3-10
· Number of
independent non-zero constants depend on crystal symmetry. For quartz (trigonal, class 32),
there are 10 independent linear
constants - 6 elastic, 2 piezoelectric and 2 dielectric. "Constants” depend
on temperature, stress, coordinate
system, etc.
· To describe
the behavior of a resonator, the differential equations for Newton's law of
motion for a
continuum, and for Maxwell's equation* must be solved, with the
proper electrical and mechanical
boundary conditions at the plate
surfaces.
· Equations are
very "messy" - they have never been solved in closed form for
physically realizable three-
dimensional resonators. Nearly all theoretical work has used
approximations.
· Some of the
most important resonator phenomena (e.g., acceleration sensitivity) are due to
nonlinear
effects. Quartz has numerous higher order constants,
e.g., 14 third-order and 23 fourth-order elastic
constants, as well as 16 third-order
piezoelectric coefficients are known; nonlinear equations are extremely
messy.
* Magnetic field effects
are generally negligible; quartz is diamagnetic, however, magnetic fields can affect the
mounting structure and electrodes.
Mathematical Description - Continued