内部含不连续的变截面梁自由振动分析

摘要
图/表
参考文献(23)
相关文章 (13)

全文: PDF (1049 KB) HTML (1 KB)
输出: BibTeX | EndNote (RIS)

摘要针对带有内弹簧的变截面欧拉梁受轴压载荷的自由振动问题，提出一种根据改进傅里叶级数和伽辽金法研究其振动特性的策略。在梁两端分别设置弹簧以模拟任意边界，包括横向位移弹簧和旋转约束弹簧。梁内部的不连续变形条件也通过设置横向及旋转弹簧的刚度系数来实现。具体计算过程中，首先建立振动位移的容许函数，其中的辅助多项式由端部函数值及导数值得到具体的形式。然后采用伽辽金法处理梁的自由振动控制微分方程，把振动问题转化为矩阵特征值问题。最后进行数值仿真验证，仿真中当边界条件和跨中弹性弹簧改变时需通过改变弹簧刚度系数，以改变系统的质量矩阵和刚度矩阵，可得到其振动特性。数值结果表明，本文的方法简便合理，具有较好的推广价值。
关键字：欧拉梁；内部不连续；改进傅里叶级数；自由振动

	服务

	把本文推荐给朋友
	加入我的书架
	加入引用管理器
	E-mail Alert
	RSS
	作者相关文章
	顾业清
	常婷婷
	鲍四元
	沈峰

关键词 ：欧拉梁, 内部不连续, 改进傅里叶级数, 自由振动

Abstract：Aiming at solving the free vibration of an Euler-Bernoulli beam with variable cross-section and internal spring subjected to axial compression load under arbitrary boundary conditions, a strategy based on the improved Fourier series and Garlerkin method is proposed to study the vibrational characteristics of beams. In order to simulate arbitrary boundaries, the transverse displacement spring and rotational spring are set at both ends of the beam. By assigning different values to the stiffness coefficient of the spring, the internal discontinuity condition is simulated. First, the displacement admissible function of the vibration displacement is established, in which the auxiliary polynomial is derived from the value of the end function or derivative. Then, the Garlerkin method is used to deal with the governing differential equation for free vibration of beams, thus the vibration problem is transformed into matrix eigenvalue problem. Finally, the numerical simulation is carried out by using the numerical calculation software. When the boundary conditions and mid-span discontinuity are changed, the mass matrix and stiffness matrix of the system can be determined by changing the spring stiffness coefficient, and the corresponding vibrational characteristics can be obtained. Numerical results show that the proposed method is simple, reasonable and of good practical significance.
Keywords: Euler-Bernoulli beam; internal discontinuity; the improved Fourier series; free vibration

Key words： Euler-Bernoulli beam internal discontinuity the improved Fourier series free vibration

收稿日期: 2021-07-27 出版日期: 2022-10-15

引用本文:

顾业清，常婷婷，鲍四元，沈峰. 内部含不连续的变截面梁自由振动分析[J]. 振动与冲击, 2022, 41(19): 164-171.
GU Yeqing, CHANG Tingting, BAO Siyuan, SHEN Feng. Free vibration analysis of variable cross-section beam with internal discontinuities. JOURNAL OF VIBRATION AND SHOCK, 2022, 41(19): 164-171.

链接本文:

http://jvs.sjtu.edu.cn/CN/ 或 http://jvs.sjtu.edu.cn/CN/Y2022/V41/I19/164

[1] Jinhee Lee. Application of Chebyshev-tau method to the free vibration analysis of stepped beams[J].International Journal of Mechanical Sciences.2015,101-102:411-420.
[2] Filippo Giunta, Alice Cicirello. On the analysis of jointed Euler-Bernoulli beams with step changes in material and cross-section under static and dynamic loads[J]. Engineering Structures. 2019(175), 66-78.
[3] 周渤,石先杰. 连续多跨梁结构振动特性分析[J].机械设计与制造,2017(08):43-46.(ZHOU Bo, SHI Xianjie. Vibration Analysis of Multi-Span Beam System[J]. Machinery Design ＆ Manufacture, 2017(08):43-46. (in Chinese)).
[4] 谢秋林,叶茂,肖峰.基于转换矩阵法的中间带弹性支撑Rayleigh梁自由振动[J].科学技术与工程, 2017,17(26):274-279.(XIE Qiu-lin, YE Mao, XIAO Feng. Free vibration analysis of Rayleigh beam with intermediate flexible constraints[J]. Science Technology and Engineering, 2017, 17(26):274-279(in Chinese)).
[5] Goranka Štimac Rončević, Branimir Rončević, Skoblar A , et al. Closed form solutions for frequency equation and mode shapes of elastically supported Euler-Bernoulli beams[J]. Journal of Sound and Vibration, 2019, 457:118-138.
[6] 王天宇,杨骁.呼吸型裂纹梁的非线性振动[J].上海大学学报(自然科学版),2020,26(03):443-455.(WANG Tianyu, YANG Xiao. Nonlinear vibration of beam with breathing crack[J]. Journal of Shanghai University(Natural Science Edition). 2020,26(03):443-455(in Chinese)).
[7] KimK wanghun, KimSok, SokKyongjin, et al. A modeling method for vibration analysis of cracked beam with arbitrary boundary condition[J]. Journal of Ocean Engineering and Science, 2018.
[8] 马一江,李园园,陈国平,等.含多条裂纹变截面简支梁的自由振动[J].振动与冲击,2019,38(19):149-154.(MA Yijiang, LI Yuanyuan, CHEN Guoping, ZHAO Yingjie. Free vibration of a variable cross-section simple supported beam with multi-crack[J]. Journal of Vibration and Shock, 2019,38(19):149-154(in Chinese)).
[9] 蒋杰,周叮.两端有裂纹固支深梁的振动特性分析[J].建筑结构学报,2018,39(S2):183-190.(JIANG Jie, ZHOU Ding. Analysis of vibration characteristics of clamped-clamped deep beams with cracks at ends[J]. Journal of Building Structures,2018,39(S2):183-190(in Chinese)).
[10] 陈得良,汪亚运,吝国胜,等.含多裂纹(群)钢筋混凝土梁的自由振动研究[J].振动与冲击,2019,38(19):109-114+154.(CHEN Deiliang, WANG Yayun, LIN Guosheng, QIU Zebin, LEI Wei. Free vibration of a RC beam with multi-crack[J]. Journal of Vibration and Shock, 2019, 38(19):109-114+154.).
[11] Caddemi S , I Caliò. Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks[J]. Journal of Sound and Vibration, 2009, 327(3-5):473-489.
[12] 陈炉云,易宏.含焊接残余应力薄圆板结构自由振动近似解[J].振动与冲击,2021,40(05):119-125.(CHEN Luyun, YI Hong. Approximate solutionto free vibration of thin circular plate structure with welding residual stress[J]. Journal of Vibration and Shock, 2021,40(05):119-125.(in Chinese)).
[13] Ariaei A，Ziaeirad S，Malekzadeh M. Dynamic response of a multi-span Timoshenko beam with internal and external flexible constraints subject to a moving mass[J].Archive of Applied Mechanics，2013，83(9)：1257-1272.
[14] Zhang H, Wang C M, Challamel N. Buckling and vibration of Hencky bar-chain with internal elastic springs[J]. International Journal of Mechanical Sciences, 2016:383-395.
[15] 张鹏,叶茂,曹文斌,等.轴向受载多跨弹性支撑连续梁的模态分析[J].科学技术与工程, 2014,14(09):122-126(ZHANG Peng,YE Mao,CAO Wen-bin,et al.Modal analysis of multi-span beam with intermediate elastic bearing under axial-loaded[J].Science Technology and Engineering,2014,14(9):122-126(in Chinese)).
[16] 陈小超,毛崎波,薛晓理. 轴向力作用不连续梁自由振动的广义函数解[J].固体力学学报,2014,35(03):319-324.(CHEN Xiaochao, MAO Qibao, XUE Xiaoli. Free vibration analysis of axial-loaded beams with different discontinuities using generalized functions[J]. Chinese Journal of Solid Mechanics. 2014,35(03):319-324.)
[17] Dimplekumar Chalishajar,Austin States,Brad Lipscomb. On Applications of Generalized Functions in the Discontinuous Beam Bending Differential Equations[J]. Applied Mathematics,2016,7(16).
[18] 张俊,李天匀,朱翔.多开口矩形板自由振动特性分析[J].振动与冲击,2020,39(14):142-147,155.(ZHANG Jun, LI Tianyun, ZHU Xiang. Free vibration characteristics analysis on a rectangular plate with multiple cutouts[J]. Journal of Vibration and Shock,2020, 39(14):142-147,155. (in Chinese)).
[19] 陆斌,陈跃华,冯志敏,等.基于Chebyshev-变分法的复杂开口形状矩形薄板弯曲振动特性分析[J].振动与冲击,2020,39(02):178-187.(LU Bin, CHEN Yuehua, FENG Zhimin, ZHANG Gang, YAN Wei, XU Qiang. Bending vibration characteristics analysis of a rectangular plate with complex opening using Chebyshev-variational method[J]. Journal of Vibration and Shock,2018, 39(S2): 183-190. (in Chinese)).
[20] W. L. LI. FREE VIBRATIONS OF BEAMS WITH GENERAL BOUNDARY CONDITIONS[J]. Journal of Sound and Vibration, 2000, 237(4):709-725.
[21] 王春玲,赵鲁珂,王涵,等.带裂纹矩形板自由振动解析解[J].计算力学学报,2019,36(06):757-762.(WANG Chunlin, ZHAO Luke, WANG Han, LI Dongbo. Analytical study of free vibration for cracked rectangular plate[J]. Chinese Journal of Computational Mechanics, 2019,36(06):757-762. (in Chinese)).
[22] 王家乐,夏桂云.Winkler地基上修正Timoshenko梁振动分析[J].振动与冲击,2020,39(03):30-37.
Wang Jiale, Xia Guiyun. Vibration analysis for a modified
Timoshenko beam on Winkler elastic foundation[J]. Journal of vibration and shock,2020,39(03):30-37(in Chinese)).
[23] Bao S.Y, Wang S. D. A generalized solution procedure for in-plane free vibration of rectangular plates and annular sectorial plates[J]. Royal Society Open Science. 2017,4(8): 170484.