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VIBRATION ANALYSIS OF A CRACKED SHAFT USING FEM A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Bachelor of Technology in Mechanical Engineering By RAJ KUMAR KISPOTTA Under the guidance of Dr. Rabindra Kumar Behera Department of Mechanical Engineering National Institute of Technology Rourkela 2009 VIBRATION ANALYSIS OF A CRACKED SHAFT USING FEM A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Bachelor of Technology in Mechanical Engineering By RAJ KUMAR KISPOTTA Department of Mechanical Engineering National Institute of Technology, Rourkela Abstract A strong interest has developed within the past several years in the dynamic behavior of turbo machinery with cracked shafts. An excellent review of the field of the dynamics of cracked rotors and of different detection procedures to diagnose fracture damage has been presented by Wauer . Vibration investigation of a damaged structure is one approach for fault diagnosis. Vibration diagnosis, as a non-destructive detection technique, has recently become of greater importance. A crack on a beam element introduces considerable local flexibility due to the strain energy concentration in the vicinity of the crack tip under load. The vibrational characteristics of a cracked Timoshenko shaft were investigated by Rajab and Sabeeh . They presented analytical expressions and derived curves relating the crack depth and location on the shaft to changes in the first few natural frequencies of the shaft. The element stiffness matrix of a beam with a crack was derived froman integration of the stress intensity factors and then a finite element model (FEM) of a cracked beam was established in reference . A similar approach based on the flexibility matrix developed by Papado-poulos and Dimarogonas is dealt with by using FEM in the present study for crack detection. National Institute of Technology Rourkela CERTIFICATE This is to certify that the thesis entitled “VIBRATION ANALYSIS OF A CRACKED SHAFT USING FEM” submitted by Sri Raj Kumar Kispotta in partial fulfillment of the requirements for the award of Bachelor of technology Degree in Mechanical Engineering at the National Institute of Technology, Rourkela (DeemedUniversity) is an authentic work carried out by him under my supervision and guidance. To the best of my knowledge, the matter embodied in the thesis has not been submitted to any other University / Institute for the award of any Degree or Diploma. Date: Dr. Rabindra Kumar Behera National Institute of Technology Rourkela-769008 ACKNOWLEDGEMENT We deem it a privilege to have been the student of Mechanical Engineering stream in National Institute of Technology, ROURKELA . Our heartfelt thanks to Dr.R K Behera , our project guide who helped us to bring out this project in good manner with his precious suggestion and rich experience. We take this opportunity to express our sincere thanks to our project guide for co-operation in accomplishing this project a satisfactory conclusion. Raj Kumar Kispotta Roll No. 10403059 Department of Mechanical Engg National Institute of technology CONTENTS SL.NO. Title Page.no. 0 Abstract 0 1 Introduction 1 2 Motions Equations of a cracked shaft 3 3 Crack Modeling 8 4 Crack Identification and Estimation method 9 4.1 Forward Problem 9 4.2 Vibration Analysis 12 4.2.1 Effect of the normalized crack location, β 12 4.2.2 Effect of the normalized crack size , α 14 4.3 Inversion Problem 15 5 Experimental Verification 16 5.1 Experimental results 19 6 Discussion and further researches 24 7 Conclusions 27 8 References 28 LIST OF FIGURES SL NO. TOPIC PAGE 1 Model of a rotor element with a transverse edge crack. 4 2 The shaft with a transverse surface crack. 8 3 Simply supported rotor system with a crack in shaft 10 4 Geometry of a cracked section in shaft. 10 5 The first three natural frequency as a function of normalized crack 13 location β for some of the normalized crack size α . 6 The first three natural frequencies as a function of normalized crack 14 size α for same of the normalized crack location. 7 Experimental setup 16 8 Frequency contour plots 23 9 Typical rotor-bearing system considered 24 LIST OF TABLES SL NO. 1 TOPIC The experimental and differently elemental results of the first three PAGE 18 natural frequencies. 2 Crack cases of shaft and identification results. 20 3 The first three natural frequencies under different bearing stiffness 25 1.INRODUCTION A method for the detection of cracks in Timoshenko shafts based in the natural frequencies is described in this on measuring changes paper . Cracks may be initiated and subsequently propagated in shaft sand structures subjected to dynamic loadings. Failure may result if the history of these cracks is not recorded and precautionary measures are not taken. At the early stages of crack growth, it is difficult to detect the existence of the crack by Visual inspection. Other more detailed techniques of non-destructive testing need to be used instead. An ultrasonic pulse technique has been used successfully to detect the positions of cracks in structures and welds . In some materials, this technique may not work due to the large attenuation of the signal at all except a particular frequency. Radiographic techniques have also been used for crack detection in structures . These Techniques , however , require higher radiation energy input for Increasing material thickness, which increases the cost of operation. In addition, crack detectability is small for a small crack Analytical width/depth ratio and for cracks not parallel to the material surface. expressions for the flexibility effect of a crack were first derived by Irwin and later used in structural analysis applications by Liebowitz .Thevibrational characteristics of a rotor containing a crack were s tudied by Henry and Okah-Avae , Mayes and Davies, Gasch , and Dimarogonas . Adams et al. used axial vibration analysis in one-dimensional structures to predict the location and the magnitude o f a defect. The effect of the defect was represented as an elastic stiffness which is determined from a knowledge of the changes in two natural frequencies of the structure. The method presented would not detect longitudinal cracks . A finite element analysis was used by Cawley and Adams to predict damage in structures. Chondros and Dimarogonas studied the effect of a crack in a welded joint on the dynamic behavior of beams with various boundary conditions. Crack depth was estimated from knowledge of changes in the system natural frequencies. A torsional elastic stiffness was used to model the effect of the crack. The torsional stiffness was determined experi- mentally from measurements of changes in the natural frequencies for various crack depth values. Dimarogonas and Massouros considered the effectof a crack on the torsional dynamic behavior of a shaft. They used the strain energy release rate to obtain an expression for local stiffness effect due to a circumferential crack. Anifantis et al. presented a nomogram for identification of a c rack on a simple beam . Ju et al. proposed a damage function relating the changes in the natural frequencies of a structureto the crack depth and location. Dimarogonas and Papadopoulos studied the dynamic behavior of a rotating shaft with a surface crack. Yuen Chen and Chen and Gounaris and Dimarogonas used finite element analysis to study the vibrational characteristics of crackedstructures. Papadopoulos and Dimarogonas considered the free and forced vibrations of coupled bending and torsional vibrations of a damaged shaft. The present work is concerned with developing analytical expressions for the compli-ance effects of a crack in a Timoshenko-type shaft for the purpose of predicting both the depth and location of the crack on the shaft. A part-through surface crack is assumed in the analysis, and changes in the shaft natural frequencies are used in the prediction of crack parameters. 2. MOTIONS EQUATIONS OF A CRACKED SHAFT The FEM model of a cracked rotor which was derived in the author’s previous work is used in present study also. A brief description of the model and equation of motion are given below. A cracked shaft loaded by three tension forces and three torsion forces is shown in Fig. 1. The equation of motion of the cracked rotor system is given by where M, C, G, K0 and Kc are the mass matrix, bearing oil-.lm damping matrix, gyroscopic matrix, whole stiffness matrix without crack and the whole stiffness decrease caused by crack, respectively . H is the transformation matrix between rotating co-ordinates Z–z and inertial co-ordinates y–z, which has where a is the angle between the two co-ordinates. F is the excitation matrix including the unbalance forces and gravity forces. The stiffness matrix of cracked where element [K]crack can be written as The .exibility matrix of cracked element can be expressed as where [c](0) is the .exibility matrix of element without crack, [c](1) is the additional .exibility matrix due to crack, which has The expressions of cij can be found in Ref. [15]. The switch function of cracks used here is same as that in Ref. [15]: where me is the mass unbalance. For the state of a crack propagating during the startup process , the stiffness of cracked shaft varies as a function of time and propagation rate. Because crack propagating is a speci.c complex process, following two assumptions are adopted in present study: (1) The crack tip growth occurs at the beginning of a cyclic. (2) The crack tip line after growth parallels to the initial crack tip line. The dynamic response can be obtained using numerical integration method for non-linear system— Newmark method [22]. The computational algorithm is as follows: According to the implicit integration formula of the Newmark integral method , we have (1) Set i =0 (2) Calculate the residual force (3) Calculate the effective stiffness matrix . (4) Solve the following equation: (5) Revise displacement, velocity, acceleration and i: and then reduplicate the above process until 3. CRACK MODELING The presence of a crack in a structure tends to modify the dynamic characteristics of the structure, such as the natural frequencies and mode shapes. This fact can be used inversely to predict the crack parameters from measurements of the changes in the natural frequencies and mode shapes of the structure once a functional relationship between the crack parameters and the changes in the structure dynamic characteristics has been determined . The crack parameters of interest are the crack depth and crack location in the structure. To this end, an analytical solution for the functional relationship between the system natural frequencies and the crack parameters is attempted for a Timoshenkotype shaft with a transverse part-through surface crack . For a shaft with a transverse surface crack and loaded with bending moment and shear force as shown in Figure 1, the displacement in the i direction is given by where Pi is the load in the same direction as the displacement and J(a) is the J-integral Figure 2 . The shaft with a transverse surface crack. 3. CRACK IDENITIFICATION AND ESTIMATION METHOD 3.1. Forward problem Friswell and Penny compared the different approaches to crack modeling, and demonstrate that for structural health monitoring using low-frequency vibration, simple models based on beam elements are able to model the effects of an open crack. Because the natural frequencies can be easily and cheaply acquired in practice and the linear rotational spring model can effectively describe open crack, the present work is based on the open transverse crack in shafts and using the first three natural frequencies to identify crack location and size. Fig. 3 shows a simply supported rotor system with a crack in shaft (suppose the crack occurred on segment L2). A transverse crack of depth δ is considered on a shaft of diameter d1 (the corresponding radius is r1) as shown in Fig. 4. The shaft has local flexibility due to the crack, in many directions, depending on the direction of the applied forces. Here, only bending deformation will be considered. Axial forces which give coupling with transverse motions of the cracked shaft will not be considered. Therefore, the shaft is bent by a pure bending moment and the additional angular deflection of the shaft end relative to the other will be computed. The local stiffness kt due to the crack is [7] …………………..(34) where E is the Young's modulus, μ is the Poisson's ratio, α=δ/2r1 denotes normalized crack size, , and the function F(η/H) can be given by the experimental formula [37]. F(η/H)=1.122-1.40(η/H)+7.33(η/H)2-13.08(η/H)3+14.0(η/H)4................................ (35) Eq. (34) is a function of normalized crack size α only and can be computed by numerical integration. β=e/L2 denotes normalized crack location. Fig. 3. Simply supported rotor system with a crack in shaft. Fig. 4. Geometry of a cracked section in shaft. The physical model and rotational spring model with stiffness kt , are shown in Fig. 5. The corresponding DOF of the left neighboring element around crack are and the corresponding DOF of the right neighboring element are wright={wj+1θj+1 }T. If we change the two entries of row and column entries of the global stiffness matrix to wleft={ θjwj}T, the corresponding and mss matrix should be exchanged their locations each other. The DOF of wleft and wright can be assembled to { θjwj+1θj+1,…}T. Hence, we can assemble cracked stiffness submatrix KS into the global stiffness matrix easily. The global mass matrix of cracked rotor system is equal to the uncracked one. From now on, the cracked rotor system finite element model is constructed by using BSWI beam element. The solution of the eigenvalue problem can then proceed as usual. For the determination of the natural frequencies ω for a given crack location (determine the location of cracked stiffness submatrix in global stiffness) and size (determine Kt), the normalized crack location β and size α are given as input. The relationship between the natural frequencies and the crack parameters is …………………………(36)Because the functions Fj(j=1,2,3,…) are unknown and the discrete values can be obtained by solving WFEM model, the surface-fitting techniques are needed for the 3D plots of Eq. (36). Then the crack identification database of forward problem for cracked rotor system has been built up. 3.2. Vibration analysis In the simulation, the rotor geometries and the material properties are as follows: , , , , , , , , μ=0.3. 13 BSWI43 Rayleigh–Euler beam and 1 BSWI43 Rayleigh–Timoshenko beam elements are employed to the vibration analysis of rotor system. Fig. 6 shows the relationship between ωi(i=1,2,3) and all possible normalized crack location β=e/L2 and size α=δ/d1 using surfacefitting techniques (here, α,β [0.1,0.9]). 3.2.1. Effects of the normalized crack location β Fig. 7 shows the first three natural frequencies as a function of normalized crack location β for some of the normalized crack size α. It can be seen from Fig. 7(a)–(c) that the change in the first three natural frequencies were affected when the crack was located at every normalized location β. For a certain normalized crack size α, Fig. 7(a) show the fundamental frequency would be decreased significantly with respect to the larger normalized crack location β. While a correlation between normalized crack location and size was given, as shown in Fig. 7(b) and (c). Taking α=0.8, for example, the second natural frequency was mostly affected when the crack was located at the center of the shaft segment L2, and the third natural frequency was mostly affected when the crack was located at β=0.25. Moreover, Fig. 7(c) shows that the third natural frequency was almost unaffected for a crack located at β=0.75; the reason for this zero influence was that the nodal point for the third mode was located at that position. Fig. 5. The first three natural frequencies as a function of normalized crack location β for some of the normalized crack size α. Unlike to the effects of the normalized crack location of simply supported shaft without mass disc, as mentioned by Dong, the changes in the first three natural frequencies of a cracked rotor system with a mass disc, would not monotonically decrease with the increment of the normalized crack location. The reason is that only the crack of shaft segment L2 is considered in the present study, and this is not affecting the crack identification results. 3.2.2. Effects of the normalized crack size α The first three natural frequencies as a function of normalized crack size α for some of the normalized crack location β are shown in Fig. 8. It is noticed that the change in the first three natural frequencies were affected when the crack was occurred at every normalized size α. For a certain normalized crack location β, Fig. 8(a)–(c) show the first three frequencies would monotonically decrease if the larger normalized crack size α was given. Fig. 6. The first three natural frequencies as a function of normalized crack size α for some of the normalized crack location From the above observations, it could be stated that the change in frequencies is not only a function of the normalized crack location β but also the normalized crack size α. In the procedures of constructing crack identification database of forward problem for cracked rotor system, the influence of the normalized crack size α and location β was considered through the computation of rotational spring stiffness kt and the location of inserting cracked stiffness submatrix into global stiffness, respectively. 3.3. Inverse problem The crack identification inverse problem can be described by ………………(37) The measurements of any two natural frequencies enable us to define the normalized location and size of a crack if Eq. (36) has been determined, i.e. the crack identification database of forward problem for cracked rotor system has been constructed. However, when we use the method of frequencies contour plots for crack identification of a rotor system, two natural frequencies contour plots may intersect at more than one point. Therefore a minimum of three frequencies is required to identify the two unknown parameters, i.e. the normalized crack location β and size α. Because the first three frequencies can be measured easily and accurately; they are usually served as inputs to solve the inverse problem in structural damage identification. Supposing the first three frequencies are known, the frequency contour plots of the crack identification database of forward problem for cracked rotor system, can be acquired and plotted on the same axes. The common intersection of all the three contour lines indicates the normalized crack location and size. This intersection becomes unique due to the fact that any cracked structural natural frequency can be represented by a frequency equation ( Eq. (36)) that is dependent on normalized crack parameters [20]. 4. EXPERIMENTAL VARIFICATION 4.1. Experimental setup Fig. 9 shows the experimental setup used for measuring the first three frequencies of the cracked rotor system with a single mass disc using the Doppler signal laser vibrometer. A Polytec Doppler laser vibrometer OFV-505/5000 was used to measure the velocities of one point in the shaft. We point out here that to measure the first three frequencies only requires one measurement point. The reason is that for the simple structure, single-input and single-output (SISO) modal analysis by using a hammer as excitation, which is a usually used method. To reduce the reflection of the laser beam and spectral noise, retro-reflective tapes were put on the measurement point in the shaft. Fig. 7. The experimental setup The laser vibrometer OFV-5000 uses the principle of the heterodyne interferometer to acquire the characteristics of mechanical vibrations. For each cracked shaft, the high-metrical frequencies can be obtained by using the standard FFT program of the software Matlab. Taking the uncracked rotor system with the same geometries and the material properties, as given by Section 3.2, and the corresponding impulse response signal and power spectrum are shown in Fig. 10. In the experimental study, the sampling frequency fs is 5000. Different elements are employed to estimate the first three frequencies, respectively, i.e.: 1. 5 BSWI43 beam elements (not consider rotatory inertia effect); 2. 200 traditional beam elements (not consider rotatory inertia effect); 3. 5 BSWI43 Rayleigh beam elements (consider rotatory inertia effect); 4. 200 BEAM3 (Software Ansys) elements (consider rotatory inertia effect). The results are shown in Table 1. Compared with the experimental frequencies, both 5 BSWI43 beam element (not consider rotatory inertia effect) and 200 traditional beam elements (not consider rotatory inertia effect) have the same computational precision. The relative errors of the second frequencies arrive at 61.5%, which can not be used to the model-based crack identification method. However, if we consider the rotatory inertia effect, both 5 BSWI43 Rayleigh beam elements (consider rotatory inertia effect) and 200 BEAM3 (Software Ansys) elements (consider rotatory inertia effect) can achieve high-computational precision. And the relative errors of the first three frequencies are less than 0.3%. Moreover, the elemental DOF of 5 BSWI43 Rayleigh beam elements (consider rotatory inertia effect) are 58, which are smaller than 402 DOF of BEAM3 elements. The results of this example indicate the accuracy and efficiency of the BSWI Rayleigh beam element, which has been constructed herein. Meanwhile, the BSWI Rayleigh beam element can be employed to build up the crack identification database for cracked rotor system. Table 1. The experimental and differently elemental results of the first three natural frequencies Computational Method frequencies/Hz f1 f2 f3 1 86.33 900.95 2 86.34 901.00 Metrical Computational frequencies/Hz errors/% f1 f2 f3 f1 f2 f3 1732.07 0.26 61.51 0.26 1732.25 0.27 61.52 0.27 86.11 557.82 1727.54 3 86.33 558.17 1728.25 0.26 0.06 0.04 4 86.33 558.17 1728.20 0.26 0.06 0.04 4.2. Experimental results We tested six cracked shafts each having an open crack at shaft with four crack cases as shown in Table 2. The material of workpiece for experiment was 40Cr steel, and the rotor geometries and the material properties are as follows: , density , , , , , Young's modulus E=2.06×1011N/m2, material , Poisson's ratio μ=0.3. The cracks actually are slots cut by a wire electrical discharge machine, the crack width is 0.2 mm, and the crack depths δ are 3.8 and 7.2 mm. In other words, the normalized crack sizes α=δ/d1 are 0.2 and 0.4. The crack locations e are 75.2, 112.8 and 150.4 mm, respectively, and the corresponding normalized crack locations β=e/L2 are 0.4, 0.6 and 0.8, respectively. Table 2. Crack cases of shaft and identification results Metrical frequencies/Hz Case β α Predicted β* (Error/%) Predicted α* (Error%) f1/Hz f2/Hz f3/Hz 1 0.4 0.2 98.2 582.04 1049.88 0.415 (3.75) 0.196 (2) 2 0.4 0.4 96.62 568.43 1016.25 0.41 (2.5) 0.42 (5) 3 0.6 0.2 98.12 583.13 1055.80 0.585 (2.5) 0.17 (14) 4 0.6 0.4 95.76 572.14 1053.12 0.605 (0.83) 0.385 (3.75) 5 0.8 0.2 97.34 583.84 1051.64 0.785 (1.88) 0.225 (12.5) 6 0.8 0.4 93.04 580.87 1032.62 0.794 (0.75) 0.425 (6.25) In most cases, however, the three lines do not accurately intersect at one point because of inaccuracies in the modeling as compared to measured results. For this purpose, the ‘zerosetting’ procedure that described by Adams [38] is used. In this procedure, Young's modulus of the structure is changed by using the undamaged natural frequencies of the structure to determine an effective value, and given by the following iterative approach: …………………………………(38) where Em is the corrected value of Young's modulus E, which can be acquired through solving Eq. (38) for each frequency. It should be noted that the physical signification for the correction of Young's modulus is not to change the value of E but to make the metrically uncracked natural frequencies match the computational ones. This procedure can greatly reduce the error between theoretical analysis and the experimental studies, which are caused by boundary conditions and material parameters. In this section, the first three metrical frequencies are employed as the inputs of inverse problem for crack quantitative identification. Fig. 11 shows the crack identification results in a shaft by using the frequency contour plots. The intersection point A of three lines indicates the normalized crack location β and size α. In the experimental studies, when the three lines do not meet exactly, the centroid of the three pairs of intersections is taken as the normalized crack location and size [13]. Table 2 shows the comparison of actual normalized crack parameters β and α and the predicted crack parameters β* and α*. For the given cases, the relative errors of β* are not more than 4% while the relative errors of α* arrive at 15%. Hence, the proposed modelbased crack identification method by using BSWI Rayleigh beam element is considered to be valid for actual application to detect open cracks in rotor systems. We point out here that the identification precisions of the crack sizes are much larger than those of the crack location. The probable reason is that the effects of the normalized crack size α are much complicated than those of the normalized crack location β. Therefore, for the identification of α, the tiny metrical errors would be significantly enlarged. Fig. 8. Frequency contour plots (- - -: 1st frequency; -.-.-: 2nd frequency; - - -: 3rd frequency): (a) case 1; (b) case 2; (c) case 3; (d) case 4; (e) case 5 and (f) case 6. 5. DISCUSSION AND FURTHER RESEARCHES Except the open crack, there are many other types of open and closed cracks and defects in rotor systems, and different types of defects have different controller equations and because damage estimation is essentially an inverse problem, it is difficult to obtain a unique or accurate estimation of cracks, especially for on-line crack identification of rotor systems. On-line structural crack detection is challenging because actual rotor system dynamic responses are always contaminated by unknown external loads as well as other harmless conditions, such as material composition uncertainly, geometric variation, sensor noise, and so on. Therefore, after the possible existence of damage is detected by an on-line damage detection method as mentioned by Green, a reliable inspection of the stopped system to find crack location and size is necessary in order to assure the existence of damage, and the model-based crack identification method by using BSWI Rayleigh beam element is intended for such use. The effect of the bearing on the fault diagnosis of the rotor system is very obvious for the rotating shafts. In this section, the bearing stiffness is considered, as shown in Fig. 12. The stiffness matrix considers the stiffness of the shaft elements including cracked shaft element and the bearing stiffness kb. Fig. 9. Typical rotor-bearing system considered. The vibration analysis has been carried out using BSWI Rayleigh beam elements for the rotor system with the same geometrical and physical parameters as shown in Section 4.2. The bearing stiffness kb is varying from 1×105 to , supposing the normalized crack location β=0.4 and crack depth α=0.4. The first three natural frequencies under different bearing stiffness are shown in Table 3. The first three natural frequencies influenced by bearing stiffness are shown in Fig. 13. To describe clearly, we adopt the logarithm coordinates. From Table 3 and Fig. 13, the influence of bearing stiffness can be neglected when kb>1×108. Both simply supported rotor system as shown in Fig. 3 and the bearing stiffness is considered as shown in Fig. 12 have the same computational results. Table 3. The first three natural frequencies under different bearing stiffness Simply Bearing stiffness kb (N/m) Frequencies supported 1×105 5×105 1×106 1×107 1×108 1×109 5×109 84.77 90.31 96.25 96.90 96.96 96.97 567.83 569.002 569.11 f1/Hz 96.97 60.25 f2/Hz 569.13 225.03 398.18 464.15 556.38 f3/Hz 1019.25 452.90 693.05 827.37 1001.13 1017.48 1019.08 1019.22 In the present experimental studies of this paper, only static (non-rotating) rotor system is studied, the bearing stiffness nearly to . Hence the simply supported model of rotor system can give reasonable results. The bearing damping and other factors are not considered. It obviously introduces tiny errors of the wavelet-based finite element model. So the ‘zero-setting’ procedure is introduced to eliminate those influences. However, it is worth to point out here that the present method is only a reliable inspection of the stopped system to find crack location and size for the sake that the bearing effect is not considered. The further research is that a more complicated rotating rotor-bearing model which includes bearing stiffness, bearing damping, gyroscopic effect, etc., will be studied for on-line crack identification use. 6. CONCLUSIONS The detection of cracks in shafts by measuring the change s in an adequate number of The natural frequencies has been considered in this paper. A crack is known to introduce local flexibility in the shaft. The local flexibility due to a crack in the presence of bending moment and shear loads is modeled by using fracture mechanics concepts . The natural frequencies of the cracked shaft are determined numerically by solving the characteristic equation of the shaft . Cracks are then predicted by measuring an adequate number of shaft natural frequencies and constructing curves similar to Figure 6. The adequate number of natural frequencies that needs to be measured depends on the number of cracks present. The first three natural frequencies need to be measured for a single crack, and each additional crack requires the measurement of two more higher natural frequencies. This may limit the application of this method to shafts with not more than three or four cracks and with no closely coupled modes. The technique presented is most suited as a preventive maintenance tool from the initial installation of the shaft through a programmed mainten- ance schedule to record the history of the cracks as they grow, at appropriate time intervals. 7. REFERENCES (1) M. D. RAJAB and A. AL SABEEH 1991 Journal of Sound and Vibration 147,465473. Vibration characteristics of cracked shafts. (2).G. R. IRWIN 1957 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 24, 361-364. Analysis of stresses and strains near the end of a crack (3)H. LIEBOWITZ, H. VANDERVELT, and D. W. HARRIS 1967 International of Journal of Solids and Structures 3, 489-500. Carrying capacity of a notched column. (4) I. W. MAYES and W. G. DAVIES 1976 Institution of Mechanical Engineers Conference Publication, Vibrations in Rotating Machinery, Paper No. C168/76. The vibrational behavior of a rotating shaft system containing a transverse crack. (5)G.-L. QIAN, S.-N. Gu and J.-S. JIANG 1990 Journal of Sound and Vibration 138, 233243. The dynamic behaviour and crack detection of a beam with a crack. (6)S. Loutridis, E. Douka and A. Trochidis, Crack identification in double-cracked beams using wavelet analysis, J. Sound Vibr. 277 (2004), pp. 1025–1039. (7)M. Kisa, J. Brandon and M. Topcu, Free vibration analysis of cracked beams by a combination of finite elements and component mode synthesis methods, Comput. Struct. 67 (1998), pp. 215–223.

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