Nonlinear Natural Frequencies and Primary Resonance of Euler-Bernoulli Beam with Initial Deflection using Nonlocal Elasticity Theory

摘要

In the present work, we study the primary responses of Euler-Bernoulli beam with initial imperfection/rise. The nonlocal elasticity theory was used to derive the mathematical model to account for the scale effect of the considered beam. One type of beams was considered in the analysis; simply supported beam. The multi-mode approach was used to obtain the reduced nonlinear temporal equations of motion that contain quadratic and cubic nonlinear terms. The method of multiple-scales was applied to obtain approximate analytical solutions for the nonlinear natural frequencies in addition to the primary and resonance responses. The obtained results were presented over a selected range of physical parameters for the two types of beams such as; beam initial rise/imperfection, scale effect parameter and excitation level.