Free vibration of a variable cross-section simple supported beam with multi-crack
MA Yijiang1, LI Yuanyuan3, CHEN Guoping2, ZHAO Yingjie2
1. School of Naval Architecture and Ocean Engineering， Jiangsu University of Science and Technology，Zhenjiang 212003，China;
2. College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;
3. School of Mechanical Engineering, University of Jinan, Jinan 250022, China
Based on the transfer matrix method, a new method was proposed to solve natural frequencies of a variable cross-section beam with multi-crack. Aiming at a kind of variable cross-section beams with their vibration modal shape functions being able to be deduced, some bending springs without mass were used to be equivalent transverse cracks, the transfer matrix for a typical segment of a beam with variable cross-section parameters and crack geometric parameters was derived. Then, according to boundary conditions of simply supported beam, the characteristic transfer matrix of the whole variable cross-section simply supported beam with multi-crack was derived to study effects of variable cross-section parameters, crack number and crack geometric parameters on natural frequencies of the beam. Results indicated that effects of cracks on vibration modes of the variable cross-section beam are very large, they can significantly reduce each order natural frequency of the variable cross-section simply supported beam.
马一江1，李园园3，陈国平2，赵颖杰2. 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 Yijiang1, LI Yuanyuan3, CHEN Guoping2, ZHAO Yingjie2. Free vibration of a variable cross-section simple supported beam with multi-crack. JOURNAL OF VIBRATION AND SHOCK, 2019, 38(19): 149-154.
[1] 叶开沅, 郑书英. 非均匀弹性基础上的非均匀变截面梁的稳定性和自由振动问题[J]. 兰州大学学报:自然科学版, 1983(S2):40-60.
K Y Ye, S Y Zheng. Problems of buckling and free vibration of non-homogenous variable cross section beams on non-homogeneous elastic foundation[J]. Journal of Lanzhou University (Natural Science), 1983(S2):40-60. (in Chinese)
[2] Lee S Y, Ke H Y. Free vibrations of a non-uniform beam with general elastically restrained boundary conditions[J]. Journal of Sound & Vibration, 1990, 136(3):425-437.
[3] Tan C A, Kuang W. Distributed Transfer Function Analysis of Cone and Wedge Beams[J]. Journal of Sound & Vibration, 1994, 170(4):557-566.
[4] Kukla S. Free Vibrations of Axially Loaded Beams With Concentrated Masses and Intermediate Elastic Supports[J]. Journal of Sound & Vibration, 1994, 172(4):449-458.
[5] Mao Q, Pietrzko S. Free vibration analysis of stepped beams by using Adomian decomposition method[J]. Applied Mathematics & Computation, 2010, 217(7):3429-3441.
[6] 张英世, 刘宗德. 轴向受载的阶梯梁的振动[J]. 工程力学, 1999, 16(3): 75-80.
S Y Zhang, Z D Liu. Vibrations of stepped beams under axial loads[J]. Engineering Mechanics, 1999, 16(3): 75-80. (in Chinese)
[7] 张元海. 一个改进的平面梁单元[J]. 计算力学学报, 2002, 19(1):109-111.
Y H Zhang. An improved plane beam finite element[J]. Chinese Journal of Computational Mechanics, 2002, 19(1): 109-111. (in Chinese)
[8] 赵斌, 王正中. 变截面梁单元力学模型的建立[J]. 工程建设与设计, 2002(5):10-12.
B Zhao, Z Z Wang. Establishment of mechanical model of variable cross section beam element[J]. Construction and Design for Project, 2002(5):10-12. (in Chinese)
[9] Gupta A K. Vibration of tapered beams [J]. Journal of Structural Engineering, 1985,111(1):19-36.
[10] Alshorbagy AE, Eltaher MA, Mahmoud FF. Free vibration characteristics of afunctionally graded beam by finite element method. Appl Math Model 2011;35:412–25
[11] Huang Y, Li XF. A new approach for free vibration of axially functionally gradedbeams with non-uniform cross-section. J Sound Vib 2010;329:2291–303.
[12] Ahamad Shahba, Soundaramoorthy Rajasekaran. Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials[J].Applied Mathenatical Modelling,2012,36(7):3094-3111.
[13] Mehmet C E,Metin A,Vedat T. Vibration of a variable cross-section beam [J]. Mechanics Research Communications,2007,34:78-84.
[14] Laura, P.A.A., Gutierrez, R.H., Rossi, R.E., 1996. Free vibration of beams of bi-linearly varying thickness. Ocean Engineering 23 (1), 1–6.
[15] Caruntu D.On onolinear vibration of non-uniform beam with recangular cross-section and parabolic thickness variation [M].Solid Mechanics and its Applications. Kluwer Academic Publishers, Dordrecht,Boston,London,2000:109-118.
[16] Ostachowicz W M, Krawczuk M. Analysis of the effect of cracks on the natural frequencies of a cantilever beam[J]. J Sound Vib,1991,150:191-201.
[17] Hu J, Liang R Y. An integrated approach to detection of cracks using vibration characteristics[J]. J Franklin Inst,1993,330(5):841-853.
[18] Shifrin E I, Ruotolo R. Natural frequencies of a beam with an arbitrary number of cracks[J], J Sound Vib,1999,222(3):409-423.
[19] Dimaronas A D, Paipeties S A. Analytical methods in rotor dynamics[M]. Elsevier applied science,London,1983.
