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Mechanical Systems

and

Signal ProcessingMechanical Systems and Signal Processing 20 (2006) 15901614

Simultaneous identification of residual unbalances and

bearing dynamic parameters from impulse responses

of rotorbearing systems

R. Tiwari, V. Chakravarthy

Department of Mechanical Engineering, Indian Institute of Technology Guwahati, 781 039, India

Received 12 June 2004; received in revised form 5 January 2006; accepted 11 January 2006

Available online 2 March 2006

Abstract

An identification algorithm for simultaneous estimation of residual unbalances and bearing dynamic parameters by

using impulse response measurements is presented for multi-degree-of-freedom (mdofs) flexible rotorbearing systems. The

algorithm identifies speed-dependent bearing dynamic parameters for each bearing and residual unbalances at predefined

balancing planes. Bearing dynamic parameters consist of four stiffness and four damping coefficients and residual

unbalances contain the magnitude and phase information. Timoshenko beam with gyroscopic effects are included in the

system finite element modelling. To overcome the practical difficulty of number of responses that can be measured, the

standard condensation is used to reduce the number of degrees of freedom (dofs) of the model. For illustration, responses

in time domain are simulated due to impulse forces in the presence of residual unbalances from a rotorbearing model and

transformed to frequency domain. The identification algorithm uses these responses to estimate bearing dynamic

parameters along with residual unbalances. The proposed algorithm has the flexibility to incorporate any type and any

number of bearings including seals. The identification algorithm has been tested with the measurement noise in the

simulated response. Identified parameters match quite well with assumed parameters used for the simulation of responses.

The response reproduction capability of identified parameters has been found to be excellent.

r 2006 Elsevier Ltd. All rights reserved.

Keywords:Rotorbearing systems; Bearing dynamic parameters; Identification

1. Introduction

High-speed rotating machineries, such as steam and gas turbines, compressors, blowers and fans, find wide

applications in engineering systems. The danger of residual unbalances in such machineries attracted attention

of researchers during quite early days[13]. From the state of the art, methods of balancing can be categorised

into two groups; the influence coefficient method, which only requires the assumption of linearity of both the

machine and measuring system, and modal balancing which in addition, requires knowledge of the modal

properties of the machine. Influence coefficient method requires less a priori knowledge of the system and

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0888-3270/$- see front matterr 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ymssp.2006.01.005

Corresponding author. Tel.: +91 361 2582667; fax: +91 361 2690762.

E-mail address: [email protected] (R. Tiwari).
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techniques have been well developed to make optimum use of redundant information[4]. The approach has a

significant disadvantage of requiring a number of test runs on site. Modal approaches require fewer test runs,

Gnielka[5] used prior knowledge of the mode shapes and modal masses and compared results to those from a

numerical model of the machine. The work of Krodkiewski et al. [6] has similar requirements and seeks to

detect changes in unbalance from running data. Both these approaches place reliance on the numerical model.

Numerical models of rotating machinery have been used to great effect over a number of years[7], and theiraccuracy and range of effectiveness have been steadily developing. Traditional turbo generators balancing

techniques require at least two run-downs, with and without the use of trial weights, respectively, to enable the

machines state of unbalance to be accurately calculated [8]. Lees and Friswell [9] presented a method to

evaluate state of unbalance of rotating machine utilising the measured pedestal vibration. Subsequently,

Edwards et al.[10]presented the experimental verification of the method [9] to evaluate the state of unbalance

of a rotating machine. From the state of the art of the unbalance estimation procedure, the unbalance could be

obtained with fairly good accuracy. Now the trend in the unbalance estimation is to reduce the number of test

runs required especially for the application of large turbogenerators where the downtime is very expensive.

Rotating machineries are supported by bearings, which play a vital role in determining the behaviour of the

rotating system under the action of dynamic loads. One of the most important factors governing the vibration

characteristics of rotating machinery is bearing dynamic parameters. The influence of bearing dynamic

characteristics on the performance of the rotorbearing system was also recognised for a long time. One of theearliest attempts to model a journal bearing was reported by Stodola[11]and Hummel[12]. They represented

the fluid film of bearings as a simple spring support, but their model was incapable of accounting for the

observed finite amplitude of oscillation of a shaft operating at a critical speed. Concurrently, Newkirk[13]and

Newkirk and Taylor[14]described the phenomenon of bearing induced instability, which they called oil whip,

and it soon occurred to several investigators that the problem of rotor stability could be related to the

properties of the bearing dynamic coefficients. Although the importance of rotor support dynamic stiffness is

generally well recognised by the design engineer, it is often the case that theoretical models available for

predicting it are insufficiently accurate, or are accurate only in very specific cases. Moreover, the stiffness and

damping characteristics are greatly dependent on many physical and mechanical parameters such as the

lubricant temperature, the bearing clearance and load, the journal speed and the machine misalignment in the

system and these are difficult to obtain accurately in actual test conditions. The uncertainties about machineparameters can make inaccurate results, obtained with the best theoretical methods aimed to study the

behaviour of fluid-film journal bearings. Owing to this, it can be very useful to determine the bearing dynamic

stiffness by means of identification methods based on experimental data and machine models. It is for this

reason that designers of high-speed rotating machinery mostly rely on experimentally estimated values of

bearing stiffness and damping coefficients in their calculations.

Several time domain and frequency domain techniques have been developed for experimental estimation of

bearing dynamic coefficients. Many works have dealt with identification of bearing dynamic coefficients and

rotorbearing system parameters using the impulse, step change in force, random, and synchronous and non-

synchronous unbalance excitation techniques. Ramsden [15] was the first to review the papers on the

experimentally obtained journal bearing dynamic characteristics. In mid-seventies, Dowson and Taylor [16]

conducted a survey in the field of bearing influence on rotor dynamics. They stressed the need for experimental

work in the field of rotor dynamics to study the influence of bearings and supports upon the rotor response, in

particular, for full-scale rotor systems. Lund [17,18] gave a review on the theoretical and experimental

methods for the determination of the fluid-film bearing dynamic coefficients. For experimental determination

of the coefficients, he suggested the necessity of accounting for the impedance of the rotor. Stone[19]gave the

state of the art in the measurement of the stiffness and damping of rolling element bearings. He concluded that

the most important parameters influencing the bearing coefficients were type of bearing, axial preload,

clearance/interference, speed, lubricant and tilt (clamping) of the rotor. Kraus et al. [20] compared different

methods (both the theoretical as well as the experimental) to obtain axial and radial stiffness of rolling element

bearings and showed a considerable amount of variation by using different methods. Someya [21]compiled

extensively both analytical as well as experimental results (the static and dynamic parameters) for various

fluid-film bearing geometries (e.g. 2-axial groove, 2-lobe, 4 and 5-pad tilting pad). Goodwin [22]reviewed the

experimental approaches to rotor support impedance measurement. He concluded that measurements made

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by multi-frequency test signals provide more reliable data. Swanson and Kirk [23] presented a survey

in a tabular form of the experimental data available in the open literature for fixed geometry

hydrodynamic journal bearings. Recently, Tiwari et al. [24,25]gave a review of the identification procedures

applied to the bearing and seal dynamic parameters estimation. The main emphasis was given to summarise

various bearing and seal models, the existing experimental techniques for acquiring measurement data from

the rotorbearing-seal test rigs, theoretical procedures to extract the relevant bearing and seal dynamicparameters and to estimate associated parameters uncertainties. They concluded that the synchronous

unbalance response, which can easily be obtained from the run-down/up of large turbomachineries, should be

exploited more for the identification of bearing dynamic parameters along with the estimation of residual

unbalance.

Until the early 1970s, the usual method to obtain the dynamic characteristics of systems was to use

sinusoidal excitation. In 1971 Downham and Woods[26]proposed a technique using a pendulum hammer to

apply an impulsive force to a machine structure. Although they were interested in vibration monitoring rather

than the determination of bearing coefficients, their work is of interest because impulse testing was thought to

be capable of exciting all the modes of a linear system. Due to the wide application of the fast Fourier

transform (FFT) algorithm and the introduction of the hardware and software signal processor, the testing of

dynamic characteristics by means of transient excitation is now common. Morton [27,28] developed an

estimation procedure for transient excitation by applying step-function forcing to the rotor. With the help of acalibrated link of known breaking load, the sudden removal of the load on the rotor in the form of a step

function (broadband excitation in the frequency domain) was used to excite the system. The Fourier transform

was used to calculate the FRFs in the frequency domain. He assumed the bearing dynamic parameters to be

independent of the frequency of excitation. The analytical FRFs, which depend on the bearing dynamic

coefficients, were fitted to the measured FRFs. He also included the influence of shaft deformation and shaft

internal damping into the estimation of dynamic coefficients of bearings. Chang and Zheng[29]used a similar

step-function transient excitation approach to identify the bearing coefficients and they used an exponential

window to reduce the truncation error in the FFT due to a finite length forcing step function. Zhang et al. [30]

used the impact method with a different fitting procedure to reduce the computation time and the uncertainty

due to phase measurement. They quantified the influence of measurement noise, the phase-measuring error

and the instrumentation reading drift on the estimation of bearing dynamic coefficients. Marsh and Yantek[31]devised an experimental set-up to identify the bearing stiffness by applying known excitation forces (e.g.

measured impact hammer blows) and measuring the resulting responses by accelerometers. They estimated the

bearing stiffness of rolling element bearings (consisting of four recirculating ball bearing elements) of a

precision machine tool using the FRFs. The tests were conducted on a specially designed test fixture (for the

non-rotating bearing case). They stressed experimental issues such as the precise location of the input and

output measurements, sensor calibration, and the number of measurements. Among the experimental

methods, the impact excitation method proposed by Nordmann and Scholhorn [32]to identify stiffness and

damping coefficients of journal bearings, is the most economical and convenient. Impulse force has an

advantage, that is, it contains many excitation frequencies simultaneously and a single impact force can excite

several modes. In this work analytical frequency response functions (FRFs), which depend on bearing

dynamic coefficients are fitted to measured responses. Stiffness and damping coefficients are the results of an

iterative fitting process. Burrows and Sahinkaya [33] showed that the frequency domain bearing dynamic

parameters identification techniques are less susceptible to noise. Zhang et al. [34]and Chan and White [35]

used the impact method to identify bearing dynamic coefficients of two symmetric bearings by curve fitting

frequency responses. Arumugam et al. [36] extended the method of structural joint parameter identification

method proposed by Wang and Liou [37] to identify the eight-linearised oil-film coefficients of tilting pad

cylindrical journal bearings utilising the experimental FRFs and theoretical FRFs obtained by finite element

modelling. Qiu and Tieu[38]used the impact excitation method to estimate bearing dynamic coefficients of a

rigid rotor system from impulse responses.

Advances in the sensor technology and increase in the computing power in terms of the amount of data

could be collected/handled and the speed at which it can be processed leads to the development of methods

that could be able to estimate residual unbalance along with bearing/support dynamic parameters

simultaneously [3943]. These methods could be able to estimate residual unbalances quite accurately but

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estimation of bearing dynamic parameters often suffers from scattering due to the ill-conditioning of the

regression matrix of the estimation equation [10,4243].

In the present paper, an algorithm for simultaneous identification of residual unbalances and bearing

dynamic parameters, by using impulse response measurements, is presented for multi-degree-of-freedom

(mdofs) flexible rotorbearing systems. Speed-dependent bearing dynamic parameters, consisting of four

stiffness and four damping coefficients for each bearing along with residual unbalances (magnitude and phase)at predefined rotor axial locations (i.e. balancing planes) are identified. The finite element method is used for

the rotor modelling through Timoshenko beam theory with gyroscopic effects. Some of the system degrees of

freedom (dofs) are eliminated by the condensation to reduce the number of measurement required for

estimation of the parameters. The impulse force is simulated in the time domain using a bell shape function

and transformed to the frequency domain using the FFT. For numerical examples, bearing responses in the

time domain are simulated for a rotorbearing model for the impulse and residual unbalance forces and

transformed in the frequency domain by the FFT. The bearing has been modelled as the short bearing for the

illustration of the method. The identification algorithm is tested with measurement noise in the simulated

response. The estimated bearing dynamic and residual unbalance parameters are found to be quite close to the

parameters assumed for the simulation of responses. The response regeneration capabilities are quite good

from identified parameters.

2. Modelling of rotorbearing systems

A general rotorbearing system can be viewed as combination of substructures namely rotor, bearings and

foundation as shown inFig. 1. For the present case the foundation is considered to be rigid. A mdofs flexible

rotorbearing system can be represented as shown inFig. 2. The model is composed of a flexible shaft, rigid

discs, and flexible bearings. The mathematical model of the shaft, discs, bearings and the impulse force are

presented in this section, from which system equations of motion are obtained in the frequency domain. The

present analysis is based on the assumption of system behaviour as linear. The shaft damping has been ignored

in the present paper.

2.1. Shaft model

The shaft is divided into finite number of elements and each element can be represented as shown inFig. 3.

The appropriate number of elements is determined depending on the order of vibration modes expected to be

known and geometry of the shaft and mounting of discs. It is assumed that for the shaft the shape of the cross-

section, dimension, and material constants are uniform in each element. The shaft is modelled by using the

Timoshenko beam element. The finite element formulation is done in real frame of reference and each element

has two nodes and at each node two translational and two rotational dofsare considered. For a shaft element

as shown inFig. 3, equations of motion is given as

Mef ugneOsGe _uf gne Kefugne ffgne, (1)

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Fig. 1. A rotorbearing-foundation system.


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wherefugne andffgne are called the elemental nodal displacement and force vectors, respectively, Os is therotor angular speed (or the rotor spin frequency) and matricesMe,Ge andKe are the elemental mass,gyroscopic and stiffness matrices, respectively (a detailed list of nomenclature is given in Appendix A) and are

expressed as

Me Mt0FMt1 F2Mt2 Mr0 FMr1 F2Mr2, (2)

Ge G0FG1F2G2, (3)

Ke K 0 FK1. (4)

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Fig. 2. A schematic diagram of flexible rotorbearings system.

l

0

z

y

xz

dz

w

v

1

1

2

2

v1

v2

w1

w2

Fig. 3. A schematic diagram of a typical shaft element.


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Details of the elemental mass, gyroscopic and stiffness matrices and the elemental nodal displacement and

force vectors are given in Appendix B.

2.2. Rigid disc model

Discs are assumed to be rigid and are modelled using the mass and mass moment of inertia terms at therespective node. The rigid disc equations of motion can be expressed as

Mdf udg OsGdf _udg ffdg, (5)where vectorsfudg andffdg are the disc displacement and force vectors, respectively, and matricesMd andGdare the disc mass and gyroscopic matrices, respectively. Details of the disc mass and gyroscopic matricesare given in Appendix B.

2.3. Bearing model

The classical linearised bearing model, with the eight spring and damping coefficients, is employed for themodelling of bearings. Bearing force at each bearing is assumed of the following form:

cxx cxy

cyx cyy

" #f _uBg

kxx kxy

kyx kyy

" #fuBg ffBg, (6)

where vectorsfuBgandffBgare the bearing displacement and force vectors, respectively. Details of the short-bearing approximation solution in the closed form for the damping and stiffness coefficients are given in

Appendix C, which is used for numerical simulations.

2.4. Impact force model

The impact force, which is to be applied on the rotor of the rotorbearing system, is simulated in the time

domain. In the present work, a bell-shape function is used to simulate the impulse force and is expressed as[38]

fimpt Fimp exp att02, (7)wheret0 is the instant at which maximum impact is applied, Fimp is the maximum impact at that instant, t is

the time instant, anda is a constant and in the present work it is taken equal to 2 2 ln 10=t20. Impact force canalso be simulated using other mathematical functions such as a half-sine wave. Bell-shape function is chosen in

particular, as it approximates the experimental impact force very well. The impact force is applied

alternatively in the horizontal and vertical directions.

2.5. Residual unbalance force model

Let the residual unbalance force vector be defined as

ffunbtg fFunbgejOs t, (8)whereOsis the spin frequency of the rotor,fFunbgis the residual unbalance force vector (elements of which arecomplex quantities and contain the amplitude and phase information) andj

ffiffiffiffiffiffiffi1

p . Often, the vibration of a

real machine is caused by further important excitations like misalignments (of bearings and couplings) and

shaft thermal bows. These excitations, which cause 1 rev. vibrations, have not been considered in the presentcase. The effect of misalignment and shaft thermal bows will reflect in estimates of the residual unbalance in

the form of an equivalent residual unbalance. However, estimates of the bearing dynamic parameters are

expected to be better, which is the main focus of the present paper.

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2.6. Equations of motion of the rotor substructure

Equations of motion of the rotor substructure, which includes the flexible shaft and rigid discs, can be

obtained by assembling the contribution of each such elemental equations of motion and are expressed, in

general, as

Mf ug OsGf _ug Kfug ffg, (9)where {u} and {f} are the rotor displacement and force vectors, respectively, and [M], [G] and [K] are the rotor

mass, gyroscopic and stiffness matrices, respectively. Static condensation is used to reduce certain dofsin finite

element equation (9), which serves to overcome the limitation of number of measurements that can be made in

practical rotors. The method essentially consists of elimination of certain dofs. The dofs eliminated in this

process are called slaves and those retained for the analysis are called masters. Generally, retained dofs

(masters) would coincide with lumped discs, bearing locations, unbalance locations along with other external

force locations. Discarded dofs (slaves) would correspond to dofs in the model, which are non-critical or

difficult to measure accurately (e.g. rotational dofs). Eq. (9) is split as subvectors and matrices relating to

master dofs and slave dofs and can be represented as

Mmm MmsMsm Mss

" # umus

( )Os Gmm GmsGsm Gss

" # _um_us

( ) Kmm KmsKsm Kss

" # um

us

( ) fmfs

( ). (10)

Subscripts m and s refer to the master and slave dofs, respectively. On assuming that no external force is

applied to slave dofs, the static transformation is given by [44]

um

us

( )

I

K1ss Ksm

" #fumg Tsfumg, (11)

where [Ts] denotes the static transformation between the full dofsvector and reduced master dofs vector. The

static condensation used to limit the number ofdofs at which the system response is analysed can cause some

important errors. It depends on the mechanical properties of the rotating machine and on the frequency range

in which the analysis is carried out. For the dynamic condensation only the form of the matrix [ Ts] will bechanged[44]. After the static transformation, Eq. (10) can be written as

MRf uRg OsGRf _uRg KRfuRg ffRg (12)with

MR TsTMTs, (13)

GR TsTGTs, (14)

KR TsTKTs, (15)

ffRg TsT

ffg, (16)wherefuRg andffRg are the rotor displacement (masters dofs) and force vectors, respectively, and matrices[MR], [KR] and [GR] are the condensed mass, stiffness and gyroscopic matrices of the rotor substructure.

2.7. Equations of motion of bearings as a substructure

Equations of motion of bearings as a substructure could be obtained by assembling equations of motion of

individual bearings (i.e. Eq. (6)), as follows:

CBf _uBg KBfuBg f0g, (17)where vector

fuB

gcontains the rotordofsat bearing locations and matrices

CB

and

KB

are, respectively, the

assembled damping and stiffness matrices for the substructure of bearings.

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2.8. System equations of motion in the frequency domain

Equations of motion of the rotor and bearings as substructures are given by Eqs. (12) and (17), respectively.

The force vector can be expressed as follows:

ff

t

g fF

gejokt, (18)

whereokis a typical excitation frequency and the vector {F} contains the amplitude and phase of forces. If the

excitation frequency is equal to the spin frequency of the shaft then {F} will contain contributions from the

impact as well as from the residual unbalance. Correspondingly, the response can be expressed as

futg fUgejokt, (19)where {U}contains the amplitude and phase of displacements. On substituting Eqs. (18) and (19) in Eqs. (12)

and (17) give, respectively, the rotor and bearings substructures governing equations in frequency domain as

ZRfURg fFRg (20)and

ZB UBf g FBf g, (21)where subscripts R and Brelate to the rotor and the bearing, respectively. The individual dynamic stiffness

matrices, Z, of each of these substructures are

ZRok; Os KR o2kMR jokOsGR (22)and

ZBok; Os KBOs jokCBOs. (23)Thedofsof the rotorbearings system (i.e. Eqs. (20) and (21)) is composed of the internal and connection dofs.

Thedofsof the rotor at bearing locations are called the connection dofs, UR,Band thedofs of the rotor other

than at bearing locations are called as the internal dofs, UR,I. Equations of motion of two substructures (i.e.

Eqs. (20) and (21)) are partitioned to the internal and connection dofs asZR;II ZR;IB

ZR;BI ZR;BB

" # UR;I

UR;B

( )

FR;I

FB;B

( ) (24)

and

ZB;BB ZB;BI

ZB;IB ZB;II

" # UR;B

UB;I

( ) FB;B

0

, (25)

wherefUB;Ig is the bearing internal dofs vector. Combining Eqs. (24) and (25) leads to general equations ofmotion for the global rotorbearing system and it can be written as

ZR;II ZR;IB 0ZR;BI ZR;BBZB;BB ZB;BI

0 ZB;IB ZB;II

264375 UR;IUR;B

UB;I

8>:

9>=>;

FR;I0

0

8>:

9>=>;. (26)

It is assumed that bearings can be modelled by using the dofs on the rotor only, so that the bearing model

does not contain any internal dofs, so Eq. (26) reduces to

ZR;II ZR;IB

ZR;BI ZR;BBZB;BB

" # UR;I

UR;B

( ) FR;I

0

. (27)

It should be noted that in Eq. (27), it is assumed that no external forces act at bearing locations. Eq. (27) will

be used for development of the simultaneous identification algorithm of residual unbalances and bearing

dynamic parameters as described in the following section.

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3. Identification algorithm

Eq. (27) describes governing equations of a general mdofflexible rotorbearing system model as shown in

Fig. 1. It is considered for developing an identification algorithm to estimate residual unbalances and bearing

dynamic parameters. The top and bottom sets of terms in Eq. (27) can be expressed as

ZR;IIfUR;Ig ZR;IBfUR;Bg fFR;Ig (28)and

ZR;BIfUR;Ig ZR;BB ZB;BBfUR;Bg f0g. (29)Eq. (28) can be written as follows:

fUR;Ig ZR;II1fFR;Ig ZR;IBfUR;Bg. (30)The vectorfFR;Igcontains superposition of unbalance forces due to residual unbalances and the impact forceapplied at the rotor substructure and can be expressed as

fFR;Ig fFunbg fFimpg, (31)where

fFunb

g and

fFimp

g are the residual unbalance and impact force vectors, respectively. On substituting

Eq. (31) in Eq. (30), we get

fUR;Ig ZR;II1fFunbg fFimpg ZR;IBfUR;Bg. (32)In Eq. (32), the vectorfUB;Ig is the connection dofs at bearing locations and can be measured in most of thepractical cases. The applied impact force can be measured, however, residual unbalances are unknown. On

substituting Eq. (32) in Eq. (29) eliminates the internal dofs vectorfUR;Ig, which is immeasurable orinaccessible in most of the practical cases. Remaining terms are arranged so that unknown terms (i.e. residual

unbalances and bearing dynamic parameters) are on the left-hand side and known terms on the right-hand

side of the expression, and are given as

ZB;BBfUR;Bg ZR;BIZR;II1fFunbg fPng (33)

with

fPnok; Osg fZR;BIZR;II1ZR;IB ZR;BBfUR;Bg ZR;BIZR;II1fFimpg, (34)where the vector,fPnok; Osg, contains terms collected at one excitation frequency ok, at a given shaft angularspeed Os, and for a given impact n (i.e. in the horizontal or vertical direction). Size offPnok; Osg is nc 1wherencis the number of connection dofs. The residual unbalance force vectorfFimbg can be represented as

fFimbg O2s feg, (35)where {e} is the residual unbalance vector and can be expressed as

feg2p1 fex1 ey1 ex2 ey2 exp eypgT, (36)where subscripts in the vector represent its size (subsequently the size of matrices will also be indicated in the

subscript). Residual unbalance vector components, es, are assumed to be present at p number of balance

planes, in two orthogonal directions to the rotor axis. Residual unbalance components at each balancing plane

contain magnitude and phase information of the residual unbalance present at that balancing plane. The term

ZB;BBfUR;Bg in Eq. (33) is then regrouped into a vector {b}, containing unknown bearing dynamicparameters and a corresponding matrix Wnok; Osnc8nb containing the related response terms at oneexcitation frequency,ok; at a given shaft angular speed,Os; and for one impact. Noting Eq. (35), Eq. (33) takes

the following form:

Wnok; OsfbOsg Rnok; Osfeg fPnok; Osg (37)with

fbOsg8nb1fk1

xx k1

xy k1

yx k1

yy k

2

xx knb

yy c1

xx c1

xy c1

yx c1

yy c

2

xx cnb

yygT

(38)

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and

Rnok; Osnc2pO2s ZR;BIZR;II1. (39)Parameters contained in the vector, {b}, depend on the form of the dynamic stiffness matrix specified for

bearings (Eqs. (6) and (21)) and ordering of these parameters may be arranged as desired. The bearing model

used for this work is specified as having the damping and stiffness matrices where each of these matricescontain direct and cross-coupled terms. The size of the matrix,Wnok; Os, and vector, bOs, in Eq. (37) are4 8nb and 8nb1, respectively, where nb is the number of bearings. The size of the matrixRnok; Os inEq. (37) is 2p2p of the order twice the number of unbalance planes, p. To show the form of the matrixWnok; Os, for the sake of illustration, if two bearings are assumed to be present in a given rotorbearingsystem, then matrices in Eq. (37) can be expressed as

Wnok; Os

x1n y1n 0 0 0 0 0 0 jokx

1n joky

1n 0 0 0 0 0 0

0 0 x1n y1n 0 0 0 0 0 0 jokx

1n joky

1n 0 0 0 0

0 0 0 0 x2n y2n 0 0 0 0 0 0 jokx

2n joky

2n 0 0

0 0 0 0 0 0 x2n y2n 0 0 0 0 0 0 jokx

2n joky

2n

2666664

3777775.

40For two bearings in the rotorbearing model Eq. (38) becomes

fbOsg f k1xx k1xy k1yx k1yy k2xx k2xy k2yx k2yy c1xx c1xy c1yx c1yy c2xx c2xy c2yx c2yygT.(41)

Eq. (37) can be written for different frequencies of excitation okwherek1, 2,y,m; and for two impacts (i.e.in the horizontal and vertical directions, alternately) at a particular rotor angular speed, Os. All such equations

are grouped and written as

WOsfbOsg ROsfeg fPOsg (42)with

WOs2ncm8nb Wx1 Wx2 . . . Wxm Wy1 Wy2 . . . WymT, (43)

ROs2ncm2p Rx1 Rx2 ::: Rxm Ry1 Ry2 ::: RymT, (44)

fPOsg2ncm1Px1 Px2 ::: Pxm Py1 Py2 ::: PymT. (45)Eq. (42) is for one rotor angular speed. Several angular speeds, Os(wheres1, 2,y,N) can be chosen and foreach angular speed corresponding impulse responses are measured. Writing Eq. (42) for each of these angular

speeds and on combining, it gives

Ag fdg (46)with

A2nc mN24nbNp

WO1 0 0 RO10 WO2 0 RO2...

.

.

..

..

.

.

....

0 0 WON RON

2666664

3777775, (47)

fgg24nbNp1 ffbO1g fbO2g fbONg feggT

(48)

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and

fdg2ncmN1 fPO1 PO2 PO3 . . . PONgT. (49)Parameters intended to identify are real numbers; however, matrices Wand R and vector P in Eqs. (47) and

(49) are in general complex, hence these matrices are separated into their real and complex parts, which leads

to a doubling of the size of these matrices. Eq. (46) takes the following form:A1fgg fd1g (50)

with

A1 Re AIm A

" # (51)

and

fd1g Re fdgIm fdg

!. (52)

The desired unknown parameters consisting of bearing dynamic parameters at speeds O1; O2;. . . ON andresidual unbalances atp planes are then estimated by least squares estimation technique by using Eq. (50). The

condition of matrices to be inverted (i.e. Eq. (51)) should be taken into account, and the condition number

may be improved by preconditioning, scaling of column/rows and/or by regularisation techniques [44,46]. In

the present work, column scaling is necessary for coefficients of stiffness and damping parameters (i.e. of

columns 1 to 16 in Eq. (40)). Regularisation can be used especially while bearings have isotropic (or nearly

isotropic) dynamic parameter characteristics [46,47], which gives unexpected spikes in the estimated

parameters.

4. Numerical simulation

A rotorbearing model as shown in Fig. 4 is considered for numerical illustrations of the presentidentification algorithm. The shaft is of steel and has 10 mm nominal diameter. The rigid discs are assumed to

have the internal diameter of 10 mm, outside diameter of 74 mm and thickness of 25 mm. The rotor model is

discretised into three two-noded elements. Details of the rotor model are given in Table 1.

A plain-cylindrical-journal bearing model is considered for bearings. For numerical illustrations purpose,

the short-bearing closed form expressions are used to generate bearing dynamic parameters (see Appendix C).

Bearing dynamic parameters consist of eight stiffness and damping coefficients, and these parameters are rotor

angular speed dependent. For cases both bearing geometries are assumed to be identical. The diameter of the

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Fig. 4. A typical rotorbearing model.


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bearing, D, is 25 mm and the length-to-diameter ratio, L/D, is 1. The radial clearance, cr, of the bearing is

0.08 mm. The kinetic viscosity,m, of the lubricant is 28 centi-Stokes at 40 1C and the specific gravity is 0.87.

Residual unbalances are created in the numerical model by placing known unbalance masses at discs 1 and

2. Unbalance masses of 2.19 g at 301 and 4.38 g at 601(angular locations are measured from a common shaft

reference point) were assumed to be present at discs 1 and 2, respectively. Both unbalance masses are assumed

to be present at 30 mm radius from the centre of discs 1 and 2. Rotor angular speeds are varied from 10 to

59 Hz in the interval of 1 Hz. The impact force model as given by Eq. (7) is considered in the numerical

illustration of the present algorithm. The instant, t0

, at which maximum impact applied is 0.006 s and the

maximum value of the impact, f0, alternatively applied in the horizontal and vertical directions are 20 and

30 N, respectively. The impulse force chosen has higher magnitude for the vertical direction as compared to

horizontal one and this difference is made, as it is easy to apply the vertical impulse force than the horizontal

one in a real situation. The impact can be applied on either of the rigid disc as shown inFig. 4. In the present

case impact is applied at disc 1. The impulse force in the time domain as shown in Fig. 5is applied to the rotor

andFig. 6shows magnitude of the impulse force in the frequency domain after performing the FFT. Fourier

transform of any function is symmetric about a vertical axis hence the amplitude frequency plots of the

impulse force are symmetric about the vertical axis (i.e. at round the excitation frequency of 130 Hz). Impulse

force contains several excitation frequencies and the range of excitation frequencies depends on the stiffness of

the contacting surface and the mass of the impact-hammer head. The stiffer the tip materials, the shorter will

be the duration of the pulse and the higher will be the frequency range covered by the impact force. It is for

this purpose a set of different hammer tips and heads are used to permit the regulation of the frequency range

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Table 1

Details of the rotor model for the numerical simulation

Station Distance from the left side (mm) Element length (mm)

1. Bearing 1 0

2. Disc 1 13 13

3. Disc 2 29.5 16.5

4. Bearing 2 42.5 13

20

15

10

0

5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035Time in seconds

30

20

10

00 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Time in seconds

ImpactinXdirectioninN

ImpactinYdirection

Fig. 5. Simulated impulse forces in the horizontal and vertical directions.


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to be encompassed[44]. From the FFT plots, it is evident that maximum useful excitation frequency range is

around 130 Hz and this range can excite first two modes (i.e. 38 and 125 Hz) of the present rotorbearing

model. For the present estimation illustrations, the excitation frequency up to 60 Hz has been considered.

Machine transient responses due to impulse excitations have been simulated in the time domain by using

Eqs. (12) and (17) in the assembled form. In practical situations the noise is always present while acquiring

bearing responses and it cannot be eliminated completely. To take care of the inherent noise present in

measured signals, simulated bearing responses are corrupted sequentially with 0%, 1%, 2% or 5% normallydistributed random noise. These corrupted bearing responses are utilised in the proposed identification

algorithm to estimate residual unbalances and bearing dynamic parameters with different level of noise. One

such numerically simulated frequency response amplitude and phase plots with 10 Hz of rotor angular speed

(that give rise to a residual unbalance force at 10 Hz) for bearing 2 in the horizontal direction due to a vertical

impulse force is shown in Fig. 7 (the dotted line). Fig. 8 shows (the dotted line) numerically simulated

frequency response amplitude and phase plots in the vertical direction for the same impulse force. In Figs. 7

and 8 the amplitude frequency plots clearly show two peaks corresponding to the first and second natural

frequencies of the rotorbearing system. These values are 38 and 125 Hz, for the present rotorbearing model.

The phase frequency plots shown in Figs. 7 and 8 show a sharp change of phase at these two natural

frequencies. Amplitude frequency plots also show a small peak at a frequency of near to the shaft angular

speed caused due to residual unbalances; however, the corresponding phase plot does not show any sharp

change as seen at natural frequencies.

On using the numerically simulated frequency responses and impulse force applied to the system in the

estimation equation (50), bearing dynamic parameters and residual unbalances are identified, simultaneously.

In the present illustration, bearing dynamic parameters of both bearings in the system and at all chosen rotor

angular speeds along with residual unbalances are identified in a single run of the computer code. However,

results are shown only for bearing 2 for brevity as shown in Figs. 9 and 10and inTables 2 and 3. Total six

different rotor speed frequency ranges are considered. In case 1: total 20 frequency steps in the frequency

range 1029 Hz, in case 2: total 20 frequency steps in the frequency range 3049 Hz, in case 3: total 20

frequency steps in the frequency range 4059 Hz, in case 4: total 30 steps in frequency range 1039 Hz, in case

5: total 40 steps in the frequency range 1049 Hz and in case 6: total 50 steps in the frequency range 1059 Hz

are considered. For example in case 1, for bearing dynamic parameters of bearings 1 and 2 in the rotor angular

speed ranges of 1029 Hz are identified along with residual unbalances at balancing planes 1 (disc 1) and 2

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15

10

0

5

0 50 100 150 200 250 300

Frequency, Hz

Amp

litudeofImpact-x

15

10

25

20

0

5

0 50 100 150 200 250 300

Frequency, Hz

AmplitudeofImpact-y

Fig. 6. The FFT of the horizontal and vertical impulse forces.


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10-3

10-4

10-5

10-60 20 40 60 80 100 120 140

Frequency (Hz)

Frequency (Hz)

6

4

2

0

-2

-40 20 40 60 80 100 120 140

Displacement(m)

Phase(rads)

estimated

simulated

estimated

simulated

Fig. 8. Estimated and simulated amplitude and phase frequency responses in the vertical direction at bearing 2 for the vertical impulse at a

rotor speed of 10 Hz (5% measurement noise).

10-3

10-4

10-5

10-60 20 40 60 80 100 120 140

Frequency (Hz)

Frequency (Hz)

6

4

2

0

-2

-40 20 40 60 80 100 120 140

Displacement(m)

P

hase(rads)

estimated

simulated

estimated

simulated

Fig. 7. Estimated and simulated amplitude and phase frequency responses in the horizontal direction at bearing 2 for the vertical impulse

at a rotor speed of 10 Hz (5% measurement noise).


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Frequency (Hz)

5

4

3

2

1

10 15 25 555045403520 30 60

Stiffnesspar

ameters(N/m)

estimated kxx

assumed kxx

estimated kyy

assumed kyy

x 105

Frequency (Hz)

8

6

4

-4

2

-2

0

10 15 25 555045403520 30 60

Stiffnessparam

eters(N/m)

estimated kxy

assumed kxy

estimated kyx

assumed kyx

x 106

Fig. 9. Assumed and estimated bearing stiffness parameters of bearing 2 for different rotor speeds (5% measurement noise).

Frequency (Hz)

2

1.9

1.8

1.7

1.5

1.6

10 15 25 555045403520 30 60Dampingparameters(Ns/m)

estimated cxx

assumed cxx

estimated cyy

assumed cyy

Frequency (Hz)

2000

0

-2000

-4000

10 15 25 555045403520 30 60Dampingparameters(Ns/m)

estimated cxy

assumed cxy

estimated cyx

assumed cyx

x 104

Fig. 10. Assumed and estimated damping parameters of bearing 2 for different rotor speeds (5% measurement noise).


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(disc 2). The size of the regression matrix [A1] is relatively large for the present algorithm (for example for case

1 the size of the regression matrix [A1] is 3200 324 (since p2, m10, nc4, N20, nb2 and thematrix size is4ncmN 24nbNp; hence, the condition number of the regression matrix before scaling ishigh. Scaling of columns 116 in Eq. (40) reduced the condition number of the regression matrix [A1]

considerably to as low as 100. Scaling factor of 107 is applied to columns 18 and scaling factor of 105 is

applied to columns 916 in Eq. (40). Tikhonov regularisation (see Appendix D) has been incorporated in the

present identification algorithm. Figs. 9 and 10 show the identified and assumed stiffness and damping

parameters, respectively, for bearing 2, in the rotor angular speed range of 1059 Hz. Identified bearing

dynamic parameters matched quite well with assumed ones at most of rotor angular speeds, even in the

presence of 5 percentage measurement noise. The identified residual unbalance at disc 1 for the measurement

noise with up to 5% is shown inTable 2for different frequency bands. The assumed magnitude of the residual

unbalance at disc 1 is 65.7 g mm and phase angle is of 301referred to a reference point on the shaft. Similarly,

the identified residual unbalance at disc 2 is shown in Table 3. The assumed magnitude of the residual

unbalance at disc 2 is 131.4 g mm and phase angle is 601. Magnitude of the identified residual unbalance at disc

1 is close to the assumed one, the deviation is up to 4.36 g mm (0.145 gm), while considering whole frequency

range in case 6. Phase of the residual unbalance at disc 1 is close to the assumed phase, and the deviation is up

to 9.561 in case 6, while considering whole frequency range. Magnitude of the identified unbalance at disc 2,

deviated from the assumed residual unbalance up to 17.91 g mm (0.597 gm) for case 6. The deviation of the

phase is up to 12.721 from the assumed phase of the residual unbalance at disc 2, for case 6. The possible

reason for the deviation is due to the ill-conditioning of the regression matrix. It should be noted that if the

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Table 2

Identified residual unbalance at balance plane 1 (disc 1) for different excitation frequency ranges and for different measurement noise levels

Rotor speed

frequency range

Number of speed

frequency points

Percentage of

noise

Residual imbalance at balance

plane 1 (g mm@degrees)

magnitude@phase

Residual imbalance % error in

estimation, magnitude@phase

10:29 20 0 [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]





10:59 50 0 [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

6.64@

12.10

The assumed unbalance at disc 1: magnitude 65.7 g mm; phase angle 301.


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balance planes are chosen too close together then they effectively act at single plane and cause inaccurate

results.Tables 2 and 3also show percent errors of magnitude and phase of the residual unbalances. However,

it is necessary to consider that the percent error of a circular function may be proved to be erroneous in some

cases. For instance, the same phase of a balance weight can be expressed as 01 or 3601. An error of 11 in the

identification of the angular position of the balance weight would cause very different percent errors if the 01

value or the 3601 value is considered. Some different techniques may be used to measure the accuracy with

which a complex parameter has been identified.

To check the response reproduction capability of the present identification algorithm, identified parameters

are substituted back in the system model, Eq. (27). The simulated response (solid line) from identified

parameters and the response simulated from assumed parameters (dotted line) are compared in the useful

excitation frequency range. The horizontal and vertical responses for bearing 2 for vertical impact are shown

inFigs. 7 and 8, respectively, and matching is found to be very good.

5. Conclusions

An identification algorithm for the simultaneous estimation of bearing dynamic parameters and residual

unbalances is presented for mdofs flexible rotorbearing systems. The identification algorithm has the

flexibility to incorporate any number of bearings and balancing planes. Residual unbalances are obtained at

predefined balancing planes. The standard condensation technique is used to reduce the number ofdofsof the

system model and hence the number of response measurements to be taken. The identification algorithm is

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Table 3

Identified residual unbalance at balance plane 2 (disc 2) for different excitation frequency ranges and for different measurement noise levels

Rotor speed

frequency range

Number of speed

frequency points

Percentage of

noise

Residual imbalance at balance

plane 2 (g mm@degrees)

magnitude@phase

Residual imbalance % error in

estimation, magnitude@phase


30:49 20 0 [email protected] [email protected]

1 [email protected] [email protected]





10:49 40 0 [email protected] [email protected]




10:59 50 0 [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] 10.68@

11.20

The assumed unbalance at disc 2: magnitude 131.4 g mm; phase angle 601.


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illustrated through a numerical rotorbearing model. The estimates of the bearing dynamic and residual

unbalance parameters have been found to be very good. The identification algorithm for simultaneous

identification of bearing dynamic parameters and residual unbalances is found to be robust against

measurement noise. The response reproduction capability from the identified bearing dynamic and residual

unbalance parameters have been found to be excellent in most of the excitation frequency range.

The identification algorithm has applicability on field to identify bearing dynamic and residual unbalanceparameters. However, it would be interesting to be seen in future use the proposed method for an in-field

analysis of the fluid-film journal bearings of large rotating machines with sensitivity analyses of estimates for

different level of excitations (i.e. impact as well as residual unbalances) with machine weight. In the paper, the

system response has been simulated using the same machine model used to identify bearing coefficients and

residual unbalances. However, in a real case, the most important cause of errors in the identified parameters is

due to the unavoidable lack of accuracy in the model of the fully assembled machine. Since the model errors are

deterministic, their effects can significantly reduce the accuracy of the identified parameters. The method

described in the paper considers only pre-established locations of the identified residual unbalances. In a real

case the location of the residual unbalances can be identified along with the magnitude and phase of each

imbalance. The effect of misalignment and shaft thermal bows will reflect in estimates of the residual unbalance

in the form of an equivalent residual unbalance. However, estimates of the bearing dynamic parameters are

expected to be better, which is the prime focus of the paper. For the present case the foundation has beenconsidered as rigid. The rigid foundation is not a restriction of the present method however it is an assumption.

The effect of the foundation and support flexibility can be incorporated in the present model by considering it

as another substructure (e.g. the foundation substructure along with the present rotor and bearings

substructures). The identification method has to be reformulated to take care of unknown foundation

parameters as well. To conclude considering all the issues discussed, it is possible (perhaps difficult; especially

exciting real turbo-machines by the impulsive force) to use the method proposed in this paper for an in-field

analysis of the fluid-film journal bearings of large rotating machines. A more practical way of acquiring the real

machine vibration data would be during the coast-up and the run-down of the machine.

Appendix A. Nomenclature

A area of cross-section of shaft

[A] regression matrix

c damping coefficient

cc number of connectiondofs

cr radial clearance of the bearing

{e} unbalance vector

E Youngs modulus

{f} nodal force vector in time domain

{F} nodal force vector in frequency domain

G modulus of rigidity

I area mass moment of inertia of the shaft cross-section

Im imaginary part

k stiffness coefficient

ksc shear factor

KB;CB stiffness and damping matrices of the bearingl shaft element length

M;K;G mass, stiffness and gyroscopic matrices, respectivelynb number of bearings

p number of balancing planes

Re real part

t0 time instant at which the impulse force is applied

[Ts] transformation matrix for the static condensation

{u} response vector in the time domain

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fURg;fUBg response vector in the frequency domainv, w linear displacements in the horizontal and vertical directions, respectively.

[Wn] matrix containing measured responses at bearings

[Z] dynamic stiffness matrix

{b} vector grouping all bearing stiffness and damping parameters

F 12EI=kscGAl2

m kinetic viscosityo excitation frequency

O angular speed of the rotor

y, f angular displacements in the horizontal and vertical directions, respectively.

Subscripts

B bearing

d disc

imp impulse

I internal

k number of excitation frequencies (1; 2;. . .; m)n impact direction (e.g. horizontal or vertical)

nc connection degrees of freedom (dofs) at bearing locations

nb number of bearings

r rotational

R rotor

s a particular angular speed (1, 2,y, N)

unb unbalance

Superscripts

b bearing number

(e) element

(ne) element nodesp number of balancing planes

T transpose of a vector or matrix

Appendix B. Timoshenko beam model

B.1. Translational mass matrix

M

t M

t0F

M

t1F2

M

t2, (B.1)

Mt0 rAl

4201F2

156

0 156 Sym

0 22l 4l222l 0 0 4l2

54 0 0 13l 156

0 54 13l 0 0 1560 13l 3l2 0 0 22l 4l2

13l 0 0 3l2

22l 0 0 4l2

266666666666664

377777777777775

, (B.2)

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Mt

1

rAl

4201F2

294

0 294 Sym

0 38:5l 7l238:5l 0 0 7l2

126 0 0 31:5l 2940 126 31:5l 0 0 2940 31:5l 7l2 0 0 38:5l 7l2

31:5l 0 0 7l2 38:5l 0 0 7l2

266666666666664

377777777777775

, (B.3)

Mt2 rAl

4201F2

140

0 140 Sym

0 17:5l 3:5l217:5l 0 0 3:5l2

70 0 0 17:5l 140

0 70 17:5l 0 0 1400 17:5l 3:5l2 0 0 17:5l 3:5l2

17:5l 0 0 3:5l2 17:5l 0 0 3:5l2

266666666666664

377777777777775

(B.4)

with

F 12EIkscGAl

2,

where ksc is shear correction factor.

B.2. Rotational mass matrix

Mr Mr0 FMr1 F2Mr2, (B.5)

Mr0 rAl

1F2

6=5l

0 6=5l Sym

0 1=10 2l=151=10 0 0 2l=15

6=5l 0 0 1=10 6=5l0 6=5l 1=10 0 0 6=5l0

1=10

l=30 0 0 1=10 2l=15

1=10 0 0 l=30 1=10 0 0 2l=15

666666666666666664

777777777777777775

, (B.6)

Mr1 rAl

1F2

0

0 0 Sym

0 1=2 l=6

1=2 0 0 l=60 0 0 1=2 0

0 0 1=2 0 0 00 1=2 l=6 0 0 1=2 l=6

1=2 0 0 l=6 1=2 0 0 l=6

666666666666666664

777777777777777775

, (B.7)

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Mr

2

rAl

1F2

0

0 0 Sym

0 0 l=3

0 0 0 l=3

0 0 0 0 00 0 0 0 0 0

0 0 l=6 0 0 0 l=3

0 0 0 l=6 0 0 0 l=3

666666666666666664

777777777777777775

. (B.8)

B.3. Stiffness matrix

K K0 FK1, (B.9)

K0 EI

1Fl3

12

0 12 Sym0 6l 4l26l 0 0 4l2

12 0 0 6l 120 12 6l 0 0 120 6l 2l2 0 0 6l 4l26l 0 0 2l2 6l 0 0 4l2

266666666666664

377777777777775, (B.10)

K1 EI

1Fl3

0

0 0 Sym

0 0 l2

0 0 0 l2

0 0 0 0 0

0 0 0 0 0 0

0 0 l2 0 0 0 l20 0 0 l2 0 0 0 l2

266666666666664

377777777777775

. (B.11)

B.4. Gyroscopic matrix

G G0FG1F2

G2, (B.12)

G0 rAr2

601F2l

0

36 0 Skew sym

3l 0 00 3l 4l2 00 36 3l 0 0

36 0 0 3l 36 03l 0 0 l2 3l 0 0

0 3l l2

0 0 3l 4l2

0

266666666666664

377777777777775

, (B.13)

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G

1

rAr2

601F2

l

0

0 0 Skew sym

15l 0 0

0 15l 5l2 0

0 0 15l 0 00 0 0 15l 0 0

15l 0 0 5l2 0 0 0

0 15l 5l2 0 15l 5l2 5l2 0

266666666666664

377777777777775

, (B.14)

G2 rAr2

60

1

F

2l

0

0 0 Skew sym

0 0 0

0 0 10l2 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 5l2 0 0 00 0 5l2 0 0 0 10l2 0

266666666666664

377777777777775

. (B.15)

B.5. Rigid disc model

Mass matrix Md

md 0 0 0

0 md 0 0

0 0 Id 0

0 0 0 Id

26664 37775, (B.16)

Gyroscopic matrix Gd

0 0 0 0

0 0 0 0

0 0 0 Ip0 0 Ip 0

266664

377775, (B.17)

Displacement vector

fq

gd

fv w y f

g. (B.18)

Appendix C. Fluid-film bearing dynamic characteristics

Fluid-film bearing stiffness and damping coefficients, direct as well as cross-coupled, can be derived from

Reynolds equation based on the short bearing approximation (i.e. pressure variation in the circumferential

direction is assumed to be negligible compared with that in the axial direction and converse applies for long

bearing approximation). The eight linearised stiffness and damping coefficients depend on the steady-state

operating conditions of the journal, and in particular upon the angular speed. For the short bearing, the

dimensionless bearing stiffness and damping coefficients, Kijkijcr=W, Cijcijcr=W, and i;jx;y, as afunction of the steady eccentricity ratio, e, of the bearing are given as [45]

Kxx4f2p2

16 p2

e4

gQe, (C.1)

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Kxypfp2 2p2e2 16p2e4gQe

effiffiffiffiffiffiffiffiffiffiffiffiffi

1e2p , (C.2)

Kyxpfp2 32p2e2 216 p2e4gQe

e ffiffiffiffiffiffiffiffiffiffiffiffiffi1 e2p , (C.3)

Kyy4fp2 32 p2e2 216 p2e4gQe

1 e2 , (C.4)

Cxx2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1e2

p fp2 2p2 8e2gQe

e , (C.5)

CxyCyx 8fp2 2p2 8e2gQe (C.6)and

Cyy2pfp2 224 p2e2 p2e4gQe

e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1e2p , (C.7)

where

Qe 1fp21e2 16e2g1=2. (C.8)

To determine stiffness and bearing coefficients of a short bearing, the Sommerfeld number

SmDLNW

r

cr

2(C.9)

is first determined, whereWis the load on the bearing, r is the bearing radius, D is the journal diameter, L is

the length of bearing, m is the viscosity of lubricant at operating temperature, O2pn the angular speed ofjournal,Nis the number of revolutions per seconds and cris the radial clearance. The eccentricity ratio of the

journal centre is defined as ee=cr, where e is the journal eccentricity.We can then determine the eccentricity ratio under steady-state operating conditions by

S L

D

2 1e

22pe

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffip21 e2 16e2

p . (C.10)Appendix D. Tikhonovs regularisation method for conditioning of ill-posed regression matrix

The linear least-squares problem of Eq. (50)

A1fgg fd1g (D.1)to determine the bearing dynamic parameters and the residual unbalances is said to be ill-disposed if thesingular values of [A1] decay gradually to zero and the ratio between the largest and smallest non-zero singular

values of [A1] is large. Ill-conditioning of matrix [A1] implies that the solution is sensitive to perturbations and

regularisation is imperative for a stabilised solution. In Tikhonovs regularisation method [47]the regularised

solution {g} is a solution to the following weighted combination of the residual norm and the side constraint:

min A1gd1 2

2l2 Lgg

22

n o, (D.2)

whereL is the identity matrix In, g is an initial estimate of the solution obtained from the assumed dynamic

parameters andl is the regularisation parameter that controls the relative minimisation of the side constraint

with respect to the residual norm. The value ofl is obtained from the L-curve which is a loglog plot of the

norm L

g

2 versus the corresponding residual norm A1gd1 2.TheL-curveaids in seeking a compromisebetween the minimisation of the two norms. In the algorithm for the numerical treatment of the Tikhonovs

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method, the singular value decomposition of matrix [A1] of the form

A1 UX

VT Xni1

uisivTi (D.3)

is used where U

u1;. . .; un

and V

v1;. . .; vn

are matrices with orthonormal columns of right and left

singular vectors ofAsuch that UTUVTV Inand P diags1;. . .; sn has non-negative singular valuesofA. Tikhonovs method produces a regularised solution

xlXni1

fiuTi b

sivi, (D.4)

where fis2is2i l2are filter factors and the filtering sets in for siol.

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