The Dynamics of Cracked Rotors - Michael I Friswell

The Dynamics of Cracked Rotors

John E. T. Penny† and Michael I. Friswell‡ † School of Engineering, Aston University, Birmingham B4 7ET, UK ‡ Department of Aerospace Engineering, Bristol University, University Walk, Bristol BS8 1TR, UK

ABSTRACT Cracked rotors are not only important from a practical and economic viewpoint, they also exhibit interesting dynamics. This paper will look at methods that have been used to model cracked rotors, and summarize the resulting dynamics. Of particular concern is the modeling of breathing cracks, which open and close due to the self-weight of the rotor, producing a parametric excitation. The paper will also review methods available to identify the presence a crack and highlight why they have limited success. Finally the paper will consider novel methods for crack detection, mainly focusing on the use of active magnetic bearings to apply force to the shaft of a machine. Consideration of the issues to be addressed to enable this approach to become a robust condition monitoring technique for cracked shafts will be given. 1 INTRODUCTION The idea that changes in a rotor’s dynamic behaviour could be used for general fault detection and monitoring was first proposed in the 1970s. Of all machine faults, probably cracks in the rotor pose the greatest danger and research in crack detection has been ongoing for the past 30 years. Current methods examine the response of the machine to unbalance excitation during run-up, run down or during normal operation. In principle, a crack in the rotor will change the dynamic behaviour of the system but in practice it has been found that small or medium size cracks make such a small change to the dynamics of machine system that they are virtually undetectable by this means. Only if the crack grows to a potentially dangerous size can they be readily detected. It is a race between detection and destruction! Detecting faults in rotating machines present certain problems that do not occur in fixed structures. First of all, in addition to cracks there are several other types of faults that can be present in a machine, such as a seal rub or a misaligned bearing. Ideally we wish to detect a crack in a machine rotor at an early stage in its development and estimate its size and location. However, compared to static structures, access to a rotor can be limited either in order to contain the working fluid or for safety reasons. When two or more machines are coupled together it is generally sufficient to be able to determine which machine rotor is cracked so that the correct stator is removed to gain access to the rotor. On the other hand, compared to a large static structure, the rotor of a machine is easily excited by the residual out-of-balance that always exists or by the use of active magnetic bearings. Crack detection methods fall into two groups, model updating and pattern recognition. In the former method, the dynamic behaviour of the rotor is used to update a model of the rotor and in the process determine both the severity and location of any crack. Clearly the crack model used must be adequate for the task. Bachschmid et al.[1] used a least squares method in the frequency domain to identify and locate machine faults (including a cracked rotor) whereas Markert et al.[2] used a least squares method in the time domain to identify and locate machine faults. If the pattern recognition approach is used, then whilst a crack model is not directly required, it is

desirable to have some idea of the dynamic behaviour that will result from a cracked rotor in order that it can be recognised in the pattern of behaviour. Again a model is required. 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 opened or closed depending on the size and location of the unbalance masses. In the case of the permanently opened crack, the rotor is then asymmetric and this condition can lead to stability problems. If the vibration due to any out-of-balance forces acting on a rotor is less than the static deflection of the rotor due to gravity the crack will open and close (or breathe) as the rotor turns. This is the situation that exists in the case of large horizontal machines and this paper will concentrate on this condition. 2 EQUATIONS OF MOTION OF A CRACKED ROTOR The analysis of a rotor-bearing system may be performed in fixed or rotating co-ordinates. If neither the bearings and foundations nor the rotor is axi-symmetric then the resulting differential equations, whether described in fixed or rotating coordinates, will be linear equations with harmonic coefficients. Typically foundations of a large machine will be stiffer vertically than horizontally and in this case the cracked rotor will not be axi-symmetric when the crack is open. Thus there is no compelling reason to used fixed or rotating coordinates for the analysis. To determine the stiffness of the rotor as the crack opens and closes it is easier to work in co-ordinates that fixed to the rotor and rotate with it. The reduction in stiffness due to a crack is then calculated in directions perpendicular to and parallel to the crack face, and these directions will rotate with the rotor. Having determined the rotor stiffness in rotating coordinates we transform the stiffness matrix to fixed coordinates and join the stiffness to the system inertia to obtain the equation of motion in fixed coordinates.

% and the reduction in stiffness due Let the stiffness matrix in rotating co-ordinates for the un-cracked rotor be K 0 % to a crack be K c (θ) , where θ is the angle between the crack axis and the rotor response at the crack location and determines the extent to which the crack is open. Thus the stiffness of the cracked rotor is % =K % −K % (θ ) . K cr 0 c

(1)

To convert this stiffness matrix from rotating to fixed co-ordinates we use the transformation matrix

 cos ( Ωt ) sin ( Ωt )  T=   − sin ( Ωt ) cos ( Ωt ) 

(2)

where Ω is the rotor speed which is assumed to be constant. Thus providing the un-cracked rotor is axisymmetrical we have T % T % K cr = T (Ωt ) K 0 T (Ωt ) − T (Ωt ) K c (θ ) T (Ωt ) = K 0 − K c (θ, t ) .

(3)

Thus fixed co-ordinates we have the stiffness of the cracked rotor given by

K cr = K 0 − K c ( θ, t ) .

(4)

Let the deflection of the system be q = q st + q dy where q st is the static deflection of the un-cracked rotor due to gravity and q dy is the dynamic deflection due to the rotating out of balance and the effects of the crack. Thus,

&& = q && dy , and the equation of motion for the rotor in fixed co-ordinates is q& = q& dy and q

(

)

&& dy + ( D + G ) q& dy + ( K 0 − K c ( θ, t ) ) q st + q dy = Qu + W Mq

(5)

where Qu and W are the out of balance forces, and the gravitational force respectively. Damping and gyroscopic effects have been included as a symmetric positive semi-definite matrix D and a skew-symmetric matrix G , although they have little direct bearing on the analysis. If there is axi-symmetric damping in the rotor then there will also be a skew-symmetric contribution to the undamaged stiffness matrix, K 0 . We refer to Equation (5) as the “full equations”.

The steady state deflection of the rotor varies over each revolution of the rotor since K c varies. However, the stiffness reduction due to the crack is usually small, and we may make the reasonable assumption that

K 0 >> K c ( θ, t ) . With this assumption the steady state deflection is effectively constant and equal to the static deflection, q st , given by

K 0q st = W .

(6)

&& dy + ( D + G ) q& dy + ( K 0 − K c ( θ, t ) ) q dy = Qu . Mq

(7)

Equation (5) then becomes

The second approximation commonly used in the analysis of cracked rotors is weight dominance. If the system is weight dominated it means that the static deflection of the rotor is much greater than the response due to the unbalance or rotating asymmetry, that is q st >> q dy . For example, for a large turbine rotor the static deflection might be of the order of 1 mm whereas at running speed the amplitude of vibration is typically 50 µm. Even at a critical speed the allowable level of vibration will only be 250 µm. In this situation that the crack opening and closing is dependent only on the static deflection and thus θ = Ωt + θ0 , where Ω is the rotor speed and θ0 is the initial angle. Thus Equation (7) becomes

&& dy + ( D + G ) q& dy + ( K 0 − K c ( t ) ) q dy = Qu Mq

(8)

where K c is now independent of θ. 3 MODELS OF A BREATHING CRACK The breathing crack was initially studied by Gasch[3, 4] who modelled the crack as a “hinge”. In this model the crack is open for one half and closed for the other half a revolution of the rotor, and the transition from open to closed (and vice-versa) occurs abruptly as the rotor turns. Mayes and Davies[5, 6] developed a similar model except that the transition from fully open to fully closed is described by a cosine function. Relating crack size to shaft stiffness is not easy. By using fracture mechanics concepts Jun et al.[7] estimated the changes in both direct and cross coupling stiffnesses due to a crack of a specific depth in a Jeffcott rotor. Papadopoulos and Dimarogonas[8] also used fracture mechanics concepts to provide a full 6 x 6 flexibility matrix for a transverse surface crack. Mayes and Davies[9] developed a simple relationship between the crack depth and the reduction in stiffness. The best crack model is still an unresolved issue and Keiner and Gadala[10] compared the natural frequencies and orbits of a simple cracked rotor using a beam model and two possible finite element models of the crack. Penny and Friswell[11] compared the response due to different crack models. Gasch[4] carried out a survey of the stability of a simple rotor with a crack and also considered the response of the rotor due to unbalance. Gasch also showed that as long as the resulting vibrations remain small the essentially non-linear equations of motion become linear with periodically time varying coefficients. Pu et al.[12] also considered a selfweight dominated rotor and solved the resulting equations using the harmonic balance method. 4 MAYES MODEL FOR A BREATHING CRACK We begin by considering a two degrees of freedom model of a Jeffcott rotor with a crack. The stiffness matrix of the rotor in rotating co-ordinates is

k ( ) 0  ɶ =  ɶx θ  K cr  0 k yɶ (θ)  

(9)

where x% and y% are the directions of the rotating co-ordinates, θ is the angle between the crack and the rotor response at the crack location and determines the extent to which the crack is open. In the Mayes model the stiffnesses in the x% and y% directions are given by

k x% (θ) = k xM % + k xD % cos (θ ) ,

k y% ( θ ) = k yM % + k yD % cos ( θ ) .

(10)

In these equations,

(

1 k yM % = 2 ku + koy%

)

(

)

1 k yD % = 2 ku − koy% ,

and

(11)

with similar expressions for the x% direction. Here ku is the un-cracked stiffness of the rotor in both the x% and y%

directions. When cos ( θ ) = 1 the crack is fully closed and k x% ( θ ) = k y% ( θ ) = ku , the un-cracked rotor stiffness. Thus

the rotor is axi-symmetric when the crack is closed. When cos ( θ ) = −1 the crack is fully open so that k x% ( θ ) = kox%

and k y% ( θ ) = koy% , where kox% and koy% represent the stiffness of the rotor in the two directions when the crack is fully open. Note that when the crack is open the rotor is asymmetric.

Converting from rotating to fixed co-ordinates using the transformation given in Equation (2) and carrying out the % defined by Equations (9), (10) and (11) gives matrix multiplications of Equation (3) with K cr

( ) ( ) 4 1 = 14 ( kox% − koy% ){sin ( 2θ ) − ( sin ( θ ) + sin ( 3θ ) )}. 2

1 k12 ( θ ) = k21 ( θ ) = 12 k xM sin ( 2θ ) + k xD % − k yM % % − k yD % ( sin ( θ ) + sin ( 3θ ) )

(12)

Also

k11 = ku − kc11 ( θ ) ,

k22 = ku − kc 22 ( θ ) ,

(13)

where 1 kc11 (θ ) = 12 (k xD ɶ + k yD ɶ ) + 2 (k xD ɶ − k yD ɶ )cos (2θ) −

(

)

(

)

1 kc 22 (θ ) = 1 k yD ɶ + k ɶxD + k ɶyD − k ɶxD cos (2θ) − 2

2

1 4

((3k xD ɶ + k yD ɶ )cos (θ) + (k xD ɶ − k ɶyD )cos (3θ))

1 4

((

)

(

)

(14)

)

3k ɶyD + k ɶxD cos (θ) + k ɶyD − k ɶxD cos (3θ)

In the above equations the trigonometric expressions have been expanded in multiple angles. Thus for the Mayes model K 0 and K c ( t ) of Equations (5) and (8) are given by

k K0 =  0 0

0 k0 

and

kc12  k K c ( t ) =  c11 .  kc 21 kc 22 

(15)

The above analysis confirms that for a breathing crack, K c (θ, t ) becomes K c ( t ) . The matrix K c ( t ) a periodic function of time only and the full non-linear Equation (5) becomes a linear parametrically excited equation, Equation (8). Also the model generates a constant term plus 1X, 2X and 3X rotor angular velocity components in the diagonal stiffness terms and 1X, 2X and 3X rotor angular velocity components in the off-diagonal stiffness terms. 5 EFFECT OF CRACK DEPTH Mayes and Davies[9] showed that the change in natural frequency due to a crack of specific depth and location is given by

( )

′′ ( s ) ∆ ω2N = − Ry N

(16)

Where y N ′′ ( s ) = d 2 y N dx 2

x= s

is the second derivative of the mode shape at the crack location, s, R is a [6]

function of the crack and rotor geometry and y N is the Nth mode shape. Mayes and Davies extended their study using dimensional analysis to describe the stress concentration factor at the crack front, leading to the simplified expression

( )

(

)(

)

∆ ω 2N = −4 EI 2 πr 3 1 − ν2 F ( µ ) y N ′′ ( sc )

(17)

where µ is a non-dimensional crack depth defined relative to the local rotor radius, and F is a universal function for a given shape of crack. Although F ( µ ) can be derived from the appropriate stress concentration factor, [6]

Mayes and Davies inferred the values of F from a series of experiments with chordal cracks. It was shown that, to a good approximation, this function is equal to the fractional change in the second moment of area of the shaft, measured or calculated at the crack face. It should be noted that when calculating this second moment of area in the plane of the crack, it must be referenced to the geometric centre of the remaining section of the rotor, rather than to the geometric centre of the original, undamaged rotor. Hence F is a very non-linear function of µ. For finite element analysis, Mayes and Davies[6] showed that a crack may be represented by reducing the second moment of area in a single element by ∆I . Using a Rayleigh type approach, it can be shown that

(

)

∆I I 0 r = 1 − ν2 F ( µ ) 1 − ∆I I0 Le

(18)

where Le is the length of the element with the reduced second moment of area. This parameter is, within a reasonable range, at the discretion of the modeller. For given values of r, ν and F ( µ ) , as Le is increased then

∆I I decreases. Thus as the element is increased in length, the reduction in the second moment of area acts over a longer section of the shaft but it is less severe. 4 ACTIVE MAGMETIC BEARINGS Active magnetic bearings (AMB) have been used in high-speed applications or where oil contamination must be prevented, although their low load capacity restricts the scope of applications. Recently a number of authors considered the use of AMBs as an actuator that is able to apply force to the shaft of a machine. Bash[13] studied the use of AMBs as an actuator for rotor health monitoring in conjunction with conventional support bearings, by applying a variety of known force inputs to a spinning rotor system. If the applied force is periodic, then the presence of the crack generates responses containing frequencies at combinations of the rotor spin speed and applied forcing frequency. The character and frequency of the force, and the dynamics of the AMB are important [14,15] [16] for the crack detection. Mani et al. and Quinn et al. studied the effect of the force from an AMB on the response of rotating machinery, including the external forcing frequency and amplitude. The excitation by unbalance and AMB forces produces combination resonances between critical speed of the shaft, the rotor spin speed and the frequency of the AMB excitation. The key is to determine the correct excitation frequency to induce a combination resonance that can be used to identify the magnitude of the time-dependent stiffness arising from the breathing mode of the rotor crack. 5 CONDITION MONITORING USING ACTIVE MAGNETIC BEARINGS The force applied on the rotor by the AMBs must be included in the equations of motion. Thus Equation (8) becomes

&& dy + ( D + G ) q& dy + ( K 0 − K c ( t ) ) q dy = Qu + Q AMB Mq

(19)

where Q AMB is the external forces applied to the rotor by the active magnetic bearing. This force will probably be chosen to be harmonic, either in one or two directions. Other waveforms would be possible if they were perceived to offer some advantage. The key aspect of the analysis is that the system has three different frequencies, namely the natural frequency (or critical speed), the rotor spin speed and the forcing frequency from the AMB. The parametric terms in the equations of motion (or non-linear terms in the full equations) cause combinational resonances in the response of the machine. Mani et al.[14,15] and Quinn et al.[16] used a multiple scales analysis to determine the conditions required for a combinational resonance, which occurs when

Ω 2 = nΩ − ωi ,

for n = ±1, ±2, ±3

(20)

where Ω is the rotor spin speed, Ω 2 is the frequency of the AMB force, and ωi is a natural frequency of the system. This analysis was based on a two degree of freedom Jeffcott rotor model with weight dominance, equivalent to that described in this paper. Mani et al. also considered the effect of detuning, that is when the excitation is close to this exact excitation frequency for resonance, and investigated the effect on the magnitude of the primary resonance close to the natural frequency of the machine. In the examples the running speed of the machine was five times higher than the natural frequency. This ratio is not practical since there is likely to be a second un-modelled resonance below the running speed. Indeed the fact that higher resonances are not modelled is a serious omission, particularly as the combinational resonances are likely to excite any higher frequency resonances. The AMB force is usually sinusoidal. Mani et al.[14] introduced the possibility of a periodic step force profile. This profile has two major disadvantages. First a step change in force will excite the higher modes of the structure making the identification of the key features that determine a cracked shaft difficult. Secondly AMBs are unable to provide such a force profile and their maximum rate of change of force is limited. 7 EXAMPLE The rotor model used for the following simulations is shown in Figure 1. The rotor is modelled using 6 elements. For all simulations the rotor speed was 960 rev/min (16 Hz). For the undamaged rotor this speed, the first and second damped natural frequencies for the forward rotating modes were 14.0 and 42.9 Hz. At this speed the out of balance at disc 2 was 101 N and, when applied, the active magnetic bearing (AMB) applied a peak harmonic force of 100 N. Figure 2 shows the Campbell diagram for the rotor system. Simulations (not shown here) verified that the results from the finite element model were not sensitive to the length of the crack element over a reasonable range. unbalance

disk 1 disk 2

bearing 1

cracked element AMB force

Figure 1. The example rotor system.

bearing 2

Damped natural frequency (Hz)

250 200 150 100 50 0

0

2000 4000 Rotor spin speed (rev/min)

6000

Figure 2. Campbell diagram.

Response magnitude (µm)


Figure 3 and 4 show the response of the rotor at disc 2 when the rotor was undamaged, and when a crack with a depth of 10% of the rotor diameter was introduced. In both cases the AMB frequency was 18.75 Hz, based on an assumed natural frequency of 13.25 Hz (slightly less than the true natural frequency of the uncracked rotor), with n = 2 in Equation (20). In the case of the cracked rotor we clearly see clusters of frequencies that should be readily detected. Figures 5 and 6 show the response of the cracked rotor when n = 1 (giving an AMB frequency of 27.5 Hz) and when n = −3 (giving an AMB frequency of 61.25 Hz). The pattern of cluster frequencies is similar to those of Figure 4, but not identical. Figures 7 and 8 show the effect of using a poor estimate of the natural frequency. Estimated natural frequencies of 14 Hz and 11 Hz are used with n = 2 in Equation (20). Although the cluster of frequencies is still present, the frequencies disappear as the estimate of natural frequency moves outside of this range.

0

10

−4

10

−8

10

0

20

40 60 Frequency (Hz)

80

Figure 3. Response for uncracked rotor (Ω2=18.75Hz).

0

10

−4

10

−8

10

0

20


80

Figure 4. Response for cracked rotor (Ω2=18.75Hz).



0

10

−4

10

−8

10

0

20


−4

10

−8

10

0

20


80

Figure 7. Response for cracked rotor (Ω2=18Hz)

−8

0

20


80

Figure 6. Response for cracked rotor (Ω2=61.25Hz). Response magnitude (µm)


Figure 5. Response for cracked rotor (Ω2=27.5Hz).

0

−4

10

10

80

10

0

10

0

10

−4

10

−8

10

0

20


80



In all of the examples so far the AMB frequency has been calculated using a natural frequency close to the first natural frequency of the rotor. In contrast, Figure 9 shows the response when the AMB frequency is close to the second natural frequency of the system, with n = 2 in Equation (20). The clusters of frequencies are now more spread out and it is difficult to recognise where one cluster ends and another one begins.

0

10

−4

10

−8

10

0

20


80


8 CONCLUSIONS A variety of approaches have been proposed to try to make use of the changes in the dynamics of a rotor to identify and possibly locate crack (and other faults) in a rotor at an early stage in their development. Several approaches have been briefly mentioned in this paper and there are others. The simulations shown suggest that the use of an auxiliary active magnetic bearing (AMB) to help identify crack in the rotor has some merit, but further work is needed to produce a robust condition monitoring technique. Applying a sinusoidal force from the AMB produces combination frequencies based on the AMB frequency, the rotational speed, and the natural frequencies, that could be used to detect cracks in the rotor. However a robust method is needed to determine the presence, location and severity of a crack from the combination frequencies in the response. Furthermore the effect of adding an extra force to the system might encourage a faster crack growth, which is obviously a disadvantage. ACKNOWLEDGMENTS Michael Friswell gratefully acknowledges the support of the royal Society through the Royal Society-Wolfson Research Merit Award and the European Commission through the Marie-Curie Excellence grant, MEXT-CT-2003002690. REFERENCES [1]

Bachschmid, N., Pennacchi, P., Tanzi, E. and Vania, A., Identification of transverse crack position and depth in rotor systems, Meccanica, Vol 35, (2000), pp. 563-582.

[2]

Markert, R., Platz, R. and Seidler, M., Model based fault identification in rotor systems by least squares fitting, International Journal of Rotating Machinery, Volume 7 (2001), Issue 5, pp. 311-321.

[3]

Gasch, R., Dynamic behaviour of a simple rotor with a cross sectional crack, International Conference on Vibrations in Rotating Machinery, Paper C178/76, IMechE, (1976), pp. 123-128.

[4]

Gasch, R., 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]

Mayes, I.W. and Davies, W.G.R., 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.

[6]

Mayes, I.W. and Davies, W.G.R., 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]

Jun, O.S., Eun, H.J., Earmme, Y.Y. and Lee, C.-W., Modelling and vibration analysis of a simple rotor with a breathing crack, Journal of Sound and Vibration, Vol. 155, No. 2, (1992), pp. 273-290.

[8]

Papadopoulos, C.A. and Dimarogonas, A. D., Coupled longitudinal and bending vibrations of a rotating shaft with an open crack, Journal of Sound and Vibration, Vol. 117, No. 1, (1987), pp. 81-93.

[9]

Mayes, I.W. and Davies, W.G.R., 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]

Keiner, K. and Gadala, M.S., Comparison of different modelling techniques to simulate the vibration of a cracked rotor, Journal of Sound and Vibration, Vol. 254, No. 5, (2002), pp. 1012-1024.

[11]

Penny, J.E.T. and Friswell, M.I., Simplified modelling of rotor cracks, ISMA 27, Leuven, Belgium, (2002), pp. 607-615.

[12]

Pu, Y.P., Chen, J., Zou, J. and Zhong, P., Quasi-periodic vibration of a cracked rotor on flexible bearings, Journal of Sound and Vibration, Vol. 251, No 5, (2002) pp 875-890.

[13]

Bash, T.J., Active Magnetic Bearings used as an Actuator for Rotor Health Monitoring in Conjunction with Conventional Support Bearings, Masters Thesis, Department of Mechanical Engineering, Virginia Tech, USA

[14]

Mani, G., Quinn, D.D. and Kasarda, M.E.F., Active health monitoring in a rotating cracked shaft using active magnetic bearings as force actuators, Journal of Sound and Vibration, Vol. 294, (2005), pp. 454465.

[15]

Mani, G., Quinn, D.D., Kasarda, M.E.F., Inman, D.J. and Kirk, R.G., Health monitoring of rotating machinery through external forcing, Proceedings of ISCORMA-3, Cleveland, Ohio, USA, 19-23 September 2005.

[16]

Quinn, D.D., Mani, G., Kasarda, M.E.F., Bash, T.J. Inman, D.J. and Kirk, R.G., Damage detection of a rotating cracked shaft using an active magnetic bearing as a force actuator - analysis and experimental verification, IEEE/ASME Transactions on Mechatronics, Vol. 10, No. 6, (2005), pp. 640-647.