1560. A series solution for the in‑plane vibration of sector plates with arbitrary inclusion angles and boundary conditions
Kaipeng Zhang1, Dongyan Shi2, Xiaoyan Teng3, Yunke Zhao4, Qian Liang5
College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, China
E-mail: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
(Received 15 September 2014; received in revised form 12 January 2015; accepted 20 January 2015)
Abstract. In this investigation, a Spectro-Geometric Method (SGM) is presented for the in‑plane vibration analysis of sector plates with an arbitrary inclusion angle, and general boundary conditions along each of its edges. Unlike in most existing studies where solutions are often developed for a particular type of boundary conditions, the current method can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures; that is, the in-plane displacement functions are invariably expressed as an accelerated trigonometric series expansion and different boundary conditions can be easily dealt with through modifying the stiffness values for restraining springs in the same way as varying other model parameters such as Young’s modulus and mass density. The expansion coefficients are considered as the generalized coordinates, and determined using the Rayleigh‑Ritz technique. The effectiveness and reliability of the current method are demonstrated by the calculated modal properties for sector plates with a range of different combinations of boundary conditions and inclusion angles up to .
Keywords: sector plates, in-plane vibration, arbitrary inclusion angles, arbitrary boundary conditions, spectro‑geometric method (SGM).
 Leissa A. W. Vibration of Plates. U. S. Government Printing Office, Washington DC, 1969.
 Bercin A. N. An assessment of the effects of in-plane vibrations on the energy flow between coupled plates. Journal of Sound and Vibration, Vol. 191, Issue 5, 1996, p. 661-680.
 Lyon R. H. In-plane contribution to structural noise transmission. Noise Control Engineering Journal, Vol. 26, Issue 1, 1985, p. 22-27.
 Onoe M. Contour vibrations of isotropic circular plates. The Journal of the Acoustical Society of America, Vol. 28, Issue 6, 1956, p. 1158-1162.
 Onoe M. Gravest contour vibration of thin anisotropic circular plates. The Journal of the Acoustical Society of America, Vol. 30, Issue 7, 1958, p. 698.
 Holland R. Numerical studies of elastic-disk contour modes lacking axial symmetry. The Journal of the Acoustical Society of America, Vol. 40, Issue 5, 1966, p. 1051-1057.
 Chen S. S. H., Liu T. M. Extensional vibration of thin plates of various shapes. The Journal of the Acoustical Society of America, Vol. 58, Issue 4, 1975, p. 828-831.
 Irie T., Yamada G., Muramoto R. Natural frequencies of in-plane vibration of annular plates. Journal of Sound and Vibration, Vol. 97, Issue 1, 1984, p. 171-175.
 Farag N. H., Pan J. Modal characteristics of in-plane vibration of circular plates clamped at the outer edge. The Journal of the Acoustical Society of America, Vol. 113, Issue 4, 2003, p. 1935‑1946.
 Park C. Il. Frequency equation for the in-plane vibration of a clamped circular plate. Journal of Sound and Vibration, Vol. 313, Issue 1-2, 2008, p. 325-333.
 Bashmal S., Bhat R., Rakheja S. In-plane free vibration of circular annular disks. Journal of Sound and Vibration, Vol. 322, Issue 1-2, 2009, p. 216-226.
 Bashmal S., Bhat R., Rakheja S. In-plane free vibration analysis of an annular disk with point elastic support. Shock and Vibration, Vol. 18, Issue 4, 2011, p. 627-640.
 Karamooz Ravari M. R., Forouzan M. R. Frequency equations for the in-plane vibration of orthotropic circular annular plate. Archive of Applied Mechanics, Vol. 81, Issue 9, 2011, p. 1307‑1322.
 Kim C. B., Cho H. S., Beom H. G. Exact solutions of in-plane natural vibration of a circular plate with outer edge restrained elastically. Journal of Sound and Vibration, Vol. 331, Issue 9, 2013, p. 2173‑2189.
 Shi X. J., Li W., Shi D. Y. Free in-plane vibrations of annular sector plates with elastic boundary supports. Proceedings of Meetings on Acoustics, Vol. 18, Issue 1, 2012, p. 1-11.
 Seok J. W., Tiersten H. F. Free vibrations of annular sector cantilever plates. Part 2: in-plane motion. Journal of Sound and Vibration, Vol. 271, Issue 3-5, 2004, p. 773-787.
 Singh A. V., Muhammad T. Free in-plane vibration of isotropic non-rectangular plates. Journal of Sound and Vibration, Vol. 273, Issue 1-2, 2004, p. 219-231.
 Jomehzadeh E., Saidi A. R. Analytical solution for free vibration of transversely isotropic sector plates using a boundary layer function. Thin-Walled Structures, Vol. 47, Issue 1, 2009, p. 82‑88.
 Leissa A. W., Mcgee O. G., Huang C. S. Vibrations of sectorial plates having corner stress singularities. Journal of Applied Mechanics – Transactions of the ASME, Vol. 60, Issue 1, 1993, p. 134‑140.
 Ramaiah G. K., Vijayaku K. Natural frequencies of circumferentially truncated sector plates with simply supported straight edges. Journal of Sound and Vibration, Vol. 34, Issue 1, 1974, p. 53‑61.
 Shi D. Y., Wang Q. S., Shi X. J., Pang F. Z. A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports. Archive of Applied Mechanics, Vol. 85, Issue 1, 2015, p. 51-73.
 Tolstov G. P. Fourier Series. Prentice Hall, Englewood Cliffs, NJ, 1965.
 Li W. L. Vibration analysis of rectangular plates with general boundary conditions. Journal of Sound and Vibration, Vol. 273, Issue 3, 2004, p. 619‑635.
 Baszenski G., Delvos F. J., Tasche M. A united approach to accelerating trigonometric expansions. Computers and Mathematics with Applications, Vol. 30, Issue 3-6, 1995, p. 33‑49.
 Shi X. J., Shi D. Y., Li W. L., Wang Q. S. A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions. Journal of Vibration and Control, 2014.
 Gorman D. J. Exact solutions for the free in-plane vibration of rectangular plates with two opposite edges simply supported. Journal of Sound and Vibration, Vol. 294, Issue 1-2, 2006, p. 131‑161.
Cite this article
Zhang Kaipeng, Shi Dongyan, Teng Xiaoyan, Zhao Yunke, Liang Qian A series solution for the in‑plane vibration of sector plates with arbitrary inclusion angles and boundary conditions. Journal of Vibroengineering, Vol. 17, Issue 2, 2015, p. 870‑882.
© JVE International Ltd. Journal of Vibroengineering. Mar 2015, Volume 17, Issue 2. ISSN 1392-8716