Related Experiment Video
Updated: Jul 19, 2026

Characterization of Full Set Material Constants and Their Temperature Dependence for Piezoelectric Materials Using Resonant Ultrasound Spectroscopy
Published on: April 27, 2016
Frequency-temperature behavior of flexural quartz resonators by means of Timoshenko's model
Hubert Hantz1, Fabrice Sthal, Roger Bourquin
1FEMTO-ST Institute, UMR CNRS 6174, Ecole Nationale Supérieure de Mécanique et des Microtechniques, Besançon, France.
Abstract:
The frequency of a flexural resonator and its frequency-temperature behavior usually are computed by Bernoulli's classical approximation. This approach is valid for beams with a large length-over-thickness-ratio. For shorter beams, the effects of shear stress and rotary inertia may play a significant role for temperature-compensated resonators. These effects have been taken into account for isotropic beams. The aim of this paper is to discuss the extension of the shear coefficient in the case of an anisotropic material and to compute the frequency-temperature characteristic of an (XYt)theta cut resonator when the shear stress and the rotary inertia have been taken into account. Comparisons between the classical approximation and this treatment are given for quartz. Furthermore, the numerical predictions obtained by means of different sets of data available for thermal sensitivities of elastic coefficients are compared.
Related Concept Videos
Flexural Stress
Hooke's Law states that within the material's elastic limits, stress is directly proportional to strain. In a member experiencing a bending moment, the strain at any point is relative to its distance...
Design Example: Underdamped Parallel RLC Circuit
Starting with a fixed...
Sound Waves: Resonance
Temperature Dependent Deformation
Characteristics of Series Resonant Circuit
IR Spectroscopy: Hooke's Law Approximation of Molecular Vibration
According to Hooke's law, the vibrational frequency is directly proportional to the...

