Linear and geometrically nonlinear free and forced vibrations of fully clamped multi-cracked beams
Mohcine Chajdi 1  
,  
Ahmed Adri 2  
,  
 
 
More details
Hide details
1
Mohammed V University in Rabat, ENSET - Rabat, MSSM, B.P.6207, Rabat Instituts, Rabat, Morocco
2
Hassan II University of Casablanca, EST - CASABLANCA, LMPGI, B.P.8012, Oasis Casablanca, Morocco
3
Mohammed V University in Rabat, EMI - Rabat, LERSIM, B.P.765 Agdal, Rabat, Morocco
CORRESPONDING AUTHOR
Mohcine Chajdi   

Mohammed V University in Rabat, ENSET - Rabat, MSSM, B.P.6207, Rabat Instituts, Rabat, Morocco
Online publish date: 2019-02-01
Publish date: 2019-02-01
Submission date: 2018-10-11
Final revision date: 2019-01-03
Acceptance date: 2019-01-23
 
Diagnostyka 2019;20(1):111–125
KEYWORDS
TOPICS
ABSTRACT
The linear and geometrically nonlinear free and forced vibrations of Euler-Bernoulli beams with multi-cracks are investigated using the crack equivalent rotational spring model and the beam transfer matrix method. The Newton Raphson solution of the transcendental frequency equation corresponding to the linear case leads to the cracked beam linear frequencies and mode shapes. Considering the nonlinear case, the beam transverse displacement is expanded as a series of the linear modes calculated before. Using the discretised expressions for the total strain and kinetic energies and Hamilton’s principle, the nonlinear amplitude equation is obtained and solved using the so-called second formulation, developed previously for similar nonlinear structural dynamic problems, to obtain the multi-cracked beam backbone curves and the corresponding amplitude dependent nonlinear mode shapes. Considering the forced vibration case, the nonlinear frequency response functions obtained numerically near to the fundamental nonlinear mode using a single mode approach show the effects of the number of cracks, their locations and depths, and the level of the concentric harmonic force. The inverse problem is explored using the frequency contour plot method to identify crack parameters, such as the crack locations and depths. Satisfactory comparisons are made with previous analytical results.
 
REFERENCES (58)
1.
Ne D. Mechanical Behaviour of Materials Engineering Methods for Deformation Fracture and Fatigue. Prence-Hall Int; 1993.
 
2.
Ye Xw, Su Yh, Han Jp. A State-Of-The-Art Review on Fatigue Life Assessment of Steel Bridges. Mathematical Problems in Engineering 2014.
 
3.
Liu Y, Xiao X, Lu N, Deng Y. Fatigue Reliability Assessment Of Orthotropic Bridge Decks Under Stochastic Truck Loading. Shock and Vibration 2016.
 
4.
Mann Jy. Bibliography on the Fatigue of Materials, Components and Structures. Elsevier; 2013.
 
5.
Schütz W. A History of Fatigue. Engineering Fracture Mechanics 1996; 54 (2): 263–300. https://doi.org/10.1016/0013-7....
 
6.
Schijve J. Fatigue of Structures and Materials in the 20th Century and the State of the Art. International Journal of Fatigue 2003; 25 (8): 679–702. https://doi.org/10.1016/S0142-....
 
7.
Doebling Sw, Farrar Cr, Prime Mb, Shevitz Dw. Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in their Vibration Characteristics: A Literature Review. Los Alamos National Lab., Nm (United States); 1996.
 
8.
Yang Xf, Swamidas Asj, Seshadri R. Crack Identification in Vibrating Beams using the Energy Method. Journal of Sound and Vibration 2001; 244 (2): 339–357. https://doi.org/0.1006/Jsvi.20....
 
9.
Yu Z, Chu F. Identification of Crack in Functionally Graded Material Beams using the P-Version of Finite Element Method. Journal of Sound and Vibration 2009; 325 (1):69–84. https://doi.org/10.1016/J.Jsv.....
 
10.
Khnaijar A, Benamar R. A New Model for Beam Crack Detection and Localization using a Discrete Model. Engineering Structures 2017; 150: 221–230. https://doi.org/10.1016/J.Engs....
 
11.
Dimarogonas Ad. Vibration of Cracked Structures: A State of the Art Review. Engineering Fracture Mechanics 1996; (5):831–857. https://doi.org/10.1016/0013-7....
 
12.
Gasch R. A Survey of the Dynamic Behaviour of a Simple Rotating Shaft with a Transverse Crack. Journal of Sound and Vibration 1993; 160 (2):313–332. https://doi.org/10.1006/Jsvi.1....
 
13.
Sabnavis G, Kirk R, Kasarda M, Quinn D. Cracked Shaft Detection and Diagnostics: A Literature Review. The Shock and Vibration Digest 2004; 36: 287–296. https://doi.org/10.1177/058310....
 
14.
Adams Rd, Cawley P, Pye Cj, Stone Bj. A Vibration Technique for Non-Destructively Assessing the Integrity of Structures. Journal of Mechanical Engineering Science 1978; 20 (2):93–100. https://doi.org/10.1243/Jmes_J....
 
15.
Sinha Jk, Friswell Mi, Edwards S. Simplified Models for the Location of Cracks in Beam Structures using Measured Vibration Data. Journal of Sound and Vibration 2002; 251(1): 13–38. https://doi.org/10.1006/Jsvi.2....
 
16.
Cerri, M.N., Vestroni, F. Identification of Damage Due to Open Cracks by Change of Measured Frequencies. 16th Aimeta Congress of Theoretical and Applied Mechanics. September 9, 2003.
 
17.
Rizos Pf, Aspragathos N, Dimarogonas Ad. Identification of Crack Location and Magnitude in a Cantilever Beam from the Vibration Modes. Journal of Sound and Vibration 1990; 138 (3): 381–388.
 
18.
B. Zastrau. Vibration of Cracked Structures, Archives of Mechanics.1985:Vol.37, Pp.731–743.
 
19.
Chati M, Rand R, Mukherjee S. Modal Analysis of a Cracked Beam. Journal of Sound and Vibration 1997; 207 (2): 249–270. https://doi.org/10.1006/Jsvi.1....
 
20.
Dimarogonas Ad, Paipetis Sa, Chondros Tg. Analytical Methods in Rotor Dynamics: Second Edition. Springer Science & Business Media; 2013.
 
21.
Papadopoulos Ca, Dimarogonas Ad. Coupled Longitudinal and Bending Vibrations of a Rotating Shaft with an Open Crack. Journal of Sound and Vibration 1987; 117 (1): 81–93.
 
22.
Friswell Mi, Penny Jet. Crack Modeling for Structural Health Monitoring. Structural Health Monitoring 2002; 1(2): 139–148. https://doi.org/10.1177/147592....
 
23.
Caddemi S, Caliò’ I. Exact Solution of the Multi-Cracked Euler–Bernoulli Column. International Journal of Solids and Structures 2008; 45 (5): 1332–1351. https://doi.org/10.1016/J.Ijso....
 
24.
Attar M. A Transfer Matrix Method for Free Vibration Analysis and Crack Identification of Stepped Beams with Multiple Edge Cracks and Different Boundary Conditions. International Journal of Mechanical Sciences 2012; 57 (1): 19–33. https://doi.org/10.1016/J.Ijme....
 
25.
Batihan Aç, Kadioğlu Fs. Vibration Analysis of a Cracked Beam on an Elastic Foundation. International Journal of Structural Stability and Dynamics 2015; 16(05): 1550006. https://doi.org/10.1142/S02194....
 
26.
Cunedioglu Y. Free Vibration Analysis Of Edge Cracked Symmetric Functionally Graded Sandwich Beams. Structural Engineering and Mechanics 2015; 56. https://doi.org/10.12989/Sem.2....
 
27.
Dado Mh. A Comprehensive Crack Identification Algorithm for Beams under Different End Conditions. Applied Acoustics 1997; 51 (4): 381–398.
 
28.
Liu Y, Xiao J, Shu D. Free Vibration of Delaminated Beams with an Edge Crack. Procedia Engineering 2014; 75:78–82. https://doi.org/10.1016/J.Proe....
 
29.
Liu Y, Shu Dw. Effects of Edge Crack on the Vibration Characteristics of Delaminated Beams. Structural Engineering and Mechanics 2015; 53:767–780. https://doi.org/10.12989/Sem.2....
 
30.
Shin Y, Yun J, Seong K, Kim J, Kang S. Natural Frequencies of Euler-Bernoulli Beam with Open Cracks on Elastic Foundations. Journal of Mechanical Science and Technology 2006; 20 (4): 467–472. https://doi.org/10.1007/Bf0291....
 
31.
Darpe Ak, Gupta K, Chawla A. Dynamics of a Two-Crack Rotor. Journal of Sound and Vibration 2003; 259 (3): 649–675. https://doi.org/10.1006/Jsvi.2....
 
32.
Douka E, Bamnios G, Trochidis A. A Method for Determining the Location and Depth of Cracks in Double-Cracked Beams. Applied Acoustics 2004; 65 (10): 997–1008. https://doi.org/10.1016/J.Apac....
 
33.
Jena Sp, Parhi Dr, Mishra D. Comparative Study on Cracked Beam with Different Types of Cracks Carrying Moving Mass. Structural Engineering And Mechanics 2015; 56(5): 797–811. https://doi.org/10.12989/Sem.2....
 
34.
Ostachowicz Wm, Krawczuk M. Analysis of the Effect of Cracks on the Natural Frequencies of a Cantilever Beam. Journal of Sound and Vibration 1991; 150 (2): 191–201.
 
35.
Sekhar As. Vibration Characteristics of a Cracked Rotor with two Open Cracks. Journal of Sound and Vibration 1999; 223(4): 497–512. https://doi.org/10.1006/Jsvi.1....
 
36.
Yoon H-I, Son I-S, Ahn S-J. Free Vibration Analysis of Euler-Bernoulli Beam with Double Cracks. Journal of Mechanical Science and Technology 2007; 21 (3):476–485. https://doi.org/10.1007/Bf0291....
 
37.
Shifrin Ei, Ruotolo R. Natural Frequencies of a Beam with an Arbitrary Number of Cracks. Journal of Sound and Vibration 1999; 222 (3): 409–423. https://doi.org/10.1006/Jsvi.1....
 
38.
Khiem Nt, Lien Tv. A Simplified Method for Natural Frequency Analysis of a Multiple Cracked Beam. Journal of Sound and Vibration 2001; 245 (4): 737–751. https://doi.org/10.1006/Jsvi.2....
 
39.
Ruotolo R, Surace C. Natural Frequencies of a Bar with Multiple Cracks. Journal of Sound and Vibration 2004; 272 (1): 301–316.
 
40.
Li Qs. Free Vibration Analysis of Non-Uniform Beams with an Arbitrary Number of Cracks and Concentrated Masses. Journal of Sound and Vibration 2002; 252 (3): 509–525. https://doi.org/10.1006/Jsvi.2....
 
41.
Binici B. Vibration of Beams with Multiple Open Cracks Subjected to Axial Force. Journal of Sound and Vibration 2005; 287 (1): 277–295. https://doi.org/10.1016/J.Jsv.....
 
42.
M. Cocchi G, Volpi M. Inelastic Analysis of Reinforced Concrete Beams Subjected to Combined Torsion, Flexural and Axial Loads. Computers & Structures - Comput Struct 1996; 61: 479–494.
 
43.
Zhou L, Huang Y. Crack Effect on the Elastic Buckling Behavior of Axially And Eccentrically Loaded Columns. Structural Engineering and Mechanics 2006; 22 (2): 169–184. https://doi.org/10.12989/Sem.2....
 
44.
Kisa M. Vibration and Stability of Multi-Cracked Beams under Compressive Axial Loading. International Journal of Physical Sciences 2011; 6 (11): 2681–2696. https://doi.org/10.5897/Ijps11....
 
45.
Cheng, S.M., Wu, X.J., Wallace, W., & Swamidas. Vibrational Response of a Beam with a Breathing Crack. Journal of Sound and Vibration 225.A.S.J 1999:201–208.
 
46.
Caddemi S, Caliò I, Marletta M. The Non-Linear Dynamic Response of the Euler–Bernoulli Beam with an Arbitrary Number of Switching Cracks. International Journal of Non-Linear Mechanics 2010; 45 (7): 714–726. https://doi.org/10.1016/J.Ijno....
 
47.
Chondros Tg, Dimarogonas Ad, Yao J. Vibration of a Beam with a Breathing Crack. Journal of Sound and Vibration 2001; 239(1): 57–67. https://doi.org/10.1006/Jsvi.2....
 
48.
Matveev Vv, Bovsunovsky Ap. Vibration-Based Diagnostics of Fatigue Damage of Beam-Like Structures. Journal of Sound and Vibration 2002; 249 (1): 23–40. https://doi.org/10.1006/Jsvi.2....
 
49.
Benamar R, Bennouna Mmk, White Rg. The Effects of Large Vibration Amplitudes on the Mode Shapes and Natural Frequencies of Thin Elastic Structures Part I: Simply Supported and Clamped-Clamped Beams. Journal of Sound and Vibration 1991; 149 (2): 179–195.
 
50.
Benamar R. Nonlinear Dynamic Behaviour of Fully Clamped Beams and Rectangular Isotropic and Laminated Plates. 1990.
 
51.
Bennouna Mm, White Rg. The Effects of Large Vibration Amplitudes on the Fundamental Mode Shape of a Clamped-Clamped Uniform Beam. Journal of Sound and Vibration 1984; 96 (3): 309–331.
 
52.
Harras B, Benamar R, White Rg. Experimental and Theoretical Investigation of the Linear and Non-Linear Dynamic Behaviour of a Glare 3 Hybrid Composite Panel. Journal of Sound and Vibration 2002; 252(2): 281–315. https://doi.org/10.1006/Jsvi.2....
 
53.
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. https://doi.org/10.1016/J.Comp....
 
54.
Adri A, Benamar R. Linear And Geometrically Non-Linear Frequencies and Mode Shapes of Beams Carrying a Point Mass at Various Locations. An Analytical Approch and a Parametric Study. Diagnostyka 2017; Vol. 18, No. 2.
 
55.
El Kadiri M, Benamar R, White Rg. Improvement of the Semi-Analytical Method, For Determining the Geometrically Non-Linear Response of Thin Straight Structures. Part I: Application to Clamped–Clamped and Simply Supported–Clamped Beams. Journal of Sound and Vibration 2002; 249 (2): 263–305. https://doi.org/10.1006/Jsvi.2....
 
56.
Azrar L, Benamar R, White Rg. Semi-Analytical Approach to the Non-Linear Dynamic Response Problem Of S–S And C–C Beams at Large Vibration Amplitudes Part I: General Theory And Application To The Single Mode Approach To Free And Forced Vibration Analysis. Journal of Sound and Vibration 1999; 224(2): 183–207. https://doi.org/10.1006/Jsvi.1....
 
57.
Tada H, Paris Pc, Irwin Gr. The Stress Analysis of Cracks Handbook, Third Edition. Ny 10016-5990: Asme; 2000.
 
58.
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. https://doi.org/10.1016/J.Jsv.....
 
eISSN:2449-5220
ISSN:1641-6414