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Stiffness and damping coefficients of an inclined journal bearing
Mechanismand Machine Theory, 1977, Vol. 12, pp. 339-355. Pergamon Press. Printed in Great Britain
Stiffness and Damping Coefficients of an Inclined Journal Bearing A. Mukherjeet and
J. S. Rao$ Abstract In the determination of the dynamic behaviour of a rotating shaft, the fluid film stiffness and damping coefficients of the bearings play an important role. The general practice is to ignore the rotational stiffnesses and damping coefficients due to the tilt of the journal in the bearing. This paper presents the stiffness and damping coefficients of such journal bearings. Using the expression for film thickness, the modified Reynolds' Equation for the tilted finite journal bearing is set up. The solution of this equation for the film pressure is obtained by using Fedor's proportionality hypothesis. The results obtained are presented in the form of non-dimensional charts. Introduction DETERMINATIONof the stiffness and damping coefficients of fluid film bearings is important in the study of the dynamic behaviour of rigid and elastic rotors. Solutions for the pressure distribution, used for the determination of stiffness and damping coefficients are given by various workers, for the case of a journal running parallel in the bearing e.g. Hummel[l], Cameron[2] for long bearings, Dubois & Ocvirk[3], Holmes[4] for short bearings and by Pinkus [5] and Strenlicht[6] for finite bearings. Detailed information about these coefficients can be seen in the books of Pinkus[7] and Smith[8]. Krishnova[9] determined the pressure distribution of the film in a bearing with tilted journal. Capriz[10] analysed the behaviour of a tilted journal at small eccentricities, in finite bearings. This work was further extended [l l] to account for larger eccentricities in short bearings. Euler's angles were used in this analysis to determine the film thickness. Kikuchi[12] used a rectangular coordinate system and determined the stiffness and damping coefficients for tilted journals in short bearings. He has shown the importance of these coefficients while determining the response of a Turbine rotor mounted on four hydrodynamic bearings. In this paper rectangular components of inclinations are used to determine the film thickness in a finite bearing with tilted journal. Using Fedor's[13] proportionality hypothesis, the film pressure is obtained, from which the stiffness and damping coefficients are determined. The coefficients thus determined are presented in nondimensional charts.
Reynolds' Equation for Tilted Journal Bearing Figure 1 shows the coordinate system adopted for the analysis of the bearing with an inclined journal. The shape of the journal cross-section at any plane parallel to X Y plane is assumed to be a circle and that the rectangular components of inclination $ and $ are small such that their second and higher order terms may be ignored. From the geometry of Fig. 1 we get u = C K i sin O, v = C K i c o s 0 u' = C K ' sin 0' = u + zd)
cos 0' = v - z~b. tMechanicalEngineeringDepartment,IndianInstituteof Technology,Kharagpur-2,India. *MechanicalEngineeringDepartment,IndianInstituteof Technology,Delhi,New Delhi,India. v' = C K '
(])
339
340
~.
Li
~.~
r A
XIw
Z L
PLANE .
JOURNAL AT ~ o PLANE
~
JOURNAL AT A-A •
Z
X
,
:N
y X-Y
PLANE
Figure 1. Rectangular c o m p o n e n t s for journal inclinations. The following expressions are derived from above equations K' - K, = C (4) sin 0 - qJ cos 0) Z
0' -
(2)
0 = / / - / / ' = ~ , , (4) cos 0 + ¢ sin 0).
Equations (1) and (2) give the expression for film thickness as (3)
h = C {1 + (K, + K2z) sin (fl - ~'z)} where K2 =
4) s i n 0 - ~ c o s
0
C and = 4) cos 0 + ¢ sin 0 CKI
(4)
With the help of eqn (4), the Reynolds' equation for a finite bearing with a tilted journal taking into account of the squeeze terms can be written as 1 0 {
R:a
_ 6/z
,30P1
¢l+K'cosp)
0{
(l+r'cos
- ~ {-K'toJ - 0') sin/3' + 2/~' cos/T}.
,3aP'~
) TQ (5)
Here 0' = 0 + ~'z a n d / ~ ' = K, +/~2z. The relation between/3 and fl', 0 and 0' and between K and K' are as given in Fig. 1. Here it may
341
be noted that in eqn (5), the film thickness is expressed in terms of/3' whereas the partial derivatives appearing on the left hand side are with respect to/3. Using the transformation of coordinates between/3 and/3' {~zl} = [10 - 1 ] {~}
(6)
eqn (5) becomes ,9 { ( I + K ' 3 , 9 P ) + ,9 f {~-~2+ ~2} ~-~7 cos/3 ,'-b-~] ~--z;z,t ( 1 + K' cos/3')3 -~zP,z}P,
cos/3)
z,
cos/3)
6tz , = ~ {-K (to - 20') sin/3'+ 2/~' cos/3'}.
(7)
The corresponding boundary conditions are 0
and
for
(8,
Silliness Coelileients For convenience of analysis Y axis is made to coincide with the line of centers on the mid-plane, then ,t, K 2 = - -~,
6 (r = C K ,
(9)
K2 = -(blc and ~- = ~ICK. Once the stiffness and damping coetlicients are analysed in this coordinate system, then their evaluation in any other coordinate system will be simpler, with the help of similarity rotation transformations. Analysis corresponding to ~b rotation Setting 4' = 0 (K2 = 0) and all time rates zero, eqn (7) becomes 1 O {(I+K, ,30P/+ O f -,)30P) R 40/3' cos/3 ) ~ J ~ ; t(1 + K, cos/3 ~z'
+,g., O, {(1 + K, cos/3,)3 a_~P,/-
,3OP~
O{ 0~,,}] 6~ + --az' (1 + K, cos/3') 3 = - ~ Klto sin/3'.
(lo)
If ~b rotation is small, the perturbation series for the pressure P, may be assumed as P = Po+~
I?'P,.
(11)
i=1
Substituting the above in eqn (10) and separating the coefficients of the same powers of ~r, we get
_ 6/x - - ~ K,oJ sin/3'
(Coeff. of I?o)
(12)
342
1 0{ ,30P1 ] O{ OPiI R~a~ 0 + / ( , c o s / 3 ) - ~ ] + o z : (l+K'c°str)3 az'j
O{
OPo)
0 f
, 30Po)
= 8/3--7 (1 + K, cos/3')3-~-z,~ +~-z, ~[(1 + K, cos ~ ) -~7~
(Coeff. of 7?').
(13)
Equation (12) is conventional Reynolds' equation for the parallel journal case, written in/3', z' coordinates, and left hand side of both the eqns (12) and (13) has Reynolds' operator. Fedor's solution for eqn (12) can be written as
6lzR2K,w (2+Klcos/3') /3'{1 P° = C-2(-(-~I(-7) (1 + K, cos/3')2 sin
cosh[(Q/R)z']] cosh(QL/2R)J"
(14)
This expression for Po, when substituted in eqn (13), with its right hand side expanded in Fourier series, one gets
, s c)2Pl - 3K,(1 + K~ OPI+ (1 + K, OZP! cos/3') 2 sin/3' "zzT~, cos/~~,.3R2 j a-~
(1 + K~ cos/3 ) 3 ~
= .,=o ~ ~" cos m/3' sinh ( Q z ' )
(15)
where Q2 = 2(I + Q,K,) ~--~-~,
K,(2 + X/(1 - K,2)) Q,= ( l + x / ( l _ K 2 ) ) 2
/~,. =/~ [~C,. +2¢,. - 2ho~,.]
(16)
with -3K1 ~
gR4w
rt=2+K2,
l~=
1
1
Q
C2cosh(.~.R) ~
8=~X1, C2=~Xz, d,. =
X , . ÷ , - X,.-i
2
m =3,4 . . . . . . .
2
X,. = K,X/(1 - K,) 2 (-1)"-'a"(m + V'(1 - K f ) 1
so' = X/(1 - K f 1
~,=(l_K2)S~2
and ~,. = 2(-1)%" and
m = 2, 3. . . . . . .
¢l,.=2(-l)"a"(l+mx/(1-K,):)
m =2,3 . . . . . .
(17)
ho = 2(1 - KI 2) Ki 2+K12 , a = l + v , ( l _ K 1 2 ) . Assuming
P, = fP, B,(z') cos (i/3') i=O
and substituting in eqn (15), we get
(D2-n2)B.+-~(D2-n2+n+2)B.+,+ .:,.
3,4 .......
(D2-n2-n+2)B._, (18)
343
where D 2 = 0 2 20 • ",
OZ~z.
Assuming further, Bi = hBi + n/~ and ./~i = .B~ sinh equating the hyperbolic terms
[TI{.B,}
(Qz'lR),
equation for .B. becomes, by
(19)
= {E,}
with gl Tt, = Q22, T,2=-~-(Q2+2)
T. = (QZ - i2),
K, Tu+, = -~- (Q2 _ i s + i + 2)
(2o)
and gl 2 Tu-1=-~(Q-i2-i+2)
for
i=2,3 . . . . . .
(21)
rest T,
are zero.
A suitable number of terms in eqn (19) can be used to obtain the required accuracy. Seven equations in this set were found to give good values, when compared with Capriz[10] and Kikuchi [ 12]. For the homogeneous part,
D2hBo+-~ (D~+ 2)hBi = 0 (D2-
n2)hB. + -~ (D ~ - n 2 + n +
+~(DZ-n2-n+2)hB._~=O
2)hB.+t n=1,2,3 ........
(22)
Using Fedor's approximation
hB1 = - a t , B o
(23)
( D 2 - a22)hBo = 0
(24)
one gets
with
s= 2Klal as 2--Klal and this gives
hBo= C, sinh (R Z') hBl = -C,a, sinh ( R Z').
(25)
344
Boundary conditions demand (26) and (27) From the boundary conditions, the two constants C1 and al in eqn (25) can be determined and then the expression for the pressure can be evaluated as 6gR:toK~(2+K, cos fl', sin [3'
+(r
.B~cosifl')sinh
c°sb ( ~ z ) l
z' + C , ( 1 - a ~ c o s O ' ) s m• h - ~ z + . . .
]
(28)
From the above, the fluid couples about X and Y axes in Fig. 1, can be obtained as Mx = - R
fL/2f~ P cos (/3' + ~rz')z' dz' dfl' J L]2 ,,lO
(29)
= ~?[R2{.B,cr2 + Cz~r3}--~--or,] R fLI2
My =
-w
a-Ln Jo P sin (B' + ~z')z' dz' dfl'
r. l~K,°', ~] ___~2/2 1 {.B,o.2+ C,+,o.3}] = ~ L c r V ( 1 - K , ~) ,=o.~.,
(30)
where .B, s|nh ( ~ - ) Ci=-
sinh [azL'~ ' \2R ] L ~ 2LR 2 [L2R2R3'~
QL
°', =i~+---~z~- k202 Q2~] tanh ( - ~ -)
LR
[QLI 2R2., /azL\
2R 2 •
{a2L] ]
tr3= L R c°sh ~ - f R ) - -Qr Smh \ g' =
(31)
12/~R3to IrK, C ~ (2 + K,2)X/(1 - K,) ~
al = - . B J . B 1 .
The corresponding tilt stiffness components are given as OMx K,,~ =
~"6 =
Nix ~=,,,c,,,,
OM~. Ky+ = Orb = MYI'~="IcK°"
(32) (33)
345
Analysis corresponding to • rotation Setting ~b and all time rates equal to zero and neglecting second and higher order terms of K2, eqn (7) becomes
= -E{K1 sin/3' + z'Kz sin/3'}- 3Kz , 2
,
(1 + KI cos/3 ) cos/3 z ~z;z'l
, dP/-I
(34)
where E _ 6toR2~ --
C 2
"
Assuming a solution of the form P =Po+~
K2"P.
(35)
and following a similar procedure for 4} rotation given before, we get for Po,
cos.( )
E(2+ K~ cos/3')K~ sin/3' Po = (2 + KI2)(1 + K1 cos/3') 2
1
(36)
cosh ( --QL ~ )
Again, as in the previous case, the equation for PI can be obtained as 02PI
-,
OP1
02P1
(1 + KI cos/3') -ff~ - 3K1 sin/~ - ~ + (1 + K1 cos/3') R 20z,2
= ,.=1 ~" D"'lz'+D"'2sinh(Q--~R)+D"'~z'c°sh(Q-~-R) sinm/3' where D,... = E { - 2~,. - 3ho(K.~',. - ~1,.)}
D,..2 = -3)(,,,
(37)
QE R c°sh ( 2 ~ )
3E { (;,,, + ho(K,(,,, - ~.,)-.x,. Q~2} Dra,3
and
(~,. = 2m(-1)"-'am/(K,~/(1 - K,2)) lq,. =
- m ( - l ) " a " ( 1 + m~/(1 - K,z)) K I ( 1 - Kl2) 3/2
1 071,,,.
~" =-30K~'
0.5R2K1 )~" = 2 + K , 2 (~"-'+ 6"+')"
(38)
346
Assuming P, = 2.~ B~ sin ifl'. i=1
(39)
Bi = .Bi + hBi ,Bi = E~z' + F~ sinh
+ H~z'cosh
.
and substituting in eqn (37), the following are obtained by equating the coefficients of ,-', and hyperbolic functions - E l + KIE2 = Dl.i - 4 E 2 - 2K1E1 = D2,1
(40)
- 9 E 3 - 2KIE4 - 5K,E2 = D~.I 2QRHI + K1RQHz = 0
(Q2 _ 1)HI + 0.5KI(Q ~+ 2)H2 = DI.3 (Q 2 - 1)F, + 0.5KI(Q 2 + 2)F~ = Di.2 (Q2 _ 4)F2 + 0.5 K,Q2F3 + 0.SKI(Q 2 - 4)FI + 2RQH2 + K , R Q H 3 + K , R Q H , = 02.5 (Q2 _ 4)/-/2 + 0.5K1Q2H3 + 0.5KI(Q 2 - 4)H, =/92.3.
(41)
Solving above eqns (40) and (41), E , Hi and F~ can be evaluated. Here again, the homogeneous part mE, will satisfy. Here (D 2 - l)hB, + 0.5K,(D2 + 2)hB2 = 0 (D z - l)hB. + 0.5KI(D z - n z + n + 2)hB.+l + 0.5K,(D z - n 2 - n + 2)hB._l = 0 n =2,3 ......
(42)
The boundary conditions (8), will give i = 1, 2 . . . . . .
(43)
Again adopting Fedor's approximation, as in the case of 4~ rotation, hB~'s may be fully evaluated. Then the expression for the pressure can be evaluated as
P = .=1 ~ E . z ' + F. sinh
+ H . z ' cosh
+ C. sinh "--R--
sin n/3'
(44)
where L _ E . ~ + F. sinh (2~_)
L
C.-
(45) sinh ( ~ )
As again in the previous case, the fluid couples and tilt stiffness can be obtained as Mx = -R
= 2R
f Ll2 f ~r P l z ' cos/3' dl3' dz' J-LI2 JO
n=2,4 I.\
+ F.al + H.o'2 + C.o'3
(46)
347
My = R
P ,z' sin/3' d/3' dz' + F~trl + Hla2+
(47)
with
trl =
0.5LR cosh ( 2 ~ ) Q
R2 sinh ( 2 ~ ) Q2
-
(0"5L)2R sinh (2-R QL)
2R
and (0.5L)2R cosh (-a-~)
sinh ( - ~ )
O/2
0/2
tr3 =
OMx I K x . = -~x = M~I
2
(48)
(49)
K2=-(I/C)
and ~M~.
. I
K , , . = - ~ - = M~,I I
(50)
K2:-(IIC)"
Damping Coefficients For determining the damping coefficients corresponding to ~b rotation, we get ~b = ~ = 0 and = 0, in eqn (7) and obtain ,3aP
=
2 a
12/~Rz . /3,. C 2 K,~z'sin
(51)
Letting E = -12~R2K1 C2 and a solution of the form
P = z'Pl(/3') + P2(/3', z')
(52)
for eqn (51), where PI is nonhomogeneous solution and P2 is the homogeneous part of the solution, is assumed. Substituting this form in eqn (51) and separating the coefficients of z and integrating for P, we get P,(/3') - ~. _E(2+ K, cos/3') sin/3' - " ( 2 + K,2)(I + K, cos fl')2" Further assuming P2(/3', z')= ~ An(z')sin (~/3'), one gets from eqn (51), n~l
(53)
348 (D
2 -
I)A,
+ 0.5K,(D 2+
(D 2 - n2)A.
+ 0.5K,(D
2 -
2)A: = 0
n 2 + n + 2)A.+,
+
0.5K,(D
2 -
n 2 -
n + 2)A. - 1 -- 0.
(54)
n=2.3.4 .... Making use of Fedor's approximation, and the boundary conditions (8), which gives
+
i
) +
P2
+-
(55)
= 0
the following is obtained for pressure P as f
E(2 + K, cos/T) sin/3' [
P = "(i-;--g
l z,
(.2L'~ | " 2 sinh \ 2R ] j
(56)
From the above the fluid couples and damping coefficients may be obtained as
m x = (r (2 + K , ~ l - K , ~ ) -i~+ RzL -2
L2R coth 2a2
_. R_ETr {L 3 M~. = -cr (2 + K,~)(I _ K 2),,2 -~+ RzL"~
L?R~coth k["2L'~2R ]J
(57)
where
az2
Cx,-
2(1 + . , K , ) K,(2 + V'(1 - K,2)) (2-.,K,) a'- (I+v'(1-K,Z)) 2
OMx [ 04a Mx
(58)
(59)
¢r=(I/CKI)
and Cy, = O~b My
(60) ~'=(I/CKO
Setting q~ = ~ = 0 and ~b = 0 in eqn (7) and following similar procedure as above, the fluid pressure for the case of q~ rotation, can be obtained as
f
6/zR 2 K2 /3,)2 z P - C-~ K , ( l + K , cos
L sinh k 2R ] 2 . . [.3L\ slnn ~ , ~
[.3z3)
(61)
where x/2K, a3 - x/(1 - K,:)"
(62)
From the above the fluid couples and stiffness coefficients can be obtained as
_6~31ii______~2 Ir fL 3 R2L LZR Mx6 2 (I - K,~) sp~-~ + ~ - ~ c ° t h
(a3L]} \2R/J
(63)
and 12/zR3/(2
My=
C~
I/" f L 3 R2L L~R . I'ot3L~) K,(l-K~)/]5+-h~-~-T~acomlTff)j~
(64)
349
C x . w = -~gMx. ~ - M, t I ,~2= .,c.
(65)
and c~My.
(66)
I
Results and Discussion The tilt stiffness and damping coefficient for ~ and • rotations are determined by using the eqns (32) and (33), (49) and (50), (59) and (60), and (65) and (66) respectively. It may be noted that Capriz's[10] solutions are valid for small eccentricity ratios and in the present analysis a2 ~ 1, with K, ~ 0, and the above mentioned equations reduce to those given by Capriz. These coefficients are evaluated for different eccentricity ratios with the help of an IBM 1620 computer, using Fortran programmes developed for this purpose. The results obtained are presented in non-dimensional form in Figs. 2-9. These results may be directly used for determining the dynamic response of a rigid or flexible rotor and the stability zones. In Figs. 2-9, Capriz's[10] results, calculated from his formula, for LID= 1.0 are also 1.0
0.(~ m
O.g-
0.7--
0-6--
0.S-
II x
0.4
0.3
• /o-2
0.2
///
\
,_/o = ,., O.
B
.Jy/
0
I 0
0.2
0.4
0-6
I
I
0.8
I.O
K i --=--e,.
Figure 2. Non-dimensional rotational stiffness vs eccentricity ratio.
MMT Vol. 12, No. 4---G
350
o_.
=o U
n
I
,~
o
n
6 0
6
~
6
~- ~ ._. 5 0
i
,
J
._
L
i
o
1
l
~
~
1.
o
A
~
i
~
l
I
o
0
0 (-4
z
0
'~"
~
._~ i1
0
~ _._~ ,,
L,
o
L
I
I
I
i
6
o
o
6
6
,
,,
a
a
,L''~I ! o ~,,'
\l\-~ o
o
m
~
6
--
~
z
O
o
2 -
14=
-~ 6
351
.03 L q'~I
|'0
L.I.
~
-02
>IX
.01
o.o
1
0.0
0.2
\
s -
I
I
I
0.4
0.6
0.8
KI
A f .0
~-
Figure 5. Non-dimensional rotational stiffness vs
eccentricity
ratio.
• I 4-
.12
• lOi-
t
• /p= 2
u~
.0|
t
• 0~
-
II
x: .04
=1
-02
O0
0.2
0-4
0.6
I
0-8
I
I
I.O
K i .~...-tm.
Figure 6. Non-dimensional
rotational d a m p i n g vs eccentricity ratio.
352
•3 2 --
.28
.24
t ~i ~
.20
.~6
I II
>iu
-12
OS
-04
I
I
1
I
I
I
I
I
.I
-2
-3
.4
.5
.6
.7
-8
I
I
.9
1.0
Figure 7. Non-dimensional rotational damping vs eccentricity ratio.
353
.11
• 10
-
--
I./D = 2.0
T u
II
•0 9
--
• 0 8
--
' 0 7
--
• 0 0
-
\ ~ \\\\\\
II >t~
.05
L/D=
1.5
\.
.04
.03
•0 2 - -
~
.01
Oo
I
I
0.2
I
F
I
0.4
CAPRIZ
I
I
0.6 KI
Flgure
-
I
o.e
I
I
~.o
•
8. Non-dimensional rotational damping vs eccentricity ratio.
354
. 11
.10
--
• 09
--
• 08
--
• 07
--
•0
/
6 --
I tl
x lU
• 04
--
• 03
--
• 02
--
• 0!
--
00,
c/o = a /
~
/D-I ~ / / / I
I 0.2
I
I
I
0.4
I 0.6
I
I 0.8
I I.(3
K
Figure9. Non-dimensionalrotationaldampingvseccentricityratio.
355 presented for comparison, for mid plane eccentricity values Km in the region 0.1 to 0.7. These results are in good agreement for 0.1 ~< K~ ~< 0.4. It may be noted here that Capriz's[10] analysis is valid for only small values of KI.
Conclusion Using the film expression in rectangular c o m p o n e n t s , the R e y n o l d s ' equation for a finite bearing with a tilted journal is set up. With the help of F e d o r ' s proportionality hypothesis, the modified R e y n o l d s ' equation is solved to determine the pressure distribution in the film, using perturbed equations. F r o m this pressure distribution, the stiffness and damping coefficients of a tilted finite journal bearing are determined and c o m p a r e d with Capriz's results. G o o d agreement is shown with Capriz's results for small eccentricity values.
Nomenclature C Cx,, C×,, etc. D K~ K K×., K×,, etc. L M×, My P R W=
radial clearance components of tilt damping coefficient diameter of the bearing eccentricity ratio at reference plane eccentricity ratio at any plane components of tilt stiffnesses length of the bearing components of fluid couple film pressure radius of the bearing load on full bearing with large LID (o angular velocity viscosity of the lubricant ~ , 6 components of journal inclination (see Fig. 1) 0,0' attitude angles (see Fig. 1) (.) On any quantity denotes its time derivative)
References 1. C. Hummel, Kritische Drehzahlen als Forge der Nachgiebigkeit des $chmiermittels im Lager, V.D.I. Forschungs, Heft 287 (1926). 2. A. Cameron, Oil whirl in journal bearing. Engng, Vol. 179, No. 4648 (1955). 3. G. B. Dubois and F. W. Ocvirk, Analytical derivation and experimental evaluation of short bearing approximation for full journal bearing. N.A.C.A. Report 1157 (1953). 4. R. Holmes, The effect of sleeve bearings on the vibration of rotating shaft. Tribology (1972). 5. O. Pinkus, Solution of Reynolds' equation for finite journal bearing. Trans. A.$.M.E. Vol. 80 (1958). 6. B. Strenlicht and F. J. Maginnis, Application of digital computer of bearing design. Trans. A.S.M.E. Vol. 75 (1957). 7. O. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication. McGraw-Hill, New York (1961). 8. D. M. Smith, Journal Bearings in Turbomachinery. Chapman and Hill, London 0969). 9. L. B. Khrisanova, Analytical and experimental investigation of the pressure in an oil film of a journal bearing with axis of the journal and bearing skewed. Fric Wear Mach. 13, 190-208 (1959). 10. G. Capriz, On some dynamic problems arisingin the theory of lubrication. Nelson Research Lab., English Electric Co. Ltd., Stafford, England. Research Report--Part I. Riv. Mat. Univ. Parma (2) l, 1-20 (1960). 11. Galletti-Manacorda and G. Capriz, Torque produced by misalignment in short bearing. Trans. A.S.M.E., J. Basic Engng Paper No. 65, Lubs-9. 12. K. Kikuchi, Analysis of unbalance vibration of rotating shaft system with many bearings and disks. Bull. J.S.M.E. Vol. 13, No. 61 (1970). 13. J. V. Fedor, A Sommerfeld solution for finite bearing with circumferential groove. 3. of Appl. Mech. Vol. 27, Series E (1%o). STEIFIGKEIT UND D A M P F U N G S K O E F F I Z I E N T E
EINES SCHR~GEN GLEITLAGERS
A. Mukherjee und J. S. Rao Kurzfassun~
-
Steifigkeit
der
Zur Festlegung
des
dynamischen
Flussigkeitsschieht
und die
Verhaltens
rotierender
Oampfungsbeiwerte
der
Wellen, Lager
eine
spielen
die
Rolle.
Es i s t ~ b l i c k , w e g e n d e r N e i g u n g d e r A c h s e i n dem L a g e r d i e s e D r e h s t e i f i g k e i t e n und D~mpfungsbeiwerte zu v e r n a c h l a s s i g e n . In d i e s e m A u f s a t z w e r d e n d t e s e S t e i f i g k e i t und D~mpfungsbeiwerte solcher
Lager
modifizierte
untersucht. Reynoldsche
Ausgehend Gleichung
y o n dem A u s d r u c k fSr
das
schrage
der
Schichtdteke
und begrenzte
wird
Gleftlager
zun~chst
die
aufgestellt.
Gebrauch Fedorscher P r o p o r t i o n a l l t ~ t s h y p o t h e s e wird dle L6sung dieser Gleichung erhalten, Ergebnisse sind in d i m e n s l o n a l o s e n Quantit~ten dargestellt.
Oureh Die
Stiffness and Damping Coefficients of an Inclined Journal Bearing A. Mukherjeet and
J. S. Rao$ Abstract In the determination of the dynamic behaviour of a rotating shaft, the fluid film stiffness and damping coefficients of the bearings play an important role. The general practice is to ignore the rotational stiffnesses and damping coefficients due to the tilt of the journal in the bearing. This paper presents the stiffness and damping coefficients of such journal bearings. Using the expression for film thickness, the modified Reynolds' Equation for the tilted finite journal bearing is set up. The solution of this equation for the film pressure is obtained by using Fedor's proportionality hypothesis. The results obtained are presented in the form of non-dimensional charts. Introduction DETERMINATIONof the stiffness and damping coefficients of fluid film bearings is important in the study of the dynamic behaviour of rigid and elastic rotors. Solutions for the pressure distribution, used for the determination of stiffness and damping coefficients are given by various workers, for the case of a journal running parallel in the bearing e.g. Hummel[l], Cameron[2] for long bearings, Dubois & Ocvirk[3], Holmes[4] for short bearings and by Pinkus [5] and Strenlicht[6] for finite bearings. Detailed information about these coefficients can be seen in the books of Pinkus[7] and Smith[8]. Krishnova[9] determined the pressure distribution of the film in a bearing with tilted journal. Capriz[10] analysed the behaviour of a tilted journal at small eccentricities, in finite bearings. This work was further extended [l l] to account for larger eccentricities in short bearings. Euler's angles were used in this analysis to determine the film thickness. Kikuchi[12] used a rectangular coordinate system and determined the stiffness and damping coefficients for tilted journals in short bearings. He has shown the importance of these coefficients while determining the response of a Turbine rotor mounted on four hydrodynamic bearings. In this paper rectangular components of inclinations are used to determine the film thickness in a finite bearing with tilted journal. Using Fedor's[13] proportionality hypothesis, the film pressure is obtained, from which the stiffness and damping coefficients are determined. The coefficients thus determined are presented in nondimensional charts.
Reynolds' Equation for Tilted Journal Bearing Figure 1 shows the coordinate system adopted for the analysis of the bearing with an inclined journal. The shape of the journal cross-section at any plane parallel to X Y plane is assumed to be a circle and that the rectangular components of inclination $ and $ are small such that their second and higher order terms may be ignored. From the geometry of Fig. 1 we get u = C K i sin O, v = C K i c o s 0 u' = C K ' sin 0' = u + zd)
cos 0' = v - z~b. tMechanicalEngineeringDepartment,IndianInstituteof Technology,Kharagpur-2,India. *MechanicalEngineeringDepartment,IndianInstituteof Technology,Delhi,New Delhi,India. v' = C K '
(])
339
340
~.
Li
~.~
r A
XIw
Z L
PLANE .
JOURNAL AT ~ o PLANE
~
JOURNAL AT A-A •
Z
X
,
:N
y X-Y
PLANE
Figure 1. Rectangular c o m p o n e n t s for journal inclinations. The following expressions are derived from above equations K' - K, = C (4) sin 0 - qJ cos 0) Z
0' -
(2)
0 = / / - / / ' = ~ , , (4) cos 0 + ¢ sin 0).
Equations (1) and (2) give the expression for film thickness as (3)
h = C {1 + (K, + K2z) sin (fl - ~'z)} where K2 =
4) s i n 0 - ~ c o s
0
C and = 4) cos 0 + ¢ sin 0 CKI
(4)
With the help of eqn (4), the Reynolds' equation for a finite bearing with a tilted journal taking into account of the squeeze terms can be written as 1 0 {
R:a
_ 6/z
,30P1
¢l+K'cosp)
0{
(l+r'cos
- ~ {-K'toJ - 0') sin/3' + 2/~' cos/T}.
,3aP'~
) TQ (5)
Here 0' = 0 + ~'z a n d / ~ ' = K, +/~2z. The relation between/3 and fl', 0 and 0' and between K and K' are as given in Fig. 1. Here it may
341
be noted that in eqn (5), the film thickness is expressed in terms of/3' whereas the partial derivatives appearing on the left hand side are with respect to/3. Using the transformation of coordinates between/3 and/3' {~zl} = [10 - 1 ] {~}
(6)
eqn (5) becomes ,9 { ( I + K ' 3 , 9 P ) + ,9 f {~-~2+ ~2} ~-~7 cos/3 ,'-b-~] ~--z;z,t ( 1 + K' cos/3')3 -~zP,z}P,
cos/3)
z,
cos/3)
6tz , = ~ {-K (to - 20') sin/3'+ 2/~' cos/3'}.
(7)
The corresponding boundary conditions are 0
and
for
(8,
Silliness Coelileients For convenience of analysis Y axis is made to coincide with the line of centers on the mid-plane, then ,t, K 2 = - -~,
6 (r = C K ,
(9)
K2 = -(blc and ~- = ~ICK. Once the stiffness and damping coetlicients are analysed in this coordinate system, then their evaluation in any other coordinate system will be simpler, with the help of similarity rotation transformations. Analysis corresponding to ~b rotation Setting 4' = 0 (K2 = 0) and all time rates zero, eqn (7) becomes 1 O {(I+K, ,30P/+ O f -,)30P) R 40/3' cos/3 ) ~ J ~ ; t(1 + K, cos/3 ~z'
+,g., O, {(1 + K, cos/3,)3 a_~P,/-
,3OP~
O{ 0~,,}] 6~ + --az' (1 + K, cos/3') 3 = - ~ Klto sin/3'.
(lo)
If ~b rotation is small, the perturbation series for the pressure P, may be assumed as P = Po+~
I?'P,.
(11)
i=1
Substituting the above in eqn (10) and separating the coefficients of the same powers of ~r, we get
_ 6/x - - ~ K,oJ sin/3'
(Coeff. of I?o)
(12)
342
1 0{ ,30P1 ] O{ OPiI R~a~ 0 + / ( , c o s / 3 ) - ~ ] + o z : (l+K'c°str)3 az'j
O{
OPo)
0 f
, 30Po)
= 8/3--7 (1 + K, cos/3')3-~-z,~ +~-z, ~[(1 + K, cos ~ ) -~7~
(Coeff. of 7?').
(13)
Equation (12) is conventional Reynolds' equation for the parallel journal case, written in/3', z' coordinates, and left hand side of both the eqns (12) and (13) has Reynolds' operator. Fedor's solution for eqn (12) can be written as
6lzR2K,w (2+Klcos/3') /3'{1 P° = C-2(-(-~I(-7) (1 + K, cos/3')2 sin
cosh[(Q/R)z']] cosh(QL/2R)J"
(14)
This expression for Po, when substituted in eqn (13), with its right hand side expanded in Fourier series, one gets
, s c)2Pl - 3K,(1 + K~ OPI+ (1 + K, OZP! cos/3') 2 sin/3' "zzT~, cos/~~,.3R2 j a-~
(1 + K~ cos/3 ) 3 ~
= .,=o ~ ~" cos m/3' sinh ( Q z ' )
(15)
where Q2 = 2(I + Q,K,) ~--~-~,
K,(2 + X/(1 - K,2)) Q,= ( l + x / ( l _ K 2 ) ) 2
/~,. =/~ [~C,. +2¢,. - 2ho~,.]
(16)
with -3K1 ~
gR4w
rt=2+K2,
l~=
1
1
Q
C2cosh(.~.R) ~
8=~X1, C2=~Xz, d,. =
X , . ÷ , - X,.-i
2
m =3,4 . . . . . . .
2
X,. = K,X/(1 - K,) 2 (-1)"-'a"(m + V'(1 - K f ) 1
so' = X/(1 - K f 1
~,=(l_K2)S~2
and ~,. = 2(-1)%" and
m = 2, 3. . . . . . .
¢l,.=2(-l)"a"(l+mx/(1-K,):)
m =2,3 . . . . . .
(17)
ho = 2(1 - KI 2) Ki 2+K12 , a = l + v , ( l _ K 1 2 ) . Assuming
P, = fP, B,(z') cos (i/3') i=O
and substituting in eqn (15), we get
(D2-n2)B.+-~(D2-n2+n+2)B.+,+ .:,.
3,4 .......
(D2-n2-n+2)B._, (18)
343
where D 2 = 0 2 20 • ",
OZ~z.
Assuming further, Bi = hBi + n/~ and ./~i = .B~ sinh equating the hyperbolic terms
[TI{.B,}
(Qz'lR),
equation for .B. becomes, by
(19)
= {E,}
with gl Tt, = Q22, T,2=-~-(Q2+2)
T. = (QZ - i2),
K, Tu+, = -~- (Q2 _ i s + i + 2)
(2o)
and gl 2 Tu-1=-~(Q-i2-i+2)
for
i=2,3 . . . . . .
(21)
rest T,
are zero.
A suitable number of terms in eqn (19) can be used to obtain the required accuracy. Seven equations in this set were found to give good values, when compared with Capriz[10] and Kikuchi [ 12]. For the homogeneous part,
D2hBo+-~ (D~+ 2)hBi = 0 (D2-
n2)hB. + -~ (D ~ - n 2 + n +
+~(DZ-n2-n+2)hB._~=O
2)hB.+t n=1,2,3 ........
(22)
Using Fedor's approximation
hB1 = - a t , B o
(23)
( D 2 - a22)hBo = 0
(24)
one gets
with
s= 2Klal as 2--Klal and this gives
hBo= C, sinh (R Z') hBl = -C,a, sinh ( R Z').
(25)
344
Boundary conditions demand (26) and (27) From the boundary conditions, the two constants C1 and al in eqn (25) can be determined and then the expression for the pressure can be evaluated as 6gR:toK~(2+K, cos fl', sin [3'
+(r
.B~cosifl')sinh
c°sb ( ~ z ) l
z' + C , ( 1 - a ~ c o s O ' ) s m• h - ~ z + . . .
]
(28)
From the above, the fluid couples about X and Y axes in Fig. 1, can be obtained as Mx = - R
fL/2f~ P cos (/3' + ~rz')z' dz' dfl' J L]2 ,,lO
(29)
= ~?[R2{.B,cr2 + Cz~r3}--~--or,] R fLI2
My =
-w
a-Ln Jo P sin (B' + ~z')z' dz' dfl'
r. l~K,°', ~] ___~2/2 1 {.B,o.2+ C,+,o.3}] = ~ L c r V ( 1 - K , ~) ,=o.~.,
(30)
where .B, s|nh ( ~ - ) Ci=-
sinh [azL'~ ' \2R ] L ~ 2LR 2 [L2R2R3'~
QL
°', =i~+---~z~- k202 Q2~] tanh ( - ~ -)
LR
[QLI 2R2., /azL\
2R 2 •
{a2L] ]
tr3= L R c°sh ~ - f R ) - -Qr Smh \ g' =
(31)
12/~R3to IrK, C ~ (2 + K,2)X/(1 - K,) ~
al = - . B J . B 1 .
The corresponding tilt stiffness components are given as OMx K,,~ =
~"6 =
Nix ~=,,,c,,,,
OM~. Ky+ = Orb = MYI'~="IcK°"
(32) (33)
345
Analysis corresponding to • rotation Setting ~b and all time rates equal to zero and neglecting second and higher order terms of K2, eqn (7) becomes
= -E{K1 sin/3' + z'Kz sin/3'}- 3Kz , 2
,
(1 + KI cos/3 ) cos/3 z ~z;z'l
, dP/-I
(34)
where E _ 6toR2~ --
C 2
"
Assuming a solution of the form P =Po+~
K2"P.
(35)
and following a similar procedure for 4} rotation given before, we get for Po,
cos.( )
E(2+ K~ cos/3')K~ sin/3' Po = (2 + KI2)(1 + K1 cos/3') 2
1
(36)
cosh ( --QL ~ )
Again, as in the previous case, the equation for PI can be obtained as 02PI
-,
OP1
02P1
(1 + KI cos/3') -ff~ - 3K1 sin/~ - ~ + (1 + K1 cos/3') R 20z,2
= ,.=1 ~" D"'lz'+D"'2sinh(Q--~R)+D"'~z'c°sh(Q-~-R) sinm/3' where D,... = E { - 2~,. - 3ho(K.~',. - ~1,.)}
D,..2 = -3)(,,,
(37)
QE R c°sh ( 2 ~ )
3E { (;,,, + ho(K,(,,, - ~.,)-.x,. Q~2} Dra,3
and
(~,. = 2m(-1)"-'am/(K,~/(1 - K,2)) lq,. =
- m ( - l ) " a " ( 1 + m~/(1 - K,z)) K I ( 1 - Kl2) 3/2
1 071,,,.
~" =-30K~'
0.5R2K1 )~" = 2 + K , 2 (~"-'+ 6"+')"
(38)
346
Assuming P, = 2.~ B~ sin ifl'. i=1
(39)
Bi = .Bi + hBi ,Bi = E~z' + F~ sinh
+ H~z'cosh
.
and substituting in eqn (37), the following are obtained by equating the coefficients of ,-', and hyperbolic functions - E l + KIE2 = Dl.i - 4 E 2 - 2K1E1 = D2,1
(40)
- 9 E 3 - 2KIE4 - 5K,E2 = D~.I 2QRHI + K1RQHz = 0
(Q2 _ 1)HI + 0.5KI(Q ~+ 2)H2 = DI.3 (Q 2 - 1)F, + 0.5KI(Q 2 + 2)F~ = Di.2 (Q2 _ 4)F2 + 0.5 K,Q2F3 + 0.SKI(Q 2 - 4)FI + 2RQH2 + K , R Q H 3 + K , R Q H , = 02.5 (Q2 _ 4)/-/2 + 0.5K1Q2H3 + 0.5KI(Q 2 - 4)H, =/92.3.
(41)
Solving above eqns (40) and (41), E , Hi and F~ can be evaluated. Here again, the homogeneous part mE, will satisfy. Here (D 2 - l)hB, + 0.5K,(D2 + 2)hB2 = 0 (D z - l)hB. + 0.5KI(D z - n z + n + 2)hB.+l + 0.5K,(D z - n 2 - n + 2)hB._l = 0 n =2,3 ......
(42)
The boundary conditions (8), will give i = 1, 2 . . . . . .
(43)
Again adopting Fedor's approximation, as in the case of 4~ rotation, hB~'s may be fully evaluated. Then the expression for the pressure can be evaluated as
P = .=1 ~ E . z ' + F. sinh
+ H . z ' cosh
+ C. sinh "--R--
sin n/3'
(44)
where L _ E . ~ + F. sinh (2~_)
L
C.-
(45) sinh ( ~ )
As again in the previous case, the fluid couples and tilt stiffness can be obtained as Mx = -R
= 2R
f Ll2 f ~r P l z ' cos/3' dl3' dz' J-LI2 JO
n=2,4 I.\
+ F.al + H.o'2 + C.o'3
(46)
347
My = R
P ,z' sin/3' d/3' dz' + F~trl + Hla2+
(47)
with
trl =
0.5LR cosh ( 2 ~ ) Q
R2 sinh ( 2 ~ ) Q2
-
(0"5L)2R sinh (2-R QL)
2R
and (0.5L)2R cosh (-a-~)
sinh ( - ~ )
O/2
0/2
tr3 =
OMx I K x . = -~x = M~I
2
(48)
(49)
K2=-(I/C)
and ~M~.
. I
K , , . = - ~ - = M~,I I
(50)
K2:-(IIC)"
Damping Coefficients For determining the damping coefficients corresponding to ~b rotation, we get ~b = ~ = 0 and = 0, in eqn (7) and obtain ,3aP
=
2 a
12/~Rz . /3,. C 2 K,~z'sin
(51)
Letting E = -12~R2K1 C2 and a solution of the form
P = z'Pl(/3') + P2(/3', z')
(52)
for eqn (51), where PI is nonhomogeneous solution and P2 is the homogeneous part of the solution, is assumed. Substituting this form in eqn (51) and separating the coefficients of z and integrating for P, we get P,(/3') - ~. _E(2+ K, cos/3') sin/3' - " ( 2 + K,2)(I + K, cos fl')2" Further assuming P2(/3', z')= ~ An(z')sin (~/3'), one gets from eqn (51), n~l
(53)
348 (D
2 -
I)A,
+ 0.5K,(D 2+
(D 2 - n2)A.
+ 0.5K,(D
2 -
2)A: = 0
n 2 + n + 2)A.+,
+
0.5K,(D
2 -
n 2 -
n + 2)A. - 1 -- 0.
(54)
n=2.3.4 .... Making use of Fedor's approximation, and the boundary conditions (8), which gives
+
i
) +
P2
+-
(55)
= 0
the following is obtained for pressure P as f
E(2 + K, cos/T) sin/3' [
P = "(i-;--g
l z,
(.2L'~ | " 2 sinh \ 2R ] j
(56)
From the above the fluid couples and damping coefficients may be obtained as
m x = (r (2 + K , ~ l - K , ~ ) -i~+ RzL -2
L2R coth 2a2
_. R_ETr {L 3 M~. = -cr (2 + K,~)(I _ K 2),,2 -~+ RzL"~
L?R~coth k["2L'~2R ]J
(57)
where
az2
Cx,-
2(1 + . , K , ) K,(2 + V'(1 - K,2)) (2-.,K,) a'- (I+v'(1-K,Z)) 2
OMx [ 04a Mx
(58)
(59)
¢r=(I/CKI)
and Cy, = O~b My
(60) ~'=(I/CKO
Setting q~ = ~ = 0 and ~b = 0 in eqn (7) and following similar procedure as above, the fluid pressure for the case of q~ rotation, can be obtained as
f
6/zR 2 K2 /3,)2 z P - C-~ K , ( l + K , cos
L sinh k 2R ] 2 . . [.3L\ slnn ~ , ~
[.3z3)
(61)
where x/2K, a3 - x/(1 - K,:)"
(62)
From the above the fluid couples and stiffness coefficients can be obtained as
_6~31ii______~2 Ir fL 3 R2L LZR Mx6 2 (I - K,~) sp~-~ + ~ - ~ c ° t h
(a3L]} \2R/J
(63)
and 12/zR3/(2
My=
C~
I/" f L 3 R2L L~R . I'ot3L~) K,(l-K~)/]5+-h~-~-T~acomlTff)j~
(64)
349
C x . w = -~gMx. ~ - M, t I ,~2= .,c.
(65)
and c~My.
(66)
I
Results and Discussion The tilt stiffness and damping coefficient for ~ and • rotations are determined by using the eqns (32) and (33), (49) and (50), (59) and (60), and (65) and (66) respectively. It may be noted that Capriz's[10] solutions are valid for small eccentricity ratios and in the present analysis a2 ~ 1, with K, ~ 0, and the above mentioned equations reduce to those given by Capriz. These coefficients are evaluated for different eccentricity ratios with the help of an IBM 1620 computer, using Fortran programmes developed for this purpose. The results obtained are presented in non-dimensional form in Figs. 2-9. These results may be directly used for determining the dynamic response of a rigid or flexible rotor and the stability zones. In Figs. 2-9, Capriz's[10] results, calculated from his formula, for LID= 1.0 are also 1.0
0.(~ m
O.g-
0.7--
0-6--
0.S-
II x
0.4
0.3
• /o-2
0.2
///
\
,_/o = ,., O.
B
.Jy/
0
I 0
0.2
0.4
0-6
I
I
0.8
I.O
K i --=--e,.
Figure 2. Non-dimensional rotational stiffness vs eccentricity ratio.
MMT Vol. 12, No. 4---G
350
o_.
=o U
n
I
,~
o
n
6 0
6
~
6
~- ~ ._. 5 0
i
,
J
._
L
i
o
1
l
~
~
1.
o
A
~
i
~
l
I
o
0
0 (-4
z
0
'~"
~
._~ i1
0
~ _._~ ,,
L,
o
L
I
I
I
i
6
o
o
6
6
,
,,
a
a
,L''~I ! o ~,,'
\l\-~ o
o
m
~
6
--
~
z
O
o
2 -
14=
-~ 6
351
.03 L q'~I
|'0
L.I.
~
-02
>IX
.01
o.o
1
0.0
0.2
\
s -
I
I
I
0.4
0.6
0.8
KI
A f .0
~-
Figure 5. Non-dimensional rotational stiffness vs
eccentricity
ratio.
• I 4-
.12
• lOi-
t
• /p= 2
u~
.0|
t
• 0~
-
II
x: .04
=1
-02
O0
0.2
0-4
0.6
I
0-8
I
I
I.O
K i .~...-tm.
Figure 6. Non-dimensional
rotational d a m p i n g vs eccentricity ratio.
352
•3 2 --
.28
.24
t ~i ~
.20
.~6
I II
>iu
-12
OS
-04
I
I
1
I
I
I
I
I
.I
-2
-3
.4
.5
.6
.7
-8
I
I
.9
1.0
Figure 7. Non-dimensional rotational damping vs eccentricity ratio.
353
.11
• 10
-
--
I./D = 2.0
T u
II
•0 9
--
• 0 8
--
' 0 7
--
• 0 0
-
\ ~ \\\\\\
II >t~
.05
L/D=
1.5
\.
.04
.03
•0 2 - -
~
.01
Oo
I
I
0.2
I
F
I
0.4
CAPRIZ
I
I
0.6 KI
Flgure
-
I
o.e
I
I
~.o
•
8. Non-dimensional rotational damping vs eccentricity ratio.
354
. 11
.10
--
• 09
--
• 08
--
• 07
--
•0
/
6 --
I tl
x lU
• 04
--
• 03
--
• 02
--
• 0!
--
00,
c/o = a /
~
/D-I ~ / / / I
I 0.2
I
I
I
0.4
I 0.6
I
I 0.8
I I.(3
K
Figure9. Non-dimensionalrotationaldampingvseccentricityratio.
355 presented for comparison, for mid plane eccentricity values Km in the region 0.1 to 0.7. These results are in good agreement for 0.1 ~< K~ ~< 0.4. It may be noted here that Capriz's[10] analysis is valid for only small values of KI.
Conclusion Using the film expression in rectangular c o m p o n e n t s , the R e y n o l d s ' equation for a finite bearing with a tilted journal is set up. With the help of F e d o r ' s proportionality hypothesis, the modified R e y n o l d s ' equation is solved to determine the pressure distribution in the film, using perturbed equations. F r o m this pressure distribution, the stiffness and damping coefficients of a tilted finite journal bearing are determined and c o m p a r e d with Capriz's results. G o o d agreement is shown with Capriz's results for small eccentricity values.
Nomenclature C Cx,, C×,, etc. D K~ K K×., K×,, etc. L M×, My P R W=
radial clearance components of tilt damping coefficient diameter of the bearing eccentricity ratio at reference plane eccentricity ratio at any plane components of tilt stiffnesses length of the bearing components of fluid couple film pressure radius of the bearing load on full bearing with large LID (o angular velocity viscosity of the lubricant ~ , 6 components of journal inclination (see Fig. 1) 0,0' attitude angles (see Fig. 1) (.) On any quantity denotes its time derivative)
References 1. C. Hummel, Kritische Drehzahlen als Forge der Nachgiebigkeit des $chmiermittels im Lager, V.D.I. Forschungs, Heft 287 (1926). 2. A. Cameron, Oil whirl in journal bearing. Engng, Vol. 179, No. 4648 (1955). 3. G. B. Dubois and F. W. Ocvirk, Analytical derivation and experimental evaluation of short bearing approximation for full journal bearing. N.A.C.A. Report 1157 (1953). 4. R. Holmes, The effect of sleeve bearings on the vibration of rotating shaft. Tribology (1972). 5. O. Pinkus, Solution of Reynolds' equation for finite journal bearing. Trans. A.$.M.E. Vol. 80 (1958). 6. B. Strenlicht and F. J. Maginnis, Application of digital computer of bearing design. Trans. A.S.M.E. Vol. 75 (1957). 7. O. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication. McGraw-Hill, New York (1961). 8. D. M. Smith, Journal Bearings in Turbomachinery. Chapman and Hill, London 0969). 9. L. B. Khrisanova, Analytical and experimental investigation of the pressure in an oil film of a journal bearing with axis of the journal and bearing skewed. Fric Wear Mach. 13, 190-208 (1959). 10. G. Capriz, On some dynamic problems arisingin the theory of lubrication. Nelson Research Lab., English Electric Co. Ltd., Stafford, England. Research Report--Part I. Riv. Mat. Univ. Parma (2) l, 1-20 (1960). 11. Galletti-Manacorda and G. Capriz, Torque produced by misalignment in short bearing. Trans. A.S.M.E., J. Basic Engng Paper No. 65, Lubs-9. 12. K. Kikuchi, Analysis of unbalance vibration of rotating shaft system with many bearings and disks. Bull. J.S.M.E. Vol. 13, No. 61 (1970). 13. J. V. Fedor, A Sommerfeld solution for finite bearing with circumferential groove. 3. of Appl. Mech. Vol. 27, Series E (1%o). STEIFIGKEIT UND D A M P F U N G S K O E F F I Z I E N T E
EINES SCHR~GEN GLEITLAGERS
A. Mukherjee und J. S. Rao Kurzfassun~
-
Steifigkeit
der
Zur Festlegung
des
dynamischen
Flussigkeitsschieht
und die
Verhaltens
rotierender
Oampfungsbeiwerte
der
Wellen, Lager
eine
spielen
die
Rolle.
Es i s t ~ b l i c k , w e g e n d e r N e i g u n g d e r A c h s e i n dem L a g e r d i e s e D r e h s t e i f i g k e i t e n und D~mpfungsbeiwerte zu v e r n a c h l a s s i g e n . In d i e s e m A u f s a t z w e r d e n d t e s e S t e i f i g k e i t und D~mpfungsbeiwerte solcher
Lager
modifizierte
untersucht. Reynoldsche
Ausgehend Gleichung
y o n dem A u s d r u c k fSr
das
schrage
der
Schichtdteke
und begrenzte
wird
Gleftlager
zun~chst
die
aufgestellt.
Gebrauch Fedorscher P r o p o r t i o n a l l t ~ t s h y p o t h e s e wird dle L6sung dieser Gleichung erhalten, Ergebnisse sind in d i m e n s l o n a l o s e n Quantit~ten dargestellt.
Oureh Die