vibrations of a rotor system with multiple coupler offsets · the whirling orbits before and after...

VIBRATIONS OF A ROTOR SYSTEM WITH MULTIPLE COUPLER OFFSETS

Chao-Yang Tsai, Shyh-Chin HuangDepartment of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan

R. O. C.

E-mail: [email protected]; [email protected]

Received October 2009, Accepted January 2011No. 10-CSME-41, E.I.C. Accession 3204

ABSTRACT

In this paper, a transfer matrix method (TMM) for rotors with multiple coupler offsets wasderived. The studies showed the coupler’s stiffness altered the rotor’s critical speeds but offsetcaused additional external excitation. The cases of two offsets in- and anti-phase in a typical rotorwere given as examples. In the in-phase case, significantly increased response amplitude occurredat lower rotational speed and the increase was linearly proportional to the offset value. As to theanti-phase case, the increased response was insignificant, implying an opposite offset would cancelout a major response of the previous offset. The whirling orbits before and after the offsetcouplers were also illustrated. The results, as expected, showed the in-phase offset displayed muchlarger radii than the anti-phase’s. The rotor’s orbits changed the whirling direction once therotation fell within a certain range and this feature seemed to be unaffected by coupler offsets.

Keywords: transfer matrix method; misalignment; unbalance; coupler offset.

VIBRATIONS D’UN SYSTEME DE ROTOR AVEC DE MULTIPLES DECALAGESDU COUPLEUR

RESUME

Dans cet article, nous elaborons une methode de transfert de matrice pour rotors avec demultiples decalages du coupleur. Les etudes ont demontre que la rigidite du coupleur modifie lesvitesses critiques du rotor, mais que le decalage cause une excitation externe additionnelle. Lescas de decalage en phase et hors phase dans un rotor typique sont donnes comme exemple.Dans le cas en phase, une reponse accrue de l’amplitude survient pendant une vitesse de rotationmoins elevee, et l’accroissement etait de facon lineaire proportionnelle a la valeur du decalage.Pour ce qui est du cas hors-phase, l’accroissement de la reponse etait insignifiant, laissantsupposer qu’un decalage oppose annulerait une reponse accrue du decalage precedent. Lemouvement des orbites avant et apres le decalage des coupleurs sont aussi illustres. Lesresultats, tels que anticipes, demontrent que le decalage en phase a deploye des rayons beaucoupplus grands que le hors-phase. Les orbites du rotor ont change la direction du mouvementtournoyant une fois que la rotation s’est trouvee a l’interieur d’un certain rayon d’action, etcette fonction ne semble pas touchee par les decalages du coupleur selon les exemples presentes.

Mots-cles : methode de transfert de matrice; desalignement; desequilibre; decalage ducoupleur.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 35, No. 1, 2011 81

1. INTRODUCTION

Rotor systems have been widely used in mechanical engineering. The dynamics of rotors havetherefore been studied for over a century. With the high speed demand of today’s machinery, itbecomes more important than ever. The approaches to vibration analysis of rotor systems can bedivided into two main streams. One is the well-used finite element method (FEM) [1–3], and theother is the relatively more traditional transfer matrix method (TMM) [4]. The main advantage ofTMM is its lower system dimension, even for a relatively complex rotor. The disadvantage is itrequires more matrix multiplications and a more sophisticated root finding algorithm.

Prohl [5], one of the pioneers, used TMM for calculating the bending critical speeds of rotorsystems. Lund and Orcutt [6] improved the transfer matrix of a shaft in a continuous fashionrather than in a discrete fashion, but neglected both rotary inertia and the gyroscopic effect.Lund [7] considered the hysteretic of a damping shaft and calculated the unbalanced response ofa general flexible rotor supported in a fluid-film journal bearing. Chao and Huang [8]introduced the modified transfer matrix in which the Euler beam and rigid disc were thefundamental elements. They obtained more accurate natural frequencies and shapes. Manyresearchers [9–11] continuously improved TMM, such as developing an oil-film bearing matrix,including rotary inertia, and the gyroscopic effects of discs.

The present study intends to examine the vibration response of a rotor system containingmultiple coupler offsets. With regard to the studies of shaft misalignment/offset, Dewell andMitchell [12] experimentally studied parallel and angular misalignment of a metallic-disk-typecoupling. By using the real time analyzer, they found the frequencies of all the integer multiples of

NOMENCLATURE

A the cross-section areaCf g offset vector

E Young’s moduluse static offset[G], [GG] shaft’s left side transfer matrix[H], [HH] shaft’s right side transfer

matrixIxx polar moment of inertiaIyy, Izz transverse moment of inertiaKB bending stiffness of the couplerKL linear stiffness of the couplerKyy, Kzz,KQQ, Khh

bearing’s stiffness coefficients

l the length of shaftMY , MZ shaft’s bending moments[M] coupler’s transfer matrixp number of offsetr dynamic offsetre eccentricity of disksSf gL left side state vector

Sf gR right side state vectorTs½ � transfer matrix of rotating

shaftTd½ � transfer matrix of an unba-

lanced diskTb½ � transfer matrix of an oil-filmed

bearing[T]i transfer matrix of i-th segment[Tu], [Tm] overall transfer matrix for

unbalance and misalignment{u} state vector of unbalanceVY, VZ shaft’s shear forcesv, w displacements in Y and Z

directionb phase relative to rotor’s refer-

enceQ, h angular displacements_QQ, _hh angular velocity in Y and Z

direction respectivelyr density of shaftd deflection of the linear springV rotational speed


rotational speed, said n6, appeared due to misalignment. They further suggested the 26 and46 components be used for misalignment diagnosis. Xu and Marangon used a universal joint tomodel an angular misalignment and employed component mode synthesis to analytically study[13] and experimentally validate [14] the calculations. In their model, the misalignment effects wererepresented by an additional bending moment of even multiple frequencies of rotational speed.They concluded the unbalance and misalignment could be characterized by the 16 and 26components, respectively. Sekhar and Prabhu [15] used higher order finite elements to study amisaligned rotor-bearing systems, in which equivalent moments and forces replaced themisalignment effect and obtained similar conclusions [13]. They also concluded the misalignmenthad little effect on the change in critical speeds. Lee and Lee [16] used FEM for misaligned rotor-bearing system, in which angular, parallel, and mixed cases were discussed extensively thoughshown whirling orbits. In that paper, coupler flexibility was not included. As a result, theyconcluded the whirling orbits did not change in parallel misalignment. Al-Hussain and Redmond[17] analytically derived the equations of two Jeffcott rotors rigidly coupled in parallelmisalignment. In their results, unlike other researchers, they did not obtain the 26 component.Al-Hussain [18] further extended the work to stability analysis. Saavedra and Ramirz [19]proposed a theoretical model for a rotor-bearing system utilizing new coupling finite elementstiffness and discussed the vibration due to shaft misalignment and residual unbalance. Theirresults showed the vibration induced by shaft misalignment was due to the variation in couplingstiffness during rotation. Experimental results [20] were presented to validate the theoretical model.Huang [21] analyzed the torsional vibration characteristic of shafts with parallel misalignment andindicated the misalignment excited the torsional vibration at 16 rotating frequency. Hili et al. [22]presented a theoretical model to analyze parallel and angular misalignment, and suggesteddiagnosing shaft misalignment from the harmonics of running frequency.

It was not until recently that the transfer matrix for an offset coupler was derived [23]. Thepresent paper applies the result of [23] and intends to derive a general formula for rotor systemswith multiple offset couplers. This paper will also examine if an anti-phase offset can adjustback the previous offset and subsequently reduce the rotor’s total vibration. Two offsets, in-phase and anti-phase, in a typical rotor system are therefore studied as examples. Mode shapesand critical speeds are also illustrated. The results show the coupler stiffness affects the rotor’scritical speeds and the offset acts as an excitation similar to an unbalance. The whirling orbitsfor two kinds of phase (in- and anti-phase) are also illustrated.

2. TRANSFER MATRIX OF AN OFFSET COUPLER

A general rotor system containing flexible shafts, discs, bearing supports and multiplecoupler offset is schematically shown in Fig. 1(a). The transfer matrices of unbalanced discs,bearings, and rotating shafts are described in detail in [23], or can be seen in the existingliterature [24], though they might differ in some respects. To make this paper stand-alone, thematrices for various elements are also given in the Appendix. The present paper focuses onderiving multiple offsets. Fig. 1(b) defines the geometric configuration of a coupler with anoffset e, i.e., the eccentricity between coupler disc centres, where a~Vtzb, V the rotationalspeed, and b the phase angle relative to rotor’s reference, usually a key phaser. r~ezd is thedynamic offset, including d, the extra radial deflection of the coupler during vibration.

A coupler is usually composed of two discs and in between there is a rigid or elastic restraint.In this paper, the mass of coupler discs is neglected because their effect is relatively small andthe elastic restraint is represented by two linear springs KL and KB. Provided rigid connection is


encountered, one can set these two stiffnesses to infinity. In Fig. 1(b), the stations (points) to theleft and right side of the jth coupler are denoted by kj and kj+1, respectively. In the paper,(X,Y,Z) denotes the absolute, non-rotating coordinate system, and (x,y,z) is the element-fixedlocal coordinate system. From the equilibrium relations and geometric configuration, thefollowing equations are derived:

vR~vLzVy

KL

ze:sin Vtzbð Þ ð1Þ

wR~wL

{1

KB

MLZ

ð2Þ

wR~wL{Vz

KL

{e:cos Vtzbð Þ ð3Þ

hR~hL{1

KB

:MLY ð4Þ

Fig. 1. Schematic diagram of a misaligned rotor, for (a) symbol of offset, and (b) displacement of offset.


where v, w and h, w are the transverse and angular displacements in the Y and Z directions,respectively. Superscripts R and L stand for the right and left side of an element. The moments,shear forces, due to neglecting the coupler’s mass and rotary inertia, are equal, i.e., MR~ML,VR~VL. The torsional vibration is not considered, so the rotation angle a~Vtzb remains allthe time. The transfer relation between the left and right side of such an offset coupler istherefore derived to be

Sf gR~ M½ � Sf gL

z Cf g ð5Þ

where {S} denotes the state vector, [M ] is the coupler’s transfer matrix, and {C} is the offsetvector. For an ordinary element, Eq. (5) usually appears in a form with no underlined term. {C}is a peculiar feature arising from the offset coupler.

The state vector Sf g is a 17 6 1 vector as

Sf g~ S1f g S2f g 1f gT ð6Þ

where

S1f g~ vc wc MZc VYc ws hs MYs VZsf gS2f g~ vs ws MZs VYs wc hc MYc VZcf g

�ð7Þ

and

v X ,tð Þ~vc xð ÞcosVtzvs xð ÞsinVt

w(X ,t)~wc xð ÞcosVtzws xð ÞsinVt

MZ X ,tð Þ~MZc xð ÞcosVtzMZs xð ÞsinVt

VY X ,tð Þ~VYc xð ÞcosVtzVYs xð ÞsinVt

w X ,tð Þ~ws xð ÞsinVtzwc xð ÞcosVt

h X ,tð Þ~hs xð ÞsinVtzhc xð ÞcosVt

MY X ,tð Þ~MYs xð ÞsinVtzMYc xð ÞcosVt

VZ X ,tð Þ~VZs xð ÞsinVtzVZc xð ÞcosVt

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð8Þ

In fact, Eq. (8) shows all the sufficient 8 independent states w, h, MY , VZ and v, w, MZ, VY inthe X-Y and X-Z planes of the system. Nevertheless, due to the whirling motion, it is necessaryand helpful to decompose these states into cosine and sine components, as denoted bysubscripts c and s, respectively.

M½ �~M1½ �8|8 M2½ �8|8 0f g8|1

M3½ �8|8 M4½ �8|8 0f g8|1

0f g1|8 0f g1|8 1

264

375 ð9Þ


In the above, [M] is the coupler transfer matrix of entries shown in the following

M1½ � ij~ M4½ � ij~

1 , when i~ j , i~1,2, � � � ,8 , j~1,2, � � � ,8{1=KB

, when i~2 , j~3ð Þ and i~6 , j~7ð Þ{1=KL

, when i~1 , j~4ð Þ and i~5 , j~8ð Þ0 , otherwise

8>>>><>>>>:

ð10Þ

M2½ � ij~ M3½ � ij~0 , when i=6 and j=3 , i~1,2, � � � ,8 , j~1,2, � � � ,8

{1=KB, when i~6 and j~3 , i~1,2, � � � ,8 , j~1,2, � � � ,8

(ð11Þ

Cf g17|1 ~ e sin b 0 0 0 e sinb 0 0 0 e cosb 0 0 0 {e cos b 0 0 0 0f gT ð12Þ

In Eq. (5), there is an augmented identity equation for calculating the response. This is awidely used technique shown in reference [24].

Equation (5) is the transfer relation of an offset coupler, where [M], similar to the otherelements, is the transfer matrix of the coupler and {C}, the offset vector, plays an exciting role.It will be seen after multiplication to the coupler’s right matrices, all components to the right ofthe offset contribute to the excitation. This means the driven parts behind the offset act as awhole excitation to the rotor. If there is no offset (e50), {C}5{0}, and Eq. (5) simply representsa transfer relation of a coupler. If a rigid coupler is used, by setting the two coupler stiffnesses toinfinity, the [M] matrix reduces to a unity matrix as expected.

3. TOTAL TRANSFER MATRIX OF A ROTOR SYSTEM

With the derived offset coupler’s TM and in conjunction with the other elements’ TM, anoverall transfer matrix for a typical rotor system, as shown in Fig. 1(a), can be derived,

Sf gRn ~ Tu½ � Sf gL

1 z Tm1½ � C1f gz Tm2½ � C2f gz � � �z Tmj½ � Cj

� �z � � �z Tmp½ � Cp

� �~ Tu½ � Sf gL

1 zXp

j~1

Tmj½ � Cj

� � ð13Þ

where Sf gL1 represents the left state of unit 1, Sf gR

n is the right state of unit n, and

Tu½ �~ T½ �n T½ �n{1::: T½ �kpz1 Mp

� �T½ �kp

::: T½ �kjz1 Mj

� �T½ �kj

::: T½ �k1z1 M1½ � T½ �k1

::: T½ �2 T½ �1Tmj½ �~ T½ �n T½ �n{1

::: T½ �kpz1 Mp

� �T½ �kp

::: T½ �kjz1z1 Mjz1

� �T½ �kjz1

::: T½ �kjz1

(ð14Þ

[T ]i denotes the ith element transfer matrix, which could be a shaft, a bearing, or a disc(Appendix). Tu½ � is the so-called overall transfer matrix yielded by consecutive multiplicationsof all element transfer matrices. Tmj½ � is the multiplications of the transfer matrices to the rightof mth

j offset, i.e., from kj+1 to n. To clearly show the formulation, p 5 3 is here taken as an


example and Eq. (13) becomes

Sf gRn ~ Tu½ � Sf gL

1 z Tm1½ � C1f gz Tm2½ � C2f gz Tm3½ � C3f g ð15Þ

where

Tu½ �~ T½ �n T½ �n{1::: T½ �k3z1 M3½ � T½ �k3

::: T½ �k2z1 M2½ � T½ �k2

::: T½ �k1z1 M1½ � T½ �k1

::: T½ �2 T½ �1Tm1½ �~ T½ �n T½ �n{1

::: T½ �k3z1 M3½ � T½ �k3

::: T½ �k2z1 M2½ � T½ �k2

::: T½ �k1z1

Tm2½ �~ T½ �n T½ �n{1::: T½ �k3z1 M3½ � T½ �k3

::: T½ �k2z1

Tm3½ �~ T½ �n T½ �n{1::: T½ �k3z1

8>>><>>>:

ð16Þ

This is the first time Eq. (13) has been derived by the TMM for rotors having multiple shaftcouplers with offsets. Applying the boundary conditions into Eq. (13), a 969 system equationwill yield

0f g8|1

1

� �~

Tu½ �’8|8 uf g8|1

0f g1|8 1

" #S’f gL

8|1

1

( )zXp

j~1

mj

� �8|4

0f g1|4

( ): C’j� �

4|1ð17Þ

with the condensed offset vector

C’j� �

4|1~ {ej sinbj, {ej sinbj, {ej cosbj, ej cosbj

� �T ð18Þ

where Tu½ �’ and unbalanced excitation uf g are the degenerated matrices of Tu½ � and mj

� �degenerates from Tmj½ �, i.e., by picking up zero rows of the right boundary and nonzerocolumns of the left boundary from Tu½ � and Tmj½ �. S’f g is the degenerated vector of {S}. Therows (right boundary) and columns (left boundary) to be picked up based on differentboundary conditions are summarized in Table 1. Further, to simplify the above equation andrearrange it, the following equations can be found

Tu½ �’ 8|8 uf g8|1

0f g1|8 1

" #S’f gL

8|1

1

( )~

Ppj~1

{ mj

� �8|4

: C’j� �

4|1

1

8<:

9=; ð19Þ

or

Tu½ �’8|8| S’f gL8|1~{ uf g8|1{

Xp

j~1

mj

� �8|4

: C’j� �

4|1ð20Þ

Equation (20) presents the rotor response due to two types of excitation. One is theunbalanced excitation {u} from the disc and the other is the offset excitation. The offset appearsto be a much more complex excitation mechanism, since all the elements behind the offset


participate in the excitation. Provided there is no offset (ej 5 0, j51,…,p.), Eq. (20) yields anunbalanced response analysis, as found by many others [3,4,6,13].

4. NUMERICAL RESULTS

In the following, examples demonstrating the influence of the offset coupler via the developedTMM are illustrated. The rotor system shown in Fig. 2, where the configuration and material ofthe system are similar to Ref. [25], consists of five bearings, five rigid discs with three unbalances(eccentricity re 5 4.061024 m), and one eleven-sectioned flexible shaft. The Young’s modulus isE520.6961010 N/m2, density r58193.0 kg/m3 and the five bearings are assumed, though notnecessarily, the same (Kyy5 2.8506107 N/m, Kzz 51.0006107 N/m, KQQ5 4150 N?m/rad andKhh5 1520 N?m/rad). Detailed dimensions of the discs and shafts are given in Table 2.

Two misaligned couplers, one at stations A, B and the other at C, D are deliberatelyassumed. Although there might be various combinations of these two offsets, only the cases of

Fig. 2. Example of a misaligned rotor.

Table 1. Entries of Tu½ �’, S’f g, uf g and mj

� �.


e1 5 e2 5 1.061023 m where in-phase and anti-phase angles, i.e., b1~b2 and b2~b1z1800 areconsidered here.

First, Fig. 3 shows how the coupler stiffness influences the first three critical speeds. In Fig. 3,the coupler stiffness is categorized into three regions, soft (I), sensitive (II), and stiff (III). Weobserve in region (II), a slight change of stiffness causes significant variation in critical speeds. Inregion (I), the coupler stiffness has relatively less influence on the critical speeds of the system,especially for the third critical speed. However, as KL falls in the rigid area (III), the critical speedsbarely change with KL. This means the combined rotor dominates the critical speeds. In thefollowing examples, the coupler stiffness is set in region (II), unless otherwise specified. The firstthree modes with and without (dashed) offset with KL in the rigid and sensitive areas areillustrated in Figs. 4 and 5, respectively. Comparing the figures, it is seen the modes basicallyremain except there are bigger jumps at the offset couplers with KL in the sensitive area. When KL

falls in the rigid area, the mode shape is close to those of no offset and the offset maintains thestatic offset. The offset is amplified in the modes because the coupler is flexible, as seen in Fig. 5.

Figure 6 shows the frequency response function (FRF) of the rotor at station 14 for threedifferent offset values. The FRF with only unbalance is overlapped as a dash curve forcomparison. Figures. 6(a) and 6(b) show the anti-phase and in-phase results, respectively.Unsurprisingly, the offset does not bring any extra peak in FRF’s, but magnifies the responseamplitude. In the anti-phase case, due to the recall of offset, i.e., e2 5 2e1, the response amplitudeis very close to the unbalance, and even doubles the offset value. This means a counter balance ofoffset indeed reduces the previous offset response. As to the in-phase case, a significantlyincreased response amplitude is noted, especially at lower speeds where the offset dominates the

Table 2. Material and geometric parameters of illustrated example.


Fig. 4. First three mode shapes for in- and anti phase cases (KL in rigid area): (a) first mode shape;(b) second mode shape; (c) Third mode shape.

Fig. 3. Influence of coupler stiffness on critical speeds.


response amplitude. In addition, the response is linearly proportional to the offset value. Eq. (18)explains this is because the offset excitation is linearly proportional to e. As the speed increases,the unbalance force becomes more significant and dominates the response.

Next, the whirling orbits under different rotational speeds are examined. The 12th (A), 15th (C)and the 13th (B), 16th (D), corresponding to the front and the rear end of the couplers, are drawn.The anti-phase and in-phase cases are shown in Figs. 7 and 8 respectively. Three plotscorresponding to (a) V,Vcr1, (b) Vcr1,V,Vcr2, and (c) Vcr2,V,Vcr3 are shown. One canobserve the whirling orbit varies with the rotational speeds, as expected. At low speed (V,Vcr1),the ends A and D basically trace very similar orbits, except with a 180u phase difference; so do theends B and C. As rotational speed increases, especially at very high speeds, A and D no longerfollow the same orbit and the C and D orbits cross over. At the rotational speed of Vcr2,V,Vcr3,it is seen the orbits switch from forward whirl to backward whirl, as shown in Figs. 7(c) and 8(c).Whirling orbits are a rather complicated phenomenon. Often, as the rotation exceeds one criticalspeed, some parts of the rotor change their phase, i.e., from forward to backward or vice versa.Figures. 7 and 8 show the changing phenomena exist as the speed falls within the second and thethird critical speeds. In these figures, the dynamic offsets show how they vary with static offset.These dynamic offsets change with coupler stiffness. If rigid couplers (d%0) are utilized, thedynamic offsets roughly equal the static offsets and the orbits are very close to circles at low speed.Comparing Figs. 7 and 8, it is seen the in-phase offset has much larger orbits. This is attributed tothe disc centrifugal force enlarging the whirling orbit in the case.

Fig. 5. First three mode shapes for in- and anti phase cases (KL in sensitive area): (a) first modeshape; (b) second mode shape; (c) third mode shape.


Fig. 6. FRF due to disk unbalance and shaft offset.


Fig. 7. Whirling orbits at anti -phase for different rotational speeds: (a) VvVcr1; (b) Vcr1vVvVcr2;(c) Vcr2vVvVcr3.


Fig. 8. Whirling orbits at in-phase for different rotational speeds: (a) VvVcr1; (b) Vcr1vVvVcr2;(c) Vcr2vVvVcr3.


5. DISCUSSION AND CONCLUSIONS

In the present paper, the transfer matrix of multiple offset couplers was developed, and theresulting equations become much more complicated than one offset. Subsequently, a generalTMM for rotors was applied to examine the influence of multiple offsets. With these multi-offset equations, rotors with various relative phases can be studied. Two offsets with in- andanti-phase cases are studied. From the shown simple cases, the phenomenon of offsetcompensation is confirmed by the present derivation, that will never be seen by one offset case.The derivation showed coupler stiffness affects the rotor’s critical speeds and the offset acts asan excitation. This is similar to the conclusion of reference [15], that the offset barely changedthe rotor’s critical speeds.

TMM derivation and numerical results in the present studies revealed the offset induced therotor’s lateral response at the same frequency as rotational speed (16) and that was unlike mostother research where multiple integer (n6) components were found. Though reference [17]obtained results similar to the present paper and concluded the absence of 26 componentsmainly due to no consideration of bearing non-linear effects and shaft asymmetries. We,however, believe the reason for components n6 disappearing in our derivation is because thecoupler’s torsional vibration was not included. Since the coupler is designed to transmit torque,and as long as the coupler’s torsional flexibility is considered, the driven shaft will fluctuate intorsion, subsequently causing a non-constant rotation. The non-constant speed in conjunctionwith the misalignment and unbalance will consequently generate cyclic forces and moment’sexcitation on lateral vibration. The cyclic forces will result in excitation of n6 components asdescribed in [13–15].

The results also showed the offsets did not change the mode shapes, except minor jumps atthe offset couplers. Dynamic response became insignificant for two offsets in the anti-phasecase, implying an offset can be compensated by another offset in the opposite direction. Thewhirling orbits appeared to have the same traits as the direction of whirl in both the in- andanti-phase cases, but the in-phase offset showed much larger orbit radii. Rotation speed fallswithin the second and third critical speed; the whirling motion switches direction and thisphenomenon is not changed by offset, either in-phase or anti-phase.

Acknowledgement

The support for this research by the National Science Council, NSC 95-2221-E-011-010, isgratefully acknowledged.

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APPENDIX

The transfer matrix of a rotating shaft

Ts½ �17|17~

Ts½ �8|8 0½ �8|8 0f g8|1

0½ �8|8 TTs

� �8|8 0f g8|1

0f g1|8 0f g1|8 1

264

375 ðA1Þ

where Ts½ � 8|8~ H½ � 8|8 G½ �{18|8, TTs

� �5 HH� �

GGh i{1

H½ � 8|8~

H1,1 H1,2 H1,3 H1,4 0 0 0 0

H2,1 H2,2 H2,3 H2,4 0 0 0 0

H3,1 H3,2 H3,3 H3,4 0 0 0 0

H4,1 H4,2 H4,3 H4,4 H4,5 H4,6 H4,7 H4,8

0 0 0 0 H5,5 H5,6 H5,7 H5,8

0 0 0 0 H6,5 H6,6 H6,7 H6,8

0 0 0 0 H7,5 H7,6 H7,7 H7,8

H8,1 H8,2 H8,3 H8,4 H8,5 H8,6 H8,7 H8,8

266666666666664

377777777777775

ðA2Þ

H1,1~cosh l1; H1,2~sinh l1; H1,3~cos l2; H1,4~sinl2;

H2,1~b1 sinhl1; H2,2~b1 coshl1; H2,3~{b2 sinl2; H2,4~b2 cosl2

H3,1~EIzzb21 coshl1; H3,2~EIzzb

21 sinhl1; H3,3~EIzzb

21 sinhl1; H3,4~EIzzb

22 sinl2

H4,1~{ EIzzb31zr Izzv

2zIyyV2

� b1

� �sinhl1

H4,2~{ EIzzb31zr Izzv

2zIyyV2

� b1

� �coshl1

H4,3~{ EIzzb32{r Izzv

2zIyyV2

� b2

� �sinl2

H4,4~ EIzzb32{r(Izzv

2zIyyV2)b2

� �cosl2


H4,5~rVv IyyzIzz

� b3 sinhl3 ; H4,6~rVv IyyzIzz

� b3 coshl3

H4,7~{rVv IyyzIzz

� b4 sinl4 ; H4,8~rVv IyyzIzz

� b4 cosl4

H5,5~coshl3; H5,6~sinhl3; H5,7~cosl4; H5,8~sinl4

H6,5~{b3 sinhl3; H6,6~{b3 coshl3; H6,7~b4 sinl4; H6,8~{b4 cosl4

H7,5~{EIyyb23 coshl3; H7,6~{EIyyb2

3 sinhl3; H7,7~EIyyb24 cosl4; H7,8~EIyyb2

4 sinl4

H8,1~rVv IyyzIzz

� b1 sinhl1; H8,2~rVv IyyzIzz

� b1 coshl1

H8,3~{rVv IyyzIzz

� b2 sinl2; H8,4~rVv IyyzIzz

� b2 cosl2

H8,5~{ EIyyb33zr Iyyv2zIzzV

2�

b3

� �sinhl3

H8,6~{ EIyyb33zr Iyyv2zIzzV

2�

b3

� �coshl3

H8,7~{ EIyyb34{r Iyyv2zIzzV

2�

b4

� �sinl4

H8,8~ EIyyb34{r Iyyv2zIzzV

2�

b4

� �cosl4

ðA3Þ

l1~lb1,l2~lb2,l3~lb3,l4~lb4, where b1,b2,b3,b4 refer to [23].

G½ � 8|8~

1 0 1 0

0 b1 0 b2

G3,1 0 G3,3 0

0 G4,2 0 G4,4

0 0 0 0

0 0 0 0

0 0 0 0

0 G4,6 0 G4,8

0 0 0 0

0 0 0 0

0 0 0 0

0 G8,2 0 G8,4

1 0 1 0

0 {b3 0 {b4

G7,5 0 G7,7 0

0 G8,6 0 G8,8

266666666666664

377777777777775

ðA4Þ

where elements of G� �

and HH� �

are derivatives of G½ � and H½ �, differences among them are

GG4,6~{G4,6

GG4,8~{G4,8

GG8,2~{G8,2

GG8,4~{G8,4

8>>>><>>>>:

HH4,5~{H4,5

HH4,6~{H4,6

HH4,7~{H4,7

HH4,8~{H4,8

8>>><>>>:

HH8,1~{H8,1

HH8,2~{H8,2

HH8,3~{H8,3

HH8,4~{H8,4

8>>><>>>:

ðA5Þ


G3,1~EIzzb21; G3,3~{EIzzb

22

G4,2~{EIzzb31{r Izzv

2zIzzV2

� b1; G4,4~EIzzb

32{r Izzv

2zIyyV2

� b2

G4,6~rVv IyyzIzz

� b3; G4,8~rVv IyyzIzz

� b4

G7,5~{EIyyb23 ; G7,7~EIyyb2

4

G8,2~rVv IyyzIzz

� b1 ; G8,4~rVv IyyzIzz

� b2

G8,6~{EIyyb33{r Iyyv2zIzzV

2�

b3 ; G8,8~EIyyb34{r Iyyv2zIzzV

2�

ðA6Þ

Transfer matrix of an unbalanced disk

Tud½ �17|17~

T�d� �

8|80½ �8|8 u1f g8|1

0½ �8|8 T��d

� �8|8

u2f g8|1

0f g1|8 0f g1|8 1

264

375 ðA7Þ

where

u1f g~ 0 0 0{mereV2 cosae 0 0 0{mereV

2 cosae

� �T

u2f g~ 0 0 0{mereV2 sin ae 0 0 0{mereV

2 sinae

� �T

8<: ðA8Þ

T�d� �

, T��d

� �are the same as Td½ � except the following elements

T��d3,6~T��d7,2~{T�d3,6~{T�d7,2~IxV2

T��d4,1~T��d8,5~T�d4,1~T�d8,5~{ mzmeð ÞV2

(ðA9Þ

Td½ �~

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 {Izv2 1 0 0 {IxvV 0 0

{mv2 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 {IxvV 0 0 0 {Ixv2 1 0

0 0 0 0 {mv2 0 0 1

266666666666664

377777777777775

ðA10Þ

Transfer matrix of an oil-filmed bearing

Tb½ �17|17~

Tb1½ �8|8 Tb2½ �8|8 0f g8|1

Tb3½ �8|8 Tb4½ �8|8 0f g8|1

0f g1|8 0f g1|8 1

264

375 ðA11Þ


where

Tb1½ �~

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 kww 1 0 0 Cwhv 0 0

kyy 0 0 1 Cyzv 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 {Chwv 0 0 0 khh 1 0

{Czyv 0 0 0 kzz 0 0 1

266666666666664

377777777777775

ðA12Þ

Tb2½ �~

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 Cwwv 0 0 0 kwh 0 0

Cyyv 0 0 0 kyz 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 khw 0 0 0 {Chhv 0 0

kzy 0 0 0 {Czzv 0 0 0

266666666666664

377777777777775

ðA13Þ

Tb3½ �~{ Tb2½ �ij , besides Tb3½ �3,2~{ Tb3½ �3, 2 , Tb3½ �4, 1~{ Tb3½ � 4, 1

Tb3½ � 7, 6~{ Tb3½ � 7, 6 , Tb3½ � 8, 5~{ Tb3½ � 8, 5

Tb4½ � ij~{ Tb1½ � ij ,besides Tb4½ � 3, 6~{ Tb1½ � 3, 6 , Tb4½ � 4, 5~{ Tb1½ � 4, 5

Tb4½ � 6, 2~{ Tb1½ � 6, 2, Tb4½ � 7, 1~{ Tb1½ �7, 1

ðA14Þ


vibrations of a rotor system with multiple coupler offsets · the whirling orbits before and after...

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