A modified incremental harmonic balance method for rotary periodic motions

Abstract

The determination of periodic solutions is an essential step in the study of dynamic systems. If some of the generalized coordinates describing the configuration of a system are angular positions relative to certain reference axes, the associated periodic motions divide into two types: oscillatory and rotary periodic motions. For an oscillatory periodic motion, all the generalized coordinates are periodic in time. On the other hand, for a rotary periodic motion, some angular coordinates may have unbounded magnitude due to the persistent circulation about their pivots. In this case, although the behaviour of the system is periodic physically, those angular coordinates are not periodic in time. Although various effective methods have been developed for the determination of oscillatory periodic motion, the rotary periodic motion can only be determined by brute force integration. In this paper, the incremental harmonic balance (IHB) method is modified so that rotary periodic motions can be determined as well as oscillatory periodic motions in a unified formulation. This modified IHB method is applied to a practical device, a rotating disk equipped with a balltype balancer, to show its effectiveness.

Pure-rotary periodic motions of a planar two-ball auto-balancer system

“…In this case, the angular coordinate of the circulating ball is non-periodic and hence cannot be expanded as a Fourier series. Lu and Lin [33] developed a modified IHB method that can remedy this problem. In the modified IHB method, the generalized coordinates are expanded into modified Fourier series as…”

“…However, only with proper modifications can this method be used to determine rotary periodic motions. In this paper, we used the modified IHB method developed in [33] to determine the periodic motions of the auto-balancer system. This paper aims to study the periodic motions of a planar two-ball auto-balancer system both numerically and experimentally.…”

Internal resonance vibration induced by nonlinearity of primary suspension system in high-speed vehicle system

“…Chen et al [4] studied the nonlinear vibration of plane structures by introducing finite element in the IHB method. Lu and Lin [13] modified IHB method so that periodic motions of rotating disk can be determined as well as oscillatory periodic motions in a unified formulation. Pun et al [16,17] analyzed the free and forced vibration behavior of an L-shaped beam with a limit stop.…”

A complementarity approach for the computation of periodic oscillations in piecewise linear systems

“…In the frequency domain, one of the most reliable methodologies is the harmonic balance. In [16], an extension of the harmonic balance method for computing the rotary and oscillatory periodic motion of a nonlinear smooth system is proposed. Mixed timefrequency-domain approaches are presented in [17,18].…”