Abstract
Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.
Highlights
Invariant manifolds are low-dimensional surfaces in the phase space of a dynamical system that constitute organizing centers of nonlinear dynamics
We address some challenges that have been hindering the computation of invariant manifolds in high-dimensional mechanical systems arising from spatially discretized partial differential equations (PDEs)
We have developed expressions that enable the computation of invariant manifolds and their reduced dynamics in physical coordinates using only the master modes associated with the invariant manifold
Summary
Invariant manifolds are low-dimensional surfaces in the phase space of a dynamical system that constitute organizing centers of nonlinear dynamics. These surfaces are composed of full system trajectories that stay on them for all times, partitioning the phase space locally into regions of different behavior. The focus of this article is to compute such invariant manifolds and their reduced dynamics accurately and efficiently in very high-dimensional nonlinear systems. Local methods approximate an invariant manifold in the neighborhood of simpler invariant sets, such as fixed points, periodic orbits or invariant tori. Global techniques generally employ numerical continuation for growing invariant manifolds from their local approximation that may be obtained from the linearized dynamics (see Krauskopf et al [48] for a survey)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
