A modified Fourier-Ritz method for free vibration of rectangular plates with elastic constrains

• Autorzy: Wu Tao , Chen Zhao Bo , Qu Jian Jun

Identyfikatory

DOI

10.15632/jtam-pl/144462

• Języki publikacji: EN

Abstrakty

EN

In this paper, a modified Fourier-Ritz method is used to study free vibration of a rectangular plate with a set of simply supported opposite sides and another set of arbitrary elastic constraints. The influence of different elastic constraint stiffness values on the modal response of the rectangular plate is also analyzed. In order to avoid that the displacement function of the rectangular plate calculated by the traditional method and its derivative may be discontinuous or non-derivable at the boundary, the displacement function is expressed in the form of the sum of standard cosine series and a periodic polynomial function. Compared with the sine series expansion, the convergence of the result is enhanced. Several sets of numerical examples with different boundary conditions are given in the article, the data shows that the results calculated by this method have good accuracy and fast convergence. In addition, this paper also analyzes the boundary conditions and discusses the influence of different spring stiffness values on the setting of boundary conditions. The results can be applied to the setting of general boundary conditions and the study of vibration control of rectangular plates.

Słowa kluczowe

EN

modified Fourier-Ritz method; rectangular plate; free vibration; boundary conditions

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2022
• Tom: Vol. 60 nr 1
• Strony: 77--89
• Opis fizyczny: Bibliogr. 28 poz., tab.

Twórcy

autor

Wu Tao

• School of Mechatronics Engineering Harbin Institute of Technology, Harbin, China

autor

Chen Zhao Bo

• School of Mechatronics Engineering Harbin Institute of Technology, Harbin, China

autor

Qu Jian Jun

• School of Mechatronics Engineering Harbin Institute of Technology, Harbin, China

Bibliografia

• 1. Abdulkerim S., Dafnis A., Riemerdes H., 2019, Experimental investigation of nonlinear vibration of a thin rectangular plate, International Journal of Applied Mechanics, 11, 267-289.
• 2. Alkhayal J., Chehab J.P., Jazar M., 2019, Existence, uniqueness, and numerical simulations of Föppl-von Kármán equations for simply supported plate, Mathematical Methods in the Applied Sciences, 42, 7482-7493.
• 3. Bahrami A., Bahrami M.N., Ilkhani M.R., 2014, Comments on “New exact solutions for free vibrations of thin orthotropic rectangular plates”, Composite Structures, 107, 745-746.
• 4. Banerjee J.R., Papkov S.O., Liu X., Kennedy D., 2015, Dynamic stiffness matrix of a rectangular plate for the general case, Journal of Sound and Vibration, 342, 177-199.
• 5. Bert C.W., Devarakonda K.K., 2003, Buckling of rectangular plates subjected to nonlinearly distributed in-plane loading, International Journal of Solids and Structures, 40, 4097-4106.
• 6. Cao Z. Y., 1989, Vibration Theory of Plates and Shells, China Railway Publishing House, Beijing.
• 7. Du J., Li W.L., Jin G., Yang T., Liu Z., 2007, An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges, Journal of Sound and Vibration, 306, 908-927.
• 8. Eisenberger M., Deutsch A., 2019, Solution of thin rectangular plate vibrations for all combinations of boundary conditions, Journal of Sound and Vibration, 452, 1-12.
• 9. Gavalas G.R., El-Raheb M., 2014, Extension of Rayleigh-Ritz method for eigenvalue problems with discontinuous boundary conditions applied to vibration of rectangular plates, Journal of Sound and Vibration, 333, 4007-4016.
• 10. Han Qingkai Z.J.Z.H., 2016, Fundamental and Numerical Simulation on Machinery Dynamics and Vibration, Wuhan University of Technology Press.
• 11. Hao Q., Zhai W., Chen Z., 2018, Free vibration of connected double-beam system with general boundary conditions by a modified Fourier-Ritz method, Archive of Applied Mechanics, 88, 741-754.
• 12. He W., Chen W., Qiao H., 2013, In-plane vibration of rectangular plates with periodic inhomogeneity: Natural frequencies and their adjustment, Composite Structures, 105, 134-140.
• 13. Huang C.S., Lin Y.J., 2016, Fourier series solutions for vibrations of a rectangular plate with a straight through crack, Applied Mathematical Modelling, 40, 10389-10403.
• 14. Ilanko S., Monterrubio L.E., Mochida Y., 2014, The Rayleigh-Ritz Method for Structural Analysis, Wiley, Somerset.
• 15. Leissa A.W., 1993, Vibration of Plates, Acoustical Society of America, Washington, D.C.
• 16. Li R., Wang H., Zheng X., Xiong S., Hu Z., Yan X., Xiao Z., Xu H., Li P., 2019, New analytic buckling solutions of rectangular thin plates with two free adjacent edges by the symplectic superposition method, European Journal of Mechanics – A/Solids, 76, 247-262.
• 17. Li W.L., 2001, Dynamic analysis of beams with arbitrary elastic supports at both ends, Journal of Sound and Vibration, 246, 751-756.
• 18. Li W.L., Daniels M., 2002, A Fourier series method for the vibrations of elastically restrained plates arbitrarily loaded with springs and masses, Journal of Sound and Vibration, 252, 768-781.
• 19. Liu X., Banerjee J.R., 2016, Free vibration analysis for plates with arbitrary boundary conditions using a novel spectral-dynamic stiffness method, Computers and Structures, 164, 108-126.
• 20. Mahapatra K., Panigrahi S.K., 2020, Dynamic response and vibration power flow analysis of rectangular isotropic plate using Fourier series approximation and mobility approach, Journal of Vibration Engineering and Technologies, 8, 105-119.
• 21. Najarzadeh L., Movahedian B., Azhari M., 2018, Free vibration and buckling analysis of thin plates subjected to high gradients stresses using the combination of finite strip and boundary element methods, Thin-Walled Structures, 123, 36-47.
• 22. Shi D., Wang Q., Shi X., Pang F., 2015, An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports, Proceedings of the Institution of Mechanical Engineers. Part C, Journal of Mechanical Engineering Science, 229, 2327-2340.
• 23. Shi X., Shi D., Li W.L., Wang Q., 2016, A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions, Journal of Vibration and Control, 22, 442-456.
• 24. Wang X., Gan L., Wang Y., 2006, A differential quadrature analysis of vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses, Journal of Sound and Vibration, 298, 420-431.
• 25. Wei Z.,Yin X.,Yu S.,Wu W., 2020, Dynamic stiffness formulation for transverse and in-plane vibration of rectangular plates with arbitrary boundary conditions based on a generalized superposition method, International Journal of Mechanics and Materials in Design, 17,1, 119-135
• 26. Xing Y.F., Xu T.F., 2013, Solution methods of exact solutions for free vibration of rectangular orthotropic thin plates with classical boundary conditions, Composite Structures, 104, 187-195.
• 27. Zhang J., Lu J., Ullah S., Gao Y., Zhao D., 2020, Buckling analysis of rectangular thin plates with two opposite edges free and others rotationally restrained by finite Fourier integral transform method, ZAMM – Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 101, 4.
• 28. Zhang J., Ullah S., Zhong Y., 2020, Accurate free vibration solutions of orthotropic rectangular thin plates by straightforward finite integral transform method, Archive of Applied Mechanics, 90, 353-368.