Combined spherical roller bearings functioning analysis.

Stirbu, Cristel ; Grigoras, Stefan ; Hanganu, Lucian Constantin 等

1. INTRODUCTION

The tendencies of spherical rolling bearings development are the increase of dynamic capacity and of the speed limit (Noronha,1990). We present an original simulation spherical roller bearings functioning method, using a combination between a theoretical model and experimental measurement of the cage speed and of oil temperature.

The matters which the paper tries to clear are:

* estimation of loading conditions of spherical rolling bearings equilibrium, based on: the inner contact loads; centrifugal roller effects; the contact interactions rollers/guiding ring and rolling bodies/cage;

* consideration of the lubrication condition in all spherical roller bearings contacts, according to the sliding speeds and using the contact loads;

* the introduction of cage speed and of lubricant temperature in the analysis programme;

* the spherical roller bearings friction losses estimation;

* the dimensional optimization of analyzed type of spherical roller bearings.

2. DYNAMIC EQUILIBRIUM OF SPHERICAL ROLLER BEARING

The spherical roller bearing radially and axially loaded was considered. The outer ring is fixed and the position of the inner ring center ([O.sub.1] in fig. 1) can be changed, under the outer load (Gupta, 1984); [[DELTA].sub.x] and [[DELTA].sub.z] are his displacements. We consider a Hertzian distribution of the contact pressure and, initially, we neglect the friction forces. The following coordinate frames have been considered: the initial frame (x, y, z) with a fixed origin, O (the center of the outer ring); the azimuthal frame ([x.sub.a], [y.sub.a], [z.sub.a]) to define the angular roller position in the bearing; the frame ([x.sub.w], [y.sub.w], [z.sub.w]) attached to the roller; the frame attached to the inner ring ([x.sub.i], [y.sub.i], [z.sub.i]). The rolling body is defined by the position angle [PSI]. We use the relation of the deformation in the contact between two convex elastic bodies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the position vectors; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]--transformation matrices and [[rho].sub.l, 2]--curvature radii of the bodies, in the contact point.

[FIGURE 1 OMITTED]

The steps of the theoretical model were: the contact deformation and the contact angles determination; the rolling body equilibrium. The last step (the inner ring equilibrium) is described by (2). The solving of (2) leads to the displacements [[DELTA].sub.x] and [[DELTA].sub.z] estimation, under combined outer load (axial - [F.sub.a] and radial - [F.sub.r]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

3. FRICTION SOURCES

Beside the value determination of the cage speed, [[omega].sub.C], the lubricant temperature measurements (Gafitanu & Stirbu, 1996), are requested in order to evaluate its viscosity. The [[omega].sub.C] value is necessary to solving (2), when the centrifugal force of the roller, ([F.sub.C] in fig. 1), determines the [Q.sub.i] inner contact load.

The roller/rolling way contact characteristics: a Hertzian pressure distribution; Newtonian lubricant behavior; the roller was conveniently cylindrically sliced; for each slice the sliding speed and the contact pressure is constant (Noronha, 1990), (Shroeder, 1994), and the elementary friction force is:

d[F.sub.f] = [eta] x [e.sup.[alpha]xp] [v.sub.sl]/[h.sub.EHD] x dA (3)

with: [eta]--lubricant viscosity (determined according to the temperature); [alpha]--pressure-viscosity coefficient; p--contact pressure, [h.sub.EHD]--film thickness for each slice and dA--elementary contact area. Integration on the inner and outer contact surfaces and the sum on all bring rollers bring about the losses for the whole spherical roller bearing. The roller/guiding ring contact is also an EHD contact. The friction losses evaluation is performed for all bearing. In the spherical roller bearings, the roller/cage and cage/inner ring contacts are lubricated in HD mode (Kleckner & Pirvics, 1982).

* The roller/cage interaction has considered as in (Houpert, 1985) and the cage pocket aspect is similar to that of the roller longitudinal section. The cage speed has the measured value.

* The cage/inner ring interaction is treated as a classical radial bearing with a known clearance, in HD lubrication mode too.

4. THE SELECTION OF SPHERICAL ROLLER BEARINGS GEOMETRY

The mains geometrical parameters of spherical roller bearings are: [R.sub.CE]--the outer raceway curvature radius; [R.sub.Ci]--the inner raceway curvature radius; [R.sub.W]--curvature radius of roller profile; [D.sub.W]--roller maximum diameter; [[alpha].sup.o]--free contact angle (modified under load). The original computer programme selects the spherical roller bearings in terms to ensure: a complete EHD lubrication; minimum friction losses and a constant dynamic capacity.

5. RESULTS

Further, a some results obtained from the 22308 C and CC type spherical roller bearing analysis is presented.

The evolution of the friction torque according to the outer load is shown in fig. 2.

The combined load brings about greater friction losses, but their increase is less strong both on the outer and inner rolling ways ([M.sub.fi,o]--friction torque on inner/outer ring; C--dynamic catalogue capacity). The preponderance of the friction torque on the inner ring against the moment on the outer ring is to be noted. Under axial load, the outer friction torque increase significantly.

Fig. 3 contents the influence of the inner contact geometry on the friction torques, under pure axial load. If the curvature radius [R.sub.CI] increases of 1.002 times towards the nominal indicated value, a decrease of inner friction torque of about 15% is obtained. The outer friction torque is much lesser influenced as well as the dynamic capacity C, fig. 3. The ratio [R.sub.W] /[R.sub.CI] (inner raceway osculation) diminishing is favorable to the reduction of the friction losses, as (Shroeder, 1994) have shown as well. The decrease of radius [R.sub.W] of about 0.9985 times (fig. 3. b) brings about a minimum of the friction losses. The effect of outer raceway osculation modifying can be neglected (R is the standard radius; R' is the modified radius).

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

6. CONCLUSIONS

The theoretical analysis model of spherical roller bearings allows to determine the friction losses, in terms to minimizing them, without reducing the basic dynamic capacity. In order to minimize the friction losses, a reduction of tolerance ranges of the curvature radii of the roller profile and of the rolling way on the inner ring, is requested.

About 75% of power losses was generated in the rolling body/rolling way contacts. The rolling body/guiding ring contacts produce 10...15[degrees]/o of the total power losses, especially under radial load. The friction losses in the roller/cage contacts and guiding ring/cage or inner ring are of more 5% of the total power losses.

The contact angles modification, especially under radial load makes possible the appearance of the skew moments (2 ... 3% of the total friction losses).

By modifying the osculation of the both raceways, one can obtain the decrease of the power losses of 10.20% of total power losses.

7. REFERENCES

Gafitanu, M., Stirbu, C. (1996). Dynamic Analysis of the Interactions between Spherical Roller Bearings Elements, Proc. of the 26-th Israel Conference on Mechanical Engineering, Haifa

Gupta, K. (1984). Advanced Dynamics of Roller Element, Springier, New York

Houpert, L. (1985). Piezoviscous--Rigid and Sliding Traction Forces. Application: the Rolling Element--Cage Pocket Contact, ASME/ASLE Lubrication Conference

Kleckner, R. J., Pirvics, J. (1982). Spherical Roller Bearing Analysis, Trans. of ASAJE, Journal of Lubrication Technology, vol. 104

Noronha, A. P. (1990., Calculated Simulation of the Operating of Spherical Roller Bearings, Ball and Roller Bearing Engineering---Industrial Engineering (FAG), no. 104.

Shroeder, W. D. (1994). Spherical Roller Bearings Basics, Power Transmission Design, June