Abstract
Dynamic analysis of piezoelectric continuum is quite complex, because it includes the coupling of electrical and mechanical terms. Complexity further increases with the consideration of three-dimensional model analysis. Implementation of the analytical methods in order to obtain a closed-form solution is a cumbersome task even in the case of a moderately complex model. As an alternative, approximation methods are used to circumvent these difficulties. In the present work, Rayleigh–Ritz method is used to obtain the eigenvectors and the eigenfrequencies for the piezoceramic cylindrical shells. Orthogonal polynomials generated using Gram–Schmidt method are used in the Rayleigh–Ritz method to formulate the eigenvalue problem. The present method has an advantage of being adapted by software with great ease and provides much accurate results, which makes it reasonably useful for analysis of more complex piezoceramic structures. The presented approximation method is validated by comparing the results with the previous works done in similar problem and by the finite element analysis.
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References
Piefort, V.: Finite element modelling of piezoelectric active structures. Dissertation, Department of Mechanical Engineering and Robotics, Université Libre de Bruxelles, Bruxelles (2001)
Pahk H.J., Lee D.S., Park J.H.: Ultra precision positioning system for servo motor-piezo actuator using the dual servo loop and digital filter implementation. Int. J. Mach. Tools Manuf. 41(1), 51–63 (2001)
Xing Z., He B., Xu K., Wang J., Dong S.: A miniature cylindrical piezoelectric motor with an asymmetric vibrator. J. Ultrason. Ferroelectr. Freq. Control 60(7), 1498–1504 (2013)
Lin S., Zhiqiang Fu Z., Zhang X., Wang Y., Hu J.: Radially sandwiched cylindrical piezoelectric transducer. J. Smart Mater. Struct. 22(1), 964 (2013)
Nishamol P.A., Ebenezer D.D.: Exact series model of Langevin transducers with internal losses. J. Acoust. Soc. Am. 135(3), 1159 (2014)
Ramamurti V., Pattabiraman J.: Free vibrations of circular cylindrical shells. J. Sound Vib. 48(1), 137–155 (1976)
Chung H.: Free vibration analysis of circular cylindrical shells. J. Sound Vib. 74(3), 331–350 (1981)
Loy C.T., Lam K.Y.: Vibration of thick cylindrical shells on the basis of three-dimensional theory of elasticity. J. Sound Vib. 226(4), 719–737 (1999)
Malekzadeha P., Faridb M., Zahedinejadc P., Karami V.: Three-dimensional free vibration analysis of thick cylindrical shells resting on two-parameter elastic supports. J. Sound Vib. 313(3–5), 655–675 (2008)
Alibeigloo A., Kani A.M.: 3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method. Appl. Math. Model. 34(12), 4123–4137 (2010)
Kapuria S., Sengupta S., Dumir P.C.: Three-dimensional solution for simply-supported piezoelectric cylindrical shell for axisymmetric load. Comput. Methods Appl. Mech. Eng. 140(1), 139–155 (1997)
Chen C.Q., Shen Y.P.: Three-dimensional analysis for the free vibration of finite-length orthotropic piezoelectric circular cylindrical shells. J. Vib. Acoust. 120(1), 194–198 (1998)
Chen W.Q., Bian Z.G., Lv C.F., Ding H.J.: 3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid. Int. J. Solids Struct. 41(3–4), 947–964 (2004)
Meirovitch L., Kwak M.K.: Convergence of the classical Rayleigh–Ritz method and the finite element method. AIAA J. 28(8), 1509–1516 (1990)
Leissa A.W.: The historical bases of the Rayleigh and Ritz methods. J. Sound Vib. 287(4–5), 961–978 (2005)
Kumar V.: Boundary characteristic orthogonal polynomials for singularly perturbed problems. Int. J. Pure Appl. Math. 75(4), 427–439 (2012)
Bhat R.B.: Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh–Ritz method. J. Sound Vib. 102, 493–499 (1985)
IEEE Std.: IEEE Standards on Piezoelectricity. The Institute of Electrical and Electronics Engineers, New York (1987)
Parashar S.K., von Wagner U., Hagedorn P.: Nonlinear shear-induced flexural vibrations of piezoceramic actuators: experiments and modeling. J. Sound Vib. 285(4), 989–1014 (2004)
Parashar S.K.: Modeling and analysis of shear-induced flexural vibrations of annular piezoceramic actuators. J. Intell. Mater. Syst. Struct. 24(13), 1572–1582 (2013)
Graham A.: Kronecker Products and Matrix Calculus with Applications. Halsted Press, Wiley, New York (1981)
Berg M., Hegedron P., Gutschmidt S.: On the dynamics of piezoelectric shells. J. Sound Vib. 274(1–2), 91–109 (2004)
Rao S.S.: Mechanical Vibrations, 4th edn. Pearson Education, Delhi (2004)
Flügge W.: Stresses in Shells, 2nd edn. Springer, Berlin (1973)
Samal M.K., Seshu P., Parashar S.K., Wagner U., Hagedorn P., Dutta B.K., Kushwaha H.S.: Nonlinear behavior of piezoceramics under weak electric fields. Part-II: numerical results and validation with experiments. Int. J. Solids Struct. 43, 1437–1458 (2006)
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Parashar, S.K., Kumar, A. Three-dimensional analytical modeling of vibration behavior of piezoceramic cylindrical shells. Arch Appl Mech 85, 641–656 (2015). https://doi.org/10.1007/s00419-014-0977-0
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DOI: https://doi.org/10.1007/s00419-014-0977-0