Abstract
Considering the quadratic nonlinear constitutive relations of piezoelectric materials, a traveling wave dynamic model for a lead zirconate titanate stator of a traveling wave ultrasonic motor is established using Hamilton’s principle and the Rayleigh–Ritz method. Applying the method of multiple scales, the second-order approximation of the primary resonance for traveling wave vibration of the stator is investigated. The second harmonic component is found in the primary response of the stator, which arises from the quadratic stiffness in the condition of weak excitation. In the region of the resonance, the two coupled modals are split and the lower-order peak bends to the left, hence a jump and delay exist in the response. In this way numerical results are given to verify the feasibility of the analytical approach. The results provide a theoretical foundation for further nonlinear dynamic analysis and design of the traveling wave ultrasonic motor.
Similar content being viewed by others
References
Qu, Z.Q.: An efficient modelling method for laminated composite plates with piezoelectric sensors and actuators. Smart Mater. Struct. 10, 807–818 (2001)
Krommer, M., Irschik, H.A.: Reissner–Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect. Acta Mech. 141, 51–69 (2000)
Tzou, H.S., Chai, W.K., Arnold, S.M.: Structronics and actuation of hybrid electrostrictive/piezoelectric thin shells. J. Vib. Acoust. 128, 79–87 (2006)
Chen, Y., Han, J.L., Liu, R.H.: Nonlinear vibration of transversely isotropic piezoelectric rectangular plate. J. Nanjing Univ. Aeronaut. Astronaut. 35(1), 18–24 (2003)
Joshi, S.P.: Nonlinear constitutive relations for piezoceramic materials. Smart Mater. Struct. 1, 80–83 (1992)
Hom, C.L., Shankar, N.A.: Dynamics model for nonlinear electrostrictive actuators. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45(2), 409–420 (1998)
Hom, C.L., Shankar, N.A.: Fully coupled constitutive model for electrostrictive ceramic materials. Intell. Mater. Syst. Struct. 5, 795–801 (1994)
von Wagner, U., Hagedorn, P.: Nonlinear effects of piezoceramics excited by weak electric fields. Nonlinear Dyn. 31, 133–149 (2003)
Parashar, S.K., von Wagner, U., Hagedorn, P.: Nonlinear shear-induced flexural vibrations of piezoceramic actuators: experiments and modeling. J. Sound Vib. 285, 989–1014 (2005)
Cao, S.Q., Gao, J.: Nonlinear dynamic model and primary resonance of piezoelectric laminated disk. J. Tianjin Univ. 40, 139–147 (2007)
Hagood, N.W., McFarland, A.J.: Modeling of a piezoelectric rotary ultrasonic motor. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(2), 210–224 (1995)
Norddin, E.G.: Hybrid modeling of a traveling wave piezoelectric motor. Ph.D. thesis, Aalborg University, Aalborg, Denmark, p. 93 (2000)
Parashar, S.K., von Wagner, U.: Nonlinear longitudinal vibrations of transversally polarized piezoceramics: experiments and modeling. Nonlinear Dyn. 37, 51–73 (2004)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Willey-Interscience, New York (1979)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gao, J., Cao, S.Q. Second-order approximation of primary resonance of a disk-type piezoelectric stator for traveling wave vibration. Nonlinear Dyn 61, 591–603 (2010). https://doi.org/10.1007/s11071-010-9673-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-010-9673-y