Browse > Article
http://dx.doi.org/10.12989/sem.2018.66.1.125

A transfer matrix method for in-plane bending vibrations of tapered beams with axial force and multiple edge cracks  

Lee, Jung Woo (Department of Mechanical System Engineering, Kyonggi University)
Lee, Jung Youn (Department of Mechanical System Engineering, Kyonggi University)
Publication Information
Structural Engineering and Mechanics / v.66, no.1, 2018 , pp. 125-138 More about this Journal
Abstract
This paper proposes a transfer matrix method for the bending vibration of two types of tapered beams subjected to axial force, and it is applied to analyze tapered beams with an edge or multiple edge open cracks. One beam type is assumed to be reduced linearly in the cross-section height along the beam length. The other type is a tapered beam in which the cross-section height and width with the same taper ratio is linearly reduced simultaneously. Each crack is modeled as two sub-elements connected by a rotational spring, and the method can evaluate the effect of cracking on the desired number of eigenfrequencies using a minimum number of subdivisions. Among the power series available for the solutions, the roots of the differential equation are computed using the Frobenius method. The computed results confirm the accuracy of the method and are compared with previously reported results. The effectiveness of the proposed methods is demonstrated by examining specific examples, and the effects of cracking and axial loading are carefully examined by a comparison of the single and double tapered beam results.
Keywords
transfer matrix method; Frobenius method; crack; axial force; tapered beam;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
연도 인용수 순위
1 Nahvi, H. and Jabbari, M. (2005), "Crack detection in beams using experimental modal data and finite element model", Int. J. Mech. Sci., 47(10), 1477-1497.   DOI
2 Neves, A.C., Simoes, F.M.F. and Pinto Da Costa, A. (2016), "Vibrations of cracked beams: Discrete mass and stiffness models", Comput. Struct., 168, 68-77.   DOI
3 Rossit, C.A., Bambill D.V. and Gilardi G.J. (2017), "Free vibrations of AFG cantilever tapered beams carrying attached masses", Struct. Eng. Mech., 61(5), 685-691.   DOI
4 Ruotolo, R. and Surace, C. (1997), "Damage assessment of multiple cracked beams: Numerical results and experimental validation", J. Sound Vibr., 206(4), 567-588.   DOI
5 Sarkar, K., Ganguli, R. and Elishakoff, I. (2016), "Closed-form solutions for non-uniform axially loaded rayleigh cantilever beams", Struct. Eng. Mech., 60(3), 455-470.   DOI
6 Sarkar, K. and Ganguli, R. (2014), "Modal tailoring and closedform solutions for rotating non-uniform euler-bernoulli beams", Int. J. Mech. Sci., 88, 208-220.   DOI
7 Skrinar, M. (2009), "Elastic beam finite element with an arbitrary number of transverse cracks", Finit. Elem. Anal. Des., 45(3), 181-189.   DOI
8 Sun, W., Sun, Y., Yu, Y. and Zheng, S. (2016), "Nonlinear vibration analysis of a type of tapered cantilever beams by using an analytical approximate method", Struct. Eng. Mech., 59(1), 1-14.   DOI
9 Vinod, K.G., Gopalakrishnan, S. and Ganguli, R. (2007), "Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements", Int. J. Sol. Struct., 44(18-19), 5875-5893.   DOI
10 Wauer, J. (1990), "On the dynamics of cracked rotors: A literature survey", Appl. Mech. Rev., 43(1), 13-17.   DOI
11 Caddemi, S. and Calio, I. (2009), "Exact closed-form solution for the vibration modes of the euler-bernoulli beam with multiple open cracks", J. Sound Vibr., 327(3-5), 473-489.   DOI
12 Attar, M. (2012), "A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions", Int. J. Mech. Sci., 57(1), 19-33.   DOI
13 Banerjee, J.R., Su, H. and Jackson, D.R. (2006), "Free vibration of rotating tapered beams using the dynamic stiffness method", J. Sound Vibr., 298(4-5), 1034-1054.   DOI
14 Yan, Y., Ren, Q., Xia, N. and Zhang, L. (2016), "A closed-form solution applied to the free vibration of the euler-bernoulli beam with edge cracks", Arch. Appl. Mech., 86(9), 1633-1646.   DOI
15 Yuan, J.H., Pao, Y.H. and Chen, W.Q. (2016), "Exact solutions for free vibrations of axially inhomogeneous Timoshenko beams with variable cross section", Mecc., 227(9), 2625-2643.
16 Zhang, K. and Yan, X. (2016), "Multi-cracks identification method for cantilever beam structure with variable cross-sections based on measured natural frequency changes", J. Sound Vibr., 387, 53-65.
17 Zhou, Y., Zhang, Y. and Yao, G. (2017), "Stochastic forced vibration analysis of a tapered beam with performance deterioration", Acta Mech., 228(4), 1393-1406.   DOI
18 Behzad, M., Ghadami, A., Maghsoodi, A. and Hale, J.M. (2013), "Vibration based algorithm for crack detection in cantilever beam containing two different types of cracks", J. Sound Vibr., 332(24), 6312-6320.   DOI
19 Broda, D., Pieczonka, L., Hiwarkar, V., Staszewski, W.J. and Silberschmidt, V.V. (2016), "Generation of higher harmonics in longitudinal vibration of beams with breathing cracks", J. Sound Vibr., 381, 206-219.   DOI
20 Caddemi, S. and Morassi, A. (2013), "Multi-cracked eulerbernoulli beams: Mathematical modeling and exact solutions", Int. J. Sol. Struct., 50(6), 944-956.   DOI
21 Chaudhari, T.D. and Maiti, S.K. (1999), "Modelling of transverse vibration of beam of linearly variable depth with edge crack", Eng. Fract. Mech., 63(4), 425-445.   DOI
22 Cheng, Y., Yu, Z., Wu, X. and Yuan, Y. (2011), "Vibration analysis of a cracked rotating tapered beam using the p-version finite element method", Finit. Elem. Anal. Des., 47(7), 825-834.   DOI
23 Chondros, T.G., Dimarogonas, A.D. and Yao, J. (1998), "A continuous cracked beam vibration theory", J. Sound Vibr., 215(1), 17-34.   DOI
24 Dimogoronas, A.D. (1996), "Vibration of cracked structures: A state of the art review", Eng. Fract. Mech., 55(5), 831-857.   DOI
25 Dona, M., Palmeri, A. and Lombardo, M. (2015), "Dynamic analysis of multi-cracked euler-bernoulli beams with gradient elasticity", Comput. Struct., 161, 64-76.   DOI
26 Lee, J.W. and Lee, J.Y. (2016), "Free vibration analysis using the transfer-matrix method on a tapered beam", Comput. Struct., 164, 75-82.   DOI
27 Fernandez-Saez, J., Morassi, A., Pressacco, M. and Rubio, L. (2016), "Unique determination of a single crack in a uniform simply supported beam in bending vibration", J. Sound Vibr., 371, 94-109.   DOI
28 Kisa, M. and Gurel, M.A. (2007), "Free vibration analysis uniform and stepped cracked beams with circular cross sections", Int. J. Eng. Sci., 45(2-8), 364-380.   DOI
29 Kundu, B. and Ganguli, R. (2017), "Analysis of weak solution of euler-bernoulli beam with axial force", Appl. Math. Comput., 298, 247-260.
30 Lee, J.H. (2009), "Identification of multiple cracks in a beam using vibration amplitudes", J. Sound Vibr., 326(1-2), 205-212.   DOI
31 Lee, J.W. and Lee, J.Y. (2017a), "A transfer matrix method capable of determining the exact solutions of a twisted bernoulli-euler beam with multiple edge cracks", Appl. Math. Model., 41, 474-493.   DOI
32 Lee, J.W. and Lee, J.Y. (2017b), "In-plane bending vibration analysis of a rotating beam with multiple edge cracks by using the transfer matrix method", Mecc., 52(4-5), 1143-1157.   DOI
33 Lee, Y.S. and Chung, M.J. (2000), "A study on crack detection using eigenfrequency test data", Comput. Struct., 77(3), 327-342.   DOI
34 Mazanoglu, K. and Sabuncu, M. (2010), "Vibration analysis of non-uniform beams having multiple edge cracks along the beam's height", Int. J. Mech. Sci., 52(3), 515-522.   DOI
35 Li, X.F., Tang, A.Y. and Xi, L.Y. (2013), "Vibration of a rayleigh cantilever beam with axial force and tip mass", J. Constr. Steel. Res., 80, 15-22.   DOI
36 Hodges, D.H. and Rutkowski, M.J. (1981), "Free-vibration ananlysis of rotating beams by a variable-order finite element method", AIAA J., 19(11), 1459-1466.   DOI
37 Loya, J.A., Rubio, L. and Fernandez-Saez, J. (2006), "Natural frequencies for bending vibrations of Timoshenko cracked beams", J. Sound Vibr., 290(3-5), 640-653.   DOI