- Email: [email protected]
Theoretical analysis on conical whirl instability of unloaded submerged oil journal bearings
309
wear, 152 (19crZ) 309-316
Theoretical analysis on conical whirl instability of unloaded submerged oil journal bearings
Abstract A nonlinear transient method is used to anaJyse the stability characteristics of an unloaded rigid rotor supported in submerged oil journal bearings undergoing conicaI whirl. The rotor has two degrees of freedom of motion in a self-excited conical whirl. The analysis considers cavitation effects, takes account of the oil film history and assumes that antis~met~ is maintained in the conical mode of vibration. The time-dependent form of the Reynolds equation (with the journal in the misaligned position) is solved by a unite-di~eren~e method with a successive over-relaxation scheme to obtain the moment components. Using these moment ~rn~onen~~ the equations of motion are solved by a fourth-arder Rung+Kutta method, to predict the transient behaviour of the rotor. Journal centre trajectories in orthogonal coordinates are obtained for different operating conditions, using a high-speed digital computer and graphics.
1. lntroductian Self-excited vibrations of oil journal bearings can be of two types, namely cy~i~dr~ca~ (transiatory) and conicai. The frequency of these vibrations is dependent on the operating conditions and is close to half the journal rotational frequency. Hirofumi [l] had given the stability criterion for a gyroscope supported in hydrodynamic grooved journal bearings. He found that the unstable region of the conical mode begins at a much lower rotor speed than it does for the cylindrical mode. He concludes that the decrease of bearing clearance and increase of moment of inertia af gimbals makes the gyroscope stable. Sternlicht and Winn [Z, 31 conclude that the relative magnitude of the transverse moment of inertia of the rotor as compared with the polar moment of inertia of the rotor and its mass can make one type of whiri predominate over the other. Conica’f whirl is common, particularly when the transverse moment of inertia is high for a rigid rotor in a single rigidly mounted bearing. In a two-bearing system, with a rigid rotor and rigidly mounted bearings, if the two bearings are closely spaced and the transverse moment of inertia is high, conical whirl occurs at a much iower speed as compared with cylindrical whirl. The two modes of vibration may occur simultaneously, causing some translatory and some conical vibrations. Pafelias and Broniarek [4] determined the elastic and damping characteristics of a partial oil journal bearing with general misalignment. Majumdar [S] obtained a perturbation solution for the torque produced by a misaligned porous gas journal bearing. Guha j6J analysed the conical whirl instability
0043-1648/921$5.00
0 1992 - Etseviet Sequoia. AU rights reserved
310
of externally pressurised porous oil journal bearings including the effect ot v~ioc~ty sbp, using a perturbation method. Guha and Majumdar f7f studied the conical whir: ~nstab~li~ of an externally pressurised porous gas journal bearing, considering vciocity slip. Yoshihiro et al. [S] carried out theoretical stability analysis of the conicat whirl mode for a symmetric rotating shaft model in which the shaft, with a rotor at its mid-point, was supported by short oil bearings. They prcscnted the variation of the stabilit! limit and eccentricity ratio for a range of values of the moment of inertia ratio from 0 to 0.49 for both rigid and flexible shafts. Self-acting plain cylindrical oil journal bearings have a relatively low thresh~~~d of stability and have been anatysed extensively for the translatory whit% Very !ittlc information is available regarding the stability limit of conical whirl, more so in the case of submerged oil journal bearings. A theoretical investigation is therefore L;II-IX~ out for the stability analysis of a conical whirl. The present paper aims at studying theoretically the conical whirl instability of an unloaded finite submerged oil journal bearing using a nonlinear transient method. This method gives the journal centre locus, from which one can ascertain the system stability. The analysis may not be of limited applications, because the whirl instability is predominant at low eccentricity ratio. The moment components are initially found for the steady-state misaligned position of the journal, by solving the Reynolds equation. The equations of motion arc then solved, using these moment components by a fourth-order Run_ge-Kut_ta method [9J, to get the angular rotations about the x- and y-axes (that is & and $,, respectively) and the derivatives of these rotations (&. and I&). These are then used in the timcdependent form of Reynolds equation to get the moment components for the next time step. The procedure is repeated to get & and I& for different time steps and a locus of the journal centre is obtained, using a high-speed digital computer and graphics.
2. Theory
Figure 1 shows a schematic diagram of an oil journal bearing together with the coordinate system used in the analysis. The journal is assumed to be unloaded and rotates with a steady angular velocity w about its axis and undergoes conical whirl when released from a misaligned position of the journal. The basic differential equation for the pressure distribution in the bearing clearance under dynamic conditions may be written as
Equation (1) is written in non-dimensional form using the fOhOWing substitutions: $=X/R, 2 = 2-r/~, & = ~v,(L/~c), & = ~~~j2~~, A = h/C, T= wt, P =P~=~w~~ (2)
The local film thickness, which is a function is given by h=C+z&
coordinates
cos f?+zi& sin B
and &, the dimensionless &=1-tf&cos
of both axial and circumferential
B+f&
sin 0
film thickness,
is given by (3)
311
tdf
tc2
Fig. 1. Bearing geometry,
coordinate
system and rotation angles of journai. (a) x-z plane; (c)
y-z plane; (d) view on plane A-B. Substituting
eqn. (J) into eqn. (2) one finds =6(-Z&
sin S+i&, cos 0)-t-12(i&
cos O+i$,, sin 0)
3. Boundary conditions The following
boundary
conditions
were satisfied
P(f?, il)=O
for OG8&27r
P(S, O)=O
for 0 ;*iBf 27r (antis~met~)
In addition, the boundaries of the cavitation zone are determined the flow continuity at the boundaries (see Fig. 2). Flow on the pressure side
by satisfying
is balanced by the flow in the cavitation zone and an additjonal flow because of boundary movement with time. The subscripts 112indicate that the quantities determined concern both the cavitation and reformation boundaries. Flow in the cavitation zone is given by
312
0 ere -1
Fig.
ICAV
l~earetl~~l
width of bubble at any lomtion withtn the c~v~latiw
2. Theoretical
model
of cavitation
zone
[lo].
4 ca” = LPUhl,2/2
Flow owing to boundary qt=W
movement
- ~~~~~~~/~~
Therefore %=4Ca”+4r substituting
Eqns.
(- - 1 CJ
f3-7li2 at
(5)-(7)
in (8)
.3$ g&
(l-p)=
2
(
non-dimensionalizing
W
1 l/Z
Eqn. (9)
(l/2 - ~~,~/~~~~ - P) =&zfr3csias>Ii*/l2
f 10)
The filling coefficient values for a given time step are determined from the flow continuity equation in the cavitation zone for that time step. The non-dimensional form of the continuity equation is available from the studies by Kicinski [lo]
a@) ; a@4 2
a0
-0
aT
The solution of Eqn. (11) indicates that the flow ph in the cavitation zone at a given time step depends on the flow /3& at the preceding time step. A numerical solution of Eqn. (4) using a unite-d~erence method with successive over-relaxation scheme satisfying the boundary conditions gives the pressure distribution. A grid of 44 points in the circumferential a?d seve,” in the axial and a time step of AT=0.25 was used in the analysis. At first & and r,$ are set equal to zero to obtain the steady-state moment components Mx and it& The moment components are computed (assuming antisymmetry is maintained) from
313
Solving the equations of motion using the above values, one can compute I,&, I$~and I& for the next time step, starting from the steady-state position. The equations of motion are
&,
~~=i~~~~z~~~~
(13)
M,=1&-1&W
(14)
For derivation of Eqns. (13) and (14) see ref. 11. The non-dimensionalized of Eqns. (13) and (14) are
form
(15) (16) Equations (15) and 16) are second-order differential equations in & and I&. They are solved by using a fourth-order Runge-Kutta method [9] for constant values of & and .?, to get &, &, I& and I,&for the next time step. These values are used in Eqns. (3) and (4) to find the pressure distribution and moment co~mp~nents.~The procedure is repeated and the solution method gives the values of $X, ctr,, Jiy and & for each time step. 4x and I,&are plotted as in Figs. 3-6 and the plots help ascertain the system stability.
4. Results and discussion
The results of the computational work have been presented in the form of plots. These plots should not be confused with the polar plots. The plots show the position of the journal centre on a plane at the edge of the bearing, at different instants of time. The cone base circle represents the base circle of the cone formed by the journal axis as it whirls conically in the bearing clearance. A point on the cone base circle COME BIISE
faBOUT
Y-A
Fig. 3. Trajectory for journal centre conical whirl of unloaded L/D= 2.00, i;=O.lO, 1=0.857.
system. r=
Fig. 4. Trajectoxy for journal centre conical whirl of unloaded L/D=l.OO, i,=O.l, f=0.857.
system. W= - 0.1, & = 0.2, I&= 0.0,
-0.1,
&=OS,
&=0.5,
ROTATION
\
indicates contact of the journal surface with the bearing inner surface. The nondim~nsional~zed form of the angular rotations are used to represent the journal centre position. The rotation about the x-axis is plotted along the vertical and the rotation about they-axis along the horizontal as in Figs. 3-6. The rotations can have a maximum value of one. The motion of the journal centre on the other side of the bearing will be a mirror image of Figs. 3-6. Figures 3 and 4 show the journal centre trajectory for a stable system. The journal is released from two different positions as indicated. In Figs. 5 and 6 the journal centre trajectory goes into a limit cycle. This indicates that the system is unstable for the parameters indicated. A typical pressure distribution within the bearing clearance is shown in Fig. 7.
315 5. Conclusions
The Reynolds equation is integrated between the boundaries determined by the flow continuity equation. This gives more realistic values of the moment components. Since the oil film history is considered, the analysis gives a more accurate description of the dynamic state of submerged bearings. (1) This analysis gives the orbital trajectory of the journal centre, which is not obtainable using a simplified perturbation theory. (2) The journal locus for a finite bearing ends in a limit cycle for an unstable system as is evident from Figs. 5 and 6. (3) An increase in the moment of inertia ratio 1, makes the journal more stable for conical whirl as found from the present analysis. (4) It was found that an increase in the transverse moment of inertia (It), with L/D and 1 being kept the same makes the journal reach its stability point faster. (5) An increase in L/D ratio with j and 1, kept constant also makes the journal reach its stability point more slowly.
The authors are grateful to the reviewers for reviewing the manuscript and making useful suggestions. Their suggestions have been incorporated in the revised manuscript.
References 1 M. Hirofumi, Stability characteristics of gyroscopes with hydrodynamic grooved rotor bearings, J. Lubric. Technd, Trans. ASME, 91 (1969) 609419. 2 B. Sternlicht and L. W. Winn, On the load capacity and stability of rotors in self-acting gas lubricated plain cyiindrical journal bearings, J. Basic Eng., Trans. ASME, Ser. D, 85 (1963) 503-512. 3 B. Sternlicht and L. W. Winn, Geometry effects on the stability threshold of half-frequency whirl in self-acting gas lubricated journal bearings, J. Basic Eng., Trans. ASME, Ser. 0, 86 (1964) 313-320. 4 T. A. Pafelias and C. A. Broniarek, Bearing system dynamics with general misalignment in the journal bearings, ASLE Trans. 24 (1981) 379-386. 5 B. C. Majumdar, Torque of misaligned gas lubricated porous journal bearings, Wear, 39 (1976) 5541. 6 S. K. Guha, Study of conical whirl instabili~ of externally pressurised porous oil journal bearings with tangential velocity slip, Trans. ASME, J. Tribot., 108 (1986) 25&26X 7 S. K. Guha and B. C. Majumdar, Study of conical whirl instability of externally pressurised porous gas journal bearing considering tangentiat velocity slip, fioc. 9th Int. Gas Beutirzg Symp., National Bureau of Standards, Washington DC, 1986. 8 T. Yoshihiro, I. Junkichi, T. Hideyuki and S. Atsuo, On the stability of a rotating elastic shaft supported by journal bearings, BUZZ.JSME, 25 (203) (1982) 856-860. 9 5. B. Scarborough, Numerical Mathematical Analysis, Oxford IBH, Calcutta, 1966, 6th edn., pp. 363-364. 10 J. Kicinski, Inffuence of the flow prehistory in the cavitation zone on the dynamic characteristics of slide bearings, Wear, 111 (1986) 289-311. 11 1. H. Shames, Engineering Mechanics: Statics and Dynamics, Prentice Half, New Delhi, 1969, pp. 652478.
316 Appendix
A: Nomenclature radical clearance journal diameter local film thickness dimensionless film thickness polar moment of inertia of journal transverse moment of inertia of journal Z,C”w/qR3L3 (dimensionless) moment of inertia ratio (1,/Z,) width of bearing restoring dynamic moment components about and y respectively M,C2/qWR3L2(dimensionless) h&C2/qwR3L2 (dimensionless) local fluid film pressure in the film region pC’/qwR’ (dimensionless) vapour prcssurc bearing radius time wt (dimensionless) wR (journal peripheral speed) circumferential coordinate, ff =X/R coordinates (xc Fig. l(c)) 7-/Z, (dimensionless)
the
orthogonal
axes x
oil gap filling coefficient absolute viscosity of oil angular velocity of shaft angular rotations of journal axis about x- and y-axes respectively qx(L/2C) (dimensionless) &(LRC) (dimensionless) angular coordinates at which the film cavitates and reforms respectively p,C2/~wR2 (dimensionless vapour pressure)
wear, 152 (19crZ) 309-316
Theoretical analysis on conical whirl instability of unloaded submerged oil journal bearings
Abstract A nonlinear transient method is used to anaJyse the stability characteristics of an unloaded rigid rotor supported in submerged oil journal bearings undergoing conicaI whirl. The rotor has two degrees of freedom of motion in a self-excited conical whirl. The analysis considers cavitation effects, takes account of the oil film history and assumes that antis~met~ is maintained in the conical mode of vibration. The time-dependent form of the Reynolds equation (with the journal in the misaligned position) is solved by a unite-di~eren~e method with a successive over-relaxation scheme to obtain the moment components. Using these moment ~rn~onen~~ the equations of motion are solved by a fourth-arder Rung+Kutta method, to predict the transient behaviour of the rotor. Journal centre trajectories in orthogonal coordinates are obtained for different operating conditions, using a high-speed digital computer and graphics.
1. lntroductian Self-excited vibrations of oil journal bearings can be of two types, namely cy~i~dr~ca~ (transiatory) and conicai. The frequency of these vibrations is dependent on the operating conditions and is close to half the journal rotational frequency. Hirofumi [l] had given the stability criterion for a gyroscope supported in hydrodynamic grooved journal bearings. He found that the unstable region of the conical mode begins at a much lower rotor speed than it does for the cylindrical mode. He concludes that the decrease of bearing clearance and increase of moment of inertia af gimbals makes the gyroscope stable. Sternlicht and Winn [Z, 31 conclude that the relative magnitude of the transverse moment of inertia of the rotor as compared with the polar moment of inertia of the rotor and its mass can make one type of whiri predominate over the other. Conica’f whirl is common, particularly when the transverse moment of inertia is high for a rigid rotor in a single rigidly mounted bearing. In a two-bearing system, with a rigid rotor and rigidly mounted bearings, if the two bearings are closely spaced and the transverse moment of inertia is high, conical whirl occurs at a much iower speed as compared with cylindrical whirl. The two modes of vibration may occur simultaneously, causing some translatory and some conical vibrations. Pafelias and Broniarek [4] determined the elastic and damping characteristics of a partial oil journal bearing with general misalignment. Majumdar [S] obtained a perturbation solution for the torque produced by a misaligned porous gas journal bearing. Guha j6J analysed the conical whirl instability
0043-1648/921$5.00
0 1992 - Etseviet Sequoia. AU rights reserved
310
of externally pressurised porous oil journal bearings including the effect ot v~ioc~ty sbp, using a perturbation method. Guha and Majumdar f7f studied the conical whir: ~nstab~li~ of an externally pressurised porous gas journal bearing, considering vciocity slip. Yoshihiro et al. [S] carried out theoretical stability analysis of the conicat whirl mode for a symmetric rotating shaft model in which the shaft, with a rotor at its mid-point, was supported by short oil bearings. They prcscnted the variation of the stabilit! limit and eccentricity ratio for a range of values of the moment of inertia ratio from 0 to 0.49 for both rigid and flexible shafts. Self-acting plain cylindrical oil journal bearings have a relatively low thresh~~~d of stability and have been anatysed extensively for the translatory whit% Very !ittlc information is available regarding the stability limit of conical whirl, more so in the case of submerged oil journal bearings. A theoretical investigation is therefore L;II-IX~ out for the stability analysis of a conical whirl. The present paper aims at studying theoretically the conical whirl instability of an unloaded finite submerged oil journal bearing using a nonlinear transient method. This method gives the journal centre locus, from which one can ascertain the system stability. The analysis may not be of limited applications, because the whirl instability is predominant at low eccentricity ratio. The moment components are initially found for the steady-state misaligned position of the journal, by solving the Reynolds equation. The equations of motion arc then solved, using these moment components by a fourth-order Run_ge-Kut_ta method [9J, to get the angular rotations about the x- and y-axes (that is & and $,, respectively) and the derivatives of these rotations (&. and I&). These are then used in the timcdependent form of Reynolds equation to get the moment components for the next time step. The procedure is repeated to get & and I& for different time steps and a locus of the journal centre is obtained, using a high-speed digital computer and graphics.
2. Theory
Figure 1 shows a schematic diagram of an oil journal bearing together with the coordinate system used in the analysis. The journal is assumed to be unloaded and rotates with a steady angular velocity w about its axis and undergoes conical whirl when released from a misaligned position of the journal. The basic differential equation for the pressure distribution in the bearing clearance under dynamic conditions may be written as
Equation (1) is written in non-dimensional form using the fOhOWing substitutions: $=X/R, 2 = 2-r/~, & = ~v,(L/~c), & = ~~~j2~~, A = h/C, T= wt, P =P~=~w~~ (2)
The local film thickness, which is a function is given by h=C+z&
coordinates
cos f?+zi& sin B
and &, the dimensionless &=1-tf&cos
of both axial and circumferential
B+f&
sin 0
film thickness,
is given by (3)
311
tdf
tc2
Fig. 1. Bearing geometry,
coordinate
system and rotation angles of journai. (a) x-z plane; (c)
y-z plane; (d) view on plane A-B. Substituting
eqn. (J) into eqn. (2) one finds =6(-Z&
sin S+i&, cos 0)-t-12(i&
cos O+i$,, sin 0)
3. Boundary conditions The following
boundary
conditions
were satisfied
P(f?, il)=O
for OG8&27r
P(S, O)=O
for 0 ;*iBf 27r (antis~met~)
In addition, the boundaries of the cavitation zone are determined the flow continuity at the boundaries (see Fig. 2). Flow on the pressure side
by satisfying
is balanced by the flow in the cavitation zone and an additjonal flow because of boundary movement with time. The subscripts 112indicate that the quantities determined concern both the cavitation and reformation boundaries. Flow in the cavitation zone is given by
312
0 ere -1
Fig.
ICAV
l~earetl~~l
width of bubble at any lomtion withtn the c~v~latiw
2. Theoretical
model
of cavitation
zone
[lo].
4 ca” = LPUhl,2/2
Flow owing to boundary qt=W
movement
- ~~~~~~~/~~
Therefore %=4Ca”+4r substituting
Eqns.
(- - 1 CJ
f3-7li2 at
(5)-(7)
in (8)
.3$ g&
(l-p)=
2
(
non-dimensionalizing
W
1 l/Z
Eqn. (9)
(l/2 - ~~,~/~~~~ - P) =&zfr3csias>Ii*/l2
f 10)
The filling coefficient values for a given time step are determined from the flow continuity equation in the cavitation zone for that time step. The non-dimensional form of the continuity equation is available from the studies by Kicinski [lo]
a@) ; a@4 2
a0
-0
aT
The solution of Eqn. (11) indicates that the flow ph in the cavitation zone at a given time step depends on the flow /3& at the preceding time step. A numerical solution of Eqn. (4) using a unite-d~erence method with successive over-relaxation scheme satisfying the boundary conditions gives the pressure distribution. A grid of 44 points in the circumferential a?d seve,” in the axial and a time step of AT=0.25 was used in the analysis. At first & and r,$ are set equal to zero to obtain the steady-state moment components Mx and it& The moment components are computed (assuming antisymmetry is maintained) from
313
Solving the equations of motion using the above values, one can compute I,&, I$~and I& for the next time step, starting from the steady-state position. The equations of motion are
&,
~~=i~~~~z~~~~
(13)
M,=1&-1&W
(14)
For derivation of Eqns. (13) and (14) see ref. 11. The non-dimensionalized of Eqns. (13) and (14) are
form
(15) (16) Equations (15) and 16) are second-order differential equations in & and I&. They are solved by using a fourth-order Runge-Kutta method [9] for constant values of & and .?, to get &, &, I& and I,&for the next time step. These values are used in Eqns. (3) and (4) to find the pressure distribution and moment co~mp~nents.~The procedure is repeated and the solution method gives the values of $X, ctr,, Jiy and & for each time step. 4x and I,&are plotted as in Figs. 3-6 and the plots help ascertain the system stability.
4. Results and discussion
The results of the computational work have been presented in the form of plots. These plots should not be confused with the polar plots. The plots show the position of the journal centre on a plane at the edge of the bearing, at different instants of time. The cone base circle represents the base circle of the cone formed by the journal axis as it whirls conically in the bearing clearance. A point on the cone base circle COME BIISE
faBOUT
Y-A
Fig. 3. Trajectory for journal centre conical whirl of unloaded L/D= 2.00, i;=O.lO, 1=0.857.
system. r=
Fig. 4. Trajectoxy for journal centre conical whirl of unloaded L/D=l.OO, i,=O.l, f=0.857.
system. W= - 0.1, & = 0.2, I&= 0.0,
-0.1,
&=OS,
&=0.5,
ROTATION
\
indicates contact of the journal surface with the bearing inner surface. The nondim~nsional~zed form of the angular rotations are used to represent the journal centre position. The rotation about the x-axis is plotted along the vertical and the rotation about they-axis along the horizontal as in Figs. 3-6. The rotations can have a maximum value of one. The motion of the journal centre on the other side of the bearing will be a mirror image of Figs. 3-6. Figures 3 and 4 show the journal centre trajectory for a stable system. The journal is released from two different positions as indicated. In Figs. 5 and 6 the journal centre trajectory goes into a limit cycle. This indicates that the system is unstable for the parameters indicated. A typical pressure distribution within the bearing clearance is shown in Fig. 7.
315 5. Conclusions
The Reynolds equation is integrated between the boundaries determined by the flow continuity equation. This gives more realistic values of the moment components. Since the oil film history is considered, the analysis gives a more accurate description of the dynamic state of submerged bearings. (1) This analysis gives the orbital trajectory of the journal centre, which is not obtainable using a simplified perturbation theory. (2) The journal locus for a finite bearing ends in a limit cycle for an unstable system as is evident from Figs. 5 and 6. (3) An increase in the moment of inertia ratio 1, makes the journal more stable for conical whirl as found from the present analysis. (4) It was found that an increase in the transverse moment of inertia (It), with L/D and 1 being kept the same makes the journal reach its stability point faster. (5) An increase in L/D ratio with j and 1, kept constant also makes the journal reach its stability point more slowly.
The authors are grateful to the reviewers for reviewing the manuscript and making useful suggestions. Their suggestions have been incorporated in the revised manuscript.
References 1 M. Hirofumi, Stability characteristics of gyroscopes with hydrodynamic grooved rotor bearings, J. Lubric. Technd, Trans. ASME, 91 (1969) 609419. 2 B. Sternlicht and L. W. Winn, On the load capacity and stability of rotors in self-acting gas lubricated plain cyiindrical journal bearings, J. Basic Eng., Trans. ASME, Ser. D, 85 (1963) 503-512. 3 B. Sternlicht and L. W. Winn, Geometry effects on the stability threshold of half-frequency whirl in self-acting gas lubricated journal bearings, J. Basic Eng., Trans. ASME, Ser. 0, 86 (1964) 313-320. 4 T. A. Pafelias and C. A. Broniarek, Bearing system dynamics with general misalignment in the journal bearings, ASLE Trans. 24 (1981) 379-386. 5 B. C. Majumdar, Torque of misaligned gas lubricated porous journal bearings, Wear, 39 (1976) 5541. 6 S. K. Guha, Study of conical whirl instabili~ of externally pressurised porous oil journal bearings with tangential velocity slip, Trans. ASME, J. Tribot., 108 (1986) 25&26X 7 S. K. Guha and B. C. Majumdar, Study of conical whirl instability of externally pressurised porous gas journal bearing considering tangentiat velocity slip, fioc. 9th Int. Gas Beutirzg Symp., National Bureau of Standards, Washington DC, 1986. 8 T. Yoshihiro, I. Junkichi, T. Hideyuki and S. Atsuo, On the stability of a rotating elastic shaft supported by journal bearings, BUZZ.JSME, 25 (203) (1982) 856-860. 9 5. B. Scarborough, Numerical Mathematical Analysis, Oxford IBH, Calcutta, 1966, 6th edn., pp. 363-364. 10 J. Kicinski, Inffuence of the flow prehistory in the cavitation zone on the dynamic characteristics of slide bearings, Wear, 111 (1986) 289-311. 11 1. H. Shames, Engineering Mechanics: Statics and Dynamics, Prentice Half, New Delhi, 1969, pp. 652478.
316 Appendix
A: Nomenclature radical clearance journal diameter local film thickness dimensionless film thickness polar moment of inertia of journal transverse moment of inertia of journal Z,C”w/qR3L3 (dimensionless) moment of inertia ratio (1,/Z,) width of bearing restoring dynamic moment components about and y respectively M,C2/qWR3L2(dimensionless) h&C2/qwR3L2 (dimensionless) local fluid film pressure in the film region pC’/qwR’ (dimensionless) vapour prcssurc bearing radius time wt (dimensionless) wR (journal peripheral speed) circumferential coordinate, ff =X/R coordinates (xc Fig. l(c)) 7-/Z, (dimensionless)
the
orthogonal
axes x
oil gap filling coefficient absolute viscosity of oil angular velocity of shaft angular rotations of journal axis about x- and y-axes respectively qx(L/2C) (dimensionless) &(LRC) (dimensionless) angular coordinates at which the film cavitates and reforms respectively p,C2/~wR2 (dimensionless vapour pressure)