浙江大学学报(工学版)  2020, Vol. 54 Issue (4): 778-786    DOI: 10.3785/j.issn.1008-973X.2020.04.017
 土木工程、交通工程

Free vibration characteristics of multi-cracked beam based on Chebyshev-Ritz method
Jia-lei ZHAO(),Ding ZHOU,Jian-dong ZHANG,Chao-bin HU*()
College of Civil Engineering, Nanjing Tech University, Nanjing 211816, China
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Abstract:

The free vibration characteristics of multi-cracked beam were analyzed based on the plane stress theory of elasticity by using Chebyshev-Ritz method. The cracked beams were divided into several sections according to their cracks. The products of boundary functions and Chebyshev polynomials were taken as the functions of the displacement, which had good convergence, making the method applicable for different geometric boundary conditions. The vibration equation of each sub-beam could be obtained by using Ritz method. The vibration characteristic equation of the whole cracked beam was established by the continuity conditions of displacements between adjacent sub-beams. The calculation results accorded well with those available from the literature and the finite element analysis. The effects of the structural parameters such as crack depth and location on the natural vibration characteristics of the beam were analyzed. As the crack depth increases, the natural frequency of the cracked beam decreases, the amplitude of the mode shape increases, and the degree of influence is affected by the location of the crack.

Key words: elasticity    Chebyshev-Ritz method    crack    free vibration    continuity conditions of displacement

 CLC: TU 311

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Jia-lei ZHAO,Ding ZHOU,Jian-dong ZHANG,Chao-bin HU. Free vibration characteristics of multi-cracked beam based on Chebyshev-Ritz method. Journal of ZheJiang University (Engineering Science), 2020, 54(4): 778-786.

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 图 1  带有2条不同深度裂纹的梁的分析模型 图 2  带有2条相同深度裂纹的梁的分析模型 图 3  带有3条不同深度裂纹的梁的分析模型 图 4  多裂纹梁的计算流程图 表 1  固支梁前8阶的频率参数Ω的收敛性 图 5  裂纹固支梁的ANSYS分析模型 表 2  固支梁计算结果与有限元分析的对比（h/L=0.1，d1/L=0.5，d2/L=0.8） 表 3  简支梁计算结果与有限元分析的对比（h/L=0.1，d1/L=0.5，d2/L=0.8） 表 4  悬臂梁计算结果与有限元分析的对比（h/L=0.1，d1/L=0.5，d2/L=0.8） 表 5  第1阶频率参数Ω1与Lourdes[15]解的对比 图 6  裂纹固支梁前8阶频率参数随裂纹深度的变化图 图 7  不同c1下固支梁W的前3阶振型 图 8  不同c2下固支梁W的前3阶振型
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