A modified transfer matrix method for the coupling lateral and torsional vibrations of symmetric rotor-bearing systems

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JOURNAL OF SOUND AND VIBRATION

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

A modiﬁed 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,

a

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30049, Taiwan, ROC

b_{Department of Mechanical Engineering, Chung Hua University, Taiwan, ROC}

Received 27 January 2004; received in revised form 9 August 2004; accepted 8 February 2005 Available online 28 April 2005

Abstract

This study develops a modiﬁed 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 ðn þ 1Þ and ðn 1Þ 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 ﬁelds, 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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system, computations of critical speeds and steady-state response at synchronous and subcritical resonances become essential for system design, identiﬁcation, diagnosis, and control.

Currently, the ﬁnite element and transfer matrix approaches are becoming two of the most prevalent methods for analyzing rotor-bearing systems. While the ﬁnite 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

Nomenclature

E, G Young’s modulus and shear modulus

A, r cross-sectional area and density of the

shaft Is_{; I}s

p transverse and polar area moment of

inertia of the shaft Id_{; I}d

p transverse and polar mass moment of

inertia of the disk

L length of the shaft element

ks Timoshenko’s shear coefﬁcient

e eccentricity

m mass

Kxx; Kyy direct axial stiffness of the bearing Kxy;Kyxcross-axial stiffness of the bearing Kyxx; Kyyy direct bending stiffness of the

bearing

Kyxy; Kyyx cross-bending stiffness of the bearing

Kj torsional stiffness of the bearing

Cxx; Cyy direct axial damping of the bearing

Cxy; Cyx cross-axial damping of the bearing

Cyxx; Cyyy direct bending damping of the

bearing

Cyxy; Cyyx cross-bending damping of the

bearing

Cj torsional damping of the bearing

O rotating speed

t time

M bending moment in ﬁxed frame

V shear force in ﬁxed frame

T axial torque

Tb torque due to bearing

x, y deﬂections of the geometric center in X

and Y directions

xc; yc deﬂections of the mass center in X and

Y directions

y angular displacements

g shear deformation angles

j angle of twist

S state variable vector

X general displacement state variable

vec-tor

F general force state variable vector

XðZÞ mode function vector of general

dis-placement

FðZÞ mode function vector of general force

½T transfer matrix

½U overall transfer matrix

Ek; Ep kineticenergy and potential energy

W work

w weight

XYZ ﬁxed frame

UVW rotating frame coincident with principal

axes of rotating element

f; y; c Euler’s angles with rotating order in rank

F spin angle of the rotating element about

the axis W

Subscript

c, s associated to cosine, sine terms

x, y components in X ; Y directions

u; v components in U ; V directions

f g; f0_g _{to be referred to as derivatives with}

respect to time and coordinate

Superscript

R, L right, left

s, d, b superscript for shaft element, disk and

bearing

h, p homogeneous solution, particular

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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 speciﬁc 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 finite element models to rotor dynamics has been highly successful. Numerous finite element procedures have attempted to generalize and improve the work of Ruhl and Booker [2]. Nelson and McVaugh [3] employed a finite 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 finite 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 first proposed by Prohl[9]. Subsequently, the effects of damping and stiffness of the fluid film bearing were included by Koenig [10], Guenther and Lovejoy [11]. Lund [12]

achieved significant 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 flexible rotors mounted on flexible 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 ﬁt 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

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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 influence of torsional–lateral coupling on the stability behavior of a simple geared system supported by oil film 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 finite element model to analyze the torsional–flexural 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 influence of incremental bending stiffness because of axial torque was insignificant. Mohiuddin and Khulief [29] presented a reduced modal form of the rotor-bearing system to find 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 ﬂexible 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 modified TMM for the coupling lateral and torsional vibrations of symmetricrotor-bearing systems. Euler’s angles are used to describe 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 flexible shaft, rigid disks, and linear bearings with respect to the fixed 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.

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2. Kinematics of rotating element

The orientation of the rotating element, in three-dimensional motion, can be completely described using Euler’s angles defined via three successive rotations to specify the relations between the principal axes of the rotating frame and the fixed frame. As shown inFig. 1(a), the rotating sequence for defining Euler’s angles is explained via the following steps: (1) rotate the initial system, parallel to fixed coordinates, into a deflected mode by an angle f about the Z-axis, (2) rotate the intermediate axes ðXYZÞ0by an angle y about the X0-axis (the so-called nodal axis) to another intermediate axes ðUVW Þ0; (3) rotate intermediate axes ðUVW Þ0 by an angle c about the W0-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 deflected in position and orientation as illustrated in Fig. 1(b), the inclined angle y of orientation is measured counterclockwise from the fixed axis Z to the spin axis W of the rotating element. In the projection description, the deflected angles (or angular displacements) are the projections of the inclined angle y; thus yx ¼y cos f and yy¼y sin f:

Additionally, the spin angle about the axis W is obtained as F ¼ f þ c from the geometric conﬁguration 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

ou ov ow 2 6 6 4 3 7 7 5 ¼ cos c sin c 0 sin c cos c 0 0 0 1 2 6 6 4 3 7 7 5 1 0 0 0 c os y sin y 0 sin y cos y 2 6 6 4 3 7 7 5 cos f sin f 0 sin f cos f 0 0 0 1 2 6 6 4 3 7 7 5 0 0 _ f 2 6 6 4 3 7 7 5 þ cos c sin c 0 sin c cos c 0 0 0 1 2 6 6 4 3 7 7 5 1 0 0 0 c os y sin y 0 sin y cos y 2 6 6 4 3 7 7 5 _y 0 0 2 6 6 4 3 7 7 5 þ cos c sin c 0 sin c cos c 0 0 0 1 2 6 6 4 3 7 7 5 0 0 _ c 2 6 6 4 3 7 7 5, Z X Y X' Y' Z' U' W' V' W U V φ θ ψ φ θ ψ φ ψ θ Z X Y 0 W X Y Z y x X' φ θ θ θ θ θ θ y x (a) _(b)

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that is,

ou¼ _y cos c þ _f sin y sin c,

ov¼ _y sin c þ _f sin y cos c,

ow¼ _c þ _f cos y. ð1Þ

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

given by

Ek ¼1₂mð _x2cþ_y2cÞ þ12ðIuo 2

uþIvo2vþIpo2wÞ. (2)

Notably the kineticenergy Ek 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 ¼ Iu¼Iv; yx ¼y cos f; yy ¼y sin f; F ¼ f þ c; and _F ¼ _f þ _c into

Eq. (2), the kineticenergy of the symmetricrotating element in the ﬁxed frame is obtained as Ek¼1₂½mð _x2cþ_y2cÞ þIpF_ 2 þIpFð_y_ x yy _yy yxÞ þI ð_y 2 xþ _y 2 yÞ. (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 constant 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

xc y_c " # ¼ x y " # þ ed x ed_y " # (4) X Y 0 V U G C e _Ω t+ R

r

whirling orbit d C : mass center G : geometric center e : eccentricityd edu d v_e ϕ

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and ed_x ed_y " # ¼ cosðOt þ jÞ sinðOt þ jÞ sinðOt þ jÞ cosðOt þ jÞ " # ed u ed_v " # . (5)

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

xc ¼x e_ dvðO þ _jÞ cosðOt þ jÞ eduðO þ _jÞ sinðOt þ jÞ, (6)

_y_c¼ _y þ ed_uðO þ _jÞ cosðOt þ jÞ ed_vðO þ _jÞ sinðOt þ jÞ. (7) Inserting Eqs. (6)–(7) and F ¼ Ot þ j into Eq. (3), the kineticenergy of the symmetricdisk is obtained by

Ek ¼1₂md½x_22 _xedvðO þ _jÞ cosðOt þ jÞ 2 _xe d

uðO þ _jÞ sinðOt þ jÞ þ _y 2

þ 2 _yed_uðO þ _jÞ cosðOt þ jÞ 2 _yed_vðO þ _jÞ sinðOt þ jÞ þ ðO þ _jÞ2ðedÞ2 þ 1₂Id_pðO þ _jÞ2þ1₂Id_pðO þ _jÞð_yxyy _yyyxÞ þ1₂Idð_y

2 xþ _y

2 yÞ.

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 ¼ wdy þ VR_xx þ MR_yyyþVRyy þ MRxyxþTRj ðVLxx þ MLyyyþVLyy þ MLxyxþTLjÞ.

Using Hamilton’s principle d

Z t2

t1

ðEkEpþW Þ dt ¼ 0 (8)

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

VR_x VL_x þmd½x þ €€ jed_v cosðOt þ jÞ þ €jed_u sinðOt þ jÞ ed_vðO þ _jÞ2 sinðOt þ jÞ

þ ed_uðO þ _jÞ2 cosðOt þ jÞ ¼ 0, ð9Þ Z X Y x TR TL MyR VR_x MLy VL_x θy ϕ θy=θyLθ R = y x L₌ R = ϕ ϕ ϕ = L= R x x Y Z X TR y Mx R θx x ML R V VL TL ϕ y ϕ θ θ ϕ= x = y = R L y θ L₌ϕ R = x x = L R (a) _(b) y y

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VR_y VL_y þmd½€y €jed_u cosðOt þ jÞ þ €jed_v sinðOt þ jÞ þ ed_uðO þ _jÞ2sinðOt þ jÞ

þ ed_vðO þ _jÞ2 cosðOt þ jÞ wd ¼0, ð10Þ where

md½je€ d_v cosðOt þ jÞ þ €jed_u sinðOt þ jÞ ed_vðO þ _jÞ2sinðOt þ jÞ þ ed_uðO þ _jÞ2cosðOt þ jÞ

and

md½je€ d_u cosðOt þ jÞ þ €jed_v sinðOt þ jÞ þ ed_uðO þ _jÞ2sinðOt þ jÞ þ ed_vðO þ _jÞ2cosðOt þ jÞ

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

The bending moment equilibrium equations in the ﬁxed coordinates are

MR_x ML_x Id€yx1₂Idpjy€ yIdpðO þ _jÞ_yy¼0, (11)

MR_y ML_y Id€yyþ1₂Id_pjy€ xþId_pðO þ _jÞ_yx¼0, (12)

where1 2I

d

pjy€ y and 1₂Idpjy€ x are the moments coupled with the twist acceleration ð €jÞ; IdpðO þ _jÞ_yy

and Id_pðO þ _jÞ_yx the gyroscopic moments coupled with the twist velocity ð _jÞ:

The torque equilibrium equations in the ﬁxed coordinates is TRTLId_pj € 1

2I d

p€yxyyþ1₂Idp€yyyxþmd½xe€ dv cosðOt þ jÞ þ €xedu sinðOt þ jÞ

€yed_u cosðOt þ jÞ þ €yed_v sinðOt þ jÞ ðedÞ2j ¼ 0,€ ð13Þ where1

2I d

p€yxyy and1₂Idp€yyyx are the torques coupled with bending angle and angular acceleration,

and

md½xe€ d_v cosðOt þ jÞ þ €xed_u sinðOt þ jÞ €yed_u cosðOt þ jÞ þ €yed_v sinðOt þ jÞ ðedÞ2j€ is the torque induced by the unbalance force.

Eqs. (9)–(13) can be simpliﬁed 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 terms1₂Id_p€yxyy and1₂Idp€yyyx 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

ksx þ md½x þ €€ jed_v cosðOt þ jÞ þ €jed_u sinðOt þ jÞ ed_vðO þ _jÞ2sinðOt þ jÞ þ ed_uðO þ _jÞ2 cos Ot þ jð Þ ¼0,

ksy þ md½€y €jed_u cosðOt þ jÞ þ €jed_v sinðOt þ jÞ þ ed_uðO þ _jÞ2sinðOt þ jÞ þ ed_vðO þ _jÞ2cosðOt þ jÞ wd ¼0,

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ks_jj Id_pj þ m€ d½xe€ d_v cosðOt þ jÞ þ €xed_u sinðOt þ jÞ €yed_u cosðOt þ jÞ þ €yed_v sinðOt þ jÞ ðedÞ2j ¼ 0,€

where ks denotes the shaft lateral stiffness and ks_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

xR¼xL; yR¼yL; yR_x ¼yL_x; yR_y ¼y_yL; jR¼jL. (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 first expanded into Fourier series with unknown coefficients. 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 coefficients of the assumed solution. The harmonicbalance 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

xðtÞ ¼ x0þ

Xn i¼1

xic cos iOt þ xis sin iOt,

yðtÞ ¼ y₀þX

n

i¼1

y_ic cos iOt þ y_is sin iOt,

yxðtÞ ¼ yx;0þ

Xn i¼1

yx;ic cos iOt þ yx;issin iOt,

yyðtÞ ¼ yy;0þ

Xn i¼1

yy;ic cos iOt þ yy;is sin iOt,

jðtÞ ¼ j₀þX

n

i¼1

j_ic cos iOt þ j_is sin iOt. (15) Other variables can be similarly expressed as

VxðtÞ ¼ Vx;0þ

Xn i¼1

Vx;ic cos iOt þ Vx;is sin iOt,

VyðtÞ ¼ Vy;0þ

Xn i¼1

Vy;ic cos iOt þ Vy;is sin iOt,

MxðtÞ ¼Mx;0þ

Xn i¼1

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MyðtÞ ¼My;0þ

Xn i¼1

My;ic cos iOt þ My;is sin iOt,

TðtÞ ¼ T0þ

Xn i¼1

Tic cos iOt þ Tis sin iOt. (16)

Using the relations

cosðOt þ jÞ ¼ cos Ot cos j sin Ot sin j cos Ot j sin Ot, sinðOt þ jÞ ¼ sin Ot cos j þ cos Ot sin j sin Ot þ j cos Ot

substituting Eqs. (15) and (16) into Eqs. (9)–(14), ignoring the nonlinear terms, and equating the coefﬁcients of the same harmonic term provides the transfer matrix equation of the disk for static, synchronous whirl and nonsynchronous whirls in the static frame:

SR 1 " # ¼ ½Td_kk S L 1 " # , (17)

where k ¼ 20n þ 11 and the state variable vector S is denoted as S ¼ X F , (18) where X ¼ ½x0 x1c xnc x1s xns y0 y1c ync y1s yns

yx;0 yx;1c yx;nc yx;1s yx;ns yy;0 yy;1c yy;nc yy;1s yy;ns j0 j1c jnc j1s jnsT

and

F ¼ ½Vx;0 Vx;1c Vx;nc Vx;1s Vx;ns Vy;0 Vy;1c Vy;nc Vy;1s Vy;ns

Mx;0 Mx;1c Mx;nc Mx;1s Mx;ns My;0 My;1c My;nc My;1s My;nsT0 T1c Tnc T1s TnsT.

4. Transfer matrix of Timoshenko shaft

As shown in Fig. 4, the ﬁnite 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 elements. Using a similar procedure to that illustrated in Section 3, the kineticenergy of the symmetricshaft element expressed in ﬁxed coordinates is

Ek ¼

1 2r

Z L 0

fA½ _x22 _xes_vðO þ _jÞ cosðOt þ jÞ 2 _xes_uðO þ _jÞ sinðOt þ jÞ

þ _y2þ2 _yes_uðO þ _jÞ cosðOt þ jÞ 2 _yes_vðO þ _jÞ sinðOt þ jÞ þ ðO þ _jÞ2ðesÞ2 þ Is_pðO þ _jÞ2þIs_pðO þ _jÞð_yxyy _yyyxÞ þIsð _y

2 xþ _y

2

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

Ep¼

1 2

Z L 0

½EIsðy0_xÞ2þEIsðy0_yÞ2þksGAðg2xþg2yÞ þGIspðj 0

Þ2dZ. (20) The work done by the external force (seeFig. 4) is

W ¼ Z L

0

rAgy dZ þ VR_xx þ MR_yyy

þ VR_yy þ MR_xyxþTRj ðVLxx þ MyLyyþVLyy þ MLxyxþTLjÞ. ð21Þ

Using Hamilton’s principle and assuming small twist angle displacement, this study obtains the force equilibrium equations of the shaft in the ﬁxed coordinates:

rA €x þ rA½ €jes_v cosðOt þ jÞ þ €jes_usinðOt þ jÞ ðO þ _jÞ2es_v sinðOt þ jÞ

þ ðO þ _jÞ2es_u cosðOt þ jÞ þ ksGAðx00y0yÞ ¼0, ð22Þ

rA €y þ rA½ €jes_u cosðOt þ jÞ þ €jes_v sinðOt þ jÞ þ ðO þ _jÞ2es_u sinðOt þ jÞ þ ðO þ _jÞ2es_v cosðOt þ jÞ þ ksGAðy0xþy

00

Þ rAg ¼ 0. ð23Þ From above equations, the twist angle ðjÞ and its derivatives can be found to emerge from the unbalance forces:

rA½ €jes_v cosðOt þ jÞ þ €jes_u sinðOt þ jÞ ðO þ _jÞ2es_v sinðOt þ jÞ þ ðO þ _jÞ2es_u cosðOt þ jÞ

and

rA½ €jes_u cosðOt þ jÞ þ €jes_v sinðOt þ jÞ þ ðO þ _jÞ2es_u sinðOt þ jÞ þ ðO þ _jÞ2es_v cosðOt þ jÞ

and inﬂuence the level of the unbalance forces.

Z X Y γy γy z x xR Z=0 Z=L L x R T MR y VxR TL My L V_xL R R ϕ R Z Y X θx y z -γ_x TR y R V x MR R y L y TL L M_x Vy L Z=0 Z=L R R ϕ R (a) (b)

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The bending moment equilibrium equations in the ﬁxed coordinates are

rIs€yxþ1₂rIpsjy€ yþrIspðO þ _jÞ_yyEIsy00xþksGAðyxþy0Þ ¼0, (24)

rIs€yy1₂rIpsjy€ xrIspðO þ _jÞ_yxEIsy00yþksGAðyyx0Þ ¼0, (25)

where1₂rIs_pjy€ yand₂1rIspjy€ x denote the moments coupled with the twist acceleration ( €j), rIspðO þ

_

jÞ_yy and rIspðO þ _jÞ_yx represent the gyroscopic moments coupled with the twist velocity ( _j).

The torque equilibrium equation in the ﬁxed coordinates is

rIs_pj þ€ 1₂rIs_p€yxyy1₂rIsp€yyyxþrA½ €xesv cosðOt þ jÞ €xesu sinðOt þ jÞ

þ €yes_ucosðOt þ jÞ €yes_v sinðOt þ jÞ þ ðesÞ2j GI€ s_pj00¼0, ð26Þ where1

2rI s

p€yxyyand1₂rIsp€yyyx are the torques coupled by bending angle and angular acceleration,

and

rA½ €xes_v cosðOt þ jÞ €xes_u sinðOt þ jÞ þ €ye_us cosðOt þ jÞ €yes_v sinðOt þ jÞ þ ðesÞ2j€ is the torque induced by unbalance force.

The natural boundary conditions are

VR_x þ ½ksGAðyyx0ÞZ¼L¼0; VLx þ ½ksGAðyyx0ÞZ¼0¼0,

VR_y þ ½ksGAðyxþy0ÞZ¼L¼0; VLy þ ½ksGAðyxþy0ÞZ¼0 ¼0,

MR_x þ ½EIsy0_x_Z¼L¼0; ML_x þ ½EIsy0_x_Z¼0 ¼0, MR_y þ ½EIsy0_y_Z¼L; ML_y þ ½EIsy0_y_Z¼0 ¼0,

TRþ ½GIs_pj0_Z¼L¼0; TLþ ½GIs_pj0_Z¼0¼0. (27) The steady-state solution of Eqs. (22)–(26) can be expressed in Fourier series form as

xðZ; tÞ ¼ x0ðZÞ þ

Xn i¼1

xicðZÞ cos iOt þ xisðZÞ sin iOt,

yðZ; tÞ ¼ y₀ðZÞ þX

n

i¼1

y_icðZÞ cos iOt þ y_isðZÞ sin iOt,

yxðZ; tÞ ¼ yx;0ðZÞ þ

Xn i¼1

yx;icðZÞ cos iOt þ yx;isðZÞ sin iOt,

yyðZ; tÞ ¼ yy;0ðZÞ þ

Xn i¼1

yy;icðZÞ cos iOt þ yy;isðZÞ sin iOt,

jðZ; tÞ ¼ j₀ðZÞ þX

n i¼1

(13)

where x0ðZÞ; y0ðZÞ; yx;0ðZÞ; yy;0ðZÞ; j0ðzÞ; xicðZÞ; xisðZÞ; yicðZÞ; yisðZÞ; yx;icðZÞ; yx;isðZÞ; yy;icðZÞ;

yy;isðZÞ; jicðzÞ and jisðzÞ are 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 XðZÞ; namely

XðZÞ ¼ ½x0ðZÞx1cðZÞ xncðZÞx1sðZÞ xnsðZÞy0ðZÞy1cðZÞ yncðZÞy1sðZÞ ynsðZÞ

yx;0ðZÞyx;1cðZÞ yx;ncðZÞyx;1sðZÞ yx;nsðZÞyy;0ðZÞyy;1cðZÞ yy;ncðZÞyy;1sðZÞ yy;nsðZÞ

j₀ðZÞj_1cðZÞ j_ncðZÞj_1sðZÞ j_nsðZÞT. ð29Þ Substituting Eq. (28) into Eqs. (22)–(26), ignoring the nonlinear terms, and equating the coefﬁcients of the same harmonic term produces 33 nonhomogeneous 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

XðZÞ ¼ XðZÞhþXðZÞp. (30) Assume the homogeneous solution XðZÞh in the forms

XðZÞh¼XhelZ, (31) where the arbitrary constant vector Xh¼ ½xh

0 xh1c xhnc xh1s xhns yh0 yh1c yhnc yh1s yhns yhx;0 yhx;1c

yh_x;nc yh_x;1s yh_x;ns yh_y;0 yh_y;1c yh_y;nc yh_y;1s yh_y;ns jh₀ jh_1c j_nch jh_1s jh_nsT 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: ðl2E2þlE1þE0ÞXh, (32)

where E2; E1and E0are matrices with size ð10n þ 5Þ ð10n þ 5Þ and are listed in Appendix B. Eq.

(32) can be rewritten as the generalized eigen-problem form l 0 E2 E2 E1 " # kk E2 0 0 E0 " # kk ( ) lXh Xh " # k1 ¼ 0 0 k1 , (33) where k ¼ 20n þ 10:

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

XðZÞh¼ X

20nþ10

i¼1

CiXhieliZ, (34)

where Ci is an undetermined constant, and Xhi is the eigenvector corresponding to li:

From Eqs. (A.3), (A.2), (A.10)–(A.12) and (A.19), the following particular solutions are obtained xp_1s¼es_v; xp_1c¼ es_u; yp_1s¼ es_u; yp_1c ¼ es_v, yp₀¼ rAg 2ksG Z2 rAg 24EIs Z 4_; _yp x0¼ rAg 6EIs Z 3_. ₍₃₅₎

(14)

Substituting Eqs. (34) and (35) into Eq. (30) yields XðZÞ ¼ ½GðZÞ C

1

, (36)

where ½GðZÞ is the matrix of the function of Z with size ð10n þ 5Þ ð20n þ 11Þ and undetermined constant vector C ¼ ½C1C2 C20nþ10T: Thus the general displacement state variable vectors can

be expressed as XR¼XðZ ¼ LÞ ¼ ½GL C 1 , (37) where ½GL ¼ ½GðZ ¼ LÞ and XL¼XðZ ¼ 0Þ ¼ ½G0 C 1 , (38) where ½G0 ¼ ½GðZ ¼ 0Þ:

The solutions of Eq. (27) can be expressed in Fourier series form as VxðZ; tÞ ¼ Vx;0ðZÞ þ

Xn i¼1

Vx;icðZÞ cos iOt þ Vx;isðZÞ sin iOt,

VyðZ; tÞ ¼ Vy;0ðZÞ þ

Xn i¼1

Vy;icðZÞ cos iOt þ Vy;isðZÞ sin iOt,

MxðZ; tÞ ¼ Mx;0ðZÞ þ

Xn i¼1

Mx;icðZÞ cos iOt þ Mx;isðZÞ sin iOt,

MyðZ; tÞ ¼ My;0ðZÞ þ

Xn i¼1

My;icðZÞ cos iOt þ My;isðZÞ sin iOt,

T ðZ; tÞ ¼ T0ðZÞ þ

Xn i¼1

TicðZÞ cos iOt þ TisðZÞ sin iOt. (39)

The mode function vector of the general force FðZÞ is deﬁned as FðZÞ ¼ ½Vx;0ðZÞVx;1cðZÞ Vx;ncðZÞVx;1sðZÞ Vx;nsðZÞ

Vy;0ðZÞVy;1cðZÞ Vy;ncðZÞVy;1sðZÞ Vy;nsðZÞ

Mx;0ðZÞMx;1cðZÞ Mx;ncðZÞMx;1sðZÞ Mx;nsðZÞ

My;0ðZÞMy;1cðZÞ My;ncðZÞMy;1sðZÞ My;nsðZÞ

(15)

By inserting Eq. (39) into (27) and using Eq. (36), the mode function vector of the general force at right FR is given by FR¼ ½HL C 1 , (41)

where ½HLdenotes a matrix with size ð10n þ 5Þ ð20n þ 11Þ; and the mode function vector of the

general force at left FL is given by

FL¼ ½H0

C 1

, (42)

where ½H0 is a matrix with size ð10n þ 5Þ ð20n þ 11Þ

Combining Eqs. (37), (38), (41) and (42) yields SR 1 " # ¼ ½ML C 1 , (43)

where ½ML is a matrix with size ð20n þ 11Þ ð20n þ 11Þ and

SL 1 " # ¼ ½M0 C 1 , (44)

where ½M0 is a matrix with size ð20n þ 11Þ ð20n þ 11Þ:

Using Eqs. (43) and (44), the transfer matrix ½Ts of the shaft can be obtained SR 1 " # ¼ ½ML C 1 ¼ ½ML½M01 SL 1 " # ¼ ½Ts S L 1 " # . (45)

The transfer matrix ½Ts; with size ð20n þ 11Þ ð20n þ 11Þ is constructed to relate two sides of a uniform and symmetric Timoshenko shaft with eccentricity for the relative 0th-order static deﬂection, synchronous whirl, and n whirl (nth order) in the staticframe.

5. Transfer matrix of the linear bearing

In the rotor system, the bearing can be simpliﬁed into a linear element. Fig. 5 illustrates the force Fb_x; Fb_y; bending moment Mb_x; Mb_y; and torque Tbacting on the shaft due to the bearing are given by Fb_x Fb_y " # ¼ Kxx Kxy Kyx Kyy " # x y " # Cxx Cxy Cyx Cyy " # _ x _y " # , Mb_x Mb_y " # ¼ Kyxx Kyxy K_yyx K_yyy " # yx yy " # Cyxx Cyxy C_yyx C_yyy " # _yx _yy " # , Tb¼ Kjj Cjj._ (46)

(16)

Hence the equilibrium relations of the force, bending moment and torque acting on the shaft can be expressed as VR_x VR_y " # ¼ VL_x VL_y " # Fb_x Fb_y " # ¼ VL_x VL_y " # þ Kxx Kxy Kyx Kyy " # x y " # þ Cxx Cxy Cyx Cyy " # _ x _y " # , MR_x MR_y " # ¼ ML_x ML_y " # Mb_x Mb_y " # ¼ ML_x ML_y " # þ Kyxx Kyxy Kyyx Kyyy " # yx yy " # þ Cyxx Cyxy Cyyx Cyyy " # _yx _yy " # , TR ¼TLþKjj þ Cjj._ (47)

Substituting the Fourier series representation of x; y; yx; yy; Vx; Vy; My; Mx and T into Eq.

(47) and equating the coefﬁcients of the same harmonic term, the transfer matrix of the linear bearing can be obtained as ½Tb

SR 1 " # ¼ ½Tb_kk S L 1 " # , (48)

where k ¼ 20n þ 11: The state variable vector S contains the total coefﬁcient of the Fourier series from staticvariables to the nth-order harmonicterm.

6. Overall transfer matrix of the whole system

Fig. 6shows that the typical system has multi-disks, bearings and a ﬂexible 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

Z X Y x TR TL MRy VR x MLy VL x θy ϕ Fxb My b x L₌ R =x x = = ϕ=ϕ θy= θy L L ϕR θRy Y Z X TR y MxR θx x ML R V VL TL ϕ y ϕ θ θ ϕ= x= y= R L y θ Lϕ R = = x x = L R Mb x Fy b (a) (b)

(17)

successively yields SR 1 " # ¼ ½U S L 1 " # ¼ ½Ts_n½Tb_j½Ts_n1½Td_q ½Td_p½Ts₂½Tb_i½Ts₁ S L 1 " # , ð49Þ

where the subscripts denote station numbers. Because a torque acts on the right end of the shaft, Eq. (49) becomes XR FR 1 2 6 4 3 7 5 ¼ U11 U11 u1 U11 U11 u1 0 0 1 2 6 4 3 7 5 XL 0 1 2 6 4 3 7 5, (50) where X ¼ ½x0 x1c xnc x1s xns y0 y1c ync y1s yns

yx;0 yx;1c yx;nc yx;1s yx;ns yy;0 yy;1c yy;nc yy;1s yy;ns

j₀ j_1c j_ncj_1s j_nsT,

FR¼ ½0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0T0T1c TncT1s TnsT,

0 is a zero vector with size ð10n þ 5Þ 1 and ui represents the excitation vector resulting from

unbalanced and unidirectional loads. The state variables of stages 0, XLand n; XR can be solved using Eq. (50), and the state variables of other stages then are obtained by multiplying transfer matrices from stage 0 of the left end stepwise until a speciﬁc stage is reached. For instance, the state variables of stage 4 (seeFig. 6) are given by

SR 1 " # ¼ ½Td_p½Ts₂½Tb_i½Ts₁ S L 1 " # , where SR comprises state variables of stage 4.

0 1 2 3 4 n-3 n-2 n-1 n

section 1 section 2 section n

K C K C

i j

disk p disk q

T

(18)

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 deﬁned as the maximum ﬂexural displacement, i.e.,

Amplitude ¼ maximum value of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xðtÞ2þyðtÞ2 q

.

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 3024 rev/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 1 when torque ¼ 5000 cos Ot 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 1490 and 2530 rev/min, and is roughly circular at 3050 rev/min (near the lateral resonant frequency 3024 rev/min).

Fig. 10 shows the response amplitudes of the components for torque ¼ 5000 cos Ot Nm: The response is composed of synchronous (i.e., 1) and 2 whirls. Notably, the synchronous component is the same as inFig. 9for T ¼ 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 1497.3 rev/min, which is a half of the lateral resonant frequency (3024 rev/min), and

K C K C

13.1 18 16 16 18 13.1

disk 1 disk 2 disk 3

unit: cm

4

T

(19)

a 1 torsional mode at 2516.7 rev/min. Fig. 11 illustrates 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 3050 rev/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 995.3 and 1258.0 rev/min, but the synchronous resonant peak is still located at 3022.6 rev/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 ¼ 5000 cos 2Ot Nm are also displayed inFig. 12. The response amplitudes and the whirling orbits of the components comprise 1 and 3 components, and are illustrated inFigs. 13 and 14, respectively. From Fig. 13, a 3 lateral mode occurs at 995.3 rev/min (around one-third of the lateral resonant frequency 3022.6 rev/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 1258.0 rev/min (half of the torsional resonant frequency 2516.7 rev/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 coefﬁcients of the shaft

E 207 109_N=m2 G 81 109N=m2 ks 0.68 r 7750 kg=m3 es u 2 105m es v 0

The coefﬁcients of the disks

md _{13.47 kg} Id_p 1020 104kg m2 Id _{512 10}4_{kg m}2 ed u of the disk 1 1 105m ed

u of the disk 2 and disk 3 0

ed

v of the disk 1, disk 2, and disk 3 0

The coefﬁcients of the bearings

Kxx; Kyy 1 107N=m

Kxy; Kyx(isotropicrotor-bearing system) 0

Kxy;Kyx(anisotropicrotor-bearing system) 5 106N=m

Kyxx; Kyyy; Kyxy; Kyyx 0

Kjof the left bearing 3 104N m=rad

Kjof the right bearing 0

Cxx; Cyy 2 103N s=m

Cxy; Cyx; Cyxx; Cyyy; Cyxy; Cyyx 0

Cjof the left bearing 1 N m s=rad

(20)

1490.0 rev/min 2530.0 rev/min 3050.0 rev/min 2640.0 rev/min Shaft rotation 1497.3 2516.7 3024.0 T=0 T=100cos Ωt T=1000cosΩt T=5000cosΩt

Rotating speed (rev/min)

Amplitude (mm)

Fig. 9. Response amplitudes and orbits ðT ¼ 5000 cos OtÞ of disk 1 (isotropicrotor-bearing system with 1 torques). Shaft

rotation 3024.0 T=0

Y X Y X Y X 0.015 0.015 orbit unit: mm 0.023 0.023 0.35 0.35 Amplitude (mm)

(21)

component. The 1 forward (excited by the unbalance force) and backward components (excited by the torque) may result in the 1 ellipse whirl orbit. If the 1 forward com-ponent is exceeding the backward comcom-ponent, the 1 whirl orbit is forward, and vice versa. Finally, the 3 whirl orbit, which is excited by the torque alone, is forward and right circular (seeFig. 14).

Fig. 15 shows the response amplitude and whirling orbits at T ¼ 5000 cos 3Ot Nm: The response amplitudes and whirling orbits of the components are illustrated in Figs. 16 and 17

respectively.Fig. 16 indicates that the response components involve the 1; 2; and 4 whirl. The response components of 2 and 4 whirls are excited by the external torque. A 2 lateral mode appears at 1468.6 rev/min, and a 4 lateral mode appears at 745.3 rev/min. 3 torsional modes appear at 838.6 and 3712 rev/min and, due to the system coupling effect, these modes appear on the 2 and 4 whirl components simultaneously. The unbalance force excites the forward 1 whirl and the torque excites the backward 2 and forward 4 whirls (seeFig. 17). At rotating speed of 3698 rev/min (near the second torsional mode), the amounts of the components are comparable, and thus the synthetic whirling orbit becomes complex.

As shown inFigs. 18–20, respectively, when the system is simultaneously excited by 1; 2; and 3 external torque (T ¼ Taðcos Ot þ cos 2Ot þ cos 3OtÞ Nm), the response amplitude, response

amplitude of components, and whirl orbits are the sum of those when the system is excited by just one of these external torques.Table 2lists the relations between the critical speeds and exciting frequency.

Case 2: Anisotropic rotor-bearing system:Fig. 21 shows the response amplitudes and orbits at T ¼ 0 and T ¼ 1000 cos Ot Nm when neglecting the eccentricity of the shaft. The response and

1X whirl

2X whirl 1x and 2X whirls

1497.3

2516.7

3024.0 T=5000cosΩt

Amplitude (mm)

(22)

Rotating speed = 1490.0 rev/min

1X whirl 2X whirl 1X and 2X whirls

X Y unit: mm 0.0018 X Y 0.05 0.05 -0.05 -0.05 0.05 0.05 -0.05 -0.05 X Y 0.0018

unit: mm X Y X X Y Y 0.06 0.06 0.07 0.07 -0.06 -0.06 -0.07 -0.07 0.012 0.012

unit: mm X Y X X Y Y 0.017 0.025 0.025 0.017 -0.017 -0.017 -0.025 -0.025 0.0067 0.0067

unit: mm X Y X X Y Y 0.22 0.25 0.22 0.001 0.25 -0.22-- -0.25 -0.22 -0.25 0.001

(23)

Amplitude (mm)

3X whirl 1X whirl

1X and 3X whirls

995.3

1258.0 3022.6

T=5000cos2 Ωt

Fig. 13. Response amplitudes of the components of disk 1 (isotropic rotor-bearing system, T ¼ 5000 cos 2Ot).

Amplitude (mm) Shaft rotation 925.0 rev/min 1300.0 rev/min 2990.0 rev/min 995.3 1258.0 3022.6 T=0 T=100cos2Ω t T=1000cos2Ω t T=5000cos2Ω t

Fig. 12. Response amplitudes and orbits ðT ¼ 5000 cos 2OtÞ of disk 1 (isotropicrotor-bearing system with 2 torques).

(24)

orbits at T ¼ 0 are denoted by solid lines agree with our previous work of Lee et al. [19]. The analytical results reveals that, due to anisotropic bearing, two synchronous ð1Þ lateral modes (2683.3 and 3082.6 rev/min) are excited and the orbits are always elliptical and synchronous for different rotating speeds. The synchronous whirling orbit reverses and becomes backward at speeds between the split critical speeds. The response and whirling orbit at T ¼ 1000 cos Ot Nm are denoted by dashed lines. Besides the synchronous lateral modes, the external torque excites

unit: mm X Y _{X X} Y Y 0.008 0.008 6.0E-4 -0.008 0.008 -0.008 -0.008 0.008 -0.008 6.0E-4

unit: mm X Y _{X X} Y _Y 0.02 0.021 0.02 0.021 -0.02 -0.021 0.0025 -0.02 -0.021 0.0025

unit: mm X Y X X Y Y 0.19 0.2 0.19 8.0E-4 0.2 -0.19 -0.2 -0.19 -0.2 8.0E-4

(25)

1X whirl

4X whirl

2X whirl 1X, 2X and 4X whirls

T=5000cos3Ωt

745.3 838.6 1468.6 3023.3 3712.0 Amplitude (mm)

Fig. 16. Response amplitudes of the components of disk 1 (isotropic rotor-bearing system, T ¼ 5000 cos 3Ot).

Amplitude (mm) 722.0 rev/min 1473.0 rev/min 2968.0 rev/min 3698.0 rev/min 745.3 838.6 1468.6 3023.3 3712.0 Shaft rotation T=5000cos3Ωt

(26)

nonsynchronous ð2Þ lateral modes (1344 and 1539.3 rev/min) and the torsional mode (2516 rev/ min), which make the orbits no longer elliptical and synchronous. Consequently, the coupling effect is disregarded and the nonsynchronous lateral modes are left out when the lateral and torsional vibration are analyzed separately.

1X whirl 2X whirl 4X whirl 1X, 2X and 4X whirls unit: mm X Y X X X Y Y Y 0.033 0.033 3.3E-4 7.3E-4 0.033 0.033 -0.033 -0.033 -0.033 -0.033 3.3E-4 7.3E-4

1X whirl 2X whirl 4X whirl 1X, 2X and 4X whirls

X Y X _X X Y Y Y unit: mm 0.021 0.025 0.0017 0.021 2.2E-4 0.025 -0.021 -0.021 -0.025 -0.025 0.0017 2.2E-4

1X whirl 2X whirl 4X whirl 1X, 2X and 4X whirls unit: mm X Y Y Y Y X X X 0.12 0.12 0.12 3.5E-5 9.0E-4 0.12 -0.12 -0.12 -0.12 -0.12 3.5E-5 9.0E-4

1X whirl 2X whirl 4X whirl 1X, 2X and 4X whirls unit: mm X X Y Y 0.018 0.04 0.012 0.012 0.018 0.04 -0.018 -0.018 -0.04 -0.04 0.012 -0.012 -0.012 Y X 0.012 -0.012 -0.012 Y X

(27)

1X whirl 4X whirl 3X whirl 2X whirl 1X, 2X, 3X and 4X whirls Ta=5000

Amplitude (mm) 745.3 838.6 1258.0 995.3 1496.0 2516.0 3024.0 3713.3

Fig. 19. Response amplitudes of the components of disk 1 (isotropic rotor-bearing system, Ta¼5000).

745.3 838.6 995.3 1258.0 1496.0 2516.0 3024.0 3713.3 Ta=5000 Ta=1000 Ta=100 Ta=10 Ta=0

Amplitude (mm)

(28)

1X whirl 2X whirl 3X whirl 4X whirl 1X, 2X, 3X and 4X whirls Rotating speed = 730.0 rev/min

X Y X X X X Y Y Y Y 0.052 0.054 3.5E-4 0.052 0.054 -0.052 -0.052 -0.054 -0.054 unit: mm 3.5E-4 0.001 0.001 0.001 0.001

1X whirl 2X whirl 3X whirl 4X whirl 1X, 2X, 3X and 4X whirls

X Y X X X X Y Y Y Y unit: mm 0.015 0.022 5.0E-4 0.003 0.015 0.022 -0.015 -0.015 -0.022 -0.022 5.0E-4 0.003 0.0045 0.0045

Rotating speed = 2505.0 rev/min unit: mm X Y X X X X Y Y Y Y 0.065 0.075 0.011 0.065 1.2E-4 0.0017 0.075 -0.065 -0.065 -0.075 -0.075 0.0017 1.2E-4 0.011

unit: mm X Y X X X X Y Y Y Y 0.3 0.3 0.3 0.0012 8.1E-4 9.0E-4 0.3 -0.3 -0.3 -0.3 -0.3 0.0012 8.1E-4 9.0E-4

Rotating speed = 3690.0 rev/min unit: mm X Y X X X X Y Y Y Y 0.015 0.032 0.015 0.008 0.001 0.0125 0.032 -0.015 -0.015 -0.032 -0.032 0.008 0.001 0.0125 -0.008 -0.008 -0.0125 -0.0125

(29)

Figs. 22–24 show the response amplitudes, component amplitudes, and whirl orbits at T ¼ Taðcos Ot þ cos 2OtÞ; respectively. Fig. 22 reveals that the synchronous resonance

occurs at 3082 and 2682.6 rev/min. The steady response is comprised of the 1; 2; and 3 whirl components (Fig. 23). Because of the effect of the system coupling, two 2 lateral modes occur at 1538.6 and 1340 rev/min, two 3 lateral modes exist at 1025.3 and 897.3 rev/min, one 1 torsional mode occurs at 2517.3 rev/min, and one 2 torsional mode exists at 1258.0 rev/min.

Fig. 24 shows the orbit shapes of the 1; 2; and 3 whirls and the syntheticwhirling orbit shape. The synthetic whirling orbit is complex because the whirl components involve the 1; 2; and 4 whirls. Table 3lists the relations between the resonant critical speeds of the anisotropic rotor-bearing system and the exciting frequency.

Table 2

Critical speeds of the isotropic rotor-bearing system External torque T (N m) Lateral natural frequency

The ﬁrst mode (rev/min) 4 subcritical speed 3 subcritical speed 2 subcritical speed ð1Þ synchronous critical speed A. Critical speeds corresponding to the lateral natural frequency

T ¼ 0 3024.0 (F)

T ¼ 5000 cos Ot 1497.3 (F) 3024.0 (F)

T ¼ 5000 cos 2Ot 995.3 (F) 3022.6 (F)

T ¼ 5000 cos 3Ot 745.3 (F) 1468.6 (B) 3023.3 (F)

T ¼ 5000ðcos Ot

þcos 2Ot þ cos 3Ot)

745.3 (F) 995.3 (F) 1496.0 (F) 3024.0 (F)

External torque T (N m) Torsional natural frequency

The ﬁrst mode (rev/min) The second mode (rev/min) 3 subcritical speed 2 subcritical speed ð1Þ critical speed 3 subcritical speed 2 subcritical speed ð1Þ critical speed B. Critical speeds corresponding to the torsional natural frequency

T ¼ 0

T ¼ 5000 cos Ot 2516.7

T ¼ 5000 cos 2Ot 1258.0

T ¼ 5000 cos 3Ot 838.6 3712.0

T ¼ 5000ðcos Ot

þcos 2Ot þ cos 3OtÞ

838.6 1258.0 2516.0 3713.3

(30)

Ta=5000 Ta=1000 Ta=0 897.3 1025.3 1258.0 1538.6 _{2517.3 2682.6} 3082.0 1340.0

Amplitude (mm)

Fig. 22. Response amplitudes of disk 1 (anisotropicrotor-bearing system with different torques).

Shaft rotation 1000 rev/min 2000 rev/min 3500 rev/min 1440 rev/min 2850 rev/min 1344.0 1539.3 2516.0 2683.3 3082.6 T=1000cosΩt T=0

orbit unit: mm X Y X Y X Y X Y X Y 5.0E-4 5.0E-4 0.003 0.003 0.01 0.01 0.0025 0.0025 0.02 0.02 Amplitude (mm)

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8. Conclusion

The main objective of this work is to offer a modified TMM for analyzing the coupling lateral and torsional vibrations of the symmetricrotor-bearing systems with an external torque. The state variables of the modified transfer matrix include the lateral deflection, angular displacements, angle of twist, shear force, bending moment, and torque. The modified transfer matrix can be used to determine the steady-state responses of synchronous and superharmonic whirls of the coupling lateral and torsional vibrations. When the unbalance force alone excites the isotropic bearing-rotor system, the whirl orbit is synchronous, forward, and right circular, and only 1 lateral mode can be excited. However, when the unbalance force and the torque with n frequency of the rotating speed excite the system simultaneously, the ðn þ 1Þ forward and ðn 1Þ backward whirls appear along with synchronous whirl. If the unbalance force alone excites the anisotropic bearing-rotor system, the whirl orbit is synchronous and elliptical. Two split 1 lateral modes appear and the synchronous whirl reverses direction and becomes backward between the split critical speeds. Like the isotropic system, if the unbalance force and the torque with n frequency of the rotating speed excite the system simultaneously, the ðn þ 1Þ and ðn 1Þ whirls appear along with synchronous whirl. In conclusion, the external torque excites the superharmonic response and affects the dynamic behavior of the rotor-bearing system. Thus, during the design stage, the effect of the external torque should be considered carefully to avoid unexpectedly damaging the rotor-bearing system.

3X whirl 2X whirl

1X whirl

1X, 2X and 3X whirls Ta=5000

897.3 1025.3 1258.0 1340.0 1538.6 2517.3 2682.6 3082.0 Amplitude (mm)

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1X whirl 2X whirl 3X whirl 1X, 2X and 3X whirls unit: mm X Y X X X Y Y Y 0.007 0.011 0.0022 0.007 0.0032 0.011 -0.007 -0.007 -0.011 -0.011 0.0022 0.0032

1X whirl 2X whirl 1X, 2X and 3X whirls unit: mm X X X X Y Y Y Y 0.028 0.045 0.022 0.028 6.0E-4 0.045 -0.045 -0.028 -0.028 -0.045 0.022 -0.022 -0.022 6.0E-4 3X whirl

1X whirl 2X whirl 3X whirl 1X, 2X and 3X whirls unit: mm X Y X X X Y Y Y 0.07 0.075 0.07 0.0035 6.5E-4 0.075 -0.07 -0.07 -0.075 -0.075 0.0035 6.5E-4

1X whirl 2X whirl 3X whirl 1X, 2X and 3X whirls unit: mm X Y X X X Y Y Y 0.13 0.13 0.13 0.0011 7.5E-4 0.13 -0.13 -0.13 -0.13 -0.13 0.0011 7.5E-4

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Acknowledgement

The authors would like to thank the National Science Council of the Republic of China, Taiwan for ﬁnancially supporting this research under Contract No. NSC 92-2212-E-009-020.

Appendix A

Substituting Eq. (28) into Eqs. (22)–(26) and equating coefﬁcients of the same harmonic terms, the following equations are obtained:

ksGAx00₀ksGAy0_y;0¼0, (A.1)

ksGAx001cksGAy0y;1cþrAO 2_x 1c1₂rAesvO 2_j 2c12rAe s uO 2_j 2srAO2esvj0 ¼ rAO2es_u, ðA:2Þ Table 3

Critical speeds of the anisotropic rotor-bearing system External torque T (N m) Lateral natural frequency

The ﬁrst mode (rev/min)

3 subcritical speed 2 subcritical speed ð1Þ synchronous critical speed A. Critical speeds corresponding to the lateral natural frequency

T ¼ 0 2683.3 (B)

3082.6 (F)

T ¼ 1000 cos Ot 1344.0 (B) 2683.3(B)

1539.3 (F) 3082.6 (F)

T ¼ 5000ðcos Ot þ cos 2OtÞ 897.3 (B) 1340.0 (B) 2682.6 (B)

1025.3 (F) 1538.6 (F) 3082.0 (F)

External torque T (N m) Torsional natural frequency

The ﬁrst mode (rev/min)

2 subcritical speed ð1Þ critical speed B. Critical speeds corresponding to the torsional natural frequency

T ¼ 0

T ¼ 1000 cos Ot 2516.0

T ¼ 5000ðcos Ot þ cos 2OtÞ 1258.0 2517.3

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ksGAx00_1sksGAy0_y;1sþrAO2x1s1₂rAesvO 2_j 2sþ12rAe s uO 2_j 2crAO2esuj0 ¼rAO2es_v, ðA:3Þ

ksGAx00_2cksGAy0_y;2cþrA4O2x2c2rAes_vO2j3c2rAesuO 2_j

3s2rAesvO 2_j

1c

þ2rAes_uO2j_1s ¼0, ðA:4Þ

ksGAx00_2sksGAy0_y;2sþrA4O2x2s2rAes_vO2j3sþ2rAesuO 2_j

3c2rAesvO 2_j

1s

2rAes_uO2j_1c¼0, ðA:5Þ

ksGAx00jcksGAy0y;jcþrAj2O2xjc1₂rAesvj2O2jðjþ1Þc12rAe s

uj2O2jðjþ1Þs

1₂rAes_vj2O2j_ðj1Þcþ1₂rAes_uj2O2j_ðj1Þs¼0, ðA:6Þ ksGAx00jsksGAy0y;jsþrAj2O2xjs1₂rAesvj2O2jðjþ1Þsþ12rAe

s

uj2O2jðjþ1Þc

1₂rAes_vj2O2j_ðj1Þs1₂rAes_uj2O2j_ðj1Þc¼0, ðA:7Þ ksGAx00ncksGAy0y;ncþrAn

2_O2_x nc1₂rAesvn 2_O2_j ðn1Þcþ12rAe s un 2_O2_j ðn1Þs ¼0, (A.8)

ksGAx00_nsksGAy0_y;nsþrAn2O2xns1₂rAes_vn2O2jðn1Þs12rAe s un

2_O2_j

ðn1Þc ¼0, (A.9)

ksGAy0x;0þksGAy000 ¼rAg, (A.10)

ksGAy0x;1cþksGAy1c00 þrAO2y1cþ12rAe s

uO2j2c12rAe s

vO2j2sþrAO2esuj0

¼ rAO2es_v, ðA:11Þ

ksGAy0x;1sþksGAy1s00 þrAO2y1sþ12rAe s

uO2j2sþ12rAe s

vO2j2crAO2esvj0

¼ rAO2es_u, ðA:12Þ

ksGAy0x;2cþksGAy002cþrA4O 2_y 2cþ2rAesuO 2_j 3c2rAesvO 2_j 3sþ2rAesuO 2_j 1c þ 2rAes_vO2j_1s ¼0, ðA:13Þ

ksGAy0x;2sþksGAy2s00 þrA4O2y2sþ2rAesuO2j3sþ2rAesvO2j3cþ2rAesuO2j1s

2rAes_vO2j_1c¼0, ðA:14Þ

ksGAy0x;jcþksGAy00jcþrAj2O2yjcþ12rAe s

uj2O2jðjþ1Þc12rAe s

vj2O2jðjþ1Þs

þ 1₂rAes_uj2O2j_ðj1Þcþ1₂rAes_vj2O2j_ðj1Þs¼0, ðA:15Þ ksGAy0x;jsþksGAy00jsþrAj2O2yjsþ12rAe

s

uj2O2jðjþ1Þsþ12rAe s

vj2O2jðjþ1Þc

(35)

ksGAy0_x;ncþksGAy00_ncþrAn2O2yncþ12rAe s un 2_O2_j ðn1Þcþ12rAe s vn 2_O2_j ðn1Þs¼0, (A.17)

ksGAy0x;nsþksGAy00nsþrAn2O2ynsþ12rAe s

un2O2jðn1Þs12rAe s

vn2O2jðn1Þc ¼0, (A.18)

EIsy00_x;0ksGAyx;0ksGAy00¼0, (A.19)

EIsy00_x;icksGAyx;icksGAy0icrIspOiOyy;isþrIsi2O2yx;ic ¼0, (A.20)

EIsy00_x;isksGAyx;isksGAy0isþrIspOiOyy;icþrIsi2O2yx;is¼0, (A.21)

EIsy00_y;0ksGAx00þksGAyy;0¼0, (A.22)

EIsy00_y;icksGAx0icþksGAyy;icrIspOiOyx;isrIsi2O2yy;ic¼0, (A.23)

EIsy00_y;isksGAx0isþksGAyy;isþrIspOiOyx;icrIsi2O2yy;is ¼0, (A.24)

GIs_pj00₀1₂rAes_vO2x1c1₂rAesuO2x1sþ1₂rAesuO2y1c12rAe s

vO2y1s¼0, (A.25)

GIs_pj00

1cþrAðesÞ2O2j1cþrIspO2j1c2rAevsO2x2c2rAesuO2x2sþ2rAesuO2y2c

2rAes_vO2y_2s ¼0, ðA:26Þ

GIs_pj00_1sþrA eð Þs 2O2j_1sþrIs_pO2j_1s2rAes_vO2x2sþ2rAesuO2x2cþ2rAesuO2y2s

þ 2rAes_vO2y_2c¼0, ðA:27Þ

1₂rAes_vO2x1cþ1₂rAesuO2x1sþ1₂rAesuO2y1cþ12rAe s

vO2y1sþGIspj 00 2c

þ 4rAðesÞ2O2j_2cþ4rIs_pO2j_2crAes_v9 2O

2_x

3crAes_u9₂O2x3s

þ rAes_u9₂O2y_3crAes_v9₂O2y_3s ¼0, ðA:28Þ 1₂rAes_vO2x1s1₂rAes_uO2x1cþ1₂rAes_uO2y1s12rAe

s vO 2_y 1cþGIspj 00 2s

þ4rAðesÞ2O2j_2sþ4rIs_pO2j_2srAes_v9 2O 2_x 3sþrAesu92O 2_x 3c þ rAes_u9 2O 2_y 3sþrAesv92O 2_y 3c¼0, ðA:29Þ GIs_pj00 jcþj 2_rAðes_Þ2_O2_j jcþj2rIspO 2_j jcrAesvðj þ 1Þ212O 2_x ðjþ1Þc rAes_vðj 1Þ21₂O2xðj1ÞcrAesuðj þ 1Þ 21 2O 2_x ðjþ1ÞsþrAesuðj 1Þ 21 2O 2_x ðj1ÞsþrAesuðj þ 1Þ 21 2O 2_y ðjþ1Þc

(36)

GIs_pj00_jsþj2rA eð Þs 2O2j_jsþj2rIs_pO2j_jsrAes_vðj þ 1Þ21₂O2xðjþ1Þs rAes_vðj 1Þ21₂O2xðj1ÞsþrAesuðj þ 1Þ212O 2_x ðjþ1ÞcrAesuðj 1Þ212O 2_x ðj1Þc

þ rAes_uðj þ 1Þ21₂O2y_ðjþ1ÞsþrAes_uðj 1Þ21₂O2y_ðj1ÞsþrAes_vðj þ 1Þ21₂O2y_ðjþ1Þc

rAes_vðj 1Þ21₂O2y_ðj1Þc ¼0, ðA:31Þ GIs_pj00_ncþn2rA eð Þs 2O2j_ncþn2rIs_pO2j_ncrAes_vðn 1Þ21₂O2xðn1Þc þrAes_uðn 1Þ21₂O2xðn1ÞsþrAesuðn 1Þ212O 2_y ðn1Þc þrAes_vðn 1Þ21₂O2y_ðn1Þs¼0, ðA:32Þ GIs_pj00_nsþn2rA eð Þs 2O2j_nsþn2rIs_pO2j_nsrAes_vðn 1Þ21₂O2xðn1Þs rAes_uðn 1Þ21₂O2xðn1ÞcþrAesuðn 1Þ212O 2_y ðn1Þs rAes_vðn 1Þ21₂O2y_ðn1Þc ¼0, ðA:33Þ where i ¼ 1; 2; 3; . . . ; n; and j ¼ 3; 4; 5; . . . ; n 1. Appendix B E2ð1; 1Þ ¼ E2ð2; 2Þ ¼ E2ð2 þ n; 2 þ nÞ ¼ E2ð3; 3Þ ¼ E2ð3 þ n; 3 þ nÞ ¼E2ð1 þ j; 1 þ jÞ ¼ E2ð1 þ n þ j; 1 þ n þ jÞ ¼ E2ð1 þ n; 1 þ nÞ ¼E2ð1 þ 2n; 1 þ 2nÞ ¼ E2ð2 þ 2n; 2 þ 2nÞ ¼E2ð3 þ 2n; 3 þ 2nÞ ¼ E2ð3 þ 3n; 3 þ 3nÞ ¼ E2ð4 þ 2n; 4 þ 2nÞ ¼ E2ð4 þ 3n; 4 þ 3nÞ ¼E2ð2 þ 2n þ j; 2 þ 2n þ jÞ ¼ E2ð2 þ 3n þ j; 2 þ 3n þ jÞ ¼ E2ð2 þ 3n; 2 þ 3nÞ ¼E2ð2 þ 4n; 2 þ 4nÞ ¼ m, E2ð3 þ 4n; 3 þ 4nÞ ¼ E2ð3 þ 4n þ i; 3 þ 4n þ iÞ ¼ E2ð3 þ 5n þ i; 3 þ 5n þ iÞ ¼ E2ð4 þ 6n; 4 þ 6nÞ ¼ E2ð4 þ 6n þ i; 4 þ 6n þ iÞ ¼ E2ð4 þ 7n þ i; 4 þ 7n þ iÞ ¼ a, E2ð5 þ 8n; 5 þ 8nÞ ¼ E2ð6 þ 8n; 6 þ 8nÞ ¼ E2ð6 þ 9n; 6 þ 9nÞ ¼ E2ð7 þ 8n; 7 þ 8nÞ ¼E2ð7 þ 9n; 7 þ 9nÞ ¼ E2ð5 þ 8n þ j; 5 þ 8n þ jÞ ¼ E2ð5 þ 9n þ j; 5 þ 9n þ jÞ ¼E2ð5 þ 9n; 5 þ 9nÞ ¼ E2ð5 þ 10n; 5 þ 10nÞ ¼ h,

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E1ð1; 4 þ 6nÞ ¼ E1ð2; 5 þ 6nÞ ¼ E1ð2 þ n; 5 þ 7nÞ ¼ E1ð3; 6 þ 6nÞ ¼ E1ð3 þ n; 6 þ 7nÞ ¼ E1ð1 þ j; 4 þ 6n þ jÞ ¼ E1ð1 þ n þ j; 4 þ 7n þ jÞ ¼ E1ð1 þ n; 4 þ 7nÞ ¼ E1ð1 þ 2n; 4 þ 8nÞ ¼ E1ð2 þ 2n; 3 þ 4nÞ ¼E1ð3 þ 2n; 4 þ 4nÞ ¼ E1ð3 þ 3n; 4 þ 5nÞ ¼E1ð4 þ 2n; 5 þ 4nÞ ¼ E1ð4 þ 3n; 5 þ 5nÞ ¼ E1ð2 þ 2n þ j; 3 þ 4n þ jÞ ¼E1ð2 þ 3n þ j; 3 þ 5n þ jÞ ¼ E1ð2 þ 3n; 3 þ 5nÞ ¼ E1ð2 þ 4n; 3 þ 6nÞ ¼ E1ð3 þ 4n; 2 þ 2nÞ ¼ E1ð3 þ 4n þ i; 2 þ 2n þ iÞ ¼ E1ð3 þ 5n þ i; 2 þ 3n þ iÞ ¼ E1ð4 þ 6n; 1Þ ¼ E1ð4 þ 6n þ i; 1 þ iÞ ¼ E1ð4 þ 7n þ i; 1 þ n þ iÞ ¼ m, E0ð3 þ 4n; 3 þ 4nÞ ¼ E0ð4 þ 6n; 4 þ 6nÞ ¼ m,

E0ð3 þ 4n þ i; 3 þ 4n þ iÞ ¼ E0ð3 þ 5n þ i; 3 þ 5n þ iÞ ¼ E0ð4 þ 6n þ i; 4 þ 6n þ iÞ

¼E0ð4 þ 7n þ i; 4 þ 7n þ iÞ ¼ m i2b, E0ð2; 2Þ ¼ E0ð2 þ n; 2 þ nÞ ¼ E0ð3 þ 2n; 3 þ 2nÞ ¼ E0ð3 þ 3n; 3 þ 3nÞ ¼ c, E0ð3; 3Þ ¼ E0ð3 þ n; 3 þ nÞ ¼ E0ð4 þ 2n; 4 þ 2nÞ ¼ E0ð4 þ 3n; 4 þ 3nÞ ¼ 4c, E0ð1 þ j; 1 þ jÞ ¼ E0ð1 þ n þ j; 1 þ n þ jÞ ¼ E0ð2 þ 2n þ j; 2 þ 2n þ jÞ ¼E0ð2 þ 3n þ j; 2 þ 3n þ jÞ ¼ j2c, E0ð1 þ n; 1 þ nÞ ¼ E0ð1 þ 2n; 1 þ 2nÞ ¼ E0ð2 þ 3n; 2 þ 3nÞ ¼ E0ð2 þ 4n; 2 þ 4nÞ ¼ n2c, E0ð6 þ 8n; 6 þ 8nÞ ¼ E0ð6 þ 9n; 6 þ 9nÞ ¼ s, E0ð7 þ 8n; 7 þ 8nÞ ¼ E0ð7 þ 9n; 7 þ 9nÞ ¼ 4s, E0ð5 þ 8n þ j; 5 þ 8n þ jÞ ¼ E0ð5 þ 9n þ j; 5 þ 9n þ jÞ ¼ j2s, E0ð5 þ 9n; 5 þ 9nÞ ¼ E0ð5 þ 10n; 5 þ 10nÞ ¼ n2s,

E0ð3 þ 4n þ i; 4 þ 7n þ iÞ ¼ E0ð3 þ 5n þ i; 4 þ 6n þ iÞ ¼ E0ð4 þ 6n þ i; 3 þ 5n þ iÞ

¼E0ð4 þ 7n þ i; 3 þ 4n þ iÞ ¼ id, E0ð2; 7 þ 9nÞ ¼ E0ð2 þ n; 7 þ 8nÞ ¼ E0ð3 þ 2n; 7 þ 8nÞ ¼ E0ð3 þ 3n; 7 þ 9nÞ ¼ E0ð5 þ 8n; 2 þ nÞ ¼ E0ð5 þ 8n; 3 þ 2nÞ ¼ E0ð7 þ 8n; 2 þ nÞ ¼E0ð7 þ 8n; 3 þ 2nÞ ¼ E0ð7 þ 9n; 2Þ ¼ E0ð7 þ 9n; 3 þ 3nÞ ¼1₂f , E0ð2 þ n; 5 þ 8nÞ ¼ E0ð3 þ 2n; 5 þ 8nÞ ¼ f , E0ð3; 8 þ 9nÞ ¼ E0ð3; 6 þ 9nÞ ¼ E0ð3 þ n; 8 þ 8nÞ ¼ E0ð3 þ n; 6 þ 8nÞ ¼ E0ð4 þ 2n; 8 þ 8nÞ ¼ E0ð4 þ 2n; 6 þ 8nÞ ¼ E0ð4 þ 3n; 8 þ 9nÞ ¼ E0ð4 þ 3n; 6 þ 9nÞ ¼ E0ð6 þ 8n; 3 þ nÞ ¼ E0ð6 þ 8n; 4 þ 2nÞ ¼ E0ð6 þ 9n; 3Þ ¼ E0ð6 þ 9n; 4 þ 3nÞ ¼ 2f ,

(38)

E0ð7 þ 8n; 4 þ nÞ ¼ E0ð7 þ 8n; 5 þ 2nÞ ¼ E0ð7 þ 9n; 4Þ ¼ E0ð7 þ 9n; 5 þ 3nÞ ¼9₂f , E0ð1 þ j; 6 þ 9n þ jÞ ¼ E0ð1 þ j; 4 þ 9n þ jÞ ¼ E0ð1 þ n þ j; 6 þ 8n þ jÞ ¼ E0ð1 þ n þ j; 4 þ 8n þ jÞ ¼ E0ð2 þ 2n þ j; 6 þ 8n þ jÞ ¼E0ð2 þ 2n þ j; 4 þ 8n þ jÞ ¼ E0ð2 þ 3n þ j; 6 þ 9n þ jÞ ¼E0ð2 þ 3n þ j; 4 þ 9n þ jÞ ¼1₂j2f , E0ð5 þ 8n þ j; 2 þ n þ jÞ ¼ E0ð5 þ 8n þ j; 3 þ 2n þ jÞ ¼ E0ð5 þ 9n þ j; 2 þ jÞ ¼E0ð5 þ 9n þ j; 3 þ 3n þ jÞ ¼1₂ðj þ 1Þ2f , E0ð5 þ 8n þ j; n þ jÞ ¼ E0ð5 þ 8n þ j; 1 þ 2n þ jÞ ¼ E0ð5 þ 9n þ j; jÞ ¼E0ð5 þ 9n þ j; 1 þ 3n þ jÞ ¼1₂ðj 1Þ2f , E0ð1 þ n; 4 þ 10nÞ ¼ E0ð1 þ 2n; 4 þ 9nÞ ¼ E0ð2 þ 3n; 4 þ 9nÞ ¼ E0ð2 þ 4n; 4 þ 10nÞ ¼1₂n2f , E0ð5 þ 9n; 2nÞ ¼ E0ð5 þ 9n; 1 þ 3nÞ ¼ E0ð5 þ 10n; nÞ ¼ E0ð5 þ 10n; 1 þ 4nÞ ¼1₂ðn 1Þ2f , E0ð2; 7 þ 8nÞ ¼ E0ð2 þ n; 7 þ 9nÞ ¼ E0ð3 þ 2n; 7 þ 9nÞ ¼ E0ð3 þ 3n; 7 þ 8nÞ ¼ E0ð5 þ 8n; 2Þ ¼ E0ð5 þ 8n; 3 þ 3nÞ ¼ E0ð7 þ 8n; 2Þ ¼ E0ð7 þ 8n; 3 þ 3nÞ ¼ E0ð7 þ 9n; 2 þ nÞ ¼ E0ð7 þ 9n; 3 þ 2nÞ ¼1₂g0, E0ð2; 5 þ 8nÞ ¼ E0ð3 þ 3n; 5 þ 8nÞ ¼ g0, E0ð3; 8 þ 8nÞ ¼ E0ð3; 6 þ 8nÞ ¼ E0ð3 þ n; 8 þ 9nÞ ¼ E0ð3 þ n; 6 þ 9nÞ ¼ E0ð4 þ 2n; 8 þ 9nÞ ¼ E0ð4 þ 2n; 6 þ 9nÞ ¼ E0ð4 þ 3n; 8 þ 8nÞ ¼ E0ð4 þ 3n; þ8nÞ ¼ E0ð6 þ 8n; 3Þ ¼ E0ð6 þ 8n; 4 þ 3nÞ ¼ E0ð6 þ 9n; 3 þ nÞ ¼ E0ð6 þ 9n; 4 þ 2nÞ ¼2g₀, E0ð7 þ 8n; 4Þ ¼ E0ð7 þ 8n; 5 þ 3nÞ ¼ E0ð7 þ 9n; 4 þ nÞ ¼ E0ð7 þ 9n; 5 þ 2nÞ ¼9₂g0, E0ð1 þ j; 6 þ 8n þ jÞ ¼ E0ð1 þ j; 4 þ 8n þ jÞ ¼ E0ð1 þ n þ j; 6 þ 9n þ jÞ ¼ E0ð1 þ n þ j; 4 þ 9n þ jÞ ¼ E0ð2 þ 2n þ j; 6 þ 9n þ jÞ ¼E0ð2 þ 2n þ j; 4 þ 9n þ jÞ ¼ E0ð2 þ 3n þ j; 6 þ 8n þ jÞ ¼ E0ð2 þ 3n þ j; 4 þ 8n þ jÞ ¼1₂j2g0, E0ð5 þ 8n þ j; 2 þ jÞ ¼ E0ð5 þ 8n þ j; 3 þ 3n þ jÞ ¼ E0ð5 þ 9n þ j; 2 þ n þ jÞ ¼E0ð5 þ 9n þ j; 3 þ 2n þ jÞ ¼1₂ðj þ 1Þ2g0,

(39)

E0ð5 þ 8n þ j; jÞ ¼ E0ð5 þ 8n þ j; 1 þ 3n þ jÞ ¼ E0ð5 þ 9n þ j; n þ jÞ ¼ E0ð5 þ 9n þ j; 1 þ 2n þ jÞ ¼1₂ðj 1Þ2g0; E0ð1 þ n; 4 þ 9nÞ ¼ E0ð1 þ 2n; 4 þ 10nÞ ¼ E0ð2 þ 3n; 4 þ 10nÞ ¼ E0ð2 þ 4n; 4 þ 9nÞ ¼1₂n2g₀, E0ð5 þ 9n; nÞ ¼ E0ð5 þ 9n; 1 þ 4nÞ ¼ E0ð5 þ 10n; 2nÞ ¼ E0ð5 þ 10n; 1 þ 3nÞ ¼1₂ðn 1Þ2g₀, where a ¼ EIs; b ¼ rIsO2; c ¼ rAO2; d ¼ rIs_pO2,

f ¼ rAes_uO2; g₀¼rAes_vO2; h ¼ GI_ps; m ¼ ksAG; s ¼ rO2AðesÞ2þrO2Isp,

i ¼ 1; 2; 3; . . . ; n; and j ¼ 3; 4; 5; . . . ; n 1.

References

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