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ISSN 1068�798X, Russian Engineering Research, 2014, Vol. 34, No. 3, pp. 131–135. © Allerton Press, Inc., 2014.Original Russian Text © A.Yu. Korneev, 2013, published in Vestnik Mashinostroeniya, 2013, No. 12, pp. 24–28.
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In the design of high�speed turbines, it is impor�tant to ensure stable rotor operation in journal bear�ings [1–4]. An effective means of estimating the sta�bility of the rotor–bearing system is the trajectorymethod, which provides information regarding theinfluence of the nonlinear reaction of the lubricantlayer and permits simulation of the rotor’s dynamicbehavior [1, 4–6]. The method is based on numericalintegration of the system of hydrodynamic equationsfor the supporting layer and the equations of rotormotion.
The rotors trajectory is the geometric locus of theposition of the (rotors) center under the action ofexternal perturbing forces and the reaction of thelubricant layer, at a specific time. The trajectory maybe ploted in both polar and Cartesian coordinates. Themethod permits study of the rotor dynamics with anyeccentricity. In conical bearings, the rotor moves notonly perpendicular to the longitudinal axis of the bear�ing but also along it. In other words, it performs com�plex three�dimensional motion. In the general case,
the rotor may be regarded as a point mass on which theexternal forces and reactions of the bearing act.
The trajectory of the center of the rotor is deter�mined by the geometric parameters and operatingconditions of the system and also by the load. We mayidentify the following basic types of plane trajectoriesand, correspondingly, states of stability of the rotorsystem [1, 4].
1. An orbitally stable state in which the (rotors)center describes repeating elliptical trajectories(Fig. 1a) typical of an unbalanced rotor performingforced oscillations under the action of a centrifugalload. The geometry of the ellipse is determined by theratio of the gravitational force and the unbalancedload [1, 4].
2. A point�stable state (focus) in which the centerof the balanced rotor describes a trajectory and stopson the movable�balance curve (Fig. 1b) [1, 4].
3. An unstable limited state of the rotor, in which acomplex open trajectory developing within a limitedplane indicates that the system contains self�exciting
Rigid�Rotor Dynamics of Conical Hydrodynamic BearingsA. Yu. Korneev
State University–Education–Science–Production Complex, Orel, Russiae�mail: [email protected]
Abstract—Equations of motion of a one�mass rotor on conical hydrodynamic bearings are proposed. Thetrajectories of bearings lubricated with water and liquid hydrogen are derived.
Keywords: rotor, conical hydrodynamic bearing, equations of motion, trajectory method
DOI: 10.3103/S1068798X14030083
0.1 0.2 0.3 0.4 e 0.1 0.2 0.3 0.4 e 0.1 0.2 0.3 0.4 e 0.1 0.2 0.3 0.4 e
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(a) (b) (c) (d)
(e) (f) (g) (h)
Fig. 1. The rotors center trajectories.
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KORNEEV
oscillations of limited amplitude due to the nonlinearproperties of the supporting layer [1, 4, 7]. In this case,the rotor remains operational.
4. Closed complex curves: a figure “eight”(Fig. 1c); a Pascal helix (Fig. 1d); a cardioid (Fig. 1e);an epicycloid (Figs. 1f and 1g); etc. In this case, therotor is subject to self�oscillation with precession fre�quency Ω and superimposed synchronous oscillationsat frequency ω due to the unbalance (biharmonicoscillations) [7].
5. An unstable unlimited state in which the rotoroperates in unstable region, while the rotors trajectoryis a developing helix that tends to the boundaries of thegap (Fig. 1h).
When the rotor is stable, any rotors trajectory withits origin at some initial point ends either at a focus ora limiting cycle; with an unstable state of the rotor, therotors trajectory either moves monotonically awayfrom the initial position, eventually reaching theboundaries of the radial gap, or gradually fills someregion without reaching the boundaries of the gap.
The dynamics of a rigid rotor (mass m) on two sym�metric conical bearings may be analyzed on the basisof Fig. 2. In the general case, the rotor may be loadedby the gravitational force, centrifugal forces (charac�terized by the unbalance Δ), and the lubricant reac�tions Ri. The equations of motion of the rigid rotortake the form [1, 8]
(1)
With a symmetric position of the rotor in two iden�tical conical bearings, the equations of motion of therigid rotor (c = ∞) in Cartesian coordinates take theform
(2)
In plotting the rotors center trajectory, Eq. (2) iswritten in polar coordinates. If we substitute the con�
mX·· ΣRX mΔω2 ωt;sin+=
mY·· ΣRY mΔω2 ωtcos mg;+ +=
mZ·· ΣRZ.=⎩⎪⎨⎪⎧
mX·· 2RSXmΔω2 ωt;sin+=
mY·· 2RSYmΔω2 ωtcos mg;+ +=
mZ·· 0.=⎩⎪⎨⎪⎧
version formulas Z = e0
into Eq. (2), we may write the dimensionless system
(3)
where Λ = = is thereduced mass characterizing the inertial properties ofthe system; h0 is the radial gap; p0 is the supply pres�sure; r0 is the length of the generatrix of the conicalsurface; t0 = 1/ω0 is the characteristic time.
Integration of the equations of motion is impossi�ble from absence of the analytical formulas for thereactions and their derivatives. Therefore, the tra�
jectories must be obtained by numerical integration ofEq. (3). An algorithm for step�by�step trajectory cal�culation was presented in detail in [8]. The system ofdifferential equations is solved by the Adams–Bash�forth method, which is of fourth�order precision [9].Comparison of different methods of solving differen�tial equations shows that the selected method is thebest [9, 10]: it is high precise and stable; and it requiresfewer computations for the right side of the equation.A disadvantage of the method is that self�starting isimpossible. Therefore, beyond the starting point, thenext three points of the trajectory are calculated by theEuler method.
In solving the equations obtained, we use a specialprogram to plot the rotors trajectory in the conical bear�ings. That is important in order to determine the stabil�ity of the system. Calculations get the family of therotors center trajectories in conical hydrodynamic bear�ings with water and liquid�hydrogen lubricant. The tra�jectories are plotted on the assumption of stable rotoroperation in the axial direction (Z = 0) without takingaccount the rotor misalignment relative to the bearing(unbalance Δ = 0). The parameters of the conicalhydrodynamic bearings are as follows: length L =53 mm; largest diameter D2 = 48 mm; cone angle α =30°. The operating conditions are as follows: lubricantsupply pressure p0 = 0.2 MPa; water temperature T0 =293 K; liquid�hydrogen temperature T0 = 20 K. Analy�sis of the trajectories illustrated the following character�istics of rotor behavior in the gap of the conical bearingswith water and liquid hydrogen lubricants:
(1) increase in rotor speed and mass increases theamplitude of the trajectory;
(2) with variation in the cone angle, the type of tra�jectory changes, but its amplitude remains the same.
The rotor speed has considerable influence on thetrajectory. In particular, as the rotor speed increases,
X ep ϕ;sin= Y ep ϕcos ;=
Λ ep'' epϕ '2–( ) 2RSX
ϕsin 2RSYϕcos+=
+ Q t ϕ–( )cos G ϕ,cos+
Λ ep''ϕ '' 2ep'ϕ '+( ) 2RSXϕ 2RSY
ϕsin–cos=
+ Q t ϕ–( )cos G ϕsin ,–
Λe0'' 0,=⎩⎪⎪⎪⎨⎪⎪⎪⎧
mh0/ p0r02t0
2( ) mh0ω02/ p0r0
2( )
RSi
Bearing axis
Rotor axisCenter of mass
X
Y
Z
mO
O2
L0
e p
L0/2
O1
Δ
Fig. 2. Rotor–bearing system.
RUSSIAN ENGINEERING RESEARCH Vol. 34 No. 3 2014
RIGID�ROTOR DYNAMICS OF CONICAL HYDRODYNAMIC BEARINGS 133
the trajectory becomes circular, and its diameter mayreach the boundaries of the gap, while the center of thetrajectory is moved to the center of the bearing onaccount of increase in the hydrodynamic reaction ofthe supporting layer (Fig. 3). When the radial gap h0 =50 μm, the rotor mass m = 3.9 kg and the speed n =10000 rpm (Fig. 3a), the trajectory is a closed curve thatis small relative to the radial gap. We may regard thesystem as stable. When n = 20000 and 30000 rpm(Figs. 3b and 3c), the trajectory is larger and its shapechanges. First, we see closed complex curves (such as
a Pascal helix) due to the superposition of synchro�nous oscillations and self�oscillations; then thosecurves begin to move away from the initial position.At n = 40000 rpm (Fig. 3d), the rotor motion in thegap of the conical bearings is unstable, since the tra�jectory is a developing helix, which tends toward theboundaries of the gap (Fig. 1h). This is especiallyviewed with increase rotor mass. Thus, with approxi�mately 25% increase in mass, the amplitude of thetrajectory is doubled, and the trajectory tends towardthe boundaries of the gap (Fig. 4).
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Fig. 3. The rotors center trajectories with water lubricant at 10000 rpm (a), 20000 rpm (b), 30000 rpm (c), and 40000 rpm (d).
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Fig. 4. The rotors center trajectories with water lubricant when the rotor mass is 3.9 kg (a) and 5 kg (b).
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Thus, we may calculate the limiting rotor speed andmass at which the trajectory reaches the boundaries ofthe radial gap in the given conditions and rotor contactthe bearing. Such contact is highly undesirable.
At specified frequency n = 20000 rpm, the trajec�tory changes when the cone angle is 15° ≤ α ≤ 60°(Fig. 5); the amplitude remains constant. When α ≤30°, the trajectory is a Pascal helix (Figs. 5a and 5b).With further increase in α, the trajectory becomes morecomplex: it consists of a set of Pascal helices and epicy�cloids; this is especially viewed at α = 45 and 60°(Figs. 5c and 5d). Hence, the rotor is subject to self�
oscillations and superimposed synchronous oscilla�tions.
Theoretical results show that, with specifiedparameters (the gap h0 = 20 μm), the rotors center tra�jectory in conical bearings in the case of liquid hydro�gen lubricant has a considerable amplitude compara�ble with the radial gap. With increase in speed of a3.9�kg rotor, the amplitude of the trajectory increases;it comes to resemble a circle which diameter compa�rable with the gap. Over time, the trajectory may reachthe boundaries of the radial gap (Fig. 6). The same sit�uation is observed with increase in rotor mass at
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(d)
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Fig. 5. The rotors center trajectories with water lubricant when the cone angle α is 15° (a), 30° (b), 45° (c), and 60° (d).
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Fig. 6. The rotors center trajectories with liquid hydrogen lubricant at 3500 rpm (a) and 4000 rpm (b).
RUSSIAN ENGINEERING RESEARCH Vol. 34 No. 3 2014
RIGID�ROTOR DYNAMICS OF CONICAL HYDRODYNAMIC BEARINGS 135
n = 5000 rpm (Fig. 7). That indicates unstable systemoperation.
Further dynamic analysis of the rotor–bearing sys�tem achieved by the investigation of families of trajec�tories obtained by variation in the speed, unbalance,rotor mass, the lubricant temperature and supply pres�sure, the bearing geometry, and so on. That allows thedetermination of the system’s stability with specifiedparameters, and hence the best operating conditionsmay be recommended.
REFERENCES
1. Korneev, A.Yu., Savin, L.A., and Solomin, O.V., Kon�icheskie podshipniki zhidkostnogo treniya (Conical Liq�uid�Friction Bearings), Moscow: Mashinostroenie�1,2008.
2. Prabhu, T.J., Ganesan, N., and Rao, B.V.A., Stabilityof vertical rotor system supported by hydrostatic thrustbearings, Proceedings of the Sixth World Congress on theTheory of Machines and Mechanisms, New Delhi, 1983,vol. 2, pp. 1339–1342.
3. Poznyak, E.L., Rotor oscillations, Vibratsii v tekhnike.T. 3. Kolebaniya mashin, konstruktsii i ikh elementov(Vibration in Engineering, Vol. 3: Vibration ofMachines, Structures, and Their Components), Mos�cow: Mashinostroenie, 1980, pp. 139–189.
4. Solomin, O.V., Vibration and stability of rotors on jour�nal bearings with lubricant boiling, Cand. Sci. Dissertation,Orel, 2000.
5. Akers, A., Michaelson, S., and Cameron, A., Boundaryof stability for radial bearings of finite length, Probl.Tren. Smazki, 1971, no. 1, pp. 170–182.
6. Artemenko, N.P., Dotsenko, V.N., and Chaika, A.I.,Trajectories of forced oscillations and self�oscillationsof high�speed rotors in hydrostatic bearings, Issledo�vanie i proektirovanie gidrostaticheskikh opor i uplotneniibystrokhodnykh mashin (Designing Hydrostatic Bear�ings and Seals of High�Speed Machines), Kharkov:KhAI, 1977, issue 4, pp. 31–35.
7. Peshti, Yu.V., Gazovaya smazka (Gas Lubrication),Moscow: MGTU im. Baumana, 1993.
8. Korneev, A.Yu., Trajectory�based dynamic model of arigid rotor on conical bearings, Izv. OrelGTU, Fund.Prikl. Probl. Tekhn. Tekhnol., 2012, no. 3�3/293, pp. 3–9.
9. Amosov, A.A., Dubinskii, Yu.A., and Kopchenova, N.V.,Vychislitel’nye metody dlya inzhenerov (NumericalMethods for Engineers), Moscow: Vysshaya Shkola,1994.
10. Samarskii, A.A. and Gulin, A.V., Chislennye metody(Numerical Methods), Moscow: Nauka, 1989.
Translated by Bernard Gilbert
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Fig. 7. The rotors center trajectories with liquid hydrogen lubricant when the rotor mass is 2.5 kg (a) and 3.9 kg (b).
