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Data-driven model order reduction for structures with piecewise linear nonlinearity using dynamic mode decomposition

Akira Saito, Masato Tanaka

Abstract

Piecewise-linear nonlinear systems appear in many engineering disciplines. Prediction of the dynamic behavior of such systems is of great importance from practical and theoretical viewpoint. In this paper, a data-driven model order reduction method for piecewise-linear systems is proposed, which is based on dynamic mode decomposition (DMD). The overview of the concept of DMD is provided, and its application to model order reduction for nonlinear systems based on Galerkin projection is explained. The proposed approach uses impulse responses of the system to obtain snapshots of the state variables. The snapshots are then used to extract the dynamic modes that are used to form the projection basis vectors. The dynamics described by the equations of motion of the original full-order system are then projected onto the subspace spanned by the basis vectors. This produces a system with much smaller number of degrees of freedom (DOFs). The proposed method is applied to two representative examples of piecewise linear systems: a cantilevered beam subjected to an elastic stop at its end, and a bonded plates assembly with partial debonding. The reduced order models (ROMs) of these systems are constructed by using the Galerkin projection of the equation of motion with DMD modes alone, or DMD modes with a set of classical constraint modes to be able to handle the contact nonlinearity efficiently. The obtained ROMs are used for the nonlinear forced response analysis of the systems under harmonic loading. It is shown that the ROMs constructed by the proposed method produce accurate forced response results.

Data-driven model order reduction for structures with piecewise linear nonlinearity using dynamic mode decomposition

Abstract

Piecewise-linear nonlinear systems appear in many engineering disciplines. Prediction of the dynamic behavior of such systems is of great importance from practical and theoretical viewpoint. In this paper, a data-driven model order reduction method for piecewise-linear systems is proposed, which is based on dynamic mode decomposition (DMD). The overview of the concept of DMD is provided, and its application to model order reduction for nonlinear systems based on Galerkin projection is explained. The proposed approach uses impulse responses of the system to obtain snapshots of the state variables. The snapshots are then used to extract the dynamic modes that are used to form the projection basis vectors. The dynamics described by the equations of motion of the original full-order system are then projected onto the subspace spanned by the basis vectors. This produces a system with much smaller number of degrees of freedom (DOFs). The proposed method is applied to two representative examples of piecewise linear systems: a cantilevered beam subjected to an elastic stop at its end, and a bonded plates assembly with partial debonding. The reduced order models (ROMs) of these systems are constructed by using the Galerkin projection of the equation of motion with DMD modes alone, or DMD modes with a set of classical constraint modes to be able to handle the contact nonlinearity efficiently. The obtained ROMs are used for the nonlinear forced response analysis of the systems under harmonic loading. It is shown that the ROMs constructed by the proposed method produce accurate forced response results.
Paper Structure (15 sections, 28 equations, 22 figures, 4 tables, 1 algorithm)

This paper contains 15 sections, 28 equations, 22 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Method
  3. Model Order Reduction
  4. Dynamic Mode Decomposition
  5. Analysis Procedure
  6. Forced response of a beam with an elastic stop with data-driven model-order reduction
  7. Snapshot extraction and dynamic mode decomposition
  8. Forced response calculation using reduced order models
  9. Forced response of a bonded shell assembly with partial debonding with hybrid data-driven model-order reduction
  10. Galerkin projection based on DMD and constraint modes
  11. Effects of debonding on the vibration modes
  12. Dynamic mode decomposition
  13. Model order reduction by DMD
  14. Forced response analysis
  15. Conclusion

Figures (22)

  • Figure 1: Overview of the forced response analysis
  • Figure 2: Cantilevered beam subject to harmonic forcing and elastic stop
  • Figure 3: Applied force and the response of the beam for both linear and nonlinear cases
  • Figure 4: Pseudo-stability diagram of the DMD eigenvalues, along with the FFT results of the displacments. ---: FFT spectra of all DOFs, ---: average spectrum, ${\bf \times}$: DMD eigenvalues.
  • Figure 5: Comparison between LNM, POD modes and DMD modes
  • ...and 17 more figures