Finite Integral Transform Solutions for Free Vibrations of Rectangular Thin Plates With Mixed Boundary Constraints

LI Yihao; XU Dian; CHEN Yiming; AN Dongqi; LI Rui

doi:10.21656/1000-0887.440051

Volume 44 Issue 9

Sep. 2023

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Article Navigation > Applied Mathematics and Mechanics > 2023 > 44(9): 1112-1121

LI Yihao, XU Dian, CHEN Yiming, AN Dongqi, LI Rui. Finite Integral Transform Solutions for Free Vibrations of Rectangular Thin Plates With Mixed Boundary Constraints[J]. Applied Mathematics and Mechanics, 2023, 44(9): 1112-1121. doi: 10.21656/1000-0887.440051

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Finite Integral Transform Solutions for Free Vibrations of Rectangular Thin Plates With Mixed Boundary Constraints

doi: 10.21656/1000-0887.440051

State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

Received Date: 2023-02-28
Rev Recd Date: 2023-04-03
Publish Date: 2023-09-01

Abstract

Abstract

Analytical solutions, with unique research value, can serve as benchmarks for empirical formulas and numerical methods, a tool for rapid parameter analysis and optimization, and a theoretical basis for experimental designs. Conventional analytical methods, e.g., the Lévy solution method, are only applicable to mechanical problems of plates and shells with opposite simply-supported edges, which, however, may fail to obtain analytical solutions for the issues with complex boundary constraints. In recent years, the finite integral transform method for plate and shell problems was developed to deal with non-Lévy-type plates and shells, but it is still infeasible to solve the mixed boundary constrains-induced complex boundary value problems of higher-order partial differential equations. Herein, for the first time, the finite integral transform method was combined with the sub-domain decomposition technique to solve the free vibrations of rectangular thin plates with mixed boundary constraints. The rectangular plate was first divided into 2 sub-domains according to the mixed boundary constraints, and the 2 sub-domains were solved analytically with the finite integral transform method. Finally, the continuity conditions were introduced to obtain the analytical solution of the original problem. Based on the side spot-welded cantilever plates commonly used in engineering, the free vibration problem of a rectangular thin plate with 1 edge subjected to clamped-simply supported constraints and the other 3 edges free, was analyzed. The obtained natural frequencies and mode shapes are in good agreement with those from the finite element method as well as the solutions in literature, thus verifying the accuracy of the proposed method. The solution procedure of the finite integral transform method can be implemented based on the governing equations without any assumption of the solution form. Therefore, this strict analytical method is widely applicable to complex boundary value problems of higher-order partial differential equations for such mechanical problems of plates and shells.
- finite integral transform method,
- mixed boundary constraint,
- rectangular thin plate,
- free vibration,
- analytical solution

(Contributed by LI Rui, M. AMM Editorial Board)

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References(24)

References

[1]	田宇. 求解矩形薄板问题的辛-叠加方法[D]. 硕士学位论文. 大连: 大连理工大学, 2018. TIAN Yu. On the symplectic superposition method for rectangular thin plates[D]. Master Thesis. Dalian: Dalian University of Technology, 2018. (in Chinese)
[2]	李若愚, 王天宏. 薄板热力耦合的屈曲分析[J]. 应用数学和力学, 2020, 41(8): 877-886. doi: 10.21656/1000-0887.400308 LI Ruoyu, WANG Tianhong. Thermo-mechanical buckling analysis of thin plates[J]. Applied Mathematics and Mechanics, 2020, 41(8): 877-886. (in Chinese) doi: 10.21656/1000-0887.400308
[3]	张磊, 张文明, 王林, 等. 基于小波Galerkin法的矩形薄板二次屈曲分析[J]. 应用数学和力学, 2023, 44(1): 25-35. doi: 10.21656/1000-0887.430097 ZHANG Lei, ZHANG Wenming, WANG Lin, et al. Secondary buckling analysis of thin rectangular plates based on the wavelet Galerkin method[J]. Applied Mathematics and Mechanics, 2023, 44(1): 25-35. (in Chinese) doi: 10.21656/1000-0887.430097
[4]	XU D, NI Z F, LI Y H, et al. On the symplectic superposition method for free vibration of rectangular thin plates with mixed boundary constraints on an edge[J]. Theoretical and Applied Mechanics Letters, 2021, 11(5): 100293. doi: 10.1016/j.taml.2021.100293
[5]	王磊, 蔡松柏. 简支任意四边形板的弯曲、稳定和振动的Navier解法[J]. 湖南大学学报, 1989, 16(1): 104-114. https://www.cnki.com.cn/Article/CJFDTOTAL-HNDX198901017.htm WANG Lei, CAI Songbo. Navier's method for solving the problem of bending, buckling and vibration of all edges simply supported quadrilateral plates[J]. Journal of Hunan University, 1989, 16(1): 104-114. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-HNDX198901017.htm
[6]	THAI H T, CHOI D H. Levy solution for free vibration analysis of functionally graded plates based on a refined plate theory[J]. KSCE Journal of Civil Engineering, 2014, 18(6): 1813-1824. doi: 10.1007/s12205-014-0409-2
[7]	丁鹏飞. 超高强度汽车钢板电阻点焊结构强度的优化分析[D]. 硕士学位论文. 上海: 华东理工大学, 2016: 1-26. DING Pengfei. Optimization on structure strength of resistance spot welding of ultra high strength steel[D]. Master Thesis. Shanghai: East China University of Science and Technology, 2016: 1-26. (in Chinese)
[8]	ZIENKIEWICZ O C, CHEUNG Y K. The finite element method for analysis of elastic isotropic and orthotropic slabs[J]. ICE Proceedings, 1964, 28(4): 471-488.
[9]	BUCZKOWSKI R, TORBACKI W. Finite element modelling of thick plates on two-parameter elastic foundation[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2001, 25(14): 1409-1427. doi: 10.1002/nag.187
[10]	CHEUNG B Y K. Finite Strip Method in Structural Analysis[M]. London: Pergamon Press, 1976.
[11]	HENWOOD D J, WHITEMAN J R, YETTRAM A L. Finite difference solution of a system of first-order partial differential equations[J]. International Journal for Numerical Methods in Engineering, 2010, 17(9): 1385-1395.
[12]	CHAKRAVORTY A K, GHOSH A. Finite difference solution for circular plates on elastic foundations[J]. International Journal for Numerical Methods in Engineering, 2010, 9(1): 73-84.
[13]	BREBBIA C A. The Boundary Element Method for Engineers[M]. New York: Halsted Press, 1978.
[14]	PEREIRA WL A, KARAM V J, CARRER J A M, et al. A dynamic formulation for the analysis of thick elastic plates by the boundary element method[J]. Engineering Analysis With Boundary Elements, 2012, 36(7): 1138-1150. doi: 10.1016/j.enganabound.2012.02.002
[15]	LIEW K M, HAN J B. Bending solution for thick plates with quadrangular boundary[J]. Journal of Engineering Mechanics, 1998, 124(1): 9-17. doi: 10.1061/(ASCE)0733-9399(1998)124:1(9)
[16]	LIU F L, LIEW K M. Differential cubature method for static solutions of arbitrarily shaped thick plates[J]. International Journal of Solids Structures, 1998, 35(28/29): 3655-3674.
[17]	ZHONG Y, SUN A M, ZHOU F L, et al. Analytical solution for rectangular thin plate on elastic foundation with four edges free by finite cosine integral transform method[J]. Chinese Journal of Geotechnical Engineering, 2006, 28(11): 2019-2022.
[18]	ZHONG Y, WANG G X, SUN A M. Vibration of a thin plate on Winkler foundation with completely free boundary by finite cosine integral transform method[J]. Journal of Dalian University of Technology, 2007, 47(1): 73-77.
[19]	ZHONG Y, YIN J H. Free vibration analysis of a plate on foundation with completely free boundary by finite integral transform method[J]. Mechanics Research Communications, 2008, 35(4): 268-275. doi: 10.1016/j.mechrescom.2008.01.004
[20]	LI R, ZHONG Y, TIAN B, et al. On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates[J]. Applied Mathematics Letters, 2009, 22(12): 1821-1827.
[21]	ZHONG Y, ZHAO X F, LI R. Freevibration analysis of rectangular cantilever plates by finite integral transform method[J]. International Journal for Computational Methods in Engineering Science and Mechanics, 2013, 14(3): 221-226.
[22]	AN D Q, XU D, NI Z F, et al. Finite integral transform method for analytical solutions of static problems of cylindrical shell panels[J]. European Journal of Mechanics A: Solids, 2020, 83: 104033.
[23]	CHEN Y M, AN D Q, ZHOU C, et al. Analytical free vibration solutions of rectangular edge-cracked plates by the finite integral transform method[J]. International Journal of Mechanical Sciences, 2023, 243: 108032.
[24]	KHALILI M R, MALEKZADEH K, MITTAL R K. A new approach to static and dynamic analysis of composite plates with different boundary conditions[J]. Composite Structures, 2005, 69(2): 149-155.