Simplified modelling of rotor cracks * † J.E.T. Penny and M.I. Friswell * School of Engineering and Applied Science, Aston University, Birmingham B4 7ET, UK † Department of Mechanical Engineering, University of Wales Swansea, Swansea SA2 8PP, UK e-mail: [email protected], [email protected] Abstract There are a number of approaches reported in the literature for modelling cracks in shafts. Although two and three dimensional finite element models may be used, the most popular approaches are based on beam models. This paper considers a number of different approaches to modelling cracks in shafts of rotating machines. Of particular concern is the modelling of breathing cracks, which open and close due to the selfweight of the rotor, producing a parametric excitation. In the simplest model the shaft stiffness is bilinear, depending on whether the crack is open or closed. More complicated functions that relate the shaft stiffness to angular position have been proposed, and two possibilities will be considered. This paper will demonstrate the influence of the crack model on the response of a Jeffcott rotor, using the harmonic balance approach. to gravity. In this case the crack will open and close (or breathe) due to the rotation of the rotor. This problem was initially studied by Gasch [3,4], Mayes and Davies [5,6], Grabowski [7] and others. Various crack opening and closing models have been proposed. In this paper the vibration response of a simple Jeffcott rotor with three different crack models is compared. 1. Introduction The notion that changes in a rotor’s dynamic behaviour could be used for on-line crack detection and monitoring was first proposed in the 1970s and much of the early work was presented in technical reports of the General Electric Co, USA. To implement this method of crack detection either a model based method or a “pattern recognition” based method is used. The pattern recognition approach looks for changes in the vibration data using some suitable analysis method, for example Prabhakar et al. [1] used wavelet transforms to detect and monitor cracks. The alternative model based approach allows vibration data from the rotor to be compared with simulated data from the rotor and crack model. The crack location and severity can then be determined by using an appropriate mathematical technique to adjust these crack model parameters to minimise the error between the actual and simulated data [2]. 2. Models of opening and closing cracks The simplest model of an opening and closing crack is the hinge model [3,4]. In this model, the crack is assumed to change from its closed to open state (and visa-versa) abruptly as the shaft rotates. The stiffness matrix in rotating coordinates (ξ, η) , due to a crack is 0 k ξ (θ ) kR = k η (θ) 0 If the vibration due to any out-of-balance forces acting on a rotor is greater than the static deflection of the rotor due to gravity, then the crack will remain either open or closed depending on the size and location of the unbalance masses. If the crack remains open, the rotor is then asymmetric and this condition can lead to stability problems. Of more importance in large machines is the situation where the vibration due any out-of-balance forces acting on a rotor is less than the deflection of the rotor due (1) Thus, in the hinge model the variable stiffnesses k ξ (θ) and k η (θ) are defined by k0 kξ ( θ ) = k ξ 607 − π π ≤θ< 2 2 3π π ≤θ< 2 2 (2a) 608 P ROCEEDINGS OF ISMA2002 - VOLUME II and of the shaft material, the depth of the crack and the lateral force. k0 kη ( θ ) = k η − π π ≤θ< 2 2 π 3π ≤θ< 2 2 (2b) where θ is the angular position of the crack, k 0 is the stiffness of the shaft when the crack is closed and k ξ and k η are the stiffnesses in the ξ and η directions when the crack is open. When the crack is closed its stiffness is equal to that of an equivalent undamaged shaft. Whilst the hinge model might adequately represent very shallow cracks Mayes and Davies [5,6] proposed a model in which the opening and closing of the crack was described by a cosine function. Thus in the Mayes model, the stiffnesses k ξ (θ) and k η (θ) are defined as follows: ( ) ( ) kξ ( θ ) = 12 k0 + kξ + 12 k0 − kξ cos θ (3a) and ( ) ( ) kη ( θ ) = 12 k0 + kη + 12 k0 − kη cos θ . (3b) Both the hinge model and Mayes model suffer from the defect that there is no direct relationship between the shaft stiffness and the depth of crack. Jun et al. [8] proposed a crack model based on fracture mechanics. The cross coupling stiffnesses, as well as the direct stiffnesses are estimated as the crack opens and closes. Thus, the stiffness matrix in rotating coordinates becomes k ξξ (θ) k ξη (θ ) kR = k ηξ (θ ) k ηη (θ) (4) To determine the stiffnesses Jun et al. assumed the crack to be at the mid-span of the shaft. The total stress intensity factor is derived from the sum of the stress intensity factors due to forces acting in the ξ and η directions. Jun assumes that the sign of total stress intensity at each point along the crack front determines whether the crack is open or closed at that point. The additional deflections in the ξ and η directions due to the open crack are then determined using Castigliano’s theorem. This involves numerical integration over the area of the proportion of the crack that is open. For a particular angle of rotation, the Jun model is a function the shaft length and diameter, the modulus of elasticity 3. Frequency components in a crack model 3.1. The hinge model The hinge model is defined by equation (2). Expanding k ξ (θ ) and k η (θ) in a Fourier series, we have k 0 + kξ kξ ( t ) = 2 C3 C5 C7 4 k 0 − kξ + + − + … C1 − π 2 3 5 7 (5a) k0 + kη kη ( t ) = 2 C3 C5 C7 4 k0 − kη + + − + … C1 − π 2 3 5 7 (5b) where C1 = cos θ , C3 = cos ( 3θ ) , etc., and θ = Ωt . Letting, k0 + kξ k + kη , k Mη = 0 k Mξ = 2 2 (6a) k0 − kξ k − kη , k Dη = 0 k Dξ = 2 , 2 (6b) and we have ( kξ ( θ ) = k M ξ + π4 k Dξ C1 − 13 C3 + 15 C5 − 71 C7 + … ) (7) ( kη ( θ ) = k M η + 4 k Dη C1 − 13 C3 + 15 C5 − 17 C7 + … π ) This is the variation of stiffness in rotating coordinates. Now the stiffness matrix in fixed coordinates, K F is given by K F = TT K R T (8) DYNAMIC MODELLING AND NUMERICAL MODELS 609 ( K F )22 = 12 ( kM ξ + kM η ) − 14 ( kξ − kη ) C2 where the transformation matrix T is S1 C T = 1 − S1 C1 ( ){( 34 kDξ + 83 kDη ) C1 + ( − 1528 kDξ + 158 kDη ) C3 92 k 8 + ( 105 Dξ − 105 k Dη ) C5 + ( − 188 k + 8 k C + …} 315 Dξ 315 Dη ) 7 + π1 (9) Thus K F = TT K R T kξ ( θ ) + kη ( θ ) = k ( θ) − k ( θ) S C 1 1 η ξ C12 S12 ( ) ( kξ ( θ ) − kη ( θ ) ) S1C1 kξ ( θ ) S12 + kη ( θ ) C12 (10) where S1 = sin θ , etc. Now, considering each of the fixed co-ordinate stiffness matrix terms in turn, beginning with the off diagonal one we have, ( K F )12 = 1 2 ( ) ( ) 1 kM ξ − k M η S2 + k Dξ − k Dη × π ( 43 S1 + 54 S3 − 214 S5 + 454 S7 − 774 S9 …) Since k Mξ − k Mη = 12 (k ξ − k η ) and k Dξ − k Dη = − 12 (k ξ − k η ) , then ( K F )12 = ( kξ − kη ){ 14 S2 − ( 1 4 S + 4S 2π 3 1 5 3 4 S + 4 S − 4 S … − 21 5 45 7 77 9 )} (11) Considering the diagonal terms: (13) 3.2. The Mayes model This model is defined, in rotating coordinates, by equations (3a) and (3b). Thus k η (θ) = k Mη + k DηC1 and k ξ (θ) = k Mξ + k Dξ C1 The stiffnesses in fixed coordinates can be developed using the same approach as for the hinge model. Thus using equation (8) we have the following stiffness expressions in fixed coordinates, (K F )12 = 12 (k Mξ − k Mη )S 2 + 14 (k Dξ − k Dη )(S1 + S 3 ) or ( K F )12 )( 83 C1 + 158 C3 − 1058 C5 + …) + 12 k M η − 12 k M ηC2 + ( 1k π Dη )( 4 C − 28 C 3 1 15 3 92 C + 105 5 −… ) and simplifying, ( K F )11 = 12 ( kM ξ + kM η ) + 14 ( kξ − kη ) C2 + π1 {( 83 kDξ + 34 kDη ) C1 + ( 158 kDξ − 1528 kDη ) C3 ( ) 8 k 188 + ( 315 Dξ − 315 k Dη ) C7 + …} 8 k 92 + − 105 Dξ + 105 k Dη C5 } ( S1 + S3 ) (14) 1 4 ( (3kDξ + kDη ) C1 + ( kDξ − kDη ) C3 ) (15) and ( K F )22 = 12 ( kM η + kM ξ ) − 14 ( kξ − kη ) C2 + 1 4 ((3kDη + kDξ ) C1 + ( kDη − kDξ ) C3 ) (16) Thus in fixed coordinates the Mayes model generates a constant term plus 1X, 2X and 3X rotor speed components in the diagonal stiffness terms and 1X, 2X and 3X rotor speed components in the off-diagonal stiffness terms. 3.3. The model of Jun et al. (12) Similarly, 1 2 ( K F )11 = 12 ( kM ξ + kM η ) + 14 ( kξ − kη ) C2 + = 12 k M ξ + 12 k M ξC2 ( ){ It may be shown that ( K F )11 = kξ ( t ) C12 + kη ( t ) S12 + π1 k Dξ ( = 14 kξ − kη S2 − In the model of Jun et al. [8] the stiffnesses k ξξ (θ) , k ηη (θ) and k ξη (θ) are determined numerically for P ROCEEDINGS OF ISMA2002 - VOLUME II a particular set of conditions. Thus the coefficients of the Fourier expansion representing the stiffnesses must be determined numerically using the least squares approach. In rotating coordinates k ξξ (θ) can be represented by the series kξξ ( θ ) = b0 + b1 cos θ + b2 cos ( 2θ ) (17) + b3 cos ( 3θ ) + b4 cos ( 4θ ) +… Normalised stiffness 610 0 180 θ (degrees) 270 360 are still cosine series and the off-diagonal stiffness, (k F )12 is still a sine series. Figures 1 and 2 show the stiffness variation with angle of rotation for a 40% crack in rotating coordinates, and also Figures 3 and 4 in stationary coordinates. The markers indicate the simulated values and the lines represent the fit of the Fourier Series. Tables 1 to 4 show the coefficients of the corresponding Fourier Series. The Fourier series provides a good fit to the functions. In these figures and tables, the stiffnesses have been normalised by dividing them by the stiffness of the undamaged shaft, k0. 1.0 1.0 0.95 k11 k22 0.9 0 90 180 θ (degrees) 270 360 Figure 3. Normalised stiffnesses (k F )11 and (k F )22 - fixed coordinates Normalised stiffness The process of converting to fixed coordinates using (8) changes the values of the coefficients in these series but the diagonal stiffnesses, (k F )11 and Normalised stiffness (18) + a3 sin ( 3θ ) + a4 sin ( 4θ ) +… Normalised stiffness 90 Figure 2: Normalised stiffness kηξ - rotating coordinates represented by the series (k F )22 0.0 -0.02 The stiffness k ηη (θ) can be represented by a similar series. The stiffness k ξη (θ) can be kξη ( θ ) = a1 sin θ + a2 sin ( 2θ ) 0.02 0.02 0.0 -0.02 0.95 kξξ k ηη 0.9 0 90 180 θ (degrees) 270 360 Figure 1: Normalised stiffnesses kξξ and kηη rotating coordinates 0 90 180 θ (degrees) 270 360 Figure 4: Normalised stiffness (k F )12 - fixed coordinates DYNAMIC MODELLING AND NUMERICAL MODELS 611 Crack depth = 10% diameter Fourier coefficient b0 b1 b2 b3 b4 b5 Crack depth = 40% diameter kξξ kηη kξξ kηη 0.998186 0.002241 0.000001 −0.000583 −0.000001 0.000203 0.999880 0.000142 0.000000 −0.000023 0.000000 −0.000006 0.953773 0.051221 0.000845 −0.005069 −0.000201 −0.000711 0.979907 0.017422 −0.000034 0.004754 0.000170 −0.001401 Table 1: Fourier coefficients for the Jun model, rotating coordinates Fourier coefficient a1 a2 a3 a4 a5 Crack depth = 10% dia Crack depth = 40% dia kηξ kηξ −0.000141 0.000000 0.000117 0.000000 −0.000078 −0.016949 −0.000364 0.006663 0.000232 0.000148 Table 2: Fourier coefficients for the Jun model, rotating coordinates Crack depth = 10% diameter Fourier coefficient b0 b1 b2 b3 b4 b5 Crack depth = 40% diameter (k F )11 (k F )22 (k F )11 (k F )22 0.999034 0.001588 −0.000847 0.000242 0.000000 −0.000019 0.999033 0.000794 0.000848 −0.000848 −0.000008 0.000215 0.967241 0.045458 −0.012870 −0.000084 0.000022 0.000198 0.966438 0.023185 0.013681 −0.000231 −0.000053 −0.002311 Table 3: Fourier coefficients for the Jun model, fixed coordinates Fourier coefficient a1 a2 a3 a4 a5 Crack depth = 10% dia. Crack depth = 40% dia (k F )12 (k F )12 0.000794 −0.000847 0.000363 0.000000 −0.000047 0.022711 −0.012859 −0.000123 0.000038 0.000497 Table 4: Fourier coefficients for the Jun model, fixed coordinates 612 P ROCEEDINGS OF ISMA2002 - VOLUME II 4. Equations of motion The analysis may be performed in fixed or rotating coordinates. If the bearings and foundations are axisymmetric then the stator dynamic stiffness will appear constant in the rotating frame, and there is some benefit in analysing the machine in rotating coordinates. Typically foundations will be stiffer vertically than horizontally, and in this case the advantage in using rotating coordinates is significantly reduced. Thus in this paper the machine is analysed in fixed coordinates. In fixed coordinates we have K (θ) = K 0 − K D (θ) where K 0 is the diagonal stiffness matrix for the undamaged beam and K D ( θ ) is the stiffness change due to damage. Let the deflection the system be q = q st + q u where q st is the static deflection and qu is the deflection due to whirl. Thus q = qu , and the equations of motion in fixed coordinates is Mqu + ( K 0 − K D ( θ ) ) ( q st + qu ) = Qu + W (19) where Qu and W are the out of balance forces and gravitational force respectively. Assuming that K 0 K D ( θ ) , then ( K 0 − K D ( θ ) ) q st ≈ K 0q st = W Fourier Series. Such parametrically excited systems can exhibit a wide variety of responses, from periodic with either super or sub harmonics, and even a chaotic response [9-12]. Indeed the steady state response may even be unstable [13]. However most detection schemes for rotor cracks look for the 2X response, and orbits from experimental rigs predominantly show super harmonic response. Thus the response will be assumed to be periodic, with the same frequency as the rotational speed. The response may then be written as a Fourier series, and the method of harmonic balance applied. The steady state response is completely defined by the coefficients of the Fourier Series, and the harmonic balance approach [14,15] generates equations for these coefficients by equating the sine and cosine coefficients in the equations of motion, at all frequencies. Since the Fourier Series is, in theory, infinite, the series is truncated, and the coefficients estimated using the lower frequency harmonic balance equations. The only slightly tricky part in applying the harmonic balance method is to simplify the product K D ( θ ) qu . Assuming that we know the Fourier Series of K D ( θ ) , K D ( θ) = K D0 ∞ + ∑ K Dci cos ( iθ ) + K Dsi sin ( iθ ) (21) i =1 and the unbalance response is written as and Mqu + ( K 0 − K D ( θ ) ) qu = Qu (20) Thus far damping and gyroscopic effects have been ignored. These effects may be included in equation (20) and the assumption made here is that the effect may be modelled as an additional velocity term with a constant coefficient matrix. Thus the skew-symmetric contribution to the stiffness matrix from damping in the rotor is ignored, and this damping is assumed to be axi-symmetric. 5. Harmonic balance analysis The equations of motion (20) for the system are linear with parametric excitation through the variable stiffness K D ( θ ) , where θ = Ωt and Ω is the rotor speed. With the crack opening and closing based on the static beam deflection the stiffness is a periodic function of time, and has been written as a qu ( θ ) = q 0 ∞ + ∑ q j =1 cj cos ( jθ ) + q sj sin ( jθ ) , (22) then K D ( θ ) qu involves products of sines and cosines that are easily simplified using the standard trigonometric formulae. Of course complex exponentials could be used instead of sines and cosines. 6. A Jeffcott rotor example A Jeffcott rotor modelled using two degrees of freedom will be used to demonstrate the effect of the different crack models on the response of a rotor. The rotor is 700 mm long, with a diameter of 15 mm, and has a 1 kg disc at the centre. The crack is assumed to be located close to the disc and only 613 responses that are symmetrical about the shaft centre are considered. The natural frequency of the rotor is approximately 42 Hz the damping ratio is taken as 1%. Gyroscopic effects are neglected. A crack of 40% of the diameter is simulated, using the model of Jun et al. [8]. This gives rise to a stiffness reduction of approximately 9% when the crack is open, and figures 1 to 4 give the stiffness variation with rotor orientation. In order to compare the crack modelling approaches, for the hinge and Mayes models the reduction in direct stiffnesses due to the open crack is taken directly from the results of the model of Jun et al. [8]. An unbalance force arising from a 10 g mass at 100 mm radius is used to excite the rotor, and at t=0 the crack is closed and the unbalance force is in the x direction. Figure 5 shows the steady state synchronous response magnitude for the three crack models. Fourier Series terms up to 21 times the running speed are included to ensure convergence of the coefficients. All the responses are very close, and only minor differences in magnitude occur near the resonance frequency, which has reduced to approximately 41 Hz. Figure 6 shows the steady state 2X response as a function of rotor speed and is clearly excited when the running speed is approximately half of the resonance frequency. Figure 7 shows the shaft orbit at 1240 rpm, where the 2X component is well excited, and shows that there is a significant quantitative difference between the models. However the form of the orbits are very similar. x response (dB) DYNAMIC MODELLING AND NUMERICAL MODELS -20 -60 -100 y response (dB) Hinge Mayes Jun et al. 0 -20 1000 2000 3000 1000 2000 Rotor speed (rpm) 3000 Hinge Mayes Jun et al. -60 -100 0 x response (dB) Figure 5: The steady state synchronous response (dB relative to 1 m) as a function of rotor speed Hinge Mayes Jun et al. -60 -80 -100 -120 y response (dB) 0 1000 2000 3000 1000 2000 Rotor speed (rpm) 3000 Hinge Mayes Jun et al. -60 -80 -100 -120 0 Figure 6: The steady state 2X response (dB relative to 1 m) as a function of rotor speed 614 P ROCEEDINGS OF ISMA2002 - VOLUME II y (mm) 1 Hinge Mayes Jun et al. 0 Machinery, Paper C178/76, IMechE, (1976), pp. 123-128. [4] R. Gasch, A survey of the dynamic behaviour of a simple rotating shaft with a transverse crack, Journal of Sound and Vibration, Vol. 160, No. 2, (1993), pp. 313-332. [5] I.W. Mayes, W.G.R. Davies, A method of calculating the vibrational behaviour of coupled rotating shafts containing a transverse crack, International Conference on Vibrations in Rotating Machinery, Paper C254/80, IMechE, (1980), pp. 17-27. -1 -1 0 x (mm) 1 Figure 7. The steady state orbits at 1240 rpm for the different crack models 7. Conclusions The three crack models examined have relatively little effect on the predicted steady state 1X response (Figure 5) but they do have some influence on the predicted whirl orbit and the steady state 2X response (Figure 6 and 7). However, in any crack identification scheme, these differences are not likely to have a significant effect. The advantage of using the model of Jun et al. to represent a crack is that although it requires some numerical integration, the stiffness variation is a function of the crack depth. This study is limited to a Jeffcott rotor and the foundations are therefore inherently isotropic. The next stage in the investigation is to incorporate the crack models into a beam type element so that more complex rotor models can be studied using finite element analysis. References [1] S. Prabhakar, A.S. Sekhar and A.R. Mohanty, Detection of cracks in a rotor-bearing system using wavelet transforms. Mechanical Systems and Signal Processing, Vol. 15, No. 2, (2001), pp. 447-450. [2] N. Bachschmid, P. Pennacchi, E. Tanzi, A. Vania, Identification of transverse crack position and depth in rotor systems, Meccanica, Vol. 35, (2000), pp. 563-582. [3] R. Gasch, Dynamic behaviour of a simple rotor with a cross sectional crack, International Conference on Vibrations in Rotating [6] I.W. Mayes, W.G.R. Davies, Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor, Journal of Vibration, Acoustics, Stress and Reliability in Design, Vol. 106, No. 1, (1984), pp. 139-145 [7] B. Grabowski, Vibration behaviour of a turbine rotor containing a transverse crack, ASME Design Engineering Technical Conference, St Louis, (1979), paper 79-DET-67. [8] O.S. Jun, H.J. Eun, Y.Y. Earmme, C.-W. Lee, Modelling and vibration analysis of a simple rotor with a breathing crack, Journal of Sound and Vibration, Vol. 155, No. 2, (1992), pp. 273290. [9] I.W. Mayes, W.G.R. Davies, The vibration behaviour of a rotating shaft system containing a transverse crack, International Conference on Vibrations in Rotating Machinery, Paper C168/76, IMechE, (1976), pp. 53-64. [10] N. Bachschmid, A. Vania, E. Tanzi, P. Pennacchi, Statical and dynamical behaviour of rotors with transversal crack - part I: comparison between different models, Proceedings of the XIV Congresso Nazionale AIMETA, Como, (1999). [11] P.C. Muller, J. Bajowski, D. Soffker, Chaotic motions and fault-detection in a cracked rotor chaotic rotor response, Nonlinear Dynamics, Vol. 5, No. 2, (1994), pp. 233-254. [12] P.N. Saavedra, L.A. Cuitino, Vibration analysis of rotor for crack identification, Journal of Vibration and Control, Vol. 8, (2002), pp. 51-67. [13] S.C. Huang, Y.M. Huang, S.M. Shieh, Vibration and stability of a rotating shaft containing a transverse crack, Journal of Sound and Vibration, Vol. 162, No. 3, (1993), pp. 387-401. DYNAMIC MODELLING AND NUMERICAL MODELS [14] N. Pugno, C. Surace, R. Ruotolo, Evaluation of the non-linear dynamic response to harmonic excitation of a beam with several breathing cracks, Journal of Sound and Vibration, Vol. 235, No. 5, (2000), pp. 749762. 615 [15] A. Tondl, T. Ruijgrok, F. Verhulst, R. Nabergoj, Autoparametric resonance in mechanical systems, Cambridge University Press, 2000. 616 P ROCEEDINGS OF ISMA2002 - VOLUME II
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