Model reduction for systems with random parameters using Spectral Submanifolds

We present a method for studying the nonlinear dynamics of systems with random parameters. We use the theory of Spectral Submanifolds (SSMs) to perform model reduction. This enables reducing the nonlinear dynamics of high-dimensional models to low-dimensional invariant manifolds. To tackle the randomness of system parameters, we compute Polynomial Chaos Expansions (PCEs) of SSMs in a purely equation-driven approach. The resulting Parametric SSMs (PSSMs) can be systematically computed for arbitrary expansion orders. This capability, besides the optimality of PCEs in probabilistic settings, enables us to study systems with moderately large parameter perturbations. For mechanical systems, we use PSSMs to obtain parametric backbone curves and frequency response curves. Further, to perform uncertainty quantification, we derive closed-form expressions for convergent statistical moments of backbone curves without the need for any simulations. We illustrate the method with examples that include a slender beam subject to random manufacturing imperfections.