Research Articles

New Finite Element Modeling Approach of a Propagating Shaft Crack

[+] Author and Article Information
Zbigniew Kulesza

Faculty of Mechanical Engineering,
Bialystok University of Technology,
15-351 Bialystok, Poland
e-mail: z.kulesza@pb.edu.pl

Jerzy T. Sawicki

Fellow ASME
Center for Rotating Machinery
Dynamics and Control (RoMaDyC),
Fenn College of Engineering,
Cleveland State University,
Cleveland, OH 44115-2214
e-mail: j.sawicki@csuohio.edu

Manuscript received March 12, 2012; final manuscript received September 26, 2012; accepted manuscript posted October 8, 2012; published online January 30, 2013. Assoc. Editor: Marc Geers.

J. Appl. Mech 80(2), 021025 (Jan 30, 2013) (17 pages) Paper No: JAM-12-1101; doi: 10.1115/1.4007787 History: Received March 12, 2012; Revised September 26, 2012; Accepted October 08, 2012

Transverse shaft cracks are one of the most dangerous malfunctions of the rotating machines, including turbo- and hydrogenerators, high-speed machine tool spindles, etc. The undetected crack may grow slowly and not disturb normal machine operation. However, if it extends to a critical depth, the immediate shaft fracture may completely damage the machine, resulting in a catastrophic accident. Therefore, in-depth knowledge of the crack propagation process is essential to ensure reliable and safe operation of rotating machinery. The article introduces a new model of the propagating shaft crack. The approach is based on the rigid finite element (RFE) method, which has previously proven its effectiveness in the dynamical analysis of numerous complicated machines and structures. The crack is modeled using several dozen spring-damping elements (SDEs), connecting the faces of the cracked section of the shaft. By controlling the exact behavior of individual SDEs, not only the breathing mechanism, but also the crack propagation process can be simply introduced. In order to accomplish this, the stress intensity factors (SIFs) along the crack edge are calculated using the novel approach based on the modified virtual crack closure technique (VCCT). Based on the SIF values, the crack propagation rate is calculated from the Paris law. If the number of load cycles is greater than the constantly updated threshold number, then the crack edge is shifted by a small increment. This way, starting from the first initially cracked SDE, the crack is extended little by little, continuously changing its shape. The approach is illustrated with numerical results, demonstrating the changes in the rotor vibration response and in the crack shape and also explaining some issues about the breathing mechanism due to the propagating shaft crack. The increasing amplitude of the 2X harmonic component is recognized as an evident propagating crack signature. The numerical results correspond well with the data reported in the literature. The RFE model of the rotor is validated by comparing the vibration responses obtained experimentally and numerically. A good agreement between these data confirms the correctness and accuracy of the proposed model. The suggested approach may be utilized for a more reliable dynamic analysis of the rotating shafts, having the potential to experience propagating transverse cracks.

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Dimarogonas, A. D., and Papadopoulos, C. A., 1983, “Vibration of Cracked Shafts in Bending,” J. Sound Vib., 91, pp. 583–593. [CrossRef]
Sinou, J. J., 2009, “Experimental Response and Vibrational Characteristics of a Slotted Rotor,” Commun. Nonlinear Sci. Numer. Simul., 14, pp. 3179–3194. [CrossRef]
Gasch, R. A., 1976, “Dynamic Behavior of a Simple Rotor With a Cross-Sectional Crack,” Proceedings of IMechE Conference on Vibrations in Rotating Machinery, London, 20 Paper No. C178/76, pp. 123–128.
Grabowski, B., 1980, “The Vibrational Behavior of a Turbine Rotor Containing a Transverse Crack,” ASME J. Mech. Des., 102, pp. 140–146. [CrossRef]
Bently, D. E., and MuszynskaA., 1986, “Detection of Rotor Cracks,” Proceedings of 15th Turbomachinery Symposium, Corpus Christi, TX, November 10–13, pp. 129–139.
Mayes, I. W., and Davies, W. G. R., 1984, “Analysis of the Response of a Multi-Rotor-Bearing System Containing a Transverse Crack in a Rotor,” ASME J. Vib., Acoust., Stress, Reliab. Des., 106, pp. 139–145. [CrossRef]
Darpe, A. K., Gupta, K., and Chawla, A., 2004, “Coupled Bending, Longitudinal and Torsional Vibrations of a Cracked Rotor,” J. Sound Vib., 269, pp. 33–60. [CrossRef]
Darpe, A. K., 2007, “Coupled Vibrations of a Rotor With Slant Crack,” J. Sound Vib., 305, pp. 172–193. [CrossRef]
Bachschmid, N., Pennacchi, P., and Tanzi, E., 2008, “Some Remarks on Breathing Mechanism, on Non-Linear Effects and on Slant and Helicoidal Cracks,” Mech. Syst. Signal Process., 22, pp. 879–904. [CrossRef]
Sawicki, J. T., Storozhev, D. L., and Lekki, J. D., 2011, “Exploration of NDE Properties of AMB Supported Rotors for Structural Damage Detection,” ASME J. Eng. Gas Turbines Power, 133, p. 102501. [CrossRef]
Sawicki, J. T., Wu, X., Baaklini, G., and Gyekenyesi, A. L., 2003, “Vibration-Based Crack Diagnosis in Rotating Shafts During Acceleration Through Resonance,” Proceedings of SPIE 5046, Nondestructive Evaluation and Health Monitoring of Aerospace Materials and Composites II, San Diego, CA, February 26–March 2. [CrossRef]
Sawicki, J. T., Friswell, M. I., Pesch, A. H., and Wroblewski, A., 2008, “Condition Monitoring of Rotor Using Active Magnetic Actuator,” Proceedings of ASME Turbo Expo 2008: Power for Land, Sea and Air, Berlin, Germany, June 9–13, ASME Paper No. GT2008-51169. [CrossRef]
Sawicki, J. T., Friswell, M. I., Kulesza, Z., Wroblewski, A., and Lekki, J. D., 2011, “Detecting Cracked Rotors Using Auxiliary Harmonic Excitation,” J. Sound Vib., 330, pp. 1365–1381. [CrossRef]
Nelson, H. D., and McVaugh, J. M., 1976, “The Dynamics of Rotor Bearing Systems Using Finite Elements,” ASME J. Eng. Ind., 98, pp. 593–600. [CrossRef]
Dimarogonas, A. D., and Paipetis, S. A., 1983, Analytical Methods in Rotor Dynamics, Applied Science Publishers, London.
Sekhar, A. S., and Balaji Prasad, P., 1997, “Dynamic Analysis of a Rotor System Considering a Slant Crack in the Shaft,” J. Sound Vib., 208(3), pp. 457–474. [CrossRef]
Andrieux, S., and Vare, C., 2002, “A 3D Cracked Beam Model With Unilateral Contact. Application to Rotors,” Eur. J. Mech. A/Solids, 21, pp. 793–810. [CrossRef]
Vare, C., and Andrieux, S., 2005, “Modeling of a Cracked Beam Section Under Bending,” Proceedings of the 18th International Conference on Structural Mechanics in Reactor Technology, SMIRT 18, Beijing, China, August 7–12, pp. 281–290.
Andrier, B., Garbay, E., Hasnaoui, F., Massin, P., and Verrier, P., 2006, “Investigation of Helix-Shaped and Transverse Crack Propagation in Rotor Shafts Based on Disk Shrunk Technology,” Nucl. Eng. Des., 236, pp. 333–349. [CrossRef]
Shih, Y. S., and Chen, J. J., 1997, “Analysis of Fatigue Crack Growth on a Cracked Shaft,” Int. J. Fatigue, 19(6), pp. 477–485. [CrossRef]
Shih, Y. S., and Chen, J. J., 2002, “The Stress Intensity Factor Study of an Elliptical Cracked Shaft,” Nucl. Eng. Des., 214, pp. 137–145. [CrossRef]
Shin, C. S., and Cai, C. Q., 2004, “Experimental and Finite Element Analyses on Stress Intensity Factors of an Elliptical Surface Crack in a Circular Shaft Under Tension and Bending,” Int. J. Fract., 129, pp. 239–264. [CrossRef]
Fonte, M., Reis, L., Romeiro, F., Li, B., and Freitas, M., 2006, “The Effect of Steady Torsion on Fatigue Crack Growth in Shafts,” Int. J. Fatigue, 28, pp. 609–617. [CrossRef]
Lissenden, C. J., Tissot, S. P., Trethewey, M. W., and Maynard, K. P., 2007, “Torsion Response of a Cracked Stainless Steel Shaft,” Fatigue Fract. Eng. Mater. Struct., 30(8), pp. 734–747. [CrossRef]
Vaziri, A., and Nayeb-Hashemi, H., 2006, “A Theoretical Investigation on the Vibrational Characteristics and Torsional Dynamic Response of Circumferentially Cracked Turbo-Generator Shafts,” Int. J. Solids Struct., 43, pp. 4063–4081. [CrossRef]
Adewusi, S. A., and Al-Bedoor, B. O., 2002, “Detection of Propagating Cracks in Rotors Using Neural Networks,” Proceedings of ASME 2002 Pressure Vessels and Piping Conference, Vancouver, BC, Canada, August 5–9, ASME Paper No. PVP2002-1518. [CrossRef]
Adewusi, S. A., and Al-Bedoor, B. O., 2002, “Experimental Study on the Vibration of an Overhung Rotor With a Propagating Transverse Crack,” Shock Vib., 9, pp. 91–104.
Mohamed, A. A., Neilson, R., MacConnell, P., Renton, N. C., and Deans, W., 2011, “Monitoring of Fatigue Crack Stages in a High Carbon Steel Rotating Shaft Using Vibration,” Procedia Eng., 10, pp. 130–135. [CrossRef]
Moës, N., Dolbow, J., and Belytschko, T., 1999, “A Finite Element Method for Crack Growth Without Remeshing,” Int. J. Numer. Methods Eng., 46, pp. 131–150. [CrossRef]
Belytschko, T., and Black, T., 1999, “Elastic Crack Growth in Finite Elements With Minimal Remeshing,” Int. J. Numer. Methods Eng., 45, pp. 601–620. [CrossRef]
Geniaut, S., Massin, P., and Moës, N., 2007, “A Stable 3D Contact Formulation for Cracks Using X-FEM,” Rev. Eur. Mec. Numer., 16, pp. 259–275. [CrossRef]
Jiang, D., and Liu, C., 2011, “Crack Growth Prediction of the Steam Turbine Generator Shaft,” J. Phys.: Conf. Ser., 305, p. 012023. [CrossRef]
Kruszewski, J., Sawiak, S., and Wittbrodt, E., 1999, Metoda Sztywnych Elementow Skonczonych w Dynamice Konstrukcji, WNT, Warszawa, Poland.
Wittbrodt, E., Adamiec-Wojcik, I., and Wojciech, S., 2006, Dynamics of Flexible Multibody Systems: Rigid Finite Element Method, Springer-Verlag, Berlin.
Kulesza, Z., and Sawicki, J. T., 2012, “Rigid Finite Element Model of a Cracked Rotor,” J. Sound Vib., 331(18), pp. 4145–4169. [CrossRef]
Rybicki, E. F., and Kanninen, M. F., 1977, “Finite-Element Calculation of Stress Intensity Factors by a Modified Crack Closure Integral,” Eng. Fract. Mech., 9(4), pp. 931–938. [CrossRef]
Krueger, R., 2004, “Virtual Crack Closure Technique: History, Approach, and Applications,” Appl. Mech. Rev., 57(2), pp. 109–144. [CrossRef]
Paris, P., and Erdogan, F., 1963, “A Critical Analysis of Crack Propagation Laws,” ASME J. Basic Eng., 85, pp. 528–534. [CrossRef]
Stephens, R. I., Fatemi, A., Stephens, R. R., and Fuchs, H. O., 2001, Metal Fatigue in Engineering, 2nd ed., Wiley, New York.
Beden, S. M., Abdullah, S., and Ariffin, A. K., 2009, “Review of Fatigue Crack Propagation Models for Metallic Components,” Eur. J. Sci. Res., 28(3), pp. 364–397.
Szata, M., and Lesiuk, G., 2009, “Algorithms for the Estimation of Fatigue Crack Growth Using Energy Methods,” Arch. Civ. Mech. Eng., IX(1), pp. 119–134. [CrossRef]
Wu, X., Sawicki, J. T., Friswell, M. I., and Baaklini, G. Y., 2005, “Finite Element Analysis of Coupled Lateral and Torsional Vibrations of a Rotor With Multiple Cracks,” Proceedings of ASME Turbo Expo 2005: Power for Land, Sea and Air, Reno-Tahoe, NV, June 6–9, ASME Paper No. GT2005-68839. [CrossRef]
Tada, H., Paris, P. C., and Irwin, G. R., 1973, The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, PA.
Bonnett, A. H., 2000, “Root Cause AC Motor Failure Analysis With a Focus on Shaft Failures,” IEEE Trans. Ind. Appl., 36(5), pp. 1435–1448. [CrossRef]
Ishida, Y., and Inoue, T., 2006, “Detection of Rotor Crack Using a Harmonic Excitation and Nonlinear Vibration Analysis,” ASME J. Vibr. Acoust., 128, pp. 741–749. [CrossRef]
Ludwig, R., and Bretchko, P., 2000, RF Circuit Design: Theory and Applications, Prentice-Hall, Upper Saddle River, NJ.
Carr, J., 2002, RF Components and Circuits, Radio Society of Great Britain, Newnes, UK.
Hynynen, K., 2011, “Broadband Excitation in the System Identification of Active Magnetic Bearing Rotor Systems,” D. Sc. thesis, Lappeenranta University of Technology, Lappeenranta, Finland.
Saavedra, P. N., and Cuitino, L. A., 2002, “Vibration Analysis of Rotor for Crack Identification,” J. Vib. Control, 8, pp. 51–67. [CrossRef]


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Fig. 1

Model of the rotor: (a) dimensions and the first discretization into 46 spring-damping elements; (b) the second discretization into 47 rigid finite elements; (c) local coordinate system ξ,η,ζ of the rotor; and local coordinate systems of individual RFEs (xr,1,xr,2,xr,3) and SDEs (yk,1,yk,2,yk,3)

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Fig. 2

Typical da/dN versus ΔKcurve

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Fig. 3

Model of the propagating crack: (a) the rotor at the location of the crack; (b) the two RFEs and several SDEs modeling the crack in the partially cracked shaft under external loads; (c) uncracked shaft cross-section at the crack location, (d) shaft cross-section at the crack location in the partially cracked shaft under external loads

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Fig. 4

Subsequent steps (a), (b), (c), (d) of crack propagation process

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Fig. 5

Cracked shaft finite element: (a) acting forces and coordinate systems; (b) cracked cross-section (based on Ref. [7])

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Fig. 6

Modified virtual crack closure method (based on Ref. [37])

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Fig. 7

Virtual crack closure method applied to the rigid finite element model of the crack

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Fig. 8

Schematic diagram of the algorithm for the numerical model of the rotor with the breathing and propagating crack

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Fig. 9

Campbell diagram of the uncracked rotor

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Fig. 10

The first four mode shapes of the uncracked rotor at 1620 rpm

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Fig. 11

Time history of the horizontal vibration of the rotor with the propagating crack

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Fig. 12

Time history of the vertical vibration of the rotor with the propagating crack

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Fig. 13

Snapshots of the growing shaft crack shapes

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Fig. 14

Breathing of the propagating 25% deep crack within the 4436th revolution

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Fig. 15

Breathing of the propagating 40% deep crack within the 4595th revolution

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Fig. 16

Frequency vibration spectra of the rotating shaft with a propagating crack

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Fig. 17

Vertical vibration of the rotor: (a) with no crack; (b) with the 25% deep crack; (c) with the 40% deep crack, solid line—experiment, dotted line—simulation

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Fig. 18

Vertical vibration spectrum of the uncracked rotor: (a) simulation; (b) experiment

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Fig. 19

Vertical vibration spectrum of the 25% cracked rotor: (a) simulation; (b) experiment

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Fig. 20

Vertical vibration spectrum of the 40% cracked rotor: (a) simulation; (b) experiment




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