Estimation of the transient response of a tuned, fractionally damped elastomeric isolator

This article addresses the frequency dependent properties of elastomeric vibration isolators in the context of lumped parameter models with fractional damping elements. A mass is placed between two fractional calculus Kelvin-Voigt elements to develop a minimal order system for the example case of a conventional elastomeric bushing typical of automotive suspension systems. Model parameters are acquired from measured dynamic stiffness spectra and a finite element model. The minimal order system model accurately predicts dynamic stiffness in both broadband resonant behavior as well as the lower-frequency regime that is controlled by damping. For transient response analysis, an inverse Laplace transform of the dynamic stiffness spectrum is taken via the Residue Theorem. Since the fractional calculus based solution is given in terms of problematic integrals, a new time-frequency domain estimation technique is proposed which approximates time-domain responses for a class of transient excitation functions. The approximation error is quantified and found to be reasonably small, and tractable closed-form transient response functions are provided along with a discussion of numerical issues.