Dynamic Analysis of Fatigue-Cracked Beams: The Nonlinear Response with the Analytical Method

Document Type : Original Article

Authors

Department of Civil Engineering, Faculty of Engineering and Technology, University of Mazandaran, Babolsar, Iran

Abstract

In this study, the bending vibration of a fatigue-cracked beam and associated constraint conditions have been solved by implementing the Homotopy Perturbation Method. A structure with a single degree of freedom, varying stiffness, and a periodic function is employed to simulate the dynamic behavior of the beam. The crack is represented as an ongoing disturbance function within the displacement field, which could be obtained from fracture mechanics. The governing equation's solution shows the super harmonics of the dominant frequency, resulting from nonlinear impacts on the dynamic vibration response of the cracked beam. The proposed method gives an analytical closed-form solution that can be easily used to analyze and design structures dynamically. The outcomes show that growing crack depth reduces the natural frequencies of a cracked beam. Moreover, increasing the severity of the crack and moving its location toward the center of the beam increases the system's damping. Perturbation methods rely on a small parameter, which is challenging to determine for real-life nonlinear problems. To overcome this shortcoming, a powerful analytical method is introduced to solve the motion equation of the cracked beam.

Keywords

Main Subjects

References

  1. Khabiri, A., Asgari, A., Taghipour, R., Bozorgnasab, M., Aftabi-Sani, A., Jafari, H. Analysis of Fractional Euler-Bernoulli Bending Beams Using Green’s Function Method. Alexandria Engineering Journal, 2024; 106: 312–327. doi:10.1016/j.aej.2024.07.023.
  2. Bahreini, A., Asgari, A., Khabiri, A., Taghipour, R. Analysis of Plane Multi-Span Frames with the Analytical Method of Force-Displacement Combination. Sharif Journal of Civil Engineering. doi:10.24200/j30.2024.64303.3318.
  3. Asgari, A., Ganji D. D., Davodi A. G. Extended tanh method and exp-function method and its application to (2+1)-dimensional dispersive long wave nonlinear equations. Journal of the Applied Mathematics, Statistics and Informatics, 2010; 6: 61–72.
  4. Ganguly, S. Methodologies for Modeling and Identification of Breathing Crack: A Review. MethodsX, 2023; 11: 102420. doi:10.1016/j.mex.2023.102420.
  5. Behvar, A., Haghshenas, M., Djukic, M. B. Hydrogen Embrittlement and Hydrogen-Induced Crack Initiation in Additively Manufactured Metals: A Critical Review on Mechanical and Cyclic Loading. International Journal of Hydrogen Energy, 2024; 58: 1214–1239. doi:10.1016/j.ijhydene.2024.01.232.
  6. Sangid, M. D. The Physics of Fatigue Crack Initiation. International Journal of Fatigue, 2013; 57: 58–72. doi:10.1016/j.ijfatigue.2012.10.009.
  7. Faverjon, B., Sinou, J. J. Identification of an Open Crack in a Beam Using an a Posteriori Error Estimator of the Frequency Response Functions with Noisy Measurements. European Journal of Mechanics, A/Solids, 2009; 28 (1): 75–85. doi:10.1016/j.euromechsol.2008.02.006.
  8. Lam, H. F., Ng, C. T., Veidt, M. Experimental Characterization of Multiple Cracks in a Cantilever Beam Utilizing Transient Vibration Data Following a Probabilistic Approach. Journal of Sound and Vibration, 2007; 305 (1–2): 34–49. doi:10.1016/j.jsv.2007.03.028.
  9. Chati, M., Rand, R., Mukherjee, S. Modal Analysis of a Cracked Beam. Journal of Sound and Vibration, 1997; 207 (2): 249–270. doi:10.1006/jsvi.1997.1099.
  10. Chondros, T. G. The Continuous Crack Flexibility Model for Crack Identification. Fatigue and Fracture of Engineering Materials and Structures, 2001; 24 (10): 643–650. doi:10.1046/j.1460-2695.2001.00442.x.
  11. Loya, J. A., Rubio, L., Fernández-Sáez, J. Natural Frequencies for Bending Vibrations of Timoshenko Cracked Beams. Journal of Sound and Vibration, 2006; 290 (3–5): 640–653. doi:10.1016/j.jsv.2005.04.005.
  12. Ke, L. L., Yang, J., Kitipornchai, S., Xiang, Y. Flexural Vibration and Elastic Buckling of a Cracked Timoshenko Beam Made of Functionally Graded Materials. Mechanics of Advanced Materials and Structures, 2009; 16 (6): 488–502. doi:10.1080/15376490902781175.
  13. Friswell, M. I., Penny, J. E. T. A simple nonlinear model of a cracked beam. In: Proceedings of the International Modal Analysis Conference; 1992 Feb 3–7; California (US).
  14. Sinha, J. K., Friswell, M. I., Edwards, S. Simplified Models for the Location of Cracks in Beam Structures Using Measured Vibration Data. Journal of Sound and Vibration, 2002; 251 (1): 13–38. doi:10.1006/jsvi.2001.3978.
  15. Kisa, M., Brandon, J. Effects of Closure of Cracks on the Dynamics of a Cracked Cantilever Beam. Journal of Sound and Vibration, 2000; 238 (1): 1–18. doi:10.1006/jsvi.2000.3099.
  16. Rytter A. Vibrational based inspection of civil engineering structures [PhD thesis]. Aalborg (DK): University of Aalborg; 1993.
  17. Abraham, O. N. L., Brandon, J. A. The Modelling of the Opening and Closure of a Crack. Journal of Vibration and Acoustics, Transactions of the ASME, 1995; 117 (3): 370–377. doi:10.1115/1.2874463.
  18. Cheng, S. M., Wu, X. J., Wallace, W., Swamidas, A. S. J. Vibrational Response of a Beam with a Breathing Crack. Journal of Sound and Vibration, 1999; 225 (1): 201–208. doi:10.1006/jsvi.1999.2275.
  19. Ariaei, A., Ziaei-Rad, S., Ghayour, M. Vibration Analysis of Beams with Open and Breathing Cracks Subjected to Moving Masses. Journal of Sound and Vibration, 2009; 326 (3–5): 709–724. doi:10.1016/j.jsv.2009.05.013.
  20. Douka, E., Hadjileontiadis, L. J. Time-Frequency Analysis of the Free Vibration Response of a Beam with a Breathing Crack. NDT and E International, 2005; 38 (1): 3–10. doi:10.1016/j.ndteint.2004.05.004.
  21. Loutridis, S., Douka, E., Hadjileontiadis, L. J. Forced Vibration Behaviour and Crack Detection of Cracked Beams Using Instantaneous Frequency. NDT and E International, 2005; 38 (5): 411–419. doi:10.1016/j.ndteint.2004.11.004.
  22. Zhang, W., Testa, R. B. Closure Effects on Fatigue Crack Detection. Journal of Engineering Mechanics, 1999; 125 (10): 1125–1132. doi:10.1061/(asce)0733-9399(1999)125:10(1125).
  23. Bovsunovsky, A. P., Surace, C. Considerations Regarding Superharmonic Vibrations of a Cracked Beam and the Variation in Damping Caused by the Presence of the Crack. Journal of Sound and Vibration, 2005; 288 (4–5): 865–886. doi:10.1016/j.jsv.2005.01.038.
  24. Curadelli, R. O., Riera, J. D., Ambrosini, D., Amani, M. G. Damage Detection by Means of Structural Damping Identification. Engineering Structures, 2008; 30 (12): 3497–3504. doi:10.1016/j.engstruct.2008.05.024.
  25. Dimarogonas, A. D. Vibration of Cracked Structures: A State of the Art Review. Engineering Fracture Mechanics, 1996; 55 (5): 831–857. doi:10.1016/0013-7944(94)00175-8.
  26. Rezaee, M., Hassannejad, R. Free Vibration Analysis of Simply Supported Beam with Breathing Crack Using Perturbation Method. Acta Mechanica Solida Sinica, 2010; 23 (5): 459–470. doi:10.1016/S0894-9166(10)60048-1.
  27. Reta, A. K., Taube, C., Morgenthal, G. Structural Condition Assessment of Cracked RC Beams Based on Digital Crack Measurements. Engineering Structures, 2025; 323: 119211. doi:10.1016/j.engstruct.2024.119211.
  28. Meiramov, D., Ju, H., Seo, Y., Lee, D. Correlation between Deflection and Crack Propagation in Reinforced Concrete Beams. Measurement: Journal of the International Measurement Confederation, 2025; 240: 115527. doi:10.1016/j.measurement.2024.115527.
  29. Dimarogonas, A. D., Paipetis, S. A. Analytical Methods in Rotor Dynamics.; Springer Science & Business Media: Dordrecht, Netherlands, 1983; doi:10.1007/978-94-007-5905-3.
  30. Chondros, T. G., Dimarogonas, A. D., Yao, J. A Continuous Cracked Beam Vibration Theory. Journal of Sound and Vibration, 1998; 215 (1): 17–34. doi:10.1006/jsvi.1998.1640.
  31. Kranz, R. L. Microcracks in Rocks: A Review. Tectonophysics, 1983; 100 (1–3): 449–480. doi:10.1016/0040-1951(83)90198-1.
  32. Shaw, S. Vibrations in mechanical systems: analytical methods and applications. Berlin (DE): Springer Science & Business Media; 1989. Vol. 31. doi:10.1137/1031116.
  33. Clough, R. W., Penzien, J. Dynamics of structures. Berkeley (US): Computers & Structures, Inc.; 1975.
  34. Chondros, T. G., Dimarogonas, A. D., Yao, J. Vibration of a Beam with a Breathing Crack. Journal of Sound and Vibration, 2001; 239(1): 57–67. doi:10.1006/jsvi.2000.3156.
Civil Engineering and Applied Solutions
Volume 1, Issue 1
June 2025
Pages 89-100

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