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We apply two recently formulated mathematical techniques, Slow-FastDecomposition (SFD) and Spectral Submanifold (SSM) reduction, to a von Karmanbeam with geometric nonlinearities and viscoelastic damping. SFD identifies aglobal slow manifold in the full system which attracts solutions at ratesfaster than typical rates within the manifold. An SSM, the smoothest nonlinearcontinuation of a linear modal subspace, is then used to further reduce thebeam equations within the slow manifold. This two-stage, mathematically exactprocedure results in a drastic reduction of the finite-element beam model to aone-degree-of freedom nonlinear oscillator. We also introduce the technique ofspectral quotient analysis, which gives the number of modes relevant forreduction as output rather than input to the reduction process.

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