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A stability analysis for a hydrodynamic three-wave journal bearing
ARTICLE IN PRESS
Tribology International 41 (2008) 434–442 www.elsevier.com/locate/triboint
A stability analysis for a hydrodynamic three-wave journal bearing Nicoleta M. Enea,, Florin Dimofteb,1, Theo G. Keith Jr.a a
Mechanical and Engineering Department, University of Toledo, 2801 W Bancroft Road, MS 312, Toledo, OH 43606, USA b University of Toledo at NASA Glenn Research Center, 21000 Brookpark Road, MS 23-3, Cleveland, OH 4413, USA Received 24 March 2006; received in revised form 2 October 2007; accepted 3 October 2007 Available online 19 November 2007
Abstract The influence of the wave amplitude and oil supply pressure on the dynamic behavior of a hydrodynamic three-wave journal bearing is presented. Both, a transient and a small perturbation technique, were used to predict the threshold to fractional frequency whirl (FFW). In addition, the behavior of the rotor after FFW appeared was determined from the transient analysis. The turbulent effects were also included in the computations. Bearings having a diameter of 30 mm, a length of 27.5 mm, and a clearance of 35 mm were analyzed. Numerical results were compared to experimental results obtained at the NASA GRC. Numerical and experimental results showed that the above-mentioned wave bearing with a wave amplitude ratio of 0.305 operates stably at rotational speeds up to 60,000 rpm, regardless of the oil supply pressure. For smaller wave amplitude ratios, a threshold of stability was found. It was observed that the threshold of stability for lower wave amplitude strongly depends on the oil supply pressure and on the wave amplitude. When the FFW occurs, the journal center maintains its trajectory inside the bearing clearance and therefore the rotor can be run safely without damaging the bearing surfaces. r 2007 Elsevier Ltd. All rights reserved. Keywords: Bearing stability; Wave bearing; Oil inlet pressure
1. Introduction The purpose of this paper is to study the dynamic characteristics and stability of a hydrodynamic wave journal bearing. A wave bearing is a journal bearing with a waved sleeve wrapped around the journal. There are two different approaches available to study the dynamic stability of a rotor supported by hydrodynamic journal bearings: linear perturbation analysis and nonlinear transient analysis. Linear perturbation theory assumes that a small perturbation is imposed on the journal center around its equilibrium position. This assumption enables the stiffness and damping coefficients, which characterize the equiliCorresponding author. Tel.: +1 419 283 3041; fax: +1 419 530 8206.
E-mail addresses: [email protected] (N.M. Ene), [email protected] (F. Dimofte), [email protected] (T.G. Keith Jr.). 1 Fax: +1 216 433 3954. 0301-679X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2007.10.002
brium position of the journal center, to be computed. The coefficients are then used to compute the critical mass and the whirl frequency. The critical mass represents the threshold of stability. However, the linear perturbation method does not predict the post whirl orbit detail. In the nonlinear method, the time-dependent Reynolds equation is solved at each journal center position. The hydrodynamic forces are obtained by integrating the pressure distribution and then motion equations are integrated until the trajectory is obtained. A considerable number of papers have been devoted to the study of the dynamic stability of journal bearings using both the approaches. Orcutt and Arwas [1] and Parkins [2] determined the stiffness and damping coefficients by numerical differentiation of Reynolds equation solutions with respect to small perturbations of displacements and velocities of the journal center. This method is very time consuming. In addition, the results significantly depend on the perturbation amplitudes.
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Nomenclature bxx, bxy, byx, byy damping coefficients (N s/m) C clearance (m) e eccentricity (m) ew wave amplitude (m) Fr radial component of fluid film force (N) Fy the resultant fluid film force perpendicular to the line of centers (N) h film thickness (m) ky, kz correction coefficients for turbulent flow kxx, kxy, kyx, kyy stiffness coefficients (N/m) m mass (kg) mcr critical mass nw number of waves p total pressure (Pa) q total flow (m2/s) Recr critical Reynolds number
Lund and Thomsen [3] and Lund [4] used a method of calculating the stiffness and damping coefficients of oil-lubricated bearings which is based on analytical differentiating of the Reynolds equation before the numerical calculation is performed. This method is more accurate than other numerical differentiating methods. Lund’s method was also used by Kostrzewsky et al. [5] to compute the dynamic characteristics of highly preloaded three-lobe journal bearings. In order to reduce the computation time, the authors assumed a polynomial form for the axial pressure distribution. Childs et al. [6] derived analytical stiffness and damping coefficient definitions in terms of an impedance vector for small motion about an equilibrium position. Ebrat et al. [7] analyzed the effects of journal misalignment and bearing structural deformation on the dynamic coefficients of a journal bearing. Rao and Sawicki [8] adapted Lund’s infinitesimal procedure so that the film content at cavitation rupture and reformation boundaries are taken into consideration for both a steady state pressure distribution and dynamic pressure gradients. Many papers have investigated the dynamic stability of journal bearings using a transient approach. One of the first transient analyses of a journal bearing was performed by Kirk and Gunter [9,10]. In their approach they determine fluid film forces by using a short bearing approximation. Majumdar and Brewe [11] presented a nonlinear transient analysis of a rigid rotor supported on oil journal bearings under unidirectional and periodic constant load and under variable load conditions. Monmousseau and Fillon [12] analyzed the transient response of a tilting-pad journal bearing to a synchronous and a non-synchronous (rotating or periodic) load.
Reeff Rel Rem t x, y z ew f g m y y0 o
435
effective Reynolds number local Reynolds number mean Reynolds number time (s) Cartesian coordinates (m) coordinate axis in axial direction (m) ew/C, wave amplitude ratio angle between starting point of the wave and the line of centers (rad) angle between starting point of the wave and the load (rad) dynamic viscosity (N s/m2) angular coordinate starting from the line of centers (rad) angular coordinate starting from the axis Ox (rad) angular velocity of the journal (rad/s)
San Andres [13] compared the transient responses of a rigid rotor supported on externally pressurized, turbulent fluid film bearings obtained using two different models: an approximate model based on constant rotordynamic coefficients and a full nonlinear model. He concluded that the approximate model provided accurate results only for small amplitude loading and for operating conditions far enough from the stability margin of the rotor bearing system. Tieu and Qiu [14] presented a comparison between the journal center trajectories of a journal bearing computed using nonlinear and linear theory. Both methods provided the same critical speed. However, under large dynamic excitation, the trajectories obtained with the two methods are significantly different. Tichy and Bou-Said [15] and Hashimoto and Wada [16] emphasized the effects of turbulence and inertia forces on the dynamic response of rotors supported in journal bearings. In this paper, the stability of a hydrodynamic wave journal bearing in the absence of any external load is analyzed using two approaches: the linear perturbation method and a nonlinear transient analysis. Because the wave bearing operates at high speeds, turbulent effects are considered. The theoretical results are compared with the experimental results obtained from tests that were performed at the NASA Glenn Research Center. 2. The wave bearing concept The wave bearing was introduced for compressible lubricants in the early 1990’s [17,18]. Since then, gas- and liquid-lubricated journal wave bearings were tested at the NASA Glenn Research Center in Cleveland, Ohio. Unlike the plain journal bearing, the wave journal bearing has a slight, but precise variation of its profile
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geometry. The variation is such that a wave profile is circumscribed on the inner bearing diameter. Numerical and experimental evaluations show that a profile with three identical waves produces the best performance. A threewave bearing is presented in Fig. 1. The wave amplitude is usually a fraction of the radial clearance. The radial clearance C is the difference between the radius of the mean circle of the waves and the shaft radius, R. In Fig. 1, both clearance and wave amplitude are drawn at an exaggerated scale so that the wave profile can be visualized. From geometrical considerations, it can be shown that the film thickness of a wave bearing is given by h ¼ C þ e cosðyÞ þ ew cos½nw ðy þ fÞ,
(1)
where nw is the number of the waves, f is the angle between the starting point of the wave and the line of centers and y is the angular coordinate starting from the line of centers (see Fig. 1). The film thickness can be also expressed in a fixed coordinate system Oxy as h ¼ C þ x cos y0 þ y sin y0 þ ew cos½nw ðy0 gÞ,
(2)
where y0 is the angular coordinate starting from the axis Ox, g is the angle between the starting point of the wave and the load and x, y are coordinates of the journal center. The load carrying capacity of the film is due to the rotation of the shaft and to the variation of the film thickness. The wave bearing has superior stability characteristics and enhanced stiffness compared to the circular journal bearing [17]. In addition, its load carrying capacity is close to that of a circular journal bearing and superior to the load carrying capacity of either the groove or the lobe bearing. Starting point of the wave
In absence of any external load, the equations of motion of a balanced rigid rotor supported on two symmetric bearings of mass m can be written in directions along and perpendicular to the line of centers, respectively, as " 2 # d2 e df m e ¼ F r þ mg cos f dt2 dt (3) 2 d f dfde m e 2 þ2 ¼ F f mg sin f dt dt dt where Fr and Ff are, respectively, the resultant reaction forces on each bearing along and perpendicular to the line of centers. The resultant reaction forces can be obtained by integrating the total pressure distribution: R L R 2p F r ¼ R 0 0 p cos y dy dz (4) R L R 2p F f ¼ R 0 0 p sin y dy dz The pressure p is computed by integrating the transient Reynolds equation written in the following form: 1 @ h3 @p @ h3 @p _ sin y þ o @h , _ þ ¼ m e cos y þ e f @z kz @z 2 @y R2 @y ky @y (5) where the film thickness is given by Eq. (1). Parameters ky and kz are correction coefficients for turbulent flow. The correction coefficients ky and kz are determined using Constantinescu’s model of turbulence [19,20]. According to this model, the correction coefficients are function of the effective Reynolds number: ky ¼ 12 þ 0:0136Re0:9 eff kz ¼ 12 þ 0:0044Re0:9 eff
'
ew
3. Nonlinear analysis
W
O e
R
R+C
y
ð6Þ
The first signs of turbulence appear when the local mean Reynolds number Rem is greater than local critical Reynolds number Recr. The flow becomes dominantly turbulent when the mean Reynolds number Rem is greater than 2Recr. With these assumptions, the effective Reynolds number is: 8 0 Rem oRecr > > < Rem Recr pRem p2Recr (7) Reeff ¼ Recr 1 Rel > > : Re Re 42Re l
Oil supply
m
cr
where
ω
! rffiffiffiffiffiffi R Recr ¼ min 41:2 ;2000 h
Sleeve
x
Line of centers
Mean circle of the waves
Fig. 1. The geometry of a three-wave journal bearing.
2rq , m roRh . Rel ¼ m
Rem ¼
ð8Þ
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y Proximity Probes
Thermocouple Thermocouple Oil Line Oil Line
45 mm
Rotor
End Bearing: Diameter: 30 mm Length: 27.5 mm Clearance: 35 μm
x Proximity Probes
End Bearing: Diameter: 30 mm Length: 27.5 mm Clearance: 35 μm
120 mm Fig. 2. Cross section of the oil wave bearing test rig.
At every time step, the intermediate equilibrium position of the journal center is characterized by variables e, f, _ _ Knowing the values of these variables and assuming and f. that the turbulence correction coefficients are equal to the coefficients corresponding to the previous time step, the transient Reynolds equation, Eq. (5), is integrated and a pressure distribution is obtained. Having the pressure distribution, new turbulence correction coefficients corresponding to every grid point are determined and the Reynolds equation, Eq. (5), is integrated again. The iterative process is repeated until the relative errors for the correction coefficients are smaller than a prescribed value equal to 105. Next, the fluid film forces are determined by integration of pressure distribution (Eq. (4)). The motion equations (Eq. (3)) are then integrated _ which and the values of the system variables (e, f, _ and f), characterize the next moment of time, are obtained. The algorithm is repeated until the orbit of the shaft is completed. The transient Reynolds equation, Eq. (5), is solved using a central differencing scheme. The resultant system of equations is solved with a successive over-relaxation method (the Gauss–Seidel method). The Reynolds boundary conditions are assumed in the cavitation region. The two ends of the bearing are considered at atmospheric pressure. In the pockets regions, the pressure is assumed to be equal to the supply pressure. A fourth-order Runge Kutta method was used to integrate the motion equations (Eq. (3)).
4. Linear perturbation analysis The bearing stability can be also analyzed by evaluating the critical mass. The critical mass represents the upper limit for stability. If the mass m of the system is smaller than the critical mass mcr, the system is stable. The system is unstable for a mass m greater than mcr. The critical mass is function of dynamic coefficients [3]: mcr ¼
ks , g2s
(9)
where ks is the effective bearing stiffness ks ¼
kxx byy þ kyy bxx kxy byx kyx bxy bxx þ byy
(10)
and gs is the instability whirl frequency: g2s ¼
ðkxx ks Þðkyy ks Þ kxy kyx bxx byy bxy byx
(11)
The stiffness and damping coefficients of the bearing are obtained by linearizing the turbulent Reynolds equation as shown in Refs. [21,22]. The resultant partial differential equations are solved using finite difference schemes. The boundary conditions are the same as for the nonlinear analysis. The pressure gradients are assumed to be zero at the two ends of the bearing, in the pocket regions and in the cavitation region. The turbulence correction coefficients are computed with the same iterative procedure described in the nonlinear analysis. A bisection algorithm
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was used to determine the equilibrium position of the bearing. 5. Experimental investigations The stability of two identical journal bearings that support a rigid symmetric rotor was evaluated during experimental investigation. A test rig, designed, fabricated and assembled at the NASA Glenn Research Center in Cleveland, Ohio, was used [23,24]. The set-up of that rig for this experimental investigation can be seen in cross section in Fig. 2. The rotor has a maximum diameter of 45 mm, a length of 120 mm and a mass of 1.65 kg. The rotor is driven by an air turbine at rotational speeds up to 60,000 rpm. Two
identical three-wave journal bearings support the rotor. All the wave bearings that were tested have a mean circle diameter of 30 mm, a length of 27.5 mm and a clearance of 35 mm. Each bearing is supplied with lubricant by three oil supply pockets. The angle between two consecutive supply pockets is 1201. The bearing temperature is read by individual thermocouples situated on each bearing. The lubricant used during the experiments is MilL-23699. Two thrust wave bearings, situated at each end of the rotor, control the axial position of the rotor. The rotor displacements in the horizontal and vertical directions are measured by proximity probes positioned in two mutually perpendicular planes. One of these proximity probes was connected to a digital oscilloscope that can
90
4 μm 60
120 3 μm 2 μm
150
30
1 μm
180
0
210
330
240
300 270
Fig. 3. Oscilloscope screen with experimental wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.305, rotating speed n ¼ 60,000 rpm). The bearing runs stably.
Fig. 5. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.305, rotating speed n ¼ 60,000 rpm, supply pressure ps ¼ 0.152 MPa). The bearing runs stably.
14 12
mcr (kg)
10 8 6 4 2
m = 0.825 kg
0 10000
20000
30000 40000 n (RPM)
50000
60000
Fig. 4. Predicted critical mass of wave bearing as function of running speed by small perturbations analysis (wave amplitude ratio ew ¼ 0.305, supply pressure ps ¼ 0.152 MPa). The critical mass is greater than the mass of the shaft related to one bearing (m ¼ 0.825 kg) for speeds up to 60,000 rpm. The bearing should run stably.
Fig. 6. Oscilloscope screen with the wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 36,000 rpm, oil supply pressure ps ¼ 0.276 MPa). The bearing runs stably.
ARTICLE IN PRESS N.M. Ene et al. / Tribology International 41 (2008) 434–442
display the wave shape and the FFT analysis of the input signal. More details about the experimental rig can be found in Ref. [24]. 6. Numerical results The overall aim of this paper is to study the influence of wave amplitude and supply pressure on the stability of a three-wave bearing having a mean circle diameter of 30 mm, a length of 27.5 mm and a clearance of 35 mm. First, a three-wave bearing with a wave amplitude ratio of 0.305 (the ratio between the wave amplitude and the
439
clearance) was tested. It was found that this bearing is stable even at rotational speeds of 60,000 rpm. The experiment was conducted at a supply pressure of 0.152 MPa. The bearing temperature was 210 1C. The oscilloscope screen containing the wave shape of the signal from one of the proximity probes and the FFT analysis for a rotational speed of 60,000 rpm is presented in Fig. 3. The FFT analysis shows the presence of an amplitude peak only at the synchronous frequency. In addition, the regular shape of the motion wave curves indicates the presence only of a synchronous motion. The numerical simulations confirm the experimental results. The small perturbation analysis reveals that the critical mass, mcr, is greater than the mass of the shaft related to one bearing (m ¼ 0.825 kg) for speeds up to 60,000 rpm. In Fig. 4, the critical mass as a
5 4.5 4
mcr (kg)
3.5 3 2.5 2 1.5
m =0.825 kg
1 0.5 0 10000 Fig. 7. Oscilloscope screen with the wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 39,000 rpm, oil supply pressure ps ¼ 0.276 MPa). An incipient sub-synchronous whirl motion is detected.
20000
30000
40000 n (RPM)
50000
60000
Fig. 9. Predicted critical mass of wave bearing as function of running speed by small perturbations analysis (wave amplitude ratio ew ¼ 0.075, oil supply pressure ps ¼ 0.276 MPa). A sub-synchronous motion should be expected for speeds greater than 40,000 rpm when the critical mass is smaller than the mass of the shaft related to one bearing (m ¼ 0.825 kg).
90
4 μm 60
120 3 μm 2 μm
150
30
1μm 180
0
210
330
240
300 270
Fig. 8. Oscilloscope screen with the wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 44,000 rpm, oil supply pressure ps ¼ 0.276 MPa). The sub-synchronous whirl motion is dominant.
Fig. 10. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.075, rotational speed n ¼ 36,000 rpm, oil supply pressure ps ¼ 0.276 MPa). The bearing runs stably.
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function of rotational speed is presented. The transient analysis shows that for running speeds up to 60,000 rpm the journal center approaches the bearing center very rapidly and then orbits around it with a very small radius, as it can be seen for in Fig. 5 for a rotating speed of 60,000 rpm. Next, a three-wave bearing with a wave amplitude ratio of 0.075 was tested. Initially, the experiment was conducted for a supply pressure of 0.276 MPa and a bearing temperature of 186 1C. For these conditions, the experiments revealed that the bearing operates stably for rotational speeds up to 36,000 rpm. The oscilloscope screen for a rotational speed of 36,000 rpm is presented in Fig. 6.
90
8 μm
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60 6 μm 4 μm
150
30
2 μm 180
0
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330
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300 270
Fig. 11. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 39,000 rpm, oil supply pressure ps ¼ 0.276 MPa). A small limit cycle appears.
90 120
At a speed of 39,000 rpm, an incipient sub-synchronous frequency is detected (see Fig. 7). The sub-synchronous frequency becomes dominant at a rotational speed of 44,000 rpm. From the FFT analysis (see Fig. 8), it can be seen that the amplitude of the sub-synchronous motion is greater than the amplitude of the synchronous motion. Also, the wave shape is very irregular. The same conclusions can be drawn from the numerical simulations. The variation of the critical mass with rotational speed is presented in Fig. 9. The critical mass is greater than the mass of the shaft related to one bearing (m ¼ 0.825 kg) for speeds smaller than 36,000 rpm. Around a rotational speed of 39,000 rpm, the critical mass is very close to m. For rotational speeds greater than 40,000 rpm, the critical mass is smaller than m. Consequently it may be concluded that the system is unstable for rotational speeds greater than 40,000 rpm. The transient analysis allows examination of post whirl orbit details. The stable trajectory of the journal center operating at 36,000 rpm is given in Fig. 10. When the journal rotates at 39,000 rpm, the journal locus reaches a limit cycle (see Fig. 11). The radius of the limit cycle increases with increasing rotational speed. For example, at the 44,000 rpm the radius of the orbit is 10.75 mm (see Fig. 12) compared to 7.55 mm at 39,000 rpm (Fig. 11). An increase of the oil supply pressure to 0.414 MPa stabilizes the bearing for velocities up to 60,000 rpm. The sub-synchronous whirl motion disappears (see Fig. 13). The critical mass becomes greater than the mass of the system (Fig. 14). Also, the limit cycles disappear and the journal approaches to the bearing center and orbits around it. As an illustration, the trajectory of the bearing rotating at 60,000 rpm is presented in Fig. 15.
15 μm 60 10 μm
150
30 5 μm
180
0
210
330
240
300 270
Fig. 12. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 44,000 rpm, oil supply pressure ps ¼ 0.276 MPa). The limit cycle is larger than the previous case (Fig. 11), but still inside the bearing clearance.
Fig. 13. Oscilloscope screen with the wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.075, rotational speed n ¼ 44,000 rpm, oil supply pressure ps ¼ 0.414 MPa). The bearing runs stably.
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Even if the wave bearing is unstable, the wave bearing maintains its orbit inside the bearing clearance. The theoretical data and the experimental results were in good agreement, which validates the developed numerical codes.
9 8 7 6 mcr (kg)
441
5 4
References
3 2 1
m= 0.825kg
0 10000
20000
30000
40000 n (RPM)
50000
60000
Fig. 14. Predicted critical mass of wave bearing as function of rotational speed by small perturbations analysis (wave amplitude ratio ew ¼ 0.075, oil supply pressure ps ¼ 0.414 MPa). The critical mass is greater than the mass of the shaft related to one bearing (m ¼ 0.825 kg) for speeds up to 60,000 rpm. The bearing should run stably.
90
4μm
120 3μm
60
2μm
150
30
1μm 180
0
210
330
240
300 270
Fig. 15. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 60,000 rpm, oil supply pressure ps ¼ 0.414 MPa). The bearing runs stably.
7. Conclusions Both a critical mass and a transient analysis were developed to investigate the dynamic behavior of a threewave bearing before, during and after the fractional frequency whirl (FFW) appears. The numerical and experimental investigation performed on a three-wave bearing (30 mm diameter, 27.5 mm length, and 35 mm radial clearance) with a wave amplitude ratio of 0.305 showed that this bearing is stable for rotational speeds up to 60,000 rpm and that the bearing operates free of FFW. The same wave bearing with a wave amplitude ratio of 0.075 experiences a sub-synchronous unstable motion at a speed which depends on the oil supply pressure.
[1] Orcutt FK, Arwas EB. The steady-state and dynamic characteristics of a full circular bearing and a partial arc bearing in the laminar and turbulent flow regimes. J Lubr Technol 1967:143–53. [2] Parkins DW. Theoretical and experimental determination of the dynamic characteristics of a hydrodynamic journal bearing. ASME J Lubr Technol 1979;101:129–39. [3] Lund JW, Thomsen KK. A calculation method and data for the dynamic coefficients of oil—lubricated journal bearing. Topics in fluid bearing and rotor bearing system design and optimization. New York: ASME; 1978. p. 1–28. [4] Lund JW. Review of the concept of dynamic coefficients for fluid film journal bearings. ASME Trans J Tribol 1987;109(1):37–41. [5] Kostrzewsky GJ, Taylor D, Flack RD, Barrett L. Theoretical and experimental dynamic characteristics of highly preloaded three lobe journal bearing. STLE Tribol Trans 1998;41:392–8. [6] Childs D, Moes H, van Leeuwen H. Journal bearing impedance descriptions for rotordynamic applications. Trans ASME 1977;99: 198–209. [7] Ebrat O, Mourelatos Z, Vlahopoulos N, Vaidyanathan K. Calculation of journal bearing dynamic characteristics including journal misalignment and bearing structural deformation. STLE Tribol Trans 2004;94(102):94–102. [8] Rao TVVLN, Sawicki JT. Linear stability analysis for a hydrodynamic journal bearing considering cavitation effects. STLE Tribol Trans 2002;45(4):450–6. [9] Kirk RG, Gunter EJ. Short bearing analysis applied to rotordynamics I: theory. ASME J Lubr Technol 1976;98:47–56. [10] Kirk RG, Gunter EJ. Short bearing analysis applied to rotordynamics II: results of journal bearing response. ASME J Lubr Technol 1976;98:319–29. [11] Majumdar BC, Brewe DE. Stability of a rotor supported on oil film journal bearing under dynamic load, NASA TM-102309, 1987 (AVSCOM TR 87-C-26). [12] Monmousseau P, Fillon M. Frequency effects on the TEHD behavior of a tilting-pad journal bearing under dynamic loading. ASME J Tribol 1999;121:321–6. [13] San Andres L. Transient response of externally pressurized fluid film bearings. STLE Tribol Trans 1997;40(1):147–55. [14] Tieu AK, Qiu ZL. Stability of finite journal bearings—from linear and nonlinear bearing forces. STLE Tribol Trans 1995;38(3):627–35. [15] Tichy J, Bou-Said B. Hydrodynamic lubrication and bearing behavior with impulsive loads. STLE Tribol Trans 1991;34(4): 505–12. [16] Hashimoto H, Wada S. Dynamic behavior of unbalanced rigid shaft supported on turbulent journal bearings—theory and experiment. ASME J Tribol 1990;112:404–8. [17] Dimofte F. Wave journal bearing with compressible lubricant—Part I: the wave bearing concept and a comparison to the plain circular bearing. STLE Tribol Trans 1995;38(1):153–60. [18] Dimofte F. Wave journal bearing with compressible lubricant—Part II: a comparison of the wave bearing with a groove bearing and a lobe bearing. STLE Tribol Trans 1995;38(2):364–72. [19] Constantinescu VN. Basic relations in turbulent lubrication and their extension to include thermal effects. Trans ASME J Lubr Technol 1973;95(2):147–53. [20] Freˆne J, Constantinescu VN. Operating characteristics of journal bearings in the transition region. In: Proceedings of the 2nd Leeds–Lyon symposium on tribology; 1975. p. 121–4.
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[21] Constantinescu VN. Lubrication in turbulent regime, US Atomic Energy Commission, 1968. [22] Hashimoto H, Wada S, Tsunoda H. Performance characteristics of elliptical journal bearings in turbulent flow regime. Bull JSME 1984;232:2265–71. [23] Dimofte F, Proctor MP, Fleming DP, Keith TG. Wave fluid film bearing tests for an aviation gearbox. In: Proceedings of the 8th international symposium on transport phenomena and dynamics of
rotating machinery, ISROMAC-8, Honolulu, HI, 26–30 March 2000, NASA/TM-2000-209766. [24] Dimofte F, Lambrulescu MI, Sawicki JT. Experimental investigations on the influence of oil inlet pressure on the stability of wave journal bearing. In: Proceedings of the 10th international symposium on transportation phenomena and dynamics of rotating machinery, Honolulu, Hi, ISROMAC10-2004-146.
Tribology International 41 (2008) 434–442 www.elsevier.com/locate/triboint
A stability analysis for a hydrodynamic three-wave journal bearing Nicoleta M. Enea,, Florin Dimofteb,1, Theo G. Keith Jr.a a
Mechanical and Engineering Department, University of Toledo, 2801 W Bancroft Road, MS 312, Toledo, OH 43606, USA b University of Toledo at NASA Glenn Research Center, 21000 Brookpark Road, MS 23-3, Cleveland, OH 4413, USA Received 24 March 2006; received in revised form 2 October 2007; accepted 3 October 2007 Available online 19 November 2007
Abstract The influence of the wave amplitude and oil supply pressure on the dynamic behavior of a hydrodynamic three-wave journal bearing is presented. Both, a transient and a small perturbation technique, were used to predict the threshold to fractional frequency whirl (FFW). In addition, the behavior of the rotor after FFW appeared was determined from the transient analysis. The turbulent effects were also included in the computations. Bearings having a diameter of 30 mm, a length of 27.5 mm, and a clearance of 35 mm were analyzed. Numerical results were compared to experimental results obtained at the NASA GRC. Numerical and experimental results showed that the above-mentioned wave bearing with a wave amplitude ratio of 0.305 operates stably at rotational speeds up to 60,000 rpm, regardless of the oil supply pressure. For smaller wave amplitude ratios, a threshold of stability was found. It was observed that the threshold of stability for lower wave amplitude strongly depends on the oil supply pressure and on the wave amplitude. When the FFW occurs, the journal center maintains its trajectory inside the bearing clearance and therefore the rotor can be run safely without damaging the bearing surfaces. r 2007 Elsevier Ltd. All rights reserved. Keywords: Bearing stability; Wave bearing; Oil inlet pressure
1. Introduction The purpose of this paper is to study the dynamic characteristics and stability of a hydrodynamic wave journal bearing. A wave bearing is a journal bearing with a waved sleeve wrapped around the journal. There are two different approaches available to study the dynamic stability of a rotor supported by hydrodynamic journal bearings: linear perturbation analysis and nonlinear transient analysis. Linear perturbation theory assumes that a small perturbation is imposed on the journal center around its equilibrium position. This assumption enables the stiffness and damping coefficients, which characterize the equiliCorresponding author. Tel.: +1 419 283 3041; fax: +1 419 530 8206.
E-mail addresses: [email protected] (N.M. Ene), [email protected] (F. Dimofte), [email protected] (T.G. Keith Jr.). 1 Fax: +1 216 433 3954. 0301-679X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2007.10.002
brium position of the journal center, to be computed. The coefficients are then used to compute the critical mass and the whirl frequency. The critical mass represents the threshold of stability. However, the linear perturbation method does not predict the post whirl orbit detail. In the nonlinear method, the time-dependent Reynolds equation is solved at each journal center position. The hydrodynamic forces are obtained by integrating the pressure distribution and then motion equations are integrated until the trajectory is obtained. A considerable number of papers have been devoted to the study of the dynamic stability of journal bearings using both the approaches. Orcutt and Arwas [1] and Parkins [2] determined the stiffness and damping coefficients by numerical differentiation of Reynolds equation solutions with respect to small perturbations of displacements and velocities of the journal center. This method is very time consuming. In addition, the results significantly depend on the perturbation amplitudes.
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Nomenclature bxx, bxy, byx, byy damping coefficients (N s/m) C clearance (m) e eccentricity (m) ew wave amplitude (m) Fr radial component of fluid film force (N) Fy the resultant fluid film force perpendicular to the line of centers (N) h film thickness (m) ky, kz correction coefficients for turbulent flow kxx, kxy, kyx, kyy stiffness coefficients (N/m) m mass (kg) mcr critical mass nw number of waves p total pressure (Pa) q total flow (m2/s) Recr critical Reynolds number
Lund and Thomsen [3] and Lund [4] used a method of calculating the stiffness and damping coefficients of oil-lubricated bearings which is based on analytical differentiating of the Reynolds equation before the numerical calculation is performed. This method is more accurate than other numerical differentiating methods. Lund’s method was also used by Kostrzewsky et al. [5] to compute the dynamic characteristics of highly preloaded three-lobe journal bearings. In order to reduce the computation time, the authors assumed a polynomial form for the axial pressure distribution. Childs et al. [6] derived analytical stiffness and damping coefficient definitions in terms of an impedance vector for small motion about an equilibrium position. Ebrat et al. [7] analyzed the effects of journal misalignment and bearing structural deformation on the dynamic coefficients of a journal bearing. Rao and Sawicki [8] adapted Lund’s infinitesimal procedure so that the film content at cavitation rupture and reformation boundaries are taken into consideration for both a steady state pressure distribution and dynamic pressure gradients. Many papers have investigated the dynamic stability of journal bearings using a transient approach. One of the first transient analyses of a journal bearing was performed by Kirk and Gunter [9,10]. In their approach they determine fluid film forces by using a short bearing approximation. Majumdar and Brewe [11] presented a nonlinear transient analysis of a rigid rotor supported on oil journal bearings under unidirectional and periodic constant load and under variable load conditions. Monmousseau and Fillon [12] analyzed the transient response of a tilting-pad journal bearing to a synchronous and a non-synchronous (rotating or periodic) load.
Reeff Rel Rem t x, y z ew f g m y y0 o
435
effective Reynolds number local Reynolds number mean Reynolds number time (s) Cartesian coordinates (m) coordinate axis in axial direction (m) ew/C, wave amplitude ratio angle between starting point of the wave and the line of centers (rad) angle between starting point of the wave and the load (rad) dynamic viscosity (N s/m2) angular coordinate starting from the line of centers (rad) angular coordinate starting from the axis Ox (rad) angular velocity of the journal (rad/s)
San Andres [13] compared the transient responses of a rigid rotor supported on externally pressurized, turbulent fluid film bearings obtained using two different models: an approximate model based on constant rotordynamic coefficients and a full nonlinear model. He concluded that the approximate model provided accurate results only for small amplitude loading and for operating conditions far enough from the stability margin of the rotor bearing system. Tieu and Qiu [14] presented a comparison between the journal center trajectories of a journal bearing computed using nonlinear and linear theory. Both methods provided the same critical speed. However, under large dynamic excitation, the trajectories obtained with the two methods are significantly different. Tichy and Bou-Said [15] and Hashimoto and Wada [16] emphasized the effects of turbulence and inertia forces on the dynamic response of rotors supported in journal bearings. In this paper, the stability of a hydrodynamic wave journal bearing in the absence of any external load is analyzed using two approaches: the linear perturbation method and a nonlinear transient analysis. Because the wave bearing operates at high speeds, turbulent effects are considered. The theoretical results are compared with the experimental results obtained from tests that were performed at the NASA Glenn Research Center. 2. The wave bearing concept The wave bearing was introduced for compressible lubricants in the early 1990’s [17,18]. Since then, gas- and liquid-lubricated journal wave bearings were tested at the NASA Glenn Research Center in Cleveland, Ohio. Unlike the plain journal bearing, the wave journal bearing has a slight, but precise variation of its profile
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geometry. The variation is such that a wave profile is circumscribed on the inner bearing diameter. Numerical and experimental evaluations show that a profile with three identical waves produces the best performance. A threewave bearing is presented in Fig. 1. The wave amplitude is usually a fraction of the radial clearance. The radial clearance C is the difference between the radius of the mean circle of the waves and the shaft radius, R. In Fig. 1, both clearance and wave amplitude are drawn at an exaggerated scale so that the wave profile can be visualized. From geometrical considerations, it can be shown that the film thickness of a wave bearing is given by h ¼ C þ e cosðyÞ þ ew cos½nw ðy þ fÞ,
(1)
where nw is the number of the waves, f is the angle between the starting point of the wave and the line of centers and y is the angular coordinate starting from the line of centers (see Fig. 1). The film thickness can be also expressed in a fixed coordinate system Oxy as h ¼ C þ x cos y0 þ y sin y0 þ ew cos½nw ðy0 gÞ,
(2)
where y0 is the angular coordinate starting from the axis Ox, g is the angle between the starting point of the wave and the load and x, y are coordinates of the journal center. The load carrying capacity of the film is due to the rotation of the shaft and to the variation of the film thickness. The wave bearing has superior stability characteristics and enhanced stiffness compared to the circular journal bearing [17]. In addition, its load carrying capacity is close to that of a circular journal bearing and superior to the load carrying capacity of either the groove or the lobe bearing. Starting point of the wave
In absence of any external load, the equations of motion of a balanced rigid rotor supported on two symmetric bearings of mass m can be written in directions along and perpendicular to the line of centers, respectively, as " 2 # d2 e df m e ¼ F r þ mg cos f dt2 dt (3) 2 d f dfde m e 2 þ2 ¼ F f mg sin f dt dt dt where Fr and Ff are, respectively, the resultant reaction forces on each bearing along and perpendicular to the line of centers. The resultant reaction forces can be obtained by integrating the total pressure distribution: R L R 2p F r ¼ R 0 0 p cos y dy dz (4) R L R 2p F f ¼ R 0 0 p sin y dy dz The pressure p is computed by integrating the transient Reynolds equation written in the following form: 1 @ h3 @p @ h3 @p _ sin y þ o @h , _ þ ¼ m e cos y þ e f @z kz @z 2 @y R2 @y ky @y (5) where the film thickness is given by Eq. (1). Parameters ky and kz are correction coefficients for turbulent flow. The correction coefficients ky and kz are determined using Constantinescu’s model of turbulence [19,20]. According to this model, the correction coefficients are function of the effective Reynolds number: ky ¼ 12 þ 0:0136Re0:9 eff kz ¼ 12 þ 0:0044Re0:9 eff
'
ew
3. Nonlinear analysis
W
O e
R
R+C
y
ð6Þ
The first signs of turbulence appear when the local mean Reynolds number Rem is greater than local critical Reynolds number Recr. The flow becomes dominantly turbulent when the mean Reynolds number Rem is greater than 2Recr. With these assumptions, the effective Reynolds number is: 8 0 Rem oRecr > > < Rem Recr pRem p2Recr (7) Reeff ¼ Recr 1 Rel > > : Re Re 42Re l
Oil supply
m
cr
where
ω
! rffiffiffiffiffiffi R Recr ¼ min 41:2 ;2000 h
Sleeve
x
Line of centers
Mean circle of the waves
Fig. 1. The geometry of a three-wave journal bearing.
2rq , m roRh . Rel ¼ m
Rem ¼
ð8Þ
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y Proximity Probes
Thermocouple Thermocouple Oil Line Oil Line
45 mm
Rotor
End Bearing: Diameter: 30 mm Length: 27.5 mm Clearance: 35 μm
x Proximity Probes
End Bearing: Diameter: 30 mm Length: 27.5 mm Clearance: 35 μm
120 mm Fig. 2. Cross section of the oil wave bearing test rig.
At every time step, the intermediate equilibrium position of the journal center is characterized by variables e, f, _ _ Knowing the values of these variables and assuming and f. that the turbulence correction coefficients are equal to the coefficients corresponding to the previous time step, the transient Reynolds equation, Eq. (5), is integrated and a pressure distribution is obtained. Having the pressure distribution, new turbulence correction coefficients corresponding to every grid point are determined and the Reynolds equation, Eq. (5), is integrated again. The iterative process is repeated until the relative errors for the correction coefficients are smaller than a prescribed value equal to 105. Next, the fluid film forces are determined by integration of pressure distribution (Eq. (4)). The motion equations (Eq. (3)) are then integrated _ which and the values of the system variables (e, f, _ and f), characterize the next moment of time, are obtained. The algorithm is repeated until the orbit of the shaft is completed. The transient Reynolds equation, Eq. (5), is solved using a central differencing scheme. The resultant system of equations is solved with a successive over-relaxation method (the Gauss–Seidel method). The Reynolds boundary conditions are assumed in the cavitation region. The two ends of the bearing are considered at atmospheric pressure. In the pockets regions, the pressure is assumed to be equal to the supply pressure. A fourth-order Runge Kutta method was used to integrate the motion equations (Eq. (3)).
4. Linear perturbation analysis The bearing stability can be also analyzed by evaluating the critical mass. The critical mass represents the upper limit for stability. If the mass m of the system is smaller than the critical mass mcr, the system is stable. The system is unstable for a mass m greater than mcr. The critical mass is function of dynamic coefficients [3]: mcr ¼
ks , g2s
(9)
where ks is the effective bearing stiffness ks ¼
kxx byy þ kyy bxx kxy byx kyx bxy bxx þ byy
(10)
and gs is the instability whirl frequency: g2s ¼
ðkxx ks Þðkyy ks Þ kxy kyx bxx byy bxy byx
(11)
The stiffness and damping coefficients of the bearing are obtained by linearizing the turbulent Reynolds equation as shown in Refs. [21,22]. The resultant partial differential equations are solved using finite difference schemes. The boundary conditions are the same as for the nonlinear analysis. The pressure gradients are assumed to be zero at the two ends of the bearing, in the pocket regions and in the cavitation region. The turbulence correction coefficients are computed with the same iterative procedure described in the nonlinear analysis. A bisection algorithm
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was used to determine the equilibrium position of the bearing. 5. Experimental investigations The stability of two identical journal bearings that support a rigid symmetric rotor was evaluated during experimental investigation. A test rig, designed, fabricated and assembled at the NASA Glenn Research Center in Cleveland, Ohio, was used [23,24]. The set-up of that rig for this experimental investigation can be seen in cross section in Fig. 2. The rotor has a maximum diameter of 45 mm, a length of 120 mm and a mass of 1.65 kg. The rotor is driven by an air turbine at rotational speeds up to 60,000 rpm. Two
identical three-wave journal bearings support the rotor. All the wave bearings that were tested have a mean circle diameter of 30 mm, a length of 27.5 mm and a clearance of 35 mm. Each bearing is supplied with lubricant by three oil supply pockets. The angle between two consecutive supply pockets is 1201. The bearing temperature is read by individual thermocouples situated on each bearing. The lubricant used during the experiments is MilL-23699. Two thrust wave bearings, situated at each end of the rotor, control the axial position of the rotor. The rotor displacements in the horizontal and vertical directions are measured by proximity probes positioned in two mutually perpendicular planes. One of these proximity probes was connected to a digital oscilloscope that can
90
4 μm 60
120 3 μm 2 μm
150
30
1 μm
180
0
210
330
240
300 270
Fig. 3. Oscilloscope screen with experimental wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.305, rotating speed n ¼ 60,000 rpm). The bearing runs stably.
Fig. 5. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.305, rotating speed n ¼ 60,000 rpm, supply pressure ps ¼ 0.152 MPa). The bearing runs stably.
14 12
mcr (kg)
10 8 6 4 2
m = 0.825 kg
0 10000
20000
30000 40000 n (RPM)
50000
60000
Fig. 4. Predicted critical mass of wave bearing as function of running speed by small perturbations analysis (wave amplitude ratio ew ¼ 0.305, supply pressure ps ¼ 0.152 MPa). The critical mass is greater than the mass of the shaft related to one bearing (m ¼ 0.825 kg) for speeds up to 60,000 rpm. The bearing should run stably.
Fig. 6. Oscilloscope screen with the wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 36,000 rpm, oil supply pressure ps ¼ 0.276 MPa). The bearing runs stably.
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display the wave shape and the FFT analysis of the input signal. More details about the experimental rig can be found in Ref. [24]. 6. Numerical results The overall aim of this paper is to study the influence of wave amplitude and supply pressure on the stability of a three-wave bearing having a mean circle diameter of 30 mm, a length of 27.5 mm and a clearance of 35 mm. First, a three-wave bearing with a wave amplitude ratio of 0.305 (the ratio between the wave amplitude and the
439
clearance) was tested. It was found that this bearing is stable even at rotational speeds of 60,000 rpm. The experiment was conducted at a supply pressure of 0.152 MPa. The bearing temperature was 210 1C. The oscilloscope screen containing the wave shape of the signal from one of the proximity probes and the FFT analysis for a rotational speed of 60,000 rpm is presented in Fig. 3. The FFT analysis shows the presence of an amplitude peak only at the synchronous frequency. In addition, the regular shape of the motion wave curves indicates the presence only of a synchronous motion. The numerical simulations confirm the experimental results. The small perturbation analysis reveals that the critical mass, mcr, is greater than the mass of the shaft related to one bearing (m ¼ 0.825 kg) for speeds up to 60,000 rpm. In Fig. 4, the critical mass as a
5 4.5 4
mcr (kg)
3.5 3 2.5 2 1.5
m =0.825 kg
1 0.5 0 10000 Fig. 7. Oscilloscope screen with the wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 39,000 rpm, oil supply pressure ps ¼ 0.276 MPa). An incipient sub-synchronous whirl motion is detected.
20000
30000
40000 n (RPM)
50000
60000
Fig. 9. Predicted critical mass of wave bearing as function of running speed by small perturbations analysis (wave amplitude ratio ew ¼ 0.075, oil supply pressure ps ¼ 0.276 MPa). A sub-synchronous motion should be expected for speeds greater than 40,000 rpm when the critical mass is smaller than the mass of the shaft related to one bearing (m ¼ 0.825 kg).
90
4 μm 60
120 3 μm 2 μm
150
30
1μm 180
0
210
330
240
300 270
Fig. 8. Oscilloscope screen with the wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 44,000 rpm, oil supply pressure ps ¼ 0.276 MPa). The sub-synchronous whirl motion is dominant.
Fig. 10. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.075, rotational speed n ¼ 36,000 rpm, oil supply pressure ps ¼ 0.276 MPa). The bearing runs stably.
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function of rotational speed is presented. The transient analysis shows that for running speeds up to 60,000 rpm the journal center approaches the bearing center very rapidly and then orbits around it with a very small radius, as it can be seen for in Fig. 5 for a rotating speed of 60,000 rpm. Next, a three-wave bearing with a wave amplitude ratio of 0.075 was tested. Initially, the experiment was conducted for a supply pressure of 0.276 MPa and a bearing temperature of 186 1C. For these conditions, the experiments revealed that the bearing operates stably for rotational speeds up to 36,000 rpm. The oscilloscope screen for a rotational speed of 36,000 rpm is presented in Fig. 6.
90
8 μm
120
60 6 μm 4 μm
150
30
2 μm 180
0
210
330
240
300 270
Fig. 11. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 39,000 rpm, oil supply pressure ps ¼ 0.276 MPa). A small limit cycle appears.
90 120
At a speed of 39,000 rpm, an incipient sub-synchronous frequency is detected (see Fig. 7). The sub-synchronous frequency becomes dominant at a rotational speed of 44,000 rpm. From the FFT analysis (see Fig. 8), it can be seen that the amplitude of the sub-synchronous motion is greater than the amplitude of the synchronous motion. Also, the wave shape is very irregular. The same conclusions can be drawn from the numerical simulations. The variation of the critical mass with rotational speed is presented in Fig. 9. The critical mass is greater than the mass of the shaft related to one bearing (m ¼ 0.825 kg) for speeds smaller than 36,000 rpm. Around a rotational speed of 39,000 rpm, the critical mass is very close to m. For rotational speeds greater than 40,000 rpm, the critical mass is smaller than m. Consequently it may be concluded that the system is unstable for rotational speeds greater than 40,000 rpm. The transient analysis allows examination of post whirl orbit details. The stable trajectory of the journal center operating at 36,000 rpm is given in Fig. 10. When the journal rotates at 39,000 rpm, the journal locus reaches a limit cycle (see Fig. 11). The radius of the limit cycle increases with increasing rotational speed. For example, at the 44,000 rpm the radius of the orbit is 10.75 mm (see Fig. 12) compared to 7.55 mm at 39,000 rpm (Fig. 11). An increase of the oil supply pressure to 0.414 MPa stabilizes the bearing for velocities up to 60,000 rpm. The sub-synchronous whirl motion disappears (see Fig. 13). The critical mass becomes greater than the mass of the system (Fig. 14). Also, the limit cycles disappear and the journal approaches to the bearing center and orbits around it. As an illustration, the trajectory of the bearing rotating at 60,000 rpm is presented in Fig. 15.
15 μm 60 10 μm
150
30 5 μm
180
0
210
330
240
300 270
Fig. 12. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 44,000 rpm, oil supply pressure ps ¼ 0.276 MPa). The limit cycle is larger than the previous case (Fig. 11), but still inside the bearing clearance.
Fig. 13. Oscilloscope screen with the wave shape of the proximity probe signal and FFT for the tested wave bearing (wave amplitude ratio ew ¼ 0.075, rotational speed n ¼ 44,000 rpm, oil supply pressure ps ¼ 0.414 MPa). The bearing runs stably.
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Even if the wave bearing is unstable, the wave bearing maintains its orbit inside the bearing clearance. The theoretical data and the experimental results were in good agreement, which validates the developed numerical codes.
9 8 7 6 mcr (kg)
441
5 4
References
3 2 1
m= 0.825kg
0 10000
20000
30000
40000 n (RPM)
50000
60000
Fig. 14. Predicted critical mass of wave bearing as function of rotational speed by small perturbations analysis (wave amplitude ratio ew ¼ 0.075, oil supply pressure ps ¼ 0.414 MPa). The critical mass is greater than the mass of the shaft related to one bearing (m ¼ 0.825 kg) for speeds up to 60,000 rpm. The bearing should run stably.
90
4μm
120 3μm
60
2μm
150
30
1μm 180
0
210
330
240
300 270
Fig. 15. Predicted orbital trajectory of the journal center of wave bearing by transient analysis (wave amplitude ratio ew ¼ 0.075, rotating speed n ¼ 60,000 rpm, oil supply pressure ps ¼ 0.414 MPa). The bearing runs stably.
7. Conclusions Both a critical mass and a transient analysis were developed to investigate the dynamic behavior of a threewave bearing before, during and after the fractional frequency whirl (FFW) appears. The numerical and experimental investigation performed on a three-wave bearing (30 mm diameter, 27.5 mm length, and 35 mm radial clearance) with a wave amplitude ratio of 0.305 showed that this bearing is stable for rotational speeds up to 60,000 rpm and that the bearing operates free of FFW. The same wave bearing with a wave amplitude ratio of 0.075 experiences a sub-synchronous unstable motion at a speed which depends on the oil supply pressure.
[1] Orcutt FK, Arwas EB. The steady-state and dynamic characteristics of a full circular bearing and a partial arc bearing in the laminar and turbulent flow regimes. J Lubr Technol 1967:143–53. [2] Parkins DW. Theoretical and experimental determination of the dynamic characteristics of a hydrodynamic journal bearing. ASME J Lubr Technol 1979;101:129–39. [3] Lund JW, Thomsen KK. A calculation method and data for the dynamic coefficients of oil—lubricated journal bearing. Topics in fluid bearing and rotor bearing system design and optimization. New York: ASME; 1978. p. 1–28. [4] Lund JW. Review of the concept of dynamic coefficients for fluid film journal bearings. ASME Trans J Tribol 1987;109(1):37–41. [5] Kostrzewsky GJ, Taylor D, Flack RD, Barrett L. Theoretical and experimental dynamic characteristics of highly preloaded three lobe journal bearing. STLE Tribol Trans 1998;41:392–8. [6] Childs D, Moes H, van Leeuwen H. Journal bearing impedance descriptions for rotordynamic applications. Trans ASME 1977;99: 198–209. [7] Ebrat O, Mourelatos Z, Vlahopoulos N, Vaidyanathan K. Calculation of journal bearing dynamic characteristics including journal misalignment and bearing structural deformation. STLE Tribol Trans 2004;94(102):94–102. [8] Rao TVVLN, Sawicki JT. Linear stability analysis for a hydrodynamic journal bearing considering cavitation effects. STLE Tribol Trans 2002;45(4):450–6. [9] Kirk RG, Gunter EJ. Short bearing analysis applied to rotordynamics I: theory. ASME J Lubr Technol 1976;98:47–56. [10] Kirk RG, Gunter EJ. Short bearing analysis applied to rotordynamics II: results of journal bearing response. ASME J Lubr Technol 1976;98:319–29. [11] Majumdar BC, Brewe DE. Stability of a rotor supported on oil film journal bearing under dynamic load, NASA TM-102309, 1987 (AVSCOM TR 87-C-26). [12] Monmousseau P, Fillon M. Frequency effects on the TEHD behavior of a tilting-pad journal bearing under dynamic loading. ASME J Tribol 1999;121:321–6. [13] San Andres L. Transient response of externally pressurized fluid film bearings. STLE Tribol Trans 1997;40(1):147–55. [14] Tieu AK, Qiu ZL. Stability of finite journal bearings—from linear and nonlinear bearing forces. STLE Tribol Trans 1995;38(3):627–35. [15] Tichy J, Bou-Said B. Hydrodynamic lubrication and bearing behavior with impulsive loads. STLE Tribol Trans 1991;34(4): 505–12. [16] Hashimoto H, Wada S. Dynamic behavior of unbalanced rigid shaft supported on turbulent journal bearings—theory and experiment. ASME J Tribol 1990;112:404–8. [17] Dimofte F. Wave journal bearing with compressible lubricant—Part I: the wave bearing concept and a comparison to the plain circular bearing. STLE Tribol Trans 1995;38(1):153–60. [18] Dimofte F. Wave journal bearing with compressible lubricant—Part II: a comparison of the wave bearing with a groove bearing and a lobe bearing. STLE Tribol Trans 1995;38(2):364–72. [19] Constantinescu VN. Basic relations in turbulent lubrication and their extension to include thermal effects. Trans ASME J Lubr Technol 1973;95(2):147–53. [20] Freˆne J, Constantinescu VN. Operating characteristics of journal bearings in the transition region. In: Proceedings of the 2nd Leeds–Lyon symposium on tribology; 1975. p. 121–4.
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[21] Constantinescu VN. Lubrication in turbulent regime, US Atomic Energy Commission, 1968. [22] Hashimoto H, Wada S, Tsunoda H. Performance characteristics of elliptical journal bearings in turbulent flow regime. Bull JSME 1984;232:2265–71. [23] Dimofte F, Proctor MP, Fleming DP, Keith TG. Wave fluid film bearing tests for an aviation gearbox. In: Proceedings of the 8th international symposium on transport phenomena and dynamics of
rotating machinery, ISROMAC-8, Honolulu, HI, 26–30 March 2000, NASA/TM-2000-209766. [24] Dimofte F, Lambrulescu MI, Sawicki JT. Experimental investigations on the influence of oil inlet pressure on the stability of wave journal bearing. In: Proceedings of the 10th international symposium on transportation phenomena and dynamics of rotating machinery, Honolulu, Hi, ISROMAC10-2004-146.