Volume 19, Issue 9 (September 2019)                   Modares Mechanical Engineering 2019, 19(9): 2139-2148 | Back to browse issues page

XML Persian Abstract Print


1- Department of Mechanical Engineering, Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran , m_rezaee@tabrizu.ac.ir
2- Department of Mechanical Engineering, Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran
Abstract:   (4987 Views)
The equations of nonlinear motion of clamped-hinged beam with an open crack were extracted and through solving them, the internal resonance in the cracked beam was studied. To this end, the crack was modeled as a torsional spring and the cracked beam was considered as two beam segments connected by a torsional spring. The equations of motion of the cracked beam were extracted considering the geometrical nonlinearity. Then, using the Galerkin’s method, these equations were changed to a set of nonlinear differential equations for vibration modes which were solved by the perturbation method. Since the mechanical energy of the beam in each mode depends on the instantaneous amplitude of vibration of the beam at the corresponding mode, so to analyze the influence of the crack on the energy exchange between the modes, the instantaneous amplitudes of the vibration modes were obtained. The results show that in the cracked beam the magnitude of the energy exchanged between the modes is less and the frequency is more than that in the intact beam. Also, by increasing the crack depth the frequency of energy exchange between the modes increases. The Vibration response obtained for the cracked beam with various amounts of the damping ratios shows that the frequency and the amplitude of energy exchange between the modes are independent of the system damping. To validate the results by the perturbation method, the equations of motions are also solved by a numerical method and the obtained results are in agreement with the results of the analytical method.
Full-Text [PDF 1050 kb]   (1783 Downloads)    
Article Type: Original Research | Subject: Vibration
Received: 2018/03/19 | Accepted: 2019/02/4 | Published: 2019/09/1

References
1. 1- Khadem SE, Rezaee M. An analytical approach for obtaining the location and depth of an all-over part-through crack on externally in-plane loaded rectangular plate using vibration analysis. Journal of Sound and Vibration. 2000;230(2):291-308. [Link] [DOI:10.1006/jsvi.1999.2619]
2. Khiem NT, Toan LK. A novel method for crack detection in beam-like structures by measurements of natural frequencies. Journal of Sound and Vibration. 2014;333(18):4084-4103. [Link] [DOI:10.1016/j.jsv.2014.04.031]
3. Mungla MJ, Sharma DS, Trivedi RR. Identification of a crack in clamped-clamped beam using frequency-based method and genetic algorithm. Procedia Engineering. 2016;144:1426-1434. [Link] [DOI:10.1016/j.proeng.2016.05.174]
4. Dahak M, Touat N, Benseddiq N. On the classification of normalized natural frequencies for damage detection in cantilever beam. Journal of Sound and Vibration. 2017;402:70-84. [Link] [DOI:10.1016/j.jsv.2017.05.007]
5. Cao M, Ye L, Zhou L, Su Z, Bai R. Sensitivity of fundamental mode shape and static deflection for damage identification in cantilever beams. Mechanical Systems and Signal Processing. 2011;25(2):630-643. [Link] [DOI:10.1016/j.ymssp.2010.06.011]
6. Nguyen KV. Mode shapes analysis of a cracked beam and its application for crack detection. Journal of Sound and Vibration. 2014;333(3):848-872. [Link] [DOI:10.1016/j.jsv.2013.10.006]
7. Dessi D, Camerlengo G. Damage identification techniques via modal curvature analysis: Overview and comparison. Mechanical Systems and Signal Processing. 2015;52-53:181-205. [Link] [DOI:10.1016/j.ymssp.2014.05.031]
8. Gelman L. The new frequency response functions for structural health monitoring. Engineering Structures. 2010;32(12):3994-3999. [Link] [DOI:10.1016/j.engstruct.2010.09.010]
9. Bandara RP, Chan THT, Thambiratnam DP. Frequency response function based damage identification using principal component analysis and pattern recognition technique. Engineering Structures. 2014;66:116-128. [Link] [DOI:10.1016/j.engstruct.2014.01.044]
10. Mohan SC, Maiti DK, Maity D. Structural damage assessment using FRF employing particle swarm optimization. Applied Mathematics and Computation. 2013;219(20):10387-10400. [Link] [DOI:10.1016/j.amc.2013.04.016]
11. Rezaee M, Fekrmandi H. Analysis of the nonlinear behavior of the free vibration of a cantilever beam with a fatigue crack using Lindstedt-Poincare's method. Amirkabir Journal of Mechanical Engineering. 2014;46(2):29-31. [Link]
12. Rezaee M, Fekrmandi H. A theoretical and experimental investigation on free vibration behavior of a cantilever beam with a breathing crack. Shock and Vibration. 2012;19(2):175-186. [Link] [DOI:10.1155/2012/563916]
13. Na C, Kim SP, Kwak HG. Structural damage evaluation using genetic algorithm. Journal of Sound and Vibration. 2011;330(12):2772-2783. [Link] [DOI:10.1016/j.jsv.2011.01.007]
14. Li J, Wu B, Zeng QC, Lim CW. A generalized flexibility matrix based approach for structural damage detection. Journal of Sound and Vibration. 2010;329(22):4583-4587. [Link] [DOI:10.1016/j.jsv.2010.05.024]
15. Tsyfansky SL, Beresnevich VI. Detection of fatigue cracks in flexible geometrically non-linear bars by vibration monitoring. Journal of Sound and Vibration. 1998;213(1):159-168. [Link] [DOI:10.1006/jsvi.1998.1502]
16. El Bikri K, Benamar R, Bennouna MM. Geometrically non-linear free vibrations of clamped-clamped beams with an edge crack. Computers & Structures. 2006;84(7):485-502. [Link] [DOI:10.1016/j.compstruc.2005.09.030]
17. Merrimi EB, El Bikri K, Benamar R. Geometrically non-linear steady state periodic forced response of a clamped-clamped beam with an edge open crack. Comptes Rendus Mécanique. 2011;339(11):727-742. [Link] [DOI:10.1016/j.crme.2011.07.008]
18. Manoach E, Samborski S, Mitura A, Warminski J. Vibration based damage detection in composite beams under temperature variations using Poincaré maps. International Journal of Mechanical Sciences. 2012;62(1):120-132. [Link] [DOI:10.1016/j.ijmecsci.2012.06.006]
19. Stojanović V, Ribeiro P, Stoykov S. Non-linear vibration of Timoshenko damaged beams by a new p-version finite element method. Computers & Structures. 2013;120:107-119. [Link] [DOI:10.1016/j.compstruc.2013.02.012]
20. Carneiro GN, Ribeiro P. Vibrations of beams with a breathing crack and large amplitude displacements. Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science. 2016;230(1):34-54. [Link] [DOI:10.1177/0954406215589333]
21. Majumder L, Manohar CS. Nonlinear reduced models for beam damage detection using data on moving oscillator-beam interactions. Computers & Structures. 2004;82(2-3):301-314. [Link] [DOI:10.1016/j.compstruc.2003.08.007]
22. Kitipornchai S, Ke LL, Yang J, Xiang Y. Nonlinear vibration of edge cracked functionally graded Timoshenko beams. Journal of Sound and Vibration. 2009;324(3-5):962-982. [Link] [DOI:10.1016/j.jsv.2009.02.023]
23. Chajdi M, Merrimi EB, ELBikri K. Geometrically nonlinear free vibration of composite materials: Clamped-clamped functionally graded beam with an edge crack using Homogenisation method. Key Engineering Materials. 2017;730:521-526. [Link] [DOI:10.4028/www.scientific.net/KEM.730.521]
24. Chajdi M, Merrimi EB, El Bikri Kh. Geometrically non-linear free and forced vibration of clamped-clamped functionally graded beam with discontinuities. Procedia Engineering. 2017;199:1870-1875. [Link] [DOI:10.1016/j.proeng.2017.09.117]
25. Nayfeh AH, Mook DT. Nonlinear oscillations. Hoboken: Wiley; 1979. [Link] [DOI:10.1115/1.3153771]
26. Meirovitch L. Analytical methods in vibrations. London: Macmillan; 1967. [Link]
27. Lin HP, Chang SC, Wu JD. Beam vibrations with an arbitrary number of cracks. Journal of Sound and Vibration. 2002;258(5):987-999. [Link] [DOI:10.1006/jsvi.2002.5184]

Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.