Small strain vibration of a continuous, linearized viscoelastic rod of expanded polymer cushion material

In this paper, the free and forced vibration response of a linearized, distributed-parameter model of a viscoelastic rod with an applied tip-mass is investigated. A nonlinear model is developed from constitutive relations and is linearized about a static equilibrium position for analysis. A classical Maxwell-Weichert model, represented via a Prony series, is used to model the viscoelastic system. The exact solution to both the free and forced vibration problem is derived and used to study the behavior of an idealized packaging system containing Nova Chemicals׳ Arcel® foam. It is observed that, although three Prony series terms are deemed sufficient to fit the static test data, convergence of the dynamic response and study of the storage and loss modulii necessitate the use of additional Prony series terms. It is also shown that the model is able to predict the modal frequencies and the primary resonance response at low acceleration excitation, both with reasonable accuracy given the non-homogeneity and density variation observed in the specimens. Higher acceleration inputs result in softening nonlinear responses highlighting the need for a nonlinear elastic model that extends beyond the scope of this work. Solution analysis and experimental data indicate little material vibration energy dissipation close to the first modal frequency of the mass/rod system.