非線性動態稀疏識別法(The sparse identification of nonlinear dynamics method，簡稱為SINDy法)為Brunton等人於近年提出的數據驅動建模技術，此技術可透過分析動態數據逆向建立動態系統的非線性運動方程式，而本研究旨在探討SINDy法於結構系統非線性建模之應用及可行性。由於SINDy法適用於自治動態系統的數據驅動建模，因此有別於傳統的系統識別方法需要穩定的激振源驅動系統，此方法能夠透過分析結構自由振動響應而逆向建立系統之非線性模型。此外，SINDy法的另一項特點是允許使用者自行選定義非線性函數作為模型的基底，並透過稀疏回歸計算各個函數的權重，因此SINDy法具有自動抑制量測誤差並濾除不適用函數的特性。本研究分別將SINDy法應用於雙自由度的Duffing振盪子與T型樑結構中，並分別以數值模擬與實驗方法驗證SINDy於上述系統的建模準確性。

The sparse identification of nonlinear dynamics (SINDy) method is a data-driven nonlinear modelling technique for dynamic systems proposed by Brunton et al. in recent years. This study is the first application of the SINDy method to construct a dynamic model of structural systems. Since the SINDy method is suitable for data-driven modeling of autonomous dynamic systems, unlike traditional system identification methods that require a stable excitation to drive the system, this method can build a nonlinear model of the system by analyzing the free vibration response of the structure. Another feature of the SINDy method is that it can use a flexible nonlinear function as the base of the model and calculate the weights of each function through sparse regression. Therefore, the SINDy method has the characteristics of automatic suppression of measurement errors and inapplicable functions. In this study, the SINDy method is applied to a two-degree-of-freedom Duffing oscillator and a T-beam structure and is validated both numerically and experimentally.

中文摘要-------------------------------------------------I
英文摘要-------------------------------------------------II
致謝-----------------------------------------------------III
目錄-----------------------------------------------------IV
圖目錄---------------------------------------------------VI
表目錄---------------------------------------------------IX
符號列表-------------------------------------------------X
第1章 緒論-----------------------------------------------1
1.1 簡介-------------------------------------------------1
1.2 研究背景與動機----------------------------------------1
1.3 論文架構----------------------------------------------3
第2章 研究方法--------------------------------------------4
2.1 稀疏識別法--------------------------------------------4
2.2 具控制之稀疏識別法------------------------------------6
2.3 最小平方稀疏回歸法------------------------------------7
2.4 研究框架---------------------------------------------9
第3章 雙自由度彈簧質塊系統之非線性模型分析------------------10
3.1 雙自由度彈簧質塊系統----------------------------------10
3.1.1 初始條件激發較豐富之響應----------------------------12
3.1.2 初始條件僅激發單一模態響應--------------------------18
第4章 具有耦合共振行為之T型樑結構系統之非線性模型分析-------24
4.1 T型樑結構耦合共振系統設計-----------------------------24
4.2 實驗儀器---------------------------------------------28
4.2.1 自由振動實驗----------------------------------------28
4.2.2 簡諧激振實驗---------------------------------------28
4.3 自由振動實驗-----------------------------------------32
4.3.1 非線性函數庫的選擇---------------------------------32
4.3.2 系統響應行為對於模型識別結果之影響-------------------41
4.3.3 量測取樣頻率對於識別模型之影響----------------------53
4.4 簡諧激振實驗----------------------------------------55
第5章 結論與未來展望-------------------------------------61
5.1 結論------------------------------------------------61
5.2 未來展望--------------------------------------------61
參考文獻------------------------------------------------62
附錄----------------------------------------------------65

[1] Kerschen, G., et al., Past, present and future of nonlinear system identification in structural dynamics. Mechanical systems and signal processing, 2006. 20(3): p. 505-592.
[2] Noël, J.-P. and G. Kerschen, Nonlinear system identification in structural dynamics: 10 more years of progress. Mechanical Systems and Signal Processing, 2017. 83: p. 2-35.
[3] Antonio, D., D.H. Zanette, and D. López, Frequency stabilization in nonlinear micromechanical oscillators. Nature communications, 2012. 3(1): p. 1-6.
[4] Amin Karami, M. and D.J. Inman, Powering pacemakers from heartbeat vibrations using linear and nonlinear energy harvesters. Applied Physics Letters, 2012. 100(4): p. 042901.
[5] Quinn, D.D., et al., Energy harvesting from impulsive loads using intentional essential nonlinearities. Journal of Vibration and Acoustics, 2011. 133(1).
[6] Tien, M.-H. and K. D’Souza, Method for controlling vibration by exploiting piecewise-linear nonlinearity in energy harvesters. Proceedings of the Royal Society A, 2020. 476(2233): p. 20190491.
[7] Wang, Y.-R., et al., Analysis of a Clapping Vibration Energy Harvesting System in a Rotating Magnetic Field. Sensors, 2022. 22(18): p. 6916.
[8] Gendelman, O.V., et al., Enhanced passive targeted energy transfer in strongly nonlinear mechanical oscillators. Journal of Sound and Vibration, 2011. 330(1): p. 1-8.
[9] Wang, Y., C. Feng, and S. Chen, Damping effects of linear and nonlinear tuned mass dampers on nonlinear hinged-hinged beam. Journal of Sound and Vibration, 2018. 430: p. 150-173.
[10] Moaveni, B. and E. Asgarieh, Deterministic-stochastic subspace identification method for identification of nonlinear structures as time-varying linear systems. Mechanical Systems and Signal Processing, 2012. 31: p. 40-55.
[11] Wang, X. and G. Zheng, Equivalent Dynamic Stiffness Mapping technique for identifying nonlinear structural elements from frequency response functions. Mechanical Systems and Signal Processing, 2016. 68: p. 394-415.
[12] Marchesiello, S. and L. Garibaldi, A time domain approach for identifying nonlinear vibrating structures by subspace methods. Mechanical Systems and Signal Processing, 2008. 22(1): p. 81-101.
[13] Masri, S. and T. Caughey, A nonparametric identification technique for nonlinear dynamic problems. 1979.
[14] Adams, D. and R. Allemang, A frequency domain method for estimating the parameters of a non-linear structural dynamic model through feedback. Mechanical Systems and Signal Processing, 2000. 14(4): p. 637-656.
[15] McKelvey, T., H. Akçay, and L. Ljung, Subspace-based multivariable system identification from frequency response data. IEEE Transactions on Automatic control, 1996. 41(7): p. 960-979.
[16] Peng, Z., et al., Feasibility study of structural damage detection using NARMAX modelling and nonlinear output frequency response function based analysis. Mechanical Systems and Signal Processing, 2011. 25(3): p. 1045-1061.
[17] Mahariq, I., et al., Identification of nonlinear model for rotary high aspect ratio flexible blade using free vibration response. Alexandria Engineering Journal, 2020. 59(4): p. 2131-2139.
[18] Brunton, S.L., J.L. Proctor, and J.N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 2016. 113(15): p. 3932-3937.
[19] Brunton, S.L. and J.N. Kutz, Data-driven science and engineering: Machine learning, dynamical systems, and control. 2022: Cambridge University Press.
[20] Cortiella, A., K.-C. Park, and A. Doostan, Sparse identification of nonlinear dynamical systems via reweighted ℓ1-regularized least squares. Computer Methods in Applied Mechanics and Engineering, 2021. 376: p. 113620.
[21] Tibshirani, R., Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B (Methodological), 1996. 58(1): p. 267-288.
[22] Brunton, S.L., J.L. Proctor, and J.N. Kutz, Sparse identification of nonlinear dynamics with control (SINDYc). IFAC-PapersOnLine, 2016. 49(18): p. 710-715.
[23] Ibrahim, A., W. Yang, and S. Towfighian. Internal resonance of T-shaped structure for energy harvesting with magnetic nonlinearity. in Active and Passive Smart Structures and Integrated Systems XII. 2018. SPIE.
[24] Li, L., H. Liu, and W. Zhang, Design and Experiment of Mass Warning Resonant Sensor Induced by Modal Coupling. IEEE Sensors Journal, 2022.
[25] Vyas, A., D. Peroulis, and A.K. Bajaj, A microresonator design based on nonlinear 1: 2 internal resonance in flexural structural modes. Journal of Microelectromechanical Systems, 2009. 18(3): p. 744-762.
[26] Ramini, A.H., A.Z. Hajjaj, and M.I. Younis, Tunable resonators for nonlinear modal interactions. Scientific reports, 2016. 6(1): p. 1-9.
[27] Sarrafan, A., B. Bahreyni, and F. Golnaraghi, Analytical modeling and experimental verification of nonlinear mode coupling in a decoupled tuning fork microresonator. Journal of Microelectromechanical Systems, 2018. 27(3): p. 398-406.
[28] Butcher, J.C., The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods. 1987: Wiley-Interscience.
[29] Dormand, J.R. and P.J. Prince, A family of embedded Runge-Kutta formulae. Journal of computational and applied mathematics, 1980. 6(1): p. 19-26.
[30] Newmark, N.M., A method of computation for structural dynamics. Journal of the engineering mechanics division, 1959. 85(3): p. 67-94.
[31] MATLAB 2022b, The MathWorks, Inc., Natick, Massachusetts, United States.