0
Research Papers: Gas Turbines: Structures and Dynamics

Stability of a Cracked Rotor Subjected to Parametric Excitation

[+] Author and Article Information
Jerzy T. Sawicki

Fellow ASME
Center for Rotating Machinery Dynamics
and Control,
Cleveland State University,
Cleveland, OH 44115-2214
e-mail: j.sawicki@csuohio.edu

Zbigniew Kulesza

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

Contributed by the Structures and Dynamics Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received September 16, 2014; final manuscript received September 30, 2014; published online December 2, 2014. Editor: David Wisler.

J. Eng. Gas Turbines Power 137(5), 052508 (May 01, 2015) (8 pages) Paper No: GTP-14-1552; doi: 10.1115/1.4028742 History: Received September 16, 2014; Revised September 30, 2014; Online December 02, 2014

It is well known that parametric vibrations may appear during the rotation of a rotor with a cracked shaft. The vibrations occur due to periodic stiffness changes being the result of the crack breathing. A parametrically excited system may exhibit parametric resonances and antiresonances affecting the stability of the system. In most cases the destabilizing effect due to parametric resonances is studied. Antiresonant cases seem to be uninteresting. However, the antiresonances have a unique property of introducing additional artificial damping to the system, thus improving its stability and reducing the vibration amplitude. Apart from different control applications, this stabilizing effect may be interesting for its probable ability to indicate the shaft crack. The possible application of the additional damping introduced by parametric excitation for the shaft crack detection is analyzed in the present paper. The approach is demonstrated with a mathematical model of a rotor with a cracked shaft. The stability analysis of the rotor is performed analytically by employing the averaging method. Stability boundaries for different frequencies of the parametric excitation and for different crack depths are derived. The results of this analysis are checked numerically by means of the Floquet's theory. Next, possible applications of the parametric excitation for the shaft crack detection are validated numerically.

FIGURES IN THIS ARTICLE
<>
Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Darpe, A. K., Gupta, K., and Chawla, A., 2004, “Coupled Bending, Longitudinal and Torsional Vibrations of a Cracked Rotor,” J. Sound Vib., 269(1–2), pp. 33–60. [CrossRef]
Gasch, R. A., 1976, “Dynamic Behavior of a Simple Rotor With a Cross Sectional Crack,” IMechE International Conference on Vibrations in Rotating Machineries (IMechE '76), Cambridge, UK, Sept. 15–17, Paper No.. C178/76, pp. 123–128.
Kulesza, Z., and Sawicki, J. T., 2012, “Rigid Finite Element Model of a Cracked Rotor,” J. Sound Vib., 331(18), pp. 4145–4169. [CrossRef]
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., 106(1), pp. 139–145. [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(10), p. 102501. [CrossRef]
Al-Shudeifat, M. A., and Butcher, E. A., 2011, “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), pp. 526–544. [CrossRef]
Kulesza, Z., and Sawicki, J. T., 2013, “New Finite Element Modeling Approach of a Propagating Shaft Crack,” ASME J. Appl. Mech., 80(2), p. 021025. [CrossRef]
Sinou, J. J., and Lees, A. W., 2005, “The Influence of Cracks in Rotating Shafts,” J. Sound Vib., 285(4–5), pp. 1015–1037. [CrossRef]
Meng, G., and Gasch, R., 2000, “Stability and Stability Degree of a Cracked Flexible Rotor Supported on Journal Bearings,” ASME J. Vib. Acoust., 122(2), pp. 116–125. [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(7), 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(2), pp. 593–600. [CrossRef]
Dimarogonas, A. D., and Paipetis, S. A., 1983, Analytical Methods in Rotor Dynamics, Applied Science Publishers, London.
Ostachowicz, W. M., and Krawczuk, M., 1992, “Coupled Torsional and Bending Vibrations of a Rotor With an Open Crack,” Arch. Appl. Mech., 62(3), pp. 191–201. [CrossRef]
Dai, L., and Chen, C., 2007, “Dynamic Stability Analysis of a Cracked Nonlinear Rotor System Subjected to Periodic Excitations in Machining,” J. Vib. Control, 13(5), pp. 537–556. [CrossRef]
Fu, Y. M., and Zheng, Y. F., 2002, “Analysis of Non-Linear Dynamic Stability for a Rotating Shaft-Disk With a Transverse Crack,” J. Sound Vib., 257(4), pp. 713–731. [CrossRef]
Huang, S. C., Huang, Y. M., and Shieh, S. M., 1993, “Vibration and Stability of a Rotating Shaft Containing a Transverse Crack,” J. Sound Vib., 162(3), pp. 387–401. [CrossRef]
Patel, T. H., and Darpe, A. K., 2008, “Influence of Crack Breathing Model on Nonlinear Dynamics of a Cracked Rotor,” J. Sound Vib., 311(3–5), pp. 953–972. [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,” ASME Paper No. GT2005-68839. [CrossRef]
Yiming, F., and Yufang, Z., 2003, “Analysis of the Chaotic Motion for a Rotor System With a Transverse Crack,” Acta Mech. Solida Sin., 16(1), pp. 74–80.
Bachschmid, N., Pennacchi, P., Tanzi, E., and Vania, A., 2000, “Identification of Transverse Crack Position and Depth in Rotor Systems,” Meccanica, 35(6), pp. 563–582. [CrossRef]
Sinou, J. J., 2008, “Detection of Cracks in Rotor Based on the 2X and 3X Super-Harmonic Frequency Components and the Crack-Unbalance Interactions,” Commun. Nonlinear Sci. Numer. Simul., 13(9), pp. 2024–2040. [CrossRef]
Kulesza, Z., and Sawicki, J. T., 2012, “Controlled Deflection Approach for Rotor Crack Detection,” ASME J. Eng. Gas Turbines Power, 134(9), p. 092502. [CrossRef]
Ishida, Y., and Inoue, T., 2006, “Detection of a Rotor Crack Using a Harmonic Excitation and Nonlinear Vibration Analysis,” ASME J. Vib. Acoust., 128(6), pp. 741–749. [CrossRef]
Kulesza, Z., and Sawicki, J. T., 2011, “Auxiliary State Variables for Rotor Crack Detection,” J. Vib. Control, 17(6), pp. 857–872. [CrossRef]
Kulesza, Z., Sawicki, J. T., and Gyekenyesi, A. L., 2012, “Robust Fault Detection Filter Using Linear Matrix Inequalities' Approach for Shaft Crack Diagnosis,” J. Vib. Control, 19(9), pp. 1421–1440. [CrossRef]
Loparo, K. A., Adams, M. L., Lin, W., Abdel-Magied, M. F., and Afshari, N., 2000, “Fault Detection and Diagnosis of Rotating Machinery,” IEEE Trans. Ind. Electron., 47(5), pp. 1005–1014. [CrossRef]
Söffker, D., Bajkowski, J., and Müller, P. C., 1993, “Detection of Cracks in Turborotors—A New Observer-Based Method,” ASME J. Dyn. Syst., Meas. Control, 115(3), pp. 518–524. [CrossRef]
He, Y., Guo, D., and Chu, F., 2001, “Using Genetic Algorithms to Detect and Configure Shaft Crack for Rotor-Bearing System,” Comput. Methods Appl. Mech. Eng., 190(45), pp. 5895–5906. [CrossRef]
Xiang, J., Zhong, Y., Chen, X., and He, Z., 2008, “Crack Detection in a Shaft by Combination of Wavelet-Based Elements and Genetic Algorithm,” Int. J. Solids Struct., 45(17), pp. 4782–4795. [CrossRef]
Litak, G., and Sawicki, J. T., 2009, “Intermittent Behaviour of a Cracked Rotor in the Resonance Region,” Chaos, Solitons Fractals, 42(3), pp. 1495–1501. [CrossRef]
Guo, D., and Peng, Z. K., 2007, “Vibration Analysis of a Cracked Rotor Using Hilbert–Huang Transform,” Mech. Syst. Signal Process., 21(8), pp. 3030–3041. [CrossRef]
Ballo, I., 1998, “Non-Linear Effects of Vibration of a Continuous Transverse Cracked Slender Shaft,” J. Sound Vib., 217(2), pp. 321–333. [CrossRef]
Sinou, J. J., 2007, “Effects of a Crack on the Stability of a Non-Linear Rotor System,” Int. J. Non-Linear Mech., 42(7), pp. 959–972. [CrossRef]
Didier, J., Sinou, J. J., and Faverion, B., 2012, “Study of the Non-Linear Dynamic Response of a Rotor System With Faults and Uncertainties,” J. Sound Vib., 331(3), pp. 671–703. [CrossRef]
Plaut, R. H., 1995, “Parametric, External and Combination Resonances in Coupled Flexural and Torsional Oscillations of an Unbalanced Rotating Shaft,” J. Sound Vib., 183(5), pp. 889–897. [CrossRef]
Hegazy, U. H., Eissa, M. H., and Amer, Y. A., 2008, “A Time-Varying Stiffness Rotor Active Magnetic Bearings Under Combined Resonance,” ASME J. Appl. Mech., 75(1), p. 011011. [CrossRef]
Kamel, M., and Bauomy, H. S., 2009, “Nonlinear Oscillation of a Rotor-AMB System With Time Varying Stiffness and Multi-External Excitations,” ASME J. Vib. Acoust., 131(3), p. 031009. [CrossRef]
Gasch, R. A., 1993, “A Survey of the Dynamic Behaviour of a Simple Rotating Shaft With a Transverse Crack,” J. Sound Vib., 160(2), pp. 313–332. [CrossRef]
Pu, Y., Chen, J., Zou, J., and Zhong, P., 2002, “The Research on Non-Linear Characteristics of a Cracked Rotor and Reconstruction of the Crack Forces,” Proc. Inst. Mech. Eng., Part C, 216(11), pp. 1099–1108. [CrossRef]
Qin, W., and Meng, G., 2003, “Nonlinear Dynamic Response and Chaos of a Cracked Rotor With Two Disks,” Int. J. Bifurcation Chaos, 13(11), pp. 3425–3436. [CrossRef]
Chen, C., and Dai, L., 2007, “Bifurcation and Chaotic Response of a Cracked Rotor System With Viscoelastic Supports,” Nonlinear Dyn., 50(3), pp. 483–509. [CrossRef]
Dohnal, F., and Verhulst, F., 2008, “Averaging in Vibration Suppression by Parametric Stiffness Excitation,” Nonlinear Dyn., 54(3), pp. 231–248. [CrossRef]
Dohnal, F., 2008, “Damping by Parametric Stiffness Excitation: Resonance and Anti-Resonance,” J. Vib. Control, 14(5), pp. 669–688. [CrossRef]
Dohnal, F., 2008, “General Parametric Stiffness Excitation—Anti-Resonance Frequency and Symmetry,” Acta Mech., 196(1), pp. 15–31. [CrossRef]
Dohnal, F., 2012, “Experimental Studies on Damping by Parametric Excitation Using Electromagnets,” Proc. Inst. Mech. Eng., Part C, 226(8), pp. 2015–2027. [CrossRef]
Ecker, H., 2009, “Beneficial Effects of a Parametric Excitation in Rotor Systems,” IUTAM Symposium on Emerging Trends in Rotor Dynamics, New Delhi, India, Mar. 23–26. [CrossRef]
Tondl, A., 1960, “The Stability of Motion of a Rotor With Unsymmetrical Shaft on an Elastically Supported Mass Foundation,” Ingenieur-Arch., 29(6), pp. 410–418. [CrossRef]
Tondl, A., and Ecker, H., 2003, “On the Problem of Self-Excited Vibration Quenching by Means of Parametric Excitation,” Arch. Appl. Mech., 72(11–12), pp. 923–932. [CrossRef]
Verhulst, F., 1990, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York.
Ecker, H., and Pumhössel, T., 2011, “Parametric Excitation of a Rotor System Due to a Periodic Axial Force,” 7th European Nonlinear Dynamics Conference (ENOC 2011), Rome, Italy, July 24–29.
Hsu, C. S., 1974, “On Approximating a General Linear Periodic System,” J. Math. Anal. Appl., 45(1), pp. 234–251. [CrossRef]
Huang, J. L., Su, R. K. L., and Chen, S. H., 2009, “Precise Hsu's Method for Analyzing the Stability of Periodic Solutions of Multi-Degrees-of-Freedom Systems With Cubic Nonlinearity,” Comput. Struct., 87(23–24), pp. 1624–1630. [CrossRef]
Dohnal, F., and Markert, R., 2010, “Stability Improvement of a Flexible Rotor in Active Magnetic Bearings by Time-Periodic Stiffness Variation,” 10th International Conference on Motion and Vibration Control (MOVIC2010), Tokyo, Japan, Aug. 17–20.

Figures

Grahic Jump Location
Fig. 1

Tested rotor: (a) finite element model and (b) shaft cross section at crack location

Grahic Jump Location
Fig. 2

Analytical stability map; dark areas are unstable

Grahic Jump Location
Fig. 3

Numerical stability map; dark areas are unstable

Grahic Jump Location
Fig. 4

Numerically calculated vertical displacement of the rotor in dependency of parametric excitation frequency η and time τ; crack depth μ=1%

Grahic Jump Location
Fig. 5

Numerically calculated vertical displacement of the rotor in dependency of parametric excitation frequency η and time τ; crack depth μ=25%

Grahic Jump Location
Fig. 6

Numerically calculated vertical displacement of the rotor in dependency of parametric excitation frequency η and time τ; crack depth μ=40%

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In