Theoretical-experimental Determination of Nonlinear Crack Parameters in the Cracked Beam under Nonlinear Low Amplitude Vibration
Author
Abstract
In the vibration of a cracked structure with small amplitude oscillations, the crack necessarily is not fully open or fully closed. Therefore, in order to provide a realistic model for the crack, one should relate the stiffness and damping at the crack location to the amount of the opening of the crack. In this study, a continuous model for vibration of a beam with a fatigue crack under low amplitude oscillations is presented in which the crack is not fully open or fully closed. By introducing a nonlinear model for the crack, the equation governing the vibration of the cracked beam is extracted. In order to consider the nonlinear behavior of the crack and to take into account the energy loss at the crack during the vibration, the bending moment at the crack location was considered as a nonlinear function of the angle of crack opening and its variations with respect to the time. The governing nonlinear equation is solved using the perturbation method. The solution reveals the dependency of the resonance frequency on the vibration amplitude. Analytical and explicit expressions are also derived for the nonlinear stiffness coefficient and the damping coefficient of the crack at the crack location. Finally, using the derived expressions for the crack parameters and experimental tests results for cracked beam, the nonlinear stiffness coefficient and the damping coefficient at the crack location is obtained.
Keywords
1. Bovsunovsky, A. P., and 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, Vol. 288, pp. 865-886, 2005.
2. Irwin, G. R., “Relation of Stresses Near a Crack to the Crack Extension Force”, 9th Congress of Applied Mechanics, Brussels, 1957.
3. Maghsoodi, A., Ghadami, A., and Mirdamadi, H. R., “Multiple-Crack Damage Detection in Multi-Step Beams by a Novel Local Flexibility-Based Damage Index”, Journal of Sound and Vibration, Vol. 332, pp. 294-305, 2013.
4. Lu, X. B., Liu, J. K., and Lu, Z. R., “A Two-Step Approach for Crack Identification in Beam”, Journal of Sound and Vibration, Vol. 332, pp. 282-293, 2013.
5. Friswell, M. I., and Penny. J. E. T., “A Simple Nonlinear Model of a Cracked Beam”, Proceedings of 10th International Modal Analysis Conference, pp. 516-521, 1992.
6. Pugno, N., Surace, C., and Ruotolo, R., “Evaluation of the Non-Linear Dynamic Response to Harmonic Excitation of a Beam with Several Breathing Cracks”, Journal of Sound and Vibration, Vol. 235, No. 5, pp. 749-762, 2000.
7. Surace, C., Ruotolo, R., and Storer, D., “Detecting Nonlinear Behavior using the Volterra Series to Assess Damping in Beam-Like Structures”, Journal of Theoretical and Applied Mechanics, Vol. 49, pp. 905-926, 2011.
8. Clark, R., Dover, W. E., and Bond, L. J., “The Effect of Crack Closure on the Reliability of NDT Prediction of Crack Size”, NDT International, Vol. 20, pp. 269-275, 1987.
9. Abraham, O. N. L., and Brandon, J. A., “Modeling of the Opening and Closure of a Crack”, Journal of Vibration and Acoustics, Vol. 117, pp. 370-377, 1995.
10. Cheng, S. M., Wu, X. J., and Wallace, A. S. J. “Swamidas, Vibration Response of a Beam with a Breathing Crack”, Journal of Sound and Vibration, Vol. 225, No. 1, pp. 201-208, 1999.
11. Rezaee, M., and Hassannejad, R., “A New Approach to Free Vibration Analysis of Beam with Breathing Crack Based on Mechanical Energy Balance Method”, Acta Mechanica Solida Sinica, Vol. 24, pp. 185-190, 2011.
12. Rezaee, M., and Fekrmandi, H., “A Theoretical and Experimental Investigation on Free Vibration Behavior of a Cantilever Beam with Breathing Crack”, Journal of Shock and Vibration, Vol. 18, pp. 1-12, 2011.
13. Bovsunovsky, A. P., Surace, C., and Ruotolo, R., “The Effect of Damping on the Non-Linear Dynamic Behavior of a Cracked Beam at Resonance and Super-Resonance Vibrations”, Key Engineering Materials, Vol. 245, pp. 97-106, 2003.
14. Rezaee, M., and Hassannejad, R., “Damped Free Vibration Analysis of a Beam with a Fatigue Crack Using Energy Balance Method, International Journal of the Physical Sciences, Vol. 5, No. 6, pp. 793-803, 2010.
15. Bovsunovsky, A. P., “The Mechanisms of Energy Dissipation in the Non-Propagating Fatigue Cracks in Metallic Materials”, Engineering Fracture Mechanics, Vol. 71, pp. 2271-2281, 2004.
16. Nayfeh, A. H., and Mook, D. T., Nonlinear Oscillations, New York: Wiley Interscience, 1979.
17. Pakdemirli, M., and Boyacy, H., “Nonlinear Vibration of a Simple-Simple Beam with Non-Ideal Support in Between”, Journal of Sound and Vibration, Vol. 268, No. 2, pp. 331-341, 2003.
2. Irwin, G. R., “Relation of Stresses Near a Crack to the Crack Extension Force”, 9th Congress of Applied Mechanics, Brussels, 1957.
3. Maghsoodi, A., Ghadami, A., and Mirdamadi, H. R., “Multiple-Crack Damage Detection in Multi-Step Beams by a Novel Local Flexibility-Based Damage Index”, Journal of Sound and Vibration, Vol. 332, pp. 294-305, 2013.
4. Lu, X. B., Liu, J. K., and Lu, Z. R., “A Two-Step Approach for Crack Identification in Beam”, Journal of Sound and Vibration, Vol. 332, pp. 282-293, 2013.
5. Friswell, M. I., and Penny. J. E. T., “A Simple Nonlinear Model of a Cracked Beam”, Proceedings of 10th International Modal Analysis Conference, pp. 516-521, 1992.
6. Pugno, N., Surace, C., and Ruotolo, R., “Evaluation of the Non-Linear Dynamic Response to Harmonic Excitation of a Beam with Several Breathing Cracks”, Journal of Sound and Vibration, Vol. 235, No. 5, pp. 749-762, 2000.
7. Surace, C., Ruotolo, R., and Storer, D., “Detecting Nonlinear Behavior using the Volterra Series to Assess Damping in Beam-Like Structures”, Journal of Theoretical and Applied Mechanics, Vol. 49, pp. 905-926, 2011.
8. Clark, R., Dover, W. E., and Bond, L. J., “The Effect of Crack Closure on the Reliability of NDT Prediction of Crack Size”, NDT International, Vol. 20, pp. 269-275, 1987.
9. Abraham, O. N. L., and Brandon, J. A., “Modeling of the Opening and Closure of a Crack”, Journal of Vibration and Acoustics, Vol. 117, pp. 370-377, 1995.
10. Cheng, S. M., Wu, X. J., and Wallace, A. S. J. “Swamidas, Vibration Response of a Beam with a Breathing Crack”, Journal of Sound and Vibration, Vol. 225, No. 1, pp. 201-208, 1999.
11. Rezaee, M., and Hassannejad, R., “A New Approach to Free Vibration Analysis of Beam with Breathing Crack Based on Mechanical Energy Balance Method”, Acta Mechanica Solida Sinica, Vol. 24, pp. 185-190, 2011.
12. Rezaee, M., and Fekrmandi, H., “A Theoretical and Experimental Investigation on Free Vibration Behavior of a Cantilever Beam with Breathing Crack”, Journal of Shock and Vibration, Vol. 18, pp. 1-12, 2011.
13. Bovsunovsky, A. P., Surace, C., and Ruotolo, R., “The Effect of Damping on the Non-Linear Dynamic Behavior of a Cracked Beam at Resonance and Super-Resonance Vibrations”, Key Engineering Materials, Vol. 245, pp. 97-106, 2003.
14. Rezaee, M., and Hassannejad, R., “Damped Free Vibration Analysis of a Beam with a Fatigue Crack Using Energy Balance Method, International Journal of the Physical Sciences, Vol. 5, No. 6, pp. 793-803, 2010.
15. Bovsunovsky, A. P., “The Mechanisms of Energy Dissipation in the Non-Propagating Fatigue Cracks in Metallic Materials”, Engineering Fracture Mechanics, Vol. 71, pp. 2271-2281, 2004.
16. Nayfeh, A. H., and Mook, D. T., Nonlinear Oscillations, New York: Wiley Interscience, 1979.
17. Pakdemirli, M., and Boyacy, H., “Nonlinear Vibration of a Simple-Simple Beam with Non-Ideal Support in Between”, Journal of Sound and Vibration, Vol. 268, No. 2, pp. 331-341, 2003.
)