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Data-Driven Reduction of the Finite-Element Model of a Tribomechadynamics Benchmark Problem

Ahmed Amr Morsy, Zhenwei Xu, Paolo Tiso, George Haller

TL;DR

The paper addresses the computational challenge of predicting nonlinear, nonsmooth tribo-mechanical dynamics in bolted joints by constructing a data-driven, smooth reduced-order model using Spectral Submanifolds (SSMs). A $4$-dimensional SSM-based ROM is learned from a few unforced FE trajectories of a $187{,}920$-DOF model and is used to predict the forced response, backbone curves, and damping observed in experiments and full FE simulations, including internal resonances. The authors employ FastSSM to learn the SSM geometry and SSMLearn to obtain a sparse polynomial normal form, enabling efficient prediction under periodic forcing via leading-order forcing terms. The results demonstrate accurate predictions (NMTE around $9.16ackslash%$ on validation data) and offer a nonintrusive, dynamics-based ROM capable of handling time-dependent forcing and providing access to joint-level stresses, with potential for parametric studies and faster surrogate evaluations.

Abstract

Bolted joints can exhibit nonsmooth and significantly nonlinear dynamics. Finite Element Models (FEMs) of this phenomenon require fine spatial discretizations, inclusion of nonlinear contact and friction laws, as well as geometric nonlinearity. Owing to the nonlinearity and high dimensionality of such models, full-order dynamic simulations are computationally expensive. In this work, we use the theory of Spectral Submanifolds (SSMs) to construct a data-driven, smoothed reduced model for a 187,920-dimensional FEM model of a broadly studied Tribomechadynamics benchmark structure with bolted joints. We train the 4-dimensional reduced model using only a few transient trajectories of the full unforced FEM model. We show that this smooth model accurately predicts the experimentally observed nonlinear forced response of the full nonsmooth benchmark problem.

Data-Driven Reduction of the Finite-Element Model of a Tribomechadynamics Benchmark Problem

TL;DR

The paper addresses the computational challenge of predicting nonlinear, nonsmooth tribo-mechanical dynamics in bolted joints by constructing a data-driven, smooth reduced-order model using Spectral Submanifolds (SSMs). A -dimensional SSM-based ROM is learned from a few unforced FE trajectories of a -DOF model and is used to predict the forced response, backbone curves, and damping observed in experiments and full FE simulations, including internal resonances. The authors employ FastSSM to learn the SSM geometry and SSMLearn to obtain a sparse polynomial normal form, enabling efficient prediction under periodic forcing via leading-order forcing terms. The results demonstrate accurate predictions (NMTE around on validation data) and offer a nonintrusive, dynamics-based ROM capable of handling time-dependent forcing and providing access to joint-level stresses, with potential for parametric studies and faster surrogate evaluations.

Abstract

Bolted joints can exhibit nonsmooth and significantly nonlinear dynamics. Finite Element Models (FEMs) of this phenomenon require fine spatial discretizations, inclusion of nonlinear contact and friction laws, as well as geometric nonlinearity. Owing to the nonlinearity and high dimensionality of such models, full-order dynamic simulations are computationally expensive. In this work, we use the theory of Spectral Submanifolds (SSMs) to construct a data-driven, smoothed reduced model for a 187,920-dimensional FEM model of a broadly studied Tribomechadynamics benchmark structure with bolted joints. We train the 4-dimensional reduced model using only a few transient trajectories of the full unforced FEM model. We show that this smooth model accurately predicts the experimentally observed nonlinear forced response of the full nonsmooth benchmark problem.
Paper Structure (13 sections, 30 equations, 13 figures, 1 table)

This paper contains 13 sections, 30 equations, 13 figures, 1 table.

Table of Contents

  1. abstract
  2. Introduction
  3. Background
  4. Linear Dynamics
  5. Nonlinear Dynamics
  6. Joints
  7. Set-up
  8. Data-Assisted Model Reduction using FastSSM and SSMLearn
  9. TRCBenchmark Test Case
  10. CONCLUSION
  11. ACKNOWLEDGEMENTS
  12. Appendix A1
  13. Appendix A2

Figures (13)

  • Figure 1: CAD Model of TRChallenge benchmark structure.darus-3147_2022
  • Figure 2: Sketch of the geometry and dynamics of full and reduced coordinates. A sample trajectory $\boldsymbol{\mathrm{\tilde{x}}}(t)$ of the full system is sketched in solid line. Its prediction using the SSM-based ROM lies on the $\mathcal{W}(E^{m})$ (in blue) and is sketched in dashed lines.
  • Figure 3: CAD model of the TRC Benchmark Structure darus-3147_2022.
  • Figure 4: Finite element model. Number of degrees of freedom = 187,920. Friction is modelled as Coulomb friction with $\mu=0.6$.
  • Figure 5: Plot (a) shows the transient time response of trajectory (1) at DOF P. Plot (b) shows its spectrogram. We note an internal resonance of the structure at high amplitudes of vibration as the second harmonic component of the first bending mode activates the torsional mode. We therefore target a 4-dimensional SSM.
  • ...and 8 more figures