The influence of phase-locking on internal resonance from a nonlinear normal mode perspective

Abstract

a b s t r a c t When a nonlinear system is expressed in terms of the modes of the equivalent linear system, the nonlinearity often leads to modal coupling terms between the linear modes. In this paper it is shown that, for a system to exhibit an internal resonance between modes, a particular type of nonlinear coupling term is required. Such terms impose a phase condition between linear modes, and hence are denoted phase-locking terms. The effect of additional modes that are not coupled via phase-locking terms is then investigated by considering the backbone curves of the system. Using the example of a twomode model of a taut horizontal cable, the backbone curves are derived for both the case where phase-locked coupling terms exist, and where there are no phase-locked coupling terms. Following this, an analytical method for determining stability is used to show that phase-locking terms are required for internal resonance to occur. Finally, the effect of nonphase-locked modes is investigated and it is shown that they lead to a stiffening of the system. Using the cable example, a physical interpretation of this is provided.

Identifying the significance of nonlinear normal modes

“…Specifically, our hypothesis is that the phase-unlocked branches may be neglected when the NNMs are used to interpret the forced responses of a system. The stability analysis presented in [29] exploits the special properties exhibited by the system due to its symmetry and hence cannot readily be generalized to asymmetric systems, such as the beam considered here. As such, a different approach is required to investigate the significance of the phase-unlocked NNMs.…”

“…As such, if an internally resonant forced response is close to an NNM, that NNM must be able to transfer energy between the modes. This explains why no internally resonant responses were seen to follow the phase-unlocked backbone curves in [29], as no energy can be transferred via the fundamental components of the response.…”

“…Figure 8 also shows that the peaks do not envelop the NNM branches precisely—they appear to be shifted to a higher frequency. This is due to the relatively high amplitude reached by the second mode away from resonance, which leads to a stiffening effect in the system, as described in [29,37]. …”

“…However, in [29]—which explores the dynamics of a taut, horizontal cable—it was observed that, for the forcing and damping cases that are considered, not all sets of NNMs relate to forced responses. This suggests that not all sets of NNMs (also referred to as NNM branches or backbone curves ) are of equal significance when considering the forced responses.…”

The Significance of Nonlinear Normal Modes for Forced Responses

“…This allows the inclusion of motions that are not in-unison, such as out-of-unison motion [11], but which exhibit the useful same properties as NNMs. However, it has been observed that not all NNMs appear to relate to the forced responses [12]. As such, the objective of this paper is to investigate the relationship between the forced responses and NNMs, in order to understand why some NNMs appear to have little influence on the forced response.…”