An Analysis of Nonlinear Beam Vibrations with the Extended ‎Rayleigh-Ritz Method

The nonlinear deformation and vibrations of beams are frequently encountered as a typical example of structural analysis as well as a mathematical problem. There have been many methods and techniques for the approximate and exact solutions of nonlinear differential equations arising from the nonlinear phenomena of elastic beam structures. One method is particularly more powerful and flexible is proposed recently as the extended Rayleigh-Ritz method (ERRM) by adding the temporal variable as another dimension of deformation formulation but eliminated through the integration over a period of vibrations. Such a procedure leads to a simple, elegant, and powerful method for the approximate solutions of nonlinear vibration and deformation problems in dynamics and structural analysis. By utilizing the usual displacement function of beams, the nonlinear vibration frequencies of Euler-Bernoulli and Timoshenko beams are obtained with the same accuracy as from other approximate solutions.