An analysis of nonlinear resonance in contoured-quartz crystal resonators

Journal of the Acoustical Society of America

Volumn 80, Issue 4, 1986, Pages 1122-1132

  Tiersten, H F ^a   Stevens, D S ^a  

^a RENSSELAER POLYTECHNIC INSTITUTE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000046760     PISSN: 00014966     EISSN: NA     Source Type: Journal    
DOI: 10.1121/1.393802     Document Type: Article

References (22)

1. D. S. Stevens and H. F. Tiersten, “An Analysis of Doubly Rotated Quartz Resonators Utilizing Essentially Thickness Modes with Transverse Variation, “ J. Acoust. Soc. Am. 79, 1811 (1986).
2. H. F. Tiersten, “Analysis of Nonlinear Resonance in Thickness-Shear and Trapped Energy Resonators, ” J. Acoust. Soc. Am. 59, 866 (1976).
3. H. F. Tiersten and A. Ballato, “Nonlinear Extensional Vibrations of Quartz Rods, ” J. Acoust. Soc. Am. 73, 2022 (1983).
4. H. F. Tiersten, “Analysis of Trapped Energy Resonators Operating in Overtones of Coupled Thickness Shear and Thickness Twist, ” J. Acoust. Soc. Am. 59, 879 (1976).
5. H. F. Tiersten and R. C. Smythe, “An Analysis of Contoured Crystal Resonators Operating in Overtones of Coupled Thickness Shear and Thickness Twist, ” J. Acoust. Soc. Am. 65, 1455 (1979).
6. R. L. Filler, “The Amplitude Frequency Effect in SC-Cut Resonators, ” in Proceedings of the 39th Annual Symposium on Frequency Control (U. S. Army Electronics Research and Development Command, Fort Monmouth, NJ and Institute of Electrical and Electronics Engineers, New York, 1985), IEEE Cat. No. 85-CH2186-5, p. 311.
7. H. F. Tiersten, “Nonlinear Electroelastic Equations Cubic in the Small Field Variables, ” J. Acoust. Soc. Am. 57, 660 (1975).
8. Since linear electric terms only are retained, the reference (or material) electric displacement vector reduces to the ordinary electric displacement vector.
9. Reference 7, Eq. (16).
10. Reference 7, Eqs. (63).
11. H. F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969), Chap. 9, Sec. 2.
12. H. F. Tiersten, “Electromechanical Coupling Factors and Fundamental Material Constants of Thickness Vibrating Piezoelectric Plates, ” Ultrasonics 8, 19 (1970).
13. A. Ballato, “Doubly-Rotated Thickness Mode Plate Vibrators, ” in Physical Acoustics, edited by W. P. Mason and R. N. Thurston (Academic, New York, 1977), Vol. 13, Sec. III.
14. H. F. Tiersten, “Analysis of Intermodulation in Thickness-Shear and Trapped Energy Resonators, ” J. Acoust. Soc. Am. 57, 667 (1975).
15. The argument is essentially the same as the one given in the beginning of the paragraph containing Eq. (106) of Ref. 3.
16. This assumption of small piezoelectric coupling being negligible when arising as a product with A2N has already been employed in writing (54), in which the linear electroelastic term is written only at ω.
17. The procedure is a straightforward iterative procedure except when the functional form resulting from the nonlinear terms is identical with that satisfying the basic linear differential equation, at which point the coefficients associated with the functional form due to the nonlinearity are treated as unknown and identical with the coefficients of the linear solution. This is the reason the sin ηN X2 term in (27) is homogeneous whereas (15) and (23) are inhomogeneous equations. This procedure is appropriate because the term on the right-hand side of (27) actually is a homogeneous term and appears to be inhomogeneous (known) only because of the nature of the formal procedure employed which tends to be misleading whenever this type of coincidence of functions occurs.
18. C. J. Wilson, “Vibration Modes of AT-Cut Convex Quartz Resonators, ” J. Phys. D7, 2449 (1974).
19. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p.786.
20. 6666is not considered to be terribly important in this work because we are interested here only in showing the typical behavior of nonlinear resonance curves predicted by the theory for different drive levels, unloaded quality factors, load resistances and order of harmonic.
21. R. Bechmann, “Elastic and Piezoelectric Constants of Alpha-Quartz, ” Phys. Rev. 110, 1060 (1958).
22. R. N. Thurston, H. J. McSkimin, and P. Andreatch, Jr., “Third Order Elastic Constants of Quartz, ” J. Appl. Phys. 37, 267 (1966).