Exact Nonlinear Model Reduction for a von Karman 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 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.

References in corpus (3)

• A Quadratic Manifold for Model Order Reduction of Nonlinear Structural Dynamics
• Generalization of Quadratic Manifolds for Reduced Order Modeling of Nonlinear Structural Dynamics
• Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations

Cited by in corpus (19)

• 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
• Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part I: Periodic response and forced response curve
• Machine Learning Approach to Model Order Reduction of Nonlinear Systems via Autoencoder and LSTM Networks
• Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response
• Explicit Backbone Curves from Spectral Submanifolds of Forced-Damped Nonlinear Mechanical Systems
• Data-driven Nonlinear Model Reduction to Spectral Submanifolds in Mechanical Systems
• Model reduction for constrained mechanical systems via spectral submanifolds
• Nonlinear Model Reduction to Fractional and Mixed-Mode Spectral Submanifolds
• An enhanced parametric nonlinear reduced order model for imperfect structures using Neumann expansion
• Exact Model Reduction for Damped-Forced Nonlinear Beams: An Infinite-Dimensional Analysis
• Hyper-reduction over nonlinear manifolds for large nonlinear mechanical systems
• Using Spectral Submanifolds for Optimal Mode Selection in Model Reduction
• Data-Assisted Non-Intrusive Model Reduction for Forced Nonlinear Finite Elements Models
• Data-driven modeling of the regular and chaotic dynamics of an inverted flag from experiments
• Active vibration control of nonlinear flexible structures via reduction on spectral submanifolds
• Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques
• Integral Equations & Model Reduction For Fast Computation of Nonlinear Periodic Response
• Exact Model Reduction and Fast Forced Response Calculation in High-Dimensional Nonlinear Mechanical Systems