Finite Strain-Based Theory for the Superharmonic and Subharmonic Resonance of Beams Resting on a Nonlinear Viscoelastic Foundation in Thermal Conditions, and Subjected to a Moving Mass Loading

We address the nonlinear free vibration, superharmonic and subharmonic resonance response of homogeneous Euler–Bernoulli beams resting on nonlinear viscoelastic foundations, under a moving mass and an abrupt uniform temperature rise. The nonlinear differential equation of motion stemming from the Hamiltonian principle and Finite Strain Theory is discretized according to a Galerkin decomposition method, and is solved by means of a multiple time scale method. A comparison between the Finite Strain theory and the Von-Karman approach is discussed, accounting for the effect of temperature rise, linear and nonlinear coefficients of the elastic foundation on the nonlinear vibration history and phase trajectory. At the same time, we check for the sensitivity of the frequency response of the system in superharmonic and subharmonic resonance for different input parameters, namely, location, velocity, and magnitude of the moving load, temperature rise and elastic foundation.