Exact nonlinear model reduction for a von Kármán beam: Slow-fast decomposition and spectral submanifolds

Abstract

We apply two recently formulated mathematical techniques, Slow-Fast Decomposition (SFD) and Spectral Submanifold (SSM) reduction, to a von Kármán beam with geometric nonlinearities and viscoelastic damping. SFD identifies a global slow manifold in the full system which attracts solutions at rates faster than typical rates within the manifold. An SSM, the smoothest nonlinear continuation of a linear modal subspace, is then used to further reduce the beam equations within the slow manifold. This two-stage, mathematically exact procedure results in a drastic reduction of the finite-element beam model to a one-degree-of freedom nonlinear oscillator. We also introduce the technique of spectral quotient analysis, which gives the number of modes relevant for reduction as output rather than input to the reduction process.

Using spectral submanifolds for optimal mode selection in nonlinear model reduction

“…References [6,7] propose a selection of modes supported by the physical understanding that a subset of axial modes should be included in the projection basis to account for the nonlinear bending-stretching coupling. Indeed, this reduction happens to result in an exact model reduction due to the presence of a slow manifold in this example, as shown in [9]. However, such physical intuition of selecting relevant axial modes is already unavailable upon a simple change in the geometry of the structure such as making the beam initially curved.…”

“…In practice, the accuracy of such a reduction procedure is dependent on an ad hoc choice of modes and hence needs to be verified on a case-by-case basis. A relevant example is an initiallystraight, nonlinear von Kármán beam [6][7][8][9], where the axial and transverse degrees-of-freedoms are coupled only by the nonlinearities. References [6,7] propose a selection of modes supported by the physical understanding that a subset of axial modes should be included in the projection basis to account for the nonlinear bending-stretching coupling.…”

Hyper-Reduction Over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

“…This is especially true for the case of thin walled structures, which exhibit a time scale separation between slow, out-of-plane; and fast, in-plane dynamics. The fast dynamics in such systems can often be enslaved statically to the slow one, in a nonlinear fashion [8]. Recent efforts [11,12] have shown that the geometrically nonlinear dynamics of structures, characterized by bending-stretching coupling (as for instance, thin walled components) can be effectively captured using a quadratic manifold (QM), constructed using model properties rather then HFM snapshots.…”

Analytic prediction of isolated forced response curves from spectral submanifolds

“…The reduced dynamics on a two-dimensional SSM serves as an exact, single-degree-offreedom reduced-order model that can be constructed for each vibration mode of the full nonlinear system (cf. [8][9][10][11][12]).…”