Internal Speeds and Motions

The seventh chapter of Rolling Bearing Analysis by Tedric Harris is entitled Internal Speeds and Motions. It discusses the velocity of the rolling element interfaces as well as the stick, slip and spin in these contacts.  This is important because unlike hydrostatic and hydrodynamic bearings, rolling element bearings have a very complex motion.  

This chapter starts with simple situations and continues to more complex areas.  It begins with calculation of cage motion and pure rolling contacts.  Pure rolling contact only occurs under three conditions which never occur in real life:

• A line contact under zero load
• A point contact under zero load
• A line contact in which the contacting bodies are of identical length

The first step when analyzing complex rolling motion is finding the generatrix.  This is the axis around which the rolling element rotates.  The rotation of the element is around this axis and can be broken down vectorially into rolling and spinning angular velocities.  The third component of the motion is lateral gyroscopic motion.  

For a ball there is typically one point with pure rolling and for a roller there are two locations.  However, these points may be actually located outside the contact area.  The rest of the contact area (excluding these one or two points) will have some slip due to the combined rolling, sliding, and gyroscopic motion.  The slip lines are drawn for several different cases in this chapter.  

Next the orbital, pivotal and spinning motions are analyzed for ball bearings.  The angular velocities of the races and ball are found for several conditions such as the inner race being fixed.  Gyroscopic pivotal motion, spin-to-roll ratio, and raceway control calculations are also carried out.

The chapter concludes with an analysis of the roller end-end flange contact.  This calculations are much simpler than the other roller motion equations.  Two types of end geometries (flat and spherical) were discussed although many additional geometries have been used in roller bearings.

Distribution of Load in Statically Loaded Bearings

The sixth chapter of Rolling Bearing Analysis by Tedric Harris is entitled Distribution of Load in Statically Loaded Bearings. In this section of the book the forces and deflections of bearings under axial, radial, and moment loading are calculated while neglecting dynamic effects.  These equations are applicable for most of the time because many applications involve steady state operation, moderate speeds and insignificant friction.  At moderate speeds the centrifugal force and gyroscopic moment can be neglected.

The chapter begins by reviewing the force-deflection relationships developed earlier.  For point contacts force is equal to a spring constant multiplied by overlap raised to the 3/2 power.  For line contacts force is proportional to overlap raised to the 10/9 power.  These relationships are used for all contacts between bearing elements.  

The derivations for loaded bearings are quite complex.  However, they are accompanied by figures and tables to aid in calculations.  Many steps of the derivations are shown as well as the final solution.

Harris begins with derivations for bearings under either a radial or thrust load.  These systems are relatively simple and lead to closed form solutions.  For the angular contact bearings the solution is a little more complex because the contact angle changes with the axial deflection.

Next combined loading is analyzed.  Combined radial and thrust loads have a simple solution, but a combined radial, thrust, and moment loading require the solution of simultaneous equations using the Newton-Raphson method.  

Finally some complex situations are studied.  The effect of misalignment on the loading, the roller crown, and non-rigid housings are all studied.  Non-rigid housings become important when the outer race of the bearing is only supported at a few points or when the shaft is hollow.  The flexible housing problem is solved using the energy method.

Contact Stress and Deformation

The fifth chapter of Rolling Bearing Analysis by Tedric Harris is entitled Contact Stress and Deformation. This chapter deals with calculating the pressure, frictional shear stress and subsurface stress in rolling element contacts.  

Contacts in rolling bearings are generally very small and consequently have high pressures.  It is not uncommon to find pressures of 3.5 GPa inside bearings.  Because of their high values the surface pressure and the near surface stress calculations are important for engineers.  Relatively low external forces lead to high pressures in bearings, so bulk material failure is generally not a concern.  Typically friction can be ignored because lubrication and sliding is negligible.  

The chapter includes derivations, equations, and chart with results from the equations.  This provides the information necessary for bearing stress calculations.  These equation in more general forms can also be found in other engineering books.

The effect of subsurface stresses on bearings were also discussed.  In a bearing the ratio of contact length to width is around 10.  For this aspect ratio the maximum shear force typically occurs at a depth of half the contact half width and the maximum von Mises stress occurs at a depth of 70 to 80 percent of the half width.  This corresponds to the area with the most material transformation.  

Common roller profile shapes were considered:  straight, fully crowned, partially crowned, and logarithmic.  The variation in contact pressure was illustrated at different load levels.  The changes are important because edge stresses occur at higher loads and depend on the roller profile.  

• If the contact length in less than the roller length, it is a typical Hertzian point contact.
• If the contact length is between 1 and 1.5 times the roller length, it is referred to as a modified line contact.
• If the contact length is longer than 1.5 times the roller length, then it is a line contact with edge loading.

 

Ball and Roller Loads

The fourth chapter of Rolling Bearing Analysis by Tedric Harris is entitled Ball and Roller Loads.  In this chapter the equations for calculating static and dynamic loads on the rolling elements in a bearing are derived.  The loads on the rolling elements will be used to find the force on other bearing components in later chapters of the book.

The one static load considered in the chapter is the normal force between the rolling element and the raceway.  Although, the element may be moving, the force can be calculated at any position as if the problem system were static.  This force is always transmitted on a line connecting the two points of contact.  The dynamic loads one the rolling elements are the body force, centrifugal force, and gyroscopic moment.  The equations in the chapter are derived for the angular contact ball bearing which is the most general case.   

The angle at which the bearings contact the races is important and dependent on loading.  A radial force applied to a bearing with a 90 degree contact angle or a thrust load applied to a 0 degree contact angle will lead to quick failure.  

Axial loads on in radial roller bearings are also discussed.  Axial loads cause roller-flange interactions which may cause heat generation and wear as well as roller skewing and tilting.  Roller tilting comes from the moment caused by the normal force from the flanges interacting with the roller ends.  Skewing is caused by the uneven distribution of pressure along the roller’s length due to tiling which leads to asymmetrical slip and friction distributions.  In most spherical roller bearings the axial load is handled using roller-raceway traction instead of end flanges.  

 

Interference Fitting and Clearance

The third chapter of Rolling Bearing Analysis by Tedric Harris is entitled Interference Fitting and Clearance.  Rolling bearings are frequently either press or shrink fitted onto the shaft or into the housing.  Having an interference fit significantly affects the diametral clearances in a bearing which in turn affect contact angle, fatigue life and more.  The operating temperature also affects the clearances.

Standards for bearing interference fits were developed by the Anti-Friction Bearing Manufacturers Association (AFBMA) and adopted by the American National Standards Institute (ANSI) and International Standards Organization (ISO).  These standards include fitting guidelines for all types of rolling element bearings.  

The effect of interference fitting on clearance of bearings was derived using elastic thick ring theory.  These equations can be used to find the reduction in clearance when a bearing is pressed into a housing of a specific thickness.  Formulas are also given for finding how much temperature increases will affect the interference of the shaft or housing if it is of a different material than the bearing (steel).  Finally, the interference will probably slightly less than predicted due to the flattening of surface asperities on the pressed surfaces, so modification factors are given to account for surface finish.  

Rolling Bearing Macrogeometry

The second chapter of Rolling Bearing Analysis by Tedric Harris is entitled Rolling Bearing Macrogeometry.  This chapter describes the large scale geometrical calculations made in rolling bearing design.  These calculations are important for determining loads, stresses, and deflections during bearing operation.  The dimensions will be altered by bearing loading, but they are simple and the starting point for more complex computations.  The critical dimensions are as follows:  

Ball Bearings

• Pitch Diameter:  the average of the inner and outer diameter of the of the raceways.
• Clearance:  the difference between 1) the difference between the inner and outer raceway diameters and 2) twice the ball diameter.
• Osculation:  the ratio between the curvature of the rolling element and the raceway.
• Contact Angle: the angle of contact between the ball and the raceway when all axial play in the bearing is taken up.
• End Play:  the maximum motion of the bearing in the axial direction.
• Free Angle of Misalignment:  the maximum angle through which the axis of the inner ring can be rotated with respect to the axis of the outer ring before contact is made.
• Curvature:  the inverse of the radius of a component.
• Relative Curvature:  The equivalent curvature when two components contact one another.

Roller Bearings

• Pitch Diameter and Clearance:  These are the same for ball and roller bearings.
• Crown: curvature of the roller direction transverse to rolling.
• Osculation:  this is the same as for ball bearings; however, the crown radius is used.
• Free Endplay:  the rollers are free in the axial direction and constrained only by the end flanges.
• Contact Angle:  this is zero for all roller bearings.
• Curvature:  the equivalent curvature between the roller contour and raceway groove is calculated.

 

Rolling Bearing Types

The first chapter of Rolling Bearing Analysis by Tedric Harris is entitled Rolling Bearing Types.  It describes different types of rolling element bearings and their strengths and weaknesses.  The bearings fit into three different categories:  ball bearings, roller bearings and linear motion bearings.  

Ball Bearings

• Radial Ball Bearings
  The single-row deep-groove conrad assembly bearing is the most popular type of roller bearing.  It is best for carrying radial loads; however, it can withstand combined radial and axial loading.  It readily accepts shields or seals.  
  There are several modified versions of this basic design.  One is using a filling-slot assembly which allows more balls to be added increasing the load carrying capacity. Another is using a double row of balls which looks similar to two bearings mounted side by side.  
  Instrument ball bearings are a special version of this type which are smaller and thinner.  These are manufactured to more stringent standards and assembled in clean rooms because their small size means even small particles or dimensional deviations will cause serious problems.
• Angular Contact Ball Bearings
  This type of bearing can support combined radial and thrust loading or high levels of thrust load because of their large contact angles.  Typically these bearings are mounted in pairs or a double row bearing is used.  Double bearings can be mounted face-to-face or back-to-back so that an axial load from either side can be supported.  Alternatively they can be tandem mounted in which case both bearings support a load from the same direction.
• Thrust Ball Bearings
  Thrust bearings support thrust loads and typically have a contact angle of 90 degrees although any contact angle over 45 degrees is classified as a thrust bearing.  These bearings can achieve high speeds and are often mounted on spherical seats to be externally aligning.  

Roller Bearings

In comparison to ball bearings roller bearings have a large load carrying capacity, have longer fatigue lives, are harder to align, are more difficult to install, and cost more.

• Radial Roller Bearings
  These bearings are capable of high speeds and have low torque friction, but they are not suitable for axial loads.  Frequently a double row of bearings is used to increase the load capacity.  
  One subset of this category is needle roller bearings.  These have very small diameter rollers in relation to their length.  This gives them greater friction and makes it harder for them to be manufactured to tight tolerances.  Sometimes these bearings do not have a cage.  
• Tapered Roller Bearings
  This type of bearing can carry combined radial and thrust loading.  Loading forces the rollers against the flange guide increasing friction, so these bearings are not suitable for high speed applications.  Tapered roller bearings can also be made in two and four row varieties.   
• Spherical Roller Bearings
  Spherical roller bearings have a raceway which is a portion of a sphere.  The crown radii on the rollers are consequently very large as well.  Because of the close osculation of the roller and raceway, these bearings have a large amount of friction but also a high load capacity.  
• Thrust Roller Bearings
  Thrust roller bearings can be made using cylindrical, spherical, tapered, or needle rollers.  All of these roller types will have limited speeds.  The spherical rollers have high load capacity, can support radial and thrust bearings and are self aligning.  Using cylindrical rollers will result in a large amount of sliding.  The loads on needle rollers need to be kept small.  

Linear Motion Bearings

• Ball bushings can be used to allow linear motion instead of sliding lubricated motion.  These bearings reduce friction and dimensional changes due to wear of the sliding components.  

Rolling Bearing Temperatures

The fifteenth chapter of Rolling Bearing Analysis by Tedric Harris is entitled Rolling Bearing Temperatures. This chapter covers two topics: calculating the temperature of a bearing and modifying the bearing system so it operates at an acceptable temperature.  The steady state temperature of a bearing system is critical and is governed by heat generation and heat dissipation.  

If the temperature becomes too high, the lubricant becomes distressed leading to bearing failure.  Bearing temperature is typically not a problem if there is a light load or low speed because not much heat is generated.  Moving air adjacent to a metal surface usually prevents overheating problems.  

Designing to ensure proper temperature has two steps: calculating the temperature and redesigning the system if necessary.  Heat generation is a function of the frictional torque and speed of a bearing.  It is equal to torque multiplied by speed and a constant.  Heat is dissipated through the three typical means:

1. Conduction:  fairly simple calculations
2. Convection:  complex because the transfer coefficient is a function of the temperature.
3. Radiation:  a non-linear equation (to the fourth power of temperature)

To solve the temperature problem a system of equations is developed using the finite difference approach which is solved iteratively.  

If the temperature in a bearing is too high, the system must be redesigned to lower the temperature or to operate at the higher temperature.  The former is the preferred approach.  The temperature can be lowered by directing moving air over the bearing structure and cooling the bearing’s oil with a heat exchanger.  The bearing can operate at high temperatures with special lubricants and if made of special materials.

Tribology Experiments

The fourteenth chapter of Principles of Tribology by Wen Shizhu and Huang Ping is titled Tribology Experiments. Tribology is a complicated phenomenon, so a lot of effort has gone into trying to standardize tribology tests.  This chapter goes over the types of tribology tests used today and the methods available for analyzing them.  

There are three levels to testing.  An investigation often begins with tests on lab test specimens.  These tests take less time, are easily controlled and yield lots of data; however, they don’t match the working conditions.  The second level of testing is simulation tests.  These use a real part and are close to the real operating environment, but the conditions are intensified.  Actual tests are long and costly but they are more reliable.  

Wear and friction tests are used to evaluate materials and lubricants.  They both use some type of a sliding, rolling or combination motion.  Three machines are especially popular.

• Four Ball Machine: three balls are stationary and the fourth rotates.  The size of the wear scar is measured.
• Timken Testing Machine or Block on Ring Machine:  A rotating pin contacts a stationary block.
• Pin on Disk Test:  a pin is pressed against a rotating disk.

Elastohydrodynamic Lubrication Tests use interferometry to measure the film thickness of the contact.  A ball rotates against a coated glass disk and the fringes caused by interference of light reflecting from the contact is used to calculate the film thickness, film shape and lubrication regime transitions.  

In wear tests there are a number of ways to characterize the wear loss or wear capacity of a part.  Weight/volume looks at the total region lost and thickness looks at the height of the worn region.  These can be determined in a number of ways:

• Weight requires small specimens and is not suitable if there is significant plastic deformation.  A long test may be required to generate a significant amount of wear.
• The length of a part is based on the size of part before and after a test.  It can show how wear varies over a surface.  A drawback is that it can be affected by deformation and temperature.
• Profiles measure the thickness over a region and represent the surface.  Wear must be greater than the surface roughness.
• Indentation requires making a mark before the test and comparing its size before and afterwards.  Frequently a Vickers hardness test is used as a mark.
• Grooving uses lines instead of points cut into the surface.  The cuts should be perpendicular to sliding to prevent debris from clogging them.  It is more accurate than indentation.
• Precipitation measure debris which are carried away by lubricating oil.  Only the total wear can be found not its distribution on a part.  
• In the Radioactive Method the material is radioactive and creates radioactive wear particles.  The radiation intensity of the oil can be used to determine the concentration of wear particles.

After a wear test it is also useful to find the structure of the surface including the topography, damage and oxide films.  The TEM and REM (Transmission Electron Microscope and Reflecting Electron Microscope) were used for this, but have been replaced by the SEM (Scanning Electron Microscope).  This can only capture a before and after snapshot of the surface.  The atomic force microscope is useful for getting surface topography.  An acoustic microscope can be used to look at subsurface damage.  

A chemical analysis of the surface can be used to find element migration and to determine if material transfer has occurred.  Energy spectrum analysis involves hitting the surface with a UV/X-ray or electron beam and reading the energy from the excited surface.  It can see if there is a surface film or whether additives have absorbed to the surface.  Electron Probe Micro Analysis (EPMA) hits the surface with an electron beam and sees what ray spectrum is emitted.

Various methods have been developed for reading the wear state during operation.  Ferrography measures the ferrous particles or wear debris from a sample of oil.  The size, shape and distribution of these factors is determined.  It proves a visual and numerical representation of the debris.  Spectral analysis can be used to find what atoms are present in the oil during operation.  Lubricant composition analysis can be used to measure lubricant parameters such as acidity, additive concentration and insoluble material during operation.  Mechanical Vibration and noise analysis is good for low speeds applications (journal bearings) but not high speed ones (gears).  This method watches for the emergence of vibrations from a baseline pattern.

Anti-Wear Design and Surface Coating

The thirteenth chapter of Principles of Tribology by Wen Shizhu and Huang Ping is titled Anti-Wear Design and Surface Coating. Local material failure can lead to the failure of an entire machine, so improving the wear resistance of critical machine elements is important.  The first part of anti-wear design is to establish a surface film (lubricant, adsorption or chemical).  The second part is to select material pairs which will wear well together.

The fluid lubricant film is generally characterized by the film thickness ratio (lambda) which is the film thickness divided by combined roughness.  There are many guidelines dependant on the contacting surfaces.  In general cases lambda should be above 1.5.  For high speeds or rough surfaces it should be above 2.  For conformal contacts it should be between 2-5 and in some cases larger.  

Viscosity is the most important factor when selecting a lubricating oil.  As a part of this process consider the average and range of pressure and temperature experienced by the contact, so the lubricant will work adequately under all operating conditions.  The second most important lubricant characteristic is oxidative stability which prevents the oil from deteriorating at elevated temperatures.  Additives can be used to help modify these properties.  

For a grease the base oil, thickener, and additives are all used to control film properties.  The thickener or densifier controls temperature resistance, water resistance, hardness and fluidity.  The base oil controls low temperature performance, viscosity and anti oxidation properties.  Additives are used similarly to in oils.  

Solid lubricants are used when a liquid or semi-solid lubricant cannot be used.  A solid lubricated part will experience more friction and wear loss than with grease or oil.  Materials used as solid lubricants should have low shear strength, thermal resistance and chemical resistance.  

Selecting a seal and filter for a lubricating environment is important as well.  Sealing is required to keep out particles which can reduce lubricant life by an order of magnitude.  Particles cause problems when they embed in the surface and begin causing abrasive wear, when they roll around and cause plastic deformation leading to fatigue wear and when they roll around creating ridges which lead to adhesive wear.  

Selecting a material depends on the primary wear mechanism.  Higher hardness is nearly always beneficial for wear prevention.  It can be increased through three different methods: mechanical processing, diffusion treatments, and coatings.

• For abrasive wear the natural hardness of material is most important because artificially increasing the hardness has a smaller effect on wear resistance.  If there is impact by particles traveling parallel to the surface, hardness is important.  If the particles are traveling perpendicularly to the surface, toughness is important.  The hardness should be more than 1.4 times the hardness of third body particles.
• For adhesive wear low solid solubility is one of the most important factors in reducing wear.  Brittle materials work better than plastic for the same reason.  High melting temperature and high crystallization temperatures also reduce adhesion.  High hardness and multiphase metals are also preferable to low hardness and single phase metals.
• For contact fatigue high hardness is important.  62 HRC is the optimal hardness because above that increased brittleness reduces life.  It is best for there to be a slight difference between material hardnesses.  Inclusions should be eliminated to have a clean material.
• For fretting conditions, materials resistant to adhesive wear perform the best.
• For corrosive wear materials which resist corrosion work best.  

Surface coatings can be made of one or multiple layers of material added to the surface.  Several coating processes are summarized briefly in this section of the book.  

• Bead welding uses the welding process to add a layer about 1 mm thick.  It is efficient but can cause cracking.
• Thermal spraying processes spray molten or semi-molten particles at the surface.  There are three types: plasma, arc and flame spraying.  This process works for very large parts.
• Slurry coatings are created by brushing onto the surface a solid-liquid mixture.  Slurry coatings can be applied at low temperatures.  One subcategory is slurry coating which consist of brushing, evaporating the liquid and then sintering.  A second is glue coatings which are painted on and then solidify.  A third is thermochemical reaction slurry coatings in which a compound in the slurry reacts chemically with the surface.
• Electro brush plating uses a graphite brush to move over the surface causing metal ions to diffuse onto the surface.  The bond strength is based on the crystal structure at the interface.
• Plating can be either chemical or physical.  Vacuum evaporation plating coating material evaporates in a  vacuum and coats the surface.  In sputtering particles bombard the coating source material releasing ions which form a thin film on the surface.  Ion plating combines melting with an ion source and create strong bonds with the surface.  
• Chemical vapor deposition has a gas which reacts with the surface and takes place at high temperatures.  

When designing a surface the material properties and microstructure of the coating must be selected while considering the operational conditions.  The coating material needs to be compatible with the substrate material.  Applying the coating must be technically feasible.  When selecting a method for applying a coating consider the material melting point, coating thickness required, bond strength and heat resistance of the substrate.  

Evaluating coatings is also difficult.  Tests should be carried out for a wide variety of properties.

1. Visual appearance should be smooth, dense, and of uniform color.  There should be no visible defects in the surface.
2. The thickness can be measured with a microscopy of a cross section, by measuring width of the part before and after coating, or by using eddy current.
3. Coating porosity is defined as the pores per unit area.  It can be measured by massing the material with and without water or by spraying a part, weighing it, grinding it to its original dimensions and weighing it again.
4. Bond strength can be measured in many different ways and can measure the adherence of the coating to itself or the coating to the substrate.
5. Hardness can be measured using static or dynamic tests.  Microhardness testing can find the hardness of a single particle.  Hoffman scratch testing provides good correlation with wear resistance.
6. Wear testing is often necessary because hardness is not a perfect predictor of wear.  
7. Fatigue can be tested using a4 point bending test
8. Residual stress is related to the thermal expansion of the coating and substrate and can be measured with X-ray diffraction or a bending curvature test.