Large amplitude free vibrations of Timoshenko beams at higher modes using coupled displacement field method

A simple but accurate continuum solution for the shear flexible beam problem using the energy method involves in assuming suitable single term admissible functions for the lateral displacement and total rotation. This leads to two non-linear temporal differential equations in terms of the lateral displacement and the total rotation and are difficult, if not impossible, to solve to obtain the large amplitude fundamental frequencies of beams as a function of the amplitude and slenderness ratios of the vibrating beam. This situation can be avoided if one uses the concept of coupled displacement field where in the fields for lateral displacement and the total rotation are coupled through the static equilibrium equation. In this paper the lateral displacement field is assumed and the field for the total rotation is evaluated through the coupling equation. This approach leads to only one undetermined coefficient which can easily be used in the principle of conservation of total energy of the vibrating beam at a given time, neglecting damping. Finally, through a number of algebraic manipulations, one gets a nonlinear equation of Duffing type which can be solved using any standard method. To demonstrate the simplicity of the method discussed above the problem of large amplitude free vibrations of a uniform shear flexible hinged beam at higher modes with ends immovable to move axially has been solved. The numerical results obtained from the present formulation are in very good agreement with those obtained through finite element and other continuum methods for the fundamental mode, thus demonstrating the efficacy of the proposed method. Also some interesting observations are made with variation of frequency Vs amplitude at different modes.