TY - JOUR

T1 - THREE-DIMENSIONAL FINITE ELEMENT SOLUTION OF THE LAGRANGIAN EQUATIONS FOR THE FREQUENCY-TEMPERATURE BEHAVIOR OF Y-CUT AND NT-CUT BARS.

AU - Yong, Y. K.

PY - 1986

Y1 - 1986

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0022917812&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0022917812&partnerID=8YFLogxK

U2 - 10.1109/freq.1986.200939

DO - 10.1109/freq.1986.200939

M3 - Conference article

AN - SCOPUS:0022917812

SN - 0161-6404

SP - 179

EP - 186

JO - Proceedings of the Annual IEEE International Frequency Control Symposium

JF - Proceedings of the Annual IEEE International Frequency Control Symposium

ER -