MEMS based oscillators have been an active area of research in the last few decades. Due to their minuscule size and lower power consumption they find a wide range of applications from accelerometers, gyroscopes, micro mirrors, gas sensors etc. to lab on chip. As the MEMS device vibrates, it will lose energy to the substrate via anchor, causing disturbance in substrate that propagates away from the oscillator in the form of stress waves. The stress wave consists of bulk acoustic waves (BAW) and surface acoustic waves (SAW). A large portion of wave energy is in the SAW (67%), that can be used to couple and synchronize the oscillators. Synchronized MEMS oscillators have potential applications in signal processing, neural computing, sensing and other fields. Synchronization of two mutually, SAW coupled MEMS oscillators, using a simplified nonlinear model is studied. Synchronization dynamics of system with various means of coupling like linear, quadratic and cubic for both dissipative and reactive are analyzed. Various solutions for different parameter values are identified and the synchronization regions mapped for each type of 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, are 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. The effect of different parameters of individual oscillator on synchronization is also studied, which provides guidance for fabrication and testing of the optimized system.
Further the concept of nonlinear normal modes (NNMs) is used to study the synchronization dynamics of the delay coupled MEMS oscillators. Shaw and Pierre’s method is used to arrive at the NNMs. It is shown that due to the synchronizing property of the system, the invariant manifold reduces to a one-dimensional closed curve which remains a subset of the two-dimensional manifold calculated by NNM computation. The obtained NNMs can be used to decouple the governing equations. This suggests a novel approach to the study of in-phase and out of phase synchronization in the delay coupled system.
In order to verify the numerical results, coupled oscillator device were fabricated after designing them through careful Finite Element Analysis.