Nonlinear natural vibration analysis of beams by selective coefficient increment

Abstract

The amplitude-frequency relation (backbone curve) of an elastic body in large amplitude natural vibration at resonance (including internal resonance) is of interest. The backbone curve is constructed by amplitude increment (decrement). Since the amplitude is given as a combination of harmonic components, the increment is performed actually on the coefficients of the harmonic components. An assumed increment is first applied to the first harmonic coefficient (the active harmonic) and a new equilibrium state is found by the Newtonian algorithm. The frequency increment is computed, all the harmonic coefficients are adjusted and the amplitude is evaluated. During the process, it is possible that one of the originally small harmonic coefficients suddenly increases in magnitude. The harmonic coefficient is selected to be the new active harmonic in the next step to find a new equilibrium state. The internal resonance of order 5 combining the first and the third linear modes of a clamped-clamped beam is predicted without difficulty. Therefore, automatic computation of combined resonance is straight forward. Matrix notation is used when possible to given concise presentation. The method is readily applicable in finite element sense.