Exact solutions for the in-plane vibrations of rectangular Mindlin plates using Helmholtz decomposition

被引:9
作者
Hashemi, Sh. Hosseini [2 ]
Moradi, A. R. [1 ]
机构
[1] Islamic Azad Univ, Arak Branch, Dept Mech Engn, Arak, Iran
[2] Iran Univ Sci & Technol, Dept Mech Engn, Tehran, Iran
关键词
Mode Shape; Rectangular Plate; Frequency Equation; Mindlin Plate; Helmholtz Decomposition;
D O I
10.1007/s00707-010-0342-5
中图分类号
O3 [力学];
学科分类号
08 ; 0801 ;
摘要
Deriving frequency equations for in-plane vibration of a rectangular plate with different boundary conditions and uniform thickness in the elastic range is the goal of this research. To derive frequency equations, the kinetic and potential energy for in-plane behavior initially are obtained by using the stress-strain-displacement expressions according to the theory of Mindlin plates in Cartesian coordinates by applying the Hamilton's principle, which leads to five sets of highly coupled differential equations for the equations of motion. Replacement of Helmholtz decomposition for the coupled differential equations creates uncoupled equations of motion. The hypothesis of a harmonic solution for the uncoupled equations lead to wave equations. The general solutions for the wave equations are obtained by using the separation of variables. Finally, the application of boundary conditions yields the frequency equations for the rectangular plate. The natural frequencies are compared and validated by finite element analysis and previously reported results.
引用
收藏
页码:345 / 361
页数:17
相关论文
共 24 条
[1]  
Achenbach JD., 1973, Wave propagation in elastic solids
[2]  
*ANSYS INC, ANSYS AC RES REL 5 4
[3]   On the free in-plane vibration of isotropic rectangular plates [J].
Bardell, NS ;
Langley, RS ;
Dunsdon, JM .
JOURNAL OF SOUND AND VIBRATION, 1996, 191 (03) :459-467
[4]  
Doyle JF., 2012, Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms
[5]   An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges [J].
Du, Jingtao ;
Li, Wen L. ;
Jin, Guoyong ;
Yang, Tiejun ;
Liu, Zhigang .
JOURNAL OF SOUND AND VIBRATION, 2007, 306 (3-5) :908-927
[6]   Modal characteristics of in-plane vibration of rectangular plates [J].
Farag, NH ;
Pan, J .
JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA, 1999, 105 (06) :3295-3310
[7]   Free and forced in-plane vibration of rectangular plates [J].
Farag, NH ;
Pan, J .
JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA, 1998, 103 (01) :408-413
[8]   Accurate analytical type solutions for the free in-plane vibration of clamped and simply supported rectangular plates [J].
Gorman, DJ .
JOURNAL OF SOUND AND VIBRATION, 2004, 276 (1-2) :311-333
[9]  
Graff KF, 1991, Wave motion in elastic solids
[10]   Parameter studies for plane stress in-plane vibration of rectangular plates [J].
Hyde, K ;
Chang, JY ;
Bacca, C ;
Wickert, JA .
JOURNAL OF SOUND AND VIBRATION, 2001, 247 (03) :471-487