Short bearing theory

5 The Short and the Long Rotor

Therefore an unbalance moment excites no resonance for a short rotor. For long rotors a resonance exists, as Fig. 5.12 shows for a rotor with I j = 1.1 Ip, 2IP and 57p.

From equations (5.26) the maximum of Vt is close to v\ — 1, that is at Q — on, where from equations (5.10) and (5-21), ¿2 = J?4. The corresponding amplification factor becomes

1


(5.27)

2


Fig. 5.12 shows that with constant damping ratio D— 0.05, the resonance curve becomes flatter with decreasing Id/IP- The dotted line is the resonance curve for a rotor with Id = 1.1 Ip and zero damping. This shows that rotors with I ^ f v Ip, where Id > Ip and with small damping have a rather flat resonance region.

The resonance described is excited by the rotating unbalance moment, that is by the centrifugal moment of constant value. This resonance is at natural frequency u4

and is in the direction of shaft rotation. A harmonically varying moment, but of constant direction can be separated into constant co-rotating and counter-rotating terms, each of half the amplitude of the harmonic moment. Hence the resonance consists of backward whirl at ,u>3 and of forward whirl at u>4. Similar relationships exist to those described in Sect. 4.4 for a harmonic force. A moment excitation, with constant direction occurs, for example, when a bearing is excited in one direction in a harmonic fashion, with 180° phase difference from the excitation at another bearing.

n._____

Pig. 5.12. Amplification factor V t. — D = 0.05,--------D - 0.

3

6 Oil-Film Bearings

Most rotors are supported in oil-film bearings or in rolling-element bearings. For maintenance free operation and to give high speeds, gas and magnetic bearings are also increasingly being used. The bearings influence the rotor vibrations to a greater or less degree by their dynamic properties in comparison to those of the rotor and the remaining components of the assembly. This influence originates essentially from the ratio of the rotor stiffness to the bearing stiffness. For a relatively elastic rotor the influence of the bearings is small, while the dynamics of a stiff rotor are determined in the main by the bearings.

This chapter will be concerned with the static and dynamic properties of oil-film bearings. This subject is very complex and in the last four decades a large number of significant contributions have appeared in the literature. To give these here would be too space consuming. We confine ourselves therefore to the essential theoretical fundamentals and consider especially the stiffness and damping coefficients, through which the oil-film bearings will be considered using conventional rotordynamic theory.

6.1 Hydrodynamic Bearing Theory

The components used in the theory of hydrodynamic bearings are the bearing-journal, the lubricant and the bearing shell. The journal rotates with angular velocity J? and is statically or dynamically loaded in the radial direction. The bearing shell is rigidly supported. The relation between the force and the dis-placement of the journal is required. The properties of the lubricant and the geometry of the clearance will play a considerable part in this relationship. We confine our consideration of the clearance geometry to the type of bearing shown in Fig. 6.1: The bore should be constant along the axis in any radial direction, that is consist of circular arcs. The cross-section should be approximately or exactly circular. For a non-circular bore we establish a reference circle of radius R. Its centre is at the midpoint of the bearing length:X and is denoted by CB -The point CB is the origin of the coordinate system X , V, Z. The bearing journal has radius r and centre C j in the A"Y-plane. It is assumed for simplicity that the journal is parallel to the bearing bore and as well as rotation undergoes only translational

motion parallel to the XY-plane. In other words, any moments arising from journal tilt are neglected.

The journal is unloaded when Cj and Cg are coincident. The radial distance between the journal surface and the bearing bore for a central journal is termed the oil-film thickness ko. In general ho differs around the circumference, that is h0

= h 0 ( i p ) . For a circular bearing bore f i Q ~ R — r, and is constant. This value is here denoted by bearing clearance

S = R - r (6.1)

for an arbitrary bearing bore.The position of the journal is denned by the eccentricity e and the angle 7 (Fig.

6.1). For an eccentric journal the oil-film thickness for e <£ r is given by

k{ip) = ho(<p) - ecos(^> - 7) (6.2)

and for a moving journal with e(t) and 7(f)

h { < p , t ) = k Q { < p ) - e { t ) z o S [ < P - 1 { t ) } . (6.3)

The expression (6.3) is called the clearance function. Further, we assume,

1. The lubricant is massless, incompressible and adheres to the bearing sur-faces.

2. The lubricant is Newtonian and its viscosity is constant in the whole of the oil-film.

1. The flow is laminar.

2. The pressure of the lubricant is constant in the radial direction.

3. Flow velocity in the radial direction is neglected.

1. Velocity gradients in the radial direction are large in relation to those in the tangential and axial directions.

2. The oil-film thickness is small compared to the journal radius.

3. The curvature of the oil-film is negligible.

9. The bearing surfaces are smooth and stiff-

Using assumption eight the oil-film clearance is developed out into one plane

(Fig. 6.2). It is therefore defined by the coordinates x, y , z and the function

/i(z,t). The bearing shell is at rest and the journal has surface velocity U = f t r ,

for angular velocity f t . The lubricant has different velocities u , v , w in x — y -

z-directions and these will depend upon position in the oil-film.For the fluid element dx, dy, dz the condition for force equilibrium is

d r ^y _ d p dr _ d p , .

d y d x ' d y d z { ' }

with p the lubricant pressure and r x y , the shear stresses.For the shear stresses from assumption two the following equations are valid,

d u d w . .

O y O y

with 7/ the dynamic viscosity. Using equations (6.5) and (6.4) we have

7 6 Oil-Film Bearingsdû _ l d p d 2 w 1 d p

d y 2 JJ d x ' d y 2 TJ d z(6.6)

6.1 Hydrodynamic Bearing Theoryand after integration,

u = r f^2 + C^ + ^ ' w ^ ^ ^ y ' + Cy + C, . (6.7)I t ] O X ¿71 O Z

U , iwww—*- w JournalvW\ W\\\\ WW s w ^

h(x.t] Oil film

ing x,u

V/////////; L

Fig- 6.2. Coordinates of oil-film.

With the boundary conditions

u — 0, w = 0, for y = 0 u — U, iv = 0 fox y = h ,

Besides the condition of force equilibrium, the continuity equation must he fulfilled. For this consider the elemental volume shown in Fig. 6.3. With

QX = J u d y

o

as the fluid flow rate in the x-direction and

hQz- j VJ d y

o

as the fluid flow rate in the z-direction, the continuity condition is

^ d x d z dt + ^~ d z d x d t + ^ d t d z d x = 0 ,o x o z a t

that is

Using equations (6.8),

8

9 6 Oil-Film Bearings

whose differentials with respect to z and z in equation (6.9) lead to


dx V dx+

d_dz T) dz

„ TT dh n dh = 6 U 7 - + 12 — ox

at (

10


the so-called Reynolds Equation.For further calculations we assume again the original geometry shown in Fig. 6.1.

With x = tpR and h(<p,t) from equation (6.3),


For r form

4- e sin (tp — 7)

sin {tp - 7) - ~ cos (<p - 7) .

R and constant viscosity 77 the Reynolds Equation

takes the following

dh 1 dhodx ~ R d<

pdh dldt —e dt

12


R2 dip I dip + h"dz2 (6.12)


6 7 } dho

Q ^ E (J2 „ 2-y) sin (tp — 7) — 2e

cos (ip — 7)

dip

14


With this equation and the appropriate boundary conditions and initial conditions, the pressure p in the oil-film is determined. Integration gives the pressure p(ip, z, t), which depends upon the position cp, z in the oil-film. For a transversely moving journal this pressure is also a function of time.

The right-hand side of equation (6.12) shows the required conditions for the formation of this oil-film pressure. In any case the existence of viscosity is essential. Furthermore, at least one term in the square brackets is required. Hence for a circular bearing whose rotating journal is centrally positioned, the oil-film force is zero. For a non-circular bore the term dh0/dip is responsible for pressure build up and hence an oil-film force even for a centrally positioned journal. For translatory motion of the journal, a dynamic oil-film pressure is in general produced. Only for e = 0 and 7 =

Í2/2 can the pressure become zero for a circular bearing.The oil-film pressure field gives the incremental oil-film force dF when integrated

over the length L and multiplied by R d tp. Thus

+ L / 2


dF = Rd.tp

16


l- L / 2

This has components

dFj — dF cos tp , dF2 = dF sin tp ,

as shown in Fig. 6.4. Integration over the circumference gives the resulting force components

2x 2x

F1=

J

dF, , F2= J

dF2 . (6.13)

The magnitude and angle ■O of the resultant oil-film force F„ are given by

F. = -JF? + FZ , tarn? - § . (6.14)

The oil-film force F3 is applied on both the bearing shell and the journal. To ensure equilibrium a corresponding force Fj must be applied externally on the journal, which is equal in magnitude and direction to F„.

For rotordynamic analysis the additional force is needed which arises due to translatory motion of the journal. In general the relation between the force and the motion is non-linear. For small displacements and velocities (Fig. 6.5) one may however linearise. With X0, Yo as the displacements of the journal centre as a result of the static load F0 = (F I Q , F2Q), the journal force F j — ( F i , F2) thus has components

Fi =F10 +

F2 — F2o + xl "i" ~~— X 2

+ ~T~ ¿1 ~f*¿2 = ^20 + AF2

(6.15)

x1 + dF1 dF l .

dxi dx2 x2 + i i + "XT" X 2

0 X 2

dF2

*1 +

dF2 dF2 dF2 .dxi dx2 x2 + dxi ¿1 + "7TT~

X 2

0 X 2

The partial differentials with respect to displacement and velocity are the stiffness and damping coefficients, respectively.

21 6 Oil-Film BearingsY x2

/ V

Fig. 6.5. Coordinates for stiffness andX FT damping

6.2 Short Bearing Theory


In order to solve the Reynolds Equation and to calculate the static and dynamic displacements a great deal of numerical work has appeared in the literature. We shall only mention here references [33, 34] due to Someya and the reviews [35] and [9] by Lund and Mitsui in Sect. 4.1 of reference [10].

As a rule calculation has to be carried out numerically. A closed algebraic solution by Dubois and Ocvirk [36], however, allows the following qualitative conclusion: For a relatively short bearing the change in pressure in the circumfer-ential direction is small in relation to the pressure change in the axial direction. Hence, in the Reynolds Equation (6.12), one may, to a reasonable approximation remove the first term. Comparison of calculated results and measurements of the displacement of the journal under static load for bearings having circular bore show pleasing agreement up to a ratio of length to diameter of unity.

The determination of stiffness and damping coefficients from this theory has been carried out by Holmes [37] and Smith [38]. In the following, calculations will be carried out with somewhat different notation and procedures from those followed in references [37, 38].

For a circular bearing the oil-film thickness is the same as the bearing clearance. Hence ho = S and thus dh0/d<p = 0, that is the clearance function (6.3) is

h(<p,t)^6-e(t)<n&[tp-y[t)~\ . (6.16)

With dp/d<p = 0, the Reynolds Equation (6.12) takes the form

1 " I^71)[e^-2^)sin(^_T)-2 e cos - T)] . (6.17)

By integration one obtains the pressure function

Pfo *, 0 = 7?[]z2 + Cxz + C2

22


and "with, the conditions

d p L~ = 0 for z - 0, v - 0 for z = ±—dz ' * 2

it follows that


p(<P, z, i) = ||[e(J?-2-r )Sin(^~7)-2é cos (y-7)] (** ~(6.18)

24


Integration over z gives the oil-film force per unit length in the circumferential direction.

q ( f , t ) = J p(<p, z, t)dz^^-^~q((p, t ) ~£/2

with (6.19)

£ (21_ 1 \ sin^ - 7) + ^ cos(^ - 7)

« (^ t) = V ^ -----------------------—fir;------------------

[1 - e cos (y> - 7)J

and

e = -c (6.20)0

as the eccentricity ratio.In the static case, with ¿ , 7 = 0 and Cl ^ 0, g is constant with time. With cr = ip

— 7 then

\ —£ sine . . 77 X3 J? . . .„ „ .

(1 - e coso-) 2 o'1

Fig. 6.6 shows the behaviour of q(cr) for several values of e. In the region 0 < <r < % q(<r) is negative. This corresponds to a tension force in the lubricant, which is impossible to sustain. One therefore assumes in this region that q(<r) = 0. Hence one obtains the pressure distribution shown in Fig. 6.7. Positive pressure occurs in the lower half as a result of the eccentricity vector e in the plane X' Y'. The resultant of forces dF — q Rd<r loads the bearing shell and the journal. The corresponding static equilibrium force on the journal is shown as FQ in Fig. 6.7.

In the dynamic case, with ¿ , 7 ^ 0 the components of the oil-film force, that is of the journal force in the X', Y'~system are

F[ = J q(<r t t)cos<r R d<r = F„ A(e, e, 7) F2 =

S afo t)sincr Rdcr = Fv f2(e, £, 7)


with

t) is obtained from equation (6.19) and <x = ip — 7. Integration gives

The force components relative to the X, Y"-system are

(6.24)

26


Fi = i ' cos 7 - F'2 sin 7 =

F1'sin7-|-i;'2C0S7 •

(6.25)


With equations (6.22) to (6.25) the explicit relations for the magnitude and direction of the journal force are obtained as functions of position £, 7 and of velocities e, 7 of the journal centre C j . In particular, equations are obtained for static loading and for stiffness and damping coefficients for any journal position.

For a horizontally supported shaft with static loading the following are the component forces

28

6 Oil-Film Bearings

F1 = 0 = f;cos7 - i^sin7 F-2 = -F0 =

F;sin7+ F^cos7(6.26)

29


with2e 2

30

6 Oil-Film Bearings

(6.27)

31


2 (1 _ £2)3/2 •

The angle between force and eccentricity is obtained from Fig. 6.8 and equations (6.27) as

-rt tt (i-£2 ) 1/2 4

F{ 4 e

and the angle 7 in Fig. 6.7 is obtained from the first of equations (6.26) as

(6.28)tana =

32

6 Oil-Film Bearingstan 7

Fi X ( l - f i 2 )l/3(6.29)

33


Finally, it follows from the second equation of (6.26) and from equations (6.27) and (6.29), putting

34

6 Oil-Film Bearings

sin 7 = tan 7

+ tan2cos 7 —

+ tan2 7

35


-r> Fig. 6.8. Special case of a static load

36

6 Oil-Film Bearings

that the relative static force FojFj, is given by

The reference force Fv can be identified as a friction force on the journal, in the following way. The friction torque for a central journal is given by

M F = FF R ,

assuming r « i£, and the friction force Fp is given by

FF = r 2-k RL = 7 } — 2t RL . (6.31)o

The reference force Fn in equation (6.23) is thus identical to the friction force Fp for a bearing with reduced clearance

f - ^ S , (6-32)

where

37


RI L

(3 = = — , the length to diameter ratio.

The force ratio FQjFv on the left-hand side of equation (6.30) is a modified Som-merfeld Number. In German literature, the Sommerfdd Number So is usually given by

2RL ~^ = !& (6"34}

and so

(6 '35)

The behaviour of S* with eccentricity £ is shown in Fig. 6.11.To calculate the stiffness and damping coefficients one needs the differentials

of Fi and Fi with respect to X j and x2 and with respect to ¿1 and x2 (see equations (6.15)). From equations (6.25) F\ and F-> are functions of £, 7 and £, 7. The required differentials are thus,

8xk d c d x k 57 d x k . .

#£ d x k 57 Sijfc ' where i

and A: are equal to 1 and 2.

The displacements dxi, dx2 in the X , y*-system correspond to the displace-ments de^ ed-y in the X', Y'-system. From Fig. 6.5 it follows that

i j j — — , the clearance ratio(6.33)

38

6 Oil-Film Bearingsd e

— d x i COS7 + d x 2 sin 7

e d y — — d x i sin7 + d x z c o s - y)

1 d ' f d - y 1d x i d i i - cos 7 , — - - X T - -

0 O X i O X i

— sin 7 e

d e 1 . d ~ f d j 1d x 2 d x 2 - sin 7 , — = — = ô 0Z2

d x 2

— cos 7 e

(6.37)

(6.38)

39

from which


With these relationships one finally obtains after transformation the following relationships for the stiffness and damping coefficients of the short circular (or cylindrical) bearing

dFi u F° d F i a a F°_=^= 7 a T , fa— (6.39)where

40

6 Oil-Film Bearings711 - [2x2 + (16-x2)£2]A(£)

nr 2 -27rV-(16- T 2 )e< A f ,712 = 4— — m

x x 2

+

( 3 2

+

x 2 )

e 2 +

( 3 2

-

2 x 2 )

e 4 A

,

' 4 s ( l ~ e 2 f *

{ £ )

(6.40)721 =

41


722 = --------------------~:r~---------------------A(e)

Ai - I (1 [x2 + (2x2-16)£2]A(

£)

P N = /32I--[2x2 + (4x2-32)£2]A(£) (6.41)

x x2 + (48-2x2)E2+xV ,

= 2-----------------------------------------A(£)

-------—i-----------^ - (6-*2)[x2 + (16-x2)e2]

The non-dimensional stiffness and damping coefficients are functions of eccentricity e .

From equation (6.30) they can also be represented as functions of Sommerfeld Number 5*. Their graphs and a discussion of their practical significance is given in Sect. 6.3.

6.3 Static and Dynamic Properties

Journal bearings can have different lubricants and different clearance geometries. Liquids are, in general, used for the lubricant and the following features relate only to such bearings. The clearance can have a completely specified geometry or be formed from movable segments. Fig. 6.9 shows some bearing types.

Bearings with prescribed clearance (Fig. 6.9 a-e) have, in principle, the same static and dynamic behaviour as the short cylindrical bearing. If we load the rotating journal with the static load Fo and the journal has an angular velocity Q = 2TTW, then the journal centre C j will take up a position denned by eccentricity vector e from the bearing centre Cs such that its direction relative to the load direction is a (Fig. 6.10). For the short cylindrical bearing this

42

6 Oil-Film Bearings

d 9

Fig. 6.9. Some bearing types

a) Segment bearing e) Unsymmetrical three-lobed bearingb) Lemon bearing f) Unsymmetrical tilt-pad bearingc) Pocket bearing g) Symmetrical tilt-pad bearingd) Three-lobed bearing

Fig. 6.19. Short cylindrical bearing. Journal centre orbits for rotating force Fr = 0.5 F0.

43

6.2 Short Bearing Theoryand deformation of the bearing shell, that is variation of the clearance space due to load and temperature differences. These and further influences have been investigated and published as a result of much numerical and experimental research. Literature on this topic, as well as many tables and diagrams of stiffness and damping coefficients can be found in references [10] and [40].

44

7 Rotors with Oil-film Bearings

In Chap. 6 relationships were given, on the basis of certain assumptions, between radial movement of the journal and the accompanying force. Accordingly, an oil-film bearing can broadly be described as a spring-damper element, with the special property that the displacement and the force in general do not occur in the same direction. In comparison with rotors with rigid bearings, a rotor on oil-film bearings thus has different natural frequencies, critical speeds and other resonance behaviour. As a result of the peculiar force relationships, the possibility of instability arises, and a large number of theoretical and experimental works appear in the literature. In the following, the essential theoretical results will be given for simple rotor models with two bearings. They correspond in the main to results obtained experimentally. In this chapter the rotor will be assumed in general to be horizontal. For a vertical rotor, special considerations hold, which will be discussed in Chap. 8. The influence of oil-film bearings on the vibration of more realistic models will be considered in parts II and III.

7.1 Equations of Motion

We consider a symmetrical rotor with two equal bearings. The rotor can be rigid or flexible and the bearings are rigidly supported. The model of the rigid rotor is shown in Fig. 7.1. The shaft turns with constant angular velocity J? and undergoes translational motion only, that is no tilt motion occurs. Thus the movement of the rotor centre W is identical to that of the journal centres C j . Using XI , x2 as the displacements from the equilibrium position ey, ex. (Fig. 7.1)1, the equations of motion are

mx1 + 2Fn ( x u x2, iT, X 2) = F i ( t ) ^

mx2Jr2Fj2{xi, x2, X -j, x2) = F2(t) .

-F/i, F j i are the components of journal force and Fi, F2 the exciting force. Equations (7.1) are non-linear, as discussed in Chap. 6 and are linearised in the usual way. For unbalance excitation these equations become, using stiffness

Wc denote here the journal eccentricity by c j t o differentiate it from the mass eccentricity.

and damping coefficients ^ and dik,

mi] + 2 { d n i i + diox2 + kux1 + ki2x2) = cos fit

mx2 + 2 ((¿21^1 + ¿22^2 + k21xi -f h22x2) = T7ieJ?3sin >f?i .(7.2)

The Jeffcott rotor will be taken as a model for the clastic rotor using the properties assumed in Chap. 3 (Fig. 7.2). The equations of the rotor centre W and of the journal centres C j are now different. They will be denoted by y\, y2 and Xi, x2, respectively, and represent displacements from an assumed equilibrium position. In Fig. 7.2, a static load in the negative 1/2-direct ion is assumed and so the two coordinate systems have their origins displaced by a distance ys for static deflection of the rotor2. Thus the equations of motion are

myi + dyi + k [ y t — Xi) = meJ?2 cos J?£

my2 + dy2 + k ( y 2 — x2) = meJ?2 sin Qt

and, for a non-linear journal force,

k { y i ~ X i ) ~ 2Fji (xl5 x2, ¿1, ¿2) k { y 2 -

x 2 ) = 2 F J 2 ( x l , x2, x:, i2) ,

(7.31

(7.4)

c.

sW y,

Fig. 7.2. Jeffcott rotor with oil-film bearings

7.2 Stability 50

or for a linear journal force

k { x 2 — 3/2) + 2 ( k 2 i x i + ¿2212) + 2 (¿3121 + ^¿2) = 0 .

7.2 Stability

Solution of the homogeneous linear equations requires the eigenvalues of the system to be found. These are conjugate complex or real and characterise the natural vibration. The imaginery part corresponds to the natural frequency in question and the real part gives the stability of the natural vibration. Fot a negative real part the vibration decays with time, and for positive real part it grows. The stability boundary of the system is reached, when the real part of an eigenvalue is zero. Shafts with oil-film bearings can become unstable due to the presence of coefficients ft12, k2:, called cross coupling stiffnesses. In practice instability must be avoided, and one must know as much as possible about the conditions and about behaviour during instability. The subject is rather complicated. We therefore first investigate the rigid rotor and then the Jeffcott rotor. The understanding obtained in this way will be useful in the investigation of large models in Parts II and III of this book.

7.2.1 Rigid Rotor

A shaft rotating at angular velocity Q is loaded in its central plane with a static force 2F0 and e = 0. As a rule 2.Fo is the dead weight G = rag. It can, however, represent other forces as for example steam force, gear tooth force or magnetic force. 2F0 rnay also be the resultant of a variety of these forces. Hence the equations

mi, + 2 { d n X i + d12x2 -f knx-i + k i 2 X 2 ) = 0 ^ ^

mi2 + 2 (d2iii + d22x2 + k2\Xi 4- k22x2) — 0

are valid and their solutions x i ( t ) and x 2 ( t ) are obtained using the substitutions

xi = ( p i e X t , x2 - tp2 e X t . (7.7)

1Chap. 3, displacements y l t y2 relate to the position of the rotor centre, when the rotor is unloaded.

2Chap. 3, displacements y l t y2 relate to the position of the rotor centre, when the rotor is unloaded.

51 7 Rotors with Oil-film Bearings

In order to obtain non-dimensional parameters, one uses the reference frequency

. or u, ~ ,/~ with 2Fa = G = mg (7.8)am V 0and non-dimensional time r ~ U> S T . Hence one obtains from equations (7.6) and after transformation, the equation

A" + 0 n A+ 7 u /?i3 A+712

021 A+72, A* + 0 M A + 7„

with 7,7.., fiik obtained from equations (6.39) and

The condition for the existence of <PI ? V; 's that the determinant of the matrix of (7.9) is zero. This leads to the characteristic equation

a.iA + o3A + a2A + ûiA + a0 = 0

(7.11)

with the coefficients

a., = f? , a3 — .43/2 , o2 = -42 + A^Q' , a! = .4i.f? , a0 = AQQ~ AQ = 7n 722 -

712721

Ai = 7U022 - 712/3=1 + 722011 - 721012 ,~ ^

¿ 2 = 01202.

¿3 = 01! +022

•4-1 = 7 n + 722 -

The roots of equation (7.11) are non-dimensional eigenvalues Ar. They are in general complex conjugate.

Ar, VT = a T ± j ZJT r = l,2 (7.13)

where ar = A T /OJ A , the non-dimensional coefficient governing growth of vibra-tion and UJ R = LOrfw*, the non-dimensional natural frequency.

The eigenvalues Ar depend upon Q and 7^, 0,-jt as can be seen from equations (7.12). In turn the coefficients 7;*, 0;* depend on £, where £ is a function of S' = Fc/Fr,. Using equation (6.23),

- w _ 2Fq6 ' =

7.2 Stability 52

where

thus

a - V? ^rr? with 2 Jo - G - m5 .

Hence the eigenvalues Ar ultimately depend upon the non-dimensional angular velocity Q and the parameter C.,, which is constant for a given system. C a i s approximately proportional to the power 1.5 of the physical size of the system.

Fig. 7.3 shows against angular velocity the natural frequencies wr and values of a T

for a rigid rotor with C\, = 1 and 10.For small values of J?, one natural frequency exists and after a certain value of Cl

there are two. Correspondingly there are the "growth coefficients" a t , a!2 and a% in the first /2-region and and a% in the second J ?~region. For & > f t t k i &\ is positive and the system is unstable. The angular velocity at the borderline Q t h has a value 2.7 w, and 2.9 w,, for C_, ~ 1 and 10, respectively while the unstable natural vibration has frequencies of 0.5 f2 t h and 0.44 Q t h , respectively.

At the stability borderline, R e A = 0. If one puts X = j u j in equation (7.11), then the following equation is obtained

53 7 Rotors with Oil-film Bearings

a.juT1 — J AÛF* — a^w2 -f J A{U> + O.Q = 0

Fig. 7.3. Rigid rotor with short cylindrical bearings. Natural frequencies and 'growth coefficients' for = I and 10.


for the stability borderline. Since its real and imaginery parts must each equal zero, the following equations are obtained

aûT1 — a2TD2 + ao — 0 (7.15)

and

(a3w2 - a j ) w = 0 . (7.16)

Solutions of equation (7.16) are u> — 0 and u>2 = oi/a3. The first solution is of no interest as it occurs at Cl = 0, from Fig. 7.3. The second solution substituted into equation (7.15) gives the condition

a^C^ — dx°2a3 *f 00^3 = 0

or using equations (7.12)

(A 2 - AÂ.A, + A 0 A-) 7f - A X A 2 A 3 ^ 0 .

Hence the non-dimensional natural frequency at the stability borderline is

T) _ ^'ft / AiA 2 A z, m .

Q l h '~ , ' iA l -AÂÂ Q Al ■

From equations (7.12), AQ, . . .A a contain the coefficients 7^., /3^, which are functions of journal eccentricity ratio e. Hence n t h is also a function of £. In Fig. 7.4 is shown this function for a rotor with short cylindrical bearings. Angular velocity is shown as abscissa and eccentricity ratio as ordinate directed downwards. This is to illustrate the behaviour of the journal position as it would appear for a shaft under gravity load and as it rises from startup.

For a system with given parameter C„ the journal centre has an eccentricity ratio of unity for f l = 0 and proceeds towards e = 0 for Q —-> co. The stability borderline

54

1-*——i—■—i—■—i—■—i—■—i—■—r-

0 2u._ Lu^61A

Fig. 7.4. Rigid rotor with short cylindrical bearings. Borderline of stability and startup curves.

7 Rotors with Oil-film BearingsQ t h = /(e) is obtained as the intersection of the startup curve with the borderline curve. For the parameters C„ of Fig. 7.4, ,f2t/Ji«s between 2.7 u t and 3 u> a .

Using Fig. 7.4 the non-dimensional stability borderline f^fc/u/, can be shown as a function of the parameter C„. Such a figure is called a stability chart. Fig. 7.5 shows such a stability chart, which was determined with data from many different types of bearing.

The required coefficients 7,^, 0^ and the static load locus function were calculated for the short cylindrical bearing from [35] and by series solution in [34]. The values from [39] were obtained from measurements. It should be noted that in general, for the same bearing shape, alternative output data exist, so that alternative results are possible for the stability borderlines. Thus, from stability charts one can only obtain an idea of trends and relationships using such figures as Fig. 7.5 and generally one cannot expect to get exact values. With this proviso the following can be concluded from Fig. 7.5:

— With regard to stability the following rank order applies:Circular bearing, pocket bearing, lemon bore bearing, three-lobe bearing. From [39], for C s < 0.7, the stability borderline of the lemon bearing is roughly the same as that of the circular bearing.

— The curves from [39] each have a distinct minimum. Hence increase in C.,can produce a change from stability to instability or vice-versa.

— The mass of the model and the bearing clearance have different exponentsin C„ and ws. Hence a discussion of the influence of these parameters on stability is rather complicated. One can thus better judge a given case, with

reference to an example.

55

Fig. 7.5. Rigid rotor with oil-film bearings. Stability chart for different types of bearing, L / D = 0.5 [34, 35, 39] (sec bibliography).


Example 7/1 A model similar to that in Fig. 7.1 is given with mass m = 10,000 kg and

bearing data D - 2 R ~ 400 mm, I - 200 mm,i; - 1.5 7<,o> V ~ 0.030 Ns/m2.We wish to find the stability onset speed and its change for a 10 % increase in

bearing clearance and for a 10 % increase in oil viscosity. It is first assumed to be a cylindrical bearing, then to be a lemon-bore bearing.

With the given data and 2F0 — mg,

C, = 1.017 and wa = 180.8 rad/s.

and from Fig. 7.5, with the curve for the cylindrical bearing [39]

Qth/<*>3 — 3.15, that is J?£/, = 569.5 rad/s,

or the borderline speed n t i l = 5438 rpm. For a 10 % larger bearing clearance

C, = 1.291 and w, = 172.4 rad/s

and hence n th — 4807 rpm, that is a 12 % lower value.The viscosity has an influence only on C„. For a 10 % greater viscosity we obtain

the borderline speed n th = 5542 rpm, that is an increase of around 1.9 % above the original value.

For lemon-bore bearings in the original condition n th = 7079 rpm.For 10 % greater bearing clearancench =7688 rpm,an increase of 8.6 %, and for 10

% greater viscosity n t/, = 6820 rpm, a decrease of 3.7 % from 7079 rpm.This example shows how different the behaviour can be. Observed borderline speed

changes occur for the two types of bearing, not onlj' to different degrees but even with different tendencies (LIS : -12 %, and +8.6 % and l.li; : +1.9 % and -3.7 %).

7.2.2 The Jeffcott Rotor

To calculate the eigenvalues and stability borderline for the Jeffcott rotor of Fig. 7.2, one assumes as for the stiff rotor, that the shaft rotating with angular velocity J?, is loaded statically with a force 2FQ in its mid plane and that the centre of mass S of the disc is coincident with W, that is e ~ 0. In addition, it will be assumed for simplicity that the disc experiences no external damping, that is d = 0. Hence from equations (7.3) the following equations are obtained

m f f 1 + f t ( f f l - * 0 = 0 (718)

my2 + k [ y 2 - x 2 ) = 0 .

These, together with equations (7.5) determine the displacements x\, x2 of the journal centre C j and y i7 y2 of the disc centre W , describing displacements from the static equilibrium position due to 2FQ .

In the usual way substitution of expressions (7.7) for x \ , x2 a.nd corresponding solutions

56

7 Rotors with Oil-film Bearingsyi- V 3 * A t , V i ^ ^ e x t (7.19)

lead to the characteristic equation for the eigenvalues. To obtain non-dimensional parameters, the natural frequency of the rotor with rigid bearings is introduced as a reference frequency namely,

57

7.2 Stability 58

m

together with non-dimensional time r — w„i. Further we put

S 2 F 0a = — , y, = -r-

y> «

or a = or - with 2 Fa = G = mg .

(7.20)

(7.21)

59 7 Rotors with Oil-film Bearings 'h = < ■ r

Hence we obtain after some manipulation the characteristic equation

% H T 4- ... 4- 6i A 4- 60 = 0 ^ (7.22)

with the coefficients

K = A2 , &c - (Ai 4- a .43)7l b4

= 1A% 4- [ a 1 -f- Au + a

A.i)7?" 63 = (2A: 4- a -43)?} ,

62 = ,4, 4. (2A0 4- a A,)7f , 5, = .4j J? , i0 = .40 J?"

and .40, - - • AA from equations (7.12).The characteristic equation is of sixth order and has, in general, the roots

X, %=% R ±J% , R - 1,2,3. (7.24)

In an analogous way to equations (7.14) we introduce

FQ 2Fa8* Cn_

(7.25)

where Cn =

7.23)

Fv r/LWR Q

7.2 Stability 60

Hence the eigenvalues Ar are obtained as functions of J? and the parameters C n and a.As an example, the natural frequencies (the imaginary parts of the eigenvalues) are

presented in Fig- 7.6 together with the "growth coefficients" (the real parts of the eigenvalues), when plotted against angular velocity of the rotor. For this purpose it was assumed that the rotor was supported on cylindrical bearings, with parameters C n = 0.3125 and y a — 0.3 S, that is a = 3.3.

There exist three natural vibrations, of which the second and the third are stable in the whole speed range considered. In contrast, the first natural vibration is unstable for Q t h — 1.2 uj n . At the stability borderline it has a frequency u»i = 0.5 Q t ^ .

The parameters at the stability borderline can be ascertained in the same way as for the rigid rotor. With A = j S, one obtains from equation (7.22) the following equations

- b6 ^ + &4 w1 - &2 5T2 + ¿0 = 0 (7.26)

[b^ - b 2 ^ 2 + b x ) v = 0 . (7.27)

From equation (7.27) the following solutions are obtained

5 = 0 and u2 = (b3 ± yjh\- 4&A) .The second of these equations together with equations (7.23), gives

(7.28)

Fig. 7.6. Natural frequencies and growth coefficients for a Jeffcott rotor with short cylindrical bearings. Cn = 0.3125, ys = 0.3 6.


The solutions w = 0 and uT ~ 1 occur for ¡7 = 0 and are therefore of no practical importance. The solution (7.28) substituted into equation (7.26) gives, after some manipulation, the following simple expression for the stability borderline

= k = = a* j A ûin w, V Ai + a -43

where Qth/^s is obtained from equation (7.17). It can be seen that the flexibility of the rotor enters only through the factor a under the root sign in equation (7.29).

Equation (7.29) gives a stability chart for the Jeffcott rotor, Fig. 7.7, which corresponds to Fig. 7.4 for the rigid rotor. Both figures relate to short cylindrical bearing supports. The startup curves are here specified by the parameter C„ and the intersection with a borderline curve gives the stability limit. By this means a stability chart such as Fig. 7.8 is obtained. This shows the stability borderline for a Jeffcott rotor with different values of ysJ5 and oil-film bearing data from measurements by Glienicke [39].

Fig. 7.7. Jeffcott rotor with short cylindrical bearings. Borderlines of stability and startup curves.

Discussion of the influence of the model parameters on the stability borderline is more complicated than for the rigid rotor because of the additional parameter yK/S, and will not be attempted here. In the same way we will not pursue further the influence of bearing type. As a rule the same procedure is valid as for the effect of bearing type on a rigid rotor. One can investigate a particular case on Figs. 7.7 and 7.S or by calculation with the given relationships.

Finally It should be mentioned that in practice differences between theoretical and practical values of ( l t h will usually be noted, If realistic models are

61

7.2 Stability 62

0.2

un 2

520

1

0.5-


0.1 0.5 5

63

7.2 Stability 64

C,

Fig. 7.8. Stability chart for a Jeffcott rotor with cylincbrical bearings. Values for7ffc, ß ik &om [39], L / D = 0.5-

analysed using these equations (see Parts II and III). For example, the minimum clearance can decrease due to thermal distortion and hence the stability borderline can be raised. A raised stability borderline can also depend on unknown damping within the system. Also thrust bearings can be stabilising by virtue of axial loads. Destabilising effects can arise due to magnetic attraction in the rotors of electrical machines or due to clearance and sealing effects in turbomachines (Chapter 10).

7.3 Unbalance Vibrations

The equations of motion for unbalance vibration of rigid and flexible rotors are given in Sect. 7.1. Solutions have been achieved by Glienicke [41], Merker [42] amongst others. In the following the most important results will be given for the horizontally-supported rotor. The behaviour of a vertical rotor will be described in Chap. S.

The unbalance forces cause the rotor axis to deflect from its equilibrium position and to move in a closed orbit. The extent, shape and position of the orbit depend upon the rotor speed. The orbit will be executed once every revolution and resonance occurs when the running speed corresponds to a natural frequency.

For a rigid rotor as in Fig. 7.1 an example will be considered from [42] in Fig. 7.9. The model is prescribed by the parameter


a j Eccentricity of the journal centreb) Rotor orbitsc) Mean orbit radius

where So is the Sommtrfdd Number from equation (6.34) and the reference frequency is given by

65

7.3 Unbalance Vibrations 109

t

Fig. 7.9. Rigid rotor with short cylindrical bearings. S,w = 0.32, e — 0.2 5. From J. Merkcr [42].

7.2 Stability 66

„„ = M = 2£ . (7.31)m^3 4m, V 8 J

Sm will be assumed to be 0.32 and the mass-eccentricity c to be 0.2 S.

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Short bearing theory