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Rotordynamic coefficients of multirecess hybrid journal bearings Part II: Fluid inertia effect
261
Wear, 129 (1989) 261 - 272
ROTORDYNAMIC COEFFICIENTS OF MULTIRECESS HYBRID JOURNAL BEARINGS PART II: FLUID INERTIA EFFECT S. K. GUHA, M. K. GHOSH and B. C. MAJUMDAR Mechanical Engineering (Received May 25,1988;
Department,
Indian Institute of Technology,
Kharagpur (India)
accepted September 22,1988)
Summary The effect of fluid inertia over the sill area on the rotordynamic coefficients of multirecess hybrid bearings has been investigated here. It has been shown that only the direct stiffness coefficients amongst the various rotordynamic coefficients are influenced by the inclusion of a fluid inertia effect over the sills. It has been found that the fluid inertia effect, when taken into account, reduces the dynamic stiffness coefficients. Therefore it manifests itself in the form of apparent mass or inertia in the system. Results are presented for a four-recess capillary-compensated hybrid journal bearing.
1. Introduction High frequency vibration characteristics of multirecess hydrostatic and hybrid journal bearings have been investigated by several researchers [ 1 71. Recent investigations highlight their use in high speed turbopumps [8 - lo]. The effect of squeeze action over the sills, the recess volume fluid compressibility effect and their operation in the turbulent regime are amongst the topics reported in recent investigations. Part I of the present investigation presented a general analysis of the combined influence of squeeze action over sills and the recess volume fluid compressibility effect on the rotordynamic coefficients of multirecess hybrid bearings. This report investigates the effect of fluid inertia over the sills on the rotordynamic coefficients of multirecess hybrid journal bearings. Fluid inertia is characterized by the Reynolds number in the dynamic Reynolds equation. The analysis and the results presented for a four-recess capillary-compensated hybrid bearing reveal that fluid inertia manifests itself in the form of apparent mass or inertia. It can be included in the dynamic direct stiffness coefficients which become dependent on the Reynolds number. Therefore only two direct stiffness coefficients are influenced by the fluid inertia from among the eight rotordynamic coefficients of the bearing. 0043-1648/89/$3.50
@ Elsevier Sequoia/Printed in The Netherlands
262
2. Theory The bearing configuration is shown schematically in Fig. 1 of Part I. For an isoviscous incompressible lubricant, the equation for the lubricant in the bearing clearance, including the effect of fluid inertia, is given in ref. 11. The journal is assumed to execute high frequency vibrations superposed over steady rotation about its own axis.
_ 2 a2(hUw a2(hW2) ayhw) + u aqhu) + u a2(hW) P ax2 p ax a.2t ax a2 a2 - a.2at The continuity -
a
J ax 0
h
udy+
Defining
equation
when integrated
across the film yields
a
h - /lL.dy+(U-U$)h=CI aZ 0
the in-plane components
(2) of the average velocity
h
U=
;judy
J
Equation
vector through ah U-_-=U-
w dy
ax
0
0
(1)
ah
ax
(2) reduces to (3)
A simplification can be made by introducing the short bearing approximation, i.e. U = 1/2Up into eqn. (3) and then we have following ref. 11 ;(wh)
=-;
(4)
The above simplification would be valid for small amplitude orbital oscillations around the steady state equlibrium position ( eo, @o) as being considered in the present theory. With the following substitutions:
p=P PS
c2w
R, = -
7)
263
eqns. (1) and (4) can be expressed in dimensionless form as
(5)
(6) Combining eqns. (5) and (6) and neglecting terms multiplied by (6/R) we obtain the Reynolds equation for the pressure distribution incorporating the fluid inertia effect
(7) where a=
A=
12/M p,(CIR )’
@a P,(CIR)~
squeeze number
bearing number
First-order perturbations can now be invoked for pressures and local film thickness with the assumption that the journal executes small amplitude harmonic oscillations with R,(elei7) and R,(tzO@lelT) along the e. and $. directions respectively p = PO+ eler7Jjl + EO&err& ii = ii, + eleiT cos 8 + eO$ieiTsin 0
(6)
where ii,=(l+EOCOSs) Introducing eqn. (8) into eqn. (7) and collecting only up to first-order terms we obtain
= ia cos e -
R,&g
A sin e -
-
12
cos e
(10)
and
R,o@,
= ia sin e + A cos e -
--
12
The boundary
conditions
(1)
&(0,-l)
=pi(e, +l) = 0
(2)
g
13) Isi (4)
.
SI~I e
for the above equations
(11) are
(e, 0) = 0 =
ji(e,
(12) at the rth recess
Pri
Z) =Di(e
+ 27r,Z)
where i = 0, 1,2 respectively for eqn. (9), eqn. (10) and eqn. (11). The steady recess pressures are evaluated following the method described in ref. 5. 2.1, Dynamic recess pressures & and fir2 The recess flow continuity equation for a capillary 6,(1
-Is,.)
= g,fi,
+ ;
(Ah,)
restrictor
121.r~ 1 a 12/.&J + c”p, p g (pVo) C’PS
is given as (13)
where “A” is the effective recess area, which includes a portion of the sill area around the recess [ 121. The dynamic recess pressures can be evaluated following small amplitude first-order perturbations of j&, a, and 5, as follows: _ irpr = ~~0 + ele prl + eO~leiT1jr2 8,
= Qro
+
fzIei7Qrl + eO@lei7&,2
h, = h,, + fzIei7 cos 8, + eO&ei’
(14) sin 8,
266
where jZrO= (I+
Ea cos 8,)
The dynamic recess pressures are obtained as below by substituting into eqn. (13) and collecting only up to the first-order terms
Q,l&.oh + vy cos 8,
Prl = -
i(wy cos 8, - Q,,l?,o@y) (I? + *2y2)
-
(X2 + WY*)
(15)
Qr2PrOh f \Ir2y sin 0, P,, = -
W-y sin 8, - &r2fi’ro~Y) (X2 + ?lr2y2)
-
(12 + Wy2)
eqn. (14)
where
Also y = v,flp, is the recess volume compressibility (oA)/R’ is the recess or frequency parameter.
parameter
and q =
2.2. dynamic load ca~~~~~~, s~~ff~e§s and ~u~~~ng coefficients The components of the restoring dynamic load along the e. and (PO directions due to pressure pr can be expressed as L/2 ( W1)Rei7
=
-2
eleiTplR cos t3 de dz
jj 0
0
.=2$ 0
(16)
2n
Li?. f %W7
277
j 0
clei7pIR sin B d@ dz
The dynamic load components of linearized stiffness, damping (W1)Rei7 = -K,,
( W,),e” = -I(,,
Ye - B,,
Ye - B,,
where the amplitude centres is given by Y, = Celeir
( W1)nei7 and ( W,),e’7 can be written and inertia coefficients as
z 2 -
E,,
E,,
of oscillation
in terms
$ (17)
s of the journal
centre
along the line of (18)
266
Substituting eqn. (18) into eqn. (17) it reduces to (in dimensionless form) (l&)a=
E
=_;!5_-i!%$ s
s
( Wl)T
* 5E$?
K,,C
(19)
E&w2 _e LDP,
. B&o
=--P---l_ LDP,E I LDP,
(W&r=
s
9
LDP,
This can be expressed in terms of the stiffness, damping and inertia coefficients as E@E=-- K#&
E,, =-;+
LDP,
s M&h (3
B,, =-
i&
=-
B
_
B,,
EfE = -
24/~L(R/c)~
= _ M’i-ii,),
EE
u
B = 24pL(&)3
R,~J-‘L CLP,D
R,rlwE+~
(20)
C&BP,
However, the stiffness and inertia coefficients can be together expressed as inertia-dependent dynamic stiffness coefficients (&A
= -R,(%hz
= (&E -&A
(&rki
= -Re(%fr
= (&e -&A
(21)
Similarly, the components of restoring dynamic load due to the dynamic pressure p2 can be expressed as L/2 (
W2)RetT = - 2s
2n
$ 0
0
~o~leiTp2R cos 0 do dz
(22) L/2 t w2)Te
2x
i7 = ss
0
0
~~~~e~Tp~R sin 8 d0 dx
It can be written in terms of dynamic coefficients as
CW2hte
iT=-K,,++,YY,,-BB,6--
f bhe
ir=-K,,Y,-Bdmdt
dY@-E,,----
d2YQ
dye
d2Y,
dt
-Eeedt2
dt2
where, Ye, the amplitude of oscillation of the journal centre in a direction perpendicular to the line of centres is given by (24)
267
Substituting
eqn. (24) into eqn. (23) it reduces to, in dimensionless
=
'---_1_+
LDP,
LDP,
=-- R&P LDP, li
-g_. Bq&w LJJP,
LDP,w&
E&d2
. Be&w
KqbC
cw2)R (W2ht
form
LDP, (25)
(W,), =
(w&r LDP,@
Spring, damping as
and inertia coefficients
+
E~~C~2 LDP,
are expressed
in dimensionless
form
K,,C =-
E@ = -R,{(i&}
LDP, q$
K~~C
= -~,{V2M
j$* =_
=
&nw72M
&qb
24/~L(R/c)~
4llC(~2M
%@
0
E,, = -
LDP,
=
0 RrpG =_
-
=
Z4~L(R/C)3
R,WJJ%I, CLDP,
Ee6 = -
%WJE@@
CLDP, Combining the stiffness and inertia dynamic stiffness coefficients G&b), = (L-& -=%i,+) = = (R,,
-E,,)
and expressing
them
as
--R,-@,)R)
(27)
-
(&&
coefficients
=
--RA~,M
3. Results and discussion Results are presented in terms of direct dynamic stiffness coefficients (Efe)d and (&,)d for a capillary-compensated bearing with the following specifications: Z = 5 = 0.5, L/D = 1.0, 6, = 7.795, in Figs. 1 - 6 for various values of y, R,, A and for eccentricity ratios of 0.2 and 0.5. It has been mentioned before that only the direct stiffness coefficients are affected by the fluid inertia. Bearing coefficients are obtained from the dynamic load due to pressures Is1 and &, as obtained from the solution of eqns. (10) and (ll), satisfying the boundary conditions given by eqns. (12) and (15). A closer examination reveals that the inertia terms appear as
’
50
1
100
’ ’ ’ ’ ’
6
200
I
1
I
i
Ilill 500
I
2000
I,
1 IO0
R, and y for EO= 0.2.
1000
’
Fig. 2. (KEE)d us. u for various values of R, and y for EC,= 0.5.
Fig. 1. (KEE)d vs. CJfor various values of
30
0.1’
l.O-
r4, I
10 -
100
:
3-
2-Y
1 __
---.= 0.01 = 0.05 y : 0.10
y
Re z5.0
-__-----R,.2.0
N:4,~~~~0.5,L/D;1.0,Eo:0.5,6,~7.795,
A i 2.5
269
x-_-_--L_/+-
2 N
d
E
270
b
271
{(R,a@)/12} cos 8 and {(R,ah$/l2} sin 0. This means that only the direct stiffness coefficients, which come from the real part of the dynamic load capacity due to pressures Is, and pz, are affected by the fluid inertia. Cross stiffness terms are due to the -A sin 8 and A cos 0 terms in eqn. (10) and eqn. (11) respectively and are therefore not affected by fluid inertia. Figures 1-- 6 show the variation of the dynamic stiffness coefficients (I(EE)d and (KeG)d with the frequency parameter o for various values of y and R,. The effect of the recess volume fluid compressibility results in the frequency dependence of the stiffness coefficients as has been observed in the results presented in Part I. The fluid inertia effect alters this stiffness coefficient dependence on the frequency parameters. Instead of plateau values for the stiffness coefficients, optimum values of the dynamic coefficients are obtained with regard to u and, for values of e higher than this, the dynamic stiffness registers a sharp fall to a lower value. The frequency at which this optimum condition occurs is dependent on y and reduces with increasing y. Therefore the bearing stiffness characteristics are u, y and R, dependent. As per the definition of R, = pC20/p and u = (12110R2)/(C2p,), the term RBu, which appears in eqns. (10) and (ll), is given by (pw2R2/p,) and is dependent on the geometry, fluid properties, supply pressure and the frequency of vibration. Thus, for a dynamic condition of operation characterized by “a”, it would be possible to select proper geometry parameters for the fluid with proper values of ~.r,p and supply pressure ps and y to obtain the optimum values of the dynamic coefficients. The bearing number (A) and eccentricity ratio (~a) effect is marginal and occurs only in the low frequency range.
4. Conclusions (1) Fluid inertia over the sill area, characterized by the Reynolds number, has a significant influence on the stiffness characteristics of multirecess hybrid journal bearings. (2) Within a useful range of the frequency parameter u, an optimum value of the dynamics stiffness coefficient may be obtained by properly choosing y (fluid compressibility parameter) with proper values of @ and p, C/R ratio and supply pressure ps. (3) The damping coefficient is not influenced by fluid inertia.
References 1 P. B. Davies and R. Leonard, The dynamic behaviour of multirecess hydrostatic journal bearings, fioc. Inst. Meek Eng., London, 284 (3L) (1969 - 1970) 139 - 147. 2 R. Leonard and W. B. Rowe, Dynamic force coefficients and the mechanism of instability in hydrostatic journal bearings, Wear, 23 (1973) 27’7 - 232. 3 G. M. Brown, Dynamic characteristic of hydrostatic thrust bearing, Int. J. Mach. Tool Des. Res., (1-2) - (1) (1961) 157.
272 4 M. K. Ghosh and B. C. Majumdar, Stiffness and damping characteristics of hydrostatic multirecess oil journal bearings, Int. J. Mach. Tool Des. Res., 18 (1978) 139 151. 5 H. Opitz, R. Bottcher and W. Effenberger, Investigation on the dynamic behaviour 10th Int. Machine Tool Design and of hydrostatic spindle bearing systems, Proc. Research Conf., 1969, Pergamon, Oxford. 6 S. M. Rhode and H. A. Ezzat, On the dynamic behaviour of hybrid journal bearings, J. Lubr. Technol., 98 (1976) 90 - 94. 7 M. K. Ghosh and N. S. Viswanath, Frequency dependent stiffness and damping coefficients of orifice compensated multirecess hydrostatic journal bearings, Int. J. Mach. Tools Manuf., 27 (3) (1987) 275 - 287. 8 S. Heller, Static and dynamic performance of externally pressurised fluid film journal bearings in the turbulent regime, J. Lubr. Technol., 95 (3) (1974) 381. 9 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. 10 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. 11 A. Z. Szeri, A. A. Raimondi and A. G. Duarate, Linear force coefficients for squeezefilm dampers, J. Lubr. Technol., 105 (1983) 326 - 334. 12 W. B. Rowe, Hydrostatic and Hybrid Bearing Design, Butterworths, London, 1983.
Appendix
A: Nomenclature
All notation the following: -‘J,,> E,,,
-%,, &
E,,,E,,,
E,,J,,
@@c)d,
_ tK,,)d
used in this paper is listed in Appendix
A of Part I except
direct inertia coefficients of the bearing, E,, = R,qwE,,/ CLDp,, Eoo = R,r)oE,,/CLDp, (dimensionless) cross inertia coefficients of the bearing, EQE = R,+JE,,/ CLDp,, EE@ = R,quE,,/CLDp, (dimensionless) dynamic direct stiffness coefficients as defined in eqns. (21) and (27) dynamic cross stiffness coefficients as defined in eqns. (21) and (27) gap Reynolds number, R, = C’w/q (dimensionless)
R, u, w UP
components of pressure-induced flow journal surface velocity due to orbital motion
6 s2
amplitude of orbital motion angular speed of the journal
Wear, 129 (1989) 261 - 272
ROTORDYNAMIC COEFFICIENTS OF MULTIRECESS HYBRID JOURNAL BEARINGS PART II: FLUID INERTIA EFFECT S. K. GUHA, M. K. GHOSH and B. C. MAJUMDAR Mechanical Engineering (Received May 25,1988;
Department,
Indian Institute of Technology,
Kharagpur (India)
accepted September 22,1988)
Summary The effect of fluid inertia over the sill area on the rotordynamic coefficients of multirecess hybrid bearings has been investigated here. It has been shown that only the direct stiffness coefficients amongst the various rotordynamic coefficients are influenced by the inclusion of a fluid inertia effect over the sills. It has been found that the fluid inertia effect, when taken into account, reduces the dynamic stiffness coefficients. Therefore it manifests itself in the form of apparent mass or inertia in the system. Results are presented for a four-recess capillary-compensated hybrid journal bearing.
1. Introduction High frequency vibration characteristics of multirecess hydrostatic and hybrid journal bearings have been investigated by several researchers [ 1 71. Recent investigations highlight their use in high speed turbopumps [8 - lo]. The effect of squeeze action over the sills, the recess volume fluid compressibility effect and their operation in the turbulent regime are amongst the topics reported in recent investigations. Part I of the present investigation presented a general analysis of the combined influence of squeeze action over sills and the recess volume fluid compressibility effect on the rotordynamic coefficients of multirecess hybrid bearings. This report investigates the effect of fluid inertia over the sills on the rotordynamic coefficients of multirecess hybrid journal bearings. Fluid inertia is characterized by the Reynolds number in the dynamic Reynolds equation. The analysis and the results presented for a four-recess capillary-compensated hybrid bearing reveal that fluid inertia manifests itself in the form of apparent mass or inertia. It can be included in the dynamic direct stiffness coefficients which become dependent on the Reynolds number. Therefore only two direct stiffness coefficients are influenced by the fluid inertia from among the eight rotordynamic coefficients of the bearing. 0043-1648/89/$3.50
@ Elsevier Sequoia/Printed in The Netherlands
262
2. Theory The bearing configuration is shown schematically in Fig. 1 of Part I. For an isoviscous incompressible lubricant, the equation for the lubricant in the bearing clearance, including the effect of fluid inertia, is given in ref. 11. The journal is assumed to execute high frequency vibrations superposed over steady rotation about its own axis.
_ 2 a2(hUw a2(hW2) ayhw) + u aqhu) + u a2(hW) P ax2 p ax a.2t ax a2 a2 - a.2at The continuity -
a
J ax 0
h
udy+
Defining
equation
when integrated
across the film yields
a
h - /lL.dy+(U-U$)h=CI aZ 0
the in-plane components
(2) of the average velocity
h
U=
;judy
J
Equation
vector through ah U-_-=U-
w dy
ax
0
0
(1)
ah
ax
(2) reduces to (3)
A simplification can be made by introducing the short bearing approximation, i.e. U = 1/2Up into eqn. (3) and then we have following ref. 11 ;(wh)
=-;
(4)
The above simplification would be valid for small amplitude orbital oscillations around the steady state equlibrium position ( eo, @o) as being considered in the present theory. With the following substitutions:
p=P PS
c2w
R, = -
7)
263
eqns. (1) and (4) can be expressed in dimensionless form as
(5)
(6) Combining eqns. (5) and (6) and neglecting terms multiplied by (6/R) we obtain the Reynolds equation for the pressure distribution incorporating the fluid inertia effect
(7) where a=
A=
12/M p,(CIR )’
@a P,(CIR)~
squeeze number
bearing number
First-order perturbations can now be invoked for pressures and local film thickness with the assumption that the journal executes small amplitude harmonic oscillations with R,(elei7) and R,(tzO@lelT) along the e. and $. directions respectively p = PO+ eler7Jjl + EO&err& ii = ii, + eleiT cos 8 + eO$ieiTsin 0
(6)
where ii,=(l+EOCOSs) Introducing eqn. (8) into eqn. (7) and collecting only up to first-order terms we obtain
= ia cos e -
R,&g
A sin e -
-
12
cos e
(10)
and
R,o@,
= ia sin e + A cos e -
--
12
The boundary
conditions
(1)
&(0,-l)
=pi(e, +l) = 0
(2)
g
13) Isi (4)
.
SI~I e
for the above equations
(11) are
(e, 0) = 0 =
ji(e,
(12) at the rth recess
Pri
Z) =Di(e
+ 27r,Z)
where i = 0, 1,2 respectively for eqn. (9), eqn. (10) and eqn. (11). The steady recess pressures are evaluated following the method described in ref. 5. 2.1, Dynamic recess pressures & and fir2 The recess flow continuity equation for a capillary 6,(1
-Is,.)
= g,fi,
+ ;
(Ah,)
restrictor
121.r~ 1 a 12/.&J + c”p, p g (pVo) C’PS
is given as (13)
where “A” is the effective recess area, which includes a portion of the sill area around the recess [ 121. The dynamic recess pressures can be evaluated following small amplitude first-order perturbations of j&, a, and 5, as follows: _ irpr = ~~0 + ele prl + eO~leiT1jr2 8,
= Qro
+
fzIei7Qrl + eO@lei7&,2
h, = h,, + fzIei7 cos 8, + eO&ei’
(14) sin 8,
266
where jZrO= (I+
Ea cos 8,)
The dynamic recess pressures are obtained as below by substituting into eqn. (13) and collecting only up to the first-order terms
Q,l&.oh + vy cos 8,
Prl = -
i(wy cos 8, - Q,,l?,o@y) (I? + *2y2)
-
(X2 + WY*)
(15)
Qr2PrOh f \Ir2y sin 0, P,, = -
W-y sin 8, - &r2fi’ro~Y) (X2 + ?lr2y2)
-
(12 + Wy2)
eqn. (14)
where
Also y = v,flp, is the recess volume compressibility (oA)/R’ is the recess or frequency parameter.
parameter
and q =
2.2. dynamic load ca~~~~~~, s~~ff~e§s and ~u~~~ng coefficients The components of the restoring dynamic load along the e. and (PO directions due to pressure pr can be expressed as L/2 ( W1)Rei7
=
-2
eleiTplR cos t3 de dz
jj 0
0
.=2$ 0
(16)
2n
Li?. f %W7
277
j 0
clei7pIR sin B d@ dz
The dynamic load components of linearized stiffness, damping (W1)Rei7 = -K,,
( W,),e” = -I(,,
Ye - B,,
Ye - B,,
where the amplitude centres is given by Y, = Celeir
( W1)nei7 and ( W,),e’7 can be written and inertia coefficients as
z 2 -
E,,
E,,
of oscillation
in terms
$ (17)
s of the journal
centre
along the line of (18)
266
Substituting eqn. (18) into eqn. (17) it reduces to (in dimensionless form) (l&)a=
E
=_;!5_-i!%$ s
s
( Wl)T
* 5E$?
K,,C
(19)
E&w2 _e LDP,
. B&o
=--P---l_ LDP,E I LDP,
(W&r=
s
9
LDP,
This can be expressed in terms of the stiffness, damping and inertia coefficients as E@E=-- K#&
E,, =-;+
LDP,
s M&h (3
B,, =-
i&
=-
B
_
B,,
EfE = -
24/~L(R/c)~
= _ M’i-ii,),
EE
u
B = 24pL(&)3
R,~J-‘L CLP,D
R,rlwE+~
(20)
C&BP,
However, the stiffness and inertia coefficients can be together expressed as inertia-dependent dynamic stiffness coefficients (&A
= -R,(%hz
= (&E -&A
(&rki
= -Re(%fr
= (&e -&A
(21)
Similarly, the components of restoring dynamic load due to the dynamic pressure p2 can be expressed as L/2 (
W2)RetT = - 2s
2n
$ 0
0
~o~leiTp2R cos 0 do dz
(22) L/2 t w2)Te
2x
i7 = ss
0
0
~~~~e~Tp~R sin 8 d0 dx
It can be written in terms of dynamic coefficients as
CW2hte
iT=-K,,++,YY,,-BB,6--
f bhe
ir=-K,,Y,-Bdmdt
dY@-E,,----
d2YQ
dye
d2Y,
dt
-Eeedt2
dt2
where, Ye, the amplitude of oscillation of the journal centre in a direction perpendicular to the line of centres is given by (24)
267
Substituting
eqn. (24) into eqn. (23) it reduces to, in dimensionless
=
'---_1_+
LDP,
LDP,
=-- R&P LDP, li
-g_. Bq&w LJJP,
LDP,w&
E&d2
. Be&w
KqbC
cw2)R (W2ht
form
LDP, (25)
(W,), =
(w&r LDP,@
Spring, damping as
and inertia coefficients
+
E~~C~2 LDP,
are expressed
in dimensionless
form
K,,C =-
E@ = -R,{(i&}
LDP, q$
K~~C
= -~,{V2M
j$* =_
=
&nw72M
&qb
24/~L(R/c)~
4llC(~2M
%@
0
E,, = -
LDP,
=
0 RrpG =_
-
=
Z4~L(R/C)3
R,WJJ%I, CLDP,
Ee6 = -
%WJE@@
CLDP, Combining the stiffness and inertia dynamic stiffness coefficients G&b), = (L-& -=%i,+) = = (R,,
-E,,)
and expressing
them
as
--R,-@,)R)
(27)
-
(&&
coefficients
=
--RA~,M
3. Results and discussion Results are presented in terms of direct dynamic stiffness coefficients (Efe)d and (&,)d for a capillary-compensated bearing with the following specifications: Z = 5 = 0.5, L/D = 1.0, 6, = 7.795, in Figs. 1 - 6 for various values of y, R,, A and for eccentricity ratios of 0.2 and 0.5. It has been mentioned before that only the direct stiffness coefficients are affected by the fluid inertia. Bearing coefficients are obtained from the dynamic load due to pressures Is1 and &, as obtained from the solution of eqns. (10) and (ll), satisfying the boundary conditions given by eqns. (12) and (15). A closer examination reveals that the inertia terms appear as
’
50
1
100
’ ’ ’ ’ ’
6
200
I
1
I
i
Ilill 500
I
2000
I,
1 IO0
R, and y for EO= 0.2.
1000
’
Fig. 2. (KEE)d us. u for various values of R, and y for EC,= 0.5.
Fig. 1. (KEE)d vs. CJfor various values of
30
0.1’
l.O-
r4, I
10 -
100
:
3-
2-Y
1 __
---.= 0.01 = 0.05 y : 0.10
y
Re z5.0
-__-----R,.2.0
N:4,~~~~0.5,L/D;1.0,Eo:0.5,6,~7.795,
A i 2.5
269
x-_-_--L_/+-
2 N
d
E
270
b
271
{(R,a@)/12} cos 8 and {(R,ah$/l2} sin 0. This means that only the direct stiffness coefficients, which come from the real part of the dynamic load capacity due to pressures Is, and pz, are affected by the fluid inertia. Cross stiffness terms are due to the -A sin 8 and A cos 0 terms in eqn. (10) and eqn. (11) respectively and are therefore not affected by fluid inertia. Figures 1-- 6 show the variation of the dynamic stiffness coefficients (I(EE)d and (KeG)d with the frequency parameter o for various values of y and R,. The effect of the recess volume fluid compressibility results in the frequency dependence of the stiffness coefficients as has been observed in the results presented in Part I. The fluid inertia effect alters this stiffness coefficient dependence on the frequency parameters. Instead of plateau values for the stiffness coefficients, optimum values of the dynamic coefficients are obtained with regard to u and, for values of e higher than this, the dynamic stiffness registers a sharp fall to a lower value. The frequency at which this optimum condition occurs is dependent on y and reduces with increasing y. Therefore the bearing stiffness characteristics are u, y and R, dependent. As per the definition of R, = pC20/p and u = (12110R2)/(C2p,), the term RBu, which appears in eqns. (10) and (ll), is given by (pw2R2/p,) and is dependent on the geometry, fluid properties, supply pressure and the frequency of vibration. Thus, for a dynamic condition of operation characterized by “a”, it would be possible to select proper geometry parameters for the fluid with proper values of ~.r,p and supply pressure ps and y to obtain the optimum values of the dynamic coefficients. The bearing number (A) and eccentricity ratio (~a) effect is marginal and occurs only in the low frequency range.
4. Conclusions (1) Fluid inertia over the sill area, characterized by the Reynolds number, has a significant influence on the stiffness characteristics of multirecess hybrid journal bearings. (2) Within a useful range of the frequency parameter u, an optimum value of the dynamics stiffness coefficient may be obtained by properly choosing y (fluid compressibility parameter) with proper values of @ and p, C/R ratio and supply pressure ps. (3) The damping coefficient is not influenced by fluid inertia.
References 1 P. B. Davies and R. Leonard, The dynamic behaviour of multirecess hydrostatic journal bearings, fioc. Inst. Meek Eng., London, 284 (3L) (1969 - 1970) 139 - 147. 2 R. Leonard and W. B. Rowe, Dynamic force coefficients and the mechanism of instability in hydrostatic journal bearings, Wear, 23 (1973) 27’7 - 232. 3 G. M. Brown, Dynamic characteristic of hydrostatic thrust bearing, Int. J. Mach. Tool Des. Res., (1-2) - (1) (1961) 157.
272 4 M. K. Ghosh and B. C. Majumdar, Stiffness and damping characteristics of hydrostatic multirecess oil journal bearings, Int. J. Mach. Tool Des. Res., 18 (1978) 139 151. 5 H. Opitz, R. Bottcher and W. Effenberger, Investigation on the dynamic behaviour 10th Int. Machine Tool Design and of hydrostatic spindle bearing systems, Proc. Research Conf., 1969, Pergamon, Oxford. 6 S. M. Rhode and H. A. Ezzat, On the dynamic behaviour of hybrid journal bearings, J. Lubr. Technol., 98 (1976) 90 - 94. 7 M. K. Ghosh and N. S. Viswanath, Frequency dependent stiffness and damping coefficients of orifice compensated multirecess hydrostatic journal bearings, Int. J. Mach. Tools Manuf., 27 (3) (1987) 275 - 287. 8 S. Heller, Static and dynamic performance of externally pressurised fluid film journal bearings in the turbulent regime, J. Lubr. Technol., 95 (3) (1974) 381. 9 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. 10 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. 11 A. Z. Szeri, A. A. Raimondi and A. G. Duarate, Linear force coefficients for squeezefilm dampers, J. Lubr. Technol., 105 (1983) 326 - 334. 12 W. B. Rowe, Hydrostatic and Hybrid Bearing Design, Butterworths, London, 1983.
Appendix
A: Nomenclature
All notation the following: -‘J,,> E,,,
-%,, &
E,,,E,,,
E,,J,,
@@c)d,
_ tK,,)d
used in this paper is listed in Appendix
A of Part I except
direct inertia coefficients of the bearing, E,, = R,qwE,,/ CLDp,, Eoo = R,r)oE,,/CLDp, (dimensionless) cross inertia coefficients of the bearing, EQE = R,+JE,,/ CLDp,, EE@ = R,quE,,/CLDp, (dimensionless) dynamic direct stiffness coefficients as defined in eqns. (21) and (27) dynamic cross stiffness coefficients as defined in eqns. (21) and (27) gap Reynolds number, R, = C’w/q (dimensionless)
R, u, w UP
components of pressure-induced flow journal surface velocity due to orbital motion
6 s2
amplitude of orbital motion angular speed of the journal