Kelvin-Voigt lumped parameter models for approximation of the Power-law Euler-Bernoulli beams

The purpose of this research is to initiate an investigation of the nonlinear material strain-rate damping effects on the amplitude and frequencies of some Euler-Bernoulli Beams. It is well known that the dynamic behaviors of most heat-treated metals can be modelled by using the power-law strain-rate dependent constitutive equations. Lumped parameter models for approximations of dynamic vibration of the power-law Euler-Bernoulli Beam subject to power-law strain-rate damping and concentrated loads are formulated. Analytic formulas of the lumped parameters, including effective the train-rate damping, are derived. The lumped-parameters are also evaluated numerically by a low-order Galerkin Method to validate and compare the lumped parameter model with another numerical model. Numerical examples made of some heat-treated aluminum and stainless-steel alloys are presented to illustrate the implications of the aforementioned lumped-parameter models on the dynamics of the beams. The results obtained in this work cover with the classical results in the literature for linear the materials as special cases. The novel lumped parameter models can provide useful insights for crashworthiness analysis of structures of heat treated metals and thermal plastics.