弹性边界下圆弧拱的自由振动分析

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振动与冲击 ›› 2016, Vol. 35 ›› Issue (21) : 120-125.

论文

弹性边界下圆弧拱的自由振动分析

赵章泳1，邱艳宇1,2，王明洋1,2，宋春明1,2，曹侃1

作者信息 +

Free vibrationanalysis of arches with elastic support boundary conditions

ZHAO Zhang-yong1,QIU Yan-yu1,2,WANG Ming-yang1,2,SONG Chun-ming1,2,CAO Kan1

Author information +

文章历史 +

摘要

工程中圆弧拱的边界不能总被简化为理想的简支或固支形式。为了研究弹性支承对圆弧拱自由振动特性的影响规律，将力、位移等变量无量纲化。根据平衡方程和坐标转换推导得出极坐标下圆弧拱在水平、竖直和转动方向支撑条件为弹性时的边界条件方程。并采用考虑弯曲和轴向变形而忽略剪切变形及转动惯量的自由振动的运动控制方程。运动方程在边界条件情况下，其解仅为关于矢跨比f,细长比s和无量纲刚度阵[K]的函数。采用Runge-Kutta法和行列式搜索法求解运动方程的特征值即无量纲频率Ωi以及特征向量即振型。通过计算发现，与理想支撑相比，弹性支承情况下细长比s对拱自振频率的影响要明显下降。理想支撑情况下圆弧拱的自振频率越高，则弹性支承对其自振频率的影响越小。与水平和竖向弹性支承相比，转动方向弹性支承仅对圆弧拱基频有较大影响。

Abstract

The boundary conditions of circular arches in practical engineering can not be always simplified as perfect hinged or fixed ends.In order to study the influence of elastic support for the free vibration characteristics of circular arch,the variables such as force and displacements werenondimensionalized.According to the balance equation and coordinate conversion,the boundary conditions of arches with vertical ,horizontal and rotational elastic supports were obtained in polar coordinates.And the flexural and axial deformations were considered while shear deformations and rotation inertia were neglected in the governing equations of motion.The solutions of the governing equations under the boundary conditions are the functions only about the rise to span ratiof，the slenderness s，and the dimensionless stiffness matrix [K] .The Runge-Kutta and determinant search method were used to obtain the eigenvalues Ωiand eigenvectors that is mode shape.By calculating different configurations, the effects of slenderness s on the natural frequencies was much smaller than perfect supports.The effects of horizontal and vertical elastic supports on natural frequencies decreases with the frequencies of perfect supports increased .And the rotational elastic supports just obviously influence the fundamental frequency.

导出引用

赵章泳1，邱艳宇1,2，王明洋1,2，宋春明1,2，曹侃1. 弹性边界下圆弧拱的自由振动分析[J]. 振动与冲击, 2016, 35(21): 120-125

ZHAO Zhang-yong1,QIU Yan-yu1,2,WANG Ming-yang1,2,SONG Chun-ming1,2,CAO Kan1. Free vibrationanalysis of arches with elastic support boundary conditions[J]. Journal of Vibration and Shock, 2016, 35(21): 120-125

参考文献

[1] Chidamparam P, Leissa A W. Vibrations Of Planar Curved Beams, Rings, And Arches[J]. Applied Mechanics Reviews, 1993, 46(9): 467–483.
[2] 项海帆. 拱结构的稳定与振动/高等结构力学丛书[M]. 人民交通出版社, 1991.
[3] Henrych J. The Dynamics Of Arches And Frames[M]. Elsevier Scientific Pub. Co., 1981.
[4] 钱七虎，孙乃光, 陈震元等. 拱型结构的自振频率计算及轴向变形对自振频率的影响[D]. 钱七虎院士论文选集, 1999.
[5] Karnovsky I A. Theory Of Arched Structures[M]. New York, Ny: Springer New York, 2012.
[6] Lee B K, Lee J Y, Choi K M等. Free Vibrations Of Arches With General Boundary Condition[J]. KSCE Journal Of Civil Engineering, 2002, 6(4): 469–474.
[7] 康婷, 白应生, 孙惠香. 水平弹性支撑圆拱的动力特性研究[J]. 力学与实践, 2013, 35(2): 50–55.
[8] Wung S. Vibration Of Hinged Circular Arches[D]. Rice University, 1967.
[9] 张相，黄东才. 结构振动力学[M]. 同济大学出版社, 1994.
[10] Al-Khafaji A W, Tooley J R. Numerical Methods In Engineering Practice[M]. Holt, Rinehart And Winston, 1986.
[11] Lee B K, Wilson J F. Free Vibrations Of Arches With Variable Curvature[J]. Journal Of Sound And Vibration, 1990, 136(1): 75–89.
[12] Lee B K, Lee T E, Ahn D S. Free Vibrations Of Arches With Inclusion Of Axial Extension, Shear Deformation And Rotatory Inertia In Cartesian Coordinates[J]. KSCE Journal Of Civil Engineering, 2004, 8(1): 43–48.
[13] Chidamparam P, Leissa A W. Influence Of Centerline Extensibility On The In-Plane Free Vibrations Of Loaded Circular Arches[J]. Journal Of Sound And Vibration, 1995, 183(5): 779–795.
[14] 金尼克. 拱的稳定性[M]. 建筑工程出版社, 1958.
[15] Fanning P J, Boothby T E, Roberts B J. Longitudinal And Transverse Effects In Masonry Arch Assessment[J]. Construction And Building Materials, 2001, 15(1): 51–60.
[16] 克拉夫, 彭津. 结构动力学[M]. 高等教育出版社, 2006.