Size-dependent nonlinear vibration analysis of nanobeam embedded in multi-layer elastic media and subjected to electromechanical and thermomagnetic loadings

Abstract In this work, magneto-electro-mechanical size-dependent nonlinear vibration analysis of nanobeam embedded in multi-layer of Winkler, Pasternak, quadratic and cubic nonlinear elastic media is presented. A nonlinear partial differential equation of motion is derived using Von Karman geometric nonlinearity, nonlocal elasticity theory, Euler-Bernoulli beam theory and Hamilton’s principle. Additionally, the efficiency of multiple scales Lindstedt-Poincare method for the strong nonlinear and large amplitude systems is presented. It is established that the results of multiple scales Lindstedt-Poincare method are in good agreements with the numerical and exact solutions for the strong nonlinear problems. However, the classical multiple scales method fails and gives results with very large discrepancies form the results of the numerical and exact solutions when the perturbation parameter is large, and the nonlinearity terms are strong. The high accuracy of the results of multiple scales Lindstedt-Poincare method and its excellent ability to produce accurate results for all values (small and large) of perturbation parameter and the nonlinearity terms show the superiority of the multiple scales Lindstedt-Poincare method over the classical multiple scales method. Further results present the effects of the model parameters on the dynamic behaviour of the nanobeam. It is hoped that the present study will advance nonlinear analysis of the engineering structures.