Model reduction for constrained mechanical systems via spectral submanifolds

• Published in: Nonlinear dynamics Vol. 111; no. 10; pp. 8881 - 8911
• Main Authors: Li, Mingwu, Jain, Shobhit, Haller, George
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.05.2023; Springer Nature B.V
• Subjects: Algebra; Automotive Engineering; Classical Mechanics; Configurations; Constraints; Control; Decomposition; Deformation; Differential equations; Dynamical Systems; Engineering; Finite element method; Kinematics; Manifolds (mathematics); Mathematical models; Mechanical Engineering; Mechanical systems; Model reduction; Multibody systems; Ordinary differential equations; Original Paper; Partial differential equations; Reduced order models; Vibration; Configuration constraints; Reduced-order models; Spectral submanifolds; Invariant manifolds; Forced response curves
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-023-08300-5

Dynamical systems are often subject to algebraic constraints in conjunction with their governing ordinary differential equations. In particular, multibody systems are commonly subject to configuration constraints that define kinematic compatibility between the motion of different bodies. A full-scale numerical simulation of such constrained problems is challenging, making reduced-order models (ROMs) of paramount importance. In this work, we show how to use spectral submanifolds (SSMs) to construct rigorous ROMs for mechanical systems with configuration constraints. These SSM-based ROMs enable the direct extraction of backbone curves and forced response curves and facilitate efficient bifurcation analysis. We demonstrate the effectiveness of this SSM-based reduction procedure on several examples of varying complexity, including nonlinear finite-element models of multibody systems. We also provide an open-source implementation of the proposed method that also contains all details of our numerical examples.