Abstract
This paper presents a method on free vibration analysis of a uniform continuous beam with an arbitrary number of cracks and spring-mass systems and a damage identification algorithm. Firstly, based on an equivalent model of a single-span beam obtained by using fictitious cracks or fictitious spring-mass systems, mode shape functions of a single-span uniform beam are derived according to fundamental solutions and recurrence formulas. Next, the motion equations and compatibility conditions of spring-mass system at intermediate supports are used to establish the characteristic equation of continuous beam with an arbitrary number of cracks and spring-mass systems. The order of determinant for the characteristic equation is lower, and that of m-spans continuous beam with an arbitrary number of cracks and T spring-mass systems is only \(3m+T-1\). Then, according to the formed free vibration analysis method, a damage identification algorithm for the uniform continuous beam with an arbitrary number of cracks and spring-mass systems is presented. This algorithm only utilizes the first-order mode to identify crack position and depth. Finally, the results of this paper are compared with those of the finite element method and other existing references to validate the proposed method. The effects of parameters for cracks and spring-mass systems on natural frequencies of continuous beams are studied. Inverse problem examples of a three-span cracked continuous beam carrying one spring-mass system are also discussed.
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Khalili, S.M.R.; Damanpack, A.R.; Nemati, N.; Malekzadeh, K.: Free vibration analysis of sandwich beam carrying sprung masses. Int. J. Mech. Sci. 52, 1620–1633 (2010)
Banerjee, J.R.: Free vibration of beams carrying spring-mass systems—a dynamic stiffness approach. Comput. Struct. 104–105, 21–26 (2012)
Wu, J.S.; Chen, D.W.: Free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems by using the numerical assembly technique. Int. J. Numer. Methods Eng. 50, 1039–1058 (2001)
Wu, J.S.; Hsieh, M.: Free vibration analysis of a non-uniform beam with multiple point masses. Struct. Eng. Mech. 9, 449–467 (2000)
Gürgöze, M.: On the eigenfrequencies of cantilevered beams carrying a tip mass and spring-mass in-span. Int. J. Mech. Sci. 38, 1295–1306 (1996)
Yesilce, Y.: Free vibrations of a Reddy–Bickford multi-span beam carrying multiple spring-mass systems. Shock Vib. 18, 709–726 (2011)
Lin, H.Y.; Tsai, Y.C.: Free vibration analysis of a uniform multi-span beam carrying multiple spring-mass systems. J. Sound Vib. 302, 442–456 (2007)
Dimarogonas, A.D.: Vibration of cracked structures: a state of the art review. Eng. Fract. Mech. 55, 831–857 (1996)
Mirzabeigy, A.; Bakhtiari-Nejad, F.: Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends. Front. Mech. Eng. 9, 191–202 (2014)
Banerjee, J.R.; Guo, S.: On the dynamics of a cracked beam. In: 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Palm Springs, CA, 4–7 (2009)
Torabi, K.; Afshari, H.; Aboutalebi, F.H.: A DQEM for transverse vibration analysis of multiple cracked non-uniform Timoshenko beams with general boundary conditions. Comput. Math. Appl. 67, 527–541 (2014)
Mazanoglu, K.; Yesilyurt, I.; Sabuncu, M.: Vibration analysis of multiple-cracked non-uniform beams. J. Sound Vib. 320, 977–989 (2009)
Zheng, T.; Ji, T.: An approximate method for determining the static deflection and natural frequency of a cracked beam. J. Sound Vib. 331, 2654–2670 (2012)
Ostachowitz, W.M.; Krawczuk, M.: Analysis of the effect of cracks on the natural frequencies of a cantilever beam. J. Sound Vib. 150, 191–201 (1991)
Shifrin, E.I.; Ruotolo, R.: Natural frequencies of a beam with an arbitrary number of cracks. J. Sound Vib. 222, 409–423 (1999)
Kisa, M.; Brandon, J.; Topcu, M.: Free vibration analysis of cracked beams by a combination of finite elements and component mode synthesis methods. Comput. Struct. 67, 215–223 (1998)
Morassi, A.: Crack-induced changes in eigenfrequencies of beam structures. J. Eng. Mech. 119, 768–803 (1993)
Kisa, M.; Gurel, M.A.: Free vibration analysis of uniform and stepped cracked beams with circular cross sections. Int. J. Eng. Sci. 45, 364–380 (2007)
Zheng, D.Y.; Kessissoglou, N.J.: Free vibration analysis of a cracked beam by finite element method. J. Sound Vib. 273, 457–475 (2004)
Chen, Q.; Fan, S.C.; Zheng, D.Y.: Natural frequency of stepped beam having multiple open cracks by transfer matrix method. In: 1st International Conference on Computational Methods, pp. 1975–1979. Springer, Dordrecht (2006). https://doi.org/10.1007/978-1-4020-3953-9_143
Attar, M.: 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, 19–33 (2012)
Caddemi, S.; Caliò, I.: Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks. J. Sound Vib. 327, 473–489 (2009)
Li, Q.S.: Free vibration analysis of non-uniform beams with arbitrary number of cracks and concentrated masses. J. Sound Vib. 252, 509–525 (2002)
Tan, G.; Wang, W.; Jiao, Y.: Free vibration analysis of a cracked simply supported bridge considering bridge—vehicle interaction. J. Vibroeng. 18, 3608–3635 (2016)
Zheng, D.Y.; Fan, S.C.: Vibration and stability of cracked hollow-sectional beams. J. Sound Vib. 267, 933–954 (2003)
Holmes, M.H.: Introduction to Perturbation Methods. Springer, Berlin (1995)
Fernández-Sáez, J.; Navarro, C.: Fundamental frequency of cracked beams in bending vibrations: an analytical approach. J. Sound Vib. 256, 17–31 (2002)
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Tan, G., Zhu, Z., Wang, W. et al. Free Vibration Analysis of a Uniform Continuous Beam with an Arbitrary Number of Cracks and Spring-Mass Systems. Arab J Sci Eng 43, 4619–4634 (2018). https://doi.org/10.1007/s13369-017-2933-0
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DOI: https://doi.org/10.1007/s13369-017-2933-0