A Modified Timoshenko Beam Theory for Nonlinear Shear-Induced Flexural Vibrations of Piezoceramic Continua

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 d15-effect is very attractive for the application, as the corresponding piezoelectric coupling coefficient is higher than that for d31- and d33-effects. In the present paper, the authors have modeled nonlinear flexural vibrations of a resonantly driven piezoceramic cantilever beam, which is excited using the d15-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 d15-effect.

[1]  T. Ikeda.  Fundamentals of piezoelectricity , 1990 .

[2]  S. Joshi.  Non-linear constitutive relations for piezoceramic materials , 1992 .

[3]  J. J. Gagnepain,et al.  Nonlinear Effects in Piezoelectric Quartz Crystals , 1975 .

[4]  Ali H. Nayfeh,et al.  On the Discretization of Distributed-Parameter Systems with Quadratic and Cubic Nonlinearities , 1997 .

[5]  T. Kaneko.  On Timoshenko's correction for shear in vibrating beams , 1975 .

[6]  P.C.Y. Lee,et al.  Piezoelectrically forced vibrations of rectangular SC-cut quartz plates , 1998 .

[7]  J. R. Hutchinson,et al.  On the Transverse Vibration of Beams of Rectangular Cross-Section , 1986 .

[8]  Jaehwan Kim,et al.  Performance test and improvement of piezoelectric torsional actuators , 2001 .

[9]  R. Bhat.  Natural frequencies of rectangular plates using characteristic orthogonal polynomials in rayleigh-ritz method , 1986 .

[10]  R. Holland,et al.  Accurate Measurement of Coefficients in a Ferroelectric Ceramic , 1969, IEEE Transactions on Sonics and Ultrasonics.

[11]  Sandeep Kumar Parashar,et al.  Nonlinear Longitudinal Vibrations of Transversally Polarized Piezoceramics: Experiments and Modeling , 2004 .

[12]  H. F. Tiersten,et al.  Linear Piezoelectric Plate Vibrations , 1969 .

[13]  Horst Beige,et al.  Electromechanical resonances for investigating linear and nonlinear properties of dielectrics , 1982 .

[14]  Utz von Wagner,et al.  Non-linear longitudinal vibrations of non-slender piezoceramic rods , 2004 .

[15]  Peter Hagedorn,et al.  Non-linear shear vibrations of piezoceramic actuators , 2005 .

[16]  A. E. Glazounov,et al.  Torsional Actuator and Stepper Motor Based on Piezoelectric d15 Shear Response , 2000 .

[17]  Ser Tong Quek,et al.  Flexural vibration analysis of sandwich beam coupled with piezoelectric actuator , 2000 .

[18]  Mustafa Arafa,et al.  On the Nonlinear Behavior of Piezoelectric Actuators , 2004 .

[19]  Thomas J. Royston,et al.  Modeling piezoceramic transducer hysteresis in the structural vibration control problem , 2000 .

[20]  Wenwu Cao,et al.  Nonlinear properties of lead zirconate–titanate piezoceramics , 2000 .

[21]  Daniel Guyomar,et al.  Piezoelectric Ceramics Nonlinear Behavior. Application to Langevin Transducer , 1997 .

[22]  Michael Krommer,et al.  On the influence of the electric field on free transverse vibrations of smart beams , 1999 .

[23]  Roger Ohayon,et al.  A Unified Beam Finite Element Model for Extension and Shear Piezoelectric Actuation Mechanisms , 1997 .

[24]  Roger Ohayon,et al.  Parametric Analysis of the Vibration Control of Sandwich Beams Through Shear-Based Piezoelectric Actuation , 1999 .

[25]  M. Kamlah,et al.  Ferroelectric and ferroelastic piezoceramics – modeling of electromechanical hysteresis phenomena , 2001 .

[26]  J. S. Yang.  Equations for the extension and flexure ofelectroelastic plates under strong electric fields , 1999 .

[27]  Nesbitt W. Hagood,et al.  Modeling of nonlinear piezoceramics for structural actuation , 1994, Smart Structures.

[28]  C. Sun,et al.  Use of thickness-shear mode in adaptive sandwich structures , 1995 .

[29]  Osama J. Aldraihem,et al.  Smart beams with extension and thickness-shear piezoelectric actuators , 2000 .

[30]  G. Altay,et al.  A non-linear rod theory for high-frequency vibrations of thermopiezoelectric materials , 2002 .

[31]  Stephen P. Timoshenko,et al.  Vibration problems in engineering , 1928 .

[32]  H. Beige.  Elastic and dielectric nonlinearities of piezoelectric ceramics , 1983 .

[33]  P.J.M. van der Hoogt,et al.  Accurate calculation methods for natural frequencies of plates with special attention to the higher modes , 1995 .

[34]  Peter Hagedorn,et al.  Nonlinear Effects of Piezoceramics Excited by Weak Electric Fields , 2003 .

[35]  Gérard A. Maugin,et al.  Nonlinear Electromechanical Effects and Applications , 1986 .

[36]  B. Houston,et al.  Modeling and measurement of nonlinear dynamic behavior in piezoelectric ceramics with application to 1-3 composites , 1998 .

[37]  Christopher S. Lynch,et al.  NONLINEAR CONSTITUTIVE BEHAVIOR OF SOFT AND HARD PZT: EXPERIMENTS AND MODELING , 1999 .

[38]  Ser Tong Quek,et al.  Analytical solution for free vibration of piezoelectric coupled moderately thick circular plates , 2002 .

[39]  Musa Jouaneh,et al.  Generalized preisach model for hysteresis nonlinearity of piezoceramic actuators , 1997 .

[40]  P. Hagedorn,et al.  PIEZO–BEAM SYSTEMS SUBJECTED TO WEAK ELECTRIC FIELD: EXPERIMENTS AND MODELLING OF NON-LINEARITIES , 2002 .

[41]  Peter Hagedorn,et al.  Investigation of nonlinear shear-induced flexural vibrations of piezoceramic actuators , 2004, SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring.

[42]  S. M. Dickinson,et al.  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 , 1986 .