DOI QR코드

DOI QR Code

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)
  • 투고 : 2017.07.04
  • 심사 : 2018.02.27
  • 발행 : 2018.04.10

초록

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.

키워드

과제정보

연구 과제 주관 기관 : Kyonggi University

참고문헌

  1. 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. https://doi.org/10.1016/j.ijmecsci.2012.01.010
  2. 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. https://doi.org/10.1016/j.jsv.2006.06.040
  3. 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. https://doi.org/10.1016/j.jsv.2013.07.003
  4. 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. https://doi.org/10.1016/j.jsv.2016.06.025
  5. Caddemi, S. and Morassi, A. (2013), "Multi-cracked eulerbernoulli beams: Mathematical modeling and exact solutions", Int. J. Sol. Struct., 50(6), 944-956. https://doi.org/10.1016/j.ijsolstr.2012.11.018
  6. 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. https://doi.org/10.1016/j.jsv.2009.07.008
  7. 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. https://doi.org/10.1016/S0013-7944(99)00029-6
  8. 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. https://doi.org/10.1016/j.finel.2011.02.013
  9. Chondros, T.G., Dimarogonas, A.D. and Yao, J. (1998), "A continuous cracked beam vibration theory", J. Sound Vibr., 215(1), 17-34. https://doi.org/10.1006/jsvi.1998.1640
  10. Dimogoronas, A.D. (1996), "Vibration of cracked structures: A state of the art review", Eng. Fract. Mech., 55(5), 831-857. https://doi.org/10.1016/0013-7944(94)00175-8
  11. Dona, M., Palmeri, A. and Lombardo, M. (2015), "Dynamic analysis of multi-cracked euler-bernoulli beams with gradient elasticity", Comput. Struct., 161, 64-76. https://doi.org/10.1016/j.compstruc.2015.08.013
  12. 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. https://doi.org/10.1016/j.jsv.2016.02.010
  13. 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. https://doi.org/10.2514/3.60082
  14. 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. https://doi.org/10.1016/j.ijengsci.2007.03.014
  15. Kundu, B. and Ganguli, R. (2017), "Analysis of weak solution of euler-bernoulli beam with axial force", Appl. Math. Comput., 298, 247-260.
  16. Lee, J.H. (2009), "Identification of multiple cracks in a beam using vibration amplitudes", J. Sound Vibr., 326(1-2), 205-212. https://doi.org/10.1016/j.jsv.2009.04.042
  17. 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. https://doi.org/10.1016/j.compstruc.2015.11.007
  18. 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. https://doi.org/10.1016/j.apm.2016.09.013
  19. 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. https://doi.org/10.1007/s11012-016-0449-4
  20. Lee, Y.S. and Chung, M.J. (2000), "A study on crack detection using eigenfrequency test data", Comput. Struct., 77(3), 327-342. https://doi.org/10.1016/S0045-7949(99)00194-7
  21. 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. https://doi.org/10.1016/j.jcsr.2012.09.015
  22. 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. https://doi.org/10.1016/j.jsv.2005.04.005
  23. 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. https://doi.org/10.1016/j.ijmecsci.2009.11.016
  24. 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. https://doi.org/10.1016/j.ijmecsci.2005.06.008
  25. 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. https://doi.org/10.1016/j.compstruc.2016.02.007
  26. 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. https://doi.org/10.12989/sem.2017.61.5.685
  27. Ruotolo, R. and Surace, C. (1997), "Damage assessment of multiple cracked beams: Numerical results and experimental validation", J. Sound Vibr., 206(4), 567-588. https://doi.org/10.1006/jsvi.1997.1109
  28. 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. https://doi.org/10.12989/sem.2016.60.3.455
  29. 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. https://doi.org/10.1016/j.ijmecsci.2014.08.003
  30. Skrinar, M. (2009), "Elastic beam finite element with an arbitrary number of transverse cracks", Finit. Elem. Anal. Des., 45(3), 181-189. https://doi.org/10.1016/j.finel.2008.09.003
  31. 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. https://doi.org/10.12989/sem.2016.59.1.001
  32. 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. https://doi.org/10.1016/j.ijsolstr.2007.02.002
  33. Wauer, J. (1990), "On the dynamics of cracked rotors: A literature survey", Appl. Mech. Rev., 43(1), 13-17. https://doi.org/10.1115/1.3119157
  34. 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. https://doi.org/10.1007/s00419-016-1140-x
  35. 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.
  36. 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.
  37. 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. https://doi.org/10.1007/s00707-016-1764-5

피인용 문헌

  1. A coupled experimental and numerical simulation of concrete joints' behaviors in tunnel support using concrete specimens vol.28, pp.2, 2021, https://doi.org/10.12989/cac.2021.28.2.189