A Study On Direct Time Integration Methods For Vibration Analysis Of Viscoelastically Damped Systems

The noise radiated by low-frequency structural vibration is the main cause of the reduced ride comfort in various transportation systems.It is significant to reduce structural vibration by the energy-dissipation mechanisms to maintain human comfort in these systems.Modeling and analysis of viscoelastic damping are thus critical in solving structural vibration problems.The underlying physical mechanism of dissipation of energy in viscoelastic damping is quite complex because of its highly frequency-dependent property,which requires suitable kernel function for the constitutive damping model to accurately describe the complicated damping characteristics of vibrating systems.The increasing use of viscoelastic damping for modeling and analyzing the mechanical systems demands to calculate the system dynamic responses with high accuracy.The system responses are calculated accurately by the mode superposition method(MSM)using extended state-space eigensolutions.The procedures adopted by the MSM for dynamic response calculations of the system involving viscoelastic damping usually require a tremendous computational cost.The numerical methods are used to calculate the dynamic system responses with high accuracy.The integration method based on multiple damping models uses many anelastic variables,which greatly increases the size of the problem.These integration methods with larger step-sizes suffer greatly to reflect well enough the initial vibration of displacement-velocity and acceleration.Other problems arise in the time integration methods are inverse matrix calculation and dynamic loading approximations,which may introduce significant computational complexities at the higher degrees-offreedom(DOFs)especially for a large-scale dynamic system.To ease these problems,an alternative symmetric formulation based on new direct time integration(DTI)method is proposed for the analysis of the systems involving an anelastic displacement field(ADF)model.The integration method built on the symmetric scheme is developed using the assumptions of linear approximations.The computational efficiency of the proposed new method is higher than that of the MSM,particularly as the number of DOF increases,implying that new method is more suitable for the vibration analysis of large-scale damped structural systems.To obtain the precise results that approach to the exact solution,investigations of the time-step-size sensitivity of the method are important for the response calculations of dynamic systems.The piecewise linear interpolation scheme is considered as a better alternative to investigate the time-step-size sensitivity of the method instead of using the assumption of linear approximations.The error propagation in the displacement and velocity responses of the damped system with the increase of time-step-size are investigated by developing a precise direct integration method(PDIM)involving the piecewise linear interpolation scheme.It is found that the direct integration linear displacement–velocity(DIM–LDV)method is sensitive compared to the proposed PDIM as time-step-size is increased on the large-scale dynamic problems.To reduce the computational complexities of inverse matrix calculation and dynamic loading approximations for a large-scale viscoelastically damped system,the improvement in the PDIM method should also be suggested.Therefore,an improvement in the proposed PDIM is presented by using the Legendre-Gauss quadrature technique to investigate the stability,accuracy,and computational efficiency of the method for a large-scale dynamic system with high DOFs.To investigate the wave dispersion and dissipation performance of locally resonant acoustic metamaterials(AMMs)in terms of their band-structure and metadamping phenomenon,the analysis as well as the application of improved PDIM for acoustic wave propagation problems in AMM is indispensable.It is demonstrated that the viscoelastic damping improves the wave dispersion performance and the metadamping phenomenon enhances the wave dissipation performance of AMM.The proposed PDIM,based on Bloch theorem,eliminates the need to account additional computation of acceleration at each time-step,enabling it advantageous over the finite element time-domain method(FETDM)in the band-structure calculations.