modal analysis of continuous rotor-bearing systems

7/28/2019 Modal Analysis of Continuous Rotor-Bearing Systems

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Journal of Sound and Vibration (1988) 126(2), 345-361

MODAL ANALYSIS OF CONTINUOUSROTOR-BEARING SYSTEMS

C.-W. LEE AND Y.-G. JEIDepartment of M echanical Engineeri ng, Korea Adv anced i nsti t ut e of Science and Technol ogy Seoul ,

Korea

(Recei ved 27 November 1987, and in revised form 5 May 1988)

Mo dal analysis is applied to continuous rotor systems with various boundary conditionswhich include isotropic and anisotropic natural boundary conditions. The rotor includesthe effects of rotary inertia and gyroscopic mom ent. In particular, the whirl speeds andmode sha pes, ba ckward and forward, of a rotating shaft are obtained as spin speed andboundary conditions vary, and the unbalance respons es are calculated by using mod alanalysis. T he effects of asymm etry in boundary conditions on the system dynamic charac -teristics are also investigated.

1. INTRODUCTIONVarious methods for transverse vibration analysis of rotor systems have been developedduring the past few decades. These m ay be divided into two major classes according tomodelling proce dure. The first is the discretization meth od, such as the FEM and TransferMatrix Method, in which a rotor system is approximated by a finite-degree-of-freedomsystem whose motions are described by ordinary differential equation s [l-4]. The secondis the analytical method in which a rotor system is treated as a distributed parametersystem whose motions are described by partial differential equations [5-g]. With therecent development of computer hardware and software, the discretization method h asbecome a popu lar method for analys is of transverse vibrations of rotor systems, since itcan be easily applied to complex rotor system s. H owever, it is often d ifficult, if notimpossible, to look into, in a systematic way, the effects of system pa rame ters such asgyroscopic mom ents, rotary inertia and boundary conditions on the whirl speeds, modeshapes and stability of a rotor. Fu rthermore, the results obtained by the discretizationmethod may not be as accurate as desired. On the other hand, although the solutiontechniques are limited to relatively simp le rotors, the analytical method often yieldsessential info rmation on the behavio rs of rotor systems.

The dynamic characteristics of distributed parameter rotor systems have been studiedby a few investigators. Gladwell and Bishop [5] performed a modal analysis of anEuler-Bernoulli beam supported by flexible bearings and solved the response problemsassociated with mass unbalance, initial curvature, and gravity. Dim entberg [6] providedan excellent review of the state-of-the-art of rotor dynam ics a nd analytical solutions tomany rotor vibration problem s, includin g the effects of gyroscopic moment and rotaryinertia. Eshlem an and Eu banks [7] investigated the effects of externally applied axialtorque on the critical speeds of a continuous rotor, inc luding the effects of shear deforma-tions. Recently, Lee et al . [9] applied modal analysis to a rotating Rayleigh beam includingthe effects of rotary ine rtia an d gyroscopic momen ts. The difficulties in modal analysisof rotor systems which include the effects of rotary inertia and gyroscopic moments arise

3450022-460X /88/200345+ 17 %03.00/O @ 1988 Academic Press Limited


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346 C.-W. LEE AND Y.-G. JEIfrom the fact that the resulting eigenvalue problem s are characterized by the presence ofskew symmetric matrices w ith differential operators as elements, due to rotation and/ordamping, resulting in non -se l f -ad jo in t e igenua fue p rob lems [9]. The resulting eigenvalueproblem becomes then a standard non-self-adjoint eigenvalue problem when the equationof motion is written in state space rather than in configuration space [9, lo]. Althoughthe boundary conditions in reference [9] were restricted to the isotropic geometricboundary conditions as in references [6,7], it was found that the forward whirl modeshapes are different from backw ard whirl mode shapes, w hich has not been observed inresults obtained by the discretization methods.In practice, rotor systems are supported by bearings which m ay not be appropriatelyrepresented by geometric boundary conditions only. The aim in this paper is to extendthe modal analysis developed in reference [9] to rotor systems having isotropic andanisotropic natural boundary conditions. The effects of rotary inertia and gyroscopicmom ents are also included as in reference [9]. The whirl speeds an d mode shapes of arotating shaft with various boundary conditions, and the unbalance responses, are com-puted. In particular, the forward and backward whirl mode shapes w ith the spin speedvaried and the effects of the asymmetry in boundary conditions on the system dynamiccharacteristics are also investigated .

2. EQUATIONS OF MOTIONConsider a flexible non-uniform rotor system consisting of D disks and B anisotropic

bearings as shown in Figure 1. For simplicity, it is assumed that discontinuities in stiffnessand inertia caused by disks and bearings are well represented by a train of delta functionsalong the shaft axis. The equations of motion including gyroscopic and rotary inertiaeffects are then expressed, in inertial co-o rdinates, as

+ kJx)y + k,,(x)z =Sy(x, ), (1 4

+ k&b+ k&b =./Xx, ~1, (lb)where O < x < l an d

pA (x )=pA e(x )+ f dS (x -xd ) ,d =l

Figure 1. General rotor-bearing system.


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CONTINUOUS ROTOR-BEARING SYSTEMS

Jr(x)= JpT(x )+ ; J 7 6(x-& j ) , Jp(x)= J; (x)+ ; J$S(PX d) ,d= l d=l

b(x) = i k;yS(x -x& k,,(x) = i +(x - xb),

347

b- b-

Lx) = it k:za(x -Xb), k.,(x) = i kf$(x - xb).b=l b=

Here PA ( X ) is the mass per unit length, &(x) the diametral mass moment of inertia,JP(x) the polar m ass moment of inertia, El(x) the flexural rigidity, m d the disk mass, Ithe length of the rotor, R the spin speed, x the position co-ordinate along the shaft, andfy(x, t),fr(~, t) are the distributed forcing fun ctions in the y and z directions, respectively.The superscript e denotes the shaft and, d and b denote the dth rigid disk located atx = xd and the bth discrete bearing located at x = xb, respectively. The associated generalboundary conditions are

(3)where the superscripts and I denote the terms located at x = 0 and 2, respectively. Theboundary conditions of equations (2) are associated with the shear force and the deflectionat both ends, and equations (3) are associated with the bending mom ent and the slopeat both ends of the rotor system considered. Equations (l), (2) and (3) constitute thebounda r y va l u e p rob l em o f t he ro t o r syst em .

It is notationally convenient to introduce the state vectorW(x, t) = {w, w, w, w, wO}T, (4)

wherew(x, t) = e, t)I I Y(X, I)u(x, r) u(x, t) = 1 I, t)

w=w(l, t), w = w(0, t), W" = i ) w/ axl , , , , do = r?rr/ax(,,,.The equations of motion (l), with the boundary conditions (2) and (3), can be rewrittenas

MW=LW+F (5)


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348where

C.-W. LEE AND Y.-G. JEI

The element matrices in equation (5) are provided in Appendix A.

3. MODAL ANALYSIS METHODThe inner product of two complex state vectors a = {a,, u*}~ and b = {b, , b2}T s definedas

(a,b)=((Il.bl)+(n2,b2)=~~~~,=(dx+ld~~L?idx (6)where the bar denotes the complex conjugate. For the two state vectors W, and W 2

(W,,W,)=(jt,,)i*>+(il,i*>+(y,,y,)+(z,,z,)+B(I,t)+B(O,r), (7)where

B(x, r) = i,(x, f)G, t) +&C T t)&(x, r) +Y,(x, f)%(X, r)+ z,(x, rP*(x, r)+9:(x, t)j$(x, r)+ii(x, Q& x, t)+yi(x, t)$(x, t)+zi(x, f)fi(X, t).

The eigenvalue problem associated with equation (5) and its adjoint are given byh,M+ ,=L+, r=l,2,3 ,..., &M *YS = L*L,, s = 1,2,3,. . . , (8)

where M * and L*, the adjoints of M and L, respectively, are found to beM* = MT, L* = LT. (9)

The eigenfunction vector +, and the adjoint eigenfunction vector W Y, re given as4, = {A&,, 4, A&I, 4,, & 4:, 44 A,& 49, &44O, d4],f* = {UJS, &, US, IL: 9LJI:, $5 w, *:, MO, $t>, (10)

where

4, and Vr, may be biorthonormalized so as to satisfy(M4 ,, Js) = a,, (L&, *S) = G% S, (11)

where 6, is the Kronecker delta. From equation (8), if k,,(x) = -k,,(x), k$ = - k t ,,, an dk : , = -k i ) , , + , and WY , atisfy the same eigenvalue problem [9], resulting in

&Jx) = K#%&) an d &Z(X ) = -G#J,(x), (12)where the constant K, is to be determined from biorthonormality conditions and isprovided in Appendix B. The biorthonormality conditions expressed in terms of eigenfunc-tions and adjoint eigenfunctions are obtained from equation (11) [9]. l/K,, a complexquantity, is a so-called moda l n o rm .


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CONTINUOUS ROTOR-BEARING SYSTEMS 34 9The distributed state u(x, 2) can be expanded in terms of the system eigenfunctions as

follows

The summ ation indices F an d B in equation (13) implicitly indicate the resonances ofthe rotor in forward and backw ard precessions, respectively. Since u is the real statevector, the complex conjug ates of + l q l always exist in the summ ation of equation (13).Substitution of equation (13) into equation (5) yields an infinite set of modal equations,

& =A ,q ,+ f , , i = F , B , r=l,2,3 ,..., (14)where

~=(W,V\Y:(x))= ;(&&+ihf,)dx.IEqu ation (14) represents an infinite set of indepen dent first-order complex ordinarydifferential equations.

4. UNIFORM CIRCULAR SHAFTIt is convenient to introduce the non-dimensional variables

x=x/& 5=yl& 77=z/l, 5=u/l, 7=0t,c2 = pA e14R2 / E I , r = R/21, Wx, 7) =_I-IPA~~~~ , (13

where u(x, t) = y(~, t) +jz(x, t), the complex displacement function, and f(x, t) =f,(x, t) +ifr(x, r) the complex forcing function. Then the equation of motion of a uniformcircular shaft w ith no disks and bearings on 0 < x < 1 can be reduced in non-dimensional-ized form, from equation (l), as

a25 2 a41g-r aX2ar2 a5-2- ax2a7>+L!%=hc2 ax4 (16)the boundary conditions (2) and (3), can also be rewritten in non-dimensionalized form.For a rotor with distributed mass center eccentricity y,(x), z,,,(x)), the equivalen tunbalance force is given as

&(x, 7) = &(x) cos T- W(X ) sin 7, h,(x, 7) = 77,(x) cos T+& (X) sin 7, (17)where the non-dimensional mass center eccentricity is

&l(x) = Ym(XVl, Ilndx) = GAxVl. (18)The non-dimensionalized displacement can be represented as (from equation l3)),

The set of modal equations (14), can be then rewritten in non-dimensionalizedc j ;=A ;oq :,+h ; , i = F, B, r = 1,2,3, . . . ,

where

h:=(h(x, 7),$&4x)>= I Gi?o,h,+k,h~) dx.0

(19)form as

(20)

The subscript 0 in equations (19) and (20) denotes a non-dimensionalized term and itwill be omitted henceforth for notationa l convenience.


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35 0 C.-W. LEE AND Y.-G. J EI4.1. UNIFORM SHAFI- WITH ISOTROPIC BOUNDARY CONDITIONS

Wh en the boundary conditions as well as the rotor are isotropic, the whirl mode shap esbecome real and planar, and the whirl motion is circular [6,9]. Therefore the forwardand backward mode shape vectors in equation (10) have the forms

(21b)The homogeneous part of equation (16) permits a solution of the form

4+(x,T) = 4 (x ) q (T ) = I& jG ejp r (22)where 5(x, 7) = 6(x, T) + jn(x, 7). The value of a characterizes the mode shapes and a isthe natural frequency. Since the whirl frequency is given by w = aR, a = 1 at the criticalspeeds where the whirl speed coincides with the spin speeds. Substitution of equation(22) into the homogeneous part of equation (16) yields

a2 - r2c2(u2 - 2a)a - c2a2 = 0. (23)Solving this equation with respect to a gives

(y =4r2C2(a2-2Q)fJ~t4c4(a2-2q)2+C2a2. (24)The expression for a shows th at, for a constan t value of c2, each value of the naturalfrequency a will result in two values of a, one always being positive and the othernegative. Therefore, the mode shap e functions 4(x) will take the form4(x) = A, cos &x + A2 sin &x + A 3 cash ax + A, sinh 4~ (25)where -p and v are the two values of a which satisfy the relation (23). Normally thepositive values 1~and v are different from each other (unlike the case of the Euler-Bernoullishaft), satisfying the relations

-/I + v = r2c2(a2 -2a), ~1.y c2a2. (26)For c2 given, equations (26) constitute two equations with three unkno wns, V, p and a.One additional constraint comes from the frequency equation, which depends on theboundary conditions specified. T wo typical isotropic (natural as well as geometric)boundary conditions will now be considered. For cases of fixed or simply supported ends(x =0, l), the frequency equations are given in reference [9].4.1.1. Can t i l e ve r s h a f t i t h a t i p disk

When a tip disk is attached at the free end of a cantilever shaft, the boundary conditionsare4(O) = 4(O) = 0, 9(1)-C2(y:a2-y~a)~(l)=0,

c#J(1)+/3c2a2~(1)+r2c2(a2-2a)~(l)=0, (27)where /3 = m / pA i , y \ = J:/ pAe13 , and y; = J , l pA l . Imposing the boundary condi-tions (27) on equation (25) gives the frequency equation as

CrCqVC2C3=0, (28)


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CONTINUOUS ROTOR-BEARING SYSTEMS 35 1where

C2 = v cos &+ p cash G- c2( y&x*- &I)(& sin &+& sinh a),C3 = -&(p cos &+ v coshG)+pca*[sin&-(&I&) sinh &I,C,= (p&sin& - v&sinhG )+Pc2a2(cos fi-cash &>.

By using the relations of (26), (27) and (28) a set of A i ( i = 1,2,3,4), v and p togetherwith a are determined. As discussed in reference [9], each natural frequency correspondsto a single mode sh ape, implying that the mode sh apes corresponding to forward andbackward precessions are different.

The whirl speeds and m ode shapes of the cantilever shaft with a tip disk are shown inFigures 2 and 3. Figure 2 indicates that the gyroscopic mom ents increase the forwardwhirl speed and decrease the backward whirl speed as the spin speed increases. Figure3 shows that the distance between the neighboring nodes tends to decrease (increase) inthe forward (backward) mode sh apes as the spin speed increases. The solid line denotesthe mode shape of non-rotating shaft. This phenomenon confirms the change in the whirlspeeds of the forward and backward precessions.

25 - 2Fzo---_1 ---___ --___ -28--_

In 15-Ef IO-

5- IFI I I 1 , 18

0 5 IO 15 20 25 30Spin speed

Figure 2. Whirl speed of cantilever shaft with a tip disk (r = 0.02, /3 = 0.7, y: = 0402, yf = 0404).

4.1.2. Sha f t w i t h so t r o p i c sp r i n g suppo r t s a t endsWhen the shaft is supported by isotropic bearings at both ends (x = 0, l), the boundaryconditions are

4(O) = #( 1) = 0, f # J ( 0 ) +K ~ (O )+~2C2 (U2 - 2~ )~ ( 0 ) = 0 ,4(l)-K+(1)+r2c2(a2-2a)4(1)=0, (29)

where K = k l - / E I = K;~ = i t ) , , , and K = k 13/ E I = K& = KS, , . Imposing the boundaryconditions of (29) on equation (25) gives the frequency equation0 -- v 0 P-vsinJ; -vcos& p sinh& k cosh&

-jL& K 0 VG K 0Q D2 03 0 4

1S

= 0, (30)


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352 C.-W. LEE AND Y.-G. JEI

Fig. 3. Mode shapes of cantilever shaft with a tip disk (c = O-40, @ = 0.7, 7: = O+W2,7; = 0@04) (a) Secondmode; (b) third mode. -.-, Forward; ----, backw ard.

whereD,=p\/;;cosfi-Ksinfi, D,=p&sin&-K1cosfi,

D,=v&cosh&-Ksinh&, D = v&sinh&-Kcosha.As before, a set of A i ( = 1,2,3,4), p and v are determined from equation (30) by usingthe relations (26). The more general problem of an anisotropic spring support is treatedin detail in the next section.4.2. UNIFORM SHAFT WITH ANISOTROPIC BOUNDARY CONDITIONS

The mode shap es of a rotor system with anisotropic boundary conditions are complexand non-planar, and the whirl motions are not circular. With the assumption thatKL, = Kit< i = 0, l), that is, orthotropic bearings, the homogeneous part of equation (16)perm its a solution of the form6(x, 7) = & (,y)q( 7) = A ejG ej , 77(x , T) = & (x)4(7) = B ej+e@. (31)Substitution of equation (31) into the homogeneous part of equation (16) gives

{Cl- a2c2( r2a + l)}A + 2jar2c2aB = 0, -2 j ar c aA + {a - a2c2( * a + l)}B = 0.(32)


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CONTINUOUS ROTOR-BEARING SYSTEMS 35 3From equations (32), the condition for existence of non-trivial A and B yields

f f2-r 2c2(a2-2a)a-c2a2=0, a2-r ZC2(u2+2a)a-c2a2=0. (3% b)Equation (33a) is identical to equation (23), whereas equ ation (33b) corresponds to theisotropic rotor rotating in the opposite direction. Solving these equations for a gives

a =fr2~2(a2-2a)*.J~ r4c4(a2-2a)2+c2aZ, (344C! =$r2c2(aZ+2a)*tJ fr4c4(c2+2u)2+ c2u2. (34b)

The expressions for a show that, for a constan t value of c2, each value of the naturalfrequency a will have four values of a, two always being positive and others negative.Therefore, C& (X) and 4,(x) will take the forms

&(x) = A , cos f i x + A2 sin fix + A , cash Gx + A4 sinh X&X + A 5 cos & ,y+ A 6 sin X&X + A , cash X&X + A s sinh X&X, (35a)

~$,(x)=B,cos~x+B~sin~~+B,cosh~~+B~sinh~~+B,cos&&+ Bs sin fix + B , cash &x + B 8 sinh X&X, (3Sb)

where Y, and -p, are the two values of a satisfying equation (33a), and v2 and -p2 thetwo values of a satisfying eq uation (33b) : i.e.,

-1, + VI= r2c2(u2 -2a), p, v, = c2a2, (364-/ .Lz+ vz= r2c2(a2+2a), p2v2 = c2a2. (36b)

For c2 given, equations (36) are four equations with five unknow ns, kl, p2, vI , v2 an da. One additional constraint comes from the frequency equation which depends on theboundary conditions specified. When a solution for a is found, then it can be readilyshown, from equations (36), that -a will automatically become another solution with CL,and vl interchanged with cc2and v2, respectively, in &(x) and 4,(x). On the other hand,the mode shapes associated with a and -a should be a complex conjugate pair. Therefore,the following relations hold

A,lii,=A,lA,=A,/A,=A,lA4=c,, A,/A,=A,/A,=A,IA,=A,lA,=c,,

B,ll?,=B,lB,=B,l~,=B,lB,=c,, &I& = B2/ & = I?,/ B, = B,/ i i , = c4(37)

Here A,, Bi are the mode shape coefficients with -a and evaluated with p, and v, beinginterchanged with p2 and v2, respectively. The CiS i = 1,2,3,4) are the complex constants.From equations (32) one also obtains the relations

B i = - j A i ( i = 1,2,3,4), B i= jA i ( i=5 ,6 ,7 ,8 ) . (38% b)Therefore &(x) and 4,(x) of equations (35) can be rewritten as

4*(x) = 4(x) + C(x), 4,(x) = -j{++(x) - d-(x)), (39)where

4(x) = A , cos 4~ + A2 sin &x + A 3 cash &x + A4 sinh Gx,4- ( x )=A ,c os& & +A 6 sin fix + A , cash J& + A B sinh &x, i-(x) = c,J(x).Here 4(x) and c#-(x) are the mode shapes associated with two different isotropic rotors,one rotating in the specified d irection and another in the opposite direction, respectively,implying that the anisotropic rotor is conceptually equivalen t to these two kind s of


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35 4 C.-W. LEE AND Y.-G. J EIisotropic rotors. In com plex notation, the non-dimensionalized forced response can bewritten as (from equations (19) and (22))

In the case when the boundary conditions becom e isotropic, one obtains for forw ardmodes from equation (21a),

$7(x) =f(i5+$&)=0, (4Ia)and for backwa rd modes from equation (21b)

9?(x) = S(#~+jG) = 0. (4Ib)The refore the non-dimensionalized forced response of an isotropic rotor becom es

5 = 5+h = 2 4TF(x)qf+ &B(x)4rS),implying that d:(x) and & LF(x ) play a role in the forced resp nse only when the systemrem ains anisotropic. In fact, it has been already shown in reference [9] that the eigenvalueproblem associated with isotropic rotor systems, when written in complex notation, yieldsa single eigenvalue corresponding to each mode. Although the complex conjugatesassociated w ith #FE(x) (d;(x)) in forward (backw ard) modes seem naturally to becomeaddition al eigensolu tions, they actually result from the rotor rotating not in the specifieddirection but in the opposite direction. They exist only as a pure math ema tical con-sequence , never contributing to the forced responses or resonances. In this respect, theobservation that half of the anisotropic rotor resonances disappears suddenly as theanisotropy becomes n ull is clearly explained.

Two typical anisotropic boundary conditions can now be considered.4.2.1. Dissimilar fixing conditions in two perpendicu lar ben ding planes

When a shaft is simply sup ported in two perpen dicular bending planes at one end(x = 0), and fixed in one ben ding plane and simp ly supported in another bending planeat the other end (x = l), the boundary conditions becom e

de(O) = MO )= &(I) = 4;(I) =O, ~,(0)=~;(0)=~,(1)=9,(I)=O. (43)the frequency equation then becomes

sin J;, sinh a, 0 0(v,+p,)sinJv, 0 ( y2 + 1~~) in fi, 0 =0 0 sin Jv, sinh G2 0. (4)

J;, cos 6, &, cash X& - 6, cos &, - fiz cash &Here each natural frequency corresponds to a single mode shape, implying that the modeshapes corresponding to forward an d backward precessions are different. D ue to theanisotropic boundary condition the mode shapes, &(x) and C& (X), are different and oneadditional resonance corresponding to each mode appears.


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CONTINUOUS ROTOR-BEARING SYSTEMS 35 54 .2 .2 . O r t ho t rop i c sp r ing suppo r t s

When a shaft is supported by orthotropic springs at both end s (x = 0, l), the boundaryconditions are

4;(0)=@(1)=0, 4;(o) = 4;(l) = 0,~;(O)+KO~~(~)~C(O)+KOC?0)9&(O)+ r2c2a2[d;(0) -.iW~MiAO)l =O,

~~ (1 ) - K~~~~ (1 )+ r2C 2U2 [~ ;(1 ) - j ( 2 / ~ )~~ ( l ) l =0 ,~ ~ (O )+KO?? (~ ) ~ ? (O )+KO~~ (~)~,(0)+~c2~[~~(~)+j(2/~)~~(0)1=0

d~(l)-K1?~?(1)+r2c2U2[~~(1)+j(2/u)d;(1)l=0, (45)where

and 8, as shown in Figure 1, is the inclination angle of the orthotropic bearing at x = 0relative to the orthotropic bearing at x = 1. By imposing the boundary conditions of (45)on equation (35) the frequency equation is found to be

IQii l = 0, (4)where the matrix elements Qij( i,j = 1,2,3,4) are as given in Appendix C. When theprincipa l axes of two orthotropic bearings coincide, i.e., K&( 6) = 0, the values of Qijbecome real, resulting in real mode sh apes. Figure 4 shows the whirl speeds a s functionsof 8 when the bearing properties are given as

K$=K ;~=80 an d K11)-K19 .0 - 1 =LM) (47)The whirl speeds change significantly as 8 varies, but are insensitive to a change in thespin speed, due to the small gyroscopic effect of the uniform slender shaft considered .For 8 = 0 and 5 , the mode shapes, &(x) and 4,(x), are shown in Figure 5. Wh en 8 is

%Ef

24 -

I I 1 1 I I0 45 90 135 180Angle kgrees)

Figure 4. Whirl speed of a shaft supported in orthotropic bearings as 0 varies (c = 20, t = 0.02).


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356 C.-W. LEE AND Y.-G. JEI

I


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CONTINUOUS ROTOR-BEARING SYSTEMS 35 70, the mode shapes are real and planar, and the relative phase angle between d,(x) and4,(x) is 90. The relative phase angle of 90 implies that the major and m inor axes ofwhirl orbit coincide with the 5, n axes. When 6 = Y, the mode shapes become complexand non-planar, and the relative phase angles are no longer 90. In general, the modeshapes &(x) and 4,(x) corresponding to the forward an d backward whirls are different,as shown in Figure 5.

When K&(X) = K$ ( x ) = 0 (i = 0, l), the mode shapes become real, K , becomes a pureimaginary value and the distributed state 5(x, T) can be rewritten, by using equation (19),as(48)

where g:= ql+i$ and g; = ql- ijl, r = 1,2,3, . . . , i = F, B, satisfying~:i+hI;iIg:=(hl-;iI)h,+2~1,,

~;i+A:~lg;=2~~~+(hl-~l)hl,, r=1,2,3 ,..., i=F,B, (49)with

To calculate the forced vibration due to the lack of mass balance, gTi and g;, may beassumed to have the forms, from equation (17),g:(x, r) = g:(x) cos r+ g:(x) sin 7, gFi(x, T) = gLi(x) cos .r+g&) sin T,

r=l,2,3 ,..., i=F,B. (50)Substitution of equations (50) into equations (49) yields

The steady state unbalance response can be obtained from equations (48), (50) and(51). For the non-d imensionalized ma ss center eccentricity given as t,,,(x) =O-02 S(x -0.25) and am = O-0, Figure 6 shows the unbalance responses of the shaftwith the bearing properties of equation (45), with 0 = 0. The unbalance responses havebeen calculated with retention of 10 modes. In Figure 6 the solid and broken lines denotethe unbalance responses observed at x = 0.25 in the n and 6 directions, respectively. Thefirst backward whirl occurs, as the spin speed increases, in the region between the first


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CONTINUOU S ROTOR-BEARING SYSTEMS 35 9APPENDIX A: ELEMENT MATRICES OF ANISOTRPOIC ROTOR SYSTEM

The element matrices of M and L are given as follows:0 0 pA-a/ ax(J , a/ax) 00 0 0 pA - a/ax(J, d/ax)m= pA-d/dx(J, a/ax) 0 0 -Rd/ax(J, d/i t x)0 pA -a/ax(J, a/ax) R ij /dx( J, a/dx) 0 I

0 0 0pA - alaw, a/ax) 0 0k= 0 - [a' / ax' (EI a2/ ax2)+ k,,] -k,:

0 -k:, - [ i ) / i i x ( El i?/ i ix*) +

where0 0 m'+J;a/ax

,'= 0 0L m+J,a/ax 0 :0 m+J;a/ax -n(J; a/ax)

0 0 mO- Jq ii /ax 0

m y= 0m - JpTa/ax 0 0 m"- $ a/ax '0 0 -n(Je, a/ax) m i=0 m"- ; a/ ax n(J; a/ ax) 0

m+J,a/ax 0 0k;= 0 m+J,dlax 0

0 0 [a/ax(EZ d2/ax2) - k_i,,]0 -k:,, [a/ax(EI a2/ax2) - ki,]

J:. 0 0 0

k;= 0 J: 00 -EI a/ax 00 0 0 -

0 1 'EI d/ax

0 0 J; 00 0 0 J ,J; 0 0 flJ(:0 J , -f 2Jo, 0

00

- bz

m-J;a/ax 0 0 0k;= 0 m-J,a/ax 0 00 0 - [a/ax( EI a/ax*) + ky.,.] -k;, 9

0 - [a/dx( EZ a2/i+x2) +1

ktz]

0 EI alax .


16/17

360 C.-W. LEE AND Y.-G. JEIAPPENDIX B: MODAL N ORM OF ANISOTROPIC ROTOR SYSTEM

APPENDIX C: FREQUENCY EQUATION OF THE ROTOR SUPPORTED BYANISOTROPIC SPRINGS

Q3,=a:S,+a:S3+a:, Q32=&+&,+~~,Q3,=u~Ss+u;S,+u~, Q34= u:S,+u:S,+u;,Q4, = a$, + a&+ a;, Q42=U~S2+U~S4+U~,Q4 3 =u $ 5 +u ; S, + u : , Q 44=u$6+u$,+a~ ,

wheresin J;;

S1=(-cosJ;;+coshJ;;;)sinh &

= -(-cosJ;;+coshJ;;):sin J11,

s3=(-cosJ;;+coshdj&sinh X&

= -(-cosJv,+coshx&)ss= sin J;; s,= - sinh X& CL 2( - c o s &+cosh a) (-cos &+cosh &) ;s, = sin J;;(-cos Jv,+cosh a) 2 sinh X&= -(-cosJv,+coshJCc,)


17/17

CONTlNUOUS ROTOR-BEARING SYSTEMSand

36 1

a i = p ,x& sinh a- K:~ cash a+ r2a2c2

a: = v& sin J;; - KX

7a6= -v,~cos~-K~~sinJY,+r2a2c2a:=p2~sinh&-tc~tcosh&+r2a2c2

a:=v,Jv,sinJ;;-K&cosJY,-r2a2c2

a! = p,& sinh \j;l;- ~1,

at = +,a cash x& - K:,, sinh t&-I- r2a2c2

a:=-v2Jv,sinJ;;;+Ki, 2cos J;;+ r2a2c2 1+- &sin 6,( >8a6= l ,J;;C0S&+Kf7n

a; = -p2& sinh a+ if,, cash a-- r2a2c2

ai = -b2& cash a+ K&, sinh a- r2a2c2

modal analysis of continuous rotor-bearing systems

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