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Parameter identification of nonlinear system via a dynamic frequency approach and its energy harvester application

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Abstract

A dynamic frequency-based parameter identification approach is applied for the nonlinear system with periodic responses. Starting from the energy equation, the presented method uses a dynamic frequency to precisely obtain the analytical limit cycle expression of nonlinear system and utilizes it as the mathematic foundation for parameter identification. Distinguished from the time-domain approaches, the strategy of using limit cycle to describe the system response is unaffected by the influence of phase change. The analytical expression is fitted with the value sets from phase coordinates measured in periodic oscillation of the nonlinear systems, and the unknown parameters are identified with the interior-reflective Newton method. Then the performance of this identification methodology is verified by an oscillator with nonlinear stiffness and damping. Besides, numerical simulations under noisy environment also verify the efficiency and robustness of the identification procedure. Finally, we apply this parameter identification method to the modeling of a large-amplitude energy harvester, to improve the accuracy of mechanical modeling. Not surprisingly, good agreement is achieved between the experimental data and identified parameters. It also verifies that the proposed approach is less time-consuming and more accuracy in identification procedure.

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Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grants 11772218 and 11872044), China-UK NSFC-RS Joint Project (Grants 11911530177 in China and IE181496 in UK), Tianjin Research Program of Application Foundation and Advanced Technology (Grant 17JCYBJC18900), and the National Key Research and Development Program of China (Grant 2018YFB0106200).

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Authors and Affiliations

  1. Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin University, Tianjin, 300350, China

    Zhiwei Zhang, Wei Wang & Chen Wang

  2. School of Computing and Engineering, Huddersfield University, Huddersfield, HD1 3DH, UK

    Wei Wang

Authors
  1. Zhiwei Zhang
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  2. Wei Wang
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  3. Chen Wang
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Corresponding author

Correspondence to Wei Wang.

Appendix A1

Appendix A1

$$\varGamma_{0,1} { = } - \left(\frac{{a_{0} \alpha_{2,0} \beta_{0,1} }}{{9\omega_{ 1 , 0}^{2} }} + \frac{{a_{0} b\alpha_{3,0} \beta_{0,1} }}{{3\omega_{ 1 , 0}^{2} }} + \frac{{2a_{0}^{3} \alpha_{4,0} \beta_{0,1} }}{{15\omega_{ 1 , 0}^{2} }} + \frac{{2a_{0} b^{2} \alpha_{4,0} \beta_{0,1} }}{{3\omega_{ 1 , 0}^{2} }} + \frac{{2a_{0}^{3} b\alpha_{5,0} \beta_{0,1} }}{{3\omega_{ 1 , 0}^{2} }} + \frac{{10a_{0} b^{3} \alpha_{5,0} \beta_{0,1} }}{{9\omega_{ 1 , 0}^{2} }}\right),$$
$$\begin{aligned} \varGamma_{ 0 , 2} { = }\frac{{a_{0}^{ 2} \alpha_{ 2,0}^{2} }}{{ 1 8\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{2} b\alpha_{2,0} \alpha_{3,0} }}{{ 3\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{2} b^{2} \alpha_{3,0}^{2} }}{{2\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{4} \alpha_{2,0} \alpha_{4,0} }}{{5\omega_{ 1 , 0}^{ 3} }} + \frac{{2a_{0}^{2} b^{2} \alpha_{2,0} \alpha_{4,0} }}{{3\omega_{ 1 , 0}^{ 3} }} + \frac{{3a_{0}^{4} b\alpha_{3,0} \alpha_{4,0} }}{{5\omega_{ 1 , 0}^{ 3} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \frac{{2a_{0}^{2} b^{3} \alpha_{3,0} \alpha_{4,0} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{4a_{0}^{6} \alpha_{4,0}^{2} }}{{25\omega_{ 1 , 0}^{ 3} }} + \frac{{6a_{0}^{4} b^{2} \alpha_{4,0}^{2} }}{{5\omega_{ 1 , 0}^{ 3} }} + \frac{{2a_{0}^{ 2} b^{4} \alpha_{4,0}^{2} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{4} b\alpha_{2,0} \alpha_{5,0} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{10a_{0}^{2} b^{3} \alpha_{2,0} \alpha_{5,0} }}{{9\omega_{ 1 , 0}^{ 3} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \frac{{3a_{0}^{4} b^{2} \alpha_{3,0} \alpha_{5,0} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{10a_{0}^{2} b^{4} \alpha_{3,0} \alpha_{5,0} }}{{ 3\omega_{ 1 , 0}^{ 3} }} + \frac{{8a_{0}^{6} b\alpha_{4,0} \alpha_{5,0} }}{{5\omega_{ 1 , 0}^{ 3} }} + \frac{{8a_{0}^{4} b^{3} \alpha_{4,0} \alpha_{5,0} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{20a_{0}^{2} b^{5} \alpha_{4,0} \alpha_{5,0} }}{{ 3\omega_{ 1 , 0}^{ 3} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \frac{{4a_{0}^{6} b^{2} \alpha_{5,0}^{2} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{10a_{0}^{4} b^{4} \alpha_{5,0}^{2} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{50a_{0}^{2} b^{6} \alpha_{5,0}^{2} }}{{9\omega_{ 1 , 0}^{ 3} }}, \hfill \\ \end{aligned}$$
$$\varGamma_{0,3} { = }\frac{{a_{0}^{3} \alpha_{4,0} \beta_{0,1} }}{{25\omega_{ 1 , 0}^{2} }} + \frac{{a_{0}^{3} b\alpha_{5,0} \beta_{0,1} }}{{5\omega_{ 1 , 0}^{2} }},$$
$$\begin{aligned} \varGamma_{ 0 , 4} { = } - \left(\frac{{a_{0}^{4} \alpha_{3,0}^{2} }}{{ 3 2\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{4} \alpha_{2,0} \alpha_{4,0} }}{{15\omega_{ 1 , 0}^{ 3} }} + \frac{{9a_{0}^{4} b\alpha_{4,0} \alpha_{3,0} }}{{20\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{6} \alpha_{4,0}^{2} }}{{10\omega_{ 1 , 0}^{ 3} }} + \frac{{9a_{0}^{4} b^{2} \alpha_{4,0}^{2} }}{{10\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{4} b\alpha_{2,0} \alpha_{5,0} }}{{3\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{6} \alpha_{3,0} \alpha_{5,0} }}{{8\omega_{ 1 , 0}^{ 3} }} \right.\hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \left. + \frac{{13a_{0}^{4} b^{2} \alpha_{3,0} \alpha_{5,0} }}{{8\omega_{ 1 , 0}^{ 3} }} + \frac{{3a_{0}^{6} b\alpha_{4,0} \alpha_{5,0} }}{{2\omega_{ 1 , 0}^{ 3} }} + \frac{{ 3 1a_{0}^{4} b^{3} \alpha_{4,0} \alpha_{5,0} }}{{6\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{8} \alpha_{5,0}^{5} }}{{8\omega_{ 1 , 0}^{ 3} }} + \frac{{15a_{0}^{6} b^{2} \alpha_{5,0}^{2} }}{{4\omega_{ 1 , 0}^{ 3} }} + \frac{{155a_{0}^{4} b^{4} \alpha_{5,0}^{2} }}{{24\omega_{ 1 , 0}^{ 3} }}\right), \hfill \\ \end{aligned}$$
$$\varGamma_{ 0 , 6} { = }\frac{{a_{0}^{6} \alpha_{4,0}^{2} }}{{ 5 0\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{6} \alpha_{3,0} \alpha_{5,0} }}{{24\omega_{ 1 , 0}^{ 3} }} + \frac{{11a_{0}^{6} b\alpha_{4,0} \alpha_{5,0} }}{{30\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{8} \alpha_{5,0}^{2} }}{{12\omega_{ 1 , 0}^{ 3} }} + \frac{{11a_{0}^{6} b^{2} \alpha_{5,0}^{2} }}{{12\omega_{ 1 , 0}^{ 3} }},$$
$$\varGamma_{0,8} { = } - \frac{{a_{0}^{8} \alpha_{5,0}^{2} }}{{72\omega_{1,0}^{3} }},$$
$$\varGamma_{ 1 , 1} { = } - \left(\frac{{a_{0}^{2} \alpha_{3,0} \beta_{0,1} }}{{16\omega_{ 1 , 0}^{2} }} + \frac{{a_{0}^{2} b\alpha_{4,0} \beta_{0,1} }}{{4\omega_{ 1 , 0}^{2} }} + \frac{{13a_{0}^{4} \alpha_{5,0} \beta_{0,1} }}{{144\omega_{ 1 , 0}^{2} }} + \frac{{5a_{0}^{2} b^{2} \alpha_{5,0} \beta_{0,1} }}{{8\omega_{ 1 , 0}^{2} }}\right),$$
$$\begin{aligned} \varGamma_{ 1 , 2} { = }\frac{{a_{0}^{3} \alpha_{2,0} \alpha_{3,0} }}{{12\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{3} b\alpha_{3,0}^{2} }}{{4\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{3} b\alpha_{2,0} \alpha_{4,0} }}{{3\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{5} \alpha_{3,0} \alpha_{4,0} }}{{10\omega_{ 1 , 0}^{ 3} }} + \frac{{3a_{0}^{3} b^{2} \alpha_{3,0} \alpha_{4,0} }}{{2\omega_{ 1 , 0}^{ 3} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \frac{{2a_{0}^{5} b\alpha_{4,0}^{2} }}{{5\omega_{ 1 , 0}^{ 3} }} + \frac{{2a_{0}^{3} b^{3} \alpha_{4,0}^{2} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{5} \alpha_{2,0} \alpha_{5,0} }}{{6\omega_{ 1 , 0}^{ 3} }} + \frac{{5a_{0}^{3} b^{2} \alpha_{2,0} \alpha_{5,0} }}{{6\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{5} b\alpha_{3,0} \alpha_{5,0} }}{{\omega_{ 1 , 0}^{ 3} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \frac{{10a_{0}^{3} b^{3} \alpha_{3,0} \alpha_{5,0} }}{{3\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{7} \alpha_{4,0} \alpha_{5,0} }}{{5\omega_{ 1 , 0}^{ 3} }} + \frac{{4a_{0}^{5} b^{2} \alpha_{4,0} \alpha_{5,0} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{25a_{0}^{3} b^{4} \alpha_{4,0} \alpha_{5,0} }}{{3\omega_{ 1 , 0}^{ 3} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \frac{{a_{0}^{7} b\alpha_{5,0}^{2} }}{{\omega_{ 1 , 0}^{ 3} }} + \frac{{20a_{0}^{5} b^{3} \alpha_{5,0}^{2} }}{{3\omega_{ 1 , 0}^{ 3} }} + \frac{{25a_{0}^{3} b^{5} \alpha_{5,0}^{2} }}{{3\omega_{ 1 , 0}^{ 3} }}, \hfill \\ \end{aligned}$$
$$\varGamma_{ 1 , 3} { = }\frac{{a_{0}^{4} \alpha_{5,0} \beta_{0,1} }}{{36\omega_{ 1 , 0}^{2} }},$$
$$\begin{aligned} \varGamma_{ 1 , 4} { = } - (\frac{{a_{0}^{5} \alpha_{3,0} \alpha_{4,0} }}{{ 2 0\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{5} b\alpha_{4,0}^{2} }}{{5\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{5} \alpha_{2,0} \alpha_{5,0} }}{{18\omega_{ 1 , 0}^{ 3} }} + \frac{{5a_{0}^{5} b\alpha_{3,0} \alpha_{5,0} }}{{12\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{7} \alpha_{4,0} \alpha_{5,0} }}{{6\omega_{ 1 , 0}^{ 3} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \frac{{11a_{0}^{5} b^{2} \alpha_{4,0} \alpha_{5,0} }}{{6\omega_{ 1 , 0}^{ 3} }} + \frac{{5a_{0}^{7} b\alpha_{5,0}^{2} }}{{6\omega_{ 1 , 0}^{ 3} }} + \frac{{55a_{0}^{5} b^{3} \alpha_{5,0}^{2} }}{{18\omega_{ 1 , 0}^{ 3} }}), \hfill \\ \end{aligned}$$
$$\varGamma_{ 1 , 6} { = }\frac{{a_{0}^{ 7} \alpha_{ 4,0} \alpha_{ 5,0} }}{{ 3 0\omega_{ 1 , 0}^{ 3} }} + \frac{{a_{0}^{ 7} b\alpha_{ 5,0}^{2} }}{{ 6\omega_{ 1 , 0}^{ 3} }}.$$

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Zhang, Z., Wang, W. & Wang, C. Parameter identification of nonlinear system via a dynamic frequency approach and its energy harvester application. Acta Mech. Sin. 36, 606–617 (2020). https://doi.org/10.1007/s10409-020-00972-1

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