Estimating turbogenerator foundation parameters: model selection

10.1098/rspa.2000.0577

Estimating turbogenerator foundation parameters: model selection and regularization By M. G. S m a rty, M. I. F ri s we ll a n d A. W. L e e s Department of Mechanical Engineering, University of Wales Swansea, Swansea SA2 8PP, UK Received 8 September 1999; accepted 7 December 1999

Estimating a model of the foundation of large machines, such as turbogenerators, is vital for tasks such as fault diagnosis and modal balancing. Unfortunately, it is rarely possible to perform a modal test on the foundations without the rotor present. This paper considers a method to estimate a foundation model using response data from a run-down, using the inherent unbalance of the rotor to excite the foundations. The method requires an accurate model of the rotor and an approximate model of the bearings. Of critical importance is the quality of the model, which may be de ned as the ability of the model to predict the response to unbalance excitations that are di¬erent from that present for the identi cation. It is shown in this paper that correct model selection and regularization are vital to produce a foundation model that meets this predictability criterion. The method is validated using simulated data and also experimental data from a test rig with a 4 m long rotor with four ®uid lm bearings. Keywords: system identi¯cation; run-down; turbogenerators; foundations; parameter estimation; rotor dynamics

1. Introduction The analysis of the vibrational behaviour of turbo machinery is a topic of great importance in most process industries and particularly in power generation. Apart from the need to design machinery to operate within acceptable limits, dynamic models are now used to great e¬ect in the diagnosis of operational di¯ culties. With the high returns from modern plants, there is an increasing need to develop reliable plant models for fault diagnosis but the models are not yet developed to the stage of being applicable with con dence across a full range of plant problems. Over the past 30 years there has been a strong trend towards the use of ®exible steel supports for large turbines, as this approach o¬ers a number of practical advantages, not least the cost of fabrication. However, this does highlight that the supporting structure has a signi cant e¬ect on the machine dynamics. Lees & Simpson (1983) discussed the modelling of this type of structure: in principle, it should be possible to develop a suitable model from nite-element techniques, but there are a number of practical di¯ culties. It is often found that similar units, y Present address: National Power plc, Windmill Hill Business Park, Whitehill Way, Swindon SN5 6PB, UK. Proc. R. Soc. Lond. A (2000) 456, 1583{1607

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® c 2000 The Royal Society

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M. G. Smart, M. I. Friswell and A. W. Lees

built to the same drawings, display substantially di¬erent vibrational behaviour. The di¬erent vibrational behaviour is mainly due to small changes in a huge number of joints, which combine to give a substantial change in the sti¬ness of the structure. With these di¯ culties, it is unlikely that the techniques of nite-element model updating (Friswell & Mottershead 1995) could be used, as there are too many uncertain parameters in the joints. The most promising avenue is to identify a model of the foundations directly from acceleration measurements at the pedestals during run-down. Lees (1988) developed a method using measurements of responses at the bearings to calculate the forces applied to the foundation at the bearings. These calculated forces were used together with the measured responses to estimate the foundation parameters in a least-squares sense. The method required models for the rotor and bearing dynamic sti¬nesses, as well as prior knowledge of the state of unbalance of the machine. He demonstrated the technique using a very simple two bearing model with speed independent characteristics. Vania (1996) followed a similar approach to Lees (1988), but used a nonlinear estimation algorithm based on Kalman lters to estimate the foundation parameters. This procedure required good starting estimates for the foundation parameters, which if not available can lead to divergent solutions. The method was tested in simulation on a rotor attached to a frame structure and no experimental results were presented. Feng & Hahn (1995) used a somewhat di¬erent approach which incorporated measurements of shaft eccentricities, thereby reducing dependence on the a priori rotor and bearing models. However, many turbines in British power stations are not equipped with proximitors and so this extra information is not always available. Feng & Hahn noted that if the foundation is ®exible enough, there will be more modes in the running range than measured degrees of freedom (DOF). In order to achieve sensible solutions, the frequency range was split and di¬erent models were estimated for each frequency range. Again, the method was tested in simulation and no experimental data were provided. Zanetta (1992) adopted the same procedure as Feng & Hahn (1995), which used measured shaft eccentricity data in addition to foundation responses. However, he used the extra information provided by the eccentricities to disregard the bearing models, preferring to estimate a dynamic model which combined the e¬ects of bearings and foundation. He also tested his method in simulation only. Smart et al . (1998) considered the estimation of parameters of turbogenerator foundations. However, the foundation model was assumed to consist of mass, damping and sti¬ness matrices, which constrained the number of modes in the foundation to equal the number of measured DOF. This paper lifts this restriction by using a frequency lter for the foundation model. This paper also discusses the selection of the order of the lter model, and regularization of the resulting equations, which are often ill conditioned.

2. Estimation of foundation forces Figure 1 is an abstract representation of a turbogenerator, whereby a rotor is connected to a ®exible foundation via oil- lm journal bearings. The foundation estimation algorithm proceeds in two steps. Firstly, known models of the rotor and Proc. R. Soc. Lond. A (2000)

Estimating turbogenerator foundation parameters

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rotor

f R, b

fu

bearing f F, b

foundation

Figure 1. The abstract representation of a turbogenerator system.

bearing are used to estimate the forces acting on the foundation, arising from a known unbalance and corresponding to a given set of foundation measurements during a run-down. Then these predicted forces are taken together with the response measurements and used to estimate dynamic sti¬ness parameters for the foundation. The dynamic sti¬ness matrix of the structure may be written as 9 8 9 2 38 ZR;ii ZR;ib 0 0 rR;i > > > fu > > > < = > < > = 6ZR;b i ZR;b b + ZB 7 Z 0 r 0 ¡ B R;b 6 7 = ; (2.1) 4 0 ZB + ZF ;b b ZF ;b i 5 > rF ;b > ¡ ZB > 0> > > : ; > : > ; 0 0 ZF ;ib ZF ;ii rF ;i 0

where Z is the dynamic sti¬ness matrix, the subscripts `i’ and `b’ refer to internal and bearing (connection) DOF, respectively, and the subscripts F, R, and B refer to foundation, rotor and bearings. r are the responses and fu the unbalance forces, which are assumed to be applied only at the rotor internal DOF. It is assumed that inertia e¬ects in the bearings are negligible, although if desired these can be included (El-Shafei 1995). The foundation dynamic sti¬ness has been partitioned into internal and bearing DOF. Since measurements are taken at the bearings and not at the internal DOF ·F can be estimated which, using of the foundation, only a reduced foundation model Z the last row of (2.1) to eliminate the internal foundation DOF, is ·F = Z F Z

;b b

¡

ZF

¡1 ;b i ZF ;ii ZF ;ib

:

Therefore, equation (2.1) may be rewritten as 9 8 9 2 38 ZR;ii ZR;ib 0 < rR;i = < fu = 4ZR;b i ZR;b b + ZB ¡ ZB 5 rR;b = 0 : : ; : ; 0 ZB + Z·F rF ;b 0 ¡ ZB Proc. R. Soc. Lond. A (2000)

(2.2)

(2.3)

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From the rst row partition of (2.3), the displacements at the internal rotor DOF are ¡1 rR;i = ZR;ii (fu ¡

ZR;ib rR;b ):

(2.4)

The displacements at the rotor connection DOF may be obtained from the second row of (2.3), using (2.4), as ¡1 ZR;b i ZR;ii (fu ¡

ZR;ib rR;b ) + (ZR;b

b

+ ZB )r R;b ¡

ZB rF

;b

= 0;

(2.5)

or, equivalently, ¡1 rR;b = ¡ P ¡1 [ZR;b i ZR;ii fu ¡

Z B rF

;b

];

(2.6)

where P = ZR;b The forces on the foundation, fF fF

;b

;b

= ZB (rR;b ¡ = ZB (P = Z1 rF

;b

¡1

b

¡1 ZR;b i ZR;ii ZR;ib :

+ ZB ¡

(2.7)

, are rF

ZB ¡

)

;b

I)rF

+ Z 2 fu ;

;b

¡

¡1 ZB P ¡1 ZR;b i ZR;ii fu

(2.8)

where the form of Z1 and Z2 is clear from (2.8). The displacement at the foundation is then given by · F rF Z

;b

= fF

;b

:

(2.9)

All the quantities in (2.8) are known, either a priori (ZR ; ZB ; fu ) or from measurement (rF ;b ). The force estimates given by (2.8) are robust, except near the natural frequencies of rotor and bearings on rigid foundations (Lees & Friswell 1996).

3. System identi¯cation There are a number of steps involved in the system identi cation process (Schoukens & Pintelon 1991); namely, model selection and parametrization, parameter estimation, model veri cation and model validation. (a) Model selection and parametrization The equations describing the system behaviour have been formulated above. This has already approximated the behaviour of the machine by, for example, assuming that the motion is linear. Even for a linear system there are, however, still choices to be made concerning the form of the model. A critical decision is whether to describe the system in the time or frequency domains. The time domain has the advantages of robustness, applicability to a wide range of input signals, no leakage problems and the possibility of on-line estimation (Schoukens & Pintelon 1991; Ljung & Glover 1981; Pintelon et al. 1994). Frequency-domain methods, on the other hand, use a much more compact dataset, can reduce noise on the data, require no estimate for the system’s initial state and can incorporate the e¬ect of out-of-band modes through residual terms. In view of these advantages, and especially in view of the Proc. R. Soc. Lond. A (2000)


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large amount of time-domain data acquired during a run-down, it was decided to restrict the analysis to frequency-domain methods only. Most existing parameter estimation methods used in modal analysis approximate the transfer function matrix with polynomial matrices (Allemang et al. 1994; Leuridan et al . 1986). Thus, from (2.9), the transfer function matrix of the foundation would be approximated by H (s) = Z·F (s)¡1 = N (s)=¢ (s)

(3.1)

for some matrix polynomial N (s) and scalar polynomial ¢ (s). The denominator is a scalar polynomial in s because the natural frequencies and damping ratios (or poles) of any structure are global quantities. Unfortunately, the data available are in terms of input force and output response rather than the transfer function matrix (equation (2.9)). Since a constant unbalance regime is used to excite the machine, the forces exerted by the rotor on the foundation via the pedestal DOF will be correlated. It is thus impossible to recover the frequency response function of the foundation from a single run-down. Indeed, the number of run-downs required would equal the number of pedestal DOF, and each run-down would have to have an independent forcing regime, which is impractical. Since the frequency response function is not available, standard methods of modal analysis cannot be applied. The alternative formulation to (3.1) is (Allemang et al . 1994) D(s)rF where the output is rF

;b

;b

= N (s)fF

and the input is fF

;b

;b

;

(3.2)

, and thus

Z·F (s) = N (s)¡1 D(s):

(3.3)

Since the number of estimated forces is the same as the number of measured responses (and equal to n), N and D are both square and have dimension n. The denominator matrix polynomial is de ned as D(s) = D0 + D1 s + D2 s2 +

+ Dq sq ;

(3.4)

+ Np s p :

(3.5)

and the numerator matrix polynomial as N (s) = N0 + N1 s + N2 s2 +

If there are exactly as many modes in the data as there are DOF in the model, then p is equal to 2n and q is equal to p ¡ 1. This is a very rare situation, and in practice the selection of correct values for p and q is one of the most important steps in system identi cation. This aspect of identi cation process will be elaborated upon later. The model parameters that must be estimated are the elements of the Nk and Dk matrices. They form a vector µ of parameters µ = [d0;11 d0;12

d0;nn

dq;11 dq;12

dq;nn

n0;11 n0;12

n0;nn

np;11 np;12

np;nn ]T ;

(3.6)

where nk;ij is the ijth element of matrix Nk and dk;ij is the ijth element of matrix Dk . It is also possible to model the foundation using a modal decomposition in place of the frequency lter expressions of (3.2){(3.5) (Zanetta 1992; Feng & Hahn 1996). Proc. R. Soc. Lond. A (2000)

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Unfortunately, the transfer function of the foundation is a highly nonlinear function of the natural frequencies and damping ratios. When such models are incorporated into identi cation schemes for foundation models, the result is a highly nonlinear optimization problem, with the associated problems of a small radius of convergence and local minima. Indeed, standard modal analysis techniques never directly optimize the residual of the measured frequency response function and its modal representation for these reasons. (b) Parameter estimation Once the model has been selected and expressed in terms of a set of parameters, these parameters must be estimated. This is accomplished by de ning an error function or residual between outputs from the system and outputs from the model, both being subjected to the same input. A cost function of the residuals is then minimized to yield estimates of the parameters. For the rational fraction model de ned above, the parameters are found by minimizing the cost function X min Jo = "T "o = rF ;b ¡ D ¡1 N fF ;b ; (3.7) o W "o ; where "o is the output error residual, W is a weighting matrix and Jo the quadratic cost function. This is a nonlinear least-squares type problem. It is clear that (3.2) may be multiplied on both sides by a scalar value and still hold true. The model as it stands therefore is not unique, and various schemes of imposing uniqueness have been proposed (Pintelon et al . 1994), such as enforcing unity coe¯ cients on one side of the equation (Nk = I or Dk = I) or setting the Euclidean norm of the parameters equal to one (kµk2 = 1). The type of constraint chosen can in®uence whether the resulting parameter estimates are biased or unbiased (Pintelon et al . 1994). Equation (3.7) is nonlinear in the parameters, and as a result iterative routines are required to solve it. These nonlinear minimization techniques require good starting estimates of the parameters, without which they may diverge or converge to a local minimum. They are generally quite robust with respect to noise. Also, in order to improve the convergence properties, it is necessary to scale the parameters, because of the di¬erent magnitudes of the elements of µ. A suitable scaling factor is the mean value of excitation frequency !. Of course, the residuals discussed above estimate a model of the foundation based on the estimated force into the foundation and the measured response of the foundations. As a second step, this model may be used to generate an initial parameter estimate for the minimization of the response residuals, given by X 2 min Jr = (3.8) k"r k ; where "r = rF ;b ¡ (Z·F ¡ Z1 )¡1 Z2 fu : (c) The linear least-squares estimator The nonlinear problem described by (3.7) may be signi cantly simpli ed if it is reformulated to be linear in the parameters. Levi (1959) suggested a method to do this for single-input single-output (SISO) systems by multiplying both sides of (3.7) Proc. R. Soc. Lond. A (2000)


1589

by the denominator. This leads to a de nition of equation error (Natke et al . 1995) X min Je = "T "e = DrF ;b ¡ N fF ;b ; (3.9) e W "e ;

where "e and Je are the equation error residual and cost function, respectively. This leads to a linear estimation problem, since (3.2) may be rewritten, for N0 = I, as [R0

R q S1 S2

Sp ]µ = fF

3

2 T fF ;b 6 0 6 Sk (s) = ¡ sk 6 . 4 .. 0

;b

;

(3.10)

where 2 T rF ;b 6 0 6 Rk (s) = sk 6 . 4 .. 0

0 rFT;b .. . 0

..

0 0 .. .

.

rF T;b

7 7 7; 5

0 fF T;b .. . 0

..

.

0 0 .. .

3 7 7 7 5

fF T;b (3.11)

and µ is de ned as µ = [d0;11 d0;12

d0;nn

dq;11 dq;12

n1;11 n1;12

n1;nn

dq;nn np;nn ]T :

np;11 np;12

(3.12)

In other words, the parameters and measurements may be separated. Equation (3.10) may be repeated for all frequency points (where the Laplace variable s is now replaced by the complex frequency j!), to give V µ = q; where

2

R0 (j!1 ) 6 R0 (j!2 ) 6 V =6 .. 4 .

.. .

R0 (j!m ) 8 9 fF ;b (j!1 ) > > > > > > < fF ;b (j!2 ) = q= : .. > > . > > > > : ; fF ;b (j!m )

Rq (j!1 ) Rq (j!2 ) .. .

S1 (j!1 ) S1 (j!2 ) .. .

Rq (j!m ) S1 (j!m )

(3.13)

.. .

3 9 Sp (j!1 ) > > > Sp (j!2 ) 7 > 7 > > 7;> .. > 5 > > . > > = S (j! ) > p

m

> > > > > > > > > > > > > ;

(3.14)

The above formulation assumes that all the parameters are independent. In practice, of course, this may not be so and certain parameters may be related to each other, for example, due to structural symmetry. This can be taken into account by means of a transformation matrix Tc , µ = Tc µc ;

(3.15)

where µc are the constrained parameters. Then Vc µc = q; Proc. R. Soc. Lond. A (2000)

(3.16)

1590


where Vc = V Tc . The subscript c will be dropped from the notation from now on, and it will be implicitly assumed, unless stated otherwise, that the parameters have had the necessary constraints applied to them. The equation error cost function is therefore Je = (V µ ¡

q)T W (V µ ¡

q);

(3.17)

which is minimized when µ = (V T W V )¡1 V T W q:

(3.18)

While (3.18) is useful from a theoretical point of view, using it to calculate parameter estimates is not the most e¯ cient in practice. It is usually better to use a singular-value decomposition or QR-decomposition approach to ensure better numerical behaviour (Golub & Van Loan 1990). (d ) Model veri¯cation and validation Once the parameters have been estimated, it is necessary to verify the model, which is normally done by examining the relative magnitudes of the residuals to determine the quality of t (Schoukens & Pintelon 1991). In the case of the output error method, the relative magnitude is "·o =

k"·o k krF ;b k

or

"·r =

k"r k : krF ;b k

(3.19)

The norm used to generate the residuals in this paper was based on the absolute value. For the equation error method, the following two de nitions are possible: "·e =

k"e k kDrF ;b k

or

"·e =

k"e k : kN fF ;b k

(3.20)

The next test of the estimated model is subjecting it and the real system to a di¬erent excitation, and comparing the residuals between system and model. This is a more rigorous check than simply checking the original residuals. (e) Selection of model order For a given frequency range, the degree of the denominator polynomial in (3.1), for a discrete model with n DOF, is 2n. This degree is essentially the number of independent variables required to determine completely the system’s response to an arbitrary input (Nise 1992), and for linear dynamic systems is equal to twice the number of physical modes of the system. Theoretical analysis of continuous structures shows that an in nite number of DOF is required to characterize their dynamic behaviour, and although it may be argued that physically all structures contain a nite number of DOF, this number will nonetheless be large. Any practical measurement will of necessity be limited both spatially (transducers cannot cover all parts of the structure) and temporally (measurements are taken within a nite-frequency bandwidth), and therefore will contain only a limited number of modes. Furthermore, the modes will be made up of both the physical modes of the structure and the noise modes due, for example, to electronic noise. It is desired Proc. R. Soc. Lond. A (2000)


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that any model of the form of (3.1) or (3.2) should be of su¯ cient order to completely describe all the physical modes of a structure, while not following the behaviour of the noise on the system (Pintelon et al . 1994). Therefore, selecting correct values of p and q for the numerator and denominator matrix polynomials is one of the most important steps in system identi cation, and a number of di¬erent approaches to this problem have been presented in the literature. If frequency response functions are available, visual inspection is an invaluable aid to estimating the approximate model order and is achieved by counting the number of peaks in the functions (Pintelon et al . 1994). The presence of close or highly coupled modes, as well as the in®uence of noise, means that this is at best a rough rst estimate of the model order. Furthermore, it is not generally valid if the inputs are correlated, because a maximum in the response may be due to a maximum in the input rather than to a physical mode. An error chart plots the residual error versus the model order. In theory, this shows a gradual decrease in error until the correct order is attained, after which there is a dramatic decrease. The presence of noise makes the selection of this cut-o¬ point quite di¯ cult (Allemang et al . 1994). As successive values are chosen for the model order, so the number of transfer matrix poles increases. These poles are directly related to estimates of the frequency and damping of the system, and these estimates may be plotted versus the model order in a format known as the stability diagram (Allemang et al . 1994; Braun & Ram 1987a; b). As the model order increases, certain poles will stabilize (in other words, remain unchanged with increasing model order), while others will ®uctuate. The latter are the noise modes, while the former modes are physical modes. Poles that are stable with respect to model order but that have positive real parts (physically unstable) are also rejected. Once again, in practice it is possible for noise modes to stabilize and be identi ed erroneously as physical modes. The rank estimation approach to model order selection relies on a singular value decomposition (SVD) of the measurement data matrix (Allemang et al . 1994). A plot of the singular values should theoretically show a sharp drop at the correct model order, but once again it is necessary for the user to exert judgement as to what is meant by `sharp’.

4. Errors in the identi¯cation the foundation model Smart et al. (1996) investigated some of the errors in estimating foundation models. The errors may be classi ed into four categories: namely, errors in the rotor model, in the bearing model, in the unbalance estimates and in the response measurements. It is assumed rstly that a good dynamic model exists for the rotor. Since rotors are manufactured to very tight tolerances, from materials whose properties are well known, it is reasonable to assume that no signi cant errors will occur here. The sti¬ness and damping matrices for the journal bearings are estimated using short bearing theory (Smith 1969). Experimental tests on journal bearings have shown reasonable agreement between theoretical and measured values (Hisa & Matsuura 1980). Good estimates are, however, dependent on an accurate knowledge of the static forces being applied to the bearings, which are di¯ cult to calculate accurately. Lees & Friswell (1996) showed that the errors in the bearings will only introduce signi cant errors in the region of frequencies of the rotor-bearing system on rigid Proc. R. Soc. Lond. A (2000)

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foundations. Provided rough estimates of the bearing dynamic sti¬ness matrices are known, these frequencies may be calculated and avoided in the estimation procedure. The state of unbalance may in theory be established from a balancing run. If two successive run-downs are performed, one due to the unknown system unbalance and one with known balance weights attached, then provided the system is linear, the response measurements may be vectorially subtracted to give the response due to the known balance weights alone. This is the basis for the balancing of industrial machines, so signi cant errors are not expected in the assumed unbalance distribution. Finally, noise on the measurements will a¬ect both input and output sides of the least-squares and nonlinear least-squares equations. This can be, for example, electrical or digitization noise (which are typically regarded as being normally distributed), or noise arising from the order-tracking routines.

5. Practical issues in foundation identi¯cation Having outlined some general system identi cation theory, and having applied this to the speci c problem of turbogenerator foundation model estimation, certain practical issues arising from this process will be discussed. Firstly, the issue of regularization of an ill-conditioned problem will be addressed. Secondly, it will be shown that identifying the foundation model using the estimated bearing forces is not optimal with respect to data weighting, and that a di¬erent formulation is required to improve robustness with respect to measurement errors. (a) Regularization Inverse problems, such as parameter estimation, are generally ill posed, which means that arbitrarily small perturbations of the data can cause arbitrarily large perturbations of the solution (Hansen 1994). There are many techniques for improving ill-posed problems, which are known collectively as regularization methods, and for least-squares type problems these may be classi ed into ve categories (Lawson & Hanson 1974); namely, weighted least squares, scaling of variables, adding additional constraints, deleting variables and singular value analysis. The residuals de ned in (3.7), (3.8) or (3.9) incorporated a weighting matrix W . This matrix must be positive de nite, and a commonly used case is where W is diagonal. Weighted least squares is essentially a row-scaling problem. Where summation is performed over many frequencies, the higher-frequency components will carry more weight than low-frequency components. This can be improved by using row scaling. If statistical uncertainty information about the data is available, then the elements of W may be de ned as the inverse of the variance of the data elements. It is frequently the case in system identi cation that the parameters in the vector µ are of widely di¬ering magnitudes. This can cause convergence and conditioning problems for the nonlinear and linear least-squares algorithms and the problem may be resolved by simply scaling the variables. Essentially, when dealing with discrete ill-posed problems, there are many solutions µ which provide an acceptably small residual ". One way of dealing with this is to impose some constraint on the norm of the parameters, and so the solution with minimum norm is chosen. If an initial estimate of the parameters is available, then Proc. R. Soc. Lond. A (2000)


1593

the solution with the minimum change in the parameters may be chosen (Ahmadian et al . 1998). Unfortunately, the a priori information available for the foundation model makes these type of constraints di¯ cult to apply. If certain variables are removed from the parameter vector µ (in other words, their values are set to zero), then the conditioning of the problem will not worsen (and may improve), while the norm of the residual vector will not decrease (and may increase) (Lawson & Hanson 1974; Friswell et al. 1998). Broadly speaking, therefore, selecting subsets of the original variables for the solution of the identi cation problem improves the conditioning at the expense of worsening the t between model and system outputs. When the number of parameters is large, then selecting the set which best represents the measured data by progressively removing parameters and examining the residual can be prohibitively expensive. Therefore, suboptimal approaches must normally be employed; for example, the best subspace approach (Lallement & Piranda 1990; Friswell et al . 1998). It may also be possible to set parameters to zero based on physical reasoning. For example, it may be assumed that two DOF at opposite ends of a machine are not coupled through the foundation. Or, analysing a machine only in the horizontal or vertical directions assumes the cross coupling is negligible. (b) The singular value decomposition In the case of a linear least-squares problem of the form of (3.13) or (3.16), if V is full rank, the unique solution which minimizes the residual is given by (3.18). If V is not full rank, then in nitely many solutions µ minimize the residual ". Writing the singular value decomposition of V as µ ¶ T § 0 V =U X ; (5.1) 0 0 where § = diag(¾1 ; ¾2 ; : : : ; ¾` ) represents the ` non-zero singular values (or, in the case of imprecise data, singular values above a speci ed numerical tolerance), and the matrices U and X are orthogonal. It can be shown (Ahmadian et al . 1998) that the parameter estimate is then µ=

` X i= 1

xi

uT i q ; ¾i

(5.2)

where ui and xi are the ith column of U and X, respectively. This enforces a unique solution (the minimum norm solution in fact) for an ill-conditioned problem. Equation (5.2) shows that the small singular values (which are a¬ected by noise and small changes in V ) have the most in®uence on the estimated parameters. Therefore, the number of retained singular values should be chosen based on noise in the data. The cut-o¬ should be around where juT i qj º j¾i j, and is known as the Picard condition (Hansen 1990; Ahmadian et al. 1998). Leuridan et al. (1986) and Lembregts & Leuridan (1990) use the SVD for data reduction in estimation problems. If rk represents measured response data at frequency !k , containing both structural and noise modes, then the autocorrelation Proc. R. Soc. Lond. A (2000)

1594


matrix of the responses is R=

m X

rk rkT ;

(5.3)

k= 1

where m is the number of discrete frequencies. If a singular value decomposition is performed on R, then the singular values below a certain tolerance represent the noisy part of the data. A transformation T may be de ned, whose columns are the left singular vectors corresponding to the retained singular values. The data are transformed by r^k = T T rk ;

(5.4)

and the estimation is carried out using the reduced dataset r^k . The inverse transformation rk = T r^k

(5.5)

recovers the predicted responses from the condensed data. (c) Data weighting: global versus local modes If equations (2.8) and (2.9) are examined, it will be seen that the foundation displacements rF ;b appear on both input and output sides of the equation. This means that peaks in the measured response data will cause corresponding peaks in the bearing forces, which do not in general coincide with the local foundation resonance frequencies. Rather, they represent global resonances of the combined rotor-bearingfoundation system. The high magnitude of data near these frequencies means that these data will not only be high quality (with a high signal{noise ratio), but will also be highly weighted in the estimation procedures. In order to obtain a reliable estimate for the foundation’s transfer function matrix, good data at the foundation’s resonance frequencies are required. However, the data at these resonances may well be away from a global resonance and the response may possibly be of low magnitude and therefore poor quality. The nonlinear estimation procedure, which works directly with the measured responses, should circumvent this problem. It is possible, however, that the estimated model will contain resonances which coincide with the global resonances in the response, rather than the true local foundation resonances. This is due to the large response near a global resonance, which, together with modelling errors, will force the estimation procedure to identify resonances at these frequencies. This should not prove a serious problem, however, since the estimated model will never be used on its own. What is desired is the contribution of the foundation dynamics to the total response of the system.

6. Description and modelling of the test rig The proposed method was tested on a small rotor test rig 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 ®exible steel pedestals bolted onto a large lathe bed which rests on a concrete foundation. The rotor itself consists Proc. R. Soc. Lond. A (2000)


1595

of two steel shafts 1.5 and 1.1 m long, each with a nominal diameter of 38 mm. At either end of the shafts are journals of diameter 100 mm, and in the centre of the shafts are machined sections which take balancing discs, three for the long rotor and two for the short rotor. The state of residual unbalance on the rotor is immaterial for the purposes of foundation identi cation, since the method assumes that two runs, one with and one without a balance weight, are subtracted to give the response due only to a known balance weight. Each balance disc was 100 mm in diameter, and had 24 10 mm holes on a pitch-circle diameter of 90 mm to take the balance bolts. The bearings are circular, have a length-to-diameter ratio of 0.3, a radial clearance of 150 m and contain oil of viscosity 0.0009 N s m¡2 . Smart (1998) gave a more detailed description of the rig and the data acquisition and processing. The rotor-bearing system is supported on pedestals which were designed to be ®exible over the running range of the rotor. They consist of rectangular steel plates, 600 £ 150 mm 2 , which have two channels cut into them, and are supported on knife edges. The vertical sti¬ness arises from the hinge e¬ect of the channels, while the horizontal sti¬ness is a result of the shaft centre tilting under an applied load. Sti¬ener bars of dimension 440 £ 24 £ 16 mm3 were attached underneath the channelled plate to increase the sti¬ness. This rig was designed to test the proposed approach on a system that is simpler than a real turbogenerator, while retaining many features. In common with real machines, the rig has journal bearings with frequency-dependent properties and signi cant foundation dynamics within the running range. The coupling between adjacent pedestals was designed to be small in the rig, and is certainly much lower than is likely in practice. The coupling between the horizontal and vertical directions within each pedestal foundation is also quite small, but this is often the case in real machines. The modelling was performed using MATLAB, and the DYNROT Toolbox (Genta 1995) was used to generate the mass and sti¬ness matrices for the rotor. Figure 2 shows a schematic of the shaft, and table 1 gives the dimensions of the shaft sections and the discs. The shaft is made of steel, and the Young’s modulus and mass density are taken as 200 GPa and 7850 kg m¡3 . Modal testing of the shafts and foundations were performed. The shafts were suspended in a ®exible sling and a roving impact hammer excitation used. Table 2 shows a comparison between the theoretical and experimental natural frequencies. Clearly, there is very good agreement between the two. Modal tests were performed to estimate natural frequencies and damping ratios of the foundation with the rotor removed, also using a roving impact hammer excitation. Tests were performed independently for the horizontal and vertical directions, with little cross-coupling occurring between the two directions. Table 3 shows the estimates of natural frequencies and damping ratios. In the horizontal direction more modes are identi ed than measured DOF. (a) Bearing model The sti¬ness and damping coe¯ cients for journal bearings may be calculated in terms of eccentricity, which depends on the physical properties of the bearing, the running speed and the static load (Smith 1969). The physical properties of the bearings on the Aston rig are given above, with the exception of the static load. In the Proc. R. Soc. Lond. A (2000)

1596 bearing 4

M. G. Smart, M. I. Friswell and A. W. Lees disk 5

disk 3

bearing 3

bearing 2

disk 2

disk 1

disk 4

Figure 2. A schematic of the rotor-bearing-foundation test rig. Table 1. Physical dimensions of the long shaft station

length (mm)

diameter (mm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

203.2 203.2 101.6 177.8 101.6 177.8 101.6 203.2 203.2 50.8 177.8 203.2 6.35 25.4 50.8 203.2 177.8 50.8 76.2 76.2 76.2 50.8 177.8 203.2

100 38.1 103 38.1 109 38.1 117 38.1 100 38.1 98.4 38.1 38.1 77.57 38.1 100 38.1 116.8 38.1 109.7 38.1 102.9 38.1 100

station

thickness (mm)

diameter (mm)

3 5 7 18 23

25.4 25.4 25.4 25.4 25.4

203.2 203.2 203.2 203.2 203.2

Proc. R. Soc. Lond. A (2000)

bearing 1


1597

Table 2. Rotor free{free natural frequencies obtained from hammer testing

mode 1 2 3 4

natural frequency (Hz) z }| { theory

experiment

error (%)

12.3 27.4 56.6 100.3

12.0 27.5 55.9 99.1

2.0 ¡0:3 1.2 1.2

Table 3. Foundation pedestal natural frequencies obtained from hammer testing horizontal }|

z

{

frequency (Hz)

damping ratio (%)

15.6 17.3 26.9 29.6 45.2 51.2 57.7

3.2 4.0 1.9 2.0 1.7 1.3 2.0

vertical }|

z

{

frequency (Hz)

damping ratio (%)

46.6 48.4 53.0 58.1

1.3 0.3 0.7 5.4

case of a single rotor supported on two bearings, the load is simply that due to gravity, but for multiple rotors, bearing alignment plays an important part in determining the load. The problem of nding the static loads acting on bearings is a nonlinear one, and DYNROT was used to estimate the loads for the rig. The static loads will depend on the sti¬ness of the foundation, but since, in practice, these will not be known a priori, a rigid foundation was assumed. The static loads acting on each of the four bearings were determined as ca. 400, 461, 486 and 221 N for bearings 1{4, respectively, and there was very little variation over the speed range considered.

7. The simulated example The theory was tested rst using simulated runs of a numerical model which was representative of the physical test rig described above. Dynamic models for the rotor and bearings were known a priori, but a suitable foundation model had to be derived. (a) Foundation model The simulated foundation model had to be broadly representative of the real machine. The mass and sti¬ness matrix elements should have the correct orders of magnitude according to the foundation static tests. The natural frequencies of the foundation should lie in the correct range. There should be more DOF in the model than can be `measured’; in other words, the foundation should have internal DOF. A foundation model ful lling these requirements was produced. No coupling was assumed between the rotational and translational DOF. The mass matrix Proc. R. Soc. Lond. A (2000)

1598

M. G. Smart, M. I. Friswell and A. W. Lees Table 4. Simulated unbalance con¯gurations unbalance (g m¡

1

case

disc

)

phase (deg)

1

1 2 3 5

0.16 0.16 0.16 0.16

0 150 45 195

2

1 2 3 5

0.16 0 0.16 0

350 0 180 0

was diagonal, with the mass of pedestals 1 and 3 taken as 50 kg and the mass of pedestals 2 and 4 as 52 kg. The moment of inertia of pedestals 1 and 3 was taken as 200 kg m2 rad¡1 , and for pedestals 2 and 4 as 210 kg m2 rad¡1 . The damping matrix was diagonal with all damping values taken as 150 N s m¡1 (or units N s m rad ¡1 for the rotational DOF). The rotational sti¬ness was taken as a diagonal matrix, with the sti¬ness at pedestals 1 and 3 taken as 2.5 MN m rad¡1 , and at pedestals 2 and 4 as 2.4 MN m rad¡1 . The translational sti¬ness included coupling between the x- and y-directions at each bearing, modelled as a spring of sti¬ness 0.1 MN m¡1 . Adjacent pedestals had their x-directions coupled with a spring of sti¬ness 0.5 MN m¡1 . Similarly, the y-directions of adjacent pedestals were coupled with a spring of sti¬ness 0.5 MN m¡1 . Each pedestal also had grounded springs of sti¬ness 0.9 MN m¡1 in the x-direction and 1 MN m¡1 in the y-direction for pedestal 1, 0.5 and 0.5 MN m¡1 for pedestal 2, 0.4 and 0.5 MN m¡1 for pedestal 3, 1 and 1 MN m¡1 for pedestal 4. It is clear that the only coupling arises in the translation part of the foundation sti¬ness matrix. (b) Simulation of the unbalance response The models of the rotor, bearing and foundation were combined using (2.1) to give the global dynamic sti¬ness matrix of the machine. Two di¬erent unbalance con gurations were used (as shown in table 4) to provide complex unbalance vectors which, when multiplied by the frequency squared, gave the excitation vector fu . The magnitude of the unbalance is similar in both cases; however, the di¬erence in phase produces di¬erent forcing regimes for each run. With the assumption of linearity, increasing the unbalance magnitude would merely increase the response proportionately. Equation (2.1) was solved over the frequency range 0.5{60 Hz, with spacing 0.5 Hz, to give the responses at the bearings. The forces at the bearings were obtained from (2.8). As discussed earlier, in practice, the identi cation process will be subject to various errors. Errors in the rotor model and unbalance forces were assumed to be negligible, leaving errors in the bearings and measurements. The former were simulated by subjecting the static forces on the bearings to uniformly distributed random errors with a standard deviation of 20% at each frequency. The measurement error was simulated by perturbing the simulated responses with uniformly distributed random noise with a standard deviation equal to 1% of the maximum absolute response. Proc. R. Soc. Lond. A (2000)

response (m)


10-

1599

5

simulated linear nonlinear

phase (deg)

200

0

- 200

0

20

40

60

frequency (Hz) Figure 3. The simulated response and reconstructed response using the linear and nonlinear estimators; bearing 3 is vertical.

(c) Estimation over the whole frequency range Equation (2.1) was solved for the rotor-bearing-foundation model described above, with a total of 16 foundation responses being calculated (two translations and two rotations at each of the four pedestals). The excitation force vector de ned by the rst case of the unbalance weight con guration (table 4) was used to provide forcing to the model. The eight translational responses were then taken and corrupted with 1% noise. These responses, together with the erroneous bearing model (obtained by perturbing the static force), were used to calculate the bearing forces, and (3.9) solved to give linear estimates for the foundation parameters. These were used as starting estimates for the nonlinear estimation routines described by (3.8). In both the linear and nonlinear cases, an output order of 2 (mass, damping and sti¬ness) and an input order of 0 was used, with symmetry of all coe¯ cient matrices being enforced and the numerator matrix N0 taken as the identity matrix. Initially, the whole frequency range from 0.5 to 60 Hz was used. Figure 3 shows a typical response, both the original noisy response used for identi cation and the reconstituted responses from the estimated foundation models (using both linear and nonlinear estimation routines). Table 5 gives the residuals of these ts, de ned in (3.19). In certain regions of the frequency range, particularly around the large peak at 45 Hz, the agreement is good, while in much of the rest of the range it is poor. The reason for this is obvious: the data contain more independent information (16 natural frequencies) than can be adequately represented by the model (maximum of eight natural frequencies). Clearly though, the nonlinear estimation method gives a better estimate of the foundation dynamics than the linear method, the latter being prone to generate spurious peaks. Therefore, from now on, only the results of the nonlinear estimation method will be shown. Proc. R. Soc. Lond. A (2000)

1600

M. G. Smart, M. I. Friswell and A. W. Lees Table 5. Relative output error norms when the whole frequency range is used

DOF

z

B1H B1V B2H B2V B3H B3V B4H B4V

34.72 64.74 35.01 59.70 28.21 39.95 48.19 46.43

linear

¹" r (%) }|

{

nonlinear 16.01 15.61 15.71 20.17 14.45 20.39 17.42 16.89

(d ) Estimation over split frequency ranges When the number of modes in a frequency range is larger than the independent DOF of a model, there are two alternatives; either split the range up into smaller ranges, each of which contains the same number of modes as DOF, or use higher-order polynomial matrix equations. In practice, the presence of noise in the data means that the number of modes in any frequency band will not be known exactly. Therefore, some combination of these alternatives will be used; in other words, the frequency range will be split up into smaller ranges, and di¬erent polynomial powers used to represent the dynamic sti¬ness of the system. Clearly, the models estimated for each di¬erent range will be quite di¬erent, since they represent condensed (and therefore frequency-dependent) approximations to the true system dynamics. The rst step in the estimation procedure was to select the frequency ranges into which the data were to be split. This was done by simply visually inspecting the responses. Some experience is required here, and a trial and error approach may be necessary (this is standard practice in modal analysis). Error charts may then be plotted for each frequency range to determine the model order. The equation error, rather than the output error, was selected, since the latter would have required a full nonlinear analysis for each di¬erent model order. An example of such charts is shown in gure 4, and the model order which yielded the minimum equation error was selected. When no error is shown against a particular model order, this indicates that the resulting set of linear equations became ill conditioned, because too high a degree of polynomial was selected. Table 6 shows the frequency ranges, and the corresponding input and output powers selected for each range. Also shown in table 6 are the residuals for each of the response measurements, de ned by (3.19), which demonstrate that a good t to the data is obtained. (e) Regularization of parameters The fact that a good t is obtained between the model and the measurements is not a su¯ cient test of the quality of the model. The responses of the model and system to a di¬erent excitation to that used in the identi cation provides a more rigorous check of the quality. Accordingly, table 6 shows the residuals of the response Proc. R. Soc. Lond. A (2000)


1601

error (%)

300 range 1–15 Hz

200 100 0

error (%)

400 range 15–22 Hz 200 0 error (%)

200 range 22–40 Hz 100 0 error (%)

60 range 40–60 Hz

40 20 0

2/0

2/1

3/0

3/1

3/2

4/0

4/1

4/2

4/3

model order (q/p) Figure 4. Equation error versus model order.

of the simulated system to the second unbalance con guration of table 4. Clearly, the predicted response to this second unbalance is somewhat poor. To improve the predictive capacity of the model, regularization was performed, using the singular value decomposition to identify the number of modes in the measured data. Figure 5 shows the singular values for the responses over each of the four ranges. Cut-o¬s were selected for the ranges 1{15 and 40{60 Hz based on a step change in the singular value. The range 15{22 Hz does appear to show a step change in the singular value, but any regularization on this range worsened rather than improved the results. The range 22{40 Hz does not show any clear cut-o¬ point. Once the cut-o¬ points were selected, equation (5.4) was used to condense the data. The equation error was again minimized to select the polynomial degrees, and the results are shown in table 6. Once the parameters were estimated, the inverse transformation (equation (5.5)) was used to recover the estimated responses. Comparing the relative output error norms provides an objective test of the improvement, and these are shown in table 6. Regularization has improved the predictability at the expense of the t to the original data. (f ) Discussion Table 6 shows that a good t is possible between the simulated system and the reduced order model used to t it, despite the presence of noise on the measurements. This result is not surprising since one is essentially using polynomials to t measured data, which, provided the data are `reasonable’, should always be possible. The crucial test is the predictive capacity of the model, when subjected to a di¬erent Proc. R. Soc. Lond. A (2000)

1602

M. G. Smart, M. I. Friswell and A. W. Lees Table 6. Fitting parameters and relative output error norms for the regularized and unregularized models (split frequency range)

range (Hz)

unregularized z }| {

0.5{15 15{22 22{40 40{60

q

p

3 4 2 2

1 2 0 0

DOF

z


10.82 12.45 8.79 14.61 10.53 16.22 12.70 11.56

¯t

regularized z }| { q

p

3 4 2 2

2 2 0 0

relative output norm, ¹" r }|

{

di® erence unbalance

¯t

di® erence unbalance

61.64 70.56 49.34 61.03 50.58 66.77 48.22 50.34

29.42 29.28 18.07 27.62 18.36 26.44 33.49 28.53

34.38 33.40 26.96 34.57 23.67 28.39 32.79 36.84

excitation from that used to estimate it. The initial results, shown in table 6, showed signi cant errors in this regard. In addition, the frequencies of the estimated foundation model do not correspond in general to those of the model used to generate the data in the rst place. The reasons for this are threefold. Firstly, the structure of the true foundation model (bandedness, connectivity) is not represented exactly by the matrix polynomial equation. Secondly, the condensed version of the foundation dynamic sti¬ness matrix is represented by the polynomial equation whose input and output orders must be estimated and are not those of the true model. Finally, if there are less physical modes in the range of interest than measured DOF of the model, then the resulting dynamic sti¬ness matrices should be rank de cient. However, noise on the measurements causes these matrices to be full rank, but ill conditioned due to the presence of spurious `noise’ modes (Friswell et al . 1997). Therefore, regularization of the estimates is required. This involves constraining the parameters in some way, thereby reducing the t between measurement and model, but producing better estimates for a di¬erent excitation. The SVD was used to estimate the number of modes contained in each frequency range. The results, shown in table 6, produced some improvement. The di¯ culty lay in selecting the e¬ective cut-o¬ point to use for the SVD. In gure 5, the singular values for the range 15{22 Hz appear to show a cut-o¬ point, but, in fact, regularizing this range worsened the resulting model estimates. In practice, selecting a cut-o¬ point below which the singular values represent noise is both di¯ cult and subjective. This merely serves to con rm that system identi cation is both Proc. R. Soc. Lond. A (2000)

Estimating turbogenerator foundation parameters 4

´ 10-

1603

8

2

1–15Hz

cut-off

0 8

´ 10-

8

15–22Hz

4 0 6

´ 10-

7

22–40Hz

3 0 ´ 10-

6

2

cut-off

40–60Hz

1 0

0

1

2

3

4

5

6

7

8

no. of singular values Figure 5. Singular values of the autocorrelation matrix for each frequency range. Table 7. Experimental unbalance con¯gurations unbalance (g m¡

1

run

disc

)

phase (deg)

1

1 2 5

0 0 0

2

1 2 5

0 1.7 1.7

0 105 180

3

1 2 5

1.7 0 1.7

225 0 315

0 0 0

an art and a science, with user intuition and interaction required. A `black-box’ approach that automatically selects the correct answer does not yet appear to be feasible.

8. Experimental results A total of three run-downs were performed, each from 55 to 5 Hz, and the rst-order response was extracted. The run-down pro le was linear as a result of a computer Proc. R. Soc. Lond. A (2000)


response (m)

1604

10-

2

10-

4

experiment model 10-

6

phase (deg)

200

0

- 200

0

20

40

60

frequency (Hz) Figure 6. The measured and estimated vertical response at bearing 3.

generated control signal. The three unbalance con gurations used for identi cation are shown in table 7, which shows the magnitude and phase angle (in degrees from the key-phasor) of the resulting unbalance vector. The magnitude of the unbalance is similar in all cases; however, the di¬erence in phase produces di¬erent forcing regimes for each run. With the assumption of linearity, increasing the unbalance magnitude would merely increase the response proportionately. The data collected during run 2 were subtracted from those collected during run 1 and were then used for estimating the foundation parameters. The frequency bandwidth was split up into four ranges, as shown in table 8, together with the optimum polynomial orders. Figure 6 shows a typical result, with good ts being obtained over the whole frequency range, although the measurements at frequencies below 15 Hz are rather noisy. The response at low frequency is very small, and thus is likely to be susceptible to noise. The use of a logarithmic scale highlights this noise. However, since the frequency range is split, these noisy data are not able to in®uence the t at higher frequencies. In practice, the poor quality data are readily recognized, and the response predictions at low frequencies would be treated with care. Figure 7 shows the results when the estimated model was excited with the unbalance con guration of run 3, compared with the measurements obtained during that run (again subtracted from run 1). There is reasonable agreement in the magnitudes of the responses, although the magnitude of the peaks predicted by the model is often quite di¬erent to that measured. Again, below 15 Hz, the measured data are rather noisy, and this lower-frequency range is responsible for a large part of the residuals, given in table 8. The results show that a good t may be obtained between the measurements and the model for a given excitation. The identi ed model shows a reasonable predictive Proc. R. Soc. Lond. A (2000)

response (m)

Estimating turbogenerator foundation parameters 10-

2

10-

4

1605

experiment 10-

model

6

phase (deg)

200

0

- 200

0

20

40

60

frequency (Hz) Figure 7. The measured and predicted vertical response at bearing 3 using a di® erent unbalance. Table 8. Fitting parameters and relative output error norms for the experimental data range (Hz)

q

p

5{18 18{27 27{40 40{55

4 4 4 4

2 2 2 0

relative output norm, ¹" r }| {

DOF

z

¯t

di® erent unbalance


8.25 6.51 6.40 6.96 6.44 7.40 7.58 10.06

82.21 63.14 63.67 46.38 48.84 36.32 49.00 50.58

capacity for the estimated model under a di¬erent excitation. Although the frequency position of the peaks are mostly correct, their magnitudes are often not. This is most probably because of an incorrect bearing model or transient e¬ects in the system. Proc. R. Soc. Lond. A (2000)

1606


9. Conclusions Using measured foundation responses, analytical rotor and bearing models and a known state of unbalance, it is possible to estimate a foundation dynamic sti¬ness model for a rotor-bearing-foundation system. The linear method, based on forces acting on the foundation, is quickly solved but su¬ers from the fact that the response contains global, rather than local, modes. The nonlinear method on the other hand provides more accurate models at the expense of longer computation times. The models showed some predictive capacity with respect to di¬erent excitations, but the magnitudes of the peaks in the predicted response were often signi cantly in error. It is believed that this is due to three things: an inaccurate bearing model, changes in the bearing characteristics from run to run and the fact that the system is not entirely stationary. Model order selection and regularization was necessary for the models to have this predictive capacity. The tools used for this were equation error plots, with the model order with a low error chosen, and the singular value decomposition, with a cut-o¬ level being speci ed below which the data were assumed to only represent noise. However, these methods are subjective and require some user judgement. Therefore, the estimation method was broadly successful, with further work required to improve the predictive capacity of the estimated models. This work forms part of a project funded by British Energy plc and BNFL (Magnox Generation) to derive methods for inferring the in° uence of ° exible turbogenerator foundations. The authors thank them for funding this work and for permission to publish this paper. M.I.F. gratefully acknowledges the support of the EPSRC through the award of an Advanced Fellowship.

References Ahmadian, H., Mottershead, J. E. & Friswell, M. I. 1998 Regularisation methods for ¯nite element model updating. Mech. Systems Signal Processing 12, 47{64. Allemang, R. J., Brown, D. L. & Fladung, W. 1994 Modal parameter estimation: a uni¯ed matrix polynomial approach. In 12th Int. Modal Analysis Conference, Honolulu, Hawaii, pp. 501{514. Braun, S. G. & Ram, Y. M. 1987a Time and frequency identi¯ cation methods in over-determined systems. Mech. Systems Signal Processing 1, 245{257. Braun, S. G. & Ram, Y. M. 1987b Structural parameter identi¯cation in the frequency domain: the use of overdetermined systems. J. Dyn. Systems, Measurement Control 109, 120{123. El-Shafei, A. 1995 Modeling ° uid inertia forces on short journal bearings for rotordynamic applications. J. Vibr. Acoust. 117, 462{469. Feng, N. S. & Hahn, E. J. 1995 Including foundation e® ects on the vibration behaviour of rotating machinery. Mech. Systems Signal Processing 9, 243{256. Feng, N. S. & Hahn, E. J. 1996 Turbomachinery foundation parameters using foundation modal parameters. In 21st Int. Seminar on Modal Analysis, Leuven, Belgium, pp. 1503{1513. Friswell, M. I. & Mottershead, J. E. 1995 Finite element model updating in structural dynamics. Dordrecht: Kluwer Academic. Friswell, M. I., Lees, A. W., Smart, M. G. & Prells, U. 1997 Identifying noise modes in estimated turbo-generator foundation models. In IoP Conf. on Modern Practice in Stress and Vibration Analysis, Dublin, Ireland, September 1997, pp. 57{64. Friswell, M. I., Mottershead, J. E. & Ahmadian, H. 1998 Combining subset selection and parameter constraints in model updating. J. Vibr. Acoust. 120, 854{859. Genta, G. 1995 DYNROT 7.0 A ¯nite element code for rotordynamic analysis. Dipartimento di Meccania, Politecnico di Torino, Italy. Proc. R. Soc. Lond. A (2000)


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Golub, G. & Van Loan, C. 1990 Matrix computation, 2nd edn. Johns Hopkins University Press. Hansen, P. C. 1990 The discrete Picard condition for discrete ill-posed problems. BIT 30, 658{ 672. Hansen, P. C. 1994 Regularisation tools: a MATLAB package for analysis and solution of discrete ill-posed problems. Numerical Algorithms 6, 1{35. Hisa, S. & Matsuura, T. 1980 Experiments on the dynamic characteristics of large scale journal bearings. In IMechE Conf. on Vibrations in Rotating Machinery, Cambridge, UK, paper C284/80, pp. 223{230. Lallement, G. & Piranda, J. 1990 Localization methods for parameter updating of ¯nite element models in elastodynamics. In 8th Int. Modal Analysis Conference, Orlando, Florida, pp. 579{ 585. Lawson, C. L. & Hanson, R. J. 1974 Solving least squares prolems. Englewood Cli® s, NJ: Prentice-Hall. Lees, A. W. 1988 The least square method applied to identify rotor/foundation parameters. In IMechE Conf. on Vibrations in Rotating Machinery, Edinburgh, paper C306/88, pp. 209{216. Lees, A. W. & Friswell, M. I. 1996 Estimation of forces exerted on machine foundations. In Int. Conf. on Identi¯cation in Engineering Systems, Swansea, March 1996, pp. 793{803. Lees, A. W. & Simpson, I. C. 1983 The dynamics of turbo-alternator foundations. In IMechE Conf. of Steam and Gas Turbine Foundations and Shaft Alignment, paper C6/83, pp. 37{44. Lembregts, F. & Leuridan, J. 1990 Frequency domain direct parameter identi¯ cation for modal analysis: state space formulation. Mech. Systems Signal Processing 4, 65{75. Leuridan, J. M., Brown, D. L. & Allemang, R. J. 1986 Time domain parameter identi¯cation methods for linear modal analysis: a unifying approach. J. Vibr. Acoust. Stress Reliability Design 108, 1{8. Levi, E. C. 1959 Complex curve ¯tting. IEEE Trans. Automatic Control 4, 37{44. Ljung, L. & Glover, K. 1981 Frequency domain versus time domain methods in system identi¯cation. Automatica 17, 71{86. Natke, H. G., Lallement, G., Cottin, N. & Prells, U. 1995 Properties of various residuals within updating of mathematical models. Inverse Problems Engng 1, 329{348. Nise, N. 1992 Control systems engineering. Redwood City, CA: Benjamin Cummings. Pintelon, R., Guillaume, P., Rolain, Y., Schoukens, J. & Van Hamme, H. 1994 Parametric identi¯ cation of transfer functions in the frequency domain|a survey. IEEE Trans. Automatic Control 39, 2245{2260. Schoukens, J. & Pintelon, R. 1991 Identi¯cation of linear systems. Oxford: Pergamon. Smart, M. G. 1998 Identi¯cation of ° exible turbogenerator foundations. PhD thesis, University of Wales, Swansea. Smart, M. G., Friswell, M. I., Lees, A. W. & Prells, U. 1996 Errors in estimating turbo-generator foundation parameters. In 21st Int. Seminar on Modal Analysis, Leuven, Belgium, pp. 1225{ 1235. Smart, M. G., Friswell, M. I., Lees, A. W. & Prells, U. 1998 Estimating turbo-generator foundation parameters. IMechE J. Engng Sci. 212, 653{665. Smith, D. M. 1969 Journal bearings in turbomachinery. London: Chapman & Hall. Vania, A. 1996 Identi¯cation of the modal parameters of rotating machine foundations. Politecnico di Milano, Dipartimento di Meccanica, Internal Report 9{96. Zanetta, G. A. 1992 Identi¯ cation methods in the dynamics of turbogenerator rotors. In IMechE Conf. on Vibrations in Rotating Machinery, Bath, paper C432/092, pp. 173{181.

Proc. R. Soc. Lond. A (2000)