CHARACTERISATION OF VISCOELASTIC LAYERS IN SANDWICH LIGHTWEIGHT PANELS THROUGH INVERSE TECHNIQUES

Transcription

1 CHARACTERISATION OF VISCOELASTIC LAYERS IN SANDWICH LIGHTWEIGHT PANELS THROUGH INVERSE TECHNIQUES L. Rouleau 1*,2, B. Pluymers 1 and W. Desmet 1 1 KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 3, BE-31, Heverlee, Belgium. bert.pluymers@kuleuven.be wim.desmet@kuleuven.be 2 currently at Structural Mechanics and Coupled Systems Laboratory, Cnam Paris, 2 Rue Conté, Paris, France. lucie.rouleau@cnam.fr ABSTRACT Constrained layer damping is a common way of passively reducing structural vibrations. To design such damping treatments, the frequency-dependent properties of the viscoelastic material are essential. Direct methods, such as dynamic mechanical analysis, may not be applicable to some polymers. The characterisation of viscoelastic materials through inverse approaches can overcome these limitations. The parameters of a viscoelastic model are identified by minimising residuals between experimental data and predictions from a finite element model. Though successfully applied to beam structures, this kind of approach has not been tested on more complex structures. In this work, the proposed inverse procedure, which identifies the parameters of a fractional derivative model over a large frequency band, is applied to frequency response functions measured on lightweight sandwich panels. 1 INTRODUCTION Lightweight materials are growingly used for design concepts in many industrial sectors in order to improve fuel economy and reduce costs of production [1]. However, this practice is frequently associated with noise and vibration problems. The application of viscoelastic materials is a common way of passively reducing structural vibrations, either in an unconstrained or a constrained configuration. In the case of constrained damping layer (CLD) treatment, the sandwich structure consists of a viscoelastic core layer bonded between two elastic layers. When this structure is

2 bending, the external layers induce high shear strains in the polymeric core, which enables the dissipation of part of the vibrational energy by heat. Numerical simulations have become an indispensable tool in the process that leads to the development of engineering structures integrating such damping treatments. In order to perform these simulations, an accurate knowledge of the material s parameters used in the numerical model are necessary. Several characterisation techniques exist to retreive the frequency-dependent viscoelastic properties from measurements [2]. Among them, the dynamical mechanical analysis (DMA) [3, 4] and the vibrating beam resonant method [5] are the most commonly used. But both techniques rely on the assumption that the material is thermo-rheologically simple, so that the timetemperature superposition principle, which enables the extrapolation of the material properties beyond the limited field of experimental frequencies, applies. However, for some polymers, such as polymeric blends or polymers with high-modulus fillers, the superposition principle does not hold [6, 7]. Considering the current development trend of polymer-based damping materials and the increasing complexity of novel materials, there is a need for a new characterisation procedure which does not rely on the thermo-rheological simplicity of materials. Recently, the inverse characterisation of the frequency-dependent linear dynamic elastic and damping properties of viscoelastic materials has gained popularity. The main idea of this approach is to estimate complex moduli (Young s, shear) from vibration tests [8 12]. To avoid making assumptions on the thermo-rheological simplicity of the material tested, measurements should be performed on a broad frequency range. Therefore, many authors estimate the complex moduli parametrically by relying on a viscoelastic model to describe the frequency dependent material properties (generalised Maxwell model [9], fractional derivative model [1 12]). The optimal parameters of the viscoelastic model are found by minimising the residuals between experimental data and predictions from a finite element model. These residuals can be defined either from modal parameters [8] or from FRFs at selected frequencies [9 12]. Though resonant frequencies contain most of the information about the material damping, the identification of the modal parameters becomes delicate when dealing with highly damped structures, due to non-negligible modal couplings. Hence, inverse characterisation techniques are usually using in part or in whole frequency response functions. Such approaches have successfully been applied to beam structures [9 12] to estimate the parameters of a viscoelastic model. However, to the author s knowledge, they have never been tested on structures with more complex geometry. In this work, the dynamic characterisation of a high damping and strong frequency-dependent viscoelastic material is performed by applying an inverse method to frequency response functions measured on sandwich lighweight panels. As advocated by Martinez-Agirre and Elejabarrieta [11], the method minimises the difference between experimental and theoretical transfer functions at certain discrete frequencies selected by the user in the neighborhood of the resonance peaks. To limit the possibility of converging toward a local minima, an initialisation procedure aiming at minimising the difference between experimental and simulated resonant frequencies is performed. The design parameters of the optimisation are the parameters of a fractional derivative model, presented in the next section, along with the finite element model and the experimental setup. Then, the proposed inverse procedure is introduced. And finally, it is applied to two sandwich panels fully or partially treated with constrained damping layers. The material properties identified from the proposed inverse method are compared to those obtained from direct measurements on a viscoanalyser. 2 NUMERICAL MODEL AND MEASUREMENTS 2.1 Fractional derivative model In general, polymers exhibit a viscoelastic behaviour due to their molecular structure. The mechanical properties of such materials are strongly-dependent on temperature and frequency, which

3 can be conveniently described by a complex G, satisfying the following constitutive equations: ˆσ(ω, T ) = G (ω, T )ˆɛ(ω, T ) (1) where ˆσ and ˆɛ are the Fourier transforms of stress and strain. The complex modulus is given by: G (ω, T ) = G (ω, T ) + ig (ω, T ) = G (ω, T ) (1 + iη(ω, T )) (2) where G and G are respectively the storage and loss modulus, and η the loss factor. In this work, the four-parameters fractional derivative model is used to describe the frequency dependency of the complex modulus at a given reference temperature T ref : G (ω, T ref ) = G + G (iωτ) α 1 + (iωτ) α (3) where G and G are respectively the relaxed and unrelaxed modulus, τ is the relaxation time and α is a fractional parameter comprised between and 1 which corresponds to the non-integer order of derivation in the σ(t) ɛ(t) relationships [13]. These four parameters must satisfy the following thermodynamical constraints: G > G >, τ > and < α < 1 (4) This viscoelastic model enables a good represention of the viscoelastic behaviour with only four parameters, which justifies its use in the proposed inverse method. 2.2 Finite element analysis of the sandwich structure The finite element modeling of the sandwich structure is challenging chiefly because of the large transverse shear deformations undergone by the constrained viscoelastic layer. One modeling strategy consists in representing the damped structure by sandwich finite elements, as initiated by Macé [14]. Many authors have contributed to the development of such finite elements, which are usually based on zig-zag theories. Due to the size of the coresponding model, this approach has been used in previous work for the characterisation of viscoelastic properties through inverse techniques [1 12]. Another modeling strategy entails the use of solid elements for the viscoelastic core and shell elements for the faces [15, 16]. This approach, which is more suited for the modeling of structure of complex geometry and directly implementable in some industrial software, is adopted in this work. Once discretised, the equations of motion can be written as: [ Kf + G (ω, T ref )K c ω 2 M ] X = F (5) where K f is the global stiffness matrix associated with the elastic faces, K c is the global stiffness matrix associated with the core layer and evaluated for a unitary shear modulus, M is the global mass matrix, G (ω, T ref ) is the complex modulus from Equation 3, and X and F are the displacement and force vectors respectively. The predicted frequency responses of the damped structure are computed by a direct method, which solves Equation 5 at each frequency. The non-linear eigenvalue problem associated with Equation 5 is solved by an iterative model strain energy [17] to compute the resonant frequencies needed for the initialisation step of the inverse procedure. 2.3 Experimental procedure The system under study is a clamped sandwich panel whose core is made of the viscoelastic material. An impact hammer is used to excite the structure and its response is measured by lightweight

4 Clamp Accelerometers (PCB-352A24) Sandwich panel PC (LMS.TestLab) Spectrum Analyser Hammer impact (PCB-86B3) Clamp Figure 1. Experimental set-up. accelerometers glued on the bare panel side by means of bee wax. Figure 1 shows a schematic diagram of the set-up. The experimental transfer function measured is the accelerance in db, defined from the ratio bewteen the transverse acceleration measured by accelerometers and the force input in the system by impacting the hammer against the sandwich structure: ( ) Â(ω) T exp (ω) = 2 log (6) ˆF (ω) where  and ˆF are the Fourier transforms of the transverse acceleration and the input force respectively. Transfer functions are measured up to 1 Hz with a frequency step of.5 Hz. The modal parameters of the damped structure are identified with LMS Polymax from the synthesised frequency response based on transfer functions measured on a grid of 7 points. This software enables an accurate estimation of the modal parameters, even for highly damped structures where modal coupling may occur. 3 PROPOSED INVERSE METHOD The goal of the inverse method is to identify the parameters of the viscoelastic model chosen to describe the viscoelastic behaviour of the material by minimising a cost function which represents the difference between a measured and a simulated response. At each step of the optimisation, the simulated response is computed from the finite element model with updated viscoelastic parameters. Therefore, the evaluation of the cost function may be time-consuming if the model contains a lot of degrees of freedom. Then, the optimisation method must be chosen so that the number of cost function evaluations is kept as low as possible. A comparative study of optimisation techniques in the context of identification of the constant mechanical properties of composite materials shows that fewer cost functions evaluations are necessary when using higher-order methods, such as gradient-based methods [18]. In this work, the BFGS (Broyden-Fletcher-Goldfarb-Shanno) method is used : only the gradient of the cost function needs to be computed since the Hessian is approximated. The optimisation problem to be solved is not convex and the BFGS method is not globally con

5 Finite element model Initialisation step Compute matrices Identification step Compute viscoelastic modulus Compute eigenfrequencies Compute FRF Compute cost function S Compute cost function S 1 2 Convergence? yes Convergence? yes END no no Compute updated parameters using BFGS Compute updated parameters using BFGS Figure 2. Flowchart of the optimisation procedure. vergent so the algorithm may converge towards a local minimum. Therefore an initialisation step is introduced in the procedure (see Figure 2). It consists in optimising the design parameters by minimising a cost function representing the difference between the measured and the simulated resonant frequencies of the structure. This approach has been used by several authors to minimise the risk of converging towards a local minima [9, 1]. 3.1 Design parameters The proposed inverse method optimises the parameters of a fractional derivative model. However, the difference in order of magnitude between these parameters is important (up to 1 9 ), which may lead to conditionning problem during optimisation. Therefore, the design parameters p = [p 1, p 1, p 3, p 4 ] taken for the optimisation are defined as follows: p 1 = 1 1 log(g ), p 2 = 1 1 log(g G ), p 3 = 1 1 log(τ), p 4 = α (7) In this way, the design parameters are of the same order of magnitude and by imposing bounds on the parameters, the thermodynamical contraints from Equation 4 are respected. 3.2 Definition of the cost function In gradient-based algorithms, the degree of regularity of the cost function is crucial : if the cost function is not smooth enough, its derivatives may be close to be discontinous, which will generate

6 instabilities in the optimisation. Several definitions of the cost function have been tested and the normalised mean square error was found to be the most appropriate. The cost function of the initialisation step ( Figure 2) S 1 is defined as follows: S 1 (p) = (ωexp k ωnum(p)) k 2 n mode (8) (ωexp) k 2 n mode where n mode is the number of observable modes, ωexp k is the k-th measured resonant frequency and ωnum(p) k is the k-th simulated resonant frequency, computed by an iterative modal strain energy (see section 2.2). The cost function of the identification step minimises the residuals between measured and simulated frequency responses: n freq (Texp k Tnum(p)) k 2 S 2 (p) = n freq (9) (Texp) k 2 where n freq is the number of discrete frequencies at which the transfer function is computed, and T k exp and T k num(p) are the measured and simulated transfer functions evaluated at a set of control frequencies ω k, composed of linearly-spaced frequency grids located around each resonance peak. 3.3 Computation of the gradient The gradient of the cost function is generally computed by the finite difference method, which may be very expensive since it requires more evaluations of the cost function. As an alternative, Kim and Lee [1] propose to apply a direct differentiation method. This approach has been adopted in this work. By differentiating Equation 8 with respect to a design parameter p j, one gets: S 1 (p) = n mode 2 ωk num(p) (ωnum(p) k ωexp) k (1) (ωexp) k 2 As previously explained, the simulated resonant frequencies ω k num are computed by an iterative modal strain energy method, which solves iteratively the following eigenfrequency problem until ω a ω k num < ɛ tol : n mode [K f + R(G (ω a, T ref, p))k v (ω k num) 2 M]Φ k num = (11) Since only the complex modulus and the eigensolutions depend on the design parameters, differentiating Equation 11 with respect to a design parameter p j leads to the following expression of the eigenfrequencies derivatives: ω k num(p) = 1 R(G (ω a, T ref, p)) (Φ k 2ωnum(p) k p num) T K cφ k num (12) j where the derivatives of the storage modulus can be obtained analytically from Equation 3. By differentiating Equation 9 with respect to a design parameter p j, the derivative of the cost

7 function S 2 is computed as: S 2 (p) = n mode 2 T num(p) k (Tnum(p) k Texp) k (13) (Texp) k 2 where k num(p) is computed from the derivative of the displacement vector X. The latter is obtained by differentiating Equation 5 with respect to a design parameter p j : n mode X(ω k, p) = R(G (ω k, T ref, p)) ( Kf + G (ω k, T ref, p)k c (ω k ) 2 M ) 1 K c X(ω k, p) (14) 4 APPLICATION TO A DAMPING MATERIAL The previously described inverse method is now applied to characterise the self-adhesive synthetic rubber of a CLD treatment from an automotive TIER supplier. The main structure studied is a panel made of aluminium 5754, of dimensions.42m.594m.3m (A2 dimensions). In order to impose boundary conditions, the dimensions of the panel are increased and two rows of holes are made, as shown on Figure 3(a), so that the panel can be mounted in a clamping frame (35 mm thick made of aluminum) by means of connecting bolts (Figure 3(b)). The quality of this clamping system have been studied in a previous work [19]. Vibration tests on the bare aluminum (a) (b) Figure 3: Dimensions of the aluminum panel tested (a) and its mounting in the clamping frame (b). panel have been conducted to update the finite element model. The dimensions and the material properties of the aluminum panel after model updating are presented in Table 1. The CLD treatment applied consists of a constraining layer of 127µm thickness made of aluminium and a layer of self-adhesive synthetic rubber of 1.5mm thickness. Two configurations are tested: in the first one (Figure 4.a), the bare panel is fully covered with the vibration-damping material, while in the second one (Figure 4.b), three patches of vibration-damping material, of dimensions.8m.15m.127m, are applied. The density of the synthetic rubber ρ c = 13 kg/m 3 is given by the supplier and a Poisson ratio of ν c =.495 is assumed. The dynamic properties of the viscoelastic core are identified from the transfer functions measured

8 Dimensions Material properties L x L y L z.429 m.62 m.3 m E f Pa ν f.3 η f.41 % ρ f 2783 kg/m 3 Table 1. Dimensions and material properties of the bare aluminum panel after model updating. Figure 4: Assembled sandwich panels: (a) unclamped fully treated panel, (b) clamped locally treated panel. at five different points of the sandwich panel up to 1 Hz. Ten control frequencies are considered around the resonances of the most excited modes (see Figures 5 and 6). The initial guess taken for the design parameters are the ones corresponding to the fractional derivative model identified from direct DMA measurements of the viscoelastic s dynamic properties, which are given in Table 2. The optimised parameters are listed in Table for both studied sandwich panels. Figures 5 and 6 show the frequency response at one of the measurement points on each sandwich panel before and after optimisation. One observes that the model considering the viscoelatic properties identified by DMA measurements does not approximate accurately the response of the sandwich panels. However, the simulated responses calculated with the parameters identified by the proposed inverse method are in good agreement with the measured transfer functions. The fractional derivative model identified from measurements on both sandwich panels are compared in Figure 7 with the DMA measurements performed on samples of the studied synthetic rubber [2]. A good agreement is observed between the master curves obtained after applying the proposed inverse method to the response of the fully and partially covered sandwich panels, which gives confidence in the identification procedure. However, they are relatively far from the master curves measured by DMA. This is in accordance with the discrepancies observed in Figures 5 and 6 between the measured transfer fucntions and the ones computed with the initial design parameters. 5 CONCLUSION To design and analyse structures treated with constrained layer damping, the dynamic properties of the viscoelastic core are essential. The four-parameters fractional derivative model generally describes accurately the frequency-dependence of viscoelastic materials. To overcome the limitations of direct approaches, inverse methods are developed to identify the parameters of the viscoelastic model such that the residuals between experimental and simulated transfer functions are minimal

9 p 1 p 2 p 3 p 4 Initial Optimised Fully covered panel Locally covered panel Table 2. Initial and optimised design parameters for both sandwich panels. 3 Initial design parameters 3 Optimised design parameters FRF 2log 1 ( a/f ) [db] Measured -2 Simulated Control frequencies FRF 2log 1 ( a/f ) [db] Measured -2 Simulated Control frequencies Figure 5: Simulated and measured transfer function of the fully treated panel before (left) and after (right) the optimisation of the viscoelastic model s parameters. Initial design parameters Optimised design parameters 4 4 FRF 2log 1 ( a/f ) [db] Measured -1 Simulated Control frequencies FRF 2log 1 ( a/f ) [db] Measured -1 Simulated Control frequencies Figure 6: Simulated and measured transfer function of the locally treated panel before (left) and after (right) the optimisation of the viscoelastic model s parameters

10 DMA measurements Initial properties Optimised properties (fully covered) Optimised properties (locally covered) Storage modulus [MPa] Loss factor Figure 7: Comparison of the master curves of the viscoelastic material measured by DMA (black crosses) and those obtained with the initial (black straight line) and optimised (dotted lines) parameters of the fractional derivative model.) In this work, the efficiency of this approach when applied to a sandwich panel is demonstrated. The dynamic properties of a synthetic rubber have been successfully identified with the proposed inverse method. A gradient-based method combined with analytic differentiation of the cost function to minimise is used to reduce the computational cost of the identification procedure. The application of reduced order methods adapted to highly damped structures would allow a further drastic reduction in the computational time of each evaluation of the cost function. ACKNOWLEDGMENTS The Research Fund KU Leuven, the MacroModelMat - M3NVH Program within the Strategic Initiative Materials in Flanders (SIM), and the European Project INTERACTIVE Marie Curie Initial IAPP are gratefully acknowledged for their financial support. REFERENCES [1] M. Rao. Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes. Journal of Sound and Vibration, 262: , 23. [2] R. Lakes. Viscoelastic measurement techniques. Review of scientific instruments, 75:797 81, 24. [3] J.D. Ferry. Viscoelastic properties of polymers. John Wiley & Sons, 198. [4] L. Rouleau, J.-F. Deü, A. Legay, and F. Le Lay. Applications of the kramers-kronig relations to time-temperature superposition for viscoelastic materials. Mechanics of Materials, 65:66 75,

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