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浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2020, Vol. 54 Issue (4): 778-786    DOI: 10.3785/j.issn.1008-973X.2020.04.017
Civil Engineering, Traffic Engineering     
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     
Received: 07 April 2019      Published: 05 April 2020
CLC:  TU 311  
Corresponding Authors: Chao-bin HU     E-mail: zhaojialei1995@njtech.edu.cn;huchaobin@njtech.edu.cn
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Jia-lei ZHAO
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Cite this article:

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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http://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2020.04.017     OR     http://www.zjujournals.com/eng/Y2020/V54/I4/778


基于Chebyshev-Ritz法分析多裂纹梁自振特性

基于弹性力学平面应力理论，利用Chebyshev-Ritz法分析多裂纹梁的自振特性. 根据裂纹情况将裂纹梁分成若干个梁段，用边界函数与第一类Chebyshev多项式的乘积构造各梁段的位移函数，具有很好的收敛性，能够适用于不同的几何边界条件. 用Ritz法得到各梁段的振动方程，根据各梁段之间的位移连续条件整合方程，建立整个裂纹梁的振动特征方程. 计算结果与有限元分析和相关文献数据吻合很好. 分析裂纹深度和位置对自振特性的影响. 随着裂纹深度的增大，裂纹梁的频率减小，振型的幅值变大，且影响的程度会受裂纹的位置影响.


关键词： 弹性力学,  Chebyshev-Ritz法,  裂纹,  自由振动,  位移连续条件 
Fig.1 Analytical model of beam with two cracks of different depths
Fig.2 Analytical model of beam with two cracks of same depth
Fig.3 Analytical model of beam with three cracks of different depths
Fig.4 Flow chart for calculation of multi-cracked beam
h/L mn Ω1 Ω2 Ω3 Ω4 Ω5 Ω6 Ω7 Ω8
0.1 40×10 0.183 7 0.488 4 0.831 2 0.987 1 1.341 8 1.774 1 1.949 6 2.397 6
0.1 50×10 0.183 3 0.488 2 0.828 5 0.986 8 1.341 2 1.769 8 1.948 7 2.397 0
0.1 50×15 0.183 3 0.488 2 0.828 5 0.986 8 1.341 1 1.769 8 1.948 7 2.397 0
0.1 60×15 0.183 1 0.488 0 0.826 7 0.986 6 1.340 7 1.766 8 1.948 1 2.396 6
0.2 40×10 0.436 8 1.055 1 1.379 3 1.611 6 2.536 7 2.620 4 3.272 4 4.115 9
0.2 50×10 0.436 3 1.054 7 1.378 7 1.608 7 2.535 7 2.617 3 3.271 2 4.113 8
0.2 50×15 0.436 2 1.054 6 1.378 6 1.608 5 2.535 7 2.617 1 3.271 1 4.113 6
0.2 60×15 0.435 8 1.054 4 1.378 2 1.606 6 2.535 0 2.614 9 3.270 3 4.112 2
0.3 40×10 0.669 8 1.483 6 1.684 9 2.128 9 3.150 1 3.347 0 4.017 4 4.273 0
0.3 50×10 0.669 2 1.482 9 1.684 4 2.125 6 3.147 7 3.344 5 4.016 5 4.271 3
0.3 50×15 0.669 2 1.482 8 1.684 3 2.125 2 3.147 4 3.344 4 4.016 4 4.271 1
0.3 60×15 0.668 8 1.482 3 1.684 0 2.123 0 3.145 7 3.342 5 4.015 7 4.270 0
Tab.1 Convergence of first-eighth frequency parameters Ω of fixed beam
Fig.5 Analytical model of cracked fixed beam in ANSYS
参数 方法 Ω1 Ω2 Ω3 Ω4 Ω5 Ω6 Ω7 Ω8
c1/h=0.2，c2/h=0.1 本文方法 0.188 1 0.491 9 0.866 2 0.993 5 1.353 7 1.825 4 1.968 2 2.407 0
c1/h=0.2，c2/h=0.1 有限元法 0.187 5 0.491 8 0.862 2 0.993 2 1.353 2 1.817 9 1.965 8 2.406 6
c1/h=0.3，c2/h=0.2 本文方法 0.183 1 0.488 0 0.826 7 0.986 6 1.340 7 1.766 8 1.948 1 2.396 6
c1/h=0.3，c2/h=0.2 有限元法 0.182 1 0.487 7 0.820 9 0.986 1 1.339 6 1.755 8 1.945 8 2.395 1
c1/h=0.4，c2/h=0.2 本文方法 0.176 9 0.487 6 0.797 4 0.986 3 1.338 1 1.693 4 1.932 7 2.389 9
c1/h=0.4，c2/h=0.2 有限元法 0.175 7 0.487 3 0.791 1 0.985 9 1.336 5 1.680 7 1.930 7 2.387 2
Tab.2 Comparison of results of fixed beam with those from FEA（h/L=0.1，d1/L=0.5，d2/L=0.8）
参数 方法 Ω1 Ω2 Ω3 Ω4 Ω5 Ω6 Ω7 Ω8
c1/h=0.2，c2/h=0.1 本文方法 0.085 6 0.336 8 0.692 1 0.978 5 1.177 9 1.668 3 1.977 9 2.259 1
c1/h=0.2，c2/h=0.1 有限元法 0.085 1 0.336 5 0.688 3 0.976 1 1.177 5 1.662 8 1.977 2 2.258 3
c1/h=0.3，c2/h=0.2 本文方法 0.081 3 0.329 0 0.655 5 0.953 6 1.169 6 1.632 8 1.954 3 2.243 9
c1/h=0.3，c2/h=0.2 有限元法 0.080 6 0.328 4 0.649 7 0.949 6 1.168 7 1.626 5 1.952 4 2.242 0
c1/h=0.4，c2/h=0.2 本文方法 0.076 3 0.328 7 0.623 6 0.926 1 1.167 4 1.599 1 1.954 1 2.237 6
c1/h=0.4，c2/h=0.2 有限元法 0.075 2 0.328 1 0.616 6 0.921 1 1.166 2 1.593 5 1.952 4 2.234 7
Tab.3 Comparison of results of simply-supported beam with those from FEA（h/L=0.1，d1/L=0.5，d2/L=0.8）
参数 方法 Ω1 Ω2 Ω3 Ω4 Ω5 Ω6 Ω7 Ω8
c1/h=0.2，c2/h=0.1 本文方法 0.031 6 0.185 2 0.493 4 0.501 0 0.886 0 1.386 9 1.476 7 1.881 9
c1/h=0.2，c2/h=0.1 有限元法 0.031 6 0.184 2 0.492 7 0.500 7 0.881 6 1.385 8 1.474 4 1.876 0
c1/h=0.3，c2/h=0.2 本文方法 0.031 3 0.177 0 0.485 2 0.493 4 0.839 7 1.359 1 1.449 0 1.840 2
c1/h=0.3，c2/h=0.2 有限元法 0.031 2 0.175 4 0.483 9 0.492 6 0.833 3 1.356 5 1.444 8 1.834 0
c1/h=0.4，c2/h=0.2 本文方法 0.030 7 0.166 8 0.475 9 0.491 9 0.812 4 1.355 6 1.420 3 1.809 4
c1/h=0.4，c2/h=0.2 有限元法 0.030 6 0.164 7 0.473 8 0.491 1 0.806 2 1.352 4 1.414 8 1.804 3
Tab.4 Comparison of results of cantilevered beam with those from FEA（h/L=0.1，d1/L=0.5，d2/L=0.8）
参数 方法 Ω1
d2/L=0.35 d2/L=0.45 d2/L=0.5
d1/L=0.25，c1/h=0.05 有限元解 0.087 8 0.087 7 0.087 7
d1/L=0.25，c1/h=0.05 本文解法 0.088 1 0.088 0 0.088 0
d1/L=0.25，c2/h=0.10 Lourdes解 0.089 7 0.089 6 0.089 6
d1/L=0.25，c2/h=0.10 有限元解 0.083 7 0.082 9 0.082 8
c1/h=0.15，c2/h=0.25 本文解法 0.084 6 0.084 0 0.083 9
c1/h=0.15，c2/h=0.25 Lourdes解 0.087 4 0.087 0 0.086 9
Tab.5 Comparison of first frequency parameter Ω1 with Lourdes’s[15] results
Fig.6 First-eighth frequency parameters of cracked fixed beams with different crack depths
Fig.7 First-third modal shapes of W of fixed beams with different c1
Fig.8 First-third modal shapes of W of fixed beams with different c2
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