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Nonlinear dynamic analysis of two-disk rotor system containing an unbalance influenced transverse crack

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Abstract

The vibration monitoring of the rotating machines is an effective tool to detect rotating element’s defects at their early stages. The dynamic characteristics such as natural frequencies, whirl orbits and 1X, 2X, 3X harmonic components of dynamic response are significantly influenced by the presence of a transverse crack in the rotating shaft. In the present work, a numerical study has been performed using the finite element method (FEM) to investigate the changes in natural frequencies and whirl orbit plots of rotating shaft system due to crack presence. The time-varying stiffness matrix of the cracked rotor system has been modeled by considering the product moment of area. A novel, simplified and competent breathing function has been proposed to approximate the actual breathing phenomena of transverse crack. The obtained equation of motion of two-disk shaft system has been solved by using the implicit Newmark time step integration algorithm. Moreover, the effect of crack depth and crack location on the natural frequencies and whirl orbits have been investigated. The whirl orbits of the cracked rotor system are found sensitive to the unbalance force direction. The size of the inner loop of whirl orbit in the neighborhood of 1/2 of critical speed provides useful information about the crack depth. The whirl orbit shapes are also affected by the location of the crack. The numerical results have closed agreement with the experimental results and literature. Hence, the proposed algorithm was found to be efficient for crack detection and can be used to identify crack depth as well as crack location.

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References

  1. Sabnavis, G., Kirk, R.G., Kasarda, M., Quinn, D.: Cracked shaft detection and diagnostics: a literature review. Shock Vib. Dig. 36(4), 287–297 (2004)

    Google Scholar 

  2. Gasch, R.: A survey of the dynamic behaviour of a simple rotating shaft with a transverse crack. J. Sound Vib. 160(2), 313–332 (1993)

    MATH  Google Scholar 

  3. Dimarogonas, A.D.: Vibration of cracked structures: a state-of-the-art review. Eng. Fract. Mech. 55(5), 831–857 (1996)

    Google Scholar 

  4. Wauer, J.R.: On the dynamics of cracked rotors: a literature survey. Appl. Mech. Rev. 43, 13–17 (1990)

    Google Scholar 

  5. Kumar, C., Rastogi, V.: A brief review on dynamics of a cracked rotor. Int. J. Rotat. Mach. 2009(2), 1–6 (2009)

    Google Scholar 

  6. Kushwaha, N., Patel, V.N.: Modelling and analysis of a cracked rotor: a review of the literature and its implications. Arch. Appl. Mech. 90(6), 15–45 (2020)

    Google Scholar 

  7. Sekhar, A.S.: Vibration characteristics of a cracked rotor with two open cracks. J. Sound Vib. 223(4), 497–512 (1999)

    Google Scholar 

  8. Sinou, J.J., Lees, A.W.: The influence of cracks in rotating shafts. J. Sound Vib. 285(4–5), 1015–1037 (2005)

    Google Scholar 

  9. Sinou, J.J.: Detection of cracks in rotor based on the 2× and 3× super-harmonic frequency components and the crack unbalance interactions. Commun. Nonlinear Sci. Numer. Simul. 13(9), 2024–2040 (2008)

    Google Scholar 

  10. Sekhar, A.S., Prabhu, B.S.: Crack detection and vibration characteristics of cracked shafts. J. Sound Vib. 157(2), 375–381 (1992)

    Google Scholar 

  11. Jun, O.S., Eun, H.J., Earmme, Y.Y., Lee, C.W.: Modelling and vibration analysis of a simple rotor with a breathing crack. J. Sound Vib. 155(2), 273–290 (1992)

    MATH  Google Scholar 

  12. Rubio, L., Fernández-Sáez, J.: A new efficient procedure to solve the nonlinear dynamics of a cracked rotor. Nonlinear Dyn. 70(3), 1731–1745 (2012)

    MathSciNet  Google Scholar 

  13. Patel, T.H., Darpe, A.K.: Influence of crack breathing model on nonlinear dynamics of a cracked rotor. J. Sound Vib. 311(3–5), 953–972 (2008)

    Google Scholar 

  14. Fu, C., Ren, X., Yang, Y., Lu, K., Wang, Y.: Nonlinear response analysis of a rotor system with a transverse breathing crack under interval uncertainties. Int. J. Non-Linear Mech. 105, 77–87 (2018)

    Google Scholar 

  15. Sinou, J.J., Lees, A.W.: A non-linear study of a cracked rotor. Eur. J. Mech. A/Solids. 26(1), 152–170 (2007)

    MATH  Google Scholar 

  16. Al-Shudeifat, M.A., Butcher, E.A., Stern, C.R.: General harmonic balance solution of a cracked rotor-bearing-disk system for harmonic and sub-harmonic analysis: analytical and experimental approach. Int. J. Eng. Sci. 48(10), 921–935 (2010)

    Google Scholar 

  17. Guo, C., Al-Shudeifat, M.A., Yan, J., Bergman, L.A., McFarland, D.M., Butcher, E.A.: Application of empirical mode decomposition to a Jeffcott rotor with a breathing crack. J. Sound Vib. 332(16), 3881–3892 (2013)

    Google Scholar 

  18. Darpe, A.K., Gupta, K., Chawla, A.: Transient response and breathing behaviour of a cracked Jeffcott rotor. J. Sound Vib. 272(1–2), 207–243 (2004)

    Google Scholar 

  19. Guo, C.Z., Yan, J.H., Bergman, L.A.: Experimental dynamic analysis of a breathing cracked rotor. Chin. J. Mech. Eng. 30(5), 1177–1183 (2017)

    Google Scholar 

  20. El Arem, S., Zid, M.B.: On a systematic approach for cracked rotating shaft study: breathing mechanism, dynamics and instability. Nonlinear Dyn. 88(3), 2123–2138 (2017)

    Google Scholar 

  21. El Arem, S.: Nonlinear analysis, instability and routes to chaos of a cracked rotating shaft. Nonlinear Dyn. 96(1), 667–683 (2019)

    MATH  Google Scholar 

  22. El Arem, S.: On the mechanics of beams and shafts with cracks: A standard and generic approach. Eur. J. Mech. A/Solids. 85, 104088 (2021)

    MathSciNet  MATH  Google Scholar 

  23. Lu, Z., Lv, Y., Ouyang, H.: A super-harmonic feature-based updating method for crack identification in rotors using a kriging surrogate model. Appl. Sci. 9(12), 2428 (2019)

    Google Scholar 

  24. Abdi, H., Nayeb-Hashemi, H., Hamouda, A.M.S., Vaziri, A.: Torsional dynamic response of a shaft with longitudinal and circumferential cracks. J. Vib. Acoust. 136(6), 1–8 (2014)

    Google Scholar 

  25. Nabian, M., Vaziri, A., Olia, M., Nayeb-Hashemi, H.: The effects of longitudinal and circumferential cracks on the torsional dynamic response of shafts. ASME Int. Mech. Eng. Cong. Exposit. 5(6), 253 (2013)

    Google Scholar 

  26. Sekhar, A.S., Prasad, P.B.: Dynamic analysis of a rotor system considering a slant crack in the shaft. J. Sound Vib. 208(3), 457–474 (1997)

    Google Scholar 

  27. Papadopoulos, C.A., Dimarogonas, A.D.: Coupled longitudinal and bending vibrations of a rotating shaft with an open crack. J. Sound Vib. 117(1), 81–93 (1987)

    Google Scholar 

  28. Dimarogonas, A.D., Papadopoulos, C.A.: Vibration of cracked shafts in bending. J. Sound Vib. 91(4), 583–593 (1983)

    MATH  Google Scholar 

  29. Chan, R.K.C., Lai, T.C.: Digital simulation of a rotating shaft with a transverse crack. Appl. Math. Model. 19(7), 411–420 (1995)

    MATH  Google Scholar 

  30. Sekhar, A.S., Prabhu, B.S.: Condition monitoring of cracked rotors through transient response. Mech. Mach. Theory 33(8), 1167–1175 (1998)

    MATH  Google Scholar 

  31. Al-Shudeifat, M.A., Butcher, E.A.: New breathing functions for the transverse breathing crack of the cracked rotor system: approach for critical and subcritical harmonic analysis. J. Sound Vib. 330(3), 526–544 (2011)

    Google Scholar 

  32. Al-Shudeifat, M.A., Butcher, E.A.: On the modeling of open and breathing cracks of a cracked rotor system. ASME, Int. Des. Eng. Tech. Conf. Comput. Informat. Eng. Conf. 44137, 919–928 (2010)

    Google Scholar 

  33. Giannopoulos, G.I., Georgantzinos, S.K., Anifantis, N.K.: Coupled vibration response of a shaft with a breathing crack. J. Sound Vib. 336, 191–206 (2015)

    Google Scholar 

  34. Spagnol, J., Wu, H., Yang, C.: Application of non-symmetric bending principles on modelling fatigue crack behaviour and vibration of a cracked rotor. Appl. Sci. 10(2), 717 (2020)

    Google Scholar 

  35. Guo, C., Al-Shudeifat, M.A., Yan, J., Bergman, L.A., McFarland, D.M., Butcher, E.A.: Stability analysis for transverse breathing cracks in rotor systems. Eur. J. Mech. A/Solids. 42, 27–34 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Friswell, M.I., Garvey, S.D., Penny, J.E.T., Lees, A.W.: Dynamics of rotating machines. Cambridge University Press, New York (2010)

    MATH  Google Scholar 

  37. Davies, W.G.R., Mayes, I.W.: The vibrational behavior of a multi-shaft, multi-bearing system in the presence of a propagating transverse crack. J. Vib. Acoust. Stress Reliab. Des. 106(1), 146–153 (1984)

    Google Scholar 

  38. Mayes, I.W., Davies, W.G.R.: Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor. J. Vib. Acoust. Stress Reliab. Des. 106(1), 139–145 (1984)

    Google Scholar 

  39. Sinou, J.J., Denimal, E.: Reliable crack detection in a rotor system with uncertainties via advanced simulation models based on kriging and Polynomial Chaos Expansion. Eur. J. Mech. A/Solids. 92(3), 104451 (2022)

    MathSciNet  MATH  Google Scholar 

  40. Chen, X.: Nonlinear responses analysis caused by slant crack in a rotor-bearing system. Journal of Vibroengineering. 18(7), 4369–4387 (2016)

    Google Scholar 

  41. Darpe, A.K., Gupta, K.: Chawla A: Coupled bending, longitudinal and torsional vibrations of a cracked rotor. J. Sound Vib. 269(1), 33–60 (2004)

    Google Scholar 

  42. Khorrami, H., Rakheja, S., Sedaghati, R.: Vibration behavior of a two-crack shaft in a rotor disc-bearing system. Mech. Mach. Theory. 113(7), 67–84 (2017)

    Google Scholar 

  43. Mobarak, H.M., Wu, H., Spagnol, J.P., Xiao, K.: New crack breathing mechanism under the influence of unbalance force. Arch. Appl. Mech. 88(3), 341–372 (2018)

    Google Scholar 

  44. Hossain, M., Wu, H.: Crack breathing behavior of unbalanced rotor system: a quasi-static numerical analysis. J. Vibroeng. 20(3), 1459–1469 (2018)

    Google Scholar 

  45. Dakel, M., Baguet, S., Dufour, R.: Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings. J. Sound Vib. 333(10), 2774–2799 (2014)

    Google Scholar 

  46. De Oliveira, L.R., de Melo, G.P.: Crack detection and dynamic analysis of a cracked rotor with soft bearings using different methods of solution. In: International Conference on Rotor Dynamics, pp. 3–17. Springer, Cham (2018)

    Google Scholar 

  47. Dong, G.M., Chen, J.: Crack identification in a rotor with an open crack. J. Mech. Sci. Technol. 23(11), 2964–2972 (2009)

    Google Scholar 

  48. Ghozlane, M.: Dynamic response of cracked shaft in rotor bearing-disk system. In: Design and Modeling of Mechanical Systems-II, pp. 615–624. Springer, Cham (2015)

    Google Scholar 

  49. Chinka, S.S.B., Adavi, B., Putti, S.R.: Effect of crack location and crack depth on natural frequencies of fixed beam using experimental modal analysis. In: Recent Advances in Material Sciences Lecture Notes on Multidisciplinary Industrial Engineering, pp. 1–12. Springer, Singapore (2019)

    Google Scholar 

  50. Karaagac, C., Öztürk, H., Sabuncu, M.: Free vibration and lateral buckling of a cantilever slender beam with an edge crack: experimental and numerical studies. J. Sound Vib. 326(1–2), 235–250 (2009)

    Google Scholar 

  51. Haji, Z.N., Oyadiji, S.O.: The use of roving discs and orthogonal natural frequencies for crack identification and location in rotors. J. Sound Vib. 333(23), 6237–6257 (2014)

    Google Scholar 

  52. He, F., Khan, M., Aldosari, S.: Interdependencies between Dynamic Response and Crack Growth in a 3D-Printed Acrylonitrile Butadiene Styrene (ABS) Cantilever Beam under Thermo-Mechanical Loads. Polymers 14(5), 982 (2022)

    Google Scholar 

  53. El-Mongy, H.H., Younes, Y.K.: Vibration analysis of a multi-fault transient rotor passing through sub-critical resonances. J. Vib. Control. 24(14), 2986–3009 (2018)

    Google Scholar 

  54. Jomezade Khazarbeygi, S., Yadavar Nikravesh, S.M.: Using a continuous mathematical model to investigate the effect of a symmetric circumferential crack on the free vibrations of a simply supported Euler-Bernoulli beam. J. Vib. Control. 109, 10775463221092836 (2022)

    Google Scholar 

  55. Xiang, J., Chen, X., Mo, Q., He, Z.: Identification of crack in a rotor system based on wavelet finite element method. Finite Elem Anal Des. 43(14), 1068–1081 (2007)

    Google Scholar 

  56. Zhou, T., Sun, Z., Xu, J., Han, W.: Experimental analysis of cracked rotor. J. Dyn. Syst. Meas. Control. 7(9), 313–320 (2005)

    Google Scholar 

  57. Wang, S., Zi, Y., Qian, S., Zi, B., Bi, C.: Effects of unbalance on the nonlinear dynamics of rotors with transverse cracks. Nonlinear Dyn. 91, 2755–2772 (2018)

    Google Scholar 

  58. Darpe, A.K., Gupta, K., Chawla, A.: Dynamics of a two-crack rotor. J. Sound Vib. 259(3), 649–675 (2003)

    Google Scholar 

  59. Li, Z., Li, Y., Wang, D., Peng, Z., Wang, H.: Dynamic characteristics of rotor system with a slant crack based on fractional damping. Chin. J. Mech. Eng. 34(1), 1–18 (2021)

    Google Scholar 

  60. Yang J.D., Zheng T., Zhang W., Yuan H., Wen.: The Complicated Response of a Simple Rotor with a Fatigue Crack, 6th International Conference on Rotating Machinery, IFToMM, Sydney, Australia (2002)

  61. Gómez-Mancilla, J., Machorro-López, J. M., Nosov, V. R.: Crack breathing mechanisms in rotor-bearings systems, its influence on system response and crack detection. ISCORMA-3, Cleveland Ohio (2008)

  62. Gasch, R.: Dynamic behaviour of the Laval rotor with a transverse crack. Mech. Syst. Signal Process. 22(4), 790–804 (2008)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Krishna School of Emerging Technology & Applied Research, KPGU, Vadodara, Gujarat Technological University, Nr. Vishwakarma Government Engineering College, Visat Three Roads, Visat–Gandhinagar Highway, Chandkheda, Ahmedabad, Gujarat, 382424, India

    Nirmal Kushwaha

  2. Mechatronics Department, G. H. Patel College of Engineering & Technology, Bakrol Road, Vallabh Vidyanagar, Anand, Gujarat, 388120, India

    V. N. Patel

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Correspondence to Nirmal Kushwaha.

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Kushwaha, N., Patel, V.N. Nonlinear dynamic analysis of two-disk rotor system containing an unbalance influenced transverse crack. Nonlinear Dyn 111, 1109–1137 (2023). https://doi.org/10.1007/s11071-022-07893-7

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