Vibration Analysis of a V-Groove Edged Cracked Cantilever Beam

 

Mr. Manoj Kumar Sahoo*, Prof. O P Sahu, Prof. Manish Verma

Department of Mechanical Engineering, Chouksey Engineering College, Bilaspur

*Corresponding Author E-mail: jkie.mksahoo@gmail.com

 

ABSTRACT:

This paper describes the vibration analysis of a cracked cantilever aluminium beam and analyzes the relation between the modal natural frequencies with crack depth, modal natural frequency with crack location. Also the relation among the crack depth, crack location and natural frequency has been analyzed. Double cracks at different depth and at different location are evaluated. And the analysis reveals a relationship between crack depth and modal natural frequency. As we know when a structure suffers from damage its dynamic property can change and it was observed that crack caused a stiffness reduction with an inherent reduction in modal natural frequencies. Consequently it leads to the change in the dynamic response of the beam. The analysis was performed using CATIA software. Modal natural frequency was found to be increasing with increase in crack depth.

 

 

INTRODUCTION:

Being very commonly used in steel construction and machinery industries, health monitoring and the analysis of damage in the form of crack in Beam structures poses a vital mean. Since long efforts are on their way to find a feasible solution for crack detection in beam structures in this regard many approaches have so far being taken place. When a structure suffers from damages, its dynamic properties can change. Crack damage leads to reduction in stiffness also with an inherent reduction in natural frequency and increase in modal damping. The paper gives a feasible relationship between the modal natural frequency and the crack depth at different location. Since free vibration analysis has frequently become a topic of many studies therefore attention is focused it only.

 

Crack localization and sizing in a beam from the free and forced response measurements method is indicated by Karthikeyan et al.¹ In the beam Timoshenko beam theory is used for modeling transverse vibrations.FEM is used for the free and forced vibration analysis of the cracked beam and open transverse crack is selected for the crack model. Being iterative in nature the iteration starts with a guess for the crack depth ratio and iteratively estimates the crack location and crack depth until the desired convergence for both is reached.

 

The amount of literature related to damage detection using shifts in resonant frequencies is quite large. Salawu and Williams 2 presents an excellent review on the use of modal frequency changes for damage diagnostics. The observation that changes in structural properties cause changes in vibration frequencies was the impetus for using modal methods for damage identification and health monitoring.

 

Fig. 1 Crack dimensions

Kim and Zhao³ proposed a novel crack detection method using harmonic response. It was concluded in their paper that slope response has a sharp change with the crack location and depth of the crack and therefore it can used as a crack  etection criterion. A fault diagnosis method based on genetic algorithms (GAs) and a model of damaged (cracked) structure is proposed by Taghi et al.⁴. In their approach the identification of the crack location and depth in the cantilever beam is formulated as an optimization problem, and binary and continuous genetic algorithms (BGA, CGA) are used to find the optimal location and depth by minimizing the cost function.

 

Fig. 2 Double cracks on the cantilever beam at 100 mm and 200mm length from the free end

 

Fig. 3 Messing

 

Ratcliffe⁵ performed the frequency and curvaturebased experiments. Orhan⁶ in his study analyzed the free and forced vibration of a cantilever beam in order to identify the crack of a cantilever beam. Single and two edged crack were mainly evaluated in his study. The investigation reveals that free vibration analysis provides suitable information for the detection of single and two cracks; whereas forced vibration can detect only the single crack condition. However, dynamic response of the forced vibration better describes changes in crack depth and location than the free vibration.

 

 

Fig.4  Mode 1

 

Fig.5  Mode 2

 

Chang and Chen⁷ presented technique for structure damage detection based on spatial wavelet analysis and the innovation of the proposed method is that both the positions and depths of multi-cracks can be estimated from spatial wavelet based method. First, the mode shapes of free vibration and natural frequencies of the multiple cracked beams are obtained. It was observed from the analysis that the positions and depths of the cracks can be predicted with acceptable precision even though there are many cracks in the beam.

 

1.      CONFIGURATIONS OF SIMULATED CRACK

In this particular approach the free vibration of a cantilever beam having V-grooved shaped edge crack are studied.

 

Fig.6  Mode 3

 

Fig.7 Mode 4

 

The length of the beam is 1000 mm and the cross-section of the same is 100 x 20 mm². As per the material properties the modulus of elasticity (E) is 70 x 10⁹ N/m² and the mass density (ρ) is 2710 kg/m³. Different crack configurations of same depth and at different locations (from different distance from the free end) are prepared to find out how the crack affects the dynamic behavior of the beam. Crack depth was kept constant at 0.2 mm and the first crack location from the free end was varied at distances of 100 mm, 110 mm, 120 mm, 130 mm, 140 mm and 150 mm and the second crack location from the free end was varied at distances of 200 mm, 210 mm, 220 mm, 230 mm, 240 mm and 250 mm simultaneously. And the effect of crack location on the natural frequencies was investigated.

 

Fig.8  Mode 5

 

 

Also the crack location from the free end was kept constant and crack depth was varied from 0.2, 0.21, 0.22, 0.23, 0.24 to 0.25mm at each step in order to investigate the effect of crack depth on natural frequencies.

 

2.      THE FINITE ELEMENT MODELING

CATIA software was used for the free vibration analysis of the uncracked and cracked beams. For this purpose a model of the uncracked and cracked beams was prepared in the CATIAV5R20 and the mesh was generated. The cantilever boundary conditions were modeled by constraining all degrees of freedom of the nodes located at the right hand side of the beam. Figure 2 shows the finite element mesh model of the beam element.

 

5 modes were selected to extract and 5 modes of natural frequencies were calculated for cracked beam. This procedure was thereafter repeated for different crack scenarios. Figures below shows different modes of natural frequencies at crack depth=2mm, first crack from free end=100mm and second crack from free end=200mm

 

3.      RESULTS:

The change in natural frequencies with crack depth for five different modes is shown in table 1, 2, 3, 4 and 5. And it was observed that in all the cases the modal natural frequencies increase with increase in crack depth.

 

It was observed that as the crack location from the free end increases, the modal natural frequency decreases. And when the crack depth at a particular crack location increases the modal natural frequency also increases.

 

Table 1: Results obtained in CATIA at d=2mm

L1(mm)	L2(mm)	FREQUENCY(Hz)
		MODE 1	MODE 2	MODE 3	MODE 4	MODE 5
100	200	16.554	81.736	103.452	288.73	291.308
110	210	16.554	81.735	103.446	288.699	291.303
120	220	16.553	81.734	103.435	288.638	291.294
130	230	16.553	81.732	103.429	288.611	291.29
140	240	16.553	81.73	103.42	288.558	291.281
150	250	16.552	81.72	103.411	288.523	291.274

 

Table 2: Results obtained in CATIA at d=2.1mm

L1(mm)	L2(mm)	FREQUENCY(Hz)
		MODE 1	MODE 2	MODE 3	MODE 4	MODE 5
100	200	16.554	81.739	103.452	288.701	291.309
110	210	16.554	81.737	103.442	288.662	291.309
120	220	16.554	81.735	103.443	288.606	291.298
130	230	16.553	81.734	103.425	288.567	291.294
140	240	16.553	81.731	103.413	288.505	291.288
150	250	16.552	81.729	103.405	288.474	291.282

 

Table 3: Results obtained in CATIA at d=2.2mm

L1(mm)	L2(mm)	FREQUENCY(Hz)
		MODE 1	MODE 2	MODE 3	MODE 4	MODE 5
100	200	16.555	81.742	103.448	288.671	291.313
110	210	16.555	81.739	103.441	288.632	291.31
120	220	16.554	81.738	103.431	288.569	291.307
130	230	16.554	81.736	103.422	288.53	291.301
140	240	16.554	81.734	103.411	288.466	291.291
150	250	16.553	81.732	103.4	288.431	291.282

 

Table 4: Results obtained in CATIA at d=2.3mm

L1(mm)	L2(mm)	FREQUENCY(Hz)
		MODE 1	MODE 2	MODE 3	MODE 4	MODE 5
100	200	16.556	81.744	103.446	288.639	291.316
110	210	16.555	81.742	103.439	288.598	291.313
120	220	16.555	81.74	103.426	288.524	291.307
130	230	16.554	81.738	103.418	288.486	291.303
140	240	16.554	81.736	103.405	288.413	291.294
150	250	16.553	81.734	103.395	288.378	291.282

 

Table 5: Results obtained in CATIA at d=2.4mm

L1(mm)	L2(mm)	FREQUENCY(Hz)
		MODE 1	MODE 2	MODE 3	MODE 4	MODE 5
100	200	16.556	81.746	103.443	288.602	291.325
110	210	16.556	81.745	103.435	288.557	291.32
120	220	16.555	81.742	103.422	288.478	291.305
130	230	16.554	81.741	103.413	288.435	291.309
140	240	16.554	81.738	103.399	288.358	291.288
150	250	16.553	81.736	103.389	288.322	291.287

 

 

4    REFERENCES:

1.       M. Karthikeyan, R. Tiwari, S. Talukdar (2006). “Crack localization and sizing in a beam based on the free and forced response measurements”. Mechanical Systems and Signal Processing 21 (2007). pp. 1362–1385

2.       Salawu, O. S. and Williams, C., 1993, “Structural Damage Detection Using Experimental Modal Analysis–International Journal of Advanced Engineering Research and Studies E-ISSN2249–8974 IJAERS/Vol. I/ Issue II/January-March, 2012/285-289 A Comparison Of Some Methods,” in Proc. of 11^th International Modal Analysis Conference, pp. 254–260

3.       Kim M-B, Zhao M. Study on crack detection of beam using harmonic responses. Proceedings of the 2004 international conference on intelligent mechatronics and automation, August 2004, Chengdu, China, p.72–6.

4.       Mohammad-Taghi Vakil-Baghmisheh, Mansour Peimani, Morteza Homayoun Sadeghi, Mir Mohammad Ettefagh (2007). “Crack detection in beam-like structures using genetic algorithms”. Applied Soft Computing 8 (2008). pp. 1150–1160.

5.       C.P. Ratcliffe, Frequency and curvature based experimental method for locating damage in structures, J. Vibration Acoustic. 122 (2000) 324–329.

6.       Sadettin Orhan (2007). “Analysis of free and forced vibration of a cracked cantilever beam”. NDT&E International 40 (2007). pp. 443–450.

7.       Chih-Chieh Chang, Lien-Wen Chen(2005). “Detection of the location and size of cracks in the multiple cracked beam by spatial wavelet based approach”. Mechanical Systems and Signal Processing. Vol.19. pp.139–155

8.       Peng, Z. K., Lang, Z. Q., Billings, S.A.(2007), “Crack Detection using nonlinear output frequency response functions”. Journal of Sound and Vibration, 301, pp. 777-788.

 

 

 

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