方奕忠,沈韩,崔新图,黄臻成,冯饶慧,廖德驹,庞晓宁

• 1:中山大学物理学院
• 2:物理学国家级实验教学示范中心(中山大学)

摘要(Abstract)：

本文在理论和实验上分别对张角为钝角、锐角，两辐射线方向边界简单支承、内外两圆弧边均悬空时水平放置的环状扇形薄板的小振动问题进行了研究。根据小挠度理论，采用极坐标系，用分离变量法求出环状扇形薄板给定边条件下不同本征频率及对应的本征振动模式，得出了通解，计算并讨论了不同本征振动模式下的圆弧状驻波波节线的半径及径向波节线的分布，求出薄板的弹性模量，给出仿真模拟驻波图，与实验上观察到的不同本征频率下的几种Chladni图形相比，理论与实验及仿真模拟结果符合得很好。

关键词(KeyWords)： 钝角、锐角环状扇形薄板;变型(或虚宗量)Bessel函数;Chladni图形;简单支承

Abstract:

Keywords:

基金项目(Foundation): 国家自然科学基金（61871410）;; 2016年广东省立项质量工程项目（精品资源共享课）（No.74130-18822540）;; 中山大学本科教学质量工程项目（教务[2021]93号）

作者(Author): 方奕忠,沈韩,崔新图,黄臻成,冯饶慧,廖德驹,庞晓宁

参考文献(References)：

• [1] CHLADNI E F F.Entdeckungen über die theory des Klanges,Breitkopf und H?rtel[M].Leipzig,1787.
• [2] RAYLEIGH L.The Theory of Sound (Vol.I) (2st Ed.revised and enlarged)[M].New York:Dover Publication,1945:358-363.
• [3] 黄炎.矩形薄板弹性振动的一般解析解[J].应用数学和力学，1988,9(11):993-1000.HUANG Y.A general analytical solution for elastic vibration of rectangular thin plate[J].Applied Mathematics and Mechanics,1988,9(11):993-1000.(in Chinese)
• [4] 严琪琪，陈彦，湖湘.自有边界条件下方形平板受迫振动模式的探究[J].大学物理，2018,37(12):11-15,25.YAN Q Q,CHEN Y,HU X.Research on forced vibration mode for a square plate with free boundary[J].College Physics,2018,37(12):11-15,25.(in Chinese)
• [5] SALIBA H T.Free vibration of simply supported general triangular thin plates:An accurate simplified solution[J].Journal of Sound and Vibration,1996,196:45-57.
• [6] ZHANG X F,LI W L.Vibration of arbitrarily-shaped triangular plates with elastically restrained edges[J].Journal of Sound and Vibration,2015,357:195-206.
• [7] WANG X,WANG Y.Free vibration analyses of thin sector plates by the new version of differential quadrature methods[J].Computer Methods in Applied Mechanics and Engineering,2004,193:3957-3971
• [8] 宋力，张平.关于弹性地基上圆环形薄板振动问题的解答(续一)[J].沈阳工业学院学报，1996,15(1):85-94SONG L,ZHANG P.Solutions to the vibration of ring-shaped thin plate on the elastic base (continuation 1)[J].Journal of Shenyang Institute of Technology,1996,15(1):85-94.(in Chinese)
• [9] 宋力，张景异.关于弹性地基上圆环形薄板振动问题的解答.金属成型工艺[J],1997,15(4):34-38.SONG L,ZHANG J Y.Solutions to the vibration of ring-shaped thin plate on the elastic base[J].Metal Forming Technology,1997,15(4):34-38.(in Chinese)
• [10] 吴伟，宋力.弹性地基上圆环形薄板振动问题的研究[J],沈阳工业学院学报，1997,16(3):46-50.WU W,SONG L.Solutions to the vibration of ring-shaped thin plate on the elastic base[J].Journal of Shenyang Institute of Technology,1997,16(3):46-50.(in Chinese)
• [11] SALIBA H T.Free vibration analysis of simply supported symmetrical trapezoidal plates[J].Journal of Sound and Vibration,1986,110:87-97.
• [12] Chladni patterns in vibrated plates[EB/OL].http://www.physics.utoronto.ca/nonlinear/chladni.html.2013-12-1.
• [13] STEIN J,ST?CKMANN H-J.Experimental determination of billiard wave functions[J].Physical Review Letters,1992,68:2867-2870.
• [14] GORMAN D J.Free vibration analysis of completely free rectangular plates by the superposition-Galerkin method[J].Journal of Sound and Vibration,2000,237:901-914.
• [15] LEISSA A W,JABER N A.Vibration of completely free triangular plates[J].International Journal of Mechanical Sciences,1992,34:605-616.
• [16] 方奕忠，王钢，沈韩，等.方形薄板二维驻波的研究[J].物理实验，2014,34(1):33-36.FANG Y Z,WANG G,SHEN H,et al.Research of 2-dimensional standing waves in square plate[J].Physics Experimentation,2014,34(1):33-36.(in Chinese)
• [17] 方奕忠，王钢，沈韩，等.圆形薄板二维驻波的研究[J].大学物理，2015,34(3):19-24.FANG Y Z,WANG G,SHEN H,et al.Research for two-dimensional standing waves on circular plate[J].College Physics,2015,34(3):19-24.(in Chinese)
• [18] 方奕忠，王钢，沈韩，等.环形薄板二维驻波的研究[J].力学学报，2015,47(4):664-671FANG Y Z,WANG G,SHEN H,et al.The research of 2-dimensional standing waves on annularplate[J].Chinese Journal of Theoretical and Applied Mechanics,2015,47(4):664-671.(in Chinese)
• [19] 方奕忠，沈韩，王钢，等.内边界固定情况下环形薄板二维驻波的研究[J].大学物理，2016,35(6):15-19.FANG Y Z,SHEN H,WANG G,et al.Two-dimensional standing waves on annular plate as the inner boundary being cramped[J].College Physics,2016,35(6):15-19.(in Chinese)
• [20] 方奕忠，沈韩，崔新图，等.张角为直角两半径边简支时扇形薄板二维驻波的研究[J].大学物理，2019,38(10):8-14.FANG Y Z,SHEN H,CUI X T,et al.Research of two-dimensional standing waves on right-angledthin sector plate with two simply-supported radial edges[J].College Physics,2019,38(10):8-14.(in Chinese)
• [21] 曹小敏，邵港萍，陆星辰，等.支撑点与振源重合的圆形薄板二维驻波的实验与仿真研究[J].大学物理，2018,37(6):57-60.CAO X M,SHAO G P,LU X C,et al.Research on experiment and simulation for two dimensional standing waves on circular plate whose support coincided with vibration source[J].College Physics,2018,37(6):57-60.(in Chinese)
• [22] LIEW K M,LIU F-L.Differential quadrature method for vibration analysis of sheardeformable annular sector plates[J].Journal of Sound and Vibration,2000,230(2):335-356.
• [23] SEOK J,TIERSTEN H F.Free vibrations of annular sector cantilever plates,Part 1:Out-of-plane Motion[J].Journal of Sound and Vibration,2004,271:757-772.
• [24] SEOK J,TIERSTEN H F.Free vibrations of annular sector cantilever plates.Part 2:In plane motion[J].Journal of Sound and Vibration,2004,271:773-787.
• [25] SHI D Y,LV X H,WANG Q S,et al.A unified solution for free vibration of ortho tropic annular sector thin plates with general boundary conditions,internal radial line and circumferential Arc supports[J].Journal of Vibroengineering,2016,18(1):361-377.
• [26] 莫尔斯P M,英格特K U.理论声学上册[M].吕如榆，杨训仁译.北京：科学出版社，1984:252-257.
• [27] 钱伟长，叶开源.弹性力学[M].北京：科学出版社，1956:285-286.
• [28] 王龙甫.弹性理论[M].北京：科学出版社，1978:357-360.
• [29] 朗道L D,栗弗席兹E M.弹性理论第五版[M].武际可，刘寄星，译.北京：高等教育出版社，2011:115-116.
• [30] HUANG C S,LEISSA A W,MCGEE O G.Exact analytical solutions for the vibrations of sectorial plates with simply-supported radial edges[J].Journal of Applied Mechanics,Transaction ASME,1993,60:478-483.
• [31] HUANG C S,MCGEE O G,LEISSA A W.Exact analytical solutions for free vibrations of thick sectorial plates with simply supported radial edges[J].International Journal of Solids Structures,1994,31:1609-1631.
• [32] 郭敦仁.数学物理方法[M].北京：人民教育出版社，1978:292-300,321-324.
• [33] 梁昆淼.数学物理方法[M].2版.北京：高等教育出版社，1978:291-292,364-365,373-374.
• [34] 吴崇试.数学物理方法[M].北京：北京大学出版社，1999:405-406,417-418,435-437.
• [35] 郑晓静.圆薄板大挠度理论及应用[M].长春：吉林科学技术出版社，1990:10-12.