从二维弹性力学出发，采用Chebyshev-Ritz法研究含单裂纹T型梁的自振特性。根据弹性理论中应变相同且总内力不变原则，利用转换截面法将T型截面梁等效为由两层不同材料特性组成的矩形截面梁。将等效后的梁沿裂纹和层界面划分成四个子域。使用Rayleigh-Ritz法得到各子域的振动特征方程，结合子域间界面处的位移连续条件导出整个T型裂纹梁的振动特征方程。采用Chebyshev多项式构建各子域的位移试函数，由Chebyshev多项式的正交完备性可获得快速收敛解。与实际T型截面梁的有限元分析结果对比验证了该方法的正确性。以两端固支T型裂纹梁为例，分析了裂纹位置和裂纹深度对结构振动特性的影响。
Abstract
Based on the 2-D elasticity theory, natural vibration characteristics of T-beams with a crack were studied using Chebyshev-Ritz method.Firstly, according to the principle of the same strain and the unchanged total internal force, the beam was equivalent to a rectangular cross-section beam composed of two material layers with different characteristics using the transfer section method.Then, the equivalent beam was divided into 4 sub-domains along the crack and layer interface.The natural vibration characteristic equation of each sub-domain was deduced using Rayleigh-Ritz method, and the natural vibration characteristic equation of the whole cracked T-beam was derived with displacement continuity conditions on interfaces of sub-domains.Chebyshev polynomials were applied to construct displacement trial functions of various sub-domains, and the fast convergence solutions were obtained using the orthogonality and completeness of Chebyshev polynomials.The correctness of the proposed method was verified by comparing its solving results with those using the finite element analysis for actual T-beams.Finally, a cracked T-beam fixed at both ends was taken as an example, effects of crack position and its depth on this beam’s natural vibration characteristics were analyzed.
关键词
T型梁 /
Chebyshev-Ritz法 /
转换截面 /
子域 /
Chebyshev多项式
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Key words
T-beam /
Chebyshev-Ritz method /
transfer section /
sub-domain /
Chebyshev polynomial
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参考文献
［1］CHONDROS T G, DIMAROGONAS A D, YAO J.Vibration of a beam with a breathing crack ［J］.Journal of Sound and Vibration, 2001, 239(1): 57-67.
［2］CHONDROS T G, DIMAROGONAS A D, YAO J.A continuous cracked beam vibration theory ［J］.Journal of Sound and Vibration, 1998, 215(1): 17-34.
［3］KIM K, KIM S, SOK K, et al.A modeling method for vibration analysis of cracked beam with arbitrary boundary condition ［J］.Journal of Ocean Engineering and Science, 2018, 11(3): 367-381.
［4］吴宁祥，谢里阳，吴克勤.含裂纹一维欧拉梁的裂纹无效位置分析［J］.应用力学学报，2007, 24(1):120-124.
WU Ningxiang, XIE Liyang, WU Keqin.Noneffective spots of crack in cracked Bernoulli-Euler beam ［J］.Journal of Applied Mechanics, 2007, 24(1): 120-124.
［5］张炜，毛崎波，聂彦平.含任意数目裂纹梁的振动分析［J］.机械设计与制造, 2012, 10(10):228-230.
ZHANG Wei, MAO Qibo, NIE Yanping.Free vibration analysis of a beam with an arbitrary number of cracks ［J］.Machinery Design & Manufacture, 2012,10(10):228-230.
［6］RICCI P, VIOLA E.Stress intensity factors for cracked T-sections and dynamic behaviour of T-beams ［J］.Engineering Fracture Mechanics, 2006, 73(1):91-111.
［7］ZENG J, MA H, ZHANG W, et al.Dynamic characteristic analysis of cracked cantilever beams under different crack types ［J］.Engineering Failure Analysis, 2017, 74:80-94.
［8］马辉，曾劲，郎自强，等.斜裂纹悬臂梁非线性振动特性分析［J］.振动与冲击，2016, 35(12):86-91.
MA Hui, ZENG Jin, LANG Ziqiang, et al.Nonlinear vibration characteristics analysis of a cantilever beam with slant crack ［J］.Journal of Vibration and Shock, 2016, 35(12):86-91.
［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, 38(增刊2): 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, 38(Sup 2): 183-190.
［10］蒋杰，周叮，胡朝斌.基于弹性力学的端部有裂缝悬臂梁的自由振动分析［J］.振动与冲击，2019, 38(15): 197-201.
JIANG Jie, ZHOU Ding, HU Chaobin.Free vibration of a cantilever beam with a crack at clamped end based on elasticity theory ［J］.Journal of Vibration and Shock, 2019, 38(15): 197-201.
［11］杨鄂川，李映辉，崔灿.基于等效刚度法的裂纹梁振动特性分析［J］.西南大学学报(自然科学版)，2013, 35(4):145-150.
YANG Echuan, LI Yinghui, CUI Can.On vibration characteristic analysis of cracked beams based on equivalent stiffness method ［J］.Journal of Southwest University (Natural Science Edition), 2013, 35(4):145-150.
［12］LEE J. Identification of a crack in a beam by the boundary element method ［J］.Journal of Mechanical Science and Technology, 2010, 24(3):801-804.
［13］ZHENG D Y, FAN S C.Natural frequency changes of a cracked Timoshenko beam by modified Fourier series ［J］.Journal of Sound and Vibration, 2001, 246(2), 297-317.
［14］ZHOU D. Three-dimensional vibration analysis of structural elements using Chebyshev-Ritz method ［M］.Beijing: Science Press, 2007.
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