  A nonlinear analysis of high-frequency thickness-shear vibrations of AT-cut
quartz crystal plates is presented with the two-dimensional finite element
method. We expanded both kinematic and constitutive nonlinear Mindlin plate
equations and then truncated them to the first-order equations as an
approximation, which is used later for the formulation of nonlinear finite
element analysis with all zeroth- and first-order displacements and electric
potentials. The matrix equation of motion is established with the first-order
harmonic approximation and the generalized nonlinear eigensystem is solved by a
direct iterative procedure. A backbone curve and corresponding mode shapes are
obtained and analyzed. The nonlinear finite element program is developed based
on earlier linear edition and can be utilized to predict nonlinear
characteristics of miniaturized quartz crystal resonators in the design
process.
