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

Akira Saito; Masato Tanaka

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

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

Panel of Figure 3 — Figure 3: Applied force and the response of the beam for both linear and nonlinear cases

Panel of Figure 4 — 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 number	Linear	Nonlinear	Deviation (%)
1	50.9	65.3	28.29
2	320	316	-1.250
3	893	886	-0.7839
4	1745	1735	-0.5731
5	2886	2874	-0.4158

Table 1: DMD eigenvalues in Hz for linear and nonlinear cases

Panel of Figure 5 — 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 (%)
1	19.01	18.93	-0.40
2	57.02	56.75	-0.48
3	114.74	113.28	-1.27
4	129.60	126.87	-2.11
5	195.78	194.11	-0.85
6	311.31	304.49	-2.19

Table 2: Natural frequencies for (A) perfectly bonded and (B) partially debonded assemblies.

Figure 11: Initial deformation of the plate assembly

Panel of Figure 12 — Figure 12: Phase portraits of the gap functions and time histories of the corresponding nodes for $0\leqslant t\leqslant 0.0625$ s

Figure 13: Results of FFT of the displacement at the forcing point

Panel of Figure 14 — 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 (%)
1	19.09	18.95	-0.74
2	56.73	56.88	0.27
3	113.25	113.64	0.34
4	126.82	127.56	0.58
5	194.05	195.07	0.52
6	304.22	306.28	0.67

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 $\sigma_{i}$

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)

Panel of Figure 21 — Figure 21: Spectrogram of displacement at the excitation point and gap function computed with FEM

Panel of Figure 22 — Figure 22: Spectrogram of displacement at the excitation point and gap function computed with DMD-ROM

	FEM	LNM	CB	DMD	POD
(A) Snapshot extraction	N/A	N/A	N/A	$6.7\times 10^{3}$	$6.7\times 10^{3}$
(B) DMD (POD) calculations	N/A	N/A	N/A	$5.9\times 10^{1}$	$1.2\times 10^{-1}$
(C) Galerkin projection	N/A	$7.2\times 10^{-3}$	$1.2\times 10^{-2}$	$5.8\times 10^{-3}$	$4.8\times 10^{-3}$
(D) Forced response calculation (per frequency)	$7.6\times 10^{3}$	$2.7\times 10^{1}$	$1.8\times 10^{1}$	$2.2\times 10^{1}$	$2.7\times 10^{1}$
Total (A)+(B)+(C)+(D) $\times n_{f}$	$9.7\times 10^{5}$	$3.5\times 10^{3}$	$2.3\times 10^{3}$	$9.6\times 10^{3}$	$1.0\times 10^{4}$

Table 4: Comparison of computational time required to complete a set of forced response calculations at the resonance (second)

Reduced-Order Modeling

Reduced-order modeling builds compact approximations of complex systems to make simulation, prediction, and control more efficient.

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Data-driven forced response analysis with min-max representations of nonlinear restoring forces

Akira Saito, Hiromu Fujita

Department of Mechanical Engineering, Meiji University, Kawasaki, Kanagawa 214-8571, Japan.

Nonlinear System Identification Dynamical Systems

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This paper discusses a novel data-driven nonlinearity identification method for mechanical systems with nonlinear restoring forces such as polynomial, piecewise-linear, and general displacement-dependent nonlinearities. The proposed method is built upon the universal approximation theorem that states that a nonlinear function can be approximated by a linear combination of activation functions in artificial neural network framework. The proposed approach utilizes piecewise linear springs with initial gaps to act as the activation functions of the neurons of artificial neural networks. A library of piecewise linear springs with initial gaps are constructed, and the contributions of the springs on the nonlinear restoring force are determined by solving the linear regression problems. The piecewise linear springs are realized by combinations of min and max functions with biases. The proposed method is applied to a Duffing oscillator with cubic stiffness, and a piecewise linear oscillator with a gap and their nonlinearities are successfully determined from their free responses. The obtained models are then used for conducting forced response analysis and the results match well with those of the original system. The method is then applied to experimentally-obtained free response data of a cantilevered plate that is subjected to magnetic restoring force, and successfully finds the piecewise linear representation of the magnetic force. It is also shown that the obtained model is capable of accurately capturing the steady-state response of the system subject to harmonic base excitation.

Abstract

Reduced-Order Modeling

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