Nonlinear Dynamics 34: 319–346, 2003. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. Identification of Nonlinear and Viscoelastic Properties of Flexible Polyurethane Foam R. SINGH, P. DAVIES, and A. K. BAJAJ Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, 140 S Intramural Drive, West Lafayette, IN 47907-2031, U.S.A. (Received: 6 August 2001; accepted: 28 May 2003) Abstract. Analysis of the steady-state response of a polyurethane foam and mass system to harmonic excitation is presented. The foam’s uni-directional dynamic behavior is modeled by using nonlinear stiffness, linear viscoelastic and velocity proportional damping components. The relaxation kernel for the viscoelastic model is assumed to be a sum of exponentials. The harmonic balance method is used to develop one- and two-term approximations to periodic solutions, and the equations developed are utilized for system identification. The identification process is based on least-squares minimization of a sub-optimal cost function that uses response data at various excitation frequencies and amplitudes. The effects of frequency range, spacing and amplitudes of the harmonic input on the results of the model parameter estimation are discussed. The identification procedure is applied to measurements of the steady-state response of a base-excited foam-mass system. Estimates of the system parameters at different levels of compression and input amplitudes are thus determined. The choice of model-order and the feasibility of describing the system behavior at several input amplitudes with a single set of parameters are also addressed. Keywords: Viscoelasticity, harmonic balance, polyurethane foam, nonlinear system identification. Nomenclature k = linear stiffness coefficient, N/m k 3 = cubic stiffness coefficient, N/m 3 k 5 = fifth power stiffness coefficient, N/m 5 c = linear velocity proportional damping coefficient, Ns/m α i = exponent associated with viscoelastic relaxation kernel, 1/s a i = coefficient associated with viscoelastic relaxation kernel, 1/s G = amplitude of input forcing, N z = amplitude of foam compression, m x = absolute displacement of the foam block, m Ŵ = viscoelastic relaxation kernel  E = suboptimal cost function 1. Introduction Polyurethane foam is a complex engineering material. As illustrated in Figure 1 its quasi-static behavior is significantly different from dynamic behavior [1–3] and it is highly nonlinear. The material properties of foam are also dependent upon temperature and humidity [4]. Vis- coelasticity is an important aspect of foam behavior. A material is called viscoelastic if the present state of response depends not only on the present state of loading, but also on the previous states [5, 6]. Commonly observed manifestations of this behavior are material creep,