Three-Dimensional Finite-Element Solution of the Lagrangean Equations for the Frequency-Temperature Behavior of Y-Cut and NT-Cut Bars

A three-dimensional finite-element matrix equation was formulated, using variational principles, from the field equations of incremental motion superposed on homogenous thermal strain. The field equations were derived from the nonlinear field equations of thermoelasticity in Lagrangean formulation. Since the equations were referred to a fixed reference frame, the element nodal coordinates and mass matrix were not 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 β ₂₂ of the thermal expansion coefficient β_ijappears in the equation. Hence for low-frequency flexural vibrations the tensor βij was assumed to be equal to β ₂₂I in the three-dimensional finite-element analysis. A hexahedral element with eight nodes and three degrees of freedom per node was used. A Guyan reduction scheme was employed to reduce the mass and stiffness matrices. Either the Householder or the full subspace iteration method was used to extract the eigenvalues for frequency calculations. 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.