Residue harmonie balance for discontinuous nonlinear oscillator with fractional power restoring force

Abstract

We solve the nonlinear ordinary differential equations which describe the motion of oscillators with discontinuous and/or fractional power restoring force by reducing the residue of harmonic balance sequentially. The new technique incorporates the salient features of both methods of harmonic balance and parameter bookkeeping. The residue is separated in two parts in each step, one conforms with the present order of approximation and the remaining part for use in the next approximation. The corrections are governed by a set of linear ordinary differential equations which can be easily solved and approximation of the 50th-order can be obtained without difficulty. Five numerical examples are given to illustrate the validity and efficiency. The approximations to the angular frequency and the limit cycle are obtained analytically. The results show that the approximations obtained are in excellent agreement with the exact solutions. For instance, the discrepancy of the 50th-order approximate frequency with respect to the exact one is low as 0.00002%. The new technique is simple in principle and can be applied to other strongly non-polynomial nonlinear oscillating systems.