The natural characteristic of a rectangular plate which is clamped at corner points has a very important role in its application and fault diagnosis. Firstly, a Lagrange equation of transverse vibration, which is based on the principle of conservation of energy and introduced the constraint conditions of the plate at support locations, is established for analyzing the free vibration problem of rectangular plates clamped at corner points. Secondly, the frequency equation of the rectangular plate is obtained using the Rayleigh-Ritz method by employing simple polynomials that is used as the modal function of the transverse vibrations of the plate. Finally, the fundamental frequencies are given for different side ratios according to theoretical results of the rectangular plates, and used to comparison with those obtained using the finite element method. The approach proposed here is used to formulate the frequency equation of rectangular plates simply-supported at corner points. Corresponding results and discussions are presented in terms of the numerical results. In order to get the fundamental frequencies conveniently, approximate expressions are given for a typical plate with certain geometrical parameters and material constants in practical engineering. The theoretical results exhibited to be much agreeable with those obtained from both the finite element method and the experimental data.
Key words
rectangular plates /
point supports /
fundamental frequency /
Rayleigh-Ritz method /
Lagrange multiplier
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
References
[1] Gorman D J. Solutions of the Levy type for the free vibration analysis of diagonally supported rectangular plates [J]. Journal of Sound and Vibration, 1979, 66:239-46.
[2] Gorman D J. Free vibration analysis of rectangular plates with symmetrically distributed point supports along the edges. Journal of Sound and Vibration, 1980, 73:563-74.
[3] Gorman D J. An analytical solution for the free vibration analysis of rectangular plates resting on symmetrically distributed point supports [J]. Journal of Sound and Vibration, 1981, 79:561-74.
[4] Huang M, Ma X Q, Sakiyama T, Matsuda H, et al. Free vibration analysis of rectangular plates with variable thickness and point supports [J]. Journal of Sound and Vibration, 2007, 300:435-52
[5] Bapat A V, Suryanarayan S. Free vibrations of rectangular plates with interior point supports [J]. Journal of Sound and Vibration, 1989, 134:291-313.
[6] Narita Y, Hodgkinson J M. Layerwise optimisation for maximising the fundamental frequencies of point-supported rectangular laminated composite plates [J]. Composite Structures, 2005, 69:127-35.
[7] 王砚, 王忠民, 阮苗. 无网格法在点弹性支承矩形薄板横向振动中的应用[J]. 计算力学学报. 2010, 27(2):238-243
WANG Yan, WANG Zhong-min, RUAN Miao. Application of meshless method in the transverse vibration of rectangular thin plate with elastic point supports [J]. Chinese Journal of Computational Mechanics, 2010, 27(2):238-243.
[8] Xu M, Cheng D. A new approach to solving a type of vibration problem [J]. Journal of Sound and Vibration, 1994, 177(4):565-571.
[9] 赵凤群, 王忠民. 点弹性支承的非保守矩形薄板的稳定性[J].西安理工大学学报, 1998, 14(4):398-403.
ZHAO Feng-qun, WANG Zhong-min. The Stability of non-conservative rectangular plate with spring attachments [J]. Journal of xi’an University of Technology, 1998, 14(4):398-40.
[10] 郭强, 沈惠申. 点支撑预应力中厚矩形板的横向振动 [J]. 工程力学. 2005, 22(4):106-111.
GUO Qiang, SHEN Hui-shen. Vibrations of initially stressed moderately thick rectangular plates with point supports [J]. Engineering Chanics. 2005, 22(4):106-111.
[11] Asku G,Felemban M B.Frequency analysis of corner point supported Mindlin plates by a finite difference energy method [J]. Journal of Sound and Vibration, 1992, 158(3):531-544.
[12] Zhou D, Cheung Y K, Kong J. Free vibration of thick, layered rectangular plates with point supports by finite layer method [J]. International Journal of Solids and Structures, 2000, 37:1483-1499.
[13] Wang C M, Wang Y C, Reddy J N. Problems and remedy for the Ritz method in determining stress resultants of corner supported rectangular plates [J]. Computers and Structures,2002,80: 145-154.
[14] Lopatin A V, Morozov E V. Fundamental frequency of an orthotropic rectangular plate with an internal centre point support[J]. Composite Structures, 2011, 93:2487-2495.
[15] 许琪楼. 四角点支承四边自由矩形板自振分析新方法 [J].振动与冲击, 2013, 32(3):83-86
XU Qi-lou. A new analysis method of free vibration of rectangular plate with 4-free-sides and 4-corner point supports [J]. Journal of Vibration and Shock, 2013, 32(3):83-86.
{{custom_fnGroup.title_en}}
Footnotes
{{custom_fn.content}}