Explicit Backbone Curves from Spectral Submanifolds of Forced-Damped Nonlinear Mechanical Systems

Abstract

Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here we extend these results to periodically or quasiperiodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two-dimensional) time dependent SSMs. A judicious choice of the parameterization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal form methods.

References in corpus (3)

• Automated Computation of Autonomous Spectral Submanifolds for Nonlinear Modal Analysis
• Exact Nonlinear Model Reduction for a von Karman beam: Slow-Fast Decomposition and Spectral Submanifolds
• Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations

Cited by in corpus (13)

• Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds
• How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models
• Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures
• Data-driven Nonlinear Model Reduction to Spectral Submanifolds in Mechanical Systems
• Nonlinear Model Reduction to Fractional and Mixed-Mode Spectral Submanifolds
• How do Conservative Backbone Curves Perturb into Forced Responses? A Melnikov Function Analysis
• Fast Computation of Steady-State Response for Nonlinear Vibrations of High-Degree-of-Freedom Systems
• Using Spectral Submanifolds for Optimal Mode Selection in Model Reduction
• Fast computation and characterization of forced response surfaces via spectral submanifolds and parameter continuation
• Data-free Non-intrusive Model Reduction for Nonlinear Finite Element Models via Spectral Submanifolds
• Stability of Forced-Damped Response in Mechanical Systems from a Melnikov Analysis
• A sub-structuring approach for model reduction of frictionally clamped thin-walled structures
• Model reduction of nonlinear time-delay systems via ODE approximation and spectral submanifolds