Nonlinear Normal Modes and Their Bifurcations for an Inertially Coupled Nonlinear Conservative System

Abstract

This work concerns the nonlinear normal modes (NNMs) of a 2 degree-of-freedom autonomous conservative springmass-pendulum system, a system that exhibits inertial coupling between the two generalized coordinates and quadratic (even) nonlinearities. Several general methods introduced in the literature to calculate the NNMs of conservative systems are reviewed, and then applied to the spring-mass-pendulum system. These include the invariant manifold method, the multiple scales method, the asymptotic perturbation method and the method of harmonic balance. Then, an efficient numerical methodology is developed to calculate the exact NNMs, and this method is further used to analyze and follow the bifurcations of the NNMs as a function of linear frequency ratio p and total energy h. The bifurcations in NNMs, when near 1:2 and 1:1 resonances arise in the two linear modes, is investigated by perturbation techniques and the results are compared with those predicted by the exact numerical solutions. By using the method of multiple time scales (MTS), not only the bifurcation diagrams but also the low energy global dynamics of the system is obtained. The numerical method gives reliable results for the high-energy case. These bifurcation analyses provide a significant glimpse into the complex dynamics of the system. It is shown that when the total energy is sufficiently high, varying p, the ratio of the spring and the pendulum linear frequencies, results in the system undergoing an order-chaos-order sequence. This phenomenon is also presented and discussed.

Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure

“…6 are the scaled variables. This bifurcation diagram is typical of multi degree of freedom systems with inertial nonlinearities near 1:2 internal resonance in two interacting modes [31]. It shows that there exist two non-trivial stable nonlinear normal modes for parameter values in the neighborhood of the internal resonance, and these arise as a result of a pitchfork bifurcation from the semitrivial nonlinear normal mode NNM1.…”

“…This is also a stable equilibrium. Since these two equilibria correspond to ϕ = 0 or ϕ = π, y 1 and y 2 vibrate in unison and we call these two motions NNM motions [31]. Representing a 1 (the amplitude of the lower frequency mode) as a function of the tip mass perturbation m , we get the bifurcation diagram shown in Fig.…”

“…For the same system (25), we now use the shooting method [31] to construct the NNMs in configuration space and the associated bifurcation diagrams.…”

“…This allows us to obtain reduced-order models that retain parametric dependence to the maximum possible extent. The nonlinear normal modes for the high fidelity multi-mode models can then be studied both by the method of multiple time scales, and by more global numerical techniques (Wang et al [31], Zhang [32,33]). In the present work, we use the shooting technique of Wang et al [31] that allows for large-amplitude nonlinear normal modes to be discovered.…”

“…The nonlinear normal modes for the high fidelity multi-mode models can then be studied both by the method of multiple time scales, and by more global numerical techniques (Wang et al [31], Zhang [32,33]). In the present work, we use the shooting technique of Wang et al [31] that allows for large-amplitude nonlinear normal modes to be discovered. Thus, bifurcations in nonlinear normal modes as a function of the amplitudes of motion (or total system energy), and variation in frequencies through internal resonance conditions can be investigated.…”