VISCO-ELASTIC ANALYSIS OF CANTILEVER BEAM USING AUGMENTED THERMODYNAMIC FIELD METHOD IN MATLAB

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1 VISCO-ELASTIC ANALYSIS OF CANTILEVER BEAM USING AUGMENTED THERMODYNAMIC FIELD METHOD IN MATLAB 1 G.N.V.V. Manikanta, 2 P. Jaikishan, 3 E. Raghavendra Yadav 1 Research Scholar, 2,3 Asst. Professor, Aditya College of Engineering & Technology, Surampalem, East Godavari Dist., A.P. ABSTRACT This work deals with the theoretical investigation of the problem of heat generation as well as heat conduction within a viscoelastic beam under prescribed loading condition. The beam is modelled with Euler-Bernoulli beam finite elements. In order to include the effect of viscoelasticity, the internal variable approach is followed where the internal variables are modelled using augmented thermodynamic field (ATF). Any viscoelastic structure, while vibrating, is associated with energy dissipation in the form of heat. This heat dissipation can be treated as a source of internal energy generation at any generic point within the structure, which is dependent on the displacement of that point during the vibration.following the internal variable approach using ATF, the temporal variation of the energy dissipation due to vibration can be found out as a difference between the elastic strain energy and the Helmholtz free energy at any point within the structure. This time-varying vibrationgenerated internal energy can be used to solve the unsteady-state heat conduction equation and to find out the temporal variation of the temperature at that point within the structure as well as the temperature profile throughout the entire structure. The plot of instantaneous temperature distribution will help to identify the location of thermally weak points. Since viscoelastic material is used for important applications like large space structures, a prior analysis of the thermal diffusion in conjunction with the correlated mechanical vibration is essential in order to avoid either of the thermal or mechanical failure of such systems. This work attempts such an analysis in the case of a viscoelastic cantilever beam subjected to transient as well as steady state vibration. I. INTRODUCTION Background and importance Viscoelasticity, as the name implies, is a property that combines elasticity and viscosity. A material, which is viscoelastic in nature, thus stores and Page No:975

2 also dissipates energies and therefore the stress in such materials is not in phase with the strain. Due to this property, it is extensively used in various engineering applications for vibration damping, control and stability. In terms of performance, higher damping can reduce steady state vibration levels and can reduce the time needed for transient vibration to settle. Vibration damping is essential to the attainment of performance goals for a variety of advanced engineering systems. In common built-up structures that operate in the atmosphere, air damping and joint damping typically dominate system damping. However material damping can also be an important contributor to overall damping in many applications, such as aerospace vehicles, large space structures, etc. Viscoelastic material damping is generally a complex function of frequency, temperature, type of deformation, amplitude and structural geometry. Several methods for incorporating the material damping into structural models have been used, and continue to be used within the engineering community. These methods include frequencydependent viscous damping, complex modulus, hysteric damping, structural damping, hereditary integrals, and modal damping. Each has some utility, but each suffers from one flaw or another. But the most satisfactory damping model is based on Internal Variable (IV). The Augmenting Thermodynamic Field (ATF), a timedomain continuum model of material damping preserves the characteristic frequency-dependent behaviour of real material (damping and modulus) and therefore is a physically motivated model compatible with current finite element structural analysis methods. In the present study the above damping model has been exploited. A cantilever beam can be considered as a basic element in many structural and machine parts. It may find wide applications particularly in the field of robotic manipulators used in nuclear power plants, microsurgery, space application, or many other precision industrial applications (Pratiher and Dwivedy (2009)) where lightweight, precision motion and temperature are of great concern. Viscoelastic material can be used in such application due to the biggest advantages of light weight and high damping. But these light weight and flexible structures suffer from the drawback of large vibration due to low stiffness. These vibrations in the beam also cause the increase in internal energy and consecutively change in the temperature of the beam. So the present work concentrates on first finding out the equations of motion of the viscoelastic beam by using Page No:976

3 augmenting thermodynamic fields (ATF) approach. The Helmholtz free energy density function is used to develop coupled material constitutive relations and partial differential equations of evolution. These equations are implemented for numerical solution within the computational frame work of finite element method. The temporal variation of the energy dissipation due to vibration can be found out as a difference between the elastic strain energy and the Helmholtz free energy at any point within the structure. This time-varying vibration- to generated internal energy can be used solve the unsteady-state heat conduction equation and to find out the temporal variation of the temperature at that point within the structure as well as the temperature profile throughout the entire structure. II. LINEAR VISCOELASTICITY The Stress-Strain Curves for a purely elastic and a viscoelastic material are shown in Figure 1. Due to loss of energy during loading and unloading time, the stress strain curve for viscoelastic material is elliptic in nature (Thompson and Dahleh(1998)). The area enclosed by the ellipse is a hysteresis loop and shows the amount of energy lost (as heat) in a loading and unloading cycle. Figure 1 The stress strain curve Some phenomena in viscoelastic materials are: (i) (ii) if the stress is held constant, the strain increases with time (creep); if the strain is held constant, the stress decreases with time (relaxation); (iii) the effective stiffness depends on the rate of application of the load; (iv) if cyclic loading is applied, hysteresis (a phase lag) occurs, leading to a dissipation of mechanical energy; (v) acoustic waves experience attenuation; (vi) rebound of an object following an impact is less than 100%; (vii) during rolling, frictional resistance occurs. III. LAYOUT OF THE PRESENT WORK From the discussion as remarked in the previous section, it is identified that the damping model has been classified in two main category. One related to non Internal Variable based models with their limitations and applications and one for the Page No:977

4 Internal Variable models and some description of major works dealing with that model. The last section includes few model which depicts the thermo mechanical behaviour of the viscoelastic material due to self heating. Based on that review, the objective and scope proposed in this work are as follows. 1. Finding out the equations of motion and Development Finite Element formulation of the viscoelastic beam by using augmenting thermodynamic fields (ATF) approach. 2. The temporal variation of the energy dissipation due to vibration is found out as a difference between the elastic strain energy and the Helmholtz free energy at any point within the structure. 3. The time-varying internal energy due to self-heating is used to solve the two dimensional unsteady-state heat conduction equation and to find out the temporal variation of the temperature at that point within the structure as well as the temperature profile throughout the entire structure using the finite difference method. IV. VISCOELASTIC BEAM AND THERMO-MECHANICAL MODELLING This chapter forms the basis of the entire work as it presents the derivation of the equations of motion of a viscoelastic beam and Development Finite Element analysis procedure. The physically significant Internal State Variable plays a central role in this work, motivating the introduction of Augmenting Thermodynamic Field (ATF) to interact with the usual mechanical displacement field of continuum structural dynamics. The techniques of nonequilibrium, irreversible thermodynamics are used to develop the coupled material constitutive equations and coupled partial differential equations of evolution. The internal heat generation term is found out by ATF approach, which is augmented with the un-steady heat conduction equation to determine the temporal variation of the temperature throughout the structure. V. CONSTITUTIVE RELATIONSHIPS USING MULTIPLE ATF APPROACH Multiple ATF or ADF are used to model the material behavior more accurately i.e. for closer approximation to the experimental data over a broad frequency range. Considering N number of ATF or ADF to interact among each other, the Helmholtz free energy density function, H is basically a potential function, which is used to derive the respective constitutive relationships. The Helmholtz free Page No:978

5 energy for multiple ATF is given by equations (1). Corresponding material constitutive relations are given by equations (2) (1) (3) The partial differential equation governing the motion of an Euler- Bernoulli beam with no distribution of external load is given by (2) In the preceding equation E is the elastic modulus, ε the mechanical strain, an independent variable, ξ the augmenting thermodynamic field (ATF) variable, the dissipation coordinate, σ, the mechanical stress, a the affinity, α a material property relating the changes in a to ξ and δ is the strength of coupling between the mechanical displacement field and the dissipation field. Equations of motion of beam The equations of motion of a uniform slender viscoelastic beam are found out by using the constitutive relationships. When the beam is deformed, it is assumed that a transverse section, originally plane, will remain plane. The stress normal to the transverse plane (parallel to X-axis) is assumed to be the main stress. The x- axis and y-axis are defined to be parallel and perpendicular to the beam axis, with and u(x, y) and v(x, y) are the respective displacement components. To this end the following approximations are used. (4) Where ρ is the density of the material, A is the cross-sectional area, σ is the stress in a fibre of infinitesimal cross-section at a distance y from the neutral axis of the section. The symbols (.) and (') stand for single partial differentiation with respect to time, and space VI. FINITE ELEMENT FORMULATIONS For the FE formulation of the spatial continuum, the time varying mechanical displacement v(x,t) and the dissipation coordinate f(x,t) are approximated (, ) = ( ), (, ) = ( ) ( ) and Where, (x) is the Hermite shape function and θ(x) is the Lagrange shapefunction. Galerkin s method by employed to the differential equations to derive the finite element form (Lesieutre and Mingori (1990)). Figure 2 illustrates a beam element and the nodal variables for the two dependent fields, where q i (i = 1 to 4) and p k (where k = 1 to 2) denote is Page No:979

6 the mechanical and ATF displacements respectively. Figure 2 A damped beam bending element (5) (6) the structure. The energy dissipation due to vibration is found out as a difference between the elastic strain energy and the Helmholtz free energy at any point within the structure. So energy dissipation is given by (7) Where, Helmholtz free energy is given by the equation (8) Substituting equations (08) and (09) in (07), we have energy dissipated (d ) due to vibration at a point in the structure. VII. HEAT GENERATION AND CHANGE IN TEMPERATURE IN THE BEAM Internal heating associated with material hysteresis can lead to large local temperature and strain increases within the material. Thus for elastomeric material the thermo mechanical modeling is very essential for prediction of internal energy due to self-heating. The internal heat generation term is found out by ATF approach, which is augmented with the un-steady heat conduction equation to determine the temporal variation of the temperature throughout (09) Heat flux at a point in a beam can be calculated by differentiating energy dissipated with respect to time (10) The expression of heat flux is rewritten as (11) Total heat flux of the element is obtained by performing the integration over the volume of element and is given as: (12) Heat conduction equation Following Ozisik (1993) the unsteady heat conduction equation is given by Page No:980

7 (13) Where = the thermal diffusivity, k is is the conductivity and c is the specific heat of the material. Finite element formulation for the heat conduction equation In finite element method the domain is broken into a set of discrete volumes or finite elements that are generally Unstructured; in 2D, they are usually triangles or quadrilaterals, while in 3D tetrahedral or hexahedra are most often used. The distinguishing feature of FE methods is that the equations are multiplied by a weight function before they are integrated over the entire domain. In the simplest FE methods, the solution is approximated by a linear shape function within each element in a way that guarantees continuity of the solution across element boundaries. Such a function can be constructed from its values at the corners of the elements. The weight function is usually of the same form. This approximation is then substituted into the weighted integral of the conservation law and the equations to be solved are derived by requiring the derivative of the integral with respect to each nodal value to be zero; this corresponds to selecting the best solution within the set of allowed functions (the one with minimum residual). The result is a set of non-linear algebraic equations. An important advantage of finite element methods is the ability to deal with arbitrary geometries. Finite element methods are relatively easy to analyze mathematically and can be shown to have optimality properties for certain types of equations Consider a body occupying the volume v, inside which heat is generated at a rate G and whose outer surface is subjected to boundary condition on portion S1. In rectangular Cartesian coordinates the governing equation is (14) The boundary conditions are In the finite element method the domain is first subdivided into a number of regions with simple polygonal shape and the temperature within each element (15) 7, is expressed as a linear combination of the nodal values T i ; i = 1;2;... r by means of finite element basis functions, i.e. (16) Where is the basis function matrix and T is the vector of nodal temperatures for i = 1;2;...M where M is the total number of nodes in the finite element mesh and N is the total number of elements. The result is Page No:981

8 a set of M linear algebraic equations for the M unknown nodal values; the finite element equation (17) Where the stiffness matrix is given by Lesieutre and Mingori (1990) of modal damping ratio of a viscoelastic cantilevered beam in bendingvibration at different frequencies. A close match validates the correctness of the code developed. In order to assure convergence of the finite element model as the element size decreases the basis functions must satisfy the standard requirements of compatibility and completeness, i.e.. the basis functions selected must provide for continuity of the temperature at the interface between any two elements as well as for the continuity of temperature and heat flux inside each element. Standard polynomials of order n satisfy the requirement. VIII. RESULTS AND DISCUSSION Validation of the finite element code The finite element code for transverse vibration of a beam is developed in and has been validatedd by comparing the results with those reported in published literature. Good matches between the computed and published results validate the correctness of the present code. Figure 3 shows the comparison between computed results and as reported by Figure 3 model damping ratio for cantilever beam using ATF approach Frequency response of a viscoelastic beam A cantilever beam made of PVC (Length L = 0.5m and diameter 0.05m) as shown in Figure 4 is considered. Following Roy (2008) the material properties of PVC at 24 C are E = 5..04e8 Pa, δ = 1390 Kg/m? and the viscoelastic parameters of PVC using double approach are β 1 = , D = ATF α 1 = ,β 2 = 3.02e4,α 2 =3.02e4, where the subscripts 1, 2 are ATFindex. The beam is subjected to harmonic excitation at the tip. The amplitude of harmonic excitation is P=1N. Page No:982

9 Figure 4 Schematic diagram of cantilever beam Figure 5 illustrates the response plot at various excitation frequencies, when the load is applied at the tip of the beam. The curve has several peaks which indicate the natural frequencies. Figure 6 and Figure 7 show the mechanical and second ATF displacement for different time span, when the beam is perturbed by initial velocity at the tip of beam. After certain instant both response decrease and reached at zero. It is due to the nature of the viscoelastic material of the beam. Because after certain instant the energy dissipates and vibration reduces. Figure 6 transient analysis Figure 5 Forced response of cantilevered beam Time response of a viscoelastic beam In this section the theoretical results of time response both for transient and steady state excitation have been analyzed. In transient analysis the beam is perturbed by an initial velocity of 0.3 m/s at the tip of the beam, whereas in the case of steady state analysis, the beam is excited by a harmonic force of amplitude 1N at the tip of the beam. Figure 7 steady state analysis Page No:983

Figure 8 heat generated for different location of beam under transient vibration Figure 9 heat generated for different location of beam under steady state vibration The heat generation and change in

Figure 8 and Figure 9 show that heat flux for different location of the beam under transient and steady state vibration.

10 Figure 8 heat generated for different location of beam under transient vibration Figure 9 heat generated for different location of beam under steady state vibration The heat generation and change in temperature Increase in internal energy as well as heat generation within the material is occurred due to bending of the beam. Figure 8 and Figure 9 show that heat flux for different location of the beam under transient and steady state vibration. Here L* indicate the non-dimensional distance, which is the ratio between the distance from the root of the beam and the total length of the beam. Like response analysis the heat flux for transient vibration gradually decreases with time but it is constant for steady state vibration. In both case the heat flux is more near root of the beam. Because the root section is subjected to more bending deformation. In this analysis the beam is considered as one dimensional long fin with convective heat loss at the tip of the beam. Heat flow occurred along the beam axis. Figure 10 show that temporal variation of the temperature for different location of the beam under transient and steady state vibration. In Figure 10 it is seen that the temperature increases with time and becomes constant. Whereas in Figure 10 the temperature increases monotonically with time. In the case of transient vibration the temperature increases due to instantaneous heat source produced by the initial perturbation. But for steady state vibration the heat source is constant. As it is case of one dimensional heat flow (heat flow is only through the beam axis), other than tip the beam surface is treated as insulated, the less heat loss occurred. So better prediction of temperature the in the entire structure two Page No:984

11 dimensional heat flow equation necessary. is multiple ATF approach to form the constitutive relationships as welll as equation of motion for the viscoelastic material. The internal heat generationn due to bending of the beam is found out by ATF approach, which is augmented with the un-steady heat equation. The temporal distribution of temperature throughout structure is obtained by solving the unsteady heat equation after discretization using finite element method. Figure 10 Time vs Temperature response along the span of the beam IX. CONCLUSION AND FUTURE SCOPE Based on the earlier discussions the following conclusions are drawn. This work applies a time domain model, useful for complete dynamic analysis as well as prediction of thermo- viscoelastic beam due to transient and mechanical behaviour of a generally steady state excitation. The continuum is discretized with finite elements, using The Augmenting thermodynamic field (ATF )approach introduces internal variables to model the viscoelastic frequency dependent property is more efficient than viscous and hysteretic representation. It is also compatible with current FINITE ELEMENT technique. Using the proposed modelling procedure in the time domain, it is possible to predict the variation of temperature at any point in the continuum with time. This may be helpful in dynamic design of polymeric structures. X. FUTURE SCOPE This study has given birth to different other possibilities which may be taken up as future research activities in this area. Page No:985

12 1. In the present work one dimensional heat conduction equation is used for simplicity. Two dimensional heat conduction equations can be used for better prediction of temperature throughout the structure. 2. The proposed model can easily be imposed in the rotating structure like rotor for correct prediction of stability limit of spin speed, where one dimensional beam model is suitable. 3. Incorporating the effects of time and temperature on viscoelastic material property by using shift factor, a non linear finite element can be developed. This model can be introduced for dynamic design of any two dimensional structure. XI. REFERENCES 1. Agrawal M., 2005, Basics of Irreversible Thermodynamics, Electrical Engineering, Stanford University, Stanford. 2. Bagley, R. L. and Torvik, P. J.,1983, Fractional calculus-a different approach to the analysis of viscoelastic damped structures, AIAA Journal, vol. 21, pp , 3. Bagley, R.L and Torvik. P.J,1985, Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures, AIAA of Journal, 23(6): Bert C.W., 1973, Material damping, an introductory review of mathematical models, measures and experimental techniques, Journal of Sound and Vibration, vol. 29(2), pp Bland, D.R., 1960, Linear Viscoelasticity, Pergamon Press, Oxford. 6. Brackbill, C. R., Lesieutre, G. A., Smith, E. C., and Govindswamy, K., 1996, Thermo-mechanical modeling of elastomeric materials, Smart Mater. Struct., vol. 5, pp Christensen, R.M., 1982, Theory of Viscoelasticity, An Introduction, Academic Press, New York. 8. Crandall S.H., 1970, The Role of Damping in Vibration Theory, Journal of Sound and Vibration, vol. 11(1), pp Ferry J.D., 1980, Viscoelastic Properties of Polymers, John Wiley & Sons, New York. 10. Golla D.F. and Hughes P.C., 1985, Dynamics of Viscoelastic Structures a Time Domain Finite Element Formulation, Journal of Applied Mechanics, Transactions of the ASME, vol. 52, pp Henwood D.J., 2002, Approximating the Hysteretic Damping Matrix by a Page No:986

13 Viscous Matrix for Modelling in the Time Domain, Journal of Sound and Vibration, vol. 254(3), pp Kaliske, M. and Rothert, H., Formulation and implementation of three-dimensional viscoelastic at small and Finite strains, Computational Mechanics, vol. 19, pp Page No:987

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