An efficient general approach to modal analysis of frame resonators with applications to support loss in microelectromechanical systems

The design of high-Q resonators such as Xylophone Bar Resonators (XBRs) capable of being fabricated using Micro-Electro-Mechanical Systems (MEMS) processes is of considerable interest in light of the widespread and rapidly growing use of systems dependent on their availability and performance. This paper is concerned with vibration analysis and Q optimisation of an XBR, with the method extending directly to other planar frames and straightforwardly to more complex structures. The Rayleigh–Ritz method is discussed in some detail, first treating the discrete case, followed by developing and applying a kinematical procedure to an L-frame structure. Attention is given to geometric interpretation of the Rayleigh–Ritz procedure and to developing an intuitive understanding the method before turning to the XBR case. Having developed an approximation for system dynamics, the results are used in conjunction with an analytical model of elastic wave propagation in the substrate to obtain an estimate for the support Q factor. Natural frequencies, mode shapes, and support Q values are presented and compared to Finite Element models of the same problem, with excellent agreement observed at substantially lower computational cost. For the first time in the literature, the geometric impedance tuning principle underlying the XBR design is validated and quantified, including sensitivity to manufacturing error.