A perturbation analysis on contoured crystal plates

The authors present a perturbation analysis on the eigenvalue problem for the resonance of crystal strip resonators with variable thickness of general shapes. Mindlin´s coupled thickness shear and flexure modes model for AT-cut crystal plates is used. A small parameter ε, representing the thickness variation, is chosen as the perturbation parameter. The frequencies and mode shapes of contoured plates are then expressed in two parts: ω² = ω ₀² + εω₁ and u_i ⁽ⁿ⁾ = u_i0⁽ⁿ⁾ + εu_il where the zeroth order terms correspond to resonance and mode shapes for uniform-thickness plates and the first-order terms are the corrections due to thickness variations, respectively. Governing equations for the zeroth and first order problem are derived. The Mindlin equations for plates of uniform thickness are naturally recovered as the zeroth order equations. Conditions for the existence of nontrivial solutions to the first order problem are established for strip resonators. With these conditions, the frequency corrections and mode shape corrections are calculated. A numerical example of the approach is given