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Research Papers

# Torsional Dynamic Response of a Shaft With Longitudinal and Circumferential Cracks

[+] Author and Article Information
H. Abdi, H. Nayeb-Hashemi

Department of Mechanical
and Industrial Engineering,
Northeastern University,
Boston, MA 02115

A. M. S. Hamouda

Department of Mechanical
and Industrial Engineering,
Qatar University,
Doha 2713, Qatar

A. Vaziri

Department of Mechanical
and Industrial Engineering,
Northeastern University,
Boston, MA 02115
e-mail: vaziri@coe.neu.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 5, 2013; final manuscript received September 17, 2014; published online October 6, 2014. Assoc. Editor: Yukio Ishida.

J. Vib. Acoust 136(6), 061011 (Oct 06, 2014) (8 pages) Paper No: VIB-13-1389; doi: 10.1115/1.4028609 History: Received November 05, 2013; Revised September 17, 2014

## Abstract

Turbo generator shafts are often subjected to cyclic torsion resulting in formation of large longitudinal cracks as well as circumferential cracks. The presence of these cracks could greatly impact the shaft resonance frequencies. In this paper, dynamic response of a shaft with longitudinal and circumferential cracks is investigated through a comprehensive analytical study. The longitudinally cracked section of the shaft was modeled as an uncracked shaft with reduced torsional rigidity. Torsional rigidity correction factor (i.e., the ratio of torsional rigidity of the cracked shaft to that of the uncracked shaft) was obtained from finite element analysis and was shown to be only a function of crack depth to the shaft radius. The resonance frequency and frictional energy loss of a shaft with a longitudinal crack were found little affected by the presence of the crack as long as the crack depth was less than $20%$ of the shaft radius even if the entire shaft is cracked longitudinally. Moreover, we showed that the longitudinal crack location could be more conveniently identified by monitoring the slope of the torsional response along the shaft. The circumferential crack was modeled as a torsional spring with a torsional damping. The torsion spring and damping constants were obtained using fracture mechanics. For a shaft with both a longitudinal crack and a circumferential crack, the resonance frequency was governed by the longitudinal crack when the circumferential crack depth was less than $30%$ of the shaft radius.

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## References

Dimarogonas, A., and Massouros, G., 1981, “Torsional Vibration of a Shaft With a Circumferential Crack,” Eng. Fract. Mech., 15(3), pp. 439–444.
Dimarogonas, A. D., 1996, “Vibration of Cracked Structures: A State of the Art Review,” Eng. Fract. Mech., 55(5), pp. 831–857.
Papadopoulos, C., and Dimarogonas, A., 1987, “Coupled Longitudinal and Bending Vibrations of a Rotating Shaft With an Open Crack,” J. Sound Vib., 117(1), pp. 81–93.
Chondros, T., 2001, “The Continuous Crack Flexibility Model for Crack Identification,” Fatigue Fract. Eng. Mater. Struct., 24(10), pp. 643–650.
Chondros, T., 2005, “Variational Formulation of a Rod Under Torsional Vibration for Crack Identification,” Theor. Appl. Fract. Mech., 44(1), pp. 95–104.
Chondros, T., and Dimarogonas, A., 1998, “Vibration of a Cracked Cantilever Beam,” ASME J. Vib. Acoust., 120(3), pp. 742–746.
Chondros, T., Dimarogonas, A., and Yao, J., 1997, “A Consistent Cracked Bar Vibration Theory,” J. Sound Vib., 200(3), pp. 303–313.
Chondros, T., Dimarogonas, A., and Yao, J., 1998, “A Continuous Cracked Beam Vibration Theory,” J. Sound Vib., 215(1), pp. 17–34.
Chondros, T., Dimarogonas, A., and Yao, J., 1998, “Longitudinal Vibration of a Bar With a Breathing Crack,” Eng. Fract. Mech., 61(5), pp. 503–518.
Chondros, T., Dimarogonas, A., and Yao, J., 1998, “Longitudinal Vibration of a Continuous Cracked Bar,” Eng. Fract. Mech., 61(5), pp. 593–606.
Chondros, T., and Labeas, G., 2007, “Torsional Vibration of a Cracked Rod by Variational Formulation and Numerical Analysis,” J. Sound Vib., 301(3), pp. 994–1006.
Christides, S., and Barr, A., 1986, “Torsional Vibration of Cracked Beams of Non-Circular Cross-Section,” Int. J. Mech. Sci., 28(7), pp. 473–490.
Dimarogonas, A. D., Paipetis, S. A., and Chondros, T. G., 2013, Analytical Methods in Rotor Dynamics, Springer, New York.
Wauer, J., 1990, “Modelling and Formulation of Equations of Motion for Cracked Rotating Shafts,” Int. J. Solids Struct., 26(8), pp. 901–914.
Gasch, R., 1993, “A Survey of the Dynamic Behaviour of a Simple Rotating Shaft With a Transverse Crack,” J. Sound Vib., 160(2), pp. 313–332.
Bicego, V., Lucon, E., Rinaldi, C., and Crudeli, R., 1999, “Failure Analysis of a Generator Rotor With a Deep Crack Detected During Operation: Fractographic and Fracture Mechanics Approach,” Nucl. Eng. Des., 188(2), pp. 173–183.
Barr, A., 1966, “An Extension of the Hu-Washizu Variational Principle in Linear Elasticity for Dynamic Problems,” ASME J. Appl. Mech., 33(2), p. 465.
Sabnavis, G., Kirk, R. G., Kasarda, M., and Quinn, D., 2004, “Cracked Shaft Detection and Diagnostics: A Literature Review,” Shock Vib. Dig., 36(4), pp. 287–296.
Shih, Y.-S., and Chung, C.-Y., 2013, “Vibration Analysis of the Flexible Connecting Rod With the Breathing Crack in a Slider-Crank Mechanism,” ASME J. Vib. Acoust., 135(6), p. 061009.
Nayeb-Hashemi, H., McClintock, F., and Ritchie, R., 1982, “Effects of Friction and High Torque on Fatigue Crack Propagation in Mode III,” Metall. Trans. A, 13(12), pp. 2197–2204.
Nayeb-Hashemi, H., McClintock, F., and Ritchie, R., 1983, “Influence of Overloads and Block Loading Sequences on Mode III Fatigue Crack Propagation in A469 Rotor Steel,” Eng. Fract. Mech., 18(4), pp. 763–783.
Nayeb-Hashemi, H., McClintock, F., and Ritchie, R., 1983, “Micro-Mechanical Modelling of Mode III Fatigue Crack Growth in Rotor Steels,” Int. J. Fract., 23(3), pp. 163–185.
Nayeb-Hashemi, H., Suresh, S., and Ritchie, R., 1983, “On the Contrast Between Mode I and Mode III Fatigue Crack Propagation Under Variable-Amplitude Loading Conditions,” Mater. Sci. Eng., 59(1), pp. L1–L5.
Ritchie, R., McClintock, F., Nayeb-Hashemi, H., and Ritter, M., 1982, “Mode III Fatigue Crack Propagation in Low Alloy Steel,” Metall. Trans. A, 13(1), pp. 101–110.
Vaziri, A., and Nayeb-Hashemi, H., 2005, “The Effect of Crack Surface Interaction on the Stress Intensity Factor in Mode III Crack Growth in Round Shafts,” Eng. Fract. Mech., 72(4), pp. 617–629.
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(14), pp. 4063–4081.
Vaziri, A., and Nayeb-Hashemi, H., 2002, “Effects of Local Energy Loss on the Dynamic Response of a Cylindrical Bar With a Penny Shape Crack,” ASME Paper No. IMECE2002-32300.
Broberg, K. B., 1999, Cracks and Fracture, Academic Press, London, UK.

## Figures

Fig. 1

Circumferential and longitudinal cracks formation in a shaft subjected to cyclic torsion

Fig. 2

(a) Finite element model of a shaft with a longitudinal crack subjected to torsion and (b) torsional rigidity correction factor of a shaft with a longitudinal crack computed by finite element technique, Eq. (3)

Fig. 3

Longitudinal energy loss factor in a shaft subjected to cyclic torsion for various crack surface interactions, Eq. (13)

Fig. 4

(a) Schematic model of a shaft with a circumferential crack and its corresponding model with torsional spring and damping and (b) schematic model of a shaft with both longitudinal and circumferential cracks

Fig. 5

(a) The effects of longitudinal crack depth located in the middle of the shaft on its first resonance frequency for various crack length and (b) the effects of longitudinal crack length on the shaft first resonance frequency for various crack depth

Fig. 6

(a) Effects of energy loss factor on the frequency response of a shaft with a longitudinal crack, (b) effects of the energy loss factor of a longitudinal crack on the first resonance frequency when 0≤α≤0.9, and (c) effects of the energy loss factor of a longitudinal crack on the first resonance frequency when 0≤a≤0.1

Fig. 10

The effects of circumferential crack depth on the shaft resonance frequency for various longitudinal crack depths when (a) Xc/L = 0.3, (b) Xc/L = 0.6, and (c) Xc/L = 0.9

Fig. 9

(a) Effects of energy loss factor on the frequency response of a shaft with a circumferential crack and (b) effects of the energy loss factor of a circumferential crack on the shaft first resonance frequency for different crack depth

Fig. 8

(a) The effects of circumferential crack depth on the shaft first resonance frequency for various crack locations and (b) the effects of circumferential crack location on the shaft first resonance frequency for various crack depth

Fig. 7

(a) First mode shape of a shaft with a longitudinal crack and (b) first derivate of the first mode shape with respect to position

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