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JOURNAL OF SOUND AND

VIBRATION

Journal of Sound and Vibration 289(2006)294–333

A modi?ed transfer matrix method for the coupling lateral and torsional vibrations of symmetricrotor-bearing systems

Sheng-Chung Hsieh a ,Juhn-Horng Chen b ,An-Chen Lee a,?

Department of Mechanical Engineering,National Chiao Tung University,1001Ta Hsueh Road,

Hsinchu 30049,Taiwan,ROC

Department of Mechanical Engineering,Chung Hua University,Taiwan,ROC

Received 27January 2004;received in revised form 9August 2004;accepted 8February 2005

Available online 28April 2005

Abstract

This study develops a modi?ed transfer matrix method for analyzing the coupling lateral and torsional vibrations of the symmetricrotor-bearing system with an external torque.Euler’s angles are used to describe the orientations of the shaft element and disk.Additionally,to enhance accuracy,the symmetric rotating shaft is modeled by the Timoshenko beam and considered using a continuous-system concept rather than the conventional ‘‘lumped system’’concept.Moreover,the harmonic balance method is adopted in this approach to determine the steady-state responses comprising the synchronous and superharmonic whirls.According to our analysis,when the unbalance force and the torque with n ?frequency of the rotating speed excite the system simultaneously,the en t1T?and en à1T?whirls appear along with the synchronous whirl.Finally,several numerical examples are presented to demonstrate the applicability of this approach.

1.Introduction

Rotor dynamics plays an important role in many engineering ?elds,such as gas turbine,steam turbine,reciprocating and centrifugal compressors,the spindle of machine tools,and so on.Owing to the growing demands for high power,high speed,and light weight of the rotor-bearing

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?Corresponding author.Tel.:+88635728513;fax:88635725372.

E-mail address:aclee@https://www.wendangku.net/doc/b28710709.html,.tw (An-Chen Lee).

S.-C.Hsieh et al./Journal of Sound and Vibration289(2006)294–333295

system,computations of critical speeds and steady-state response at synchronous and subcritical resonances become essential for system design,identi?cation,diagnosis,and control. Currently,the?nite element and transfer matrix approaches are becoming two of the most prevalent methods for analyzing rotor-bearing systems.While the?nite element method(FEM) formulates rotor-bearing systems by second-order differential equations directly utilized for control design and estimation,the transfer matrix method(TMM)solves dynamic problems in the

frequency domain.The TMM utilizes a marching procedure,starting with the boundary conditions at one side of the system,and successively marching along the structure to the other side of the system.The satisfaction of the boundary conditions at all boundary points provides the basis for solution location.The state of the rotor system at a speci?c point is transferred between successive points through transfer matrices.This method is particularly suitable for ‘‘chainlinked’’structures such as rotor systems.The primary advantage of the TMM is that it does not require the storage and manipulation of large system arrays [1].

The application of ?nite element models to rotor dynamics has been highly successful.Numerous ?nite element procedures have attempted to generalize and improve the work of Ruhl and Booker [2].Nelson and McVaugh [3]employed a ?nite element model to formulate the dynamicequation of a linear rotor system and determine the stability and steady-state responses.

Moreover,Nelson [4]and O zgu ven and O

zkan [5]further improved the ?nite element model by including the effects of rotary inertia,gyroscopic moments,shear deformation and internal damping.

Genta [6]proposed a scheme for investigating the parametric vibration and instability of an asymmetricrotor-bearing system via FEM without giving the general formulation of the motion equation.Genta thus failed to investigate the effects of asymmetry on the motion of rotor-bearing systems.The effects of deviatoric stiffness of shaft and bearing owing to asymmetry on steady-state responses was investigated by Kang et al.[7]and transient responses under acceleration was investigated by Lee et al.[8].

The TMM was ?rst proposed by Prohl [9].Subsequently,the effects of damping and stiffness of the ?uid ?lm bearing were included by Koenig [10],Guenther and Lovejoy [11].Lund [12]achieved signi?cant advances in the TMM by considering the effects of gyroscopic,internal friction and aerodynamic cross-coupling forces.Bansal and Kirk [13]applied the TMM in modal analysis for calculating the damped natural frequencies and examining the stability of ?exible rotors mounted on ?exible bearing supports.Lund [14]presented a scheme for estimating the sensitivity of the critical speeds of a rotor to change the design factors.The use of TMM on the rotors being exposed to a constant axial force and torque was considered by Yim et al.[15].In the above works,the shaft is modeled using a lumped-system sense to relate the state variables of the two ends of the segment via transfer matrix.Because the lumped mass is concentrated at each end of the section,the shaft must be divided into numerous sections to yield accurate results.Consequently,considerable computing time is required.

Lund and Orcutt [16]constructed the shaft transfer matrix in a continuous-system sense analytically and examined the unbalance vibrations experimentally.Furthermore,Inagaki et al.[17]devised a TMM scheme for determining the steady-state response of asymmetric rotor-bearing systems by considering only the effect of transverse inertia,while ignoring the effects of rotary inertia and gyroscopic moment.However,their study only considered a single harmonic component for the synchronous whirl.Additionally,David et al.[18]showed that the harmonic balance technique incorporating the TMM can be applied to analyze parametric systems.Moreover,Lee et al.[19]improved the TMM of the continuous-systems sense to ?t the synchronous elliptical orbits of the linear rotor-bearing systems by doubling the number of state variables to 16.Their study also considered the rotary inertia,gyroscopic and transverse shear effects.Furthermore,the utilization of TMM for continuous systems was extended to the unbalancing shaft [20]and asymmetricrotors [21].All of the above studies assumed that the

S.-C.Hsieh et al./Journal of Sound and Vibration 289(2006)294–333

296

S.-C.Hsieh et al./Journal of Sound and Vibration289(2006)294–333297 rotating shaft in the axial direction is rigid.However,the values of the transverse amplitudes calculated based on this assumption may differ markedly from the actual values.

Regarding the torsional analysis using the TMM,Pestel and Leckie[22]provided a thorough reference for applying the transfer matrix to determine the natural frequencies and mode shape for torsional systems.Moreover,Pilkey and Chang[23]presented a generalized method for applying the boundary conditions to a torsional transfer matrix model that is useful in developing an algorithm for accomplishing the desired analysis.Sankar[24]presented one multi-shaft torsional transfer matrix approach.This method built the transfer matrix for each branch separately,applied compatibility relations at the junction,and then used the boundary conditions to obtain the characteristic determinant of the system.Finally,Rao[25]employed the TMM to analyze the free vibration,transient response,critical speed,and instability of the torsional rotor system.

Schwibinger and Nordmann[26]examined the in?uence of torsional–lateral coupling on the stability behavior of a simple geared system supported by oil?lm bearings.Schwibinger and Nordmann found that the classical eigenvalue analysis ignoring the coupling of torsional and lateral vibrations in gears might cause serious errors in the stability prediction,such as the critical speeds and natural modes.Qin and Mao[27]developed a new?nite element model to analyze the torsional–?exural characteristics of the rotor system.Additionally,Rao et al.[28] investigated the lateral transient response of geared rotors raised by torsional excitation.Rao et al.concluded that even if the critical speed of the rotor did not approach the running speed, the lateral response at a multiple of the spin speed and the torsional response were very large,and the in?uence of incremental bending stiffness because of axial torque was insigni?cant. Mohiuddin and Khulief[29]presented a reduced modal form of the rotor-bearing system to ?nd the transient responses owing to different excitations using the FEM.Al-Bedoor[30] presented a dynamic model for a typical elastic blade attached to a disk mounted on a shaft which was?exible in the torsional direction.The resulting model and simulation results exhibited strong dependence and energetic interaction between the shaft torsional deformations and the blade bending deformations.Additionally,Al-Bedoor[31]presented a model for interpreting the coupled torsional and lateral transient vibrations of the simple Jeffcott rotor.His analysis demonstrated the existence of inertial coupling and nonlinear interaction between the torsional and lateral vibrations.Lee[32]formulated the coupled equations of motion in a lateral bending–torsion for an unbalanced disk of the simple Jeffcott rotor for analyzing the instabilities.

This work develops a modi?ed TMM for the coupling lateral and torsional vibrations of symmetricrotor-bearing systems.Euler’s angles are used to desc ribe the orientations of the shaft elements and disks.First,Hamilton’s Principle and Newton’s second law are used to derive the motion equations of the?exible shaft,rigid disks,and linear bearings with respect to the?xed coordinate,and second,the transfer matrices of the elements are established using the harmonic balance method.Third,the state variables of the element matrices are related in stepwise fashion from the left end to the right end to obtain the overall transfer matrix of the rotor system.The overall transfer matrix can be used to determine the steady-state responses of synchronous and superharmonic whirls of the coupling lateral and torsional vibrations.Finally,several numerical examples are presented to demonstrate the applicability of the approach.

2.Kinematics of rotating element

The orientation of the rotating element,in three-dimensional motion,can be completely described using Euler’s angles de?ned via three successive rotations to specify the relations between the principal axes of the rotating frame and the ?xed frame.As shown in Fig.1(a),the rotating sequence for de?ning Euler’s angles is explained via the following steps:(1)rotate the initial system,parallel to ?xed coordinates,into a de?ected mode by an angle f about the Z -axis,(2)rotate the intermediate axes eXYZ T0by an angle y about the X 0-axis (the so-called nodal axis)to another intermediate axes eUVW T0;(3)rotate intermediate axes eUVW T0by an angle c about the W 0-axis to produce the principal coordinates UVW :The Euler’s angles f ;y ;and c fully characterize the orientation of the rotating element at any given instant.

When a rotating element is de?ected in position and orientation as illustrated in Fig.1(b),the inclined angle y of orientation is measured counterclockwise from the ?xed axis Z to the spin axis W of the rotating element.In the projection description,the de?ected angles (or angular displacements)are the projections of the inclined angle y ;thus y x ?y cos f and y y ?y sin f :Additionally,the spin angle about the axis W is obtained as F ?f tc from the geometric con?guration of the rotating element with a very small oblique angle y :

Through the coordinate transformation,the components of the angular velocities in the directions of principal axes can be found to be

o u o v o w 2

3775?cos c sin c 0

àsin c cos c 0001

266437751000c os y sin y 0àsin y cos y 26643775cos f sin f 0àsin f cos f 00012664377500_f 2664377

5tcos c sin c 0àsin c cos c 00012

6437751000c os y sin y 0àsin y cos y 2664

3775_y

26643775tcos c sin c 0àsin c cos c 00012664377500_c 26643775,

(a)

(b)

Fig.1.Orientation of the rotating element.

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298

that is,

o u ?_y

cos c t_f sin y sin c ,o v ?à_y

sin c t_f sin y cos c ,o w ?_c

t_f cos y .e1T

Using principal axes,the kinetic energy E k of a rotating element moving in three-dimensions is

given by

E k ?12m e_x 2c t_y 2c Tt12eI u o 2u tI v o 2v tI p o 2w T.

(2)

Notably the kineticenergy E k of Eq.(2)includes two parts,one associated with the motion of the

mass center,and the other associated with the angular velocities of the rotating element.

Substituting Eq.(1),I ?I u ?I v ;y x ?y cos f ;y y ?y sin f ;F ?f tc ;and _F

?_f t_c into Eq.(2),the kineticenergy of the symmetricrotating element in the ?xed frame is obtained as

E k ?12

?m e_x 2c t_y 2c TtI p _F 2tI p _F e_y x áy y à_y y áy x TtI e_y 2x t_y 2y T .(3)

The kineticenergy in the form of Eq.(3),was used by Greenhill et al.[33]to investigate rotor-bearing systems with a symmetricshaft and symmetricdisks at a c onstant speed.3.Transfer matrix of the rigid disk

The disk is assumed to be rigid,thin,and symmetric.Fig.2shows the whirling orbit of the disk

with mass imbalance.The geometric relations yield

x c y c "#?x

"#te d x e d y "#

(4)

t +

C : mass center

G : geometric center ?Fig.2.Whirling orbit of the disk.

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299

and

e d x e d y

cos eO t tj Tàsin eO t tj Tsin eO t tj T

cos eO t tj T

#e d u e d v

(5)

Substituting Eq.(5)into Eq.(4)and differentiating it,following relations are obtained:

_x c ?_x àe d v eO t_j Tcos eO t tj Tàe d u eO t_j Tsin eO t tj T,(6)_y c ?_y te d u eO t_j Tcos eO t tj Tàe d v eO t_j

Tsin eO t tj T.(7)

Inserting Eqs.(6)–(7)and F ?O t tj into Eq.(3),the kineticenergy of the symmetricdisk is

obtained by

E k ?1m d ?_x 2à2_xe d v eO t_j Tcos eO t tj Tà2_xe d u eO t_j Tsin eO t tj Tt_y

2t2_ye d u eO t_j Tcos eO t tj Tà2_ye d v eO t_j

Tsin eO t tj TteO t_j T2ee d T2 t12I d p eO t_j T2t12I d p eO t_j Te_y x y y à_y y y x Tt12

I d e_y 2x t_y 2y T.Fig.3illustrates that the work done by the disk weight,bending moments,shear forces,and the

torque on the left and right of the disk is

W ?àw d y tV R x x tM R y y y tV R y y tM R x y x tT R j àeV L x x tM L y y y tV L y y tM L x y x tT L

j T.

Using Hamilton’s principle

t 2t 1

eE k àE p tW Td t ?0

(8)

and assuming small twist angle displacement,the force equilibrium equations of the disk in the ?xed coordinates can be obtained as follows:

V R x àV L x tm d ?à€x t€j e d v cos eO t tj Tt€j e d u sin eO t tj Tàe d

v eO t_j

T2sin eO t tj Tte d u eO t_j

T2cos eO t tj T ?0,e9T

X (a)

(b)

Fig.3.Forces,moments,and torques acting on the disk.

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300

V R

y àV L

tm d?à€yà€j e d

coseO ttjTt€j e d

sineO ttjTte d

eOt_jT2sineO ttjT

te d

eOt_jT2coseO ttjT àw d?0,e10Twhere

m d?€j e d

v coseO ttjTt€j e d

sineO ttjTàe d

eOt_jT2sineO ttjT

te d

eOt_jT2coseO ttjT and

m d?à€j e d

u coseO ttjTt€j e d

sineO ttjTte d

eOt_jT2sineO ttjT

te d

eOt_jT2coseO ttjT

are the unbalance forces of the disk in the x and y directions,respectively.The twist angleejTand its derivatives affect the level of the unbalance force of the disk.

The bending moment equilibrium equations in the?xed coordinates are

M R

x àM L

àI d€y xà1

I d

€jy yàI d

eOt_jT_y y?0,(11)

M R

y àM L

àI d€y yt1

I d

€jy xtI d

eOt_jT_y x?0,(12)

where1

2I d

€jy y and1

I d

€jy x are the moments coupled with the twist acceleratione€jT;I d

eOt_jT_y y

and I d

p eOt_jT_y x the gyroscopic moments coupled with the twist velocitye_jT:

The torque equilibrium equations in the?xed coordinates is

T RàT LàI d

p €jà1

I d

€y

y yt1

I d

€y

y xtm d?€xe d

coseO ttjTt€xe d

sineO ttjT

à€ye d

u coseO ttjTt€ye d

sineO ttjTàee dT2€j ?0,e13T

where1

2I d

€y

y y and1

I d

€y

y x are the torques coupled with bending angle and angular acceleration,

and

m d?€xe d

coseO ttjTt€xe d u sineO ttjTà€ye d u coseO ttjTt€ye d v sineO ttjTàee dT2€j

is the torque induced by the unbalance force.

Eqs.(9)–(13)can be simpli?ed into motion equations of the simple Jeffcott rotor[31,32].In the simple Jeffcott rotor,the unbalanced disk is located at the middle of the shaft,and only its lateral and torsional motion is allowed.Gyroscopic and rotary inertia effects are neglected,i.e.,Eqs.(11)

and(12)vanish.The coupling terms1

2I d

€y

y y and1

I d

€y

y x in Eq.(13)also disappear.If the shear

forces and torques are replaced by the lateral stiffness forces and torsional stiffness torques of the shaft,respectively,Eqs.(9)–(10)and(13)become

k s xtm d?à€xt€j e d

v coseO ttjTt€j e d

sineO ttjTàe d

eOt_jT2sineO ttjT

te d

u eOt_jT2cos O ttj

eT ?0,

k s ytm d?à€yà€j e d

coseO ttjTt€j e d v sineO ttjTte d ueOt_jT2sineO ttjT

te d

v eOt_jT2coseO ttjT àw d?0,

S.-C.Hsieh et al./Journal of Sound and Vibration289(2006)294–333301

k s j jàI d

€jtm d?€xe d

coseO ttjTt€xe d

sineO ttjT

à€ye d

coseO ttjTt€ye d

sineO ttjTàee dT2€j ?0,

where k s denotes the shaft lateral stiffness and k s

j represents the shaft torsional stiffness.The

above motion equations are the same as those in Refs.[31,32].

The compatible relations between the two sides of the disk are given by

x R?x L;y R?y L;y R

x ?y L

;y R

?y L

;j R?j L.(14)

For a nonlinear differential equation,Hayashi[34]introduced the harmonic balance method for obtaining the solution of a higher approximation as follows.The solution was?rst expanded into Fourier series with unknown coef?cients.The assumed solution was then inserted into the original equation,and the sine and cosine terms of the respective frequencies were set to zero. Solving the simultaneous equations thus obtained can identify the unknown coef?cients of the assumed solution.The harmonicbalanc e method has been utilized by Kang et al.[7,21].

Using the harmonic balance method,the steady-state responses of Eqs.(9)–(14)can each be expressed in Fourier series form as

xetT?x0t

X n

i?1

x ic cos i O ttx is sin i O t,

yetT?y0t

X n

i?1

y ic cos i O tty is sin i O t,

y xetT?y x;0t

X n

i?1

y x;ic cos i O tty x;is sin i O t,

y yetT?y y;0t

X n

i?1

y y;ic cos i O tty y;is sin i O t,

jetT?j0t

X n

i?1

j ic cos i O ttj is sin i O t.(15) Other variables can be similarly expressed as

V xetT?V x;0t

X n

i?1

V x;ic cos i O ttV x;is sin i O t,

V yetT?V y;0t

X n

i?1

V y;ic cos i O ttV y;is sin i O t,

M xetT?M x;0t

X n

i?1M x;ic cos i O ttM x;is sin i O t,

S.-C.Hsieh et al./Journal of Sound and Vibration289(2006)294–333 302

M y et T?M y ;0tX n i ?1M y ;ic cos i O t tM y ;is sin i O t ,

T et T?

T 0tX n i ?1

T ic cos i O t tT is sin i O t .(16)

Using the relations

cos eO t tj T?cos O t cos j àsin O t sin j %cos O t àj sin O t ,sin eO t tj T?sin O t cos j tcos O t sin j %sin O t tj cos O t

substituting Eqs.(15)and (16)into Eqs.(9)–(14),ignoring the nonlinear terms,and equating the

coef?cients of the same harmonic term provides the transfer matrix equation of the disk for static,synchronous whirl and nonsynchronous whirls in the static frame:

S R

1"#??T d

k ?k S L 1"#,(17)where k ?20n t11and the state variable vector S is denoted as

S ?X F ,

(18)

where

X ??x 0x 1c ááx nc x 1s ááx ns y 0y 1c ááy nc y 1s ááy ns

y x ;0y x ;1c ááy x ;nc y x ;1s ááy x ;ns y y ;0y y ;1c ááy y ;nc y y ;1s ááy y ;ns j 0j 1c ááj nc j 1s ááj ns T

and

F ??V x ;0V x ;1c ááV x ;nc V x ;1s ááV x ;ns V y ;0V y ;1c ááV y ;nc V y ;1s ááV y ;ns

M x ;0M x ;1c ááM x ;nc M x ;1s ááM x ;ns M y ;0M y ;1c ááM y ;nc M y ;1s ááM y ;ns T 0T 1c ááT nc T 1s ááT ns T .

4.Transfer matrix of Timoshenko shaft

As shown in Fig.4,the ?nite shaft element can be considered to comprise numerous small rotating elements.Thus the total kineticenergy of the shaft element is the sum of these kinetic energies of the rotating https://www.wendangku.net/doc/b28710709.html,ing a similar procedure to that illustrated in Section 3,the kineticenergy of the symmetricshaft element expressed in ?xed c oordinates is

E k ?1

2r Z L 0

f A ?_x 2à2_xe s v eO t_j Tcos eO t tj Tà2_xe s u eO t_j Tsin eO t tj Tt_y 2t2_ye s u eO t_j Tcos eO t tj Tà2_ye s v eO t_j Tsin eO t tj TteO t_j T2ee s T2 tI s p eO t_j T2tI s p eO t_j Te_y x y y à_y y y x TtI s e_y 2x t_y 2y

Tg d Z .e19T

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303

The total potential energy according to the bending and shear deformations can be expressed in ?xed coordinates as in Kang et al.[6]

E p ?12Z

L 0?EI s ey 0x T2tEI s ey 0y T2tk s GA eg 2x tg 2y TtGI s p ej 0T2

d Z .(20)Th

e work done by the external force (see Fig.4)is

W ?Z L

àr Agy d Z tV R x x tM R

y y y

tV R y y tM R x y x tT R j àeV L x x tM L y y y tV L y y tM L x y x tT L j T.

e21T

Using Hamilton’s principle and assuming small twist angle displacement,this study obtains the

force equilibrium equations of the shaft in the ?xed coordinates:

àr A €x tr A ?€j e s v cos eO t tj Tt€j e s u sin eO t tj TàeO t_j T2e s v sin eO t tj T

teO t_j T2e s u cos eO t tj T tk s GA ex 00

ày 0y T?0,

e22T

àr A €y tr A ?à€j e s u cos eO t tj Tt€j e s v sin eO t tj TteO t_j T2e s u sin eO t tj T

teO t_j T2e s v cos eO t tj T tk s GA ey 0x ty 00Tàr Ag ?0.

e23T

From above equations,the twist angle ej Tand its derivatives can be found to emerge from the

unbalance forces:

r A ?€j e s v cos eO t tj Tt€j e s u sin eO t tj TàeO t_j T2e s v sin eO t tj TteO t_j T2e s u cos eO t tj T

and

r A ?à€j e s u cos eO t tj Tt€j e s v sin eO t tj TteO t_j T2e s u sin eO t tj TteO t_j T2e s v cos eO t tj T

and in?uence the level of the unbalance forces.

Y R

(a)(b)

Fig.4.Forces,moments,and torques acting on the ?nite shaft element.

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304

The bending moment equilibrium equations in the?xed coordinates are

r I s€y xt1

r I s p€jy ytr I s peOt_jT_y yàEI s y00xtk s GAey xty0T?0,(24) r I s€y yà1r I s p€jy xàr I s peOt_jT_y xàEI s y00ytk s GAey yàx0T?0,(25)

where1

2r I s

€jy y and1

r I s

€jy x denote the moments coupled with the twist acceleration(€j),r I s

eOt

_jT_y y and r I s

p eOt_jT_y x represent the gyroscopic moments coupled with the twist velocity(_j).

The torque equilibrium equation in the?xed coordinates is

r I s

p €jt1

r I s

€y

y yà1

r I s

€y

y xtr A?à€xe s

coseO ttjTà€xe s

sineO ttjT

t€ye s

u coseO ttjTà€ye s

sineO ttjTtee sT2€j àGI s

j00?0,e26T

where1

2r I s

€y

y y and1

r I s

€y

y x are the torques coupled by bending angle and angular acceleration,

and

r A?à€xe s

v coseO ttjTà€xe s

sineO ttjTt€ye s

coseO ttjTà€ye s

sineO ttjTtee sT2€j

is the torque induced by unbalance force. The natural boundary conditions are

V R

x t?k s GAey yàx0T Z?L?0;V L

t?k s GAey yàx0T Z?0?0,

V R

y t?àk s GAey xty0T Z?L?0;V L

t?àk s GAey xty0T Z?0?0,

M R

t?àEI s y0x Z?L?0;M L xt?àEI s y0x Z?0?0,

M R

t?àEI s y0y Z?L;M L yt?àEI s y0y Z?0?0,

T Rt?àGI s

j0 Z?L?0;T Lt?àGI s p j0 Z?0?0.(27)

The steady-state solution of Eqs.(22)–(26)can be expressed in Fourier series form as

xeZ;tT?x0eZTt

X n

i?1

x iceZTcos i O ttx iseZTsin i O t,

yeZ;tT?y0eZTt

X n

i?1

y iceZTcos i O tty iseZTsin i O t,

y xeZ;tT?y x;0eZTt

X n

i?1

y x;iceZTcos i O tty x;iseZTsin i O t,

y yeZ;tT?y y;0eZTt

X n

i?1

y y;iceZTcos i O tty y;iseZTsin i O t,

jeZ;tT?j0eZTt

X n

i?1j iceZTcos i O ttj iseZTsin i O t,(28)

S.-C.Hsieh et al./Journal of Sound and Vibration289(2006)294–333305

where x 0eZ T;y 0eZ T;y x ;0eZ T;y y ;0eZ T;j 0ez T;x ic eZ T;x is eZ T;y ic eZ T;y is eZ T;y x ;ic eZ T;y x ;is eZ T;y y ;ic eZ T;y y ;is eZ T;j ic ez Tand j is ez Tare the mode functions of the relative 0th order and n ?harmonicwhirl with respect to the static frame of the shaft.For convenience,the mode function vector of general displacement are denoted as X eZ T;namely

X eZ T??x 0eZ Tx 1c eZ Tááx nc eZ Tx 1s eZ Tááx ns eZ Ty 0eZ Ty 1c eZ Tááy nc eZ Ty 1s eZ Tááy ns eZ T

y x ;0eZ Ty x ;1c eZ Tááy x ;nc eZ Ty x ;1s eZ Tááy x ;ns eZ Ty y ;0eZ Ty y ;1c eZ Tááy y ;nc eZ Ty y ;1s eZ Tááy y ;ns eZ Tj 0eZ Tj 1c eZ Tááj nc eZ Tj 1s eZ Tááj ns eZ T T .

e29T

Substituting Eq.(28)into Eqs.(22)–(26),ignoring the nonlinear terms,and equating the coef?cients of the same harmonic term produces 33nonhomogeneous differential equations as listed in Appendix A.The general solutions of the nonhomogeneous system,Eqs.(A.1)–(A.33),can be represented by the sum of the homogeneous and the particular solutions,namely

X eZ T?X eZ Th tX eZ Tp .

(30)

Assume the homogeneous solution X eZ Th in the forms

X eZ Th ?X h e l Z ,

(31)

where the arbitrary constant vector X h ??x h 0x h 1c ááx h nc x h 1s ááx h ns y h 0y h 1c ááy h nc y h 1s ááy h

ns y h x ;0y h x ;1c á

áy h x ;nc y h x ;1s ááy h x ;ns y h y ;0y h y ;1c ááy h y ;nc y h y ;1s ááy h y ;ns j h 0j h 1c ááj h nc j h 1s ááj h ns T

and l is the characteristic value,with respect to a nature mode.

Substituting Eq.(31)into Eqs.(A.1)–(A.33)yields the following characteristic equation:

el 2E 2tl E 1tE 0TX h ,

(32)

where E 2;E 1and E 0are matrices with size e10n t5T?e10n t5Tand are listed in Appendix B.Eq.(32)can be rewritten as the generalized eigen-problem form

0E 2E 2E 1"#k ?k

àE 200àE 0"#k ?k

()l X

h X h

"#k ?1

?00 k ?1,(33)where k ?20n t10:

By solving Eq.(33),the eigen-value l and corresponding eigenvector X h are obtained.Hence the homogeneous solution is

X eZ Th

X 20n t10i ?1

C i X h i e

l i Z

,(34)

where C i is an undetermined constant,and X h i is the eigenvector corresponding to l i :

From Eqs.(A.3),(A.2),(A.10)–(A.12)and (A.19),the following particular solutions are obtained

x p 1s ?e s

v ;

x p 1c ?àe s

u ;

y p 1s ?àe s

u ;

y p 1c ?àe s

v ,

y p 0?

r Ag 2k s G Z 2à

r Ag 24EI s

Z 4

;y p x 0?

r Ag 6EI s

Z 3

.(35)

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306

Substituting Eqs.(34)and(35)into Eq.(30)yields

XeZT??GeZT

,(36)

where?GeZT is the matrix of the function of Z with sizee10nt5T?e20nt11Tand undetermined constant vector C??C1C2áááC20nt10 T:Thus the general displacement state variable vectors can be expressed as

X R?XeZ?LT??G L

,(37)

where?G L ??GeZ?LT and

X L?XeZ?0T??G0

,(38)

where?G0 ??GeZ?0T :

The solutions of Eq.(27)can be expressed in Fourier series form as

V xeZ;tT?V x;0eZTt

X n

i?1

V x;iceZTcos i O ttV x;iseZTsin i O t,

V yeZ;tT?V y;0eZTt

X n

i?1

V y;iceZTcos i O ttV y;iseZTsin i O t,

M xeZ;tT?M x;0eZTt

X n

i?1

M x;iceZTcos i O ttM x;iseZTsin i O t,

M yeZ;tT?M y;0eZTt

X n

i?1

M y;iceZTcos i O ttM y;iseZTsin i O t,

TeZ;tT?T0eZTt

X n

i?1

T iceZTcos i O ttT iseZTsin i O t.(39) The mode function vector of the general force FeZTis de?ned as

FeZT??V x;0eZTV x;1ceZTááV x;nceZTV x;1seZTááV x;nseZT

V y;0eZTV y;1ceZTááV y;nceZTV y;1seZTááV y;nseZT

M x;0eZTM x;1ceZTááM x;nceZTM x;1seZTááM x;nseZT

M y;0eZTM y;1ceZTááM y;nceZTM y;1seZTááM y;nseZT

T0eZTT1ceZTááT nceZTT1seZTááT nseZT T.e40TS.-C.Hsieh et al./Journal of Sound and Vibration289(2006)294–333307

By inserting Eq.(39)into (27)and using Eq.(36),the mode function vector of the general force at right F R is given by

F R ??H L C

1 ,(41)

where ?H L denotes a matrix with size e10n t5T?e20n t11T;and the mode function vector of the general force at left F L is given by

F L ??H 0 C

1 ,(42)

where ?H 0 is a matrix with size e10n t5T?e20n t11TCombining Eqs.(37),(38),(41)and (42)yields

S R

1"#??M L C 1

,(43)

where ?M L is a matrix with size e20n t11T?e20n t11Tand

S L

"#??M 0 C 1

(44)

where ?M 0 is a matrix with size e20n t11T?e20n t11T:

Using Eqs.(43)and (44),the transfer matrix ?T s of the shaft can be obtained

S R 1"#??M L C 1 ??M L ?M 0

à1S L

1"#??T s

S L 1

"#.(45)

The transfer matrix ?T s ;with size e20n t11T?e20n t11Tis constructed to relate two sides of a

uniform and symmetric Timoshenko shaft with eccentricity for the relative 0th-order static de?ection,synchronous whirl,and n ?whirl (n th order)in the staticframe.5.Transfer matrix of the linear bearing

In the rotor system,the bearing can be simpli?ed into a linear element.Fig.5illustrates the

force F b x ;F b y ;bending moment M b x ;M b y ;and torque T b

acting on the shaft due to the bearing are given by

F b x F b y

"#?àK xx K xy K yx K yy "#x y "#à

C xx C xy C yx C yy "#_x

_y "#

,M b x

M b y

?à

K y xx K y xy K y yx

K y yy

"#y x y y

"#àC y xx C y xy C y yx

C y yy

"#_y x _y

y "#

,T b ?àK j j àC j _j

.(46)

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308

Hence the equilibrium relations of the force,bending moment and torque acting on the shaft can be expressed as

V R x V R y "#?V L x V L y "#àF b x F b

"#?V L x V L y "#tK xx K xy K yx K yy "#x y "#tC xx C xy C yx C yy "#_x _y "#

,M R x M R y

"#?M L x M L y

"#àM b x M b y

"#?M L x M L y

"#tK y xx K y xy K y yx

K y yy

"#y x y y

"#tC y xx C y xy C y yx

C y yy

"#_y x _y

y "#

,T R ?T L tK j j tC j _j

.(47)

Substituting the Fourier series representation of x ;y ;y x ;y y ;V x ;V y ;M y ;M x and T into Eq.

(47)and equating the coef?cients of the same harmonic term,the transfer matrix of the linear bearing can be obtained as ?T b

S R 1"#??T b

k ?k S L 1"#,(48)where k ?20n t11:The state variable vector S contains the total coef?cient of the Fourier series

from staticvariables to the n th-order harmonicterm.6.Overall transfer matrix of the whole system

Fig.6shows that the typical system has multi-disks,bearings and a ?exible shaft with a torque at the right end.The overall transfer matrix of the rotor system is the relation between the two ends of the shaft,and can be derived by stepwise relationship of the state vectors from the left end to the right end.The multiplication of the matrices of all elements from the left to the right end

Fig.5.Forces,moments,and torques acting on the node of the bearing.

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309

successively yields

#??U

??T s n ?T b j ?T s n à1?T d q ááá?T d p ?T s 2?T b i ?T s 1

e49T

where the subscripts denote station numbers.Because a torque acts on the right end of the shaft,Eq.(49)becomes

R F R 1264375?U 11U 11u 1U

11U 11u 10012

64375X L 01264375,

(50)where

X ??x 0x 1c ááx nc x 1s ááx ns y 0y 1c ááy nc y 1s ááy ns

y x ;0y x ;1c ááy x ;nc y x ;1s ááy x ;ns y y ;0y y ;1c ááy y ;nc y y ;1s ááy y ;ns j 0j 1c ááj nc j 1s ááj ns T ,

F R ??00áá00áá000áá00áá000áá00áá000áá00áá0T 0T 1c ááT nc T 1s ááT ns T ,

0is a zero vector with size e10n t5T?1and u i represents the excitation vector resulting from

unbalanced and unidirectional loads.The state variables of stages 0,X L and n ;X R can be solved using Eq.(50),and the state variables of other stages then are obtained by multiplying transfer matrices from stage 0of the left end stepwise until a speci?c stage is reached.For instance,the state variables of stage 4(see Fig.6)are given by

S R

1"#??T d p ?T s 2?T b i ?T s

1S L 1"#,where S R comprises state variables of stage 4.

Fig.6.A general rotor-bearing system.

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310

7.Numerical examples

To demonstrate the applicability of this approach and show the effects of mass unbalance and external torque on the steady-state vibration,a rotor-bearing system with the symmetricshaft is used,as illustrated in Fig.7.The response amplitude is de?ned as the maximum ?exural displacement,i.e.,

Amplitude ?maximum value of ?????????????????????????

x et T2ty et T2q .The systems supported by the isotropicand anisotropicbearings are analyzed individually.Table 1lists the details of the rotor-bearing systems.

Case 1:Isotropic rotor-bearing system :If no external torque but only the unbalance force is acting on the system,the whirling orbit is forward,synchronous,and right circular (Fig.8).A synchronous lateral mode occurs at 3024rev/min and the amplitude becomes increases at this rotating speed.Fig.9illustrates the response amplitudes excited by the different 1?torques along with the unbalance force,and the orbits of disk 1when torque ?5000cos O t N m :Two peaks other than synchronous resonance clearly appear.With increasing the amount of external torque,the amplitudes of the added resonant peaks increase,and the positions of the resonant peaks become irrelevant to the amount of the external torque.This behavior implies that the amount of external torque cannot alter the rotor nature frequency.The whirling orbits excited by both the unbalance force and external torque are forward but not necessarily synchronous and right circular.The whirling orbit is double-looped at 1490and 2530rev/min,and is roughly circular at 3050rev/min (near the lateral resonant frequency 3024rev/min).

Fig.10shows the response amplitudes of the components for torque ?5000cos O t Nm :The response is composed of synchronous (i.e.,1?)and 2?whirls.Notably,the synchronous component is the same as in Fig.9for T ?0:Accordingly,the unbalance force,with 1?exciting frequency,is known to excite the synchronous component.The torque excites the torsional vibration with torsional exciting frequency and,under the system coupling effect,also stimulates the lateral vibration whose whirly frequency is that of the torque plus or minus the rotating speed.Thus,owing to the coupling effect of the rotor system,the 1?torque excites a 2?lateral mode at 1497.3rev/min,which is a half of the lateral resonant frequency (3024rev/min),and

unit: cm

Fig.7.System for the numerical examples.

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311

a 1?torsional mode at 2516.7rev/min.Fig.11illustrates the orbits of 1?;2?;and synthetic whirls.The orbits of the 1?and 2?components are all forward,so that the synthetic orbit is also forward.At the rotating speed of 3050rev/min (near the lateral resonant frequency),the amplitude of 1?whirl component exceeds that of the 2?whirl,and therefore the resulting synchronous orbit is right circular.

When 1?external torques are replaced by 2?ones,the nonsynchronous resonant peaks are located at 995.3and 1258.0rev/min,but the synchronous resonant peak is still located at 3022.6rev/min (see Fig.12).The positions of the resonant peaks are also irrelevant to the amplitude of the external torque.The whirling orbits at T ?5000cos 2O t Nm are also displayed in Fig.12.The response amplitudes and the whirling orbits of the components comprise 1?and 3?components,and are illustrated in Figs.13and 14,respectively.From Fig.13,a 3?lateral mode occurs at 995.3rev/min (around one-third of the lateral resonant frequency 3022.6rev/min)since,under the system coupling effect,the 2?torque excites the 3?forward and 1?backward whirls.Furthermore,a 2?torsional mode occurs at 1258.0rev/min (half of the torsional resonant frequency 2516.7rev/min)and,because of the system coupling effect,these modes appear on the 1?and 3?whirl components simultaneously.The unbalance force excites the 1?forward

Table 1

Details of the three-disk rotor system The coef?cients of the shaft E 207?109N =m 2G 81?109N =m 2k s 0.68

r 7750kg =m 3e s u 2?10à5m e s v

The coef?cients of the disks m d 13.47kg

I d p

1020?10à4kg m 2I d 512?10à4kg m 2e d u of the disk 11?10à5m e d u of the disk 2and disk 30e d v of the disk 1,disk 2,and disk 30

The coef?cients of the bearings K xx ;K yy

1?107N =m K xy ;K yx (isotropicrotor-bearing system)0

K xy ;K yx (anisotropicrotor-bearing system)5?106N =m K y xx ;K y yy ;K y xy ;K y yx 0

K j of the left bearing 3?104N m =rad K j of the right bearing 0

C xx ;C yy

2?103N s =m C xy ;C yx ;C y xx ;C y yy ;C y xy ;C y yx 0

C j of the left bearing 1N m s =rad C j of the right bearing

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312

Rotating speed (rev/min)

A m p l i t u d e (m m )

Fig.9.Response amplitudes and orbits eT ?5000cos O t Tof disk 1(isotropicrotor-bearing system with 1?

torques).

Rotating speed (rev/min)

A m p l i t u d e (m m )

Fig.8.Synchronous whirling orbits of disk 1(isotropic rotor-bearing system without torque).

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313