Identification Procedures as Tools for Fault Diagnosis of Rotating Machinery

System identification procedures offer the possibility to correct erroneous models, based on measurement data. Recently, this conventional field of application is being extended to fault detection and system diagnosis. In contrast to conventional approaches, identification procedures try to establish an unequivocal relation in between the damage and specific mechanical parameters, based on a suitable model. Furthermore, they can be employed during normal operation of the machinery. In this paper, several identification procedures on the basis of the Extended Kalman Filter are introduced and employed for model-based fault detection. Their feasability is proved by several examples. First, it is shown that the crack depth of a simulated Jeffcott-rotor can be calculated correctly. Then, the procedures are utilized to determine the crack depth of a rotor test rig. Finally, it is proved that identification procedures can be employed for the determination of unbalances without having to apply test masses.

INTRODUCTION onitoring and diagnosis of machinery is a field which is receiving increasing interest. Current practices and trends are described e.g. by Eshleman [1990], Randall [1990], Jonas [1992] and E1-Shafei [1993]. Very often, special attention is being focused on the detection of cracks: Wauer [1990] gives a concise overview about the research performed in the area of modeling the dynamics of cracked rotors and detection procedures. A possibility to detect a crack is e.g. the monitoring of the rotor vibrations and the comparison to the results of a suitable analytical model, Muszynska [1982], or to a reference signal, Imam [1983].
In contrast to approaches utilizing signal analysis, model-based procedures are being developed, which try to establish an unequivocal relation in between the damage (not necessarily a crack) and specific mechanical parameters. To achieve this, a-priori knowledge like e.g. a mathematical description of the system dynamics is employed. Waller and Schmidt [1990] describe a modal observer suitable for the application to MDOF systems. Another method is presented by S6ffker and Bajkowski [1991], which makes use of the state observer for nonlinear systems developed by Mfiller [1990]. Structural similarities to the approach of Fritzen and Seibold 1990] can be shown, where the Extended Kalman Filter is applied to shaft crack detection.
In this paper, time domain identification algorithms, based on the Extended Kalman Filter (EKF) and the Instrumental Variables method (IV), and a modification of the EKF in analogy to Mtller 1990] are presented and applied to fault diagnosis of rotating machinery. Specific advantages are the direct processing of time domain measurements, without the necessity of a transformation to frequency domain, and the establishing of a physically meaningful relation in between the damage and a certain parameter. Furthermore, unknown states are recon-S. SEIBOLD AND C.E FRITZEN structed and possible parameter changes can be monitored during normal operation of the machinery. These model-based procedures are sensible supplementals of the approved tools for fault diagnosis and their feasability is explained by two examples: a simulated Jeffcottrotor, where the crack depth is identified, and a rotor test rig, where the depth of a crack and the unbalance are determined.

THE INSTRUMENTAL VARIABLES METHOD (IV)
The IV-method is a parameter identification procedure which yields consistent and asymptotically unbiased estimates, requiring only little information about the noise characteristics. For a general description see Durbin [1954]. Wong and Polak [1967] and Young 1970] apply the IV-method as a purely statistical procedure to control problems. The IV-method can be derived from the well known Least Squares normal equations A being the coefficient matrix. Eq. (1) can be established for problems that are linear in the parameters. The estimated parameters /0LS, which result from (1), are biased, and the expected values of/LS are e E LS} P0 + E{(ATA) -1 ATe} (2) Pk+l [I--/kA'+l]Pk" (6) In the case of a mechanical system, assuming that displacements x are measured, we need the complete information about x and its derivatives X and X and to determine the coefficient matrix A and the Instrumental Variables matrix W. These derivatives could be calculated by means of numerical differentiation. But, this might lead to bad results, especially if the level of measurement noise is high. Another possibility is to apply spline approximation, which is a better choice. Of course, this might cause problems if stochastic signals are dealt with. In this paper, it will be shown how the Extended Kalman Filter can be employed, cf. .

THE EXTENDED KALMAN FILTER (EKF)
The EKF is based on the Kalman Filter (KF), which was derivated by Kalman [1960] based on the concept of orthog0nal projection, as an optimal filter for linear systems. But, as for many nonlinear problems cannot be linearized globally, suitable algorithms have to be developed. Jazwinski [1970] describes, how the KF has to be modified so that it can be applied as EKF to nonlinear systems. The basic idea is to linearize at each time step around a reference trajectory. In this way, the global nonlinearity is maintained.
The motion of a mechanical system can be described by the differential equation Po are the true parameters and is the error vector. The bias EI(ATA) -1 ATewill not vanish, because A and are usually correlated. Nevertheless, unbiased estimates can be produced if an Instrumental Variable matrix W with certain properties is introduced. According to Young [1970], the elements of W should be highly correlated with the unobservable noise-free responses of the system, but totally uncorrelated with the noise, so that E lwTfi --0 and EIWTA} is existing and nonsingular.
Then, unbiased estimates/3iv are be obtained by Wong and Polak [1967] and Young [1970] give a recursive version of the IV-method (RIV): M (x,p,t) g(x,2,u,p,t), M being the mass matrix which might depend on the displacements x and time t. g is the vector of generalized forces and moments, including elastic, damping, gyroscopic forces etc., and u is the input vector.
The EKF is a recursive time domain procedure suitable for the identification of nonlinear systems. System disturbances as well as measurement noise can be considered and a state space formulation of the differential equations of motion is required. In the case of mechanical systems, the state space vector z consists of the displacements x and the velocities . .F or the purpose of simultaneous state and parameter estimation, z is extended by the unknown parameters p: T T zT= (XT, )T, pT) (Zx T, ZV, Zp). (8) Applying state space notation, the vibrations of a mechanical system can be described by the following equation, assuming time constant parameters: Considering discrete time steps tk, the relation in between the observed measurement data Yk and the state variables zk at time t is (1 O) In our case, C is a constant matrix and n k denotes the measurement noise. The EKF consists of a predictor and a corrector part, with the prediction (11) tk basing on the model equations f. The corresponding covariance matrix is Employing new measurements Yk+l, the prediction (11) is corrected by and the corrected covariance matrix is The Kalman gain matrix Kg is calculated by The initial values are normally distributed, with mean Zo and covariance Po. Ak is the time discrete system matrix and results from a local linearization of the system equations f around the estimated trajectory at each time step tk: It is important to note that in this way, the global nonlinearity is maintained. Q*k is the discrete system noise covariance matrix and has to be calculated at each time step tk, tOO. R k is the covariance matrix of the measurement noise n k. Therefore, the confidence in the initial values Zo, in the system equations f as well as in the measurements Yk can be expressed by means of the covariance matrices P, Q and R, respectively.

THE COMBINED ALGORITHM REKFIV
The EKF is very time consuming and might, in certain cases, show poor convergence properties if it is used for combined state and parameter estimation, Ljung [1979]. But, being used exclusively as a state estimator, the EKF will perform well and provide us with the advantage of the reconstruction of unknown states and dofs. Therefore, the idea is to combine EKF and RIV in a way that the EKF is used as a state estimator to produce the Instrumental Variables for the RIV. We propose to proceed in the following manner, see figure 1:

THE MODIFIED EXTENDED KALMAN FILTER (MEKF)
The EKE (11)-(17), was developed for stochastic dynamic systems and shows a structural similarity to the state observer, which is known from control theory.
MOiler 1990] designed a state observer for deterministic systems in order to determine unknown nonlinearities. These are interpreted as "external disturbances", which are characterized by a suitable model. In analogy, the EKF is modified in such a manner that it can reconstruct these "external disturbances", Seibold et al. [1993], without the necessity to model their dynamics, S6ffker et al. [1994]. Essentially, the state space vector z has to be extended by the "external disturbances" and their time derivatives (I'T,1)T): z w (xT,ff,vT,9T).
The EKF-equations can be used like being described in (11)-(17). The coupling in between the linear part of the system (xT,ff) and the "exogeneous system" (vT,T) is done via the prediction (11), where the linear part of the system has to be supplemented by the exogeneous system, which was calculated in the preceding time step. In our first example, the identification procedures will be used to detect a transverse crack in a Jeffcott-rotor. The crack is considered by the breathing crack model of Gasch 1976]. The system rotates with circular frequency to, is viscously damped and loaded by unbalance and gravitational forces, figure 2. As a consequence of the opening and closing due to the crack, the stiffness k switches in between the two values kl and kz, depending on the actual displacements x and x 2 (resp. {), figure   3. The stiffness of the uncracked shaft is ko and for small cracks, we can assume k2 ko k.
cf. Fritzen, Seibold [1990], and the decreasing of the stiffness due to the crack is described by the crack factor r; r -1 -1.
r+l The linear part of the equations of motion is on the left hand side of (24), and on the right hand side are the excitations f('r) and the nonlinear part v('r), which can be interpreted as additional forces due to the crack. The dynamic behaviour is simulated by numerical integration of (22). The parameters are set to D 0.011, v 0.45, a" 0.01, e 0.01 and [3 0. The relation of the crack factor r to the crack depth can be derived e.g. according to Mayes and Davies [1984]: r 0.01 corresponds to a crack depth in percentage of the diameter of the shaft of 19%. On the basis of measured displacements X and x 2, EKF as state and parameter estimator, REKFIV and MEKF are employed to detect the depth of the crack.
For the application of EKF and REKFIV, the knowledge of the crack model v('r), eq. (22), is assumed. EKF and REKFIV calculate a crack factor r 0.00995 resp. r 0.01005, figure 4. The initial parameters were set to r o 0, which implies that no crack is present. In contrast to this, the calculations with the MEKF can be performed without the knowledge of a crack model, i.e. the exogeneous system v('r) does not have to be structured. The result of the MEKF is shown in figure 5, where it is compared to the true values v l('r) according to eq. (22). This exogeneous system can be interpreted as nonlinear forces due to the crack. Except for small The EKF can be modified to estimate time variable parameters, i.e. in this case time dependent stiffnesses, so that a crack model is not a prerequisite for the calculations. In the EKF-equation (12), the covariance matrix of system noise Q has to be set to adequate values. Figure   6 shows the identification of one of the time varying stiffness parameters. Like in figure 5, the opening (Cl < 1) and closing (Cll 1) of the crack can be seen. The results correspond to the true crack factor r 0.01 and are compared to the true slope of the stiffness c cos 2 v'rA, see eq. (22). The two curves plottet in figure   6 are almost identical.

Rotor Test Rig" Identification of the Crack Depth
The rotor test rig, figure 7, consists of a long, thin shaft of 18 mm diameter on hinged supports. A disc is mounted in the middle. Initiated by a slot with a depth of 2 mm, a crack of mm is introduced, so that the total depth of the damage is 3 mm respectively 17% of the  Rotor Test Rig: Identification of the Unbalance without Test Masses initiating slot, which prevents a complete closing of the In the last example, EKF and REKFIV are utilized to crack surface. Figure 9 shows the identification of r by determine the unbalance of the test rig described in the REKFIV. The oscillations around the final value of r preceding chapter, with the modification that the diameter of the (undamaged) shaft is 12 mm. Again, the horizontal and vertical vibrations of the disc are measured during stationary operation. The system was modeled according to eq. (22), setting v(a') 0. Table I  0.012 may be motivated by the fact, that the relatively long and thin shaft is somewhat prebent. The calculations of the MEKF, figure 10, again provide us with a quantative result (r 0.013; this corresponds to a crack depth of 20%) and a qualitative result: the crack opens in the range v l('r) > 0 and the assumption that the crack surface really does not close completely is confirmed, because there is no range V l(a-) 0 (compare to figure5).

CONCLUSIONS
System identification procedures offer the possibility to correct erroneous models, based on measurement data. Recently, this conventional field of application is being extended to fault detection and system diagnosis. In contrast to conventional approaches, identification procedures try to establish an unequivocal relation in between the damage and specific mechanical parameters, based on a suitable model. Furthermore, they can be employed during normal operation of the machinery. In this paper, three system identification procedures were presented:  Mtiller [1990]. The feasability of the procedures for model-based fault detection of rotating machinery was proved by several examples. First, the crack depth of a simulated Jeffcott-rotor was calculated correctly. In addition, the results of the MEKF yielded a qualitative assertion about the character of the damage. Then, the three procedures were successfully applied to a rotor test rig, in order to detect the crack depth. A supplementary information was obtained: the crack surface did not close completely due to the initiating slot. Finally, it was shown that system identification procedures can be employed for the determination of unbalances without test masses.
In future projects, the algorithms will be extended in such a manner that they can be applied to larger structures modeled by finite elements.