A 3D MODEL OF THE RIGID ROTOR SUPPORTED BY JOURNAL BEARINGS)
Jiří TŮMA1, Radim KLEČKA2, Jaromír ŠKUTA3 and Jiří ŠIMEK4 1 VSB – Technical University of Ostrava, Ostrava, Czech Republic, [email protected] 2 VSB – Technical University of Ostrava, Ostrava, Czech Republic, [email protected] 3 VSB – Technical University of Ostrava, Ostrava, Czech Republic, [email protected] 4 Techlab s.r.o., Sokolovská 207, 190 00 Praha 9, [email protected]
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1. Introduction
It is known that the journal bearing with an oil film becomes instable if the rotor rotation speed crosses a certain value, which is called the Bently-Muszynska threshold [1]. To prevent the rotor instability, the active control can be employed. The arrangement of proximity probes and piezoactuators in a rotor system is shown in figure 1. It is assumed that the bushings (carrier ring), inserted into pedestals with clearance, is a movable part in two perpendicular directions while rotor is rotating.
Im(r) Bushing Proximity probes
Re(r) Ω Re(u)
Journal Piezoactuators
Im(r)
Fig. 1. Journal coordinates and arrangement The research work supported by the GAČR (project no. 101/07/1345) is aimed at the design of the journal bearing active control based on the carrier ring position manipulation by the piezoactuators according to the proximity probe signals, which are a part of the closed loop including a controller. The effect of the feedback on the rotor stability is analyzed by [5]. The test stand of the TECHLAB design [1] is shown in figure 2.
Fig. 2. Test stand for the journal bearing active control
2. Simulation model
There are many ways how to model a rotor system, but this paper prefers an approach, which is based on • the concept developed by Muszynska [2] and supported by Bently Rotor Dynamics Research Corporation. • the lubricant flow prediction using a FE method for Reynolds equation solution [4]
The reason for using Muszynska approach is that this concept offers an effective way to understand the rotor instability problem and to model a journal vibration active control system by manipulating the sleeve position by actuators [1], which are a part of the closed loop composed of proximity probes and a controller. The solution of the Reynolds equation gives more precise rotor dynamic characteristics including rotor stability. To verify the active journal bearing control by simulation, the simplified mathematical model is required. Till now, it has been developed the model, which is describing the displacement of the journal centerline inside a bearing housing equipped by an inner bushing. It is assumed that the bushing position inside bearing is controlled by piezoactuators as it is shown in figure 1. The simulation model contains equations predicting the response in the journal position to the bushing displacement. The basic model is suitable for a long flexible shaft with a fly wheel, which is positioned close to the journal bearing. But the test stand is equipped by the relatively short rigid shaft supported by two journal bearings. There is a mutual influence of the control systems acting to the bushing position inside of two bearing housings. The response of the bushing displacement at the opposite journal bearing is transmitted by the gyroscopic forces of the shaft, which is rotating at the high speed. The topic of this section is focused to extending the actual model by the gyroscopic effect influencing the shaft behavior during rotation with the simultaneous bushing displacements. The control system does not contain two independent two-parameter control loops (for X and Y directions at each journal bearing) but one four- parameter control system. Both the internal forces, such as the spring, damping and tangential forces, and the external forces, such as the gravity, gyroscope and unbalance forces, are acting on the rotor. As Muszynska has stated these bearing forces can be modeled as a rotating spring and damper system at the angular velocity λΩ in rad/s, where λ is a dimensionless parameter, which is slightly less than 0.5, and Ω is the shaft angular velocity in rad/s. The parameter λ is denominated by Muszynska as the fluid averaged circumferential velocity ratio. The fluid pressure wedge is the actual source of the fluid film stiffness in a journal bearing and maintains the rotor in equilibrium. To simplify modeling in Matlab-Simulink, the quantities like force, velocity and displacement, are position vectors of complex numbers. The letter r designates the position of the journal centre line in the bearing housing while the letter u designates the position of the inner movable bushing as it is shown in figure 3. Fluid forces acting on the rotor in coordinates rotating at the same angular frequency as the spring and damper system are given by the formula
Frot = K (rrot − urot )+ D(r&rot − u& rot ), (1) where the parameters, K and D, specify proportionality of stiffness and damping to the relative position of the journal centre displacement vector rrot − urot and velocity vector
r&rot − u& rot , respectively. To model the rotor system, the fluid forces have to be expressed in the stationary coordinate system, in which the rotor centre-line displacement and velocity vectors are designated by r − u and r& − u& , respectively.
0, 0 ………. coordinates of the bearing housing center x(t), y(t) …... coordinates of the journal center Y (Im) journal ux(t), uy(t) … coordinates of the bushing center housing Y (Im) bushing bushing center (ux,uy) u Ω X (Re) (0,0) X (Re) housing r r center (x,y) journal center
Fig. 3. Coordinate system
Substitution into the fluid force equation results in the following formula
F = K ()r − u + D(r& − u& )− jDλΩ (r − u), (2) where the complex term jDλΩ (r − u) has the meaning of the force acting in the perpendicular direction to the vector r − u and this force is called tangential. As the rotor angular velocity increases, this force can become very strong and can cause instability of the rotor behavior. The parameters K and D, specifying oil film stiffness and damping, are a function of the journal centerline position vector, namely the oil film thickness. It was proved that the closer position of the journal to the bearing wall and simultaneously the thinner oil film, the greater value of both these parameters. Some authors, such as Muszynska [3], assume that it is possible to approximate these functions by formulas
2 3 2 2 2 1 5 K = K0 ()1− ()r e , D = D0 (1− (r e) ) , λ = λ0 (1− (r e) ) (3) where e is a journal bearing clearance. The authors of this paper analyzed the other formula structure as well [5]. Due to the fact that the shaft is considered as a rigid body, the ends of the position vectors lie on a straight line, which is identical with the shaft axis, see figure 4. Angles φRe and φIm designate the inclinations of the shaft axis from the bearing housing axis, which is forming an intersection of two perpendicular planes serving for projection of the shaft axis. The plane, which is coinciding with the real axis, is horizontal while the other plane, which is coinciding with the imaginary axis, is vertical. The angle φRe specifies the inclination of the shaft axis projection into the horizontal plane from the bearing housing axis while the angle φIm specifies the inclination of the shaft axis projection into the vertical plane from the bearing housing axis.
Y (Im) shaft gravity r2 center bearing bearing housing (0, 0) housing 1 2 r1 r X (Re) bearing housing axis shaft axis
φIm φRe
Fig. 4. Shaft inclinations
There are two bearings supports of the rotating shaft. Let r be a position vector of the shaft center of gravity and l1 and l2 is the distances of the center of gravity from the journal bearings. The force F has to be indexed according to the journal bearings
F1 = K ()r1 − u1 + D(r&1 − u& 1 )− jDλΩ (r1 − u1 ) (4) F2 = K ()()()r2 − u2 + D r&2 − u& 2 − jDλΩ r2 − u2 ,
Let the angels φRe and φIm be combined into the complex variable Φ = ϕRe + jϕIm . The position vectors of the shaft ends in both the journal bearings are as follows r = r + l ()sin()ϕ + j sin(ϕ ) ≈ r + l Φ = X1+ jY1 1 1 Re Im 1 (5) r2 = r − l2 ()sin()ϕRe + j sin ()ϕIm ≈ r − l2Φ = X 2 + jY 2,
The first derivation of the variables r1 and r2 with respect to time results
r&1 ≈ r& + l1(ϕ&Re + jϕ&Im ) = r& + l1Φ& (6) r&2 ≈ r& − l2 ()ϕ&Re + jϕ&Im = r& − l2Φ& ,
The equation of motion in stationary coordinates is obtained
2 M &r& = M g + mruω exp( j(ω t + δ ))− F1 − F2 , (7) where M is the total rotor mass and g is acceleration of gravity. The unbalance force, which is produced by unbalance mass m mounted at a radius ru, acts in the radial direction and has a phase δ at time t = 0 . The shaft rotating at the high rotation speed can be considered as a gyroscope [6]
AΦ&& = l2F2 − l1F1 + jC ΩΦ& , (8) where A is a moment of inertia of the shaft about its axis and C is a moment of inertia of the same shaft about the axis, which is perpendicular to the shaft axis.
3. Simulation study of the model behavior during run-up
As it was stated before the numeric solution of the journal equation of motion is obtained by using Matlab-Simulink. The block diagram of the rotor system is shown in figure 5. To test the model response, the following values of the parameters were employed. The bushing position u2 = 0, while the variable position u1 corresponds to the rotation at the speed of 40 rad/s with the amplitude of 20 μm. The moments of inertia corresponds a short rigid hollow shaft of the outer diameter of 30 mm and the inner diameter of 20 mm, which was made of steel. The rotational speed runs up from 0 to 400 rad/s during 10 s. The simulation results are shown in figure 5. The rotation of the bushing in the journal bearing 1 does not influence the position of the shaft in the journal bearing 2. M = 2.38 kg; rotor mass lam0 = 0.475; fluid averaged circumferential velocity ratio (lambda) K0 = 4000 N/m; oil film stiffness D0 = 2000 Ns/m; oil film damping coefficient e = 0.0001 m; clearance in the journal bearing (100 μm) MMR = 0.00001 kg m product of the unbalance mass m mounted at a radius ru L1 = 0.1 m; distance of the rotor gravity center from the journal bearing 1 L2 = 0.1 m; distance of the rotor gravity center from the journal bearing 2 A = 0.0006 kg m2; a moment of inertia of the shaft about its axis C = 0.008 kg m2; moment of inertia of the same shaft about the axis, which is perpendicular to the shaft axis
1 MMR*u^2 |.| -i*M*9.81 -K- 4 OMEGA Unbalance1 Add1 dr/dt RPM Mag-Angle Static force1 Gain4 1 to Complex2 r s 1/M dx/dt x dx/dt x 1 Integrator1 r In Gain2 Integrator 6 Integrator 5 2 In1 Out1 L1 Add2 r1 In2 F1 dFi/dt Fi 3 Force F1 In Out1 L2 1/A dx/dt x dx/dt x In1 Out2 3 In F2 In2 Out3 r2 In1 Out1 Gain1 Integrator 4 Integrator 3 In3 Out4 2 4 2 In Subsystem1 3 u1 Force F2 3 In u2 In1 Out1 2 4 i * C * In Out2 OMEGA In3 du/dt i*C 4 1 Subsystem 5 Gain3 du/dt 2
Fig. 5. Block diagram of the rotor system supported on two journal bearings
The rotational speed runs up from 0 to 400 rad/s during 10 s. The simulation results are shown in Fig. 6. The rotation of the bushing in the journal bearing 1 does not influence the position of the shaft in the journal bearing 2.
4. Conclusion
Test stand for investigation of possibilities to influence rotor behavior through external excitation of sliding journal bearings was designed and partly tested. Lumped parameter model of the rigid rotor system including gyroscopic effect was created and simulation study of rotor behavior during run-up was carried out. As it was find out for the rigid rotor, the displacement of the bushing 1 does not affect the displacement of the bushing 2.
-4 -4 x 10 x 10 1 1
0.5 0.5
0 0 Y1 Y2
-0.5 -0.5
-1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 X1 -4 X2 -4 x 10 x 10 Fig. 6. Orbit plots
Acknowledgement
This work was supported by the Czech Science Foundation under project No. 101/07/1345 “Active control of journal bearings aimed at suppressing the rotor instability”.
References
[1] Šimek, J. - Tůma, J. - Svoboda, R: Test stand for investigation of external excitation influence on behaviour of rotor supported in sliding journal bearings. Colloquium Dynamics of Machines 2008, Prague 2008, p.169-174 [2] Muszynska, A.: Whirl and Whip – Rotor / Bearing Stability Problems. Journal of Sound and Vibration (1986) 110 (3), pp 443-462. [3] Muszynska, A.: Rotordynamics, Taylor & Francis Group, New York 2005, ISBN 0- 8247-2399-6. [4] Svoboda, R.: Theoretical analysis of influence of bearing bushing external excitation on dynamic properties of the rotor, Technical report TECHLAB No. 07-301, 2007 (in Czech) [5] Tůma, J - Bilošová, A. - Šimek, J. - Svoboda, R.: A simulation study of the rotor vibration. Colloquium Dynamics of machines 2008. Prague, 2008, p. 1-8. [6] Kožešník, J : Základy teorie přístrojů, SNTL Praha 1987. (in Czech)