Free and Forced Vibration Analysis of Non-Local Euler-Bernoulli Beam Resting on Nonlinear Foundation

In this study, the primary response of an Euler-Bernoulli beam resting on nonlinear elastic foundation is investigated. The beam is subjected to thermal and magnetic axial loads. The nonlocal Eringen’s elasticity theory is used to derive the mathematical model to account for the scale effect of the beam. A simply supported beam is considered in the analysis, and the multi-mode approach is used to obtain the reduced nonlinear temporal equations of motion that contain quadratic and cubic nonlinear terms. The method of multiple-scales is applied to obtain approximate analytical solutions for the nonlinear natural frequencies in addition to the primary resonance response curves. Moreover, the effective nonlinearity is obtained as a function of the natural frequencies and the coefficients of the elastic foundation. The results reveal that the scale parameter has a significant effect on the frequencies and amplitudes of the beam. The obtained results are presented over a selected range of physical parameters such as the scale effect parameter, foundation parameters, thermal and magnetic loads, and the excitation level. Time responses, phase planes and Poincaré maps are generated for the beam under consideration.