Nonlinear Dynamics of a Beam on an Elastomeric Foundation: Coupled Effects of Inertia, Shear, and Nonlinear Interactions

This study presents a nonlinear dynamic analysis of a beam resting on a viscoelastic elastomeric foundation, incorporating shear deformations, viscoelastic damping, foundation inertia, and nonlinear strains to capture the full elasto-dynamic interaction. Conventional models that prioritize stiffness and damping are contrasted with the present study's comprehensive framework, which considers the material-dependent response and the influence of excitation amplitude on stability and bifurcation. The governing equations are solved in the frequency domain using a continuation-based method, revealing a strong dependence of the system behavior on both the material properties of the foundation and the amplitude of the external excitation. For low-amplitude base inputs, the system manifests approximately linear behavior, with resonance peaks following classical vibrational characteristics. However, as the excitation amplitude increases, the response deviates significantly due to nonlinear interactions, leading to complex phenomena such as period-doubling bifurcations, Neimark-Sacker bifurcations, branch-point cycles, and limit-point cycles. The veracity of the frequency response findings, in certain instances, is substantiated by time domain response, FFT analysis, phase portrait trajectories, and Poincaré section maps. The findings demonstrate that, contingent on material characteristics, system response can vary not only quantitatively but also qualitatively due to substantial nonlinearities, resulting in bifurcation and instability within specific frequency ranges. These findings underscore the imperative for judicious material selection and meticulous control of excitation amplitude to avert undesirable dynamic behavior. The knowledge gained from this work is crucial for the design of wearable sensors, MEMS devices, and vibration isolation systems to ensure stable and predictable performance under dynamic loading conditions.