Machine Design

Bearing simulation gets real

The Beast preprocessor displays a model of a deep-groove ball bearing (SKF Type 6309) used for the simulations.

Simulation of a deep-groove ball bearing running at 600 rpm under a 1-kN radial load. Ball azimuth zero is at the bearing top (12 o'clock). A 0.5 ball azimuth represents the 6 o'clock position.

The instrumented cage for testing ball-cage forces with a calibration ball in the cage pocket sans the outer ring. Axial cuts in the cage bar boost strain-gage signals and help to mechanically isolate one pocket from another.

A comparison of plots from experiments and simulations show good agreement. Note that test signals changing sign in the misaligned case are an instrumentation artifact and do not reflect real forces.

The pressure distribution of all balls on an inner ring segment at an arbitrary point for the misaligned test case is represented as a parametric surface. The circumferential direction is from bottom left to top right.

The test cage with a steel rim affixed to the backbone. The rim provides a "readable" surface for the proximity sensors.

Simulating bearing dynamics is highly specialized and beyond the capabilities of most off-the-shelf analysis packages. That is why bearing maker SKF developed its own Bearing Simulation Tool (Beast) for internal use.

Beast's multibody scheme models the dynamic behavior of all bearing components under load — balls, rollers, inner and outer rings, and cages. The model includes forces on the cage, its motion, as well as skew and tilt of rollers, and ball skidding. Earlier rolling bearing simulation codes either ignored cages or described them with simplified 2D models. Beast, in contrast, uses a 3D model for the task. Engineers specify all boundary conditions, applied loads, and parts geometry, especially important metrics for modeling cages.

To verify the simulation's accuracy, SKF engineers ran tests on deep-groove ball bearings in a rig dubbed Catriona and compared results to Beast simulations of the same style bearings.

The rig uses a high-precision hydrostatic spindle to locate and drive an inner-bearing ring. The outer ring attaches to a yoke fitted with pneumatic force actuators; one in the radial direction and three in the axial direction. This arrangement makes various load combinations possible. Of interest are the forces between the cage and balls (cage-pocket force) and cage motion.

A special cage fitted with strain gages on selected bars measures cage-pocket forces. Individual balls reside in pockets formed by two bars. Instrumented bars are split in half axially to boost signal strength and better mechanically isolate the two bar sides. The cage slips on a low-friction aerostatic spindle and inserts between the bearing inner and outer rings. The model predicts negligible braking torque on the cage from the spindle which agrees well with experiments. The program also models spindle inertia and damping.

The first tests radially load a bearing with a 1,000 N force acting upward while spinning the inner ring at 600 and 6,000 rpm.

In the 600-rpm case, the model predicts a series of declining impacts on the front and rear cage bars. For reference, the front cage bar is ahead of the advancing ball, and the rear behind it.

Consider one complete rotation of a ball around the bearing. A ball starts at the bearing top or 12 o'clock position, 180° from the load zone. Gravity accelerates the ball downward as it crests the top until it impacts the front cage bar and bounces with decaying amplitude. This happens because the speed is low and the ball is unloaded.

The ball remains close to the front cage bar as it moves into the load zone because the loaded balls ahead of it determine cage speed. Gravitational force and rolling losses on the outer ring slow the ball as it exits the load zone and heads uphill. Here, the ball-cage speed difference is larger than when the ball is heading downhill, so impact forces are proportionately larger on the rear cage bar than on the front bar. However, these effects are not readily apparent from experiments. In this case the simulations are merely predicting a behavior.

Not so with the 6,000-rpm case. Both simulation and experimental results closely agree on force-peak timing and impact magnitudes. Here, balls no longer have time to bounce to rest on the front cage bar as in the low-speed case. Moreover, ball impacts are mostly on the rear cage bar because of a larger speed difference between the ball and cage.

Experiment also agrees with simulation when introducing a horizontal misalignment. Such misalignment loads balls unevenly making them run at different tangential speeds. Balls may then assume a tangential position that exceeds available cage-pocket clearance so they remain in contact with the cage longer. A ball contacts the cage twice per ball revolution around the bearing, once at the front cage bar and once at the rear. Force peaks start with impacts but are mostly friction driven.

A second test series studies the movement of a standard polymer (glass-fiber-reinforced polyamide 6,6) snap cage. The only modification to the cage is a thin steel collar mounted on the cage backbone to provide a "readable" surface for the proximity sensors. Cage motion is measured with three proximity sensors in the axial direction and four in radial directions. Cage motion in the bearing plane is of most interest.

The balls are loaded axially and equally so they all run at the same speed. This typically produces a steady whirl motion of the cage. At start-up, gravity forces the balls into different positions relative to each other. This mechanism has a stochastic component that produces different whirl radii and center positions. For example, two experiments with identical axially loads but with different rotation directions, gave two completely different whirl radii. Again, simulation and experimental results are in accord.

Contact-force calculations are probably the most important part of bearings simulations. These calculations require a detailed geometric description of bearing-contact surfaces because features on the order of 0.1 micron can impact roller stability or contact-pressure distributions. Local geometric deviations are specified while the program's tribology model handles small-scale variations such as surface roughness. The program's elastic-contact model includes 3D elastic effects of arbitrary geometry and truncation. FE methods and experiments verify that the model gives an exact solution for Hertzian contacts.

An elasto-hydrodynamic lubrication (EHL) contact algorithm simulates lubrication. It uses a well-established pressure-viscosity relationship and a compressible oil assumption. Runtime EHL calculations are too slow for simulation purposes so the program instead uses central-film thickness, pressure moment, and pressure-distribution formulas derived from multiple EHL calculations covering a wide range of conditions.

In a rolling-element bearing, the lubrication mode can vary from boundary to fully separated surfaces (EHL). A formula describes when the bearings transitions between the two regimes. The separation parameter requires a user-specified film thickness, L = h/s, where h = calculated film thickness and s = root mean square of the surface roughness of the contacting surfaces.

A smoothed variant of the limiting shear-stress model finds shear stress when in the fully separated mode. Users apply a coefficient of friction for the boundary regimen. Also input is the amount of oil in the contact zone described by the combined oil-layer thickness of the two contacting surfaces. The film-thickness routine then automatically figures the degree of oil starvation. Other topological effects, such as grinding-roughness orientation, may also influence film build-up and, consequently, traction. However, until this phenomenon is better understood, the program uses an application-specific factor.

Semiempirical models for bearing materials and squeeze damping simulate all contacts. The models are calibrated against experiments that bounce balls on plates of various materials and with different oil-layer thickness'.

The connection between a bearing and machine is probably as important as what's going on inside the bearing. Here, connections between bodies are described with three linear motions and three rotations about the same axes. Several connection types include a linear spring and an adjustable damper. Bearing models can also have nonlinear stiffness, clearance and damping, and arbitrary load and motion functions. These mechanisms can be used individually or combined in any way. Beast outputs animated movements and forces (with vectors) as well as power loss, film thickness, pressure distribution, slip-speed distribution, and wear. Some parameters, such as contact pressure or slip velocities, are displayed as 3D images on the bodies or on parametric surfaces while 2D graphs provide further detail.

Lars-Erik Stacke and Dag Fritzson, SKF Group Manufacturing Development Centre, Geteborg, Sweden, and Bengt Rydell, SKF Engineering & Research Centre B.V. (ERC), Nieuwegein, the Netherlands, provided information for this article.

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