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Predicting Generalized Steady States in Aperiodically Forced Mechanical Systems

Roshan S. Kaundinya, Isabella Thiel, Bálint Kaszás, Shobhit Jain, George Haller

Abstract

The existence of generalized steady states (GSSs) in nonlinear mechanical systems under moderate temporally aperiodic forcing has only been shown recently. Here we derive systematic expansions for such GSSs and construct a numerical algorithm that yields explicit and arbitrarily refinable approximations for GSSs without the need for an initial convergence period. This is to be contrasted with a direct numerical integration of the system, whose convergence is hard to assess or is even undefined for short, transient forcing. When at least the linear part of the equations of motion is known, our GSS algorithm outperforms available data-driven neural-network-based techniques for predicting forced response in structural dynamics problems. In a fully equation-driven setting, our GSS computations are shown to be faster than a direct numerical integration of forced nonlinear finite-element models of beams and shells.

Predicting Generalized Steady States in Aperiodically Forced Mechanical Systems

Abstract

The existence of generalized steady states (GSSs) in nonlinear mechanical systems under moderate temporally aperiodic forcing has only been shown recently. Here we derive systematic expansions for such GSSs and construct a numerical algorithm that yields explicit and arbitrarily refinable approximations for GSSs without the need for an initial convergence period. This is to be contrasted with a direct numerical integration of the system, whose convergence is hard to assess or is even undefined for short, transient forcing. When at least the linear part of the equations of motion is known, our GSS algorithm outperforms available data-driven neural-network-based techniques for predicting forced response in structural dynamics problems. In a fully equation-driven setting, our GSS computations are shown to be faster than a direct numerical integration of forced nonlinear finite-element models of beams and shells.
Paper Structure (25 sections, 1 theorem, 93 equations, 17 figures, 5 tables)

This paper contains 25 sections, 1 theorem, 93 equations, 17 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Generalized steady state
  3. Setup
  4. GSS uniqueness and formal expansion
  5. Automated computation of the GSS expansion
  6. Iterated recursion via multi-index notation
  7. Evaluation of integrals in the GSS formulas
  8. Case 1: General damping
  9. Case 2: Structural damping
  10. Modal truncation
  11. Vector Padé approximations for the GSS
  12. Examples
  13. Example 1: Axially moving beam under chirp excitation
  14. Example 2: Randomly excited oscillator chain
  15. Example 3: Partially data-driven computation of the GSSs for the nonlinear oscillator chain
  16. ...and 10 more sections

Key Result

Theorem 1

Consider a $\delta$-size ball $B_{\delta} \subset V$ centered about $\mathbf{z} = 0$, within this open-ball and for all $t \in \mathbb{R}$ assume the following: Then there exists a unique uniformly bounded generalized steady state (GSS) $\mathbf{z}^*(t)$ in $B_{\delta}$. The GSS is asymptotically stable and is as smooth in any parameter as system (eq:first_order_system).

Figures (17)

  • Figure 1: (a) The axially moving beam setup adopted from li22a with simply supported boundary conditions. (b) Linear chirp forcing signal profile.
  • Figure 2: (a) Time evolution of the first Galerkin mode $u_1(t)$ under chirp base excitation for the full model (black), the $O(1)$ linear GSS Taylor approximation (blue) and the $O(10)$ GSS approximation (red). (b) Snapshot of the beam profile at $t \approx 4.99 \text{ [s]}$. (c) Snapshot of the beam profile at $t = 7.80 \text{ [s]}$.
  • Figure 3: (a) Oscillator chain setup. (b) Gaussian random forcing signals with zero mean and standard deviation $\sigma = 1.54$
  • Figure 4: (a) Normalized displacement trajectories of the first (top) and last (bottom) masses in the chain for the full model (black), the $O(10)$ GSS (red) and the autoeconder+LTSM model (green). (b) Same as in (a) but for a previously unseen random forcing realization generated from the same distribution used in the training of the autoencoder+LSTM model.
  • Figure 5: (a) Normalized displacement trajectories of the first (top) and last (bottom) masses in the chain under a modified Gaussian distribution with zero mean and standard deviation $\sigma = 1.8$, for the full model (black), the $O(10)$ GSS (red) and the autoeconder + LTSM model (green). (b) Snapshot of the oscillator chain at $t = 80 \text{ }\mathrm{[s]}$. The middle mass block represents the center of mass position of the rest of the $16$ masses in the chain.
  • ...and 12 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1: Unique generalized steady state (GSS)
  • proof