On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems

Abstract

Numerical computation of Nonlinear Normal Modes (NNMs) for conservative vibratory systems is addressed, with the aim of deriving accurate reduced-order models up to large amplitudes. A numerical method is developed, based on the center manifold approach for NNMs, which uses an interpretation of the equations as a transport problem, coupled to a periodicity condition for ensuring manifold's continuity. Systematic comparisons are drawn with other numerical methods, and especially with continuation of periodic orbits, taken as reference solutions. Three different mechanical systems, displaying peculiar characteristics allowing for a general view of the performance of the methods for vibratory systems, are selected. Numerical results show that invariant manifolds encounter folding points at large amplitude, generically (but not only) due to internal resonances. These folding points involve an intrinsic limitation to reduced-order models based on the center manifold and on the idea of a functional relationship between slave and master coordinates. Below that amplitude limit, numerical methods are able to produce reduced-order models allowing for a precise prediction of the backbone curve.

Numerical Computation of Nonlinear Normal Modes of Nonconservative Systems

“…The 2N-2 equations are quasilinear first-order hyperbolic partial differential equations that are similar to flow equations encountered in fluid dynamics. This interpretation of the PDEs is identical to the interpretation made by Touzé and its co-workers [11] who interpreted the PDEs as a transport problem. The vector V can therefore be interpreted as the velocity vector of the master coordinates flow.…”

“…The manifold-governing PDEs are solved in modal space using a Galerkin projection with the NNM motion parametrized by amplitude and phase variables. In a recent contribution, Touzé and co-workers [11] also tackled the PDEs in modal space. They show that these PDEs can be interpreted as a transport equation, which, in turn, can be discretized using finite differences.…”

Lyapunov Equivalent Representation Form of Forced, Damped, Nonlinear, Two Degree-of-Freedom Systems

“…In an attempt to avoid this situation, the modal system (4) was next replaced by a truncated fifth-order power series expansion that contained nonlinear damping terms [13,38,[51][52][53]. As usual, driving forces were assumed to remain constant during the transformation approach.…”

“…First, we focused on studying the unforced dynamic system (30) and plotted the amplitude-time curves by using the equivalent cubic, and quintic expressions: (29), (53) Figure 16 shows the time-amplitude curves for the two modes of the system. Notice that as a consequence of the fifth-order and nonlinear decay terms of the quintic approach, the equivalent representation forms (53) and (54) provide an improvement in the RMSE values. Next, the driving force magnitude of f 1 = 0.25 was considered and then, the frequency-amplitude response, as well as the LLE curves were plotted.…”

“…An improvement was achieved in the frequency-amplitude response curves obtained from the quintic approach since the estimated curves match the numerical results well; however, one can notice from Figure 18, that when the quintic approach is used to compute the LLE values, the RMSE value is slightly higher than that computed by using the cubic model, which is mainly due to the numerical procedure used to compute these values rather than to the decoupling proposed approach. Here, the black, solid lines describe the numerical integration solutions of Equation (30), while the blue, dashed lines represent the numerical solutions obtained from the nonlinear equations of motion, (53) and (54). (29), (30), (53) and (54).…”