Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Abstract

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The "locked pendulum" mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.

Global dynamics of an autoparametric spring–mass–pendulum system

“…In this regard, one-term harmonic solutions were assumed for the approximate solution, which is reasonable for small amplitude harmonic excitation. See [4] for a comprehensive comparison between harmonic balance solutions and those obtained by numerical integration. Therefore, the solutions are assumed in the following form:…”

“…In the nontrivial solution, both Y 3 and B are nonzero. In the frequency interval (region A) between the points denoted as "pitchfork bifurcations," it is known that the semi-trivial solution has larger amplitude of response compared to the response amplitude of the third mode in the nontrivial solution [3,4,11]. However, at larger mistunings (regions B) the amplitude of the semi-trivial solution is smaller than that in the nontrivial solution.…”

“…As the excitation frequency is changed, there arise changes in stability in steady-state solution branches either through a pitchfork bifurcation or through a saddlenode bifurcation (turning point). These results are mentioned only for completeness, and typical results for this class of systems, based on averaged equations, can be found in [4] as well as in [12]. For harmonic balance solutions, Floquet multipliers of the system linearized about the periodic orbit may be used to indicate stability.…”

“…It is noteworthy that this system has a strong dependence on its parameters R, ζ 2 , ζ 1 , β c , and ε. Many parametric studies have been performed and details are available in [4] and [11].…”

“…Examples of nonlinear modal coupling due to 1:2 internal resonance include pitch and roll motion in ships and surface waves in fluid containing cavities. Others [3][4][5][6][7] have studied these systems with a view to using the pendulum as a vibration absorber or neutralizer. To enhance the effectiveness of the absorber, a computer-controlled mass sliding over a collar on an oscillating rod was used in [8] as a tunable vibration absorber, and active vibration control laws have been also developed on the basis of this concept (e.g., see [9]).…”