An Investigation on the Nonlinear Free Vibration Analysis of Beams with Simply Supported Boundary Conditions Using Four Engineering Theories

Abstract

The objective of this study is to present a brief survey on the geometrically nonlinear free vibrations of the Bernoulli-Euler, the Rayleigh, shear, and the Timoshenko beams with simple end conditions using the Homotopy Analysis Method HAM . Expressions for the natural frequencies, the transverse deflection, postbuckling load-deflection relation to, and critical buckling load are presented. The results of nonlinear analysis are validated with the published results, and excellent agreement is observed. The effects of some parameters, such as slender ratio, the rotary inertia, and the shear deformation, are examined as other parameters are fixed.

Natural Frequency Analysis of Two Nonlinear Panels Coupled with a Cavity Using the Approximate Elliptic Integral Solution and the Method of Harmonic Residual Minimization

“…Some studies (e.g., [16][17][18][19][20][21][22][23][24]) are related to the nonlinear structural vibrations, but their focuses are different from that in this paper. In fact, there are numerous theories used for solving various nonlinear vibration problems (e.g., [25][26][27][28][29][30][31][32][33][34][35]). In some approaches for solving the problems of large amplitude structure or nonlinear oscillation (e.g., [36][37][38][39]), they require a significant effort in the tedious eigenvalue solution steps.…”