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Rotordynamic coefficients of multirecess hybrid journal bearings part I
Wear, 129 (1989) 245 - 259
245
ROTORDYNA~IC COEFFICIENTS JOURNAL BEARINGS PART I
OF ~ULTIR~CESS
HYBRID
M. K. GHOSH, S. K. GUHA and B. C. MAJUMDAR Mechanicai
Engineering
(Received May 25,1988;
Department,
Indian Institute
of Technology,
Kharagpur
(India)
accepted September 22,1988)
Summary Rotordyn~ic coefficients of a multirecess hybrid bearings have been evaluated following first-order perturbation of the dynamic Reynolds equation. Frequency effects on the rotordynamic coefficients are observed for high frequency vibrations for bearings with deep recesses. Results are presented for a capillary-compensated four-recess journal bearing.
1. Introduction High frequency vibration characteristics of multirecess hydrostatic journal bearings have attracted much attention recently because of their increased application as load support elements in high speed t~bomach~e~. Current applications include their use as high speed long life LH, hybrid bearings for reusable rocket engine turbopumps and in precision machine tools. High rigidity and stable operation are their main advantages in use. A recent investigation by Braun et al. [l] dealt with a thermohydraulic model for a cryogenic hydrostatic journal bearing incorporating variable fluid properties. However, Redecliffe and Vohr [2], Heller [3] and Artiles et al. [4] dealt with the problem in the turbulent operational regime. Kocur and Aliaire [5] recently highlighted further applications of these bearings as guide bearings in the main coolant pumps in nuclear power plants and in steam turbines of helium-cooled reactors and they adopted a finite element method for the solution. A general and excellent analysis of the dynamic behaviour of multirecess hydrostatic journal bearings was presented by Davies and Leonard [6] for bearings with short sill dimensions. The possibility of whirl instability was also investigated by Leonard and Rowe [7]. Ghosh [8] presented a perturbation analysis using a finite difference method to evaluate the dynamic coefficients. It was shown recently by Ghosh and Viswanath f9) that the recess volume fluid compressib~ity drastically alters the dynamic characteristics of these bearings in the high frequency range. In this paper a general analysis has been presented to evaluate the rotordynamic coefficients of a capillary-compensated multirecess hybrid 0043-1648/89/$3.50
@ Elsevier Sequoiaf~inted
in The Netherlands
246
bearing with reported.
deep
recesses.
The
effects
of high frequency
vibrations
are
2. Theory The bearing configuration is shown in Fig. 1. The journal undergoes rotation about its own axis and the bearing system is subjected to a small amplitude high frequency excitation from an external source. For an isoviscous incompressible lubricant, the equation of flow of the lubricant in the bearing clearance is written as
(1) With the following jg
g
substitution:
e=;
z=
z
U=RS2
p=P
L/2
PS
(2)
7=cdt
eqn. (1) can be expressed
in dimensionless
form as (3)
SUPPLY
LINE
CHECK
VALVE
PRESSURE RESSURE
GAUGE CONTROL
pkggSUMP Fig. 1. Multirecess
hybrid
journal
bearing system.
VALVE
247
For the journal centre executing small amplitude vibrations around an equilibrium position (Q, $a) first-order perturbations can be used to express dynamic pressures and film thickness as Jj = pIs,+ e,ei7j1 + ea@,ei7& ii = ho + e1ei7 cos B + eO&eir sin B
(4)
where fi* = 1+ Eg cos 8 Introducing obtain
eqn. (4) into eqn. (3) and neglecting higher order terms we
= io cos 8 - A sin 8
(6)
and
=iosin8+AcosB
(7)
Appropriate boundary conditions for the above equations are expressed as (1) pi(0,--1)
=pj(@, +l) = 0
(2)
~(e,o)=o
t3)
Pi
=Pri
(4) &(O,Z)=pi(l)
at the rth recess f 2n,Z)
where i = 0, 1, 2 respectively for eqn. (5), eqn. (6) and eqn. (7). The steady recess pressures (pro) were evaluated following ref. 3. The above equations were written in finite difference form and solved using a high speed digital computer following the Gauss-Siedel iterative procedure using the overrelaxation factor.
248
2.1. Dynamic recess pressures j& and & The recess flow continuity equation bearing is given below as
a,( 1 - &) = &,p, + $
for
a
capillary-compensated
12/.lo + C”PS
(Ah,)
(9)
where “A” is the effective recess area which includes a portion of the sill area around the recess [lo]. The dynamic recess pressures can be evaluated following small amplitude first-order perturbations for p,, a, and h, as follows: p, = Pro +
-
ele’7& + EO@le'T1jr2
-
4,. = Qra + Ele
i7 -
Qrl + %he
i7 -
(10)
Qr2
h, = hro + Elei7 cos 13~+ c0@rei7 sin 8,
where
iz,o=(1
+ fe cos e,)
The dynamic recess pressures PrI and jr2 are obtained by substituting (10) into eqn. (9) and collecting only first-order terms as below
Prl = -
QrlprO~ + \k2y cos er (X2 + \k2y2) Q,2p,oX + Jr2y sin
i(\ky
cos
eqn.
8, - Q,&.o*ky)
(X2 + -,k2y2)
8,
(11)
i(\ky sin 4. - Qr2Pr0W)
Pr2 = (h2+
*2y2)
-
(X2 + \k2y2)
where
6, = wo
h = (8,, + 6,)
Also, y = vo/3p, is the recess volume compressibility (oA)/R2 is the recess or frequency parameter.
parameter
and *
=
2.2. Stiffness and damping characteristics The components of the restoring dynamic load along the e. and @o directions due to the perturbed film pressure eIeiTpl can be written as follows: L/2 (W,),ei7
(W,),e"
=
=
-2
2
2n
eIei7p,R
ss 0
0
L/2
2n
JV 0
cos 8
de dz
e,ei7pp,R sin 8 de dz 0
(12)
(13)
249
Further, the restoring dynamic load resulting from the small periodic displacement e”pr can be written in terms of linearized spring and damping coefficients as ( %)Re i7 = -I&y,
-B,,
( W,),eiT = --Kcpe~ e
-
dY, dt
B,,
(14)
dY,
(15)
dt
where the amplitude of oscillation of eccentricity is given by
of the journal
centre
yE = CeleiT
along the direction
(16)
Substituting
eqn. (16) into eqns. (14) and (15) we obtain B&o
(wI)R (%)R = -
=----_kc LDP,~ I LDP,
LDP, _ i B&o LDP,
(17)
(18)
where K,, and K,, are the direct and cross stiffness respectively and B,, and B,, are the direct and cross damping terms. Stiffness and damping can be expressed in nondimensional form as
LC
K,, = --R,{( &)R} = -
LDP,
K,,C
I-& = -R,{(
ivii,),} = -
LDP,
jj
EE
0
(19)
B
= _ Lmt(%)R)
= 24/AL&C)T
%, = 24/.~L(R/c)~ Similarly, the components of the restoring dynamic load along the eO and @,,directions due to the perturbed film pressure eo@leiTp, can be written as follows: L/2 (
W2)Rei7 = -2s
( W2),ei7
= q-
0
277
e,@,ei7p2R
0
$ 0
L/2
277
J
0
EO@le”p,R
cos 8 de de
sin 8 de dz
(26)
(21)
250
The restoring dynamic load can be expressed spring and damping coefficients as ( W2)Re17 = - GOY o -
W2hei7 = -K,,yG
in terms of the linearized
dy0
4,
(22)
dt
dY@
-B,,
(23)
dt
where the amplitude of oscillation of the journal perpendicular to that of eccentricity is given as
centre along the direction
y$ = CeO@leiT Substituting
(24)
eqn. (24) into eqns. (22) and (23) we get (25)
OQ, =
(W&r
=-- K@oC B,,Cw
= --Re{(W,)R)
K@@= -%d%),)
I
LDP,
K,, and K,, are the spring constants constants of the bearing. The spring and damping constants form as &I
(26) \
-l-
LDP,
LDP,Eo@ I
and B,$ and B,,
are the damping
can be expressed
in dimensionless
L-W =
-
LDP, =
Wa)
%dC -
LDP, ~~~ = _
u E,,
=-
B &J
&I{(~,)R)
M(K),> 0
= 24/_~L(R/c)~
(27b) R00 = 24/..1L(R/c)~
3. Results and discussion Results are presented for a four-recess capillary-compensated bearing, with specifications ti = 6 = 0.5, L/D = 1.0, 6, = 7.795, in Figs. 2 - 13. The variation of the direct stiffness coefficients zcE and j?e@ with the frequency parameter u is shown in Figs. 2 - 4 for various values of the recess volume parameter y and the speed parameter A. It is observed that, within a frequency range of 100 < u < 1000, the stiffness coefficient increases with u by an order of magnitude or more before it levels off to a
251 102
N.L,;.i.O.5,
.c,=o.z
------
E,:O.5
I __
y : 0.01
2 -y
= 0.05
3 -y
= 0.10
h=2,5
/
-/ 04 30
L/o=l.o,6c=7.795,
~
'
' ' "'I 50 100
I 200
I
I11111 500
1000
1 2000
I 4
10
u vs. u for various
Fig. 2. &,
'._ F
N=L,
a=b=0.5, -A=
50
6,:
7.795,
E,= 0.2
5.0
1-y
= 0.01
2-y
= 0.05
3-y=
30
~1.0,
2.0
-----_A=
0.66
L/D
values of Ed and y.
100
0.10
200
500
1000
2000
1000
o-
Fig. 3. &,o
vs. u for various
values of A and y for e. = 0.2.
Fig. 3. &,G
vs. u for various
values of A and y for EO = 0.2.
252
L/D=
N.Lrii=i-0.5, -----*~ r---y
30
1.0 , 6,=
7.795 ,Eo=
0.5
A : 2.0 5.0 = 0.01
2---y
= 0.05
3-y
= 0.10
50
100
200
500
1000
2000
5000
u
Fig. 4. &@
us. (5 for various
values of A and y for e0 = 0.5.
,.-rN=L,~;~:0.5,L/D=l0,6~:7.795.h;
2,5
---------
u Fig. 5. BEE us. u for various
values of fo and y,
253
0.01’ ’ ’ ’ ’ ” ’ 30
50
100
t
1
200
500
1000
2004
5m
o-
Fig. 6. gGe vs. u for various values of /t and y for EO= 0.2.
0.14
NIL,
8=$.0.5,L/O=l.O,
‘&=7.795, E,,‘o.s
O.O~o~ 100
500
200
1000
2000
5000
u
Fig. 7. &@ vs. u for various values of A and y for co= 0.5.
constant value. This behaviour is influenced by the recess volume parameter y. The constant high frequency value of the stiffness coefficient decreases with increasing y. The speed parameter effect is relatively small. Variation of the direct damping coefficients BEE and EGO with (Tis shown in Figs. 5 - 7 for various values of y and A. Direct damping coefficients are seen to decrease with increasing frequency parameter u prior to attaining a constant value at
254
’ ’
-4.0
30
50
“If’ 100
i I
200
I
b
II
,bIk,
500
1000
2000
/
I
WOO
CT
Fig. 8. I?,@ and I?$,,
us. u for various
values
of A and y for e. = 0.2.
high frequencies. An increase in the recess volume parameter y decreases the damping coefficients. The recess volume fluid compressibility effect is manifested through eqn. (11) for dynamic recess pressures. When the recess volume fluid compressibility effect is large, the bearing behaves more like a liquid spring as is marked in the frequency response of direct stiffness and damping coefficients. The speed parameter A effect is observed for Eea, and effect in the B,,, particularly at e. = 0.5. This is due to a hydrodynamic pressure distribution over the sills and the modifications in the perturbed flow ori and a,.2 in eqn. (II). Variation of the cross stiffness coefficients EEch and KG, with (I is shown in Figs. 8 - 9 for various values of y, A and co. Cross stiffness coefficients are generated owing to hydrodynamic and rotational effects and are visible over the entire frequency range. At low frequencies (u < 100) the recess volume effect is negligible and therefore it is due to purely rotational effects. It increases with increasing speed parameter A. Positive and negative values of zEG and EQE respectively are typical of hybrid bearings. Figures 10 - 13 depict the variation of cross damping coefficients BEG and &,, with e for different values of y, A and eo. BGf increases with u and attains a constant value at high u. It increases with increasing r, A and co.
255 3.0 EO’0.5 ___ A i 2.0 -----_*~~.O
2.0-
1-y: G;s 1.0 -
0.01
z--y=
0.05
3-y=
0.10
0.0
Fig. 9. Keo and & 2.5
-3.5 30
vs. u for various values of A and 7 for fg = 0.5.
N~&,fi;6=0.5,
1 I@ill’ 50
~/~.,.~.6,=7.7~~,
1 I I ,t,,,l 100
200
500
E,'O.2
I 1000
2000
t, 5000
cr
Fig. 10. &
vs. a for various values of i\ and y for e. = 0.2.
256 --. N;C,0;~.0.5,~1D=l.O,6,~7.795 ,E,=o.5
4.0
-h=
2.0
I
-l
-----A..S.O 1---y=
3.0
0.01
2--y:o.o5 3---y=
0.10
~~
/ //
3
1.0 1'
0.0 '
50
/
/I
1' /
'
/ / A-c
100
/
%2
,'
_----
30
/ / /
,'
,'
do-
-+=
/' '
//
'
/ //
2.0
X103
i /--
200
/
5oa
1000
2000
5( 10
o-
Fig. 11. &
5.0
US. D for various
values of A and y for &o = 0.5.
N;~,~~b:0.5.L/D=l.0,6c=7~795,~o=0.2 -A
= 2.0
_---__n:5.0 L.0 -
1---y=
0.01
z--y=o.o5 3-y;
Fig. 12. &Q
0.10
US. u for various values of A and 7 for Q, = 0.2.
In contrast, BP@ decreases with increasing CJuntil it reaches a datum value at high frequencies. It also shows a decrease with increasing y and A. However, cross damping coefficients are small in magnitude. The dynamic response of the bearing due to either unbalance or external dynamic loading can be determined using these rotordynamic coefficients. The bearing is likely to exhibit whirl instability in the low frequency regime of operation.
257
30
50
100
200
MO
1000
2000
5000
o-
Fig. 13. Bee
vs. u for various
values of A and 7 for EO = 0.5.
4. Conclusions (1) The rotordynamic coefficients of multirecess hybrid bearings are frequency dependent for bearings with deep recesses. (2) The dynamic force response of the bearing would be dependent on the frequency of vibration, the recess volume compressibility parameter and the speed parameter for a given bearing geometry. (3) Whirl instability may arise in the low frequency range where the recess volume compressiblity effect is negligible.
Acknowledgment The Industrial
authors express their gratitude to the Council of Scientific and Research, India, for financial support during this research work.
References 1 M. J. Braun, R. L. Wheeler, III and R. C. Hendricks, A fully coupled variable properties thermohydraulic model for a cryogenic hydrostatic journal bearing, J. Tribal., 109 (3) (1987) 405. 2 J. M. Redecliffe and J. H. Vohr, Hydrostatic bearings for cryogenic rocket engine turbopumps, J. Lubr. Technol., 91 (3) (1969) 557 - 575. 3 S. Heller, Static and dynamic performance of externally pressurised fluid film journal bearings in the turbulent regime,J. Lubr. Technol., 95 (3) (1974) 381.
258 4 A. Artiles, J. Walowit and W. Shapiro, Analysis of hybrid fluid film journal bearings with turbulence and inertia effects, Advances in Computer Aided Bearing Design,
American Society of Mechanical Engineers, New York, 1982. 5 J. A. Kocur and P. E. Allaire, Finite element analysis of turbulent lubricated hydrostatic journal bearings for static and dynamic conditions, ASLE Trans., 29 (1986) 126. 6 P. B. Davies and R. Leonard, The dynamic behaviour journal bearings, Proc. Inst. Mech. Eng., London, 184 (3L)
7 8 9
10
of multirecess hydrostatic (1969 - 1970) 139. R. Leonard and W. B. Rowe, Dynamic force coefficients and the mechanism of instability in hydrostatic journal bearings, Wear, 23 (1973) 277. M. K. Ghosh, Dynamic characteristics of multirecess externally pressurised oil journal bearings, J. Lubr. Technol., 100 (4) (1978) 467 - 471. M. K. Ghosh and N. S. Viswanath, Frequency dependent stiffness and damping coefficients of orifice compensated multirecess hydrostatic journal bearings, fnt. J. Mach. Took Manuf., 27 (3) (1987) 275. W. B. Rowe, Hydrostatic and Hybrid Bearing Design, Butterworths, London, 1983.
Appendix A: Nomenclature axial length of the recess, ti = a/L (dimensionless) effective recess area i; = Nb/rD (dimensionless) direct damping coefficients of fluid film B,, = Be,/ 24gL(R ,‘Ct3, BQQ,= R&~~PL(R/C)~ (dimensionless) cross damping coefficients of fluid film & = Bee/ (dimensionless) 24~.tL(R/c)~, & = &,,/24~L(R/c)~ radial clearance diameter of capillary tube recess depth diameter of journal eccentricity, E = e/c (dimensionless) steady state eccentricity, e. = e,/C (dimensionless) perturbed eccentricity, et = e,/C (dimensionless) film thickness, h = h/C (dimensionless) steady state film thickness, ho = ho/C (dimensionless) film thickness at the centre of the rth recess, &, = h,/C (dimensionless) (-1)“2 direct stiffness coefficients of fluid film BEE= I(,$/ Lll)p,, I!$,+ = K~~C/LDp~ {dimensionless) cross stiffness coefficients of fluid film KE+= K,&/ LDp,, E$e = K,,C/LDp, (dimensionless) length of capillary tube length of bearing number of recesses supply pressure of fluid film thickness, p = p/p, (dimensionless) steady state film thickness, PO= po/ps (dimensionless)
259
Prl, Pr2,PrlvPr2 ::;?i., R t, 7 u ~;~
C~,,,,
(WI),,
(W2)Ft
x9 Y, 2
YE
Y A
dynamic film pressures, fir = plIps, p2 = p2/ps (dimensionless) pressure at the rth recess, & = p,/p, (dimensionless) steady state recess pressure at the rth recess pro = pro/ps (dimensionless) dynamic recess pressures at the rth recess, & = prl/ps, jr2 = pr2/ps (dimensionless) flow at the rth recess, Q, = 12pQ,/C3p, (dimensionless) steady state flow at the rth recess, Q,a = 12pQ,,/C3p,, (dimensionless) journal radius time, T = wt (dimensionless) shaft surface, velocity recess volume, V, = Ad,, v,, = V,/AC (dimensionless) radial components of restoring dynamic loads due to dynamic film pressures p1 and p2 (WI), = (WI),/ LDpg,, (v,), = ( W2),/LDp,~,,4, (dimensionless) transverse components of restoring dynamic loads due to dynamic film pressures p1 and p2 (E,), = (IV,),/ LDp,e, , ( W2)T = ( W2)T/LDps~O@I (dimensionless) Cartesian coordinates amplitude of oscillation of journal centre along the direction of eccentricity amplitude of oscillation of journal centre along the direction perpendicular to that of eccentricity fluid compressibility parameter angular coordinate angular coordinate at the centre of rth recess steady state attitude angle perturbed attitude angle absolute viscosity of lubricant density of lubricant kinematic viscosity of lubricant, 7) = p/p squeeze number or frequency parameter, c = 12pw/ P,((C/R)~ (dimensionless) recess frequency parameter, \k = aAIR (dimensionless) frequency of vibration of journal angular speed of journal capillary design parameter, 6, = 3nd:/32C31, (dimensionless) recess volume fluid compressibility parameter, y = vJ3p, (dimensionless) bearing number, A = 6puS2/p,(C/R)2 (dimensionless)
245
ROTORDYNA~IC COEFFICIENTS JOURNAL BEARINGS PART I
OF ~ULTIR~CESS
HYBRID
M. K. GHOSH, S. K. GUHA and B. C. MAJUMDAR Mechanicai
Engineering
(Received May 25,1988;
Department,
Indian Institute
of Technology,
Kharagpur
(India)
accepted September 22,1988)
Summary Rotordyn~ic coefficients of a multirecess hybrid bearings have been evaluated following first-order perturbation of the dynamic Reynolds equation. Frequency effects on the rotordynamic coefficients are observed for high frequency vibrations for bearings with deep recesses. Results are presented for a capillary-compensated four-recess journal bearing.
1. Introduction High frequency vibration characteristics of multirecess hydrostatic journal bearings have attracted much attention recently because of their increased application as load support elements in high speed t~bomach~e~. Current applications include their use as high speed long life LH, hybrid bearings for reusable rocket engine turbopumps and in precision machine tools. High rigidity and stable operation are their main advantages in use. A recent investigation by Braun et al. [l] dealt with a thermohydraulic model for a cryogenic hydrostatic journal bearing incorporating variable fluid properties. However, Redecliffe and Vohr [2], Heller [3] and Artiles et al. [4] dealt with the problem in the turbulent operational regime. Kocur and Aliaire [5] recently highlighted further applications of these bearings as guide bearings in the main coolant pumps in nuclear power plants and in steam turbines of helium-cooled reactors and they adopted a finite element method for the solution. A general and excellent analysis of the dynamic behaviour of multirecess hydrostatic journal bearings was presented by Davies and Leonard [6] for bearings with short sill dimensions. The possibility of whirl instability was also investigated by Leonard and Rowe [7]. Ghosh [8] presented a perturbation analysis using a finite difference method to evaluate the dynamic coefficients. It was shown recently by Ghosh and Viswanath f9) that the recess volume fluid compressib~ity drastically alters the dynamic characteristics of these bearings in the high frequency range. In this paper a general analysis has been presented to evaluate the rotordynamic coefficients of a capillary-compensated multirecess hybrid 0043-1648/89/$3.50
@ Elsevier Sequoiaf~inted
in The Netherlands
246
bearing with reported.
deep
recesses.
The
effects
of high frequency
vibrations
are
2. Theory The bearing configuration is shown in Fig. 1. The journal undergoes rotation about its own axis and the bearing system is subjected to a small amplitude high frequency excitation from an external source. For an isoviscous incompressible lubricant, the equation of flow of the lubricant in the bearing clearance is written as
(1) With the following jg
g
substitution:
e=;
z=
z
U=RS2
p=P
L/2
PS
(2)
7=cdt
eqn. (1) can be expressed
in dimensionless
form as (3)
SUPPLY
LINE
CHECK
VALVE
PRESSURE RESSURE
GAUGE CONTROL
pkggSUMP Fig. 1. Multirecess
hybrid
journal
bearing system.
VALVE
247
For the journal centre executing small amplitude vibrations around an equilibrium position (Q, $a) first-order perturbations can be used to express dynamic pressures and film thickness as Jj = pIs,+ e,ei7j1 + ea@,ei7& ii = ho + e1ei7 cos B + eO&eir sin B
(4)
where fi* = 1+ Eg cos 8 Introducing obtain
eqn. (4) into eqn. (3) and neglecting higher order terms we
= io cos 8 - A sin 8
(6)
and
=iosin8+AcosB
(7)
Appropriate boundary conditions for the above equations are expressed as (1) pi(0,--1)
=pj(@, +l) = 0
(2)
~(e,o)=o
t3)
Pi
=Pri
(4) &(O,Z)=pi(l)
at the rth recess f 2n,Z)
where i = 0, 1, 2 respectively for eqn. (5), eqn. (6) and eqn. (7). The steady recess pressures (pro) were evaluated following ref. 3. The above equations were written in finite difference form and solved using a high speed digital computer following the Gauss-Siedel iterative procedure using the overrelaxation factor.
248
2.1. Dynamic recess pressures j& and & The recess flow continuity equation bearing is given below as
a,( 1 - &) = &,p, + $
for
a
capillary-compensated
12/.lo + C”PS
(Ah,)
(9)
where “A” is the effective recess area which includes a portion of the sill area around the recess [lo]. The dynamic recess pressures can be evaluated following small amplitude first-order perturbations for p,, a, and h, as follows: p, = Pro +
-
ele’7& + EO@le'T1jr2
-
4,. = Qra + Ele
i7 -
Qrl + %he
i7 -
(10)
Qr2
h, = hro + Elei7 cos 13~+ c0@rei7 sin 8,
where
iz,o=(1
+ fe cos e,)
The dynamic recess pressures PrI and jr2 are obtained by substituting (10) into eqn. (9) and collecting only first-order terms as below
Prl = -
QrlprO~ + \k2y cos er (X2 + \k2y2) Q,2p,oX + Jr2y sin
i(\ky
cos
eqn.
8, - Q,&.o*ky)
(X2 + -,k2y2)
8,
(11)
i(\ky sin 4. - Qr2Pr0W)
Pr2 = (h2+
*2y2)
-
(X2 + \k2y2)
where
6, = wo
h = (8,, + 6,)
Also, y = vo/3p, is the recess volume compressibility (oA)/R2 is the recess or frequency parameter.
parameter
and *
=
2.2. Stiffness and damping characteristics The components of the restoring dynamic load along the e. and @o directions due to the perturbed film pressure eIeiTpl can be written as follows: L/2 (W,),ei7
(W,),e"
=
=
-2
2
2n
eIei7p,R
ss 0
0
L/2
2n
JV 0
cos 8
de dz
e,ei7pp,R sin 8 de dz 0
(12)
(13)
249
Further, the restoring dynamic load resulting from the small periodic displacement e”pr can be written in terms of linearized spring and damping coefficients as ( %)Re i7 = -I&y,
-B,,
( W,),eiT = --Kcpe~ e
-
dY, dt
B,,
(14)
dY,
(15)
dt
where the amplitude of oscillation of eccentricity is given by
of the journal
centre
yE = CeleiT
along the direction
(16)
Substituting
eqn. (16) into eqns. (14) and (15) we obtain B&o
(wI)R (%)R = -
=----_kc LDP,~ I LDP,
LDP, _ i B&o LDP,
(17)
(18)
where K,, and K,, are the direct and cross stiffness respectively and B,, and B,, are the direct and cross damping terms. Stiffness and damping can be expressed in nondimensional form as
LC
K,, = --R,{( &)R} = -
LDP,
K,,C
I-& = -R,{(
ivii,),} = -
LDP,
jj
EE
0
(19)
B
= _ Lmt(%)R)
= 24/AL&C)T
%, = 24/.~L(R/c)~ Similarly, the components of the restoring dynamic load along the eO and @,,directions due to the perturbed film pressure eo@leiTp, can be written as follows: L/2 (
W2)Rei7 = -2s
( W2),ei7
= q-
0
277
e,@,ei7p2R
0
$ 0
L/2
277
J
0
EO@le”p,R
cos 8 de de
sin 8 de dz
(26)
(21)
250
The restoring dynamic load can be expressed spring and damping coefficients as ( W2)Re17 = - GOY o -
W2hei7 = -K,,yG
in terms of the linearized
dy0
4,
(22)
dt
dY@
-B,,
(23)
dt
where the amplitude of oscillation of the journal perpendicular to that of eccentricity is given as
centre along the direction
y$ = CeO@leiT Substituting
(24)
eqn. (24) into eqns. (22) and (23) we get (25)
OQ, =
(W&r
=-- K@oC B,,Cw
= --Re{(W,)R)
K@@= -%d%),)
I
LDP,
K,, and K,, are the spring constants constants of the bearing. The spring and damping constants form as &I
(26) \
-l-
LDP,
LDP,Eo@ I
and B,$ and B,,
are the damping
can be expressed
in dimensionless
L-W =
-
LDP, =
Wa)
%dC -
LDP, ~~~ = _
u E,,
=-
B &J
&I{(~,)R)
M(K),> 0
= 24/_~L(R/c)~
(27b) R00 = 24/..1L(R/c)~
3. Results and discussion Results are presented for a four-recess capillary-compensated bearing, with specifications ti = 6 = 0.5, L/D = 1.0, 6, = 7.795, in Figs. 2 - 13. The variation of the direct stiffness coefficients zcE and j?e@ with the frequency parameter u is shown in Figs. 2 - 4 for various values of the recess volume parameter y and the speed parameter A. It is observed that, within a frequency range of 100 < u < 1000, the stiffness coefficient increases with u by an order of magnitude or more before it levels off to a
251 102
N.L,;.i.O.5,
.c,=o.z
------
E,:O.5
I __
y : 0.01
2 -y
= 0.05
3 -y
= 0.10
h=2,5
/
-/ 04 30
L/o=l.o,6c=7.795,
~
'
' ' "'I 50 100
I 200
I
I11111 500
1000
1 2000
I 4
10
u vs. u for various
Fig. 2. &,
'._ F
N=L,
a=b=0.5, -A=
50
6,:
7.795,
E,= 0.2
5.0
1-y
= 0.01
2-y
= 0.05
3-y=
30
~1.0,
2.0
-----_A=
0.66
L/D
values of Ed and y.
100
0.10
200
500
1000
2000
1000
o-
Fig. 3. &,o
vs. u for various
values of A and y for e. = 0.2.
Fig. 3. &,G
vs. u for various
values of A and y for EO = 0.2.
252
L/D=
N.Lrii=i-0.5, -----*~ r---y
30
1.0 , 6,=
7.795 ,Eo=
0.5
A : 2.0 5.0 = 0.01
2---y
= 0.05
3-y
= 0.10
50
100
200
500
1000
2000
5000
u
Fig. 4. &@
us. (5 for various
values of A and y for e0 = 0.5.
,.-rN=L,~;~:0.5,L/D=l0,6~:7.795.h;
2,5
---------
u Fig. 5. BEE us. u for various
values of fo and y,
253
0.01’ ’ ’ ’ ’ ” ’ 30
50
100
t
1
200
500
1000
2004
5m
o-
Fig. 6. gGe vs. u for various values of /t and y for EO= 0.2.
0.14
NIL,
8=$.0.5,L/O=l.O,
‘&=7.795, E,,‘o.s
O.O~o~ 100
500
200
1000
2000
5000
u
Fig. 7. &@ vs. u for various values of A and y for co= 0.5.
constant value. This behaviour is influenced by the recess volume parameter y. The constant high frequency value of the stiffness coefficient decreases with increasing y. The speed parameter effect is relatively small. Variation of the direct damping coefficients BEE and EGO with (Tis shown in Figs. 5 - 7 for various values of y and A. Direct damping coefficients are seen to decrease with increasing frequency parameter u prior to attaining a constant value at
254
’ ’
-4.0
30
50
“If’ 100
i I
200
I
b
II
,bIk,
500
1000
2000
/
I
WOO
CT
Fig. 8. I?,@ and I?$,,
us. u for various
values
of A and y for e. = 0.2.
high frequencies. An increase in the recess volume parameter y decreases the damping coefficients. The recess volume fluid compressibility effect is manifested through eqn. (11) for dynamic recess pressures. When the recess volume fluid compressibility effect is large, the bearing behaves more like a liquid spring as is marked in the frequency response of direct stiffness and damping coefficients. The speed parameter A effect is observed for Eea, and effect in the B,,, particularly at e. = 0.5. This is due to a hydrodynamic pressure distribution over the sills and the modifications in the perturbed flow ori and a,.2 in eqn. (II). Variation of the cross stiffness coefficients EEch and KG, with (I is shown in Figs. 8 - 9 for various values of y, A and co. Cross stiffness coefficients are generated owing to hydrodynamic and rotational effects and are visible over the entire frequency range. At low frequencies (u < 100) the recess volume effect is negligible and therefore it is due to purely rotational effects. It increases with increasing speed parameter A. Positive and negative values of zEG and EQE respectively are typical of hybrid bearings. Figures 10 - 13 depict the variation of cross damping coefficients BEG and &,, with e for different values of y, A and eo. BGf increases with u and attains a constant value at high u. It increases with increasing r, A and co.
255 3.0 EO’0.5 ___ A i 2.0 -----_*~~.O
2.0-
1-y: G;s 1.0 -
0.01
z--y=
0.05
3-y=
0.10
0.0
Fig. 9. Keo and & 2.5
-3.5 30
vs. u for various values of A and 7 for fg = 0.5.
N~&,fi;6=0.5,
1 I@ill’ 50
~/~.,.~.6,=7.7~~,
1 I I ,t,,,l 100
200
500
E,'O.2
I 1000
2000
t, 5000
cr
Fig. 10. &
vs. a for various values of i\ and y for e. = 0.2.
256 --. N;C,0;~.0.5,~1D=l.O,6,~7.795 ,E,=o.5
4.0
-h=
2.0
I
-l
-----A..S.O 1---y=
3.0
0.01
2--y:o.o5 3---y=
0.10
~~
/ //
3
1.0 1'
0.0 '
50
/
/I
1' /
'
/ / A-c
100
/
%2
,'
_----
30
/ / /
,'
,'
do-
-+=
/' '
//
'
/ //
2.0
X103
i /--
200
/
5oa
1000
2000
5( 10
o-
Fig. 11. &
5.0
US. D for various
values of A and y for &o = 0.5.
N;~,~~b:0.5.L/D=l.0,6c=7~795,~o=0.2 -A
= 2.0
_---__n:5.0 L.0 -
1---y=
0.01
z--y=o.o5 3-y;
Fig. 12. &Q
0.10
US. u for various values of A and 7 for Q, = 0.2.
In contrast, BP@ decreases with increasing CJuntil it reaches a datum value at high frequencies. It also shows a decrease with increasing y and A. However, cross damping coefficients are small in magnitude. The dynamic response of the bearing due to either unbalance or external dynamic loading can be determined using these rotordynamic coefficients. The bearing is likely to exhibit whirl instability in the low frequency regime of operation.
257
30
50
100
200
MO
1000
2000
5000
o-
Fig. 13. Bee
vs. u for various
values of A and 7 for EO = 0.5.
4. Conclusions (1) The rotordynamic coefficients of multirecess hybrid bearings are frequency dependent for bearings with deep recesses. (2) The dynamic force response of the bearing would be dependent on the frequency of vibration, the recess volume compressibility parameter and the speed parameter for a given bearing geometry. (3) Whirl instability may arise in the low frequency range where the recess volume compressiblity effect is negligible.
Acknowledgment The Industrial
authors express their gratitude to the Council of Scientific and Research, India, for financial support during this research work.
References 1 M. J. Braun, R. L. Wheeler, III and R. C. Hendricks, A fully coupled variable properties thermohydraulic model for a cryogenic hydrostatic journal bearing, J. Tribal., 109 (3) (1987) 405. 2 J. M. Redecliffe and J. H. Vohr, Hydrostatic bearings for cryogenic rocket engine turbopumps, J. Lubr. Technol., 91 (3) (1969) 557 - 575. 3 S. Heller, Static and dynamic performance of externally pressurised fluid film journal bearings in the turbulent regime,J. Lubr. Technol., 95 (3) (1974) 381.
258 4 A. Artiles, J. Walowit and W. Shapiro, Analysis of hybrid fluid film journal bearings with turbulence and inertia effects, Advances in Computer Aided Bearing Design,
American Society of Mechanical Engineers, New York, 1982. 5 J. A. Kocur and P. E. Allaire, Finite element analysis of turbulent lubricated hydrostatic journal bearings for static and dynamic conditions, ASLE Trans., 29 (1986) 126. 6 P. B. Davies and R. Leonard, The dynamic behaviour journal bearings, Proc. Inst. Mech. Eng., London, 184 (3L)
7 8 9
10
of multirecess hydrostatic (1969 - 1970) 139. R. Leonard and W. B. Rowe, Dynamic force coefficients and the mechanism of instability in hydrostatic journal bearings, Wear, 23 (1973) 277. M. K. Ghosh, Dynamic characteristics of multirecess externally pressurised oil journal bearings, J. Lubr. Technol., 100 (4) (1978) 467 - 471. M. K. Ghosh and N. S. Viswanath, Frequency dependent stiffness and damping coefficients of orifice compensated multirecess hydrostatic journal bearings, fnt. J. Mach. Took Manuf., 27 (3) (1987) 275. W. B. Rowe, Hydrostatic and Hybrid Bearing Design, Butterworths, London, 1983.
Appendix A: Nomenclature axial length of the recess, ti = a/L (dimensionless) effective recess area i; = Nb/rD (dimensionless) direct damping coefficients of fluid film B,, = Be,/ 24gL(R ,‘Ct3, BQQ,= R&~~PL(R/C)~ (dimensionless) cross damping coefficients of fluid film & = Bee/ (dimensionless) 24~.tL(R/c)~, & = &,,/24~L(R/c)~ radial clearance diameter of capillary tube recess depth diameter of journal eccentricity, E = e/c (dimensionless) steady state eccentricity, e. = e,/C (dimensionless) perturbed eccentricity, et = e,/C (dimensionless) film thickness, h = h/C (dimensionless) steady state film thickness, ho = ho/C (dimensionless) film thickness at the centre of the rth recess, &, = h,/C (dimensionless) (-1)“2 direct stiffness coefficients of fluid film BEE= I(,$/ Lll)p,, I!$,+ = K~~C/LDp~ {dimensionless) cross stiffness coefficients of fluid film KE+= K,&/ LDp,, E$e = K,,C/LDp, (dimensionless) length of capillary tube length of bearing number of recesses supply pressure of fluid film thickness, p = p/p, (dimensionless) steady state film thickness, PO= po/ps (dimensionless)
259
Prl, Pr2,PrlvPr2 ::;?i., R t, 7 u ~;~
C~,,,,
(WI),,
(W2)Ft
x9 Y, 2
YE
Y A
dynamic film pressures, fir = plIps, p2 = p2/ps (dimensionless) pressure at the rth recess, & = p,/p, (dimensionless) steady state recess pressure at the rth recess pro = pro/ps (dimensionless) dynamic recess pressures at the rth recess, & = prl/ps, jr2 = pr2/ps (dimensionless) flow at the rth recess, Q, = 12pQ,/C3p, (dimensionless) steady state flow at the rth recess, Q,a = 12pQ,,/C3p,, (dimensionless) journal radius time, T = wt (dimensionless) shaft surface, velocity recess volume, V, = Ad,, v,, = V,/AC (dimensionless) radial components of restoring dynamic loads due to dynamic film pressures p1 and p2 (WI), = (WI),/ LDpg,, (v,), = ( W2),/LDp,~,,4, (dimensionless) transverse components of restoring dynamic loads due to dynamic film pressures p1 and p2 (E,), = (IV,),/ LDp,e, , ( W2)T = ( W2)T/LDps~O@I (dimensionless) Cartesian coordinates amplitude of oscillation of journal centre along the direction of eccentricity amplitude of oscillation of journal centre along the direction perpendicular to that of eccentricity fluid compressibility parameter angular coordinate angular coordinate at the centre of rth recess steady state attitude angle perturbed attitude angle absolute viscosity of lubricant density of lubricant kinematic viscosity of lubricant, 7) = p/p squeeze number or frequency parameter, c = 12pw/ P,((C/R)~ (dimensionless) recess frequency parameter, \k = aAIR (dimensionless) frequency of vibration of journal angular speed of journal capillary design parameter, 6, = 3nd:/32C31, (dimensionless) recess volume fluid compressibility parameter, y = vJ3p, (dimensionless) bearing number, A = 6puS2/p,(C/R)2 (dimensionless)