Bucher and Ewins ROS

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I4 THE ROYAL10.1098/rsta.2000.0714 WJ SOCIETY

Modal analysis and testing of rotating structuresBY I. BUCHER1 AND D. J. EWINS2

1Faculty of Mechanical Engineering, Technion,Haifa 32000, Israel ([email protected])2Department of Mechanical Engineering,Imperial College of Science, Technology and Medicine,London SW7 2BX, UK ([email protected])This paper surveys the state of the art of modal testing or experimental modal analy-sis of rotating structures. When applied to ordinary, non-rotating structures, modaltesting is considered to be well established. Rotating structures, on the other hand,impose special difficulties when one seeks to obtain the parameters of the dynamicalmodel experimentally. This paper focuses on the necessary experimental techniquesand their relationship to the current state of the existing theory. Existing modalanalysis methods, models and techniques, and their advantages, limitations and rel-evance are outlined and compared. In addition, some new developments allowing usto circumvent some of the above-mentioned difficulties are presented.Rotating machines appear in almost every aspect of our modern life: cars, aero-planes, vacuum cleaners and steam-turbines all have many rotating structures whosedynamics need to be modelled, analysed and improved. The reliability, stability andthe response levels of these machines, predicted by analytical models, are generallynot satisfactory until validated by experimentally obtained data. For this purpose,modal testing has to be employed and further advance is essential in order to over-come the difficulties in this area.In this paper, the differences between the mathematical models used for dynamicanalysis of non-rotating and rotating structures are clarified. The implications of themodel structure, in the latter case, on the application of modal testing are presented,as this is a point of great importance when experimental modal analysis is employedfor rotating structures. Models with different degrees of complexity are being usedfor different types of rotating machines. A classification of such models is outlinedin this work and the underlying assumptions and features are described in termsof a hierarchical complexity. Several applications of modal testing are reported hereand some experimental evidence to support the validity of the theory is presented.Desired future activities, which are required to advance the theory and practice ofthis field, are summarized in conclusion.

Keywords: rotating structures; modal testing; speed-dependent parameters;whirling mode shapes

1. IntroductionRotating machines appear in almost every aspect of modern life, ranging from domes-tic appliances to automobiles, power plants and aeroplanes. The tendency to createPhil. Trans. R. Soc. Lond. A (2001) 359. 61-96 ? 2001 The Royal Society

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I. Bucher and D. J. Ewinsfaster and better rotating machines, fuelled by the desire to increase their productiv-ity, increasingly requires deep knowledge of their dynamical behaviour and thereforeaccurate models are needed. The efficiency of rotating machines is greatly improvedby maintaining small gaps between rotating and non-rotating parts. This can beachieved only when the designers of the machines have great confidence in the dynam-ical models of these machines, thus knowing that no failure due to excessive vibrationlevels may occur. The most reliable means of obtaining a dynamical model describingthe dynamic behaviour of a machine involves an experimental procedure as it betterreflects reality. However, machines that contain rotating elements are often difficultto model reliably using theory, mainly due to inherent uncertainties as to the operat-ing and boundary conditions and, in these cases, properly conducted in situ testingis by far the most reliable approach. In modal testing, we seek to extract the vibra-tion modes (i.e. natural frequencies, mode shapes and damping coefficients) frommeasured data, and this allows us to describe the dynamical behaviour effectively.In this survey, we pursue the general methodology of modal testing and discusssome aspects of applying this method to structures having rotating elements. Thepresent paper is meant to serve three main purposes.

(1) To provide an initial guide or road map for newcomers and a summary forpractitioners.(2) To outline some of the assumptions and experimental procedures currentlybeing used.(3) To present some new results in modal testing of rotating structures.The structure of the paper is as follows. Section 1 provides an introduction to thesubject and outlines briefly the basic theory of modal analysis with special refer-ence to rotating structures. In this section, both shaft- and disc-related equations ofmotion are presented and the role of stiffness isotropy and of damping in the modaldecomposition are outlined. Section 2 is devoted to experimental methods. In ?3some excitation techniques are compared, the required and recommended signal-processing procedures are described and some experimental results presented anddiscussed. The final section concludes the paper and summarizes the current stateof the art of this emerging topic.

(a) Rotating structures: structural dynamics models and their modal decompositionIn our introduction of the various mathematical models of rotating structures, weconfine ourselves to models which describe 'small' vibrations. Testing and modellingthe dynamics of rotating machines is a very broad area and the current text con-centrates on a fraction of this discipline. Within this subset of models we furtherrestrict ourselves to linear models formulated in terms of a finite number of degreesof freedom (DOFs) (i.e. to a discretized model) which are an approximation of thegoverning partial differential equation of the model. Such a system of equations isgenerally time varying (Yakubovich & Starzhinskii 1975) as, due to the rotation,some properties may change periodically with time,

M(t)i(t) + C(t)q(t) + K(t)q(t) = f(t), q(t), f(t) e RNX1 (1.1)Phil. Trans. R. Soc. Lond. A (2001)

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Modal analysis and testing of rotating structuresload

/ xternal \ 4 discdrivig force anisotropic,driving Ocrackedelement

bearings transmission

foundation, K'~N F-l bearings rFigure 1. A schematic description of a machine containing rotating components.

Such time variations can be a result of non-isotropic shaft segments (Genta 1988)or cracked shafts (Gasch & Pfutzner 1975; Gasch 1976). A hypothetical but repre-sentative system is depicted schematically in figure 1. This system contains drivingelements, shafts, motion-coupling devices, bearings, gears and transmission elements.In addition, there could be discs (possibly bladed) and some external loads, whichmay affect the dynamics.A common assumption which is often made is that the structure under test containsonly isotropic rotating elements (i.e. it is assumed that both inertia and stiffnessproperties are not a function of the instantaneous angle of rotation). When theformer assumptions hold and when the measurements are performed in an inertialcoordinate system, we obtain an additional simplification under which the matricesin (1.1) are no longer time dependent but are only speed dependent (Lalanne &Ferraris 1990),

M(Q)q(t) + C(Q)q(t) + K(t)q(t) = f(t). (1.2)For a constant speed of rotation 97, equation (1.2) represents a general linear time-invariant (LTI) system and, as a result, commonly available tools can be usedto analyse the dynamic behaviour of such a system. The differential equation ofmotion (1.2), which represents a rotating structure, is said to be non-self-adjoint(NSA). Passive non-rotating structures are generally self-adjoint and so their fre-quency response and their system matrices are symmetric (Meirovitch 1980; Han& Zu 1995). The effect of rotation on components gives rise to a non-symmetric(and NSA) equation of motion and to a non-symmetric frequency-response func-tion (FRF) matrix. Models of self-adjoint structures can be completely expanded interms of a single set of eigenvectors (modes) and eigenvalues. On the other hand,NSA structures require two sets of eigenvectors with a set of eigenvalues to describetheir dynamic behaviour fully.The main differences between static and rotating structures stem from the follow-ing facts.

(i) Rotating structures have speed- (and possibly time-) dependent properties (seefigure 2) (Genta 1988).

(ii) The response and the excitation parameters often operate in different (sta-tionary or rotating) frames of reference. This may give rise to a complicatedPhil. Trans. R. Soc. Lond. A (2001)

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I. Bucher and D. J. Ewins

250- (a) 22 A (b) (200-

, 100-50 o *-

0 2000 4000 6000 8000 H(Q)speed of rotation, Q2

Figure 2. Variation of structure's dynamics with rotation speed. (a) The variation of naturalfrequencies with speed of rotation for a typical rotating structure. (b) The variation of FRF withfrequency and speed of rotation (shown at two distinct speeds of rotation).

relationship between the frequencies of the excitation and those appearing inthe response (see Tobias & Arnold 1957; Lee 1991; Irretier & Reuter 1994).(iii) Rotating structures are non-self-adjoint in general. This gives rise (unlike non-rotating structures) to non-symmetric matrices in the equation of motion (seeNordmann 1983; Geradin & Rixen 1994; Xu & Gasch 1995).(iv) Damping and stiffness for shafts, splines, press-fits in fluid-film bearings cangive rise to 'non-conservative stiffness' due to internal energy dissipation in arotating element. These effects may convert rotational energy into vibratoryresponse which may be unstable (see Ehrich 1992; Childs 1993; Kramer 1993).(v) As rotating structures are mostly axisymmetric, they possess almost identi-cal pairs of natural frequencies and therefore special measures are needed foraccurate extraction of their modes.

(vi) A considerable amount of kinetic energy is stored in high-speed rotating ele-ments. This energy may be coupled to the vibratory response giving rise tophenomena not directly caused by the applied excitation during an experimen-tal procedure (Gasch & Pfutzner 1975).(vii) Nonlinear effects that are most notable when the response levels are high arecommonly found in bearings and coupling elements (Tondl 1965; Gasch 1976).

If one wishes to obtain a model to describe the dynamics of such a structureexperimentally, in a way that allows us to predict the structural response to anarbitrary excitation, all the aforementioned effects need to be taken into account. Inparticular, the effects of rotation, as demonstrated in figure 2, cannot be ignored forhigh-speed rotating elements.Figure 2 illustrates the dependence of the natural frequencies and the FRFs uponthe speed of rotation.As mentioned earlier, application of the existing methodology of modal testing isexplored here and the experimental extraction of mode shapes will be described in ?2.Phil. Trans. R. Soc. Lond. A (2001)

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Modal analysis and testing of rotating structuresIn the following section, a brief theoretical account of modal analysis is provided toassist with the description of the experimental methods.

(b) Modal decomposition: some theoretical aspects affectingthe experimental procedureIn this section, equation (1.2) is expanded in a modal series. The right- and left-hand eigenvectors are then used to express the FRFs. This expansion serves as thebasis for the modal testing procedure, which is addressed later. Several models andthe accompanying assumptions are developed in a diminishing (legree of complexity.Let the free response solution of (1.2) at a constant speed of rotation, Q, beexpressed asq(t) = ext. q(t) C RN. (1.3)

Substituting (1.3) in the homogeneous version of (1.2), we obtainX(A = o0. (1.4)

where we use the definitionX(A) = (A2M() + AC(Q) + K(Q)) (1.5)

for X(A), which is the so-called dynamic stiffness or lambda matrix (Lancaster 1977;Muzynska 1993; Lee 1993).The 2N solutions of the equation det X(Ar) = 0 comprise the eigenvalues A,, forwhich there are corresponding right-hand eigenvectors (right latent vectors) main-taining X(A,r)r - 0, and left-handed eigenvectors r,, which are defined as the solu-tions of (;rXY(A) = .The inverse of the dynamic stiffness y(A)---the FRF matrix H(u)-can be decom-posed into a modal series which is expressed in terms of the eigenvalues and the(right- and left-) eigenvectors as follows:X-1(i:) = (-W2M(j) + iC(Q) + K(Q))-1.

H(uj) (see Appendix A for the fill derivation) can be expressed asN /,T +Or(/T'rH(w,) - C r ,1, H(w) R NxN (1.6)-L'U -- \ iLAJ Ar

where a, are scaling coefficients that can be eliminated by proper scaling of /,.and (r,.If we now consider the implications of this formula for the process of model identi-fication via a modal test, equation (1.6) reveals that both a row and a column of thefrequency-response matrix must be known in order to be able to estimate both V'rand 0,r. This means in practical terms that both the response sensor and the excita-tion device need to be moved to every point on the structure. Naturally, many pointson the structure cannot be accessed with an excitation device and so, practically, afull model of a general rotating structure cannot be obtained in the conventional wayof modal testing.Several researchers (see, for example, Lancaster 1977; Meirovitch 1980; Genta1988; Zhang et al. 1987; Wang & Kirkhope 1994: Geradin & Rixen 1994) havePhil. Trans. R. Soc. Lond. A (2001)

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I. Bucher and D. J. Ewinsconsidered some simplified theoretical versions of (1.2). In their formulations, themass, gyroscopic/damping and stiffness matrices can take some special forms, result-ing in some special relationship between the left- and the right-hand eigenvectors.These relationships can potentially simplify the experimental procedure and thusrequire fewer measurements, necessitating less accessibility to the tested structure.Zhang et al. (1987) have shown that the left-eigenvectors can be deduced from theright-eigenvectors in some special cases even when the stiffness matrix is asymmetric(e.g. when the rotor is supported by fluid-film bearings or seals (Childs 1993)). Thisresult has mainly a theoretical value, as real systems would generally not obey theconstraints indicated in his work. We shall confine ourselves here to the (admittedlyrestricted) case of bearings exhibiting symmetric stiffness (and damping) matrices(e.g. systems supported on anti-friction, ball- and active magnetic-bearings). In someof these cases, we can show that the application of an excitation force at a few points,or even a single point, on the structure can still yield a complete model.

(c) Undamped gyroscopic systemsUndamped gyroscopic vibrating structures (see Meirovitch (1980) for a definition)differ from non-gyroscopic ones by an extra speed-(of rotation)-dependent term.In order to isolate effects attributed to the gyroscopic terms we can further sim-plify (1.2). Assuming that there is no damping and that the mass and stiffnessmatrices are speed independent, one obtains a simplified equation,

Mq(t) + QGq(t) + Kq(t) = f(t). (1.7)Here, G describes the gyroscopic effects and it can be shown (Meirovitch 1980) that

M =MT, K =KT, G=-GT. (1.8)For this system, the left-hand eigenvectors can be computed directly from the right-hand eigenvectors (see Appendix C for proof) and consequently the FRF matrixH(w), which is still non-symmetric, has a known form,

H(W) ZEarra.rvr T1 AJ Ar ig - AXr=1(iw

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Modal analysis and testing of rotating structures

Figure 3. Foundation and bearing supporting a rotating shaft segment.

(d) Shaft-bending dynamicsShaft or rotor vibration is one of the most important factors affecting rotatingmachines. In this section, we concentrate on some special cases of shaft-bendingvibration, where we seek further simplifications of the model and a correspondinglysimplified testing procedure.As most rotating elements (e.g. shafts and discs) are axisymmetric, the bound-ary conditions (foundations or the bearings) have a significant effect on the overalldynamical behaviour. Typically, a shaft is mounted in some type of bearing thatusually resides on a foundation, as shown in figure 3. The motion of the shaft ismeasured in the x- and y-directions and the vector of DOFs can be convenientlydivided (Lee 1993; Joh & Lee 1993) as

q(t) ((t)) (1.10)where x(t) and y(t) may include both linear and angular DOFs.In this section, we will discuss the influence of the foundation and, in particular,the influence of its stiffness matrix (neglecting the damping matrix) on the requiredmodel. It will prove convenient to use a numerical example where a specific rotorrepresented by a finite-element (FE) model (Genta 1994) is considered.

Example 1.1 (rotor FE model). In this example (see figure 4), the bearingshave an elastic stiffness supporting the shaft in both the x- and y-directions. Theshaft has a total length of 600 mm, and is symmetric around the centre bearing. Theshaft is divided into 12 equal elements having a length of 50 mm. The shaft materialproperties are taken as: Young's modulus E = 70 GPa and density 3200 kg m-3;Poisson's ratio v = 0.3, while the rigid discs have a density of 7800 kg m-3. Thismodel will be used to illustrate both the free and the forced response properties ofrotating structures having various types of supports, thus illustrating various effects.

(i) Perfectly axisymmetric shaft, with isotropic bearings: undamped caseThe first model we shall focus on is the simplest. In this model, the foundationsexhibit isotropic stiffness behaviour and have no damping. Consequently, the rotorPhil. Trans. R. Soc. Lond. A (2001)

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I. Bucher and D. J. Ewinsdisc element

15 min

U

shaft clement- 126 mm dia.80 mm~/~~ ~L dia.5 mm 13 mmdia. r di

bearingelementstiffness anddampingFiglure4. Simiple rotor FEiiodel with elastic slupl)ortsat the b)earings.

Figure 5. Isotropic slupporl (J\A = Kr) shlowingtile directiols oftle appliedl force and( tlle response.

(n 1)0 allalyse('d ill 011 directioll (say, .1) froml wllicli the response in the other diirec-tioil (y) can 1)e (directly computed. This type of structure will have Ilo(le sliapes inwhih each('point traces plerfect circular orl)its (see figllre 6). Isotropic slll))orts arecllaracterized( })y the fact that any displacemlent will be exactly in the salme directionlof tlhe applied force (see figure 5).Ill the isotropic case (tlie imodel in figllre 4). we use three idenltical bearings liavillgall i(lentical stiffness AK KB of 106 N m1111n both the x- and(Iy-directions. One of themlo(de shapes of tllis imodel, computed for a rotation speed( of 3000 rpm. is shown infigure 6.Rotors iimoullted(on isotropic suppIorts have mode shapIes that are always containedin one )lalle. tils plane is rotating (either forward of backward) at a freqllencywhich is equal to the iatulral frequency of the specific mode. Indlee.d figure 6, whiclshows tlhe deflected shaft and a surface conllecting the benlt shaft to its unbent state,illlstrates thie imotion of tihe shaft dllrilg a complete rotation (at speed( of rotationof 3()()0 rlpm). In this case, eac p)Oillt on the shaft traces a perfectly circullar orbit inthe xty-plane il each cycle.For an isotropic rotor system (shaft and bearings), thle mass, gyroscopic and stiff-lness imatrices have a special forml (as described il Appendix C). Furthermlore, it liasbieen shown (see, for examlple, Lee 1993) that all appropriate selection of coor(dinates(in accordance witli (1.10)) yields a special form of these matrices (Lee & Youllg-DonPhil. 7TIan.S.R. Soc. Loud. A (2001)

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/ 1

Figure 6. A circular backward whirling Lmodle hape (6th miode, natural frequency at 172 Hz) ofa rotatingrotor (at 3000rpm) having isotropic siipports.The thick line indicates the locationof the shaft centreat t = 0.

1993) (see also (1.8)),o G 0 Go K- Ko 0 IR2unx2n0 Alo= _-GT 0 0 K C I

q -- C iixl ' E .-x Iq = {} e 221x f ={ } R2nx1(1.11)

Here, llo and Ko represent the motion in one (lirection (either x or y) and Gorepresents the gyroscopic coupling between the x- and y-planes.Another property of this selection of coordinates is mlathematically expressed asAMo Ao AI o = Ao. Go-= Go RC ' (1.12)

As stated above, the y-direction part of the nmodeshape is lagging (for a backwardwhirling miode) or leading (for a forward whirling mode) the .-direction part by 90?and therefore we can write, in this case.

(i x(. a ER 2nXI (1.13)where a e RN/2 is real.

The FRF matrix is conveniently partitioned, as shown in Appendix C, and thepart solely related to the x-direction (both in terms of excitation and response, andhence one quarter of the full matrix H((c)) is obtained by substitution of (1.13)Phil. Trans. R. Soc. Lond. A (2001)

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Tz lulsol? Ibrg#2 KyL x 4 x X

Figure 7. A rotating system with only gyroscopic cross-coupling and a special type of bearings.into (1.9) (see also Wang & Kirkhope 1994, eqn (51)),

N/2 fiaraaT/2 T (1.14)r=1

where the constants 3r are real and, as shown in Appendix C, so is Hxx(w).A similar argument leads to the conclusion that the response in the x-directionto a force in the orthogonal y-direction consists of a purely imaginary FRF (again,provided there is no damping present),N/2 irwaraT

Ha)=i Elf_J2rr (1.15)xy(w) = iSz 2:a (1.15)r=l rIt is now clear (for this case) that the complete modal base can, theoretically, beobtained from n = N/2 measured FRFs in one direction, while using a single exci-tation point. This point was noted by Zhang et al. (1987) and Lee (1993).Some complication arises when the supports are not isotropic or when the dampingcannot be neglected. Lee (1993) has proposed the use of a perturbed modal set and,indeed, this approach is only suitable for small deviations from isotropy. In a paperby Wang & Kirkhope (1994), a method for the efficient calculation of eigen-propertieswas proposed and the model considered there was exploited for the approach devel-oped here. This approach is described in the following section (see also Bucher &Ewins 1996).(ii) Axisymmetric shaft mounted on a special type of undamped non-isotropicbearings

The particular model presented here considers the case where the bearings havegreater stiffness in one direction than the other but where the principal axes ofstiffness coincide for all the bearings on which the rotor is mounted. Such a situationis depicted schematically in figure 7.The distance of each point on the ellipse (in figure 7) from its the centre indicatesthe value of stiffness in the direction indicated by a line connecting the centre andthe specific point on the ellipse. It has been shown (Wang & Kirkhope 1994) that,

Phil. Trans. R. Soc. Lond. A (2001)

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KA 10 5)105 N m"K NKB=( - 105 N m-10( 1%

lO 0 1 0 m

Figure 8. Rotor mounted on a general (non-isotropic) type of foundation.in this case (assuming, without loss of generality, that the x-axis coincides with oneof the principal axes of stiffness), the mode shapes can be expressed as

()r +ib)a r =rP r=1,...,N. (1.16)This special case gives rise to mode shapes in which the deflected shaft is alwayscontained in a plane which is rotating at a rate corresponding to the specific naturalfrequency of vibration. Such motion is essentially described in figure 6, except that,in the present case, each point on the shaft traces an elliptical orbit, rather than thecircle in the isotropic case. The FRF for this case can expressed by the followingexpression (Bucher & Ewins 1996):

N/2 - _ TH(w) r2(a ar )+ i (- 0rab ) /(3w2_-_w2) (1.17)N/2)= - brbT) -b2aTr=l -It is evident from (1.17) (see also Appendix C) that the real part of the FRF rep-resents the in-plane response while the imaginary part represents the cross-couplingbetween the x- and y-planes. Cross-coupling is caused by the gyroscopic effects,but the damping (which is excluded from (1.17)) may be another reason for thecross-coupling. It is worth mentioning that, when the damping is very small, equa-tion (1.17) provides a reasonable estimate of the FRF, particularly in a region away

from the natural frequencies. This will be illustrated later by some experimentalresults.Theoretically, a single column of the FRF matrix will suffice in order to derive thecomplete modal model. In this case, a single exciter (say, in the x-direction, DOFnumber j) can be used, and the measured FRFs can be modelled, by

N/2- 2Hj(w)- = E rj (-iwwrb) -( 2 ). (1.18)r=1 -

Due to the rotation of the rotor, or more precisely due to the gyroscopic couplingthat it produces, we can extract the information in a direction which is orthogonal tothe excitation. In this case, the rotation simplifies some aspects of the experimentalprocedure.Phil. Trans. R. Soc. Lond. A (2001)

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Figure 9. A mode shape at Q = 3000 rpm having a natural frequency at 144.9 Hz. Due to thenon-isotropic bearings, the Imotionis iion-planar.(iii) General anisotropic bearings

In general, it cannot be guaranteed that the bearings will always comnplywith theregular niodlels presented in ? 1 d (i) and 1 d (ii). A more general type of foundation.as illustrated in figure 8, gives rise to non-planar mode shapes. In order to illustratethe effect of such anisotropic foundations on the modal properties. the FE model fromfigure 4 was combined with the bearing properties of figure 8 to yield the mode shapedepicted in figure 9. In this case. the bearing foundations are no longer isotropic andtheir principal axes do not coincide.The model in figure 6 was combined with the foundation in figure 8 to conpuitethe mode shapes. One mode shape is depicted in figure 9, where it can b)e observedthat this modle is no longer planar.A rotor with anisotropic bearings and, in particular, a structure that incliudesdamping, cannot be filly modelled from measuremenlts using excitation in just oneor two DOFs. Measurement of a complete row and column (of the FRFs) are needed.The only possibility in this case is to use perturbed models, under the assumptionthat the deviation fromi the isotropic miodel, or, alternatively, from the undampedmiodel, is small.(iv) Models with small anisotropy or light damping

It has been shown by Lee (1991) that a smiall deviation from isotropy can berepresented in terms of perturbed eigenvectors. Here, a slightly more general model(the one presenited in ?1 d (ii)) will be used as a basis. To this iiodel, a small amountPhil. Trn7s. R. Soc. Lond. A (2001)

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Modal analysis and testing of rotating structuresof damping is added and it will be shown that it is still possible to extract a completemodel given two columns of the FRF matrix. The two columns are to be measuredusing an excitation at one point along the shaft in both the x- and y-directions. Theperturbed mode shapes of this system are expressed as (Bucher & Ewins 1996)

-- (t)r =.(a[b-ib),a 07~- r A,.--?w,1-=/Cl-/(2cW (1.19)where ag, b, are small perturbation vectors of the rth mode representing the devia-tion from the model in ? 1 d (iii), which is represented by

al-ibThe subscript r is omitted from ihereon, but it should be understood that both aand ac, for example, are vectors related to the rth mode.Substituting (1.19) into (A 11) in the appendix (see also Bucher & Ewins 1996),one obtains

;N W2D,, + iwFrH(w) = E1 ,2 (1.20)() - 2 - 2 + 2iCr,rW'where a first-order approximation leads to (again omitting the indices for brevity)

??( ba^aT+ baT bbT ,D aaT abT - a -bTDr baT ba +ba bbTFr (r kr baT + baT bbT (1.21)

1 (agaT +aaT abT1r (- -baT bEbT + bbT .By applying an excitation in both the x- and y-directions at a single point along theshaft, one obtains 4n complex equations from which the 4n entries in the vectorsa, b, ag, b C Rn 1 (in addition to the eigenvalues) can be extracted for every mode(Bucher & Ewins 1996). Equations (1.20) and (1.21) are revisited in ?4b (i), wheresome experimental results are shown and analysed.

(e) Anisotropic rotor mounted on anisotropic bearingsPractical engineering structures sometimes contain both anisotropic rotating ele-ments (i.e. rectangular cross-section, Hooke joints) and anisotropic supports. Suchstructural elements are schematically depicted schematically in figure 10.The anisotropic rotor resting on anisotropic supports (as shown in figure 10a)possesses time-varying inertia properties, whether observed in stationary or rotat-ing coordinates. Similarly, the gear-system shown in figure 1Obhas angle-dependentstiffness, which depends on the instantaneous state of meshing. Such systems should

be described by (1.1), where, for constant speed of rotation, the mass and stiffnessmatrices are periodic functions of time. Very few experimental results are reportedfor such systems, and Nordmann & Schwibinger (1989) describe one of the very fewPhil. Trans. R. Soc. Lond. A (2001)

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iXA

(2

(a) (b)Figure 10. Structural elements possessing angle-dependent properties.

attempts. These systems need to be described using Floquet's theory (Genta 1988;Yakubovich & Starzhinskii 1975), where the response to a pure sinusoidal excita-tion Fo cos wt gives rise to a large number of frequencies in the response spectrum,i.e. n2Q? pw, n,p integers (Bucher et al. 1994). In this case, the natural frequenciesand the mode shapes are themselves periodic functions of time, and a general pro-cedure with which one could obtain the modal parameters experimentally still doesnot exist.2. Disc dynamics

Rotating discs, including bladed discs and impellers, are often dealt with separatelyfrom shaft dynamics. This fact is partly justified as most disc modes, apart from the0- and 1-nodal diameters modes are decoupled from the shaft bending (Bienzo &Grammel 1954; Ewins 1976, 1980).The discussion here is mostly confined to models related to testing of rotatingdiscs in situ (Tobias & Arnold 1957; Radcliffe & Mote 1983). In this case, the excita-tion means are limited and one usually has to rely on the natural excitation existingwhile operating in normal working conditions. Rotating discs, being mostly decou-pled from shaft dynamics, need to be directly excited (i.e. not through the shaft),usually by a stationary exciter (Radcliffe & Mote 1983). The response is measuredin the stationary (Irretier & Reuter 1994) or rotating (Bucher et al. 1994) framesof reference (see ?3 for a description of measurement methods). The use of differentframes of reference alters the apparent frequencies that are observed in each coordi-nate system. A stationary sensor, for example, will measure many frequencies otherthan the frequency of excitation, partly due to the relative motion of vibrating disc,with respect to the sensor. An additional complication is caused by the fact thatsome excitation forms are stationary while others may be rotating and thus a largenumber of frequencies may compose the measured dynamic response of the disc.A complete derivation of the rather lengthy expressions involved in rotating discdynamics is beyond the scope of this paper and the reader is referred to Irretier &Reuter (1995) and Khader & Loewy (1991) for this material. Only a brief accountof the required mathematics is provided here in order to facilitate the application ofmodal testing methods.A schematic description of an experimental system and a photograph of this sys-tem are shown in figures 11 and 12. Both the electrodynamic actuators and theelectromagnetic exciter (see figure 11) are being used to excite the system. Figure 11assists in defining the coordinate systems, which will be used in the presented math-ematical formulation. A photograph of the actual experimental system is presentedas figure 12.Phil. Trans. R. Soc. Lond. A (2001)

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Modal analysis and testing of rotating structuressensor -bb,

A iA

electromagneticexciter 0

x

Q

Figure 11. Disc (i.e. rotating) coordinate system, the excitation and measurement device.

Figure 12. An experimental system showing a flexible disc with shaft and disc excitationdevices (see Bucher et al. 1994; Bucher & Ewins 1996).Some experimental results from the rig shown in figure 12 are provided later inthis paper.

(a) A brief account of modal analysis theory for rotating discsIn modal analysis of rotating discs, one usually assumes that the disc is perfectlyaxisymmetric (or cyclically symmetric in the case of a bladed disc). In such cases, itPhil. Trans. R. Soc. Lond. A (2001)

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I. Bucher and D. J. Ewinsmay be assulmed that the mode shapes can be expressed in polar coordinates as

COSr) = RC() cos(0), , (r, 0) = Rs (r) sii(n(H). (2.1)where r and 0 are polar coordinates attached to the disc (as described in figure 11)and the indices n and s denote the number of nodal diameters an(ld odal circles.respectively.In this type of structure, both sine- and cosine-related mode shapes exist at eaclnatural frequency, except for the case of modes where n = 0.It has been shown (see, for example, Tobias & Arnold 1957: Irretier & Ruleter1995) that the response of a perfectly axisymmetric disc, as measured by a stationaryobserver (such as a proximity probe at a fixed location), to a harmonic excitation ofthe forlm f(t) = Foei(t-3). also applied by a stationary device, can be expresse(l as(Irretier & Reuter 1995)

n=O(2.2)

whereH ) (-Rsin (A)Rsn (E) + RcS (rA)Rq?s( E)) ,Hn ( ) - sin c-A++ cos(-t _ 2) e

and we use the definition for Asin(a), which is a function of ca- ?: n?. as(,s5) a+ 2i(,sin ,Sill (2.3)

We use the definition of coordinates shown in figure 11, where rA and rE denote theradial locations of the response measurement and excitation DOFs, respectively.Equation (2.2) shows that the measured response is composed of waves travellingin a co-rotating and counter-rotating direction. The rotation gives rise to a frequencyshift of +2nQ for a co-rotating wave and -2mni for a counter-rotating wave. Curi-ously enough, the disc resonates (the denominators in (2.2) attain a minimum) atseveral frequencies maintaining wu n- a n,s, while the measured response will beat completely different frequencies from the frequency of excitation. Equation (2.2)illustrates one of the difficulties which the experimenter faces when measuring thedynamic response of a rotating disc. The many modes contributing to the totalresponse to a fixed harmonic excitation give rise to a multitude of response frequen-cies. In addition, for every frequency component existing in the excitation, a largenumber of frequency terms are observed. Indeed, examining (2.2), we can see thata single excitation frequency cwresults in an infinite number of frequency terms inthe response. These frequencies are all functions of the excitation frequency and thespeed of rotation; to be more specific, one may expect all combinations of w + 2n?7.n - 0, ?l, ?2, ?3....In Irretier & Rueter (1994), a method for extracting the modal parameters isdescribed. Due to the large number of spectral lines, generated by a single frequencyexcitation, appearing in the response, an effective method for eliminating most of theharmonics is to choose an excitation frequency close to resonance, e.g. w n? Q w7i(Bucher et al. 1994). This makes one mode dominant and reduces the complexityPhil. Trans. R. Soc. Lond. A (2001)


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Modal analysis and testing of rotating structuresof the measured signal considerably. Other aspects of the testing procedure are dis-cussed in ?3.When the perfect angular symmetry assumption is relaxed (i.e. to accommodatedefects, residual stresses or simply due to geometric imperfection), one is forced toassume a more general form of (still periodic) mode shapes (Okubo et al. 1995;Irretier & Rueter 1995).

cos c0) = E Cns() + u)n,S?r. 0) R> (r) cos(u(o + 0 t(r/)))ix; (2.4)

/sin(r, 0) E Rlsi (r) sin(u( + 0Snll ((r)))., s n, s, n, S, u '0 )Further complications may be encountered, especially when a non-axisymmetric

structure rests on non-isotropic foundations. In this case, the equations of motion,whether represented in stationary or rotating coordinates, give rise to time-varyingcoefficients (Genta 1988; Nordmann & Schwibinger 1989). Consequently, a sinusoidalexcitation, whether applied in stationary or in rotating coordinates, will give riseto a more general expression for the anticipated frequency terms in the response,i.e. ?+p + (?n ? m)Q. n, , ,p = 0, 1, 2, 3....

3. Test methods: selection of excitation method measurementapproach and curve fitting

Modal testing is basically composed of two steps: (i) acquisition of response data(typically, FRFs); and (ii) curve fitting the measured response functions to obtainan assumed parametric (modal) model (which was described in ? 1).The test methods can be divided into several groups, each one dealing with somespecific parts of the rotating machine. Here, we describe the methods that are associ-ated with the shaft and disc dynamics separately. The reader is referred to the litera-ture for a more comprehensive treatment of bearing characteristics (see, for example,Childs 1993; Kramer 1993; Ehrich 1992), and to Ewins (1980) for studies of bladeddiscs. Gears, transmissions, etc., are generally treated in the machine dynamics ormachine diagnostics related literature and overall machine and foundation dynamicsare discussed in Kramer (1993).

(a) Experimental methods for extracting shaft dynamicsShaft-dynamic models can be obtained by using the standard approach as in ordi-nary modal testing (Ewins 1984; Lee & Hong 1993; Joh & Lee 1993; Gahler & F6rch1994; Irretier & Renter 1994; Irretier 1999). In the general approach, a known forceis applied at one or more locations and the response is simultaneously measured atseveral locations along the shaft in the x- and y-directions. Such an arrangement is

shown schematically in figure 13.Special excitation methods are needed for each category of structural component,e.g. shafts, discs and bearings.Phil. Trans. R. Soc. Lond. A (2001)

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I. Bucher and D. J. Ewinssensors

?2 ?2 rFr

FI.x ~ Fi,x

F, actuators F,Figure 13. A rotatingshaft showingthe appliedforcesand the measuredresponse.

(i) Excitation methods for extracting rotating shaft modelsDue to the nature of rotating machines, the application of a controlled or mea-

surable force is far more difficult than in the non-rotating case. In this section, acomparative summary of excitation methods that have been used in the past is pro-vided.Controlled unbalance in situ

Within this group of methods, several researchers have tried to estimate the exist-ing unbalance (Lee & Hong 1993; Irretier & Kreuzing-Janik 1998), while others haveapplied a known amount of unbalance (see Muzynska & Bently 1993; Iwatsubo et al.1988). By measuring the response, an attempt can be made to estimate the FRFs.A typical element which was used for such an excitation source is a freely spinningdisc (spinner) (Muzynska & Bently 1993). A known mass m, which is attached at aradius r, generates a rotating force having the magnitude F = w2mr, where w is theindependently controlled speed of rotation of the spinner.Advantages. Relatively simple mechanical construction. Fixed force amplitude forsmall vibration amplitude and constant speed of rotation. Easy to apply rotatingforces (forward or backwards).Disadvantages. Difficult to maintain constant frequency. A tracking filter is needed toextract the vibration component synchronous to the spinner's rotation. Not possibleto apply a point force in the fixed coordinate system.Impulsive excitation

The traditional hammer testing method (Ewins 1984; Nordmann & Schwibinger1989; Zhang et al. 1987), often used in modal testing, has been applied to rotatingshafts.Advantages. Easy to apply. No additional hardware is required apart from a suitablyinstrumented hammer (Ewins 1984). Broadband excitation (excites several modessimultaneously).Disadvantages. Force amplitude and direction is not easily repeatable. May con-tain uncontrolled tangential components. Categorized as a broadband excitation andtherefore possesses inferior signal/noise ratio.Phil. Trans. R. Soc. Lond. A (2001)

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motor

force|shaker transducersacc. sensors

shaftFigure 14. The experimentalsystem showingthe shaft, discs, motor and exciters.

Electrodynamic shakers connected via auxiliary bearingIn this approach (Rogers & Ewins 1989; BRITE 1996; Stanbridge & Ewins 1996)(see figures 12, 14), one or two electrodynamic shakers are attached to the rotatingshaft via a low friction (e.g. ball) bearing. This arrangement allows one to vary theamplitude, frequency and the spatial direction of the force at the bearings, which areapplied at a specific location on the shaft.

Advantages. Force amplitude and frequency can be controlled from a standard signalgenerator. Forward and backward rotating forces can be generated and a good signalto noise ratio (sine excitation) is obtained.Disadvantages. Requires a special attachment to the shaft. The applied forces canbe greatly affected by shaft motion (see ?4 for a discussion of signal-processing andexternal effects).Active magnetic bearings (AMBs)

These rather sophisticated, emerging, devices, which replace conventional supportbearings in some high-speed machines, can be used to apply and to measure forcesto/on the shaft while maintaining their primary role of supporting the shaft (Lee etal. 1995; BRITE 1996; Gahler & Mohler 1996).Advantages. A relatively small effort is required (mainly software) to add an abilityto generate a controlled sinusoidal (or other chosen) excitation to a system whichis already supported by AMBs. Typically, several forces (two per bearing) can beapplied simultaneously, giving rise to a truly multiple-input experimental system.Disadvantages. Requires a considerable amount of knowledge to operate. Can causedamage to the tested machine as these devices usually support the machine. Occa-sionally provides only the 'free-free' dynamics of the shaft as forces are measured atthe interface between the bearings and the shaft. The electromagnetic force is greatlyaffected by shaft dynamics, thereby giving rise to feedback in the system that mayresult in biased estimates of the FRFs (see ?4 d (i)).Phil. Trans. R. Soc. Lond. A (2001)

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I. Bucher and D. J. EwinsOrdinary excitation methods applied to the foundation

This approach seems natural as it may, in many cases, be the only possibleapproach due to accessibility limitations (Kramer 1993).Advantages. Straightforward to apply using standard modal testing techniques andequipment.Disadvantages. Many. Most of the shaft and disc dynamics cannot be easily excitedfrom the foundation and so much information may be missed. On the other hand.modes of the foundation that do not affect the rotating parts may dominate and thusthe modes related to the rotating part might be masked by much larger signals.

(b) Identification of modal parameters of rotating structures: curve fittingFitting a modal model to the measured FRFs completes the modal testing process.As rotating machines present many sources of energy, it is customary to assume thata frequency-domain approach is more suitable for testing rotating structures, asthis allows one to apply engineering judgement in separating the many processesthat coexist in a rotating machine. In the frequency domain, the modal expansionassumes the following form (Ewins 1984)

N A ArH(wj)- i A + A- + Eo+ 2 (3.1)--- \w - \r 2

where Ar are the residue matrices (see (1.7)) and E0, E2 are the matrices of residualterms, which compensate for out-of-range modes.The main difference in the model shown in (3.1) for the rotating and the non-rotating cases is the fact that Ar is not symmetric in the rotating case and itsnumerical rank could be higher than 1 (see Balmes 1994; Gahler & Mohler 1996).The higher rank results from the axisymmetric nature typical of rotating machinecomponents, giving rise to close (nearly repeated) natural frequencies. Thus each A,may represent more than a single mode (Balmes 1994).One of the methods that was used by the authors (BRITE 1996) is an iterativetwo-stage approach due to Balmes (1994), with special constraints imposing thestructure of (1.17) or (1.20) where appropriate. In step (i) of this method, the polesare estimated and, in step (ii), equation (3.1) is solved for the residue matrices Ar.

Steps (i) and (ii) are repeated, using a similar approach to the one presented inBalmes (1994), until convergence is obtained. It is worth restating that. unlike in thenon-rotating case, residue matrices Ar are not symmetric and thiis a suitable rankestimation scheme should be used (e.g. singular values decomposition (SVD)).A successful estimation of Ar is followed by a solution for the required modeshapes where (1.17) or (1.20) may be used for this purpose (Bucher & Ewins 1996).

(c) Signal processing aspects and the analysis of transient responseA topic that is often overlooked, due to the large number of details involved withthe dynamics of rotating structures, is the associated signal-processing procedure.A rotating machine, as mentioned before, has many sources of energy where eachsource and every structural element may exhibit vibrations at different frequen-cies. The rotation adds some modulation effects (Bucher et al. 1994) which further


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Modal analysis and testing of rotating structuresincrease the spectral complexity. Another reason for the existence of a large numberof spectral lines in the response measured on real machines is the fact that that thestructural supports and the connecting elements often have some degree of nonlinearbehaviour. Indeed, it is often possible to notice that numerous harmonics (multiplesof rotation speed) exist in measured response signals during rotation. Several workshave dealt with the proper selection of excitation signals. For example, Muszynska& Bently (1993) used harmonic excitation created by unbalance, while Lee et al.(1995) presented a special broadband signal which generated a forward or backwardrotating force. Gahler & Mohler (1996) and BRITE (1996) have used sinusoidal exci-tation and have demonstrated its advantages over broadband excitation. In general,an electro-mechanical excitation device may be strongly coupled to the mechanicalvibrating part (due to nonlinear feedback). In this case, a multi-harmonic excitation(BRITE 1996) will provide better signal-to-noise ratio than a simple sinusoidal ora broadband excitation. A short discussion (see Bucher (1998) for a complete dis-cussion) of coupling mechanisms between the excitation device and the machine ispresented below.(i) Effect of feedback on accuracy of the estimated FRF

It is well known that the forces exerted by electrodynamic and electromagneticexcitation devices can depend on the response as well as on the input currents. Thisphenomenon can be explained by Faraday's law,.1Rmi(t) + ddt

which relates the induced voltage v to the current i, ohmic resistance Rm and mag-netic flux ?. The change rate of the magnetic flux,do q) 9i 90aOx- :dt Oi Ot Ox t '

depends on the relative motion of the excited structure (Ox/Ot, where x symbol-izes a spatial coordinate) and the magnetic coils. The force which is expressed byf(t) = ac)2 (a constant) has a nonlinear dependence upon the dynamic response x(t).The term 90/0x, often addressed as the instantaneous back EMF coefficient, givesrise to a force which depends on the motion of the structure. and is often affectedby other sources than the induced current I (To & Ewins 1991; Bucher 1998).An excitation scheme where the force is also a function of the response (feedback)is illustrated in figure 15. Here, Fnm(w) s the applied force vector, X(() the responsevector and H(w) the FRF matrix of the structure under test. Also, I(w) is a vector ofinput signals, Fe(w) is a disturbance force vector (due to internal or internal sources)and Kf is a feedback matrix term which determines the amount of force dependencyupon the response.It has been shown that a standard estimator for the FRF which uses the cross-and auto-spectrum estimates of the actually applied force and the measured response(see, for example, To & Ewins 1991; Bendat 1996; Bucher 1998),HI (w) - S (3.2)Suu


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lI() --- Ki V H(w) +X

Fe I H,()

Figure 15. Block diagram of a response-dependent excitation scheme in the frequency domain.is biased and is related to the exact FRF H(w) by (Bucher & Ewins 1996)

He (w)SUF,HI(w) = H(w) + He (3.3)Suuwhere the error term

He(W)SUF _ KflHe(w)2 SFeFeSuu 1 - KfH* (w) SFmFm

depends upon the feedback term and upon the significance of the external distur-bances on the response.It is well known that the magnitude of the disturbances (due to unbalance andother sources) is often larger for rotating structures than is the directly appliedforce. Therefore, it can be concluded from (3.3) that the presence of feedback maycause a significant deviation of the estimated FRF from the true value. When sineexcitation is being used, unlike random excitation, frequencies different from thatof the excitation can be ruled out and therefore, in such cases, stepped-sine is thepreferred method.(ii) Sine excitation for rotating structures

Although stepped-sine excitation techniques have been used for decades, someparticular features are worth mentioning in the case of rotating structures. Amongthese are (i) the consideration of slightly nonlinear behaviour, (ii) the additionalinput due to unbalance and (iii) the assumptions behind the applied least-squarestechnique by which the FRFs are extracted from the measurements (Bjorck 1996).The sine-testing procedure extracts the FRFs at a number of discrete frequenciesby stepping through the desired frequency range. At each frequency of interest, theamplitudes and phases of the 2q forces and 2p responses are measured. The mea-surements of the excitation and the response are curve fitted in the time domain toform vectors of sine and cosine coefficients,f = (fT fT )T

= (f,cos Jsin)= (fl,cos f2,cos ... fj,sin ... f2p,cos) E R X , T (3.4)T=(T,T,,2pxlrl,cos cos sin 2pcos)(ri- 0 r'0 . r,~ C R12px'l,


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Modal analysis and testing of rotating structureswhere, for example, the force and response at the jth DOF would be

fj = fj,cos cos wt + fj,sin sin wt, rj =rjj, cosO +t+ rj, sinint. (3.5)It can be shown, by using standard linear systems theory, that f and r are relatedby means of the sought frequency-response matrix, i.e.

rcos - irsin = H(U)(fcos -ifsin). (3.6)It is clear that H(w) C WRXqan be extracted by a combination of q independent setsof excitation vectors, f (and therefore requires q vector measurements per frequency).This approach has been universally used and, indeed, using the unbalance pertur-bation approach, Muszynska (1994) and Muszynska & Bently (1993) have used fourforces (q = 2) by rotating the unbalance in co- and counter-rotating directions inseeking a linear model of the rotating structure.At this point it is worth mentioning that a linearized (rather than linear) model isusually sought, while behind the direct usage of (3.6) is the assumption that the realand imaginary parts of the FRF are related (see Ruh 1981; Pirard 1989). Indeed, thesteady-state response of a linear system to a sinusoidal excitation,

f (t) = fcos cOswt + fsin sin t,gives rise to r(t) r=rCScos Wt+ rsinsint,which can be shown to yield

Trcos_ HR () HI(w) (fcosrsin --HI(w) HR(w)_ \fsin ()

where H(w) = HR(w) + iHi(w).But the special structure of a skew-symmetric matrix such as that shown in (3.7)cannot be guaranteed to exist, even for slightly nonlinear systems. Consequently, itis necessary to estimate four independent parameters instead of two (the real andimaginary parts of the FRF).In order to be able to accommodate some nonlinear behaviour, the higher-orderFRF approach (Ruh 1981) can be adopted. Assuming a polynomial model (the termis defined in Ruh 1981), one can include several harmonics in both the excitationand the response, i.e.Q Q

f(t) = f fcos c nwt+ fs sin nt, r(t) cos t + r sin nwt,n=l n=l(3.8)

where the amplitudes of the sine and cosine coefficients in both the force and theresponse are related (as a consequence of a series expansion (Bucher 1998)) by amatrix representing the higher-order FRFs at a single frequency (Storer & Tomlinson1993),/' 1 -\i~A1^1 A1,1 * A1,^ ~\Nnl\rsi ANnl ,1^1-l 1 A4/

- scon, cos , sin cos,sin -__'sin,sin sin,sin (39)

N~nfNnIsin,1 A 1 ...N l,Nn inl_sin,cos sin,sinPhil. Trans. R. Soc. Lond. A (2001)

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+f(Hz) A

-f(Hz) I

forward

backspeed (rpm)

Figure 16. Double-sided Campbell diagraml separating the forward from the backward whirl.

Here, the ith harmonic of the response is related to the jth harmonic of the excitationby A';j 5r'/foyAc's,Slll = OFcos/OfsillEquation (3.9) presents a true linearized FRF that provides higher accuracv inexchange for a more elaborate experimental procedure.(iii) Solving for the FRF

As one wishes to obtain the coefficients from which the FRF can be estimated.(As sin), one needs to apply 2qQ independent sets of forces (see (3.8)). where q isthe number of excited degrees of freedom and Q is the number of harmonics to beconsidered. Performing the required number of measurements, one can augment theestimated force and response coefficients to form

R= AF. (3.10)where

- ( r1 Ncos1sinR =

r51

fNnl\ cos / i

( r j NCossinl 1

sin1 22qQ..

A direct solution of (3.10) for A may yield a biased estimate as both R and F areformed from measured quantities and thus contain some noise. One should assessthe amount of noise in the response and force measurements and form a solutionmethod that takes this into account. A possible strategy could make use of the totalleast-squares approach (see Bjorck 1996; Golub & Van Loan 1996), which assumesthat there is some uncertainty in both the right- and left-hand sides of (3.10).Phil. Trans. R. Soc. Lond. A (2001)

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Modal analysis and testing of rotating structuressensors..

backward)rward

Figiire 17. Vil)ratioll patteiril dolll)le-si(lc(l C(Iliplcll (liagraiii.

11= 1.~ ~ ~ - . ?I

+ tforwalrd .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~." . r~.~ C C5 s,'

_ _ "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

. . ' '- '..

. .

11=2 _. 11=3r(Z'VLj? .. -? ..?dr

t 1 -,? C:rr'` 'I??--,?I ?h?'' *e' r,r L-C-. ?? r*-rj ir ?ictclrr

speed QFigure 18. Three double-sided spectrogralms for three different wavelelgtlls

(the considered numbers of nodal dialneters are n = 1, 2, 3).Note that by making use of the pseudo-inverse of F (which is often implementedas A = RFT(FFT)-1) we inherently assume that all the uncertainty is confined tothe response measurements r, and hence R.It can therefore be concluded that the method presented in Muzynska (1994), forexample, is illlherenltly imited and cannot truly estimate a linearized FRF, since onlytwo possible excitation sets (instead of the required four) can be generated with arotating-unbalance excitation.

(d) Estimation of operation deflection shapes andnatural frequencies from transientsAny machine is occasionally exposed to run-up and run-down conditions where thespeed of rotation is varied until the normal operating conditions are reached. DuringPhil. Trans. R. Soc. Lond. A (2001)

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I

fc

r, ,.,,,-.i


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I. Bucher and D. J. Ewins60 - (a) force singular, values0- 0-^


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Modal analysis and testing of rotating structureshas the advantage that it allows one to observe the effects of rotation upon naturalfrequencies and to visualize the extent by to each mode participates in the dynamicresponse.An extended approach has been developed for discs for which a diagram that hasseveral dimensions can be constructed. The additional dimensions separate the discmodes according to the different numbers of nodal diameters (wavelengths). Fig-ure 12 shows an arrangement in which a number of sensors are distributed as shownin figure 17. This arrangement allows one to process the signals from all sensor posi-tions simultaneously and to separate the response into components associated withindividual numbers of nodal diameters. As illustrated in figure 17, an array of sensorsat a fixed radius will measure the variation of the disc deflection so that applying theFourier transform in both time and space (Bucher & Ewins 1997; Schmiechen 1997)yields separations as depicted in figure 18. These measurements were performed onthe rig photographed in figure 12.The presented signal-processing method allows one to display the spectral contentsof the response measured from a rotating machine in considerable detail. This methodprovides a qualitative estimate of the natural frequencies and allows one to isolatethe contribution of a particular mode shape from the total response. The methodis suitable for any type of rotating structure and is not restricted to time-invariantmodels. Through examination of these diagrams, the existence of periodically varyingcoefficients can be detected.

4. Illustrative examples and case studiesSeveral examples for the application of modal testing to rotating structures are pro-vided in this section. Special features of rotating machines are emphasized and rotat-ing versus non-rotating conditions are compared.

(a) Application of active magnetic bearingsThe test rig which is shown in figure 20 and described in BRITE (1996) andGahler & Mohler (1996) was used to measure the FRFs by applying a stepped-sineexcitation approach (Bucher & Ewins 1996). This test rig contained two magneticbearings, each capable of applying a force in two perpendicular directions, giving riseto a total of four excited DOFs. Since the shaft itself was very lightly damped, themethod which was described in ?? 1 d and 1 c could be applied in this case.A typical measured FRF is depicted in figure 19, with some additional informationas explained below.Figure 19 shows a typical FRF measured at speeds of 0 and 1000 rpm. These mea-surements illustrate the wide dynamic range which was obtained with the stepped-sine excitation method. The anticipated difficulties which were mentioned in ?3 c (i)can be observed in this case by inspection of the singular values of the force matrix(F in (3.10)) which is shown in figure 19a for each measured frequency. In thisplot, the number of independent force vectors drops near each natural frequency andtherefore (3.10) becomes ill-conditioned. This singularity is more dominant for thenon-rotating case (figure 19a) and can explain the deterioration in the accuracy ofthe FRF near resonance. The singular values that were computed for the responsematrix R show that a rank of 2 is preserved for the whole frequency range. This


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Figulre 20. Experimental rig showing the mlotor and the miagnetic bearings, witli tc Illmeasure(andIlpredicted forward whirling mode shape ovcrlaid. at 3000 rpim.

(a) (b)-60 -60

-100 -100-dB \ dB

-140'

- ^--140-140-180

-220r0 100 200 300 0 100 200 300frequencyHz) frequencyHz)

Figure 21. Singular values of the 4 x 4 FRF matrix at (a) 0 and (b) 2400 rpm.fact strengthens the assumption that at least two measurements need to be takenfor complete determination of the response (and hence the FRF) at each node or asecond measurement.

Indeed, this system provided sufficiently accurate data to estimate the whirlingmode shapes, one of which is depicted in figure 20.Although this method generally proved successful, one must take into accountthat the reported experiments were conducted under laboratory conditions, where aconsiderable effort was needed to obtain the reported results.

(b) Validation of the theoryA closer inspection of the results shows that the potential problems associatedwith feedback (see ?3 c (i)) do arise, as can be deduced from the singular values plotin figure 19a. The selection of the four perturbation vectors needed at each measured


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dB

-100 frequency response

300 310 320 330 340 350frequency (Hz)

Figlll 22. Sillglar values of the 4 x 4 FRF matrix at 0 rplll an(d a typical FR'.frequencly was therefore adjusted il order to minilnize this phenomenon. An FRFlllatrix was lneasire(l b)yprocessing four excitatiol and four response measurements,all taken at the mIagletic bearings in the x- and y-directions. An SVD algorithmdecomllosed this FRF matrix (at each frequiency) in or(derto reveal its rank, whichis a result of the number of participating lmodes at every fieqiuency. The results arelel)ictedl ill figure 21.

Figure 21 sllows the four singular values of the 4 x 4 FRF imatrix as a fiuctionof frequelncy. This plot clearly iml(licatesthat tlle rank of the FRF mlatrix does inot(1rop)below 2 in tle Illeasure(l frequency range for thle lnon-rotating case. Il tllerotatillg case, on tlle othler lhalld, one llod(le prevails, especially ill thle vicinity of tllenatural flrequencies. It call tllls be concliuded tllat a sinlgle shaker would not havebeenl sufficient for extracting the FRF lmatrix correctly in the rotating case, wliiletlis is llot true for the nion-rotating one.A closer look (at figure 21) arolnd the naturlal fi'equencies reveals a few mioredetails. and these are described below.The irregularity in tlle curves in figure 22 is assumiled o be due to the large feedbackeffect arollll(l resollance. This feedback (which exists in all clectrodylalmiic shakersclose to resoinamce)gives rise to a singularity and therefore to a numllerically ensitivecomplll)ltatiollprocess (see ?3 c). Ill this case, thle loisy estimate was inldeed close toresollnanceall(1 onle lmay suspect that a Ineasurelment systemllof lesser quality thantlce one sllown ill figure 20 mlliglithave lead to severely deteriorated results.

(i) Fitting equation (1.20) to the measurementsTlle mleasured FRF llmatrixwas used ill order to fit a miiodel aiid extract tlleresi(llues. A typical result, wllicll was llmeaslred at a speed of 1800 rpmI was fittedI'il. Tran7s.R. So:. Lond. A (2001)

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I. Bucher and D. J. Ewinsaccording to (A 11), and is as follows:

(-61.2 -50 5.3 5.3 \e brT) = 103 -53.8 -43.9 5.3 5.2Rp~c(/~ie /JrT)= 10_3 -5 3.87.6 -6.1 -64.5 -51.4

-5.6 -4.6 -56.8 -45.3/ ~,> (4.1)/ 1.5 1.2 21.3 16.9\i2RerT) = io-5 1.5 1.1 18 142 Re( r?0Tr) _ 10 1.-19.5 -15.9 2.9 2.6

\-17.1 -13.9 2.2 2.0/Evidently, the structure of the matrices in equation (4.1) agrees closely with (1.17)and (1.21). The accuracy, which is no better than 10%, due to the above-mentionedsingularity, could have been improved significantly if the assumptions in (1.21) had

been used to reduce the number of parameters sought. Still, it is evident that theassumed model provides a good approximation for this type of system, and thismakes the proposed experimental identification procedure feasible.

5. Conclusions and outlookThis paper has surveyed the field of modal testing of rotating machinery structures.Some models and methods were discussed and some practical, as well as theoretical,recommendations and pitfalls outlined. Although much progress has been made inthe instrumentation and measurement techniques for modal testing, basic limitationsinhibit the experimenter from obtaining a full model for a rotating structure (e.g. thelack of ability to excite rotating machines at a sufficient number locations). Practi-cal machines, which contain transmissions and anisotropic elements, possess time-varying coefficients in their equation of motion and thus give rise to behaviour whichcannot be treated rigorously using existing experimental modal analysis methods.However, a considerable amount of information can be extracted by using a suitablesignal-processing approach. It has been shown that the application of experimentalmodal testing to rotating structures is a viable option as long as the appropriatemodels are being used. More progress in the theory and in the experimental proce-dures should be achieved before experimental modal analysis (EMA) can be routinelyapplied to any type of rotating machine.

Appendix A. Transformation into state-space formUsing

z (t) (z (t)one can convert (1.2) (omitting the explicit dependency upon speed of rotation Q?)into

Az(t) + Bz(t) =(f() (Al)


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Modal analysis and testing of rotating structureswhere, for convenience, we define

M 0 C KA K B=- K (A2)- 0 K K (A 2)Equation (A1) transforms (1.1) into a first-order vector equation. This particularselection of matrices (equation (A2)) gives rise to a symmetric A and a skew-symmetric B, which proves numerically convenient for the undamped or lightlydamped case (Meirovitch & Ryland 1979).

(a) Right-eigenvectorsExpressing the free response of (A 1) as

z(t) =- peAP (Ap'Ip eApt (A 3)and substituting in (A 1), we obtain an equation (Geradin & Rixen 1994; Meirovitch& Ryland 1979; Lee 1993) from which we are able to compute 2N pairs of eigenvaluesand eigenvectors, Ap, )p,

ApACp+ Bp -0, (A 4)or, in matrix form,

AiA + B = O, (A 5)where t- = [1 ?2 ... 2N], A = diag(Al, A2,..., 2N)-

(b) Left-eigenvectorsThe left-eigenvector matrix,

< = [01 02 ... 02N],diagonalizes (A 5) and each matrix individually (Geradin & Rixen 1994),

OTA diag(a a a2)T=diag(a , ), diag(b1, b2,..., b2N) (A6)

leading to 2N uncoupled modal equations, Apap+b = 0, p = 1,...,2N.The columns of P can be scaled to obtain (TAI = I. The relationship betweenthe left- and the right-eigenvectors can therefore be obtained directly (Lancaster1977) fromAT= . (AA 7)

It can be shown that the steady-state response to a sinusoidal excitation f(t)foeiwt can be expressed asK(ut))V=1it r or (A8)/"(t) = H() foe- i - (A 8)\?ct,l ~~~~~r-1 w,


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I. Bucher and D. J. EwinsGiven (A 8). we are able to express the FRF matrix H(Cw)explicitly.

x(t) = H(w)'feiwt ( + foT ) eflw (A 9)1o - \r UiJ- A\lNote that

H(&) = X-l(i~:) = (-:2M(9) + iwC(Q) + K(Q))-1. (A 10)This FRF can be reformulated using a second-order denominator (for which we haveAr, =-(,rg ? i\/1 - C2 ). as is customary in vibration analysis (Bucher &( Ewils1996).

w Re( 'i,1. -iw Re(t,.) (A. )HK) ^ 2 - &2 + 2i(WLL(1

Appendix B. Eigenvalues of undamped gyroscopic systemsLet tr, = a + ib. substitute it into (1.7) and multiply by {4/}T to the left. to obtain

{Il}T(A2M + XAG + K){'} = (a -ib)T (XM + XAG + K)(a + ib). (B )This results in a scalar equation.

A2'n, + 2iASg + k = 0. (B 2)where

in =- a Ma + bMb. k =- a Ka + b Kb, ( = bTGa.Here, the symlmetry of AM and K, as well as the skew-symmletry of G. have beenused.

Solving (B 2) for A yields a purely imaginary solution,A = i(-Qgq ? /229g2 + mk)/ m,. (B 3)

Since 7n > 0. k > 0, g is real and A is pllrely imaginary.It is worth mentioning that A is a fuinction of two parameters 2, = A/, alllt/ = g/in. with which A takes the form

A = i(-1 -V/( /7)2 + 2).and this clearly demonstrates the dependency of the natural freqiuencies liponl thespeed of rotation and illustrates the 'gyroscopic split of natural frequencies'.

Appendix C. Frequency response functions ofan undamped isotropic rotorThe following matrices can describe the dynamlic behaviour of all isotropic rotor:

A[M I Ko I(22 (x 11[nMl [G] -G jo [K] 0R A - o (C: )...... Go 0 IPhil. Trans. R. Soc. Lond. A (200)1)

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Modal analysis and testing of rotating structuresThe FRF matrix, which is defined by (A 10), can be partitioned into x- and y-relatedterms as

H Hxx HxyH yx H yy_It call be observed that the various sllb-matrices have the following form:

zxxZyx

(C 2)

(C 3)(C4)

HXX- (Z,x Z,~Zuxy Zyx).Zxy _ Ko - w AIo i0GoZyy --iJGo Ko - 2AI()

Using the previous identities, we can show thatH,X = (Ko - 21o - 2Go(Ko - 2Alo)-'Go)-

is a purely real matrix.Denotillg ZO0= = K - o w2A10o, e obtainH = ( Z0 ?- 0 Go (Z O)0-1Go 1^xx - "{ O ~ -1O G -0)

(C 5)

(C 6)

H - i_ ,Z70=0G(Zo7=0 2Goo:7Q=O) - -1and also

Hx - ,i0 tZo Go(Zo =0 2Go (Zo=O)-1 Go-1x - -l/o jQ(o - Uoo /Oare both purely imaginary.

Appendix D. Nomenclature

(C 7)

mass, viscous damping plus gyroscopic and stiffness matricesgyroscopic matrixvector of coordinates (DOF) describing the response(displacement)vector of external forcesresponse amplitude vectors multiplying to cosine and sinetime functionsforce amplitude vectors imultiplying to cosine and sinetime functionsresponse and force (respectively) amplitude matrices fora single frecquencyspeed of rotation (rpmlor rad s-1)right-eigenvectorcomplex conjugate of Ceigenvalue (A, the rth eigenvalue)dynamic stiffiessleft-eigenvectorscaling coefficients related to the rth modecosine and sine related (respectively) eigenfunctions(in polar coordinates)

and

M(t). C(t), K(t)Gq(t)f(t)/cos sinlfcos fsiniR, F

,(1,A,(A)/30,3-, a,r(,,C.o o) ,,'l (r. 0)71-~ , r n, s

Phil. Traris. R. Soc. Lond. A (2001)

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I. Bucher and D. J. EwinsMo, K0, Go mass, stiffness and gyroscopic matrices defined per direction(plane), e.g. x or yNSA non-self-adjointFRF frequency-response functionSVD singular values decomposition

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