# Exact Nonlinear Model Reduction for a von Karman beam: Slow-Fast Decomposition and Spectral Submanifolds

## TL;DR

This paper presents a mathematically exact reduction method for a nonlinear von Karman beam using Slow-Fast Decomposition and Spectral Submanifolds, significantly simplifying complex models to a single nonlinear oscillator.

## Contribution

It introduces a novel two-stage reduction combining SFD and SSM techniques, including spectral quotient analysis, for precise model reduction of nonlinear beams.

## Key findings

- Reduction from finite-element model to one-degree-of-freedom oscillator
- Global slow manifold identified that attracts solutions rapidly
- Spectral quotient analysis determines relevant modes for reduction

## Abstract

We apply two recently formulated mathematical techniques, Slow-Fast Decomposition (SFD) and Spectral Submanifold (SSM) reduction, to a von Karman 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.