Abstract
The application of piezoceramics excited near resonance frequencies using weak electric fields, such as in ultrasonic motors, has led to close investigations of their behavior at this state. Typical nonlinear effects, such as a softening behavior, have been observed in resonantly driven piezoceramic structures, which cannot be adequately defined by linear theories. In piezoceramic actuators the d 15-effect is very attractive for the application, as the corresponding piezoelectric coupling coefficient is higher than that for d 31- and d 33-effects. In the present paper, the authors have modeled nonlinear flexural vibrations of a resonantly driven piezoceramic cantilever beam, which is excited using the d 15-effect. First, a linear description of the problem is investigated. A modified Timoshenko beam theory is formulated and an appropriate description of the electric field is included. The exact analytical solution of the linear field equations along with the associated boundary conditions is obtained. As an alternative approach, the Rayleigh–Ritz energy method is also used to obtain the eigenfrequencies and the eigenvectors. Series comprising orthogonal polynomial functions, generated using the Gram-Schmidt method, are used in the Rayleigh–Ritz method. To ascertain the efficacy of the Rayleigh–Ritz method for piezoceramic continua, the results obtained from it are compared with the exact analytical solution. To model the observed nonlinear effects, the electric enthalpy density is extended including higher-order conservative terms. The constitutive relations are correspondingly extended and additional nonlinear damping terms are adjoined. Hamilton's principle via the Ritz method is used to derive the discretized nonlinear equations of motion of a piezoceramic cantilever beam. The eigenfunctions of the linear case are used as shape functions in the Ritz method for discretization. The approximate solution of the nonlinear equation of motion is obtained using the perturbation method. Nonlinear parameters are determined by comparing the theoretical and the experimental responses. The modeling technique and the nonlinear effects described in this paper should be helpful in optimizing the existing applications and developing new applications based on the d 15-effect.
Similar content being viewed by others
References
Kamlah, M., 'Ferroelectric and ferroelastic piezoceramics-modeling of electromechanical hysteresis phenomena', Continuum Mechanics and Thermodynamics 13, 2001, 219–268.
Ge, P. and Jounaeh, M., 'Generalized Preisach model for hysteresis nonlinearity of piezoceramic actuators', Precision Engineering 20, 1997, 99–111.
Royston, T. J. and Houston, B. H., 'Modeling and measurement of nonlinear dynamic behavior in piezoelectric ceramics with application to 1-3 composites', Journal of Acoustical Society of America 104(5), 1998, 2814–2827.
Lee, S. H. and Royston, T. J., 'Modeling piezoceramic transducer hysteresis in the structural vibration control problem', Journal of Acoustical Society of America 108(6), 2000, 2843–2853.
Chan, K. H. and Hagood, N.W., 'Modeling of nonlinear piezoceramics for structural actuation', in Proceedings of the SPIE: Smart Structures and Intelligent Systems, Vol. 2190, 1994, pp. 194–205.
Fan, J., Stoll, W. A., and Lynch, C. S., 'Nonlinear constitutive behavior of soft and hard PZT: Experiments and modeling', Acta Materialia 47(17), 4415–4425.
Holland, R. and Eer Nisse, E. P., 'Accurate measurement of coefficients in a ferroelectric ceramic', IEEE Transactions on Sonics and Ultrasonics SU-16(4), 1969, 173–181.
Beige, H. and Schmidt, G., 'Electromechanical resonances for investigating linear and nonlinear properties of dielectrics', Ferroelectrics 41, 1982, 39–49.
Beige, H., 'Elastic and dielectric nonlinearities of piezoelectric ceramics', Ferroelectrics 51, 1983, 113–119.
Joshi, S. P., 'Non-linear constitutive relations for piezoceramic materials', Smart Materials and Structures 1, 1992, 80–83.
McMahon, D. H., 'Acoustic second-harmonic generation in piezoelectric crystals', Journal of Acoustical Society of America 44(4), 1968, 1007–1013.
Jiang, W. and Cao, W., 'Nonlinear properties of lead zirconate-titanate piezoceramics', Journal of Applied Physics 88(11), 2000, 6684–6689.
Maugin, G. A., Nonlinear Electromechanical Effects and Applications, World Scientific, Singapore, 1985.
Altay, G. A. and Dökmeci, M. C., 'A non-linear rod theory for high-frequency vibrations of thermopiezoelectric materials', International Journal of Non-Linear Mechanics 37, 2002, 225–243.
Guyomar, D., Aurelle, N., and Eyraud, L., 'Piezoelectric ceramics nonlinear behavior. Application to Langevin transducer', Journal de Physique III 7(6), 1997, 1197–1208.
Arafa, M. and Baz, A., 'On the nonlinear behavior of piezoelectric actuators', Journal of Vibration and Control 10, 2004, 387–398.
Parashar, S.K. and vonWagner, U., ' Nonlinear longitudinal vibrations of transversally polarised piezoceramics: Experiments and modeling', Nonlinear Dynamics 37(1), 2004, 51–73.
von Wagner, U. and Parashar, S. K., 'Nonlinear longitudinal vibrations of transversally polarised piezoceramics', in Proceedings of the Fourth International Symposium of Continuous Systems, Keswick, UK, 2003, pp. 51–53.
vonWagner, U., 'Non-linear longitudinal vibrations of non-slender piezoceramic rods', International Journal of Non-Linear Mechanics 39, 2004, 673–688.
vonWagner, U., 'Nonlinear effects of piezoceramics excited by weak electric fields', Nonlinear Dynamics 31, 2003, 133–149.
Parashar, S. K., DasGupta, A., von Wagner, U., and Hagedorn, P., 'Nonlinear shear vibrations of piezoceramic actuators', International Journal of Non-Linear Mechanics, 2004 (in press).
Parashar, S. K., DasGupta, A., and Hagedorn, P., 'Investigation of nonlinear shear induced flexural vibrations of piezoceramic actuators', SPIE 11th Annual International Symposium on Smart Structures and Materials, San Diego, California, March 14-18, 2004.
Sun, C. T. and Zhang, X. D., 'Use of thickness-shear mode in adaptive sandwich structures', Smart Materials and Structures 4, 1995, 202–206.
Benjeddou, A., Trindade, M. A., and Ohayon, R., 'A unified beam finite element model for extension and shear piezoelectric actuation mechanisms', Journal of Intelligent Material Systems and Structures 8, 1997, 1012–1025.
Trindade, M. A., Benjeddou, A., and Ohayon, R., ' Parametric analysis of the vibration control of sandwich beams through shear-based piezoelectric actuation', Journal of Intelligent Material Systems and Structures 10, 1999, 377–385.
Aldraihem, O. J. and Khdeir, A. A., 'Smart beams with extension and thickness-shear piezoelectric actuators', Smart Materials and Structures 9, 2000, 1–9.
Kim, J. and Kang, B., 'Performance test and improvement of piezoelectric torsional actuators', Smart Materials and Structures 10, 2001, 750–757.
Glazounov, A. E., Zhang, Q. M., and Kim, C., 'Torsional actuator and stepper motor based on piezoelectric d15 shear response', Journal of Intelligent Material Systems and Structures 11, 2000, 456–468.
Dong, S., Kim, H. W., Strauss, M. T., Uchino, K., and Viehland, D., 'A piezoelectric shear-shear mode ultrasonic motor', in Proceedings of the 8th International Conference on New Actuators, Bremen, Germany, 2002, pp. 126–129.
Lee, P. C. Y. and Lin, W. S., 'Piezoelectrically forced vibrations of rectangular SC-cut quartz plates', Journal of Applied Physics 83(12), 1998, 7822–7833.
Wang, Q. and Quek, S. T., 'Flexural vibration analysis of sandwich beam coupled with piezoelectric actuator', Smart Materials and Structures 9, 2000, 103–109.
Krommer, M. and Irschik, H., 'On the influence of the electric field on free transverse vibrations of smart beams', Smart Materials and Structures 8, 1999, 401–410.
Yang, J. S., 'Equations for the extension and flexure of electroelastic plates under strong electric fields', International Journal of Solids and Structures 36, 1999, 3171–3192.
Liu, X., Wang, Q., and Quek, S. T., 'Analytical solution for free vibration of piezoelectric coupled moderately thick circular plates', International Journal of Solids and Structures 39, 2002, 2129–2151.
Timoshenko, S. P., Young, D. H., and Weaver (Jr), W., Vibration Problems in Engineering, 4th edn., Wiley, New York, 1975.
IEEE Std., IEEE Standards on Piezoelectricity, The Institute of Electrical and Electronics Engineers, New York, 1987.
Tiersten, H. F., Linear Piezoelectric Plate Vibrations, Plenum Press, New York, 1969.
Mindlin, R. D. and Deresiewicz, H., 'Timoshenko shear coefficient for flexural vibrations of beam', in Proceedings of 2nd U.S. National Congress of Applied Mechanics, ASME, New York, 1954, pp. 175–178.
Kaneko, T., 'On Timoshenko's correction for shear in vibrating beams', Journal of Physics D: Applied Physics 8, 1975, 1927–1936.
Hutchinson, J. R. and Zillmer, S. D., ' On the transverse vibration of beams of rectangular cross-section', Journal of Applied Mechanics 53, 1986, 39–44.
Bhat, R. B., 'Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method', Journal of Sound and Vibration 102, 1985, 493–499.
Dickinson, S. M. and Di Blasio, A., 'On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates', Journal of Sound and Vibration 108(1), 1986, 51–62.
Oosterhout, G. M., Van Der Hoogt, P. J. M., and Spiering, R. M. E. J., 'Accurate calculation methods for natural frequencies of plates with special attention to the higher modes', Journal of Sound and Vibration 183(1), 1995, 33–47.
Gagnepain, J. J. and Besson, R., 'Nonlinear effects in piezoelectric quartz crystals', in Physical Acoustics Principles and Methods, W. P. Mason, and R. N. Thurston (eds.), Academic Press, New York, 1975.
Nayfeh, A. H. and Lacrabonara, W., 'On the discretization of distributed parameter systems with quadratic and cubic nonlinearities', Nonlinear Dynamics 13, 1997, 203–220.
Pakdemirli, M. and Boyaci, H., 'Comparison of direct-perturbation methods with discretization-perturbation methods for non-linear vibrations', Journal of Sound and Vibration 186(5), 1995, 837–845.
Ikeda, T., Fundamentals of Piezoelectricity, Oxford University Press, Oxford, 1990.
von Wagner, U. and Hagedorn, P., 'Piezo-beam system subjected to weak electric field: Experiments and modelling of non-linearities', Journal of Sound and Vibration 256(5), 2002, 861–872.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
vonWagner, U., 'Nichtlineare Effekte bei Piezokeramiken unter schwachem elektrischem Feld: Experimentelle Untersuchung und Modellbildung', Habilitation thesis, Technische Universität Darmstadt, GCA-Verlag, Herdecke, 2003.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Parashar, S.K., Von Wagner, U. & Hagedorn, P. A Modified Timoshenko Beam Theory for Nonlinear Shear-Induced Flexural Vibrations of Piezoceramic Continua. Nonlinear Dynamics 37, 181–205 (2004). https://doi.org/10.1023/B:NODY.0000044678.78930.cb
Issue Date:
DOI: https://doi.org/10.1023/B:NODY.0000044678.78930.cb