The effect of cubically varying contours on the thickness-shear and flexural vibrations of quartz plates

Thickness-shear and flexural vibrations of crystal plates are studied by Mindlin´s first-order equations and by assuming that the vibration is independent of x₃ coordinate. Exact solutions in terms of infinite power series are obtained for the coupled thickness-shear and flexural vibration equations of quartz plates with cubically varying thickness in the x₁ direction. The effect of contouring on the resonance frequencies and mode shapes are examined in detail and compared with those of the plates with linearly varying contours