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Parameter identification of a rotor with an open crack
European Journal of Mechanics A/Solids 23 (2004) 325–333
Parameter identification of a rotor with an open crack G.M. Dong ∗ , J. Chen, J. Zou The State Key Laboratory of Vibration, Shock and Noise, Shanghai Jiao Tong University, Shanghai 200030, PR China Received 22 May 2003; accepted 14 November 2003
Abstract A continuous model is presented for vibration analysis and parameter identification of a static (non-rotating) rotor with an open crack, which is based on two assumptions that the cracked rotor is an Euler–Bernoulli beam with circular cross-section and that the cracked region is modeled as a local flexibility with fracture mechanics methods. Through numerical analysis, the effects of the location and depth of the crack on the changes in the eigenfrequencies and mode shapes of the cracked rotor are investigated, and the ratios of the changes in the first two eigenfrequencies are discussed for rotors with shallow cracks. A crack identification algorithm, which makes use of the deflections of the first mode shape at two symmetric points and the contour diagram of crack location versus crack depth for the first two normalized eigenfrequencies, is proposed to predict the crack location and depth on the rotor, and two illustrative examples demonstrate the validity and availability of the proposed algorithm. 2003 Elsevier SAS. All rights reserved. Keywords: Parameter identification; Continuous model; Vibration analysis; Cracked rotor; Open crack; Crack location; Crack depth; Eigenfrequency; Mode shape
1. Introduction One form of damage that can lead to catastrophic failure if undetected is fatigue cracking of the shaft, so early detection of cracked rotor in engineering practices is of significant importance to the reliability and durability of large rotating machinery, which makes fault diagnosis and condition monitoring of cracked rotor paid more and more attention (Dimarogonas, 1996; Dimarogonas and Papadopoulos, 1983; Gasch, 1993; Pu et al., 2002a, 2002b; Wauer, 1990; Zou et al., 2002, 2003a, 2003b). For the time being, the research on the cracked rotor is still at the theoretical stage, and most of the previous research is only involved in detecting the crack in a rotor and not capable of determining the crack location and depth. A static (non-rotating) rotor with an open crack can be considered as a simply supported beam, as a consequence, the research in relation to non-rotating structures such as beams and columns is useful for locating and estimating the severity of the crack on a rotor (Armon et al., 1994; Liang et al., 1992a, 1992b; Ostachowicz and Krawczuk, 1991; Qian et al., 1990; Shen and Pierre, 1990). A crack in a structure introduces a local flexibility, which affects the dynamic behavior of the whole structure to a considerable degree and results in reduction of natural frequencies and changes in mode shapes of vibration. In the paper of Chondros and Dimarogonas (1980), the effect of a crack on the deformation of a beam has been considered similar to that of an elastic hinge. Dimarogonas and Papadopoulos (1983) modeled the crack as a local flexibility and computed the equivalent stiffness using fracture mechanics methods. By perturbation theory and transfer matrix analysis, Gudmundson (1982, 1983) calculated the variation of eigenfrequencies due to the crack on a cantilever beam. Following Gudmundson (1982, 1983), Gounaris and Dimarogonas (1988) proposed a finite element model using transfer matrix method and local flexibility theorem; * Corresponding author.
E-mail address: [email protected] (G.M. Dong). 0997-7538/$ – see front matter 2003 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2003.11.003
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in order to consider the discontinuity deformation due to the crack on the beam, they adopted two different shape functions for two segments separated by the crack. The dynamic response of the cracked structures and the changes in the natural frequencies and mode shapes between the cracked and the uncracked structures contain information about the location and dimension of the damage. Using the receptance technique and Taylor series expansion, Adams et al. (1978), Cawley and Adams (1979) have shown that the ratio of the frequency changes in two modes is only a function of the damage location respectively. Rizos et al. (1990) estimated the crack location and depth with satisfactory accuracy from the result of an analytical solution of a cracked beam, and from experimental data of the measured amplitudes at two points of the structure vibrating at one of its natural modes and the respective vibration frequency. For a rotor removed from service, Inagaki et al. (1982) used natural vibration and static deflection analysis to find the crack size and location. Hu and Liang (1993) proposed a two-step procedure to identify cracks in beam structures: they used the effective stress concept coupled with Hamilton’s principle to derive a formulation relating the changes in the natural frequencies to the member stiffness changes. The elements that contain cracks could be identified based on the formulation and then a spring damage model was used to quantify the location and depth of the crack in each damaged element. On the basis of the research of Gudmundson (1982, 1983) and Gounaris and Dimarogonas (1988), Nikolakopoulos et al. (1997) presented the dependency of the structural eigenfrequencies on crack depth and location in contour graph form: to identify the location and depth of a crack, the intersection points of the superposed contours that correspond to the measured eigenfrequency variations caused by the presence of the crack should be determined. With the model from Nikolakopoulos et al. (1997), Suh et al. (2000) presented a detection method that uses hybrid neuro-genetic technique to identify the location and depth of a crack on a structure. In this paper, a continuous model, which is based on that the dimension of the cracked rotor satisfies the Euler–Bernoulli theory and that the cracked region is modeled as a local flexibility, is presented for vibration analysis and crack identification of a static (non-rotating) rotor with an open crack. Through numerical analysis, the effects of the location and depth of the crack on the changes in the eigenfrequencies and mode shapes of the cracked rotor are investigated, and the ratios of the changes in the first two eigenfrequencies are discussed for rotors with shallow cracks. Then a crack identification algorithm, which makes use of the deflections of the first mode shape at two symmetric points and the contour diagram of crack location versus crack depth for the first two normalized eigenfrequencies, is proposed to predict the crack location and depth on the rotor. At last, two illustrative examples are demonstrated to assess the validity of the proposed algorithm.
2. Continuous model of rotor with an open crack The continuous model of the problem is shown in Fig. 1, which includes the schematic diagram of a non-rotating rotor with a single open crack of depth ad and location Z1 . The cross-sectional radius of the rotor is R and the length of the rotor is L. The vibration frequency equation of the rotor is written in the well-known form ∂ 4 y(x, t) ∂ 2 y(x, t) + ρA = 0, (1) ∂4x ∂2t where E is Young’s modulus, ρ is material density, I is the geometrical moment of inertia of the cross-section, A is crosssectional area of the rotor and x is the coordinate along the rotor. For vibration in natural modes, the deflection of the rotor is EI
y(x, t) = Y (x) sin(ωt).
(2)
Substituting Eq. (2) into Eq. (1), after simple algebraic transformation, one has Y iv (β) − λ4 Y (β) = 0,
(3)
where β = x/L, λ4 = ρAω2 L4 /(EI ).
Fig. 1. Rotor with a single open crack simply supported on both ends.
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The general solution of Eq. (3) is given by Y1 (β) = A1 cosh(λβ) + A2 sinh(λβ) + A3 cos(λβ) + A4 sin(λβ),
β ∈ [0, β1 ),
(4a)
Y2 (β) = B1 cosh(λβ) + B2 sinh(λβ) + B3 cos(λβ) + B4 sin(λβ),
β ∈ (β1 , 1],
(4b)
where β1 = Z1 /L, Z1 is crack location of the rotor, Ai and Bi (i = 1, 2, 3, 4) are constants to be determined from the boundary conditions. The boundary conditions for the cracked rotor under consideration are Y1 (β)|β=0 = 0,
Y1 (β)|β=0 = 0,
Y2 (β)|β=1 = 0,
Y2 (β)|β=1 = 0.
(5)
The continuity at the crack position is as follows Y1 (β)|β=β1 = Y2 (β)|β=β1 ,
Y1 (β)|β=β1 = Y2 (β)|β=β1 ,
Y1 (β)|β=β1 = Y2 (β)|β=β1 .
(6)
The compatibility condition due to the local flexibility is Y2 (β)|β=β1 − Y1 (β)|β=β1 =
cEI Y (β)|β=β1 L 2
(7)
where c is the local flexibility due to the crack on a rotor with a circular cross-section shown in Fig. 2. The dimensionless local flexibility c¯ is calculated with fracture mechanics method proposed by Dimarogonas and Papadopoulos (1983) and c¯ varying with the relative crack depth a¯ d = ad /R is shown in Fig. 3. Combining Eqs. (4)–(7), the characteristic equation is derived
1 1 cosh(λβ1 ) cosh(λβ ) 1 det sinh(λβ1 ) sinh(λβ ) 1 0
0
1
0
0
0
−1
0
0
sinh(λβ1 )
cos(λβ1 )
sin(λβ1 )
− cosh(λβ1 )
sinh(λβ1 )
− cos(λβ1 )
− sin(λβ1 )
− cosh(λβ1 )
cosh(λβ1 )
sin(λβ1 )
− cos(λβ1 )
cosh(λβ1 )
− sin(λβ1 )
cos(λβ1 )
− sinh(λβ1 ) cEI λ cosh(λβ ) − sinh(λβ ) 1 1 L
0
0
0
cosh λ
0
0
0
cosh λ
0 0
0
0
0
− sinh(λβ1 )
− cos(λβ1 )
− sinh(λβ1 )
cos(λβ1 )
− cosh(λβ1 )
− sin(λβ1 )
cEI λ sinh(λβ ) − cosh(λβ ) 1 1 L
− cEI L λ cos(λβ1 ) + sin(λβ1 )
sinh λ
cos λ
sinh λ
− cos λ
0
− sin(λβ1 ) sin(λβ1 ) cos(λβ1 ) cEI − L λ sin(λβ1 ) − cos(λβ1 ) sin λ 0
− sin λ
= 0.
(8)
8×8
From Eq. (8), the characteristic roots λi are determined and used for the calculation of the eigenfrequencies ωi =
λi 2 EI . L ρA
(9)
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Fig. 2. Geometry of a cracked section of a shaft.
Fig. 3. Dimensionless local flexibility.
3. Vibration analysis In the simulation, the rotor geometries and the material properties are as follows: L = 3 m, E = 2.07 × 1011 N/m2 , ρ = 7.7 × 103 kg/m3 and Poisson ratio v = 0.3. At the same time, the rotor geometries and the crack parameters are expressed in dimensionless forms for convenience. 3.1. Effects of the relative crack depth a¯ d = ad /R For a rotor with slenderness ratio L/D = 8 (D = 2R is the diameter of the rotor) and crack location Z1 /L = 0.45, a parametric study of the effect of the crack depth on the vibration characteristics of the rotor is carried out by varying the relative crack depth a¯ d = ad /R. The variations of the first three normalized eigenfrequencies ωci /ωni (i = 1, 2, 3) with the relative crack depth are shown in Fig. 4, where ωc1 , ωc2 , ωc3 are the first three eigenfrequencies of cracked rotor and ωn1 , ωn2 , ωn3 are the first three eigenfrequencies of uncracked. It can be seen from Fig. 4 that the change in the first normalized eigenfrequency with crack present is significant and the change in the second normalized eigenfrequency is quite small because the crack location is close to the antinodal point of the first mode, which is otherwise the nodal point of second mode. It can also be seen from Fig. 4 that for a given crack location, the changes in eigenfrequencies of the cracked rotor monotonically increase with the increment of the crack depth, so if the crack location is known in advance, the crack depth can be read out from the changes in eigenfrequencies.
Fig. 4. Variations of normalized eigenfrequencies with relative crack depth for a rotor with slenderness ratio L/D = 8 and crack location Z1 /L = 0.45.
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(a)
329
(b)
Fig. 5. Variations of the normalized eigenfrequency with crack location for cracked rotor with slenderness ratio L/D = 8 and relative crack depth a¯ d = ad /R = 0.4. (a) First normalized eigenfrequency. (b) Second normalized eigenfrequency.
(a)
(b)
Fig. 6. Three-dimensional diagram of normalized eigenfrequency of cracked rotor versus crack location and relative crack depth (slenderness ratio L/D = 8). (a) Variation of first normalized eigenfrequency. (b) Variation of second normalized eigenfrequency.
3.2. Effects of the crack location Z1 /L For a cracked rotor with slenderness ratio L/D = 8 and relative crack depth a¯ d = 0.4, the variations of eigenfrequencies with crack locations Z1 /L on the rotor are shown in Fig. 5. It is noticed that the changes in normalized eigenfrequencies ωci /ωni (i = 1, 2) show the symmetric property because of the symmetry of the rotor considered. It can also be seen from Fig. 5 that the changes in normalized eigenfrequencies depend on how close the crack is to that mode shape node, that is, the reduction in the eigenfrequency of a mode is larger if the crack is near to the antinodal point of that mode shape. Taking the effects of the crack depth and location into consideration, a detailed calculation on the first two normalized eigenfrequencies ωci /ωni (i = 1, 2) of a cracked rotor with slenderness ratio L/D = 8 has been carried out. The results are shown in Fig. 6, which provide the basis for identification of crack parameters. 3.3. Comparison of the mode shapes between the cracked rotor and the uncracked For a rotor with slenderness ratios L/D = 8, crack location Z1 /L = 0.45 and relative crack depth a¯ d = 0.4, the corresponding mode shapes of the first two eigenfrequencies are shown in Fig. 7. From Fig. 7, the changes in the first mode shape at crack location Z1 /L = 0.45 can be observed clearly, so for a rotor with a single crack, the first mode shape is helpful for identifying the location of the crack.
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(a)
(b)
Fig. 7. Mode shapes of the rotor with slenderness ratio L/D = 8, crack location Z1 /L = 0.45 and relative crack depth a¯ d = ad /R = 0.4. (a) First mode, (b) second mode.
Table 1 Normalized eigenfrequencies for different slenderness ratio L/D (a¯ d = ad /R = 0.6, Z1 /L = 0.45) ωc1 /ωn1 ωc2 /ωn2 ωc3 /ωn3
L/D = 4
L/D = 6
L/D = 8
L/D = 10
L/D = 12
0.9064 0.9911 0.9332
0.9347 0.9937 0.9515
0.9498 0.9952 0.9619
0.9593 0.9961 0.9687
0.9657 0.9967 0.9734
3.4. Effects of the slenderness ratios L/D The rotors with various slenderness ratios L/D are also considered, and the normalized eigenfrequencies ωci /ωni (i = 1, 2, 3) for different L/D are tabulated in Table1. It can be seen clearly from Table 1 that the changes in eigenfrequencies due to a single crack are appreciable in case of shafts with low slenderness ratios. 3.5. Discussion on shallow cracks For shallow cracks, the ratios of the changes in the first two eigenfrequencies between the cracked rotor and the uncracked ω2 /ω1 for a rotor with slenderness ratios L/D = 8 and crack location Z1 /L = 0.45 are tabulated in Table 2, where ω1 = ωn1 − ωc1 and ω2 = ωn2 − ωc2 . It can be noticed from Table 2 that the ratio of the frequency changes in the first two modes is nearly invariant as the relative crack depth a¯ d increases, which corresponds with the deduction of Adams et al. (1978) and Cawley and Adams (1979) that the ratio of the frequency changes in two modes is only a function of the damage location. The variations of the ratios of the changes in the first two eigenfrequencies ω2 /ω1 with different crack locations for rotors with different slenderness ratios are shown in Fig. 8. From Fig. 8, it is obvious that the influence of the slenderness ratio on the ratios of the changes in the first two modes is also very small.
Table 2 Ratios of the changes in the first two eigenfrequencies between the cracked rotor and the uncracked ω2 /ω1 for a rotor with slenderness ratios L/D = 8 and crack location Z1 /L = 0.45 ω1 ω2 ω2 /ω1
a¯ d = 0.05
a¯ d = 0.1
a¯ d = 0.15
a¯ d = 0.2
a¯ d = 0.25
0.0748 0.0293 0.3917
0.4051 0.1586 0.3915
1.0717 0.4194 0.3913
2.1184 0.8284 0.3910
3.5735 1.3963 0.3907
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Fig. 8. Variations of the ratios of the changes in the first two eigenfrequencies with crack location for the rotor with different slenderness ratios.
4. Crack identification algorithm Based on the above analysis, a crack identification algorithm is proposed, which uses the deflections of the first mode at two symmetric points of the rotor and the contour diagram of crack location versus crack depth for first two normalized eigenfrequencies to estimate crack location and depth. Firstly, the contour diagram of crack location versus crack depth for the first two normalized eigenfrequencies is used to calculate two possible cracks with accurate depths and locations. Secondly, the deflections of the first mode at two symmetric points of the rotor are used to erase the symmetric property and determine the actual crack location: if the deflection of the point on the left side of the rotor is larger than that of the point on the right side, the crack is located on the left side of the rotor, else the crack is located on the right side. The first crack identification problem is stated as follows: the slenderness ratio of the aforementioned cracked rotor is L/D = 8, the first two normalized eigenfrequencies are ωc1 /ωn1 = 0.9980, ωc2 /ωn2 = 0.9959, and the deflections of two symmetric points on the left and right sides are 0.7106, 0.7068. Because the deflection of the point on the left side of the rotor is larger than that of the point on the right side, the crack is located on the left side of the rotor. The contour diagram of crack location versus crack depth for the given normalized eigenfrequencies is shown in Fig. 9, from Fig. 9, it can be read out that the
Fig. 9. Contour diagram of crack location versus crack depth for first two normalized eigenfrequencies ωc1 /ωn1 = 0.9980, ωc2 /ωn2 = 0.9959 of the cracked rotor.
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Fig. 10. Contour diagram of crack location versus crack depth for first two normalized eigenfrequencies ωc1 /ωn1 = 0.9181, ωc2 /ωn2 = 0.9411 of the cracked rotor. Table 3 Comparison between the predicted and actual results of the first example
Z1 /L a¯ d
Predicted results
Actual results
Errors (%)
0.246 0.2005
0.25 0.2
1.6 0.25
Table 4 Comparison between the predicted and actual results the second example
Z1 /L a¯ d
Predicted results
Actual results
Errors
0.35 0.8
0.35 0.8
– –
two possible crack locations are Z1 /L = 0.246, Z1 /L = 0.754, and the relative crack depth is a¯ d = 0.2005. Because the crack is located on the left side of the rotor, the crack location Z1 /L = 0.246 and the relative crack depth a¯ d = 0.2005 are the final predicted results. The second crack identification problem is stated as follows: the slenderness ratio of the aforementioned cracked rotor is L/D = 8, the first two normalized eigenfrequencies are ωc1 /ωn1 = 0.9181, ωc2 /ωn2 = 0.9411 and the deflections of two symmetric points on the left and right sides are 1.0002, 0.9006. Because the deflection of the point on the left side of the rotor is larger than that of the point on the right side, the crack is located on the left side of the rotor. The contour diagram of crack location versus crack depth for the given normalized eigenfrequencies is shown in Fig. 10, from Fig. 10, it can be read out that the two possible crack locations are Z1 /L = 0.35, Z1 /L = 0.65, and the relative crack depth is a¯ d = 0.8. Because the crack is located on the left side of the rotor, the crack location Z1 /L = 0.35 and the relative crack depth a¯ d = 0.8 are the final predicted results. The predicted crack depth and location are compared with actual results in Tables 3 and 4. From Tables 3 and 4, it can be seen that the error is very small, illustrating the validity of the proposed crack identification algorithm.
5. Conclusions A continuous model is introduced for vibration analysis and crack identification of a static (non-rotating) rotor with an open crack, assuming that the cracked rotor dimensions satisfy the Euler–Bernoulli theory and the cracked region is modeled as a local flexibility.
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Through vibration analysis, some helpful results are obtained: (1) The changes in eigenfrequencies of the cracked rotor depend on how close the crack is to one of that mode shape nodes, and for a given crack location these changes monotonically increase with the increment of the crack depth; (2) Due to the symmetry of the rotor considered, the changes in eigenfrequencies with different crack locations show the symmetric property: if the first two eigenfrequencies are known, two possible cracks can be predicted at the symmetric locations on the rotor; (3) The changes in eigenfrequencies due to the crack are appreciable in cases of the rotors with low slenderness ratio; (4) The changes in the first mode shape at crack location can be observed clearly, so the first mode shape is helpful for identifying the location of the crack; (5) For shallow cracks, the ratio of the change in the first two eigenfrequencies between the cracked rotor and the uncracked ω2 /ω1 is only a function of the damage location, the influence of the relative crack depth and the slenderness ratio on ω2 /ω1 is very small. Based on above analysis, a crack identification algorithm is proposed, which uses the deflections of the first mode shape at two symmetric points and the contour diagram of crack location versus crack depth for first two given normalized eigenfrequencies. Two illustrative examples demonstrate the availability and validity of the proposed crack identification algorithm. Acknowledgements This project is supported in part by National Fundamental Research and Development Project (approved No. G1998020321), by National Natural Science Foundation and Institute of Engineering Physics of China (approved No. 10176014), also by the State Key Laboratory of Vibration, Shock & Noise (approved No. VSN-2002-03), the authors are grateful to their financial support. References Adams, R.D., Cawley, P., Pye, C.J., 1978. A vibration technique for non-destructively assessing the integrity of structures. J. Mech. Engrg. Sci. 20, 93–100. Armon, D., Haim, Y.B., Braun, S., 1994. Crack detection in beams by rank-ordering of eigenfrequency shifts. Mech. Systems Signal Process. 8, 81–91. Cawley, P., Adams, R.D., 1979. The location of defects in structures from measurements of natural frequencies. J. Strain Anal. Engrg. Design 14, 49–57. Chondros, T.G., Dimarogonas, A.D., 1980. Identification of cracks in welded joints of complex structures. J. Sound Vib. 69, 531–538. Dimarogonas, A.D., 1996. Vibration of cracked structures: a state of the art review. Engrg. Fracture Mech. 55, 831–857. Dimarogonas, A.D., Papadopoulos, C.A., 1983. Vibration of cracked shafts in bending. J. Sound Vib. 91, 583–593. Gasch, R., 1993. A survey of the dynamic behavior of a simple rotating shaft with a transverse crack. J. Sound Vib. 160, 313–332. Gounaris, G., Dimarogonas, A., 1988. A finite element of a cracked prismatic beam for structural analysis. Comput. & Structures 28, 309–313. Gudmundson, P., 1982. Eigenfrequency changes of structures due to cracks, notches or other geometrical changes. J. Mech. Phys. Solids 30, 339–353. Gudmundson, P., 1983. The dynamic behavior of slender structures with cross-sectional cracks. J. Mech. Phys. Solids 31, 329–345. Hu, J.L., Liang, R.Y., 1993. An integrated approach to detection of cracks using vibration characteristics. J. Franklin Inst. 330, 841–853. Inagaki, T., Hanki, H., Shiraki, K., 1982. Transverse vibrations of a general cracked-rotor bearing system. Trans. ASME J. Mech. Design 104, 345–355. Liang, R.Y., Hu, J.L., Choy, F., 1992a. Theoretical study of crack-induced eigenfrequency changes on beam structures. J. Engrg. Mech. 118, 384–396. Liang, R.Y., Hu, J.L., Choy, F., 1992b. Quantitative NDE technique for assessing damages in beam structures. J. Engrg. Mech. 118, 1468–1487. Nikolakopoulos, P.G., Katsareas, D.E., Papadopoulos, C.A., 1997. Crack identification in frame structures. Comput. & Structures 64, 389–406. Ostachowicz, W.M., Krawczuk, M., 1991. Analysis of the effect of cracks on the natural frequencies of a cantilever beam. J. Sound Vib. 150, 191–201. Pu, Y.P., Chen, J., Zou, J., 2002a. Quasi-periodic vibration of cracked rotor on flexible bearings. J. Sound Vib. 215, 875–890. Pu, Y.P., Chen, J., Zou, J., 2002b. The research on non-linear characteristics of a cracked rotor and reconstruction of the crack forces. In: Proceedings of IMechE, Part C. J. Mech. Engrg. Sci. 216, 1099–1108. Qian, G.L., Gu, S.N., Jiang, J.S., 1990. The dynamic behavior and crack detection of a beam with a crack. J. Sound Vib. 138, 233–243. Rizos, P.F., Aspragathos, N., Dimarogonas, A.D., 1990. Identification of crack location and magnitude in a cantilever beam from the vibration modes. J. Sound Vib. 138, 381–388. Shen, M.H.H., Pierre, C., 1990. Natural modes of Bernoulli–Euler beams with symmetric cracks. J. Sound Vib. 138, 115–134. Suh, M.W., Shim, M.B., Kim, M.Y., 2000. Crack identification using hybrid neuro-genetic technique. J. Sound Vib. 238, 617–635. Wauer, J., 1990. On the dynamics of cracked rotors: a literature survey. Appl. Mech. Rev. 43, 13–17. Zou, J., Chen, J., Pu, Y.P., 2002. On the wavelet time-frequency analysis algorithm in identification of a cracked rotor. J. Strain Anal. Engrg. Design 37, 239–246. Zou, J., Chen, J., Niu, J.C., 2003a. Study on transient response and wavelet time-frequency feature of a cracked rotor passage through subcritical speed. J. Strain Anal. Engrg. Design 38, 269–276. Zou, J., Chen, J., Niu, J.C., 2003b. Discussion on the local flexibility due to the crack in a cracked rotor system. J. Sound Vib. 262, 365–369.
Parameter identification of a rotor with an open crack G.M. Dong ∗ , J. Chen, J. Zou The State Key Laboratory of Vibration, Shock and Noise, Shanghai Jiao Tong University, Shanghai 200030, PR China Received 22 May 2003; accepted 14 November 2003
Abstract A continuous model is presented for vibration analysis and parameter identification of a static (non-rotating) rotor with an open crack, which is based on two assumptions that the cracked rotor is an Euler–Bernoulli beam with circular cross-section and that the cracked region is modeled as a local flexibility with fracture mechanics methods. Through numerical analysis, the effects of the location and depth of the crack on the changes in the eigenfrequencies and mode shapes of the cracked rotor are investigated, and the ratios of the changes in the first two eigenfrequencies are discussed for rotors with shallow cracks. A crack identification algorithm, which makes use of the deflections of the first mode shape at two symmetric points and the contour diagram of crack location versus crack depth for the first two normalized eigenfrequencies, is proposed to predict the crack location and depth on the rotor, and two illustrative examples demonstrate the validity and availability of the proposed algorithm. 2003 Elsevier SAS. All rights reserved. Keywords: Parameter identification; Continuous model; Vibration analysis; Cracked rotor; Open crack; Crack location; Crack depth; Eigenfrequency; Mode shape
1. Introduction One form of damage that can lead to catastrophic failure if undetected is fatigue cracking of the shaft, so early detection of cracked rotor in engineering practices is of significant importance to the reliability and durability of large rotating machinery, which makes fault diagnosis and condition monitoring of cracked rotor paid more and more attention (Dimarogonas, 1996; Dimarogonas and Papadopoulos, 1983; Gasch, 1993; Pu et al., 2002a, 2002b; Wauer, 1990; Zou et al., 2002, 2003a, 2003b). For the time being, the research on the cracked rotor is still at the theoretical stage, and most of the previous research is only involved in detecting the crack in a rotor and not capable of determining the crack location and depth. A static (non-rotating) rotor with an open crack can be considered as a simply supported beam, as a consequence, the research in relation to non-rotating structures such as beams and columns is useful for locating and estimating the severity of the crack on a rotor (Armon et al., 1994; Liang et al., 1992a, 1992b; Ostachowicz and Krawczuk, 1991; Qian et al., 1990; Shen and Pierre, 1990). A crack in a structure introduces a local flexibility, which affects the dynamic behavior of the whole structure to a considerable degree and results in reduction of natural frequencies and changes in mode shapes of vibration. In the paper of Chondros and Dimarogonas (1980), the effect of a crack on the deformation of a beam has been considered similar to that of an elastic hinge. Dimarogonas and Papadopoulos (1983) modeled the crack as a local flexibility and computed the equivalent stiffness using fracture mechanics methods. By perturbation theory and transfer matrix analysis, Gudmundson (1982, 1983) calculated the variation of eigenfrequencies due to the crack on a cantilever beam. Following Gudmundson (1982, 1983), Gounaris and Dimarogonas (1988) proposed a finite element model using transfer matrix method and local flexibility theorem; * Corresponding author.
E-mail address: [email protected] (G.M. Dong). 0997-7538/$ – see front matter 2003 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2003.11.003
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in order to consider the discontinuity deformation due to the crack on the beam, they adopted two different shape functions for two segments separated by the crack. The dynamic response of the cracked structures and the changes in the natural frequencies and mode shapes between the cracked and the uncracked structures contain information about the location and dimension of the damage. Using the receptance technique and Taylor series expansion, Adams et al. (1978), Cawley and Adams (1979) have shown that the ratio of the frequency changes in two modes is only a function of the damage location respectively. Rizos et al. (1990) estimated the crack location and depth with satisfactory accuracy from the result of an analytical solution of a cracked beam, and from experimental data of the measured amplitudes at two points of the structure vibrating at one of its natural modes and the respective vibration frequency. For a rotor removed from service, Inagaki et al. (1982) used natural vibration and static deflection analysis to find the crack size and location. Hu and Liang (1993) proposed a two-step procedure to identify cracks in beam structures: they used the effective stress concept coupled with Hamilton’s principle to derive a formulation relating the changes in the natural frequencies to the member stiffness changes. The elements that contain cracks could be identified based on the formulation and then a spring damage model was used to quantify the location and depth of the crack in each damaged element. On the basis of the research of Gudmundson (1982, 1983) and Gounaris and Dimarogonas (1988), Nikolakopoulos et al. (1997) presented the dependency of the structural eigenfrequencies on crack depth and location in contour graph form: to identify the location and depth of a crack, the intersection points of the superposed contours that correspond to the measured eigenfrequency variations caused by the presence of the crack should be determined. With the model from Nikolakopoulos et al. (1997), Suh et al. (2000) presented a detection method that uses hybrid neuro-genetic technique to identify the location and depth of a crack on a structure. In this paper, a continuous model, which is based on that the dimension of the cracked rotor satisfies the Euler–Bernoulli theory and that the cracked region is modeled as a local flexibility, is presented for vibration analysis and crack identification of a static (non-rotating) rotor with an open crack. Through numerical analysis, the effects of the location and depth of the crack on the changes in the eigenfrequencies and mode shapes of the cracked rotor are investigated, and the ratios of the changes in the first two eigenfrequencies are discussed for rotors with shallow cracks. Then a crack identification algorithm, which makes use of the deflections of the first mode shape at two symmetric points and the contour diagram of crack location versus crack depth for the first two normalized eigenfrequencies, is proposed to predict the crack location and depth on the rotor. At last, two illustrative examples are demonstrated to assess the validity of the proposed algorithm.
2. Continuous model of rotor with an open crack The continuous model of the problem is shown in Fig. 1, which includes the schematic diagram of a non-rotating rotor with a single open crack of depth ad and location Z1 . The cross-sectional radius of the rotor is R and the length of the rotor is L. The vibration frequency equation of the rotor is written in the well-known form ∂ 4 y(x, t) ∂ 2 y(x, t) + ρA = 0, (1) ∂4x ∂2t where E is Young’s modulus, ρ is material density, I is the geometrical moment of inertia of the cross-section, A is crosssectional area of the rotor and x is the coordinate along the rotor. For vibration in natural modes, the deflection of the rotor is EI
y(x, t) = Y (x) sin(ωt).
(2)
Substituting Eq. (2) into Eq. (1), after simple algebraic transformation, one has Y iv (β) − λ4 Y (β) = 0,
(3)
where β = x/L, λ4 = ρAω2 L4 /(EI ).
Fig. 1. Rotor with a single open crack simply supported on both ends.
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The general solution of Eq. (3) is given by Y1 (β) = A1 cosh(λβ) + A2 sinh(λβ) + A3 cos(λβ) + A4 sin(λβ),
β ∈ [0, β1 ),
(4a)
Y2 (β) = B1 cosh(λβ) + B2 sinh(λβ) + B3 cos(λβ) + B4 sin(λβ),
β ∈ (β1 , 1],
(4b)
where β1 = Z1 /L, Z1 is crack location of the rotor, Ai and Bi (i = 1, 2, 3, 4) are constants to be determined from the boundary conditions. The boundary conditions for the cracked rotor under consideration are Y1 (β)|β=0 = 0,
Y1 (β)|β=0 = 0,
Y2 (β)|β=1 = 0,
Y2 (β)|β=1 = 0.
(5)
The continuity at the crack position is as follows Y1 (β)|β=β1 = Y2 (β)|β=β1 ,
Y1 (β)|β=β1 = Y2 (β)|β=β1 ,
Y1 (β)|β=β1 = Y2 (β)|β=β1 .
(6)
The compatibility condition due to the local flexibility is Y2 (β)|β=β1 − Y1 (β)|β=β1 =
cEI Y (β)|β=β1 L 2
(7)
where c is the local flexibility due to the crack on a rotor with a circular cross-section shown in Fig. 2. The dimensionless local flexibility c¯ is calculated with fracture mechanics method proposed by Dimarogonas and Papadopoulos (1983) and c¯ varying with the relative crack depth a¯ d = ad /R is shown in Fig. 3. Combining Eqs. (4)–(7), the characteristic equation is derived
1 1 cosh(λβ1 ) cosh(λβ ) 1 det sinh(λβ1 ) sinh(λβ ) 1 0
0
1
0
0
0
−1
0
0
sinh(λβ1 )
cos(λβ1 )
sin(λβ1 )
− cosh(λβ1 )
sinh(λβ1 )
− cos(λβ1 )
− sin(λβ1 )
− cosh(λβ1 )
cosh(λβ1 )
sin(λβ1 )
− cos(λβ1 )
cosh(λβ1 )
− sin(λβ1 )
cos(λβ1 )
− sinh(λβ1 ) cEI λ cosh(λβ ) − sinh(λβ ) 1 1 L
0
0
0
cosh λ
0
0
0
cosh λ
0 0
0
0
0
− sinh(λβ1 )
− cos(λβ1 )
− sinh(λβ1 )
cos(λβ1 )
− cosh(λβ1 )
− sin(λβ1 )
cEI λ sinh(λβ ) − cosh(λβ ) 1 1 L
− cEI L λ cos(λβ1 ) + sin(λβ1 )
sinh λ
cos λ
sinh λ
− cos λ
0
− sin(λβ1 ) sin(λβ1 ) cos(λβ1 ) cEI − L λ sin(λβ1 ) − cos(λβ1 ) sin λ 0
− sin λ
= 0.
(8)
8×8
From Eq. (8), the characteristic roots λi are determined and used for the calculation of the eigenfrequencies ωi =
λi 2 EI . L ρA
(9)
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Fig. 2. Geometry of a cracked section of a shaft.
Fig. 3. Dimensionless local flexibility.
3. Vibration analysis In the simulation, the rotor geometries and the material properties are as follows: L = 3 m, E = 2.07 × 1011 N/m2 , ρ = 7.7 × 103 kg/m3 and Poisson ratio v = 0.3. At the same time, the rotor geometries and the crack parameters are expressed in dimensionless forms for convenience. 3.1. Effects of the relative crack depth a¯ d = ad /R For a rotor with slenderness ratio L/D = 8 (D = 2R is the diameter of the rotor) and crack location Z1 /L = 0.45, a parametric study of the effect of the crack depth on the vibration characteristics of the rotor is carried out by varying the relative crack depth a¯ d = ad /R. The variations of the first three normalized eigenfrequencies ωci /ωni (i = 1, 2, 3) with the relative crack depth are shown in Fig. 4, where ωc1 , ωc2 , ωc3 are the first three eigenfrequencies of cracked rotor and ωn1 , ωn2 , ωn3 are the first three eigenfrequencies of uncracked. It can be seen from Fig. 4 that the change in the first normalized eigenfrequency with crack present is significant and the change in the second normalized eigenfrequency is quite small because the crack location is close to the antinodal point of the first mode, which is otherwise the nodal point of second mode. It can also be seen from Fig. 4 that for a given crack location, the changes in eigenfrequencies of the cracked rotor monotonically increase with the increment of the crack depth, so if the crack location is known in advance, the crack depth can be read out from the changes in eigenfrequencies.
Fig. 4. Variations of normalized eigenfrequencies with relative crack depth for a rotor with slenderness ratio L/D = 8 and crack location Z1 /L = 0.45.
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(a)
329
(b)
Fig. 5. Variations of the normalized eigenfrequency with crack location for cracked rotor with slenderness ratio L/D = 8 and relative crack depth a¯ d = ad /R = 0.4. (a) First normalized eigenfrequency. (b) Second normalized eigenfrequency.
(a)
(b)
Fig. 6. Three-dimensional diagram of normalized eigenfrequency of cracked rotor versus crack location and relative crack depth (slenderness ratio L/D = 8). (a) Variation of first normalized eigenfrequency. (b) Variation of second normalized eigenfrequency.
3.2. Effects of the crack location Z1 /L For a cracked rotor with slenderness ratio L/D = 8 and relative crack depth a¯ d = 0.4, the variations of eigenfrequencies with crack locations Z1 /L on the rotor are shown in Fig. 5. It is noticed that the changes in normalized eigenfrequencies ωci /ωni (i = 1, 2) show the symmetric property because of the symmetry of the rotor considered. It can also be seen from Fig. 5 that the changes in normalized eigenfrequencies depend on how close the crack is to that mode shape node, that is, the reduction in the eigenfrequency of a mode is larger if the crack is near to the antinodal point of that mode shape. Taking the effects of the crack depth and location into consideration, a detailed calculation on the first two normalized eigenfrequencies ωci /ωni (i = 1, 2) of a cracked rotor with slenderness ratio L/D = 8 has been carried out. The results are shown in Fig. 6, which provide the basis for identification of crack parameters. 3.3. Comparison of the mode shapes between the cracked rotor and the uncracked For a rotor with slenderness ratios L/D = 8, crack location Z1 /L = 0.45 and relative crack depth a¯ d = 0.4, the corresponding mode shapes of the first two eigenfrequencies are shown in Fig. 7. From Fig. 7, the changes in the first mode shape at crack location Z1 /L = 0.45 can be observed clearly, so for a rotor with a single crack, the first mode shape is helpful for identifying the location of the crack.
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(a)
(b)
Fig. 7. Mode shapes of the rotor with slenderness ratio L/D = 8, crack location Z1 /L = 0.45 and relative crack depth a¯ d = ad /R = 0.4. (a) First mode, (b) second mode.
Table 1 Normalized eigenfrequencies for different slenderness ratio L/D (a¯ d = ad /R = 0.6, Z1 /L = 0.45) ωc1 /ωn1 ωc2 /ωn2 ωc3 /ωn3
L/D = 4
L/D = 6
L/D = 8
L/D = 10
L/D = 12
0.9064 0.9911 0.9332
0.9347 0.9937 0.9515
0.9498 0.9952 0.9619
0.9593 0.9961 0.9687
0.9657 0.9967 0.9734
3.4. Effects of the slenderness ratios L/D The rotors with various slenderness ratios L/D are also considered, and the normalized eigenfrequencies ωci /ωni (i = 1, 2, 3) for different L/D are tabulated in Table1. It can be seen clearly from Table 1 that the changes in eigenfrequencies due to a single crack are appreciable in case of shafts with low slenderness ratios. 3.5. Discussion on shallow cracks For shallow cracks, the ratios of the changes in the first two eigenfrequencies between the cracked rotor and the uncracked ω2 /ω1 for a rotor with slenderness ratios L/D = 8 and crack location Z1 /L = 0.45 are tabulated in Table 2, where ω1 = ωn1 − ωc1 and ω2 = ωn2 − ωc2 . It can be noticed from Table 2 that the ratio of the frequency changes in the first two modes is nearly invariant as the relative crack depth a¯ d increases, which corresponds with the deduction of Adams et al. (1978) and Cawley and Adams (1979) that the ratio of the frequency changes in two modes is only a function of the damage location. The variations of the ratios of the changes in the first two eigenfrequencies ω2 /ω1 with different crack locations for rotors with different slenderness ratios are shown in Fig. 8. From Fig. 8, it is obvious that the influence of the slenderness ratio on the ratios of the changes in the first two modes is also very small.
Table 2 Ratios of the changes in the first two eigenfrequencies between the cracked rotor and the uncracked ω2 /ω1 for a rotor with slenderness ratios L/D = 8 and crack location Z1 /L = 0.45 ω1 ω2 ω2 /ω1
a¯ d = 0.05
a¯ d = 0.1
a¯ d = 0.15
a¯ d = 0.2
a¯ d = 0.25
0.0748 0.0293 0.3917
0.4051 0.1586 0.3915
1.0717 0.4194 0.3913
2.1184 0.8284 0.3910
3.5735 1.3963 0.3907
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Fig. 8. Variations of the ratios of the changes in the first two eigenfrequencies with crack location for the rotor with different slenderness ratios.
4. Crack identification algorithm Based on the above analysis, a crack identification algorithm is proposed, which uses the deflections of the first mode at two symmetric points of the rotor and the contour diagram of crack location versus crack depth for first two normalized eigenfrequencies to estimate crack location and depth. Firstly, the contour diagram of crack location versus crack depth for the first two normalized eigenfrequencies is used to calculate two possible cracks with accurate depths and locations. Secondly, the deflections of the first mode at two symmetric points of the rotor are used to erase the symmetric property and determine the actual crack location: if the deflection of the point on the left side of the rotor is larger than that of the point on the right side, the crack is located on the left side of the rotor, else the crack is located on the right side. The first crack identification problem is stated as follows: the slenderness ratio of the aforementioned cracked rotor is L/D = 8, the first two normalized eigenfrequencies are ωc1 /ωn1 = 0.9980, ωc2 /ωn2 = 0.9959, and the deflections of two symmetric points on the left and right sides are 0.7106, 0.7068. Because the deflection of the point on the left side of the rotor is larger than that of the point on the right side, the crack is located on the left side of the rotor. The contour diagram of crack location versus crack depth for the given normalized eigenfrequencies is shown in Fig. 9, from Fig. 9, it can be read out that the
Fig. 9. Contour diagram of crack location versus crack depth for first two normalized eigenfrequencies ωc1 /ωn1 = 0.9980, ωc2 /ωn2 = 0.9959 of the cracked rotor.
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Fig. 10. Contour diagram of crack location versus crack depth for first two normalized eigenfrequencies ωc1 /ωn1 = 0.9181, ωc2 /ωn2 = 0.9411 of the cracked rotor. Table 3 Comparison between the predicted and actual results of the first example
Z1 /L a¯ d
Predicted results
Actual results
Errors (%)
0.246 0.2005
0.25 0.2
1.6 0.25
Table 4 Comparison between the predicted and actual results the second example
Z1 /L a¯ d
Predicted results
Actual results
Errors
0.35 0.8
0.35 0.8
– –
two possible crack locations are Z1 /L = 0.246, Z1 /L = 0.754, and the relative crack depth is a¯ d = 0.2005. Because the crack is located on the left side of the rotor, the crack location Z1 /L = 0.246 and the relative crack depth a¯ d = 0.2005 are the final predicted results. The second crack identification problem is stated as follows: the slenderness ratio of the aforementioned cracked rotor is L/D = 8, the first two normalized eigenfrequencies are ωc1 /ωn1 = 0.9181, ωc2 /ωn2 = 0.9411 and the deflections of two symmetric points on the left and right sides are 1.0002, 0.9006. Because the deflection of the point on the left side of the rotor is larger than that of the point on the right side, the crack is located on the left side of the rotor. The contour diagram of crack location versus crack depth for the given normalized eigenfrequencies is shown in Fig. 10, from Fig. 10, it can be read out that the two possible crack locations are Z1 /L = 0.35, Z1 /L = 0.65, and the relative crack depth is a¯ d = 0.8. Because the crack is located on the left side of the rotor, the crack location Z1 /L = 0.35 and the relative crack depth a¯ d = 0.8 are the final predicted results. The predicted crack depth and location are compared with actual results in Tables 3 and 4. From Tables 3 and 4, it can be seen that the error is very small, illustrating the validity of the proposed crack identification algorithm.
5. Conclusions A continuous model is introduced for vibration analysis and crack identification of a static (non-rotating) rotor with an open crack, assuming that the cracked rotor dimensions satisfy the Euler–Bernoulli theory and the cracked region is modeled as a local flexibility.
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Through vibration analysis, some helpful results are obtained: (1) The changes in eigenfrequencies of the cracked rotor depend on how close the crack is to one of that mode shape nodes, and for a given crack location these changes monotonically increase with the increment of the crack depth; (2) Due to the symmetry of the rotor considered, the changes in eigenfrequencies with different crack locations show the symmetric property: if the first two eigenfrequencies are known, two possible cracks can be predicted at the symmetric locations on the rotor; (3) The changes in eigenfrequencies due to the crack are appreciable in cases of the rotors with low slenderness ratio; (4) The changes in the first mode shape at crack location can be observed clearly, so the first mode shape is helpful for identifying the location of the crack; (5) For shallow cracks, the ratio of the change in the first two eigenfrequencies between the cracked rotor and the uncracked ω2 /ω1 is only a function of the damage location, the influence of the relative crack depth and the slenderness ratio on ω2 /ω1 is very small. Based on above analysis, a crack identification algorithm is proposed, which uses the deflections of the first mode shape at two symmetric points and the contour diagram of crack location versus crack depth for first two given normalized eigenfrequencies. Two illustrative examples demonstrate the availability and validity of the proposed crack identification algorithm. Acknowledgements This project is supported in part by National Fundamental Research and Development Project (approved No. G1998020321), by National Natural Science Foundation and Institute of Engineering Physics of China (approved No. 10176014), also by the State Key Laboratory of Vibration, Shock & Noise (approved No. VSN-2002-03), the authors are grateful to their financial support. References Adams, R.D., Cawley, P., Pye, C.J., 1978. A vibration technique for non-destructively assessing the integrity of structures. J. Mech. Engrg. Sci. 20, 93–100. Armon, D., Haim, Y.B., Braun, S., 1994. 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