The spreading residue harmonic balance study on the vibration frequencies of tapered beams

• Vibration frequencies of tapered beams are studied.
• A spreading residue harmonic balance is presented and employed.
• The presented second order approximations are more accurate than the existing results.
• The presented solutions yield a thoughtful and insightful understanding of the effect of system parameters.

A vast amount of work can be found in the field of beam vibration dealing with analytical and numerical techniques. The solution procedure of a spreading residue harmonic balance for approximating the periodic behavior of a tapered beam is presented in this paper. In this solution procedure, no small parameter is assumed, all former residual errors are introduced in the present approximation to improve the accuracy. The obtained results are compared with the other existing solutions and the exact ones which are shown in graphs. It is evident that the method considered here is quickly convergent and only the first-order analytical approximation leads to high accuracy of the solutions. In addition, the effect of system parameter and vibration amplitude on the nonlinear frequency is evaluated and compared, and excellent agreement between approximate frequencies and exact ones. It is predicted that the solution method can be found wide application in the other nonlinear oscillation problems.