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.
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.
Fig.1Analytical model of beam with two cracks of different depths
Fig.2Analytical model of beam with two cracks of same depth
Fig.3Analytical model of beam with three cracks of different depths
Fig.4Flow 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.1Convergence of first-eighth frequency parameters Ω of fixed beam
Fig.5Analytical 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.2Comparison 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.3Comparison 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.4Comparison 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.5Comparison of first frequency parameter Ω1 with Lourdes’s[15] results
Fig.6First-eighth frequency parameters of cracked fixed beams with different crack depths
Fig.7First-third modal shapes of W of fixed beams with different c1
Fig.8First-third modal shapes of W of fixed beams with different c2
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