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On suppression of chaotic motion of a nonlinear MEMS oscillator

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Abstract

This work investigates the behavior of the linear and nonlinear stiffness terms and damping coefficient related to the dynamics of a microelectromechanical resonator. The system is controlled by forcing it into an orbit obtained from the analytical solution of the harmonic balance method. The control techniques considered are the polynomial expansion of Chebyshev, the Picard interactive method, Lyapunov–Floquet, OLFC control, and SDRE controls. Additionally, in order to study the thermal effects, the effect of damping with fractional-order was implemented. To analyze the behavior of the system in fractional-order, the wavelet-based scale index test was carried out. In addition, the control robustness is investigated analyzing the parametric errors, and the sensitivity of the fractional derivative variation.

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Acknowledgements

The authors acknowledge support from CNPQ, CAPES, FAPESP and FA, all Brazilian research funding agencies.

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  1. Department of Electric Engineering, Federal Technological University of Paraná, UTFPR, Ponta Grossa, PR, 84016-210, Brazil

    Angelo M. Tusset, Jose M. Balthazar, Rodrigo T. Rocha, Mauricio A. Ribeiro & Wagner B. Lenz

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  1. Angelo M. Tusset
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Appendix

Appendix

Variables a, b, c, d,  and e for Eq. (5).

$$\begin{aligned}&-\,4a\omega ^{2}-4\alpha -9a^{2}\alpha -6b^{2}\alpha +\,4c\alpha +24a^{2}c\alpha +6b^{2}c\alpha -6c^{2}\alpha +6c^{3}\alpha +6ad\alpha \nonumber \\&\quad -\,36acd\alpha -6d^{2}\alpha +12cd^{2}\alpha +36abe\alpha -\,6e^{2}\alpha +12ce^{2}\alpha +4ak_l \nonumber \\&\quad +\,3\left( a^{3}-a^{2}d+b^{2}d-c^{2}d+2bce +\,2a\left( {b^{2}+c^{2}+d^{2}+e^{2}} \right) \right) k_{nl} =0 \end{aligned}$$
(A.1)
$$\begin{aligned}&\frac{1}{2}\left( \begin{array}{l} -8b\omega ^{2}-3b\left( {-2a\left( {-1+c} \right) +d+2cd} \right) \alpha +9b^{2}e\alpha \\ +\,\left( {2+9a^{2}-3c+3c^{2}+3d^{2}} \right) e\alpha +3e^{3}\alpha -4c\omega \mu \\ \end{array} \right) \nonumber \\&\quad +\,bk_l +\frac{3}{4}\left( 2a^{2}b+2a\left( {bd+ce} \right) +\,b\left( {b^{2}+c^{2}+2\left( {d^{2}+e^{2}} \right) } \right) \right) k_{nl} =0 \end{aligned}$$
(A.2)
$$\begin{aligned}&\begin{array}{l} \dfrac{1}{2}\left( {\begin{array}{l} -8c\omega ^{2}+4a^{3}\alpha +9c^{2}\left( {a-d} \right) \alpha -2d\alpha -9a^{2}d\alpha -3b^{2}d\alpha -3d^{3}\alpha -3be\alpha \\ -3de^{2}\alpha +3c\left( {-2a+d+2be} \right) \alpha +a\left( {2+3b^{2}+6d^{2}+6e^{2}} \right) \alpha +4b\omega \mu \\ \end{array}} \right) \\ \quad +\,ck_l +\frac{3}{4}\left( 2a^{2}c+a\left( {-2cd+2be} \right) +\,c\left( {b^{2}+c^{2}+2d^{2}+2e^{2}} \right) \right) k_{nl} =0 \\ \end{array} \end{aligned}$$
(A.3)
$$\begin{aligned}&\begin{array}{l} -\,36d\omega ^{2}+3a^{2}\alpha -3b^{2}\alpha -4c\alpha -18a^{2}c\alpha -\,6b^{2}c\alpha +3c^{2}\alpha -6c^{3}\alpha \\ \quad -\,12ad\alpha +24acd\alpha -18cd^{2}\alpha +\,12bde\alpha -6ce^{2}\alpha -12e\omega \mu +4dk_l \\ \quad +\,\left( -a^{3}+3a\left( {b^{2}-c^{2}} \right) +\,6a^{2}d+3d\left( {2b^{2}+2c^{2}+d^{2}+e^{2}} \right) \right) k_{nl} =0 \\ \end{array} \end{aligned}$$
(A.4)
$$\begin{aligned}&\begin{array}{l} \dfrac{1}{2}\left( {\begin{array}{l} 3b^{3}\alpha +b\left( {2+9a^{2}-3c+3c^{2}+3d^{2}} \right) \alpha +9be^{2}\alpha \\ -6e\left( {3\omega ^{2}+\left( {a-2ac+cd} \right) \alpha } \right) +6d\omega \mu \\ \end{array}} \right) +ek_l \\ \quad +\frac{3}{4}\left( {2abc+2a^{2}e+e\left( {2b^{2}+2c^{2}+d^{2}+e^{2}} \right) } \right) k_{nl} =0 \\ \end{array} \end{aligned}$$
(A.5)

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Tusset, A.M., Balthazar, J.M., Rocha, R.T. et al. On suppression of chaotic motion of a nonlinear MEMS oscillator. Nonlinear Dyn 99, 537–557 (2020). https://doi.org/10.1007/s11071-019-05421-8

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