THREE-DIMENSIONAL FINITE ELEMENT SOLUTION OF THE LAGRANGIAN EQUATIONS FOR THE FREQUENCY-TEMPERATURE BEHAVIOR OF Y-CUT AND NT-CUT BARS.

Using variational principles a three-dimensional finite element matrix equation was formulated from the field equations of incremental motion superposed on homogeneous thermal strain. The field equations were derived from the nonlinear field equations of thermoelasticity in the Lagrangian formulation. Since the equations were referred to a fixed reference frame, the element nodal coordinates and mass matrix did not have to be updated with changes in temperature. Only the stiffness matrix must be updated. An anisotropic plate equation was derived and it was observed that only one term, beta //2 //2, of the thermal expansion coefficient, beta //i //j, appears in the equation. Hence, for low-frequency flexural vibrations, the tensor beta //i //j was assumed to be equal to beta //2 //2 I in the three-dimensional finite-element analysis. Results using the finite-element method were compared with the analytical and experimental results for a Y-cut plate, NT-cut bars and tuning forks.