Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping

This paper investigates the frequency dependent viscoelastic dynamics of a multifunctional composite structure from finite element analysis and experimental validation. The frequency-dependent behavior of the stiffness and damping of a viscoelastic material directly affects the system's modal frequencies and damping, and results in complex vibration modes and differences in the relative phase of vibration. A second order three parameter Golla–Hughes–McTavish (GHM) method and a second order three fields Anelastic Displacement Fields (ADF) approach are used to implement the viscoelastic material model, enabling the straightforward development of time domain and frequency domain finite elements, and describing the frequency dependent viscoelastic behavior. Considering the parameter identification a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Agreement between the curve fits using both the GHM and ADF and experiment is within 0.001 percent error. Continuing efforts are addressing the material modulus comparison of the GHM and the ADF model. There may be a theoretical difference between viscoelastic degrees of freedom at nodes and elements, but their numerical results are very close to each other in the specific frequency range of interest. With identified model parameters, numerical simulation is carried out to predict the damping behavior in its first two vibration modes. The experimental testing on the layered composite beam validates the numerical predication. Experimental results also show that elastic modulus measured from dynamic response yields more accurate results than static measurement, such as tensile testing, especially for elastomers.     

Nonlinear vibration of viscoelastic sandwich beams by the harmonic balance and finite element methods

This paper deals with geometrically nonlinear vibrations of sandwich beams with viscoelastic materials. For this purpose, a new finite element formulation has been developed, in which a zig-zag model is used to describe the displacement field. The viscoelastic behaviour is handled by using hereditary integrals and their relationships with complex moduli. An efficient solution procedure based on the harmonic balance method is also developed. To demonstrate its abilities, various problems of nonlinear vibrations of sandwich beams are considered. First, the results derived from the proposed approach are compared with those of nonlinear dynamic analyses using direct time integration and to experimental data. Then, the influence of the vibration amplitude on the damping properties of sandwich beams is investigated. The effect of an initial axial strain is also examined.     

Forced response of FRP sandwich panels with viscoelastic materials

In this paper a new analytical model is presented that accurately predicts the forced response of fibre reinforced plastic (FRP) sandwich plates subjected to transverse applied loads. It is based on Reddy's refined high order shear deformation theory and offers the feasibility of accounting for the viscoelastic properties of the constitutive materials without restriction to the steady state motion. This is achieved by modelling the viscoelastic behaviour of the constitutive materials using the Golla Hughes McTavish mathematical tool. Validation of the new approach is achieved by comparing results under harmonic loading conditions against data obtained using the proposed new analytical model. Subsequently, predicted responses for a given FRP sandwich plate under various transverse applied loads are presented. The results outline the importance of being able to account for the viscoelastic properties of the constitutive materials when modelling the dynamic behaviour of sandwich structures.     

Frequency-domain acoustic pressure formulation for rotating source in uniform subsonic inflow with arbitrary direction

This paper presents a frequency-domain formulation for predicting noise radiated from the rotating thickness and loading sources in uniform subsonic inflow with arbitrary direction. The proposed frequency-domain formulation is an extension of the recently published frequency-domain formulation for the stationary medium. It avoids the singular integral and numerical interpolation problems encountered in the time-domain numerical method. Three test cases, i.e., noise radiation from the rotating monopole and dipole point sources and the Isom thickness noise of a transonic rotor in the subsonic uniform flow, have been carried out to validate the proposed formulation. Both the acoustic pressure spectrum and directivity pattern computed with the present frequency-domain method are in good agreement with those obtained from the time-domain method, thus validating the correctness of the present formulation. Furthermore, the numerical results indicate that the frequency-domain formulation is suitable for tonal noise prediction, while it is inefficient for broadband noise prediction.     

Finite element reduced order model for noise and vibration reduction of double sandwich panels using shunted piezoelectric patches

This work concerns the control of sound transmission through double laminated panels with viscoelastic core using semi-passive piezoelectric shunt technique. More specifically, the system consists of two laminated walls, each one composed of three layers and called sandwich panel with an air cavity in between. The external sandwich panel has a surface-mounted piezoelectric patches. The piezoelectric elements, connected with resonant shunt circuits, are used for the vibration damping of some specific resonance frequencies of the coupled system. Firstly, a finite element formulation of the fully coupled visco-electro-mechanical-acoustic system is presented. This formulation takes into account the frequency dependence of the viscoelastic material. A modal reduction approach is then proposed to solve the problem at a lower cost. In the proposed technique, the coupled system is solved by projecting the mechanical displacement unknown on a truncated basis composed by the first real short-circuit structural normal modes and the pressure unknown on a truncated basis composed by the first acoustic modes with rigid boundaries conditions. The few initial electrical unknowns are kept in the reduced system. A static correction is also introduced in order to take into account the effect of higher modes. Various results are presented in order to validate and illustrate the efficiency of the proposed finite element reduced order formulation.     

Assessment of sandwich models for the prediction of sound transmission loss in unidirectional sandwich panels

The consistent higher-order approach and the two-parameter foundation formulation are used for the derivation of sound transmission loss in symmetric unidirectional (infinitely wide) sandwich panels with isotropic face sheets. In both models, transmission loss is calculated using decoupled equations representing symmetric and anti-symmetric motions of a sandwich panel. The closed-form expressions for impedances and transmission coefficient of a symmetric sandwich panel with an isotropic core are derived for the two-parameter foundation model. A comparison between the numerical predictions based on the two sandwich models and available experimental data shows that the consistent higher-order formulation can be used to predict the transmission loss in symmetric sandwich panels with both honeycomb and isotropic cores. For prediction of transmission loss of symmetric sandwich panels with an isotropic core, the two-parameter foundation model is more convenient, while the consistent higher-order approach is more accurate.     

Linear and nonlinear vibrations analysis of viscoelastic sandwich beams

In this paper, numerical models are proposed for linear and nonlinear vibrations analyses of viscoelastic sandwich beams with various viscoelastic frequency dependent laws using the finite element based solution. Real and various complex eigenmodes approaches are investigated as Galerkin bases. Based on harmonic balance method, simplified and general approaches are developed for nonlinear vibration analysis. Analytical frequency-amplitude and phase-amplitude relationships are elaborated based on the numerically computed complex eigenmodes. The equivalent loss factors and frequencies as well as the forced harmonic response and phase curves are performed for sandwich beams with various boundary conditions and frequency dependent viscoelastic laws.     

An efficient multi-domain spectral algorithm for the simulation of dispersive metallic structures in the time domain

In this article, we present a multi-domain formulation of the spectral time domain algorithm for the simulation of dispersive materials. We propose a leap-frog time-stepping scheme similar to the finite-difference time domain method in order to minimize memory usage. Dispersive material behavior is modelled in the frequency domain and used in our time-domain algorithm by introducing auxiliary differential equations for the macroscopic polarization. Absorbing boundary conditions are presented that can be used with dispersive materials. The numerical investigation of structures with material parameters fitted to experimental data shows excellent agreement with theoretical calculations.     

Finite-element modeling of lead magnesium niobate electrostrictive materials: dynamic analysis

A finite-element model is proposed for the time-domain analysis of electrostrictive materials. Hom's material model, developed for lead magnesium niobate (PMN) ceramics, is used. It includes the quadratic dependence of strain with polarization, the saturation of polarization, assumes constant temperature, and excludes hysteresis. The theoretical formulation is justified by the principle of virtual works. The numerical model is obtained after discretization in space and time. The validation is performed by comparing numerical results with semianalytical results for an electrostrictive spherical shell subjected to a step in voltage or in charge. From these results, a method to compute the coupling coefficient of electrostrictive materials, based on Ikeda's definition, is proposed and applied to a bar with parallel electric field.     

Investigation and modelling of damping in a plate with a bonded porous layer

Behavior of a poro-elastic material bonded onto a vibrating plate is investigated in the low-frequency range. From the analysis of dissipation mechanisms, a model accounting for damping added by the porous layer on the plate is derived. This analysis is based on a 3-D finite element formulation including poro-elastic elements based on Biot displacement theory. First, dissipated powers related to thermal, viscous and viscoelastic dissipation are explicited. Then a generic configuration (simply-supported aluminium plate with a bonded porous layer and mechanical excitation) is studied. Thermal dissipation is found negligible. Viscous dissipation can be optimized as a function of airflow resistivity. It can be the major phenomenon within soft materials, but for most foams viscoelastic dissipation is dominant. Consequently an equivalent plate model is proposed. It includes shear in the porous layer and only viscoelasticity of the skeleton. Excellent agreement is found with the full numerical model.     

Finite element analysis of the dynamic behavior of a laminated windscreen with frequency dependent viscoelastic core

This paper presents numerical and experimental validation of results obtained by a shell finite element, which has been developed for modeling of the dynamic behavior of sandwich multilayered structures with a viscoelastic core. The proposed shell finite element is very easy to implement in existing finite element solvers, since it uses only the displacements as degrees of freedom at external faces and at inter-layer interfaces. The displacement field is linearly interpolated in the thickness direction of each layer, and analytical integration is made in the thickness direction in order to avoid meshing of each sandwich layer by solid elements. Only the two dimensional mid-surface of reference is meshed, facilitating the mesh generation task. A simplified modal approach using a real modal basis is also proposed to efficiently calculate the dynamic response of the sandwich structure. The proposed method reduces the memory size and computing time and takes into account the frequency-dependence of the polymer core mechanical properties. Results obtained by the proposed element in conjunction with the simplified modal method have been numerically and experimentally validated by comparison to results obtained by commercial software codes (MSC/nastran and ESI/rayon-vtm), and to measurements done on automobile windscreens.     

Transient nearfield acoustic holography based on an interpolated time-domain equivalent source method

Transient nearfield acoustic holography based on an interpolated time-domain equivalent source method (ESM) is proposed to reconstruct transient acoustic fields directly in the time domain. Since the equivalent source strengths solved by the traditional time-domain ESM formulation cannot be used to reconstruct the pressure on the source surface directly, an interpolation function is introduced to develop an interpolated time-domain ESM formulation which permits one to deduce an iterative reconstruction process. As the reconstruction process is ill-conditioned and especially there exists a cumulative effect of errors, the Tikhonov regularization is used to stabilize the process. Numerical examples of reconstructing transient acoustic fields from a baffled planar piston, an impulsively accelerating sphere and a cube box, respectively, demonstrate that the proposed method not only can effectively reconstruct transient acoustic fields in the time domain, but also can visualize acoustic fields in the space domain. And, in the first numerical example, the cumulative effect of errors and the validity of using the Tikhonov regularization to suppress the errors are described.     

Viscoelastic analysis of multi-body systems using the finite element method

A study of the effect of viscoelastic material damping on the dynamic response of multibody systems, consisting of interconnected rigid, elastic and viscoelastic components, is presented. The motion of each elastic or viscoelastic body is identified by using three sets of modes: rigid body, reference and normal modes. Rigid body modes describe translation and large angular rotation of a body reference. Reference modes are the result of imposing the body-axis conditions. Normal modes define the deformation of the body relative to the body reference. Constraints between different components are formulated by using a set of non-linear algebraic equations that can be introduced to the dynamic formulation by using a Lagrange multiplier technique or can be utilized to eliminate dependent co-ordinates by partitioning the constraint Jacobian matrix. In developing the system equations of motion of the viscoelastic component, an assumption of a linear viscoelastic model is made. A Kelvin-Voigt model is employed, wherein the stress is assumed to be proportional to the strain and its time derivative. The formulation yields a constant damping matrix and the damping forces depend only on the local deformation; thus, no additional coupling between the reference and elastic co-ordinates appears in the formulation when considering the viscoelastic effects. It is demonstrated, by a numerical example, that the viscoelastic material damping can have a significant effect on the dynamic response of multibody systems.     

Dynamics of gas bubbles in viscoelastic fluids. II. Nonlinear viscoelasticity

The nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear viscoelastic media are examined. The constitutive equation [Upper-Convective Maxwell (UCM)] used for the fluid is suitable for study of large-amplitude excursions of the bubble, in contrast to the previous work of the authors which focused on the smaller amplitude oscillations within a linear viscoelastic fluid [J. S. Allen and R. A. Roy, J. Acoust. Soc. Am. 107, 3167-3178 (2000)]. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of viscoelastic constitutive equations into bubble dynamics equations. The numerical method used to solve the governing system of equations (one integrodifferential equation and two partial differential equations) is outlined. An energy balance relation is used to monitor the accuracy of the calculations and the formulation is compared with the previously developed linear viscoelastic model. Results are found to agree in the limit of small deformations; however, significant divergence for larger radial oscillations is noted. Furthermore, the inherent limitations of the linear viscoelastic approach are explored in light of the more complete nonlinear formulation. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that tissue viscoelasticity may be an important consideration for the risk assessment of potential cavitation bioeffects.     

FINITE ELEMENT FORMULATION AND ACTIVE VIBRATION CONTROL STUDY ON BEAMS USING SMART CONSTRAINED LAYER DAMPING (SCLD) TREATMENT

This work deals with the active vibration control of beams with smart constrained layer damping (SCLD) treatment. SCLD design consists of viscoelastic shear layer sandwiched between two layers of piezoelectric sensors and actuator. This composite SCLD when bonded to a vibrating structure acts as a smart treatment. The sensor piezoelectric layer measures the vibration response of the structure and a feedback controller is provided which regulates the axial deformation of the piezoelectric actuator (constraining layer), thereby providing adjustable and significant damping in the structure. The damping offered by SCLD treatment has two components, active action and passive action. The active action is transmitted from the piezoelectric actuator to the host structure through the viscoelastic layer. The passive action is through the shear deformation in the viscoelastic layer. The active action apart from providing direct active control also adjusts the passive action by regulating the shear deformation in the structure. The passive damping component of this design eliminates spillover, reduces power consumption, improves robustness and reliability of the system, and reduces vibration response at high-frequency ranges where active damping is difficult to implement. A beam finite element model has been developed based on Timoshenko's beam theory with partially covered SCLD. The Golla-Hughes-McTavish (GHM) method has been used to model the viscoelastic layer. The dissipation co-ordinates, defined using GHM approach, describe the frequency-dependent viscoelastic material properties. Models of PCLD and purely active systems could be obtained as a special case of SCLD. Using linear quadratic regulator (LQR) optimal control, the effects of the SCLD on vibration suppression performance and control effort requirements are investigated. The effects of the viscoelastic layer thickness and material properties on the vibration control performance are investigated.     

Time-domain sweeplets for acoustic measurements

This paper proposes new types of time-domain generated sine sweeps for impulse response measurements. A general time-domain analytical formulation method, combined with numeric phase alignment and frequency-domain inverse filtering is presented. It is applied to derive three new families of controllable spectrum sine sweeps called sweeplets, capable of matching 1/f^β background noises, producing finite band defocusing and single frequency focusing shapes. Mathematical properties concentrating on practical control of the signal shapes are examined. Effects of various perturbations, such as stationary and transient background noise, harmonic distortion and finite length are presented. Applicability of the proposed method is experimentally verified by a room acoustic measurement example.     

Time-domain simulation of acoustic wave propagation and interaction with flexible structures using Chebyshev collocation method

A time-domain Chebyshev collocation (ChC) method is used to simulate acoustic wave propagation and its interaction with flexible structures in ducts. The numerical formulation is described using a two-dimensional duct noise control system, which consists of an expansion chamber and a tensioned membrane covering the side-branch cavity. Full coupling between the acoustic wave and the structural vibration of the tensioned membrane is considered in the modelling. A systematic method of solution is developed for the discretized differential equations over multiple physical domains. The time-domain ChC model is tested against analytical solutions under two conditions: one with an initial state of wave motion; the other with a time-dependent acoustic source. Comparisons with the finite-difference time-domain (FDTD) method are also made. Results show that the time-domain ChC method is highly accurate and computationally efficient for the time-dependent solution of duct acoustic problems. For illustrative purposes, the time-domain ChC method is applied to investigate the acoustic performance of three typical duct noise control devices: the expansion chamber, the quarter wavelength resonator and the drum silencer. The time-dependent simulation of the sound-structure interaction in the drum silencer reveals the delicate role of the membrane mass and tension in its sound reflection capability.     

Time-domain calculations of sound interactions with outdoor ground surfaces

A time-domain formulation for sound propagation in rigid-frame porous media, including waveform attenuation and dispersion, is developed. The new formulation is based on inversion of the relaxation functions from a previous model [Wilson DK, Ostashev VE, Collier SL. J Acoust Soc Am 2004;116:1889-92], thereby casting the convolution integrals in a form amenable to numerical implementation. Numerical techniques are developed that accurately implement the relaxational equations and transparently reduce to previous results in low- and high-frequency limits. The techniques are demonstrated on calculations of outdoor sound propagation involving hills, barriers, and ground surfaces with various material properties. We also compare the relaxation formulation to a widely applied phenomenological model developed by Zwikker and Kosten. The two models can be made equivalent if the resistance constant, structure constant, and compression modulus in the ZK model are allowed to be weakly frequency dependent. But if the ZK parameters are taken to be constant, as is typically the case, the relaxation model provides more accurate calculations of attenuation by acoustically soft porous materials such as snow, gravel, and forest litter.     

Vibro-acoustic study of a viscoelastic sandwich ring immersed in water

This work deals with the reduction of the system composed of a sandwich structure with a viscoelastic core coupled to fluids. Two reduction methods are proposed in this paper to solve this problem in the frequency domain. The first one consists in developing added mass operators to take into account the fluid. The second one is the use of iterative methods to calculate the coupled complex modes of the dissipative problem. This numerical strategy is applied to the response of a bidimensional sandwich ring coupled to internal and external fluids.