Synchronization of surface acoustic wave (SAW)-based delay-coupled self-oscillating MEMS

Abstract

Coupled limit cycle oscillators have potential usage in neurocomputing, pattern recognition, signal processing units such as multilayer perceptrons and stable timekeeping devices. Here, we study the synchronization dynamics of parametrically excited coupled dome-shaped MEMS (microelectromechanical system), exhibiting limit cycle oscillations and hysteresis, which have previously been shown to synchronize with an external reference signal [1, 2]. The coupling considered here is through surface acoustic wave (SAW), emanating in the substrate near each oscillator, due to periodic forcing of the same, which acts as a base vibration term for the forced oscillator. Dynamics of the individual MEMS can be adequately modeled as Mathieu–van der Pol–Duffing (MVDPD) oscillator [3], while the effect of SAW can be added as a delay coupling term, because of finite time involved in travel of this signal, resulting in a set of delay differential equations (DDEs). Various solutions for different parameter values are identified and the synchronization regions mapped under the case of a general dissipative type coupling. Numerical integration of the DDE for the system as well as the ODE obtained by applying the Krylov–Bogoliubov averaging method to the same is used for detailed analysis. The averaged equations are further reduced to get a phase equation which can be used to map the regions of synchronization at a much smaller computational budget. It is found that in general the delay terms lead to an increase in the synchronization region, while a cubic coupling term would be more effective. Effect of different parameters of individual oscillator on synchronization is also studied, which provide guidance for fabrication and testing of the optimized system.

Appendices

Appendix:-1D slow flow equation for linear coupled system

$$\begin{aligned} {\dot{\theta }}= & {} \varDelta +\frac{3\beta }{8\omega } {\frac{-16\,{A}^{4}\cos \left( \omega \,\tau -\theta \right) \lambda \,B _{{D}}+16\,{A}^{4}\cos \left( \omega \,\tau +\theta \right) \lambda \,B_{ {D}}+8\,{A}^{2}\cos \left( 2\,\omega \,\tau +2\,\theta \right) {B_{{D}}} ^{2}-8\,{A}^{2}\cos \left( 2\,\omega \,\tau -2\,\theta \right) {B_{{D}}} ^{2}}{ \left( \left( 3\,{A}^{2}-4 \right) \lambda +4\,B_{{D}} \right) ^{2}}} \nonumber \\&+\frac{4B_D}{8}\bigg ({\frac{ \left( {A}^{2}\lambda +2\,B_{{D}}\cos \left( \omega \,\tau + \theta \right) \right) \sin \left( \omega \,\tau -\theta \right) }{{A}^ {2}\lambda +2\,B_{{D}}\cos \left( \omega \,\tau -\theta \right) }}-{ \frac{ \left( {A}^{2}\lambda +2\,B_{{D}}\cos \left( \omega \,\tau - \theta \right) \right) \sin \left( \omega \,\tau +\theta \right) }{{A}^ {2}\lambda +2\,B_{{D}}\cos \left( \omega \,\tau +\theta \right) }} \bigg ) \end{aligned}$$

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Appendix:-1D slow flow equation for cubic coupled system

$$\begin{aligned} \begin{aligned} {\dot{\theta }}=&\varDelta +\frac{3\beta }{8\omega } \left[ \left( \frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau +\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18cos(\omega \tau +\theta )-3cos(2\omega \tau +2\theta )-15]}\right) ^2\right. \\&-\left. \left( \frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau -\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18cos(\omega \tau -\theta )-3cos(2\omega \tau -2\theta )-15]}\right) ^2 \right] \\&+\frac{3\omega ^2 B_{DC} }{8}\left[ \left( \frac{\left( \frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau +\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18\cos (\omega \tau +\theta )-3\cos (2\omega \tau +2\theta )-15]}\right) ^3}{\left( \frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau -\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18cos(\omega \tau -\theta )-3cos(2\omega \tau -2\theta )-15]}\right) }\right. \right. \\&+\left( \frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau -\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18\cos (\omega \tau -\theta )-3\cos (2\omega \tau -2\theta )-15]}\right) \\&\left. \left( \frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau +\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18\cos (\omega \tau +\theta )-3\cos (2\omega \tau +2\theta )-15]}\right) \right) \sin (\omega \tau -\theta )\\&-\left( \frac{(\frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau -\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18\cos (\omega \tau -\theta )-3\cos (2\omega \tau -2\theta )-15]})^3}{(\frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau +\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18\cos (\omega \tau +\theta )-3\cos (2\omega \tau +2\theta )-15]})}\right. \\&+\left( \frac{2A^3[3 \omega ^2 B_{DC}(cos(\omega \tau -\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18\cos (\omega \tau -\theta )-3\cos (2\omega \tau -2\theta )-15]}\right) \\&\left. \left( \frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau +\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18\cos (\omega \tau +\theta )-3\cos (2\omega \tau +2\theta )-15]}\right) \right) \sin (\omega \tau +\theta )\\&+\left( \frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau -\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18\cos (\omega \tau -\theta )-3\cos (2\omega \tau -2\theta )-15]}\right) ^2 \sin (2\omega \tau +2\theta )\\&\left. -\left( \frac{2A^3[3 \omega ^2 B_{DC}(\cos (\omega \tau +\theta )-1)-\lambda ]}{\lambda (4-3A^2)+A^2\omega ^2 B_{DC} [18\cos (\omega \tau +\theta )-3\cos (2\omega \tau +2\theta )-15]}\right) ^2 \sin (2\omega \tau -2\theta )\right] \end{aligned} \end{aligned}$$

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