Automated computation of autonomous spectral submanifolds for nonlinear modal analysis

Abstract

We discuss an automated computational methodology for computing two-dimensional spectral submanifolds (SSMs) in autonomous nonlinear mechanical systems of arbitrary degrees of freedom. In our algorithm, SSMs, the smoothest nonlinear continuations of modal subspaces of the linearized system, are constructed up to arbitrary orders of accuracy, using the parameterization method. An advantage of this approach is that the construction of the SSMs does not break down when the SSM folds over its underlying spectral subspace. A further advantage is an automated a posteriori error estimation feature that enables a systematic increase in the orders of the SSM computation until the required accuracy is reached. We find that the present algorithm provides a major speed-up, relative to numerical continuation methods, in the computation of backbone curves, especially in higher-dimensional problems. We illustrate the accuracy and speed of the automated SSM algorithm on lower-and higher-dimensional mechanical systems.

How do conservative backbone curves perturb into forced responses? A Melnikov function analysis

“…i are reported in Table 1. Since no resonance holds among the latter, the system features six families of periodic orbits emanating from the origin by the Lyapunov subcenter manifold theorem [25]. Using numerical continuation starting from small-amplitude linearised periodic motions, we compute the conservative backbone curve for each mode a shown in Figure 4a.…”

Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems

“…Applications of this model reduction approach appear in Jain et al [10] and Szalai et al [27]. Ponsioen et al [22] provide an automated computation package for two-dimensional SSMs of a general autonomous, nonlinear mechanical system.…”

Explicit third-order model reduction formulas for general nonlinear mechanical systems

“…For conservative systems, this third-order model reduction is performed onto Lyapunov Subcenter Manifolds (LSMs) [1], whereas for damped-forced systems, we perform the reduction onto Spectral Submanifolds (SSMs) [2], which are mathematically rigorous versions of the nonlinear normal modes proposed first by Shaw and Pierre [3]. To obtain higher-order LSM-and SSM-reduced models, one may use the automated numerical reduction procedure developed in [4].…”