Signature analysis of cracked cantilever beam
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Signature analysis of cracked cantilever beam

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    Signature analysis of cracked cantilever beam Signature analysis of cracked cantilever beam Document Transcript

    • International Journal of Advanced in Engineering and Technology (IJARET)International Journal of Advanced Research Research in Engineeringand Technology (IJARET), ISSN 0976 – 6480(Print) IJARETISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEMEISSN 0976 – 6499(Online) Volume 1Number 1, May - June (2010), pp. 105-117 © IAEME© IAEME, http://www.iaeme.com/ijaret.html SIGNATURE ANALYSIS OF CRACKED CANTILEVER BEAM Sharad V. Kshirsagar Asst. Professor, Mechanical Engineering Department Sinhgad College of Engineering Pune, E-mail: sharadkshirsagar@gmail.com Dr. Lalit B. Bhuyar Mechanical Engineering Department Prof. Ram Meghe Institute of Technology & Research Badnera, MaharashtraABSTRACT Beams are more widely used in the machine-structures. Fatigue-type of loading ofsuch engineering parts is likely to introduce cracks at the highly stressed regions and leadto damage and deterioration during their service life. Cracks are a main cause ofstructural failure. Once a crack is initiated, it propagates and the stress required forpropagation is smaller than that required for crack initiation. After many cycles operatingstresses may be sufficient to propagate the crack. The crack propagation takes place overa certain depth when it is sufficient to create unstable conditions and fracture take place.The sudden failure of components is very costly and may be catastrophic in terms ofhuman life and property damage. Forced vibration analysis of a cracked cantilever beamwas carried out and the results are discussed in this paper. An experimental setup wasdesigned in which a cracked cantilever beam excited by an exciter and the signature wasobtained using an accelerometer attached to the beam. To avoid non-linearity, it wasassumed that the crack remain always open.Keywords: Crack detection, forced vibrations, signature analysis.1. INTRODUCTION Literature on Fault detection and condition monitoring was focused on thevibration-based method which can be classified into modal-based and signature-basedmethods. In modal based techniques data can be condensed from the actual measured 105
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEMEquantities like resonant frequencies, mode shape vectors and quantities derived fromthese parameters for the crack detection [1, 3, 4, 6]. In signature based methods the vibration signature of cracked machinery structurecan be useful for the fault diagnosis and condition monitoring. Thus, the development ofcrack detection methods has received increasing attention in recent years. Among thesetechniques, it is believed that the monitoring of the global dynamics of a structure offersfavorable alternative if the on-line (in service) damage detection is necessary. In order toidentify structural damage by vibration monitoring, the study of the changes of thestructural dynamic behavior due to cracks is required for developing the detectioncriterion. [2, 5, 7-13].2. GOVERNING EQUATIONS OF FORCED VIBRATION The equation of motion for the beam element without crack can be written asfollows from [14]: -------------------------------- (1) (e ) (e ) where [M ] is the element mass matrix, [K wc ] is the element stiffness matrix,{F (t )}(e ) is the element external force vector, {q(t )} (e ) is the element vector of nodaldegree of freedoms and t is the time instant. The subscript wc represents without crack,the superscript e represents element and dot represents the derivative with respect to thetime. The crack is assumed to affect only the stiffness. Hence the equation of motion of acracked beam element can be expressed as ------------------------------- (2) (e ) where {qc (t )} is the nodal degrees of freedom of the cracked element, the (e )subscript c represents the crack and [K c ] is the stiffness matrix of the cracked elementand is given as −1 [K c ](e ) = [T ][C ](e ) [T ]T ------------------------------------------- (3)with [C ](e ) = [C0 ](e ) + [Cc ](e ) ------------------------------------------ (4) 106
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME (e ) (e ) where [C0 ] is the flexibility matrix of the uncracked beam element, [Cc ] is the (e )flexibility matrix of the crack, and [C ] is the total flexibility matrix of the crackedbeam element. Equations of motion of the complete system can be obtained by assembling thecontribution of all equations of motion for cracked and uncracked elements in the system.Then the system equation of motion becomes --------------------------------------- (5) where [M ] is the assembled mass matrix, [K ] is the assembled stiffness matrix,{F (t )} is the assembled external force vector, and {q(t )} is the assembled vector of nodaldofs of the system. Let the force vector be defined as {F (t )} = {F }e jwt , ----------------------------------------------- (6) {} where w is the forcing frequency, F is the force amplitude vector (elements ofwhich are complex quantities) and j = −1 . Thus, the response vector can be assumed as {q(t)} ={q}ejwt, ------------------------------------------------ (7) {} where q is the response amplitude vector and their elements are complexquantities. Using Eqs. (6) and (7) for modal frequency, the system governing equation asfollows: (− w [M ] + [K ]){q}= {F }. ------------------------------------------ (8) 2 For a given system properties (i.e. [M ] and [K ] the response can be simulatedfrom Eq. (8) corresponding to a given force F . {}3. SIMULATION In the finite element simulation, a cantilever beam with rectangular edge crack isconsidered. The length and cross-sectional area of the beam are 800 mm, and 50x6 mm2,respectively. As for the material properties the modulus of elasticity (E) is 0.675 1011N/m2, the density (ρ) 27522.9 kg/m3 and the Poisson’s ratio (µ ) is 0.33. 107
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME3.1. Generation of Cracked Beam Model A 8-node three-dimensional structural solid element under SOLID 45 wasselected to model the beam. The beam was discretized into 11859 elements with 54475nodes. Cantilever boundary conditions modeled by constraining all degrees of freedomsof the nodes located on the left end of the beam. APDL PROGRAMMING is used tocreate 135 cracked beam models by varying the crack depth from 5 mm to 45 mm andcrack location from 50 mm to 750 mm. Figure 1 show the finite element mesh model ofthe beam generated in Ansys (12). Figure 2 Finite element mesh model.3.2 Harmonic Analysis Full Solution Method, Reduced Solution Method, Mode superposition Method arethe methods to be used to solve the harmonic equation.Mode Superposition Method is used to solve in the current analysis.[M] {ü} + [C] {ú} + [K] {u} = {Fa} __________________________(9)where:[M] = structural mass matrix[C] = structural damping matrix[K] = structural stiffness matrix{ü} = nodal acceleration vector{ú} = nodal velocity vector{u} = nodal displacement vector 108
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME{Fa} = applied load vector All points in the structure are moving at the same known frequency, however, notnecessarily in phase. Also, it is known that the presence of damping causes phase shifts.Therefore, the displacements may be defined as:{u} = {umax ei Φ }ei t --------------------------------------------- (10)where:umax = maximum displacementi = square root of -1 = imposed circular frequency (radians/time) = 2πff = imposed frequency (cycles/time)t = timeΦ = displacement phase shift (radians){ Fa } = {Fmax ei ψ }ei t-------------------------------------------(11)where:Fmax = force amplitudeψ = force phase shift (radians) The dependence on time (ei t) is the same on both sides of the equation and maytherefore be removed. Figure 2 show the boundary condtions for harmonic analysis. Figure 2 Boundary conditions for harmonic analysis. 109
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME4. EXPERIMENTATION A number of carefully designed experiments were carried out on a CantileverBeam. Figure 3 shows the components of this experimentation. Vibration signals werecollected for both uncracked and several cracked beam conditions. The excitationfrequency was set at approximately 40 Hz. With the sensor mounted on the beam at freeend, vibration signals were measured for various fault conditions by on-line monitoringwhen beam was under stationary excitation. Table 1 show the comparison of theexperimental results with the simulated results. Figure 3 Experimental setup Table 1 Comparison of simulated and experimental results Crack Crack Mode 1 Mode 2 Mode 3 case C/L a/h Simulated Expt. Simulated Expt. Simulated Expt. 0.1 0.9901 1 0.9945 1 0.9973 1 1 1/16 0.2 0.9614 0.97 0.9792 0.9867 0.9899 0.9904 0.1 0.9935 1 0.9999 1 0.9987 1 2 3/16 0.4 0.8929 0.9118 0.9972 1 0.9769 0.9856 0.3 0.9636 0.9708 0.9863 0.99 0.9589 0.9604 3 5/16 0.4 0.9315 0.9433 0.9747 0.9780 0.9265 0.9394 0.2 0.9917 0.9987 0.9804 0.9890 0.9962 1 4 7/16 0.3 0.9805 0.9898 0.9559 0.9623 0.9912 1 0.2 0.9944 1 0.9767 0.9901 0.9999 1 5 8/16 0.5 0.9537 0.9611 0.8477 0.8602 0.9986 1 0.2 0.9990 1 0.9857 0.9945 0.9753 0.9790 6 11/16 0.4 0.9951 1 0.9363 0.9456 0.9046 0.9200 7 14/16 0.5 1 1 0.9917 0.9989 0.9514 0.9654 110
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME5. RESULTS AND DISCUSSIONS Before the experiments were carried out, the first three natural frequencies of thebeams were simulated by FEA. From the results obtained, it was decided that using afrequency range upto 1.2 kHz for experimental measurements would be sufficient toinclude the first three natural frequencies. The frequency response functions obtained were curve-fitted. The simulated datafrom the curve-fitted results were tabulated and plotted in the form of frequency ratio(ratio of the natural frequency of the cracked beam that of the uncracked beam) versusthe crack depth ratio (a/h) [the ratio of the depth of a crack (a) to the thickness of thebeam (h)] for various crack location ratios (C/L) (ratio of the location of the crack to thelength of the beam). Figure 4 to 6 show the plots of the first three frequency ratios as a function ofcrack depths for some of the crack positions considered for each set of boundaryconditions (fifteen locations for each set of boundary conditions). Figure 7 to 9 shows thefrequency ratio variation of three modes in terms of crack position for various crackdepth. 1 1 Frequency Ratio 0.9 2 1st Mode 0.8 3 0.7 4 0.6 6 5 0.5 0.1 0.3 0.5 0.7 0.9 Crack Depth Ratio Figure 4 Fundamental natural frequency ratio in terms of crack depth for various crack positions C / l (1→15/16; 2→9/16; 3→7/16; 4→5/16; 6→2/16). 111
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME 1 1 0.9 2 3 Frequency Ratio 0.8 2nd Mode 0.7 4 0.6 5 0.5 0.4 6 0.3 0.1 0.3 0.5 0.7 0.9 Crack Depth Ratio Figure 5 Second natural frequency ratio in terms of crack depth for various crack Positions C / l (1→15/16; 2→2/16; 3→5/16; 4→6/16; 5→7/16; 6→10/16). 1 1 0.9 2 Frequency Ratio 3rd Mode 0.8 0.7 3 5 0.6 4 0.5 0.4 0.1 0.3 0.5 0.7 0.9 Crack Depth RatioFigure 6 Third natural frequency ratio in terms of crack depth for various crack positions C / l (1→2/16; 2→15/16; 3→9/16; 4→14/16; 5→13/16). 1.00 Frequency Ratio 0.80 1st Mode 0.60 0.40 0.20 0.0625 0.3125 0.5625 0.8125 Crack Location Ratio 0.1 0.3 0.5 0.7 0.9 Figure 7 1st Mode frequency ratio in terms of crack position for various crack depths. 112
    • International Journal of Advanced Research in Engineering and Technology (IJARET) ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME Frequency Ratio 0.90 2nd Mode 0.70 0.50 0.30 0.0625 0.3125 0.5625 0.8125 Crack Location Ratio 0.1 0.3 0.5 0.7 0.9 Figure 8 2nd Mode frequency ratio in terms of crack position for various crack depths. Frequency Ratio 0.90 3rd Mode 0.70 0.50 0.30 0.0625 0.3125 0.5625 0.8125 Crack Location Ratio 0.1 0.3 0.5 0.7 0.9 Figure 9 3rd Mode frequency ratio in terms of crack position for various crack depths. From the results and plots the following observations were made for al1 the cases considered: i. Natural frequencies were reduced due to presence of crack.ii. Effects of cracks were high for the small values of crack location ratioiii. The second natural frequency was greatly affected at the C / l = 11/16 for all crack depths.iv. The third natural frequency was almost unaffected for the crack locations ( C / l = 2/16 and 8/16); the reason for this influence was that the location of nodal point was located at that point on the beam.v. Due to shifts in the nodal positions (as a consequence of cracking) of the second and the third modes, the changes in the higher natural frequencies depended on how close the crack location was to the mode shape nodes. Consequently, it was be observed from 113
    • International Journal of Advanced Research in Engineering and Technology (IJARET) ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME the results that the trend of changes in the second, and the third frequencies are not monotonic, as we have in the first natural frequency.vi. From the results obtained, it is observed, for example, that when the crack depth ratio is 0.9, the third natural frequency was comparatively much less affected than the first and second frequencies for a crack located at C / l =8/16 but, it is highly affected for other crack locations. This could be explained by the fact that decrease in frequencies is greatest for a crack located where the bending moment is greatest. It appears therefore that the change in frequencies is not only a function of crack depth and crack location, but also of the mode number.vii. For various cases considered, the frequencies decreased rapidly with the increase in the crack depths for all three modes. As stated earlier, the decrease in the fundamental natural frequency was greatest when the crack occurred closer to the fixed point. This could be explained by the fact that the bending moment was the largest at that point (where the amplitude of the first mode shape is greatest) for the first mode, thereby, resulting in a greater loss of bending stiffness due to crack. However, the second and third modes were less affected at this location. The frequencies decreased by about 23.7% & 10.2%, and 49.7% for the first, second and third modes, respectively, as the crack grew to half of the beam depth (for crack at 50 mm from fixed end). 35 30 25 Area Ratio 20 15 10 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Crack Depth 0.0625 0.2500 0.3125 0.5000 0.5625 Figure 10 Area under the frequency response curve as a function of crack position. 114
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME 14 12 10 Area Ratio 8 6 4 2 0 0.0625 0.1875 0.3125 0.4375 0.5625 0.6875 0.8125 0.9375 Crack Location 0.4 0.5 0.7 0.9 Figure 11 Area under the frequency response curve as a function of crack depth.CONCLUSIONS Based on the experimental data, and plots, and the observations above, numerousinferences could be made such as follows: a) For of the cases considered, the dopes of frequency ratio versus crack depth curves were very small for small crack depth ratios. This implies that small cracks have little effects on the sensitivities of natural frequencies. Hence, using only results based on frequency changes alone for identifying cracks in most practical problems may be misleading as it is very unlikely to have large cracks. b) For a particular mode, the decrease in frequency and change in mode shape become noticeable as the crack grew bigger. c) For a given crack depth ratio, the location of the crack greatly affects the dynamic response of the cracked beam. d) Investigating the mode of vibration at some crack location may indicate a pure bending mode for small crack depth ratios, but, as the crack grows in size. The ending mode may contain a significant influence of longitudinal vibration mode also (occurrence of coupling). e) Fatigue crack alters the local stiffness which changes dynamic response. From Fig. 10 -11 it is seen that area under the frequency response curve can be used as one of the elements of crack detection. 115
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEMEREFERENCES[1] T.G.Chondros, A.D.Dimarogonas and J.Yao, A Continuous Cracked Beam Vibration Theory, Journal of Sound and Vibration, 215(1), 1998, pp.17-34[2] George D. Gounaris, Chris A. Papadopoulos, Crack Identification in Rotating Shafts by Coupled Response Measurements, Engineering Fracture Mechanics, 69, 2002, pp.339-352[3] G.M. Owolabi, A.S.J. Swamidas, R. Seshadri, Crack Detection in Beams using Changes in Frequencies and Amplitudes of Frequency Response Functions, Journal of Sound and Vibration, 265, 2003, pp.1-22[4] Shuncong Zhong, S. Olutunde Oyadiji, Analytical Predictions of Natural Frequencies of Cracked Simply Supported Beams with a Stationary Roving Mass, Journal of Sound and Vibration, 311 ,2008, pp.328-352[5] Jiawei Xiang, Yongteng Zhong, Xuefeng Chen, Zhengjia He, Crack Detection in a Shaft by Combination of Wavelet-Based Elements and Genetic Algorithm, International Journal of Solids and Structures, 45, 2008, pp.4782-4795[6] Marta B. Rosales , Carlos P. Filipich, Fernando S. Buezas, Crack Detection in Beam-Like Structures, Engineering Structures, 31 ,2009, pp.2257-2264[7] R. K. C. Chan and T. C. Lai, Digital Simulation Transverse Crack, Appl. Math. Modelling, 19, 1995, pp.411-420[8] Menderes Kalkat, Sahin Yildirim, Ibrahim Uzmay, Design of Artificial Neural Networks for Rotor Dynamics Analysis of Rotating Machine Systems, Mechatronics, 15 ,2005, pp.573-588[9] Weixiang Sun, Jin Chen, Jiaqing Li, Decision Tree and PCA-Based Fault Diagnosis of Rotating Machinery, Mechanical Systems and Signal Processing, 21 ,2007, 1300-1317[10] A.K.Darpe, K.Gupta, A.Chawla, Dynamics of a Bowed Rotor with a Transverse Surface Crack, Journal of Sound and Vibration, 296, 2006, pp.888-907[11] Ashish K. Darpe, A Novel Way to Detect Transverse Surface Crack in a Rotating Shaft, Journal of Sound and Vibration, 305 ,2007, pp.151-171 116
    • International Journal of Advanced Research in Engineering and Technology (IJARET)ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME[12] K.M.Saridakis, A.C.Chasalevris, C.A. Papadopoulos, A.J. Dentsoras, Applying Neural Networks, Genetic Algorithms and Fuzzy Logic for the Identification of Cracks in Shafts by using Coupled Response Measurements, Computers and Structures, 86 ,2008, pp.1318-1338[13] Robert Gasch, Dynamic Behaviour of the Laval Rotor with a Transverse Crack, Mechanical Systems and Signal Processing, 22 ,2008, pp.790-804[14] N. Dharmaraju, R. Tiwari, S. Talukdar,(2004), Identification of an Open Crack Model in a Beam Based on Force–Response Measurements, Computers and Structures 82, pp.167-179 117