References

1

Allik H. and Hughes T.J.R., "Finite element method for piezoelectric vibration," International Journal for Numerical Methods in Engineering 2, 151-157 (1970)

2

Shaw E.A.G, "On the resonant vibrations of thick barium titanate disks," The Journal of the Acoustical Society of America 28(1), 38-50 (January 1956)

3

Auld B.A., Acoustic fields and waves in solids Vol I, (John Wiley and Sons Inc., New York, 1973)

4

Auld B.A., Acoustic fields and waves in solids Vol II, (John Wiley and Sons Inc., New York, 1973)

5

Bathe H.J., Finite element procedures in engineering analysis, (Prentice-Hall, Inc., New Jersey, 1982)

6

Kunkel H.A.,Locke S. and Pikeroen B., "Finite-element analysis of vibrational modes in piezoelectric ceramic disks," IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 37(4), 316 (1990)

7

Locke S., Kunkel H.A. and Pikeroen B., "Finite element modelling of piezoelectric ceramic disks," (Ultrasonics Symposium, Denver, Colorado, 701-706, 1987)

8

Guo N., Nanyang Technical University, Singapore, Privat kommunikasjon, ved P. Lunde ved CMR og M. Vestrheim ved UiB (1994). Har hatt tilgang til simulerte data (fil på diskett) gjennom CMR.

9

Guo N., Nanyang Technical University, Singapore, Privat kommunikasjon, ved P. Lunde ved CMR, M. Vestrheim ved UiB og J. Kocbach ved UiB (23. januar 1996 med svar 15. mars 1996).

10

Guo N., The vibration characteristics of piezoelectric discs, (Doctoral Thesis, department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, London, SW7, 1989)

11

Burnett D.S., Finite element analysis - from concepts to applications, (AT& T Bell Laboraties, Whippany, New Jersey, 1987)

12

Lerch R., "Berechnung des Schwingungsverhaltens piezoelektrischer kørper mit einem Vektorprozessor," Siemens Forschungs- und Entwicklungsberichte, Springer-Verlag 15(5), 234-238 (1986)

13

Ostergaard D.F. and Pawlak T.P., "Three-dimensional finite elements for analyzing piezoelectric structures," Proceedings of the IEEE Ultrasonics Symposium, Williamsburg, VA, 639-642 (1986)

14

Kagawa Y. and Yamabuchi T., "Finite element approach for a piezoelectric circular rod", IEEE Transactions on Sonics and Ultrasonics SU-23, 379-385 (1976)

15

Lerch, R., "Simulation of piezoelectric devices by two- and three-dimensional finite elements," IEEE-Transactions on Ultrasonics, Ferroelectrics and Frequency Control 37(2), 233-247 (1990)

16

Jungermann R.L. and Kino G.S., "Measurment of normal surface displacement for the characterization of rectangular acoustic array elements", Journal of the Acoustical Society of America 76, 516-524 (1984)

17

EerNisse E.P., "Variational method for electroelastic vibration analysis", IEEE Transactions on Sonics and Ultrasonics SU-14 (4), 153-160 (1967)

18

Holland R. and EerNisse E.P., Design and analysis of piezoelectric resonant devices, (The M.I.T. press, Cambridge, Massachsetts and London, England, 1969)

19

Reinen T.A. and Berg T., "Tredimensjonal modellering av piezoelektriske svingarar. Sluttrapport, Rapport nummer STF40 A89032", ELAB-RUNIT, 7034 Trondheim, Norway (1989)

20

Paul H.S., "Vibrations of circular cylindrical shells of piezoelectric silver iodide crystals", The Journal of the Acoustical Society of America 40 1077-1080 (1966)

21

Lanceleur P., Francois de Belleval J. and Mercier N, "Modeling of transient deformation of piezoelectric ceramics", IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency control 39 (2), 293-301 (1992)

22

Berlincourt D., Currran D.R. and Jaffe H., "Piezoelectric and piezomagnetic materials and their function in transducers," (In: Physical Acoustics, Principles and Methods. Volume 1 - Part A., Edited by W. P. Mason (Academic Press, New York, 1964), Chapter 3

23

Jensen, H., "Calculations for piezoelectric ultrasonic transducers, report R-536", Risø National Laboratory, P.O. Box 49, DK-4000 Roskilde, Denmark, (1986)

24

Coursant R.H., Mequio C. and Pesque P., "Simulation of the acousto-electric response of ultrasonic narrow strip transducers with mechanical losses", 1983 Ultrasonics symposium, 414-419, (1983)

25

Naillon M., Coursant R.H. and Besnier F., "Analysis of piezoelectric structures by a finite element method", Acta Electronica 25 4, 341-362 (1983)

26

Boucher D., Lagier M. and Maerfeld C., "Computation of the vibrational modes for piezoelectric array transducers using a mixed finite element-pertubation method", IEEE Transactions on Sonics, Ultrasonics and Frequency control SU-28, 318-330 (Sept. 1981)

27

Allik H., Webman K.M. and Hunt J.T, "Vibrational response of sonar transducers using piezoelectric finite elements," Journal of the Acoustical Society of America 56, 1782-1791 (1974)

28

NAFEM, The standard NAFEM Benchmarks, (1990)

29

Abaqus/Standard Verification Manual, Version 5.4, (Hibbit, Karlsson & Sorensen, 1994)

30

Abaqus/Standard Users Manual, Version 5.4, (Hibbit, Karlsson & Sorensen, 1994)

31

Reitz, Milford and Christy, Foundation of Electromagnetic Theory, (Addison-Wesley, 1993)

32

Ellingsen, T. Klassisk Mekanikk, (Matematisk Institutt, Universitet i Bergen, 1987)

33

Raoelijaona F. and Dulmet B., "Finite element analysis using trigonometric interpolations for quasi-thickness piezoelectric resonators", 1994 Ultrasonic symposium, 965-968 (1994)

34

Ghali A. and Neville A.M., Structural Analysis : A unified classical and matrix approach, Second Edition,( Chapman & Hall Ltd, 1978)

35

Vestrheim M., Akustiske målesystemer, Forelesningsnotater i FYS373, akustiske målesystemer ved Fysisk Institutt, Universitetet i Bergen, foreløpig utgave, (Fysisk Institutt, Universitetet i Bergen, Bergen, 1994)

36

Vestrheim M., Forelesningsnotater i FYS272, akustiske transdusere ved Fysisk Institutt, Universitetet i Bergen, foreløpig utgave, (Fysisk Institutt, Universitetet i Bergen, Bergen, 1994)

37

Cook, R.D., Concepts and Applications of Finite Element Analysis, 2d ed., (Wiley, New York, 1981)

38

Barlow, J., "Optimal stress locations in finite element models," International Journal of Numerical Methods in Engineering 10 243-251 (1976)

39

IEEE Standard on Piezoelectricity - Std. 176 - 1978, (The Institute of Electrical and Electronics Engineers, Inc, New York, 1978)

40

IEEE Standard on Piezoelectricity - Std. 176 - 1987, (The Institute of Electrical and Electronics Engineers, Inc, New York, 1987)

41

Guide to dynamic measurments of piezoelectric ceramics with high coupling, (International Electrotechnical Commission, Publication 483, 1976)

42

Zienkewitz O.C., The Finite Element Method, (McGraw-Hill, New York, 1977)

43

Challande P., "Application de la methode des elements finis a l'etide de cavites piezoelectriques: influence de la geometrie sur les modes de vibration et leur coefficient de couplage", Jornal de Mecanique theorique et appliquee 7(4), 461-477 (1988)

44

Mercer C.D., Reddy B.D. and Eve R.A, Finite element method for piezoelectric media, Technical report UCT/CSIR Applied Mechanics Research Unit No. 92, (Cape Town , University of Cape Town, 1987)

45

Sigmon K., Matlab primer, (Boca Raton , CRC Press, 1994)

46

Abderson E. et. al., LAPACK user's guide, (Philadelphia, Pa. , Society for Industrial and Applied Mathematics, 1992)

47

Kinsler L.E and Frey A.R., Fundamentals of acoustics 2nd ed., (New York , Wiley, c1962)

48

Naillon M., Besnier F. and Coursant R.H. "Finite element analysis of narrow piezoelectric parallepiped vibrations. Energetical coupling modeling", IEEE 1983 Ultrasonics Symposium, Atlanta, 773-777 (1983)

49

Guo N. and Cawley P., "The finite element analysis of the vibration characteristics of piezoelectric discs," Journal of Sound and Vibration, 159(1), 115-138 (1992)

50

Guo N. and Cawley P., "Measurment and prediction of the frequency spectrum of piezoelectric disks by modal analysis," Journal of the Acoustical Society of America, 92(6), 3379-3387 (1992)

51

Guo N. and Cawley P., "Transient response of piezoelectric discs to applied voltage pulses," Ultrasonics 1991, 29, 208-216 (1992)

52

Kagawa Y., Tsuchiya T. and Kataoka T., "Finite element simulation of responses of piezoelectric actuators", Journal of Sound and Vibration,191(4), 519-538 (1996)

53

Lunde P. and Vestrheim M., "Piezoelectric transducer modelling, Part I: Basic equations for poled ferroelectric ceramics," Chr. Michelsen Institute, Bergen, CMI Report No. CMI-91-A10010 (December, 1991), (Confidential)

54

Mason W.P., Electromechanical transducers and wave filters, Second edition, (D.Can. Norstrand Company Inc., 1948)

55

Krimholtz R., Leedom D.A. and Matthaei G.L., "New equivalent circuits for elementary piezoelectric transducers," Electronic Letters 6(13), 398-399 (1970)

56

Tiersten H.F, Linear piezoelectric plate vibrations: Elements of the linear theory of piezoelectricity and the vibrations of piezoelectric plates, (New York , Plenum Press, 1969)

57

Mindlin R.D and Medick M.A., "Extensional vibrations of elastic plates", ASME Journal of Applied Mechanics 26, 561-569 (1959)

58

Aggarwal R.R., "Axially Symmetric vibrations of a finite isotropic disk. I", Journal of the Acoustical Society of America 24, 463-467 (1952)

59

Aggarwal R.R., "Axially Symmetric vibrations of a finite isotropic disk. II", Journal of the Acoustical Society of America 24, 663-666 (1952)

60

Ueha S., Sakuma S. and Mori E., "Measurment of vibration velocity distributions and mode analysis in thick disks of ", Journal of the Acoustical Society of America 73, 1842-1847 (1983)

61

Ikegami S., Ueda L. and Kobayashi S., "Frequency spectra of resonant vibration in disk plates of piezoelectric ceramics", Journal of the Acoustical Society of America 55, 339-344 (1974)

62

Piezoelectric ceramic data Bulletin 66011/F, (Vernitron Piezoelectric Division, 232 Forbes Road, Bedford, OH 44146, 1976)

63

Piezoceramics, High quality components and materials for the electronics industry, (Ferroperm, Stubbeled 7, DK-Vedbæk, Denmark, 1992)

64

Meitzler A.H., O'Bryan H.M.Jr. and Tiersten H.F., "Definition and measurment of radial mode coupling factors in piezoelectric ceramic materials with large variations in poisson's ration", IEEE Transactions on Sonics and Ultrasonics, SU-20 (3), 233-239 (1973)

65

Gindre M., Urbach W., Coursant R.H. and Fink M., "Mechenical displacement induced in a piezoelectric structure: Experimental measurement by laser interferometry and simulation by a finite element method", Journal of the Acoustical Society of America 84(1), 11-19 (1988)

66

Challande P., "Optimizing ultrasonic transducers based on piezoelectric composites using a finite-element method", IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 17(2), 135-140 (1990)

67

Davis P., Rabinowitz P., Methods of numerical integration, (New York , Academic Press, 1975)

68

Du Toit H.F., George P.L., Laug P., Pate P., Steer D., Vidrascu M., An introduction to MODULEF: MODULEF user guide no. 1, INRIA, France, 1991

69

PZflex: Flex Support, Weidlinger Associates, Inc., 4410 El Camino Real, Suite 110, Los Altos, CA 94022, USA

70

Atila: Magsoft Corporation, 1223 Peoples Avenue , Troy, NY 12180

71

Ansys revision 5.0 , technical description of capabilities, (Swanson Analysis Systems, Inc., Johnson Road, P.O. Box 65, Houston, PA 15342-0065, USA, 1992)