Where Next for Condition Monitoring of Rotating Machinery?

Where Next for Condition Monitoring of Rotating Machinery? A. W. Lees 1 and Michael I. Friswell 2 1 School of Engineering, University of Wales, Swansea, Singleton Park, Swansea, SA2 8PP, UK a.w.lees@swansea.ac.uk 2 Department of Aerospace Engineering, Bristol University, Queen s Building, University Walk, Bristol BS8 1TR, UK m.i.friswell@bristol.ac.uk Abstract Condition monitoring has progressed for many years with significant but fairly incremental improvements to algorithms for data analysis. Pattern recognition methods for the condition monitoring of rotating machinery are well established and can successfully detect and classify many types of faults. However, some faults are difficult to distinguish and locating faults requires a spatial model of the machine. To be useful, these models must be an accurate representation of the machine. Of course, as with any model, the required accuracy depends on the use to which the model is put. This paper considers models of various faults and machines, highlights the use of estimation methods, and discusses the required model fidelity. Examples of faults such as cracks, unbalance, bends and misalignment are given. Consideration is given to the information that may be established about an operational machine, and how it might be implemented in future condition monitoring systems to significantly enhance their value. Keywords: Condition monitoring, Vibration, Rotating machinery, Models, Fault diagnosis 1 Introduction Large machines, such as turbogenerators, have a high capital cost and hence the development of condition monitoring techniques for such rotating machines is important. Condition monitoring has a wide variety of interpretations, ranging from a subjective judgement from relatively crude measurements to the use of advanced signal processing on plant data to form the basis of maintenance scheduling. It seems inevitable and certainly desirable, that the trend towards advanced signal processing will continue for high value equipment. The basic goal is to gain the maximum information from the measurement data available, although this raises questions about the quantity and quality of data required. Condition monitoring has always applied; engineers have taken out of service machines that exhibit excessive noise and/or vibration or showing a marked deterioration in performance. Significant effort has been aimed at the detection and diagnosis of incipient faults and the minimization of false alarms, and the increase in computing power has played an important role [1, 2]. Many operational parameters play a role in condition monitoring but at present there is some difficulty in merging operational information with the variations of vibration amplitudes under nominally steady running conditions, and indeed this is a feature of the inadequacy of the models currently in use. The monitoring may be either online or offline, or based on statistics or physics, and extensive experience has been gained within each of these classifications. For a large turbo-alternator, the practice for many ADVANCES IN VIBRATION ENGINEERING, 5(4) 2006 KRISHTEL emaging SOLUTIONS PRIVATE LIMITED

264 A. W. LEES AND MICHAEL I. FRISWELL / ADVANCES IN VIBRATION ENGINEERING, 5(4) 2006 years has been to gather data during machine run-downs. This data is rich in information as a number of resonances of the machine may be excited as the rotor traverses the appropriate speed range. In addition to these run-downs, on-load data is also available. Since many machines, such as turbo-alternators, operate at constant speed the data is not rich in information, although the quantity of data is large. The simplest possible scheme would be classified as alarm triggering. Such a scheme would monitor online levels with a pre-set level that triggers an investigation, together with some higher level at which the machine in question would be removed from service. Most, if not all, of the published standards on vibration levels are based on this type of concept. The prescribed levels for different courses of action (investigate or abort) are based on a wealth of experience with a variety of machines. The vibration-based detection of faults such as rotor unbalance, rotor bent, crack, rub, misalignment, fluid induced instability are well-developed and widely used in practice [3, 4], however the quantification parts (the extent of identified faults and their locations) have been active areas of research for many years. Over the past thirty years, theoretical models have played an increasing role in the rapid resolution of problems in rotating machinery. In spite of the success of model-based estimation of faults, a complete mathematical model is still not available for condition monitoring in many cases. The models required for condition monitoring fall into two categories, namely models of faults and models of machines. Essentially the faults will cause the response behavior to change, but these changes will be filtered by the dynamics of the machine. The well-established characteristics that are used to diagnose machine faults are based on the dynamics of the faults. How these characteristics are affected by the machine is often very subtle and difficult to determine without a model of the machine. The other major question concerns model fidelity and accuracy. Through simulation it may be possible to distinguish small changes in response due to different faults or locations. However, the model will contain errors and the measurements will be contaminated by noise. To robustly detect and locate faults the change in response must be larger than these model errors and measurement noise. This paper will discuss many of the general issues, but will concentrate on a number of models as examples. The uncertain part of the machine model is assumed to be the foundation. Faults considered include cracked shafts, unbalance, bends and misalignment. 2 Models and Identification Methods Dynamic models for structures and rotating machines can take many different forms and contain different parameters. Model updating in structural dynamics [5] considers, predominantly, finite element models with a small number of uncertain physical parameters, for example for stiffness modeling of joints. On the other hand experimental modal analysis of structures or rotating machines [6] considers a model based on the properties of modes. Although not based on the physical properties of the machine, the resulting model does include spatial information, provided sufficient measurement points are used (which is often difficult due to lack of access). To obtain the best model, it is important to use whatever information is available on the system. Thus, if part of the system, for example the rotor, can be modeled accurately, then it is important that the identification concentrates on those parts of the system that are difficult to model. This will be discussed more in detail later in this paper. 2.1 Estimation methods When the parameters of a model are unknown, they must be estimated using measured data. Usually the measured response will be a nonlinear function of the parameters. In these cases, minimizing the

WHERE NEXT FOR CONDITION MONITORING OF ROTATING MACHINERY? 265 error between the measured and predicted response will produce a nonlinear optimisation problem, with the usual questions about convergence and local minima. The most common approach is to linearise the residuals, obtain a least squares solution and iterate. If the identification problem is well-posed then this simple approach will be adequate. The usual response to problems encountered in the optimisation is to try more advanced algorithms, but often the issue is that the estimation problem has not been posed correctly, and including some physical insight into the problem provides a much better solution. Probably the most important difficulty in parameter estimation is ill-conditioning. In the worst case this can mean that there is no unique solution to the estimation problem, and many sets of parameters are able to fit the data. Many optimization procedures result in the solution of linear equations for the unknown parameters. The use of the singular value decomposition (SVD) [7] for these linear equations enables ill-conditioning to be identified and quantified. The options are then to increase the available data, which is often difficult and costly, or to provide extra conditions on the parameters. These can take the form of smoothness conditions (for example, the truncated SVD), minimum norm parameter values (Tikhonov regularization) or minimum changes from the initial estimates of the parameters [8, 9]. 2.2 Foundation models Power station turbogenerators may be considered to consist of three major parts; the rotor, the fluid journal bearings and the foundations. In many modern plants, these foundation structures are flexible and have a substantial influence on the dynamic behavior of the machine. Often a good finite element model of the rotor and an adequate model of the fluid bearing may be constructed. Indeed several finite element based software packages are available for such modeling. However, a reliable model for the foundation is difficult, if not impossible, to construct due to a number of practical difficulties. Inclusion of the foundation model is very important as the dynamics of a flexible foundation also contributes significantly to the dynamics of the complete machine. Many research studies have suggested methods to estimate the foundation models directly from the measured machine responses during run-down [10, 11], although more research is needed on their practical application. Regularization is often vital to obtain accurate results. Row and column scaling of the resulting matrices is required because of the difference in magnitude between the mass and stiffness parameters, and also because of the large frequency range. Other physically based constraints may be applied to the foundation model to improve the conditioning. For example, the mass, damping and stiffness matrices of the foundation may be assumed to be symmetric, therefore reducing the number of unknown foundation parameters. Other constraints could be introduced, such as a diagonal mass or damping matrix, or block diagonal matrices if the bearing pedestals do not interact dynamically. Some methods require the inversion of the dynamic stiffness matrix of the rotor where the degrees of freedom at the bearing locations have been fixed (that is the rotor is pinned at the bearings). This matrix is almost singular at frequencies close to the natural frequencies of the pinned rotor. The ill-conditioning is particularly bad for lightly damped rotors, and the errors introduced result in meaningless parameter estimates. One option is to remove the run-down data within a small frequency band close to these problem frequencies. Although the foundation estimation problem is ill-conditioned, and so it is difficult to obtain a physically meaningful model of the foundation, in some instances this may not be a problem. For example, the estimation of unbalance seems robust to ill-conditioning in the foundation estimation, and this will be considered further later.

266 A. W. LEES AND MICHAEL I. FRISWELL / ADVANCES IN VIBRATION ENGINEERING, 5(4) 2006 2.3 Neural networks and other methods Black box methods are often not considered as model based approaches. However, any simulation of an input-output relationship must make some assumptions about the underlying process, and hence essentially has an underlying model. For example, a neural network is essentially a very sophisticated curve fitting algorithm, and ill-conditioning is a major problem, evidenced by over-fitting and a lack of generalization. The advantage of neural networks is that the class of input-output relationships that may be fitted is huge. However, better results will always be obtained if physical insight is used to guide the modeling and estimation process. Indeed there is often a need to reduce the number of input nodes to present to a neural network, and understanding is vital to obtain the correct feature extraction and data reduction. Another use of physical models is the generation of training or test data for these identification schemes. Typically experimental data for a sufficient range of events is difficult or expensive to obtain. Since running a model many times is relatively easy and cheap, these simulations may be used to increase the test data. However, it is vital that this simulated data correctly reproduces the important features of the real machine, and hence requires a validated and, if necessary, updated model. 3 Models for Crack Detection and Location There are a number of approaches reported in the literature for modeling cracks in shafts. 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. The situation where the vibration due to any out-of-balance forces acting on a rotor is less than the deflection of the rotor due to gravity of more importance in large machines. In this case the crack will open and close (or breathe) due to the rotation of the rotor. Friswell and Penny gave more detail of breathing cracks in static structures [12] and in rotating shafts [13]. Sinou and Lees [14] considered the influence of a breathing crack, particularly where the rotor speed is approximately half a machine resonance frequency. Lees and Friswell [15] considered the dynamics of machines with asymmetric rotors with cracks. Although two and three dimensional finite element models may be used, the most popular approaches are based on beam models. How refined does the model have to be to detect and locate cracks in shafts? 3.1 Models of breathing cracks The simplest model of an opening and closing crack is the hinge model [16, 17]. In this model, the crack is assumed to change from its closed to open state (and vice-versa) abruptly as the shaft rotates. 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 [18, 19] proposed a model in which the opening and closing of the crack was described by a cosine function. 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. Stiffnesses, as well as the direct stiffnesses were estimated as the crack opens and closes. Numerical integration over the area of the proportion of the crack that is open is required. For a particular angle of rotation, the Jun model is a function of the shaft length and diameter, the modulus of elasticity of the shaft material, the depth of the crack and the lateral force.

WHERE NEXT FOR CONDITION MONITORING OF ROTATING MACHINERY? 267 Fig. 1 Normalized direct and cross stiffnesses in rotating coordinates for the Jun et al. [20] model It is instructive to consider the frequency components of the different crack models as the shaft rotates in fixed coordinates. This may be obtained using a Fourier Series of the stiffness variation in rotating coordinates and then transforming to fixed coordinates [13]. Because of the instantaneous switching of the stiffness the hinge model contains higher frequency components. However, the Mayes model only contains components up to 3X. The Jun model also contains higher frequency components. Figure 1 shows the direct and cross stiffness terms in rotating coordinates for a 40% crack using the Jun model, and should be compared to a step for the hinge model, or a cosine function for the Mayes model (in both cases the cross terms are zero). 3.2 The effect of breathing cracks A breathing crack opens and closes as the shaft rotates, leading to equations of motion with parametric excitation. These may be solved using the Harmonic Balance Method to estimate the steady state response of the machine. A Jeffcott rotor modeled 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 center. The crack is assumed to be located close to the disc and only responses that are symmetrical about the shaft center 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, and this gives rise to a stiffness reduction of approximately 9% when the crack is open. In order to compare the crack modeling 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. [20]. 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 2 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 3 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 4 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. The three crack models examined have relatively little effect on the predicted steady state 1X response but they do have some influence on the predicted whirl orbit and the steady state 2X response. However, in any crack

268 A. W. LEES AND MICHAEL I. FRISWELL / ADVANCES IN VIBRATION ENGINEERING, 5(4) 2006 Fig. 2 The steady state synchronous response Fig. 3 The steady state 2X response Fig. 4 The steady state orbits at 1240 rpm for the different crack models identification scheme, these differences are not likely to have a significant effect and simple models are more readily used. 4 Balancing Rotor unbalance is the most common cause of vibration in rotating machinery, and over the last 40 years various balancing techniques have been introduced to reduce such vibration [21, 22]. Generally speaking, no matter what method is selected, employing the measurement data from multiple transducers is able to balance rotors in multi-planes, where trial and correction masses can be added. For the influence coefficient, the measurements are often taken at a single speed. 4.1 Influence coefficient methods viewed as an identification procedure Single- and multi-plane balancing techniques are essentially estimation approaches. For single plane balancing, measurements are taken at the single balancing plane and an addition run with a trial unbalance

WHERE NEXT FOR CONDITION MONITORING OF ROTATING MACHINERY? 269 is performed. In the as found condition the measured response, r = r 0 e j t, depends on the inherent unbalance through Z( )r 0 = mε 2 e jβ (1) where Z( ) is the dynamic stiffness at the balance disk, and mε and β are the magnitude and phase of the unbalance. For identification it is important which quantities are unknown, and in this case it is the dynamic stiffness and the unbalance. Clearly there is not sufficient information as it stands to estimate these from a single response, and so a trial unbalance is added producing synchronous response r = r 1 e j t. Now there are two measured complex quantities, r 0 and r 1, for the two complex unknowns, Z( ) and mεe jβ, and thus a solution may be found in the usual way. The estimation requires the division by r v = r 1 r 0, and if this quantity is small, then noise will cause the estimates to be inaccurate. Thus, for a well-conditioned estimation problem, we require that the trial unbalance causes a significant change in the response (compared to the measurement noise and other uncertainties). Two plane balancing is similar, although now the unknown dynamic stiffness is a 2 2 matrix and the equivalent inherent unbalance is estimated at the two balance planes. This requires run-downs with two trial unbalance mass distributions on the two balance planes, in addition to the run-down in the as found condition. The standard approach uses the measured responses at balance planes at the running speed. Counting the unknown quantities, we have four dynamic stiffness coefficients and the equivalent unbalance at the two planes, giving six complex quantities. We have two complex response measurements from three run-downs, and so we should in theory have sufficient information for the estimation. The estimation of the dynamic stiffness matrix requires the inversion of the matrix [ r v1 r v2 ] = [ (r 1 r 0 ) (r 2 r 0 ) ] (2) where r 0 is the as found response at the two balance planes, and r 1 and r 2 are the responses with the trial unbalance masses. Both of the trial unbalance configurations must have a significant effect on the response of the machine, otherwise one or more columns of the matrix in (2) will be small. Furthermore, it is well-known that the balance planes must be carefully positioned so that the trial unbalance masses are able to give different responses. The problem is that the matrix in (2) must be well conditioned, which means that the columns must be as close to orthogonal as possible. This not only requires choosing the trial unbalance mass positions carefully, but also requires that the balance planes are sufficiently well separated in the machine. More trial unbalance runs may be incorporated, and in this case the matrix in (2) has more columns. For example, with three trial unbalance runs [ r v1 r v2 r v3 ] = [(r 1 r 0 ) (r 2 r 0 ) (r 3 r 0 )]. (3) The estimation of the dynamic stiffness matrix is now over-determined and the solution may be obtained using weighted least squares or the pseudo inverse. The extra measurements will help to average any noise present in the measurements, and the use of three trial unbalance configurations mean that the chances of choosing a poor set of trial unbalances is reduced. However, if the balance planes are poorly chosen then the estimation will still be ill-conditioned. The extension to three or more balance planes or multiple shaft speeds is straight-forward [23].

270 A. W. LEES AND MICHAEL I. FRISWELL / ADVANCES IN VIBRATION ENGINEERING, 5(4) 2006 4.2 Robust balancing and general methods The last example has indicated a more general approach where the unbalance is estimated that minimizes the errors in the predicted responses in some least squares sense. The measurements may be located anywhere on the machine, and be many more than the number of balance planes. Equally the response may be measured at more than one shaft speed. The question is whether these increased number of measurements help in the estimation process. Clearly adding more measurement locations will require the estimation of more dynamic stiffness terms. Measuring at multiple shaft speeds will require the estimation of multiple dynamic stiffness matrices, unless a physical model of the machine is built (this is discussed more fully in the next section). Of course, we have a good model of the forces generated by the unbalance as a function of shaft speed, and the number of balance disks is assumed fixed. Thus, the number of parameters related to the inherent unbalance do not change with the number of measurements, leading to the potential to improve the conditioning of the problem. In all of this sight should not be lost of the objectives of the balancing, and this affects the choice of residuals to minimize during the estimation. Garvey et al. [24] introduced the concept of robust balancing, which is particularly suitable for rotors that are balanced in balancing machines. Of particular concern is when the balancing machine is not dynamically similar to the stator of the machine in which the rotor is ultimately deployed. The approach combines the expected distribution of rotor properties, the expected distribution of stator and bearing properties and a cost function reflecting the relative importance of various balancing objectives, to determine an optimal balance correction. The objective function is based on the mean square response at a number of measurement locations, but now also includes variations in parameters (for example stator stiffness) that are weighted according to how likely they are. Minimization of this objective function to produce the optimum values for the balance masses requires a model to incorporate the uncertainty. 4.3 Unbalance estimates from a single run-down In a series of papers Lees, Friswell and co-workers [11, 25 27] have developed an approach to estimate faults on a machine when the stator or foundation is flexible and unknown. The equations of motion of the machine may be written as Z R,ii Z R,ib 0 Z R,bi Z R,bb + Z B Z B 0 Z B Z B + Z F r R,i r R,b r F,b = where Z is the dynamic stiffness matrix, the subscripts b and i refer to internal and bearing (connection) degrees of freedom respectively, and the subscripts F,R, and B refer to the foundation, the rotor and the bearings. r are the responses and f u are the force vectors, which are assumed to be applied only at the rotor internal degrees of freedom. The dynamic stiffness matrix of the foundation, Z F, is defined only at the degrees of freedom connecting the bearings and the foundation. In practice, this will be a reduced order model, where the internal foundation degrees of freedom have been eliminated. The dynamic stiffness matrix of the bearings is given by Z B. It has been assumed that the inertia effects within the bearings are negligible, although these could be included if required. Solving (4) to eliminate the unknown response of the rotor gives Z F r F,b + Z B P 1 Z R,bi Z 1 R,ii f u = Z B [P 1 Z B I]r F,b (5) f u 0 0 (4)

WHERE NEXT FOR CONDITION MONITORING OF ROTATING MACHINERY? 271 where P = Z R,bb + Z B Z R,bi Z 1 R,ii Z R,ib. For the identification it remains to determine which quantities are known and which are to be estimated. Of course all component models in (5) are functions of the shaft speed, and the measurements are the synchronous responses during a run-down. It is assumed that good models for the rotor and the bearings, Z R and Z B, are known a priori and that the response r F,b is measured. Here, we assume that the synchronous force vector (unbalance or bend), f u, foundation model parameters, Z F, are unknown. The foundation model will be modeled using mass, damping and stiffness matrices, and the frequency range will be split if required. The unbalance is estimated at a number of pre-determined balance planes. The number of planes should be considered carefully based on the influence of the modes in the operating speed range of interest. If there are more balance planes than modes of interest, then a unique unbalance estimate will not be obtained, irrespective of the balancing method chosen, although the modal unbalance in the lower modes will be similar. Equation (5) then gives a linear set of equations for the unbalance and foundation parameters at each shaft speed. The unbalance parameters consist of a magnitude and angular position at each balance plane, and the dependence on shaft speed squared is used to reduce the number of parameters to be estimated. A bend is very similar to an unbalance except that the force is constant with shaft speed, and this difference allows bends and unbalance to be distinguished. The method was applied to a flexible rotor mounted on four fluid bearings, with a flexible foundation, located at Aston University, Birmingham. The rig consists of a solidly coupled, two-shaft system mounted on four oil lubricated journal bearings. The bearings sit on flexible steel pedestals bolted onto a large lathe bed which rests on a concrete foundation. The rotor itself consists of two steel shafts of 1.56 m and 1.175 m long, each with nominal diameter of 38 mm. Accelerometers are mounted at each bearing measuring in the horizontal and vertical directions. A finite element model was created for the rotor with 51 two-noded beam elements. Short bearing theory was used to obtain values for the bearing stiffness and damping. Lees et al. [28] gave more detail. The machine was run-down from 55 Hz to 5 Hz in 210 frequency steps. The first order responses for the horizontal and vertical acceleration at the bearing pedestals were extracted. The static load at the bearings was estimated to be 221 N, 486 N, 461 N and 400 N at bearings 1 to 4 respectively. Three runs were performed, the first with the residual unbalance and the second and third cases with the addition of different unbalance weight distributions. Since the residual unbalance was unknown, the unbalance was estimated in the as found condition and also with the unbalance weights added. The unbalance estimates were subtracted and compared to the actual mass added. The frequency range was split into four bands, namely 5 17 Hz, 17 28 Hz, 28 40 Hz and 40 55 Hz. Table 1 gives the estimated results. The amplitude of the unbalance is consistently estimated more accurately than the phase. This may be because of errors in the bearing model, or maybe small phase shifts in the measurement system. Investigations are continuing in methods to estimate bearing Table 1 Unbalance estimate from the experimental run-down data by subtracting unbalance Unbalance Estimated (Amplitude (gm) @ phase (deg.)) Actual Unbalance Runs Planes First Run Second Run Added Unbalance Added (gm @ deg.) 1 and 2 Disk 2 0.958 @ 210.11 0.742 @ 85.60 1.508 @ 54.03 1.70 @ 105 Disk 5 0.860 @ 6.90 1.067 @ 153.44 1.847 @ 168.31 1.70 @ 180 1 and 3 Disk 1 1.053 @ 209.65 2.283 @ 266.36 1.919 @ 293.67 1.70 @ 225 Disk 5 0.883 @ 11.71 2.498 @ 41.40 1.785 @ 56.43 1.70 @ 315

272 A. W. LEES AND MICHAEL I. FRISWELL / ADVANCES IN VIBRATION ENGINEERING, 5(4) 2006 models from measured data [29]. The fit of the estimated responses to the measured responses is not particularly good, because of the ill-conditioning of the foundation model parameters, and the fact that an equation error rather than an output error approach is used. However it is encouraging that the unbalance estimation is still excellent despite these problems, showing that the unbalance estimation is robust. 5 Misalignment and Static Load Estimation In machines with many bearings, the alignment is critical for smooth operation but is difficult to measure. One major effect of misalignment is to alter the load taken by individual bearings, and because fluid bearing clearances are small, even small changes in alignment can lead to significant differences in load. The static load on a bearing alters the stiffness and bearing properties of the bearing and this effect may be predicted by a simplified bearing model, for example short bearing theory [30]. Since the bearing stiffness and bearing alters the machine response the approach adopted for unbalance estimation above, may be extended to estimate static load and hence misalignment. For a given bearing model, the dynamic stiffness Z B, will be a nonlinear function of load. Minimizing (5) for the bearing loads is then a nonlinear optimization problem, although the bearings will be relatively low in number. An alternative approach for couplings is to model the misalignment as a static force and moment [27]. This force and moment may be related to linear and angular displacements using the coupling stiffness. The procedure is demonstrated on a small rig at the University of Wales Swansea, shown in Fig. 5. Each foundation of this rig consists of a horizontal beam (500 mm 25.5 mm 6.4 mm) and a vertical beam (322 mm 25.5 mm 6.4 mm) made of steel. The horizontal beam is bolted to the base plate and the vertical beam to the bearing assembly as seen in the photograph. A layer of acrylic foam (315 mm 19 mm 1 mm) was bonded to the vertical beam and thin layers of metal sheet (315 mm 19 mm 50 µm) added to increase the damping. Modal testing confirms that the damping of the foundation increased from 0.6% to 1.4% for the first lateral mode, due to this damping layer. A steel shaft of 12 mm outside diameter and 980 mm length is connected to these flexible supports and is coupled to the motor through a Fig. 5 Photograph of the experimental rig at the University of Wales Swansea

WHERE NEXT FOR CONDITION MONITORING OF ROTATING MACHINERY? 273 Table 2 Estimation of both the rotor unbalance and misalignment from the experimental run-down data for the small rig Added Unbalance Actual Estimated Estimated Misalignment (g @ deg.) Unbalance Unbalance y z θ z θ y With respect With respect Run Disk (g @ deg.) (g @ deg.) (mm) (mm) (deg.) (deg.) to Run 1 to Run 4 1 A Residual 1.84 @ 130 0.10 0.21 0.40 0.69 B Residual 1.46 @ 350 4 A Residual 1.99 @ 127 0.09 0.18 0.32 0.20 B Residual 1.44 @ 341 2 A 0.76 @ 180 2.48 @ 144 0.14 0.20 0.75 0.45 0.82 @ 181 0.82 @ 190 B 0.76 @ 0 2.33 @ 354 0.88 @ 4 0.98 @ 14 3 A 1.52 @ 180 2.75 @ 158 0.14 0.18 0.42 0.16 1.41 @ 191 1.46 @ 202 B 1.52 @ 0 2.96 @ 1 1.55 @ 11 1.67 @ 18 5 A 0.76 @ 45 2.54 @ 114 0.08 0.21 0.24 0.17 0.95 @ 79 0.76 @ 76 B 0.76 @ 225 1.34 @ 323 0.66 @ 236 0.45 @ 228 flexible coupling. The translational stiffness and rotational stiffness of the flexible coupling are estimated to be 27 kn/m and 25 Nm/rad respectively [27]. The flexible foundations are connected at 15 mm and 765 mm from the right end of the shaft through self-lubricating ball bearings. The shaft also carries two identical balancing disks of 75 mm outside diameter and 15 mm thickness and placed at 140 mm and 640 mm from the right end of the shaft. Disk A is defined to be the one nearest to the motor. Different run-down experiments were performed with the rotor speed reducing from 2500 rpm to 300 rpm for different combinations of added mass to the balance disks A and B listed in Table 2. Runs 1 and 4 were residual run-downs i.e., without any added mass to the disks. Order tracking was performed in the frequency range from 5.094 Hz to 40.969 Hz in steps of 0.125 Hz. Table 2 shows that the estimated unbalance is excellent and close to the actual values and the estimated misalignment in the rotor at the coupling is quite consistent for each run. The order of the estimated misalignment (approximately 0.1 and 0.2 mm in the horizontal and vertical directions and their related angular misalignment is 0.4 and 0.2 respectively) is very small and such a misalignment is quite possible during the rig assembly. The small deviation in the estimation may be because of noise in the measurements, however the estimation seems to be quite robust. Hence this experimental example confirms that an accurate estimation of both the rotor unbalance and misalignment is possible using measured responses from a single run-down of a machine. 6 Bearing Models Bearings, including rolling element bearings, fluid-film bearings (journal and thrust, hydrostatic, hydrodynamic, hybrid, gas-lubricated and squeeze-film), magnetic and foil bearings, are crucial components of large machinery. All bearings may be characterized by their dynamic properties at a limited number of connection degrees of freedom between the rotor and stator. In actual test conditions obtaining reliable estimates of the bearing operating conditions is difficult and this leads to inaccuracies in the well-established theoretical bearing models. To reduce the discrepancy between response measurements

274 A. W. LEES AND MICHAEL I. FRISWELL / ADVANCES IN VIBRATION ENGINEERING, 5(4) 2006 and predictions, physically meaningful and accurate parameter identification is required in actual test conditions. Such experiments may also be used to validate bearing models. The equations of motion are identical to (4), except that the experiments are performed on foundations that are assumed to be effectively rigid [29]. Then, [ ZR,ii Z R,ib Z R,bi Z R,bb + Z B ]{ rr,i r R,b } = { fu 0 }. (6) The second block row may be used to estimate the bearing dynamic parameters, by Z B r R,b = [K B ( ) 2 M B ( ) + j C B ( )] r R,b = Z R,bi r R,i Z R,bb r R,b. (7) where the bearing matrices have been written in terms of mass, damping and stiffness matrices. These matrices are frequency dependent, and in practice the real part gives the dynamic stiffness and the imaginary part gives the damping. We assume that a model of the rotor is available and the rotor response at the bearing is measured. Then if the unbalance force is known the response at the internal degrees of freedom can be estimated from the first block row of (6). Thus, the only unknowns in (7) are the bearing properties, and these speed dependent bearing parameters may be estimated using ordinary least squares estimation techniques. Tiwari et al. [29] highlighted that these estimation equations become ill-conditioned in the presence of circular orbits, and suggested two ways of regularizing the solution, namely by assuming isotropic bearing and by obtaining extra data from the machine operated in reverse (although of course this second solution would not work with fluid bearings). 7 Implementation of Model Based Condition Monitoring Although the approaches detailed in this paper might appear to be fairly complex, it is purely algorithmic and could be readily automated. This would allow a significant quantity of information to be gained following each machine run-down. A possible procedure is as follows: (1) The run-down is automatically processed as described above. There is a rather modest overhead in doing this. The rotor for each class of machines would have to be modeled, but for a power station with four machines, there would probably be a single rotor model. This point has been discussed in several previous papers [10, 31]. The variations that exist between nominally identical machines usually arise because of their supporting structure and these differences are accounted for in the above methods. Ideally the rotor models should be validated using a free-free vibration test. (2) Trend the estimated unbalance against previous values. Differences in operating condition, for example between hot and cold run-downs, must be considered. (3) Compare foundation parameters with previous values. With the automated calculations, separate records could be assembled to monitor foundation parameters, the state of balance, rotor bends, bearing (static) loads and hence the alignment. (4) Alignment changes monitored. (5) Estimate bearing characteristics. If proximity probe data is available on the unit then the bearing characteristics may be inferred and may be indicative of wear within the bearing. This is possible by the derivation of the forces at the bearings by the use of a good rotor model.

WHERE NEXT FOR CONDITION MONITORING OF ROTATING MACHINERY? 275 Given such a system a great deal of insight into a machine s condition can be gained with virtually no additional equipment or effort above that which is expended currently. Basic research has now developed techniques for the calculation of the parameters mentioned above, although there is still discussion among several research groups with regard to the most efficient methods. When these issues have been resolved significant software development will be required to implement these approaches. However, prototype schemes could be installed in the very near future. A further intriguing possibility arises. After the analysis a full knowledge of the rotor s position in the bearings exists and the unbalance forces have been estimated. Thus the information required to estimate the deflected shape of the rotor is available and may be used to highlight any areas of potential rubbing. The discussion has considered the implementation of what may be loosely termed information technology and signal processing applied to otherwise conventional machines. However the greatly enhanced knowledge of the underlying system is the natural route to consider the introduction of smart machines. It is unlikely that all machine faults will be identified uniquely, however it is feasible to estimate probabilities for different types of fault. Bachschmidt et al. [32] presented some work it this area but further steps are possible with refined plant models. This may be taken a step further with the introduction of actuators that apply forces to mitigate some of the effects of faults. The most obvious approach is through the use of magnetic bearings introducing variable stiffness parameters within a control circuit. Other possibilities include piezo-electric actuators, pressurized bearings and magnetic dampers. Some of these technologies will, no doubt, emerge at the smaller end of the machine range and may be expected to play an increasing important role. 8 Conclusions This paper has considered some of the general issues in obtaining and using models of rotating machines for condition monitoring. Of crucial importance is a good physical understanding of the dynamics of the machine. This allows an improved estimation of the model by concentrating only on those parts that are uncertain, and can lead to improved diagnosis of faults. The importance of model based condition monitoring will increase as machines run faster and the need to minimize down-time increases. Improvements in data acquisition and sensing devices mean that measuring the dynamic response (and other variables) on a machine is more cost effective. The challenge is to better utilize this increased information, through better modeling of the machines and potential faults, to make better maintenance decisions. Acknowledgements Michael Friswell gratefully acknowledges the support of the Royal Society through a Royal Society- Wolfson Research Merit Award and the European Commission through the Marie Curie Excellence Grant, MEXT-CT-2003-002690. References [1] Davies, W. G. R., Lees, A. W., Mayes, I. W. and Worsfold, J. H., Vibration problems in power stations, I. Mech. E Conference on Vibrations in Rotating Machinery, Cambridge, September 1976.

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