math.DSMar 18, 2026

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

Authors: Akira Saito, Masato Tanaka

Organizations: Department of Mechanical Engineering, Meiji University, Kawasaki, Kanagawa 214-8571, Japan. · Toyota Research Institute of North America Toyota Motor North America, Inc. Ann Arbor, Michigan, 48105, USA

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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.

Figures & tables

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.
Mode numberLinearNonlinearDeviation (%)
150.965.328.29
2320316-1.250
3893886-0.7839
417451735-0.5731
528862874-0.4158
Table 1: DMD eigenvalues in Hz for linear and nonlinear cases
Figure 5: Comparison between LNM, POD modes and DMD modes
Figure 6: Principal angles between the subspaces spanned by LNMs, POD modes, and DMD modes for different number of modes
Figure 7: Forced response results of full-order FEM, POD, and DMD
Figure 8: Phase portrait of the tip movement at the resonance
Figure 9: Bonded shell assembly with partial debonding
Figure 10: Linear normal modes of the assembly
Mode Number(A) (Hz)(B) (Hz)Deviation (%)
119.0118.93-0.40
257.0256.75-0.48
3114.74113.28-1.27
4129.60126.87-2.11
5195.78194.11-0.85
6311.31304.49-2.19
Table 2: Natural frequencies for (A) perfectly bonded and (B) partially debonded assemblies.
Figure 11: Initial deformation of the plate assembly
Figure 12: Phase portraits of the gap functions and time histories of the corresponding nodes for 0⩽t⩽0.06250\leqslant t\leqslant 0.0625 s
Figure 13: Results of FFT of the displacement at the forcing point
Figure 14: Pseudo-stability diagram of the DMD eigenvalues, along with the average Fourier spectrum of the displacement of all DOFs. — : FFT spectra of all DOFs, — : average spectrum, ×{\bf\times} : DMD eigenvalues.
No.Linear (Hz)Nonlinear (Hz)Deviation (%)
119.0918.95-0.74
256.7356.880.27
3113.25113.640.34
4126.82127.560.58
5194.05195.070.52
6304.22306.280.67
Table 3: DMD frequencies extracted from the free response of the shell assembly by initial displacement
Figure 15: Mode shapes of DMD modes ordered based on frequencies of the corresponding DMD eigenvalues
Figure 16: Mode shapes of linear normal modes ordered based on frequencies
Figure 17: Mode shapes of POD modes ordered based on the magnitude of the corresponding singular values σi\sigma_{i}
Figure 18: Comparison of errors in the natural frequencies for the ROMs compared
Figure 19: Results of forced response analysis
Figure 20: Time histories of gap functions at the resonance (307 Hz)
Figure 21: Spectrogram of displacement at the excitation point and gap function computed with FEM
Figure 22: Spectrogram of displacement at the excitation point and gap function computed with DMD-ROM
 FEMLNMCBDMDPOD
(A) Snapshot extractionN/AN/AN/A6.7×1036.7\times 10^{3}6.7×1036.7\times 10^{3}
(B) DMD (POD) calculationsN/AN/AN/A5.9×1015.9\times 10^{1}1.2×10−11.2\times 10^{-1}
(C) Galerkin projectionN/A7.2×10−37.2\times 10^{-3}1.2×10−21.2\times 10^{-2}5.8×10−35.8\times 10^{-3}4.8×10−34.8\times 10^{-3}
(D) Forced response calculation (per frequency)7.6×1037.6\times 10^{3}2.7×1012.7\times 10^{1}1.8×1011.8\times 10^{1}2.2×1012.2\times 10^{1}2.7×1012.7\times 10^{1}
Total (A)+(B)+(C)+(D) ×nf\times n_{f}9.7×1059.7\times 10^{5}3.5×1033.5\times 10^{3}2.3×1032.3\times 10^{3}9.6×1039.6\times 10^{3}1.0×1041.0\times 10^{4}
Table 4: Comparison of computational time required to complete a set of forced response calculations at the resonance (second)
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