K(s) is a frequency dependent stiffness matrix, K is a constant stiffness contribution and G(s) = (K(s) K ) /s is the damping matrix (frequency dependent). q(s) and F (s) are vectors containing the degrees of freedom and forcing terms respectively. The transfer function matrix corresponding to this system is given by H(s) = D(s), (F.3) where D(s) is known as the dynamic stiffness. The FRF matrix is found by evaluating this equation for s = iω. Computing the frequency response function in this manner is referred to as the exact method. To compute FRFs between certain forces and degrees of freedom using this direct approach, a matrix inversion is required at each frequency of interest. Although exact in nature, it becomes very inefficient and computationally expensive especially for large scale systems (large N q ). In modeling the box structure the number of degrees of freedom is in the order N q = O(5 4 ). Model reduction techniques could be employed to reduce computation effort. These are not considered in this study as applying it correctly for damping problems can be a study in itself. Modal superposition employing the residue theorem is commonly used in finding FRFs. For any damped linear system, the frequency response function matrix can be expressed by the contribution of different modes as [59] with H(iω) = N r k= U k U T k m k ω2 + N q k=n r+ [ γ k U k U T k iω λ k m k = U T k MU k and γ k = + γc k U c ku H k iω λ c k U T k D(s) s ] + p k=2n q+. U k s=λk γ k U k U T k iω λ k In the expression for H(iω), the first part corresponds to the so-called elastic modes which are subcritically damped (the eigenvalues come in complex conjugate pairs). The second contribution correspond to the nonviscous modes with real eigenvalues. These are generally difficult to find. The modal superposition approach is usually approximated by truncating the summation to only the first r modes, excluding the non-viscous mode contributions. This truncated FRF computation method will be referred to as the Approximate Modal Superposition Method (AMSM) [64]. (F.4) (F.5) 25

150 Appendix F: FRF method comparison Several methods have been proposed in literature to improve the accuracy of the truncated modal superpositions approach. The Mode Acceleration Method (MAM) [64] H(iω) = and Improved Approximation Method (IAM) [64] r k= iωγ k U k U T k (iω λ k ) λ k + K (F.6) H(iω) = r k= (iωλ k + iω λ k ) γ k U k U T k (iω λ k ) λ 2 k + K iωk G K, (F.7) with G = lim s G(s) are examples of these improved methods both making use of a Neumann expansion [64]. The two methods will be considered in an accuracy comparison in the next section. Two other techniques are considered, which will be briefly explained next. In 24, Li et al. presented a method using the lower mode contributions, first Neumann expansion term and second Neumann expansion term to form a projection basis for the system [62]. This can be applied to reduce the system in computing dynamic response or frequency response function. The method will be referred to here as Li24. A projection matrix T p is employed which is composed as with and T 3 = T 2 = r k= T p = [ T T 2 T 3 ] T = r k= r k= (F.8) (F.9) γ k U k U T k F (iω) iω λ k, (F.) γ k U k U T k F (iω) λ k γ k U k U T k F (iω) λ 2 k + K F (iω) iωk G K F (iω). The projection is applied to the system resulting in the reduced equations (F.) (F.2) D R (iω) q R (iω) = F R (iω) (F.3) with D R (iω) = T T p D(iω)T p, q(iω) = T p q R (iω) and F R (iω) = T T p F (iω) (F.4) The reduced [3 3] system can be used to approximate the frequency response function of interest. Note that this technique requires a projection step for each frequency of interest. For more detail on this method, see [62]. The final FRF computation method considered in this appendix is the IMPS method. The approach is proposed in this study, the derivation and relevant equations of which are given in Section It is based on interpolation of certain eigenvalues and residues in order to include the effect of the frequency dependent behavior of the viscoelastic material. It basically incorporates theses modes contribution without having to find these specific modes. F.2 FRF comparison The methods for computing/approximating the frequency response functions described above (i.e. AMSM, MAM, IAM, Li24 and IMPS) will be compared to the exact method (inverse of D for all frequencies) in this section. Two test cases are considered, i.e. a four-mass-spring system and a cantilever beam. For both, viscoelastic material is incorporated to add damping. 26

151 Section F.2: FRF comparison F.2. Case : four-mass-spring system As a first case for testing the accuracy of the different FRF computation methods, a simple four degree of freedom system is considered (see Figure F.). It consists of four masses connected in series and to the fixed world by means of springs with stiffness k e and blocks of viscoelastic material (VEM). Norsorex R is the material considered for the VEM, see Figure 4.9 for the frequency dependency of this materials mechanical properties. q q q q k k k k k e e e e e m m m m VEM F F F F Figure F.: A four degree of freedom viscoelastically damped system (inspired by [64]). The mass matrix of the system with degree of freedom vector [ q q 2 q 3 q 4 ] T is m M = m 2 m 3 m 4. (F.5) The stiffness and damping function matrix are given by K = (k e + aev()) K v and G(s) = E v(s) Ev() K s v respectively, where a is a factor capturing the geometric effects on stiffness of the VEM blocks and 2 K v = (F.6) (F.7) The frequency response function between a force F (see Figure F.) and translation q is computed using the different approaches for m = kg, m 2 = 2 kg, m 3 = 3 kg, m 4 = 4 kg and k e = 5, N/m. These values are taken from [64] and a is set to. m. The results are depicted in Figure F.2, where only the three pair of eigenvalues with lowest frequency are considered (elastic modes). These eigenvalues are found using the iteration scheme shown in Figure 4.. For the IMPS approach, 5 interpolation points are used. It can be seen that the mode truncation mainly affects the anti-resonances (AMSM compared to the exact solution). The MAM approach improves the approximation for the frequencies up to the third resonance. The IAM approach appears to be inaccurate. A clear reason for this deviation is not found, but it can possibly be linked to the damping model. The results found in [64] are accurate for the IAM approach and the only difference with this case study in the damping model considered. The best approximation is found using the method proposed in [62] (referred to here as Li24). Even the fourth resonance peak is found whereas only the first three eigenvalue pairs are included in the computation. The IMPS approach gives similar accuracy as AMSM. This is because the method only improves the frequency dependency incorporation of the Young s modulus, a property that does not vary much of the frequency range considered. Truncation has similar effects for IMPS as for AMSM. 27

152 Appendix F: FRF method comparison 4 H(iω) [m/n] 6 8 exact AMSM MAM IAM Li24 IMPS 2 phase [deg] 2 Figure F.2: FRF function computed using different methods for a four degree of freedom viscoelastically damped system. F.2.2 Case 2: Cantilever beam As a second case, a system for which the number of degrees of freedom is an order higher is taken. A cantilever beam with a viscoelastic damper connected at the end is considered (see Figure F.3). The block of VEM consists of the Norsorex R compound. Beam VEM Figure F.3: Cantilever beam with viscoelastic damping element connected near the end. The beam (with length L b ) is modeled using the finite element method. The structure is divided in m b onedimensional elements of length e b each with a rotation θ and transversal displacement degree of freedom v for each node. The elementary mass and stiffness matrix are [] M b,e = ρ bw t b e b e b 54 3e b 4e 2 b 3e b 3e 2 b 56 22e b sym 4e 2 b (F.8) 28

153 Section F.2: FRF comparison and K b,e = E bw t 3 b 2e 3 b 2 6e b 2 6e b 4e 2 b 6e b 2e 2 b 2 6e b sym 4e 2 b (F.9) respectively, with ρ b and E b the mass density and Young s modulus of the beam. W and t b are the width and thickness of the beam respectively. These beam elements are commonly referred to as the Euler-Bernoulli beam elements. The different elements are assembled to compose a full mass and stiffness matrix corresponding to the degrees of freedom [ ] T v θ... v m+ θ m+. The viscoelastic material is incorporated by adding a stiffness term to the matrix element corresponding to the transversal degree of freedom at the end of the beam. This stiffness is k v (s) = Ev(s)A v /L v, (F.2) with A v and L v the cross sectional area and length of the block respectively. For the methods where a matrix G is required, this contribution can be split in a damping term and static stiffness as mentioned in Section F.. The different FRF computation techniques are employed to approximate the driving point FRF between a force and transversal displacement at the location of the damping element. The parameter values used can be found in Table F.. 5 interpolation points are used for the IMPS approach. The resulting FRFs when only considering the first 5 eigenvalue pairs, r =, (for IMPS r = 5) are depicted in Figure F.4. The estimate for the IMPS method is shown in a different plot since otherwise it would overlap with Li24 and the exact method over a wide frequency range. Table F.: Parameter values used in FRF method accuracy comparisons for a damped cantilever beam. Symbol Description Value W Width of the beam.2 m t b Thickness of the beam.2 m L b Length of the beam.5 m ρ b Mass density of the beam 78 kg/m 3 E b Young s modulus of the beam 2 GPa m b Number of elements composing the beam 3 A v Cross sectional are of the VEM element 4 4 m 2 L v Length of the VEM element. m Figure F.4 shows that the AMSM, MAM and IAM are inaccurate in estimating the frequency response function using a truncated modal superposition approach. The FRFs for the IMPS method and Li24 are similar and overlap with the exact results for frequencies up to the fourth resonance peak. The fifth peak is approximated well too. Both can be considered accurate for frequencies sufficiently below the next eigenfrequency (i.e. Im {λ r+ }). It is found that some extra peaks may appear under specific conditions for the Li24 method (see Figure F.4, especially the phase). This effect was more prominent when employing the technique for computing FRFs of the box structure. A clear explanation for this is not found, but it might be linked to matrix conditioning and scaling. A conditioning step as mentioned in [62] is not included in this study. It is concluded that the IMPS approach gives the best results for the damping considered. The method requires that a sufficient number of modes is considered such that the modal superposition truncation effect is minimal. 29

154 Appendix F: FRF method comparison H(iω) [m/n] 5 exact AMSM MAM IAM Li phase [deg] (a) 6 H(iω) [m/n] 8 exact IMPS phase [deg] (b) Figure F.4: Driving point FRF for a cantilever beam at the location of the damper computed using different methods. 3

155 Appendix G Additional FRFs undamped box The frequency response functions (accelerance) between an actuation force and corresponding translation is computed for different degrees of freedom. The translational degree of freedom perpendicular to the wall at the location of the sensor (see Figure G.) is used for the response identifier. The actuation force points (also perpendicular to the different walls) are illustrated in Figure G.. In this appendix some of the FRFs for the undamped box structure found using the Matlab model and from experiments are shown, see Figures G.2 to G.6. The corresponding coherences are depicted as well. The IMPS method is used for computing the FRFs where the modal superposition truncation order is set to r = and for the interpolation, N f = 5 frequency points are considered Wall Wall 4 Wall Wall Sensor position Wall Hammer position Figure G.: Fold-out of the box structure, illustrating the sensor placement and hammer excitation points. The blue squares indicate the actuation points for which FRFs are shown below. 3

156 Appendix G: Additional FRFs undamped box Magnitude [m/s 2 / N] 2 2 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure G.2: Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] 2 2 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure G.3: Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 32

157 Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure G.4: Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure G.5: Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 33

158 Appendix G: Additional FRFs undamped box Magnitude [m/s 2 / N] 2 2 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure G.6: Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 34

159 Appendix H Additional FRFs damped box The frequency response functions (accelerance) between an actuation force and corresponding translation is computed for different degrees of freedom. The translational degree of freedom perpendicular to the wall at the location of the sensor (see Figure G.) is used for the response identifier. The actuation force points (also perpendicular to the different walls) are illustrated in Figure G.. In this appendix some of the FRFs for the damped box structure found using the Matlab model and from experiments are shown. In Section H. a number of FRFs for the box with discrete dampers are depicted. Section H.2 contains FRFs for the box with the different constrained layer damping configurations. The corresponding coherences are depicted for each measured FRF as well. The IMPS method is used for computing the FRFs where the modal superposition truncation order is set to r = and for the interpolation, N f = 5 frequency points are considered. H. Discrete dampers Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.: Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 35

160 Appendix H: Additional FRFs damped box Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.2: Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] 2 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.3: Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b). 36

161 Section H.2: Constrained layer dampers Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.4: Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.5: Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). H.2 Constrained layer dampers This section contains some additional frequency response functions for the box with the different constrained layer damping configurations. A distinction is made between the fully covering constrained layer dampers (Section H.2.) and the box with partially covering constrained layer patches (Section H.2.2). 37

162 Appendix H: Additional FRFs damped box H.2. Thick VEM layer Fully covering CL dampers Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.6: Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.7: Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 38

163 Section H.2: Constrained layer dampers Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.8: Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.9: Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 39

164 Appendix H: Additional FRFs damped box Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.: Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). Thin VEM layer Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.: Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 4

165 Section H.2: Constrained layer dampers Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.2: Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] 2 model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.3: Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b). 4

166 Appendix H: Additional FRFs damped box Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.4: Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.5: Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 42

167 Section H.2: Constrained layer dampers H.2.2 Partially covering CL dampers Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.6: Frequency response function between force at point 4 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.7: Frequency response function between force at point 9 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 43

168 Appendix H: Additional FRFs damped box Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.8: Frequency response function between force at point 34 and out of plane deflection at the sensor location (driving point FRF) from measurements and model (a) and corresponding coherence (b). Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.9: Frequency response function between force at point 49 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 44

169 Section H.2: Constrained layer dampers Magnitude [m/s 2 / N] model experiments Coherence [ ] Phase [degrees] (a) (b) Figure H.2: Frequency response function between force at point 73 and out of plane deflection at the sensor location from measurements and model (a) and corresponding coherence (b). 45

170

171 Appendix I Experiment hardware specifications This appendix includes the data sheets with the specifications for the 3D accelerometer and modal hammer used in experiments. The specifications of the measurement equipment can be found on the following pages. 47

172 Appendix I: Experiment hardware specifications Model 65 Isotron accelerometer Features NEW! 65-R available as replacement sensors Triaxial, low-impedance output Small size (-mm cube, 5 gram) High output ( mv/g), high resolution (Mili-g) Ideal for structural analysis Shock-proof, overload-protected Single connector, flexible cable Description The high sensitivity and high resolution of Endevco s model 65 distinguish this triaxial accelerometer from comparable products. The Endevco model 65 is packaged in a -mm cube of welded titanium construction. Interface to the model 65 is via a Microtech 4-pin connector. Temporary petro-wax adhesive and a ten-foot cable assembly with BNC connectors are provided as standard accessories. The model 65 s excellent frequency response, both amplitude and phase, provide the user with a triaxial accelerometer ideally suited for structural and component testing, drop tests and general laboratory vibration work. The reduced size of this accelerometer enables the test engineer or technician to measure the accelerations of three orthogonal axes of vibration simultaneously on lightweight structures. A model number suffix indicates output sensitivity in mv/g; i.e., - features mv/g. Endevco signal conditioner models 33, 2792B, 2793 or 446B are recommended for use with this accelerometer. TYPICAL TEMPERATURE RESPONSE 2 5 (WHT) (BLK) % DEVIATION (RED) (GRN) (-5) (-7) (38) TEMPERATURE o F o (82) (27) ( C) FREQUENCY IN HERTZ

173 APPLIES TO CALIFORNIA FACILITY Model 65 Isotron accelerometer Specifications The following performance specifications conform to ISA-RP-37.2 (964) and are typical values, referenced at +75 F (+24 C) and Hz, unless otherwise noted. Calibration data, traceable to National Institute of Standards and Technology (NIST), is supplied. Dynamic characteristics Units - - Range g ±5 ±5 Voltage sensitivity, typical mv/g Frequency response See typical amplitude response Amplitude response ±db typical [] Hz.4 to.5 to 6 Resonance frequency Hz 6 45 Transverse sensitivity % < 5 Temperature response See typical curve Amplitude non-linearity % < Output characteristics Output polarity See arrows on outline drawing DC output bias voltage [3] Vdc +2.3 to +3.5 Output impedance Ω Full scale output voltage Vpk ±5 Resolution (2 Hz to khz broadband) g rms.6.4 Grounding Signal ground connected to case Power requirement Complient voltage Vdc +23 to +3 Supply current ma +2 to +2 Warm-up time (to reach 9% of final bias) sec < 2 Environmental characteristics Temperature range -67 F to 257 F (-55 C to +25 C) Humidity Welded construction Sinusoidal vibration limit g pk ±5 ±2 Shock limit [4] g pk Base strain sensitivity at 25 µstrain eq. g/µstrain <. <. Thermal transient sensitivity eq. g/ F (/ C). (.2). (.2) Physical characteristics Dimensions See outline drawing Weight gm (oz) 5 (.7) Case material Titanium, commercially pure Connector [5] 4 pin Microtech style side mounted Mounting [6] Adhesive or m2.5 thread Mounting torque lbf-in (Nm) 7 (.8) Calibration Supplied, each axis: Voltage sensitivity mv/g Maximum transverse sensitivity % Frequency response % 2 to 6 Accessories Product Description R 327AM3-2 Triaxial cable, 85 C, 3 BNC s at Included Optional instrumentation end EH755 Screw, cap m2.5 x.45 x 6 mm Included Included EH76 Screw, set m2.5 x.45 x 6 mm Included Included Mounting wax Included Optional 327AVM3-2 Triaxial cable, 2 C (transducer extension Optional Optional cable, mates with Model 327AM3) 3849 DAAK deluxe accelerometer adhesive kit Optional Optional 65M version Electrical isolation, case [2] Optional Optional Model 33, 3-channel signal conditioner Notes:. Frequency response calibration on x and y may be limited by the mounting fixture of the calibration system. Actual frequency responses of axis x and y are similar to axis z. 2. Case isolation available as model 65M-XXX. Must be specified at time of order Vdc minimum must be available to the accelerometer to ensure full-scale operation at the temperature extremes. A B Continued product improvement necessitates that Endevco reserve the right to modify these specifications without notice. Endevco maintains a program of constant surveillance over all products to ensure a high level of reliability. This program includes attention to reliability factors during product design, the support of stringent Quality Control requirements, and compulsory corrective action procedures. These measures, together with conservative specifications have made the name Endevco synonymous with reliability. ENDEVCO CORPORATION. ALL RIGHTS RESERVED 37 RANCHO VIEJO ROAD, SAN JUAN CAPISTRANO, CA92675 USA (8) (949) fax (949) applications@endevco.com 6 49

174 Appendix I: Experiment hardware specifications Modal Hammer Model 232 Model 232-5, -, -5, - Four Ranges 3 Replaceable Tips Low impedance (ISOTRON ) Output Acceleration Compensated Ergonomically Designed Grip DESCRIPTION Scaled modal models require a precise force measurement. This can be achieved by electrodynamic and servohydraulic exciters controlled by a signal generator via a power amplifier. A more convenient and economical excitation method is a hammer fitted with a high-quality piezoelectric force transducer. In applications where a high crest factor and a limited ability to shape the input force spectrum is of no concern, impact hammer testing is an ideal source of excitation. Impact hammers are highly portable for field work and provide no unwanted mass loading to the structure under test. The Modal Hammer excites the structure with a constant force over a frequency range of interest. Three interchangeable tips are provided which determine the width of the input pulse and thus the bandwidth. Typical force spectra produced with different tips are shown on the right. For larger structures, an optional head extender is available to increase the head s mass. The hammer structure is acceleration compensated to avoid glitches in the spectrum due to hammer structure resonances. The ergonomically designed handle grip helps the user optimize control and reduce the possibility of "double hits". The hammer features an ISOTRON impedance converter providing an IEPE output which is compatible with most FFT analyzers and data acquisition systems. ENDEVCO s 446B single channel signal conditioner or the model 33 three channel conditioner are recommended for use with the 232. To excite larger structures, see Endevco Model 233, 234 and 235 sledge hammers. db/div. Not actual size rubber aluminum plastic KHz 2 KHz ENDEVCO MODEL 232 Multiple-Range Impact Hammer Adaptor/ Charger Connecting Cable Signal Conditioner Directly into FFT Analyzer IEPE Input Analyzer (Not Included) APPLIES TO CALIFORNIA FACILITY 5

175 ENDEVCO MODEL 232 Modal Hammer SPECIFICATIONS The following performance specifications are typical values, referenced at +75 F (+24 C), 4 ma, and Hz, unless otherwise noted. MODEL 232 MODAL HAMMER Units RANGE, full scale lbf (N) (4448) 5 (22) (445) 5 (22) SENSITIVITY, typical mv/lbf (mv/n) 5 (.4) (2.27) 5 (.4) (22.7) MAXIMUM FORCE, typical lbf (N) (4448) RESONANCE FREQUENCY khz 5 FREQUENCY RANGE, max. khz 8 HEAD MASS grams HEAD DIAMETER inches (mm).75 (9) IMPACT TIP DIAMETER inches (mm).25 (6.4) DC OUTPUT BIAS Vdc 9 to OUTPUT IMPEDANCE Ohms < FULL SCALE OUTPUT V ±5 SUPPLY VOLTAGE Vdc 8 to 24 SUPPLY CURRENT ma 2 to TEMPERATURE RANGE F ( C) -67 to 257 (-55 to 25) OVERALL LENGTH in (mm) 8.76 (223) SENSOR MATERIAL 7-4 PH Stainless Steel HANDLE MATERIAL Fiberglass with rubber grip CONNECTOR BNC ACCESSORIES P/N EHM 28 P/N EHM 29 P/N EHM 2 OPTIONAL ACCESSORIES P/N EHM653 IMPACT TIP, POLYURETHANE IMPACT TIP, DELRIN IMPACT TIP, ALUMINUM CARRYING CASE 4 GRAM HEAD EXTENDER NOTES. Only 2-gram tips supplied with the hammer set should be used. Heavier or lighter tips may affect acceleration compensation. 2. To prevent damage to mounting threads, do not use excessive torque when installing/changing impact tips. 3. Maintain high levels of precision and accuracy using Endevco's factory calibration services. Call Endevco s inside sales force at for recommended intervals, pricing and turn-around time for these services as well as for quotations on our standard products. Endevco complete Modal Front End System Continued product improvement necessitates that Endevco reserve the right to modify these specifications without notice. Endevco maintains a program of constant surveillance over all products to ensure a high level of reliability. This program includes attention to reliability factors during product design, the support of stringent Quality Control requirements, and compulsory corrective action procedures. These measures, together with conservative specifications have made the name Endevco synonymous with reliability. ENDEVCO CORPORATION, 37 RANCHO VIEJO ROAD, SAN JUAN CAPISTRANO, CA USA (8) (949) fax (949) applications@endevco.com 5