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.
Figures & tables
Figure 1: Schematic diagram of a single layer perceptron
Figure 2: Schematic diagram of multiple piecewise springs acting on the mass
Figure 3: Convergence of the approximated function with respect to the number of functions. p2=10 . The regression was conducted for data in the filled region ( −5⩽x⩽5 ).
Figure 4: Spring constants versus the initial gap values
Figure 5: Time history of the oscillator with the original cubic nonlinearity and with the min-max representations
Figure 6: Comparison of the time-frequency relationships using scalogram of the response
(M,N)
(8,8)
(32,32)
(128,128)
RMSE (m)
2.38
0.497
0.215
Table 1: RMSE values for the approximations
Figure 7: Comparison of the forced response results with single cubic stiffness and multiple PWL springs, along with classical approximate solutions and backbone curve
Figure 8: Schematic diagram of the piecewise linear system
Figure 9: Convergence of the approximated function with respect to the number of functions. p2=10 . The regression was conducted for data in the filled region ( −5⩽x⩽5 ).
Figure 10: Spring constants versus the initial gap values for the piecewise linear stiffness ( L=0.5 )
Figure 11: Comparison of the free response results with a single piecewise linear stiffness and multiple PWL stiffnesses.
(M,N)
(8,8)
(32,32)
(256,256)
RMSE (m)
0.554
0.229
0.028
Table 2: RMSE values for the approximations
Figure 12: Comparison of the forced response results with a single piecewise linear stiffness and multiple PWL springs
Figure 13: Phase portraits and Fourier spectra at primary and secondary resonances
Figure 14: Test equipment for free vibration test. The plate is in equilibrium state.
Figure 15: Time response of the displacement of the plate tip and backbone curve obtained from the free response
Figure 16: Identified potential and its associated magnetic force
Figure 17: Spring constants versus the initial gap values for the min-max functions
Figure 18: Results of free response ( M=32 ). The regression was conducted for data in the filled region ( 0⩽t⩽3 ).
Figure 19: Convergence study with respect to the number of ϕ(x,gi) (Fit duration=50%)
Figure 20: Convergence study with respect to the fit duration to total duration ( M=32 )
Figure 21: Test equipment for forced vibration test
Figure 22: Comparison of the measured forced response time histories and those computed with the proposed approach at 15Hz.
Figure 23: Comparison of the measured forced response results and those computed with the proposed approach
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.
Akira Saito, Masato Tanaka
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
Nonlinear contacts and friction strongly influence the vibration response of assembled structures, but their accurate numerical treatment is computationally demanding. The harmonic balance method is widely used to compute periodic steady-state responses, yet the required alternating frequency-time scheme becomes costly for nonsmooth and hysteretic nonlinearities and must be repeated throughout the nonlinear solution process. Here we show that this procedure can be replaced by neural networks that directly map displacement Fourier coefficients to nonlinear force coefficients and provide the corresponding Jacobian through automatic differentiation. The surrounding solver and continuation algorithms remain unchanged for the computation of frequency response curves. The neural networks exclusively learn individual nonlinear elements rather than complete system responses. Physics-based nondimensionalization and phase normalization facilitate the learning process and enable a single trained network to cover a wide range of parameter combinations. Building on the cubic spring, unilateral spring, and Jenkins elements considered here, the approach points toward a reusable library of nonlinear-element surrogates that can be combined in arbitrary number and location within a mechanical system. By bypassing the iterative force evaluation in time domain, the method offers favorable computational scaling for high-resolution analyses and systems with many nonlinear elements.
Miriam Goldack, Johann Groß, Malte Krack +1
Technische Universität Berlin, Chair of Cyber-Physical Systems in Mechanical Engineering, Straße des 17. Juni 135, 10623 Berlin, Germany · University of Stuttgart, Institute of Aircraft Propulsion Systems, Pfaffenwaldring 6, 70569 Stuttgart, Germany
In this paper, the problem of data-driven discovery of nonlinear ordinary differential equations (ODEs) is recast, and a new interpretable machine learning (ML) method is proposed. The proposed method aims to learn the unknown vector field of nonlinear dynamics without prior knowledge of the system's physics from only one single state trajectory's data. The proposed method has two fundamental differences with existing methods: 1) the formulation presented in this method is derived based on Functional Analysis and Operator Theory, and 2) the cost function is constructed in the function space as a distance between two functions as an integral, instead of the discrete-sum of errors used in existing ML approaches. An incremental learning algorithm is proposed to learn the unknown vector field to handle new training samples in an online manner. The proposed method can discover the unknown vector field from both forced and unforced autonomous and non-autonomous (or time-varying) dynamical systems. The proposed method is able to simultaneously discover unknown external forces as a function of time and unknown underlying dynamics. Finally, numerical examples are given to demonstrate the advantages of the proposed method.
Seyyed Shaho Alaviani, Yongzhi Qu, Gregory W. Vogl
Department of Mechanical Engineering University of Minnesota-Twin Cities Minneapolis, MN 55455, USA · Department of Mechanical Engineering University of Utah Salt Lake City, UT 84112, USA · National Institute of Standards and Technology Gaithersburg, MD 20899, USA