Types of Bearings

2/6/2002© 2000 Alexander Slocum 10-1

Topic 10Types of Bearings

Topics• What is a Bearing?• Robust Design• Concurrent Engineering• Centers of Action• Constraint• Preload• Load/Life• Lubrication• Error Motions


What is a Bearing?

• Bearing is defined by Webster’s to be “a support or supporting part”– In machine design, a bearing is a component that allows for relative motion

between two bodies• Your skeleton is the central structure that supports your body and its modules,

your• Your joints are bearings that allow different body modules to move with

respect to each other• Bearings allow machines to move

• Bearings can have many forms, but only two types of motions– Linear motion or rotary motion

• In all bearings, cleanliness and surface finish are most important• There are many different types of bearings

• Sliding• Rolling• Flexing• Fluid Film (hydrodynamic)

– All are designed using the same philosophy of understanding the flow of forces in the machine, and the mechanical constraints used to mount them


Robust Design

• The bearings are the machine elements that allow components to move with respect to each other with a minimum amount of friction and wear.

• When designed, manufactured, and used properly, bearings will work great!• The biggest killers of bearings are:

– Overloading originates ouchies!– Overconstraint ordains overloading!– Maligned moments make a morass!– Dirt decries disaster!– Lubrication loss lessens life!


Concurrent Engineering

• Bearings: For the concepts you are envisioning, how will you support moving components by linear or rotary motion?

• Concurrent engineering requires us to consider:– Structure: How can you use symmetry and monolithic features to provide support

and ease installation?

– Kinematics: Do the bearings have the required range of motion, without jamming?

– Actuators, Sensors & Controls: Will be bearings withstand the actuator loads and speeds?

– Manufacturing: How do you make sure you can make it?

– Always think ahead to how will you make what you design!


Centers of Action

• A body behaves as if all its mass in concentrated at its center of mass• A body supported by bearings, behaves as if all the bearings are concentrated at the

center of stiffness– The point at which when a force is applied to a locked-in-place axis, no angular motion

of the structure occurs– It is also the point about which angular motion occurs when forces are applied

elsewhere on the body– Found using a center-of-mass type of calculation (K is substituted for M)– To find the X location of the center of stiffness with respect to an arbitrary coordinate

system:

∑

∑

=

== N

ii

N

iii

stiffnessofcenter

X

KXX

1

1__

center of stiffness axis


Centers of Action• Moments are what cause bending and binding in the system, so if they can be

minimized, this is good!• If the actuation force acts through the center of mass:

– There is no inertial moment on the body• There are no reaction forces on the bearings

• The center of mass and the center of stiffness do not necessarily have to be located at the same point

– However, for stacked multi axis structures:• The centers of mass of the axes move• Locate the point of actuation at the nominal center of mass

• If the force is located between the center of friction and the center of stiffness:– There will be no moment acting on the system– The center of friction and the center of stiffness do not necessarily have to be

located at the same point


Constraint

• EVERY element has six degrees of freedom• You MUST let move what you need to move

– E.g., large degrees of freedom, your overall Design Parameters

• You MUST restrain what needs to be restrained– E.g., resist wheel and linkage forces, such as thrust forces that try to pry the wheel

off the shaft

• You MUST NOT restrain natural error motions that exist to allow for misalignment between elements!

– E.g., a car’s drive shaft has universal joints at the ends and a spline (linear sliding) connection

• As the car flexes, these elements accommodate relative motion between the axle and the transmission

– E.g., shafts that bend a lot have their bearings in spherical mounts, or they use spherical roller bearings


Constraint: Design Process

• From your FRs, you should be able to determine whether or not the bearing needs to resist bi-directional loads

– This will require the bearing to either be one or two-sided– It can ride on top of a rail and resist pulling or tipping loads by gravity, magnets, or

vacuum– It can wrap around a rail and resist pulling or tipping loads by geometry

• Make sure you have constrained what you want to constrain!– If the body is to have N degrees of freedom free to move, there has to be 6 – N

bearing reaction points!– Remember, to resist translation, a force is required.– To resist rotation, a moment, or two forces acting as a couple, is required!

X

Y

Z

θ

θ

θ

Y

X

Z


Constraint: Design Process

• Sketch the component to be supported by the bearings and the applied loads• Sketch the bearing contact points

– Is the problem statically determinate?– E.g., can a simple free body diagram enable you to calculate the loads at each of the

bearing contact points?

• Imagine the system gets hot and expands• Can your bearings accommodate the expansion of system components?

• Do expanding components act like presses to squoonch the bearings?

• Imagine the system is heavily loaded• Can your bearings accommodate deflections in the structures?

• Imagine the system’s dimensions are WAY off• Can your bearings accommodate the improperly manufactured parts?


Constraint: Stick Figures

• Exercise: Draw a shaft and add radial and thrust bearing surfaces and other features (e.g., bearings, support structure) to constrain the shaft– Use the coordinate system, with its six degrees of freedom arrows as a reminder

that bearing points must be selected to work individually or in pairs to restrain five DOF

X

Y

Z

θ

θ

θ

Y

X

Z


Constraint: Saint-Venant

• St. Venant: Linear Bearings:– Linear motion:

• L/D>1, 1.6:1 very good. 3:1 super ideal

• St. Venant: Rotary Bearings:– Rotary motion:

• L/D>3 if you are to have the bearings “build the shaft into a wall”

• IF L/D<3, BE careful that slope from shaft bending does not KILL the bearing!

Wheel

Shaft

Sliding bearing in structure

!!Non Optimal!! Wheel

Shaft

Sliding bearing in structure

!!Optimal!!


Constraint: Rotary Motion

• Every rotary motion axis has one large degree of freedom, and five small error motions

• 5 degrees of freedom are typically constrained with one thrust bearing and two radial bearings


Constraint: Rotary Motion

• Rotary motion bearings’ inner races are mounted to a shaft, and the outer races fit within a bore

• Thinking of constraints is the key (once again)

Sometimes the inner and outer races need to be constrained, and sometimes only the inner race


Constraint: Linear Motion

• Every linear motion axis has one large degree of freedom, and five small error motions

– 5 degrees of freedom are typically constrained with various forms of bearing surfaces (bearing pads)

– Typical preloaded machine tool carriages have pairs of preloaded bearing pads in vertical and horizontal directions at each of 4 corners


Constraint: Golden Rectangle• Sketch the geometry of the system (3D boxes)• Use the Golden Rectangle as a starting point:

– This usually yields structurally stiff and aesthetically pleasing designs.– You must watch the movie (available on video) Donald Duck in

Mathemajicland!• Spread the bearings out as much as possible

– The greater the ratio of the longitudinal to latitudinal (length to width) spacing:• The smoother the linear motion will be and the less the chance of walking (yaw

error)– First try to design the system so the ratio of the longitudinal to latitudinal spacing of

bearing elements is about 2:1– For the space conscious, the bearing elements can lie on the perimeter of a golden

rectangle (ratio about 1.618:1)

– Make sure the length to width is greater than 1:1!1.618

1.000

1.618:1 1:1


Constraint: Jamming• When will a drawer jam?

– When (F2+F4)µ>FA, the slide jams!– To be safe and conservative, create designs where the L/D ratio is greater than 1,

and ideally L/D =3 when there is sliding friction!

FL

FA

L

w

Y

X

F4

F2

b

a

µ F4

µ F2

F = 0 = 2F − 4F − LF∑

M = 0 = 2F a + 4F a∑ − µ 2F b + µ 4F b − LF L − AF w

2F =LF L + a + µb) + AF w( )

2a

4F =LF L − a + µb) + AF w( )

2a


Constraint: Rail Parallelism

• Parallelism errors between bearing rails are one of the biggest sources of over-constraint– Strategy:

• Accurate rail placement ($)

• System compliance (loss of accuracy)

• Controlled compliance:

• Clearance in one of the bearings

• Flexure support of one of the bearings

• Abbe support– The center of the flexure (or ball joint) is a distance H above the round shaft center– Center distance errors (δ) between the round shafts are accommodated by roll (θ) of the

bearing carriage.– Vertical error motion (∆) of the hemisphere is a second order effect– Example: δ = 0.1”, H = 4”, θ = 1.4 degrees, and ∆ = 0.0012”

( )( )

2

arcsin

1 cos2

H

HH

δθ

δθ

=

∆ = − ≈


Constraint: Structures

• A bearing is only as good as the structure that supports it– Utilize symmetry whenever possible

• Asymmetric structures often have internal gradients, which are an indicator of potential problems

• Start at the tool tip or workplace with estimates of forces and acceleration requirements

– Work backward through the structural system and determine forces and moments on members

• Minimize the structural loop and use closed sections whenever possible• Large plate sections should be stiffened with ribs or other means to keep them

from vibrating like drumheads• Bearings’ stiffness should be on the order of the structure: Use superposition!


Constraint: Forced Geometric Congruance

• IF the axis of motion is misaligned from the axis of the bearing, the two will fight each other

– Everything has finite stiffness!• F=kx• The resulting forces can overload and kill the bearing!• Either more accurate components and assembly is required, or compliance, or

clearance (pin in oversized hole) must be provided between the parts


Preload

• Preload allows for bi-directional loading– If not careful, it can lead to over-constraint

• Preload maximizes stiffness• Preload deflection is small, so preload can be easily lost by manufacturing

error or wear– Preload loss via wear is avoided with the use of spring loaded preload systems

• Spring loaded preload systems accommodate rail thickness variations without a large change in preload force

– Spring loaded preload systems have limited force and moment capability– Springs can be disk washers, or the deformation of the structure


Preload: Mechanics

• Sum of the forces:

• From this and the relation Fload = Ktotalδ:

• Careful of preload forces not being overcome by load, or stiffness falls toKlower pad

No preload Preloaded Preloaded with force applied

F load - (F preload + K upper pad δ ) + ( F preload - K lower pad δ ) = 0

K total = K upper pad + K lower pad


Preload: Pitch Stiffness• Assume a preloaded bearing with translational stiffness K

– The deflection of the carriage will be described by δ = θx + δend

• The equilibrium equations are:

• Using the above to substitute first for δend and then θ, the slope θ of the carriage and the deflection δend at the end are respectively:

• The translational stiffness of the carriage at its centroid is:

• The moment applied to the carriage about its centroid is simply F(λ - L/2), so the rotational stiffness of the carriage about its centroid is:

F = KL

θx + δenddx0

L

= K θL2

+ δend

Fλ = KL

θx + δendxdx0

L

= K θL2

3 + δendL

2

θ = 12F(λ - L/2)

L2K δend = F

K1 -

6 λ - L/2L

Ktranslational @ centroid = K

Krotational @ centroid = KL2

12


Preload: Pitch Stiffness

• For four discrete bearings (e.g., linear motion guides' blocks) are mounted at the corners of a carriage

– It will be assumed that the carriage and structure the rails are mounted to behave like a rigid body

– Assume each bearing block has stiffness of KbY and KbZ, and the moment stiffness is insignificant

– Assume the X and Z distances between the bearing blocks are LX and LZ respectively

• The translational stiffness of the system at the center of stiffness will be:

• The moment stiffness at the bearing center of stiffness will be:

X (i direction)

Y

x , 0, z1 1

x , 0, z4 4

x , 0, z3 3

x , 0, z2 2

F

x , y , zX

FX FX FX

F

x , y , zY

FY FY FY

F

x , y , zZ

FZ FZ FZ

x , 0, zi,j i,j

For generic model

Z (j direction)

KY = 4KbY KZ = 4KbZ

KθX = KbYLZ2

KθY = KbZLX2

KθZ = KbYLX2


Preload: Rolling Elements

• In order to maximize stiffness and resistance to impact loads, rolling element bearings must be “preloaded”

– To be preloaded, all the rolling elements must be under load, often one element loaded against another

– When preloaded, even the rolling elements “in tension” act as springs to provide stiffness


Preload: Angular Contact Bearings

• Angular contact bearings are the mainstay of industry– They are preloaded in a back-to-back configuration to prevent thermal growth

overload

• Constraint management is the key!


Load/life

• Your machines will have to withstand at most 6 actual runs on the contest table, BUT dozens and dozens of trial runs

• You need to design and engineer the components for long life– Structural fatigue– Bearing wear

• Sketch the component to be supported by the bearings– Estimate and sketch the direct loads applied to the component– Sketch the bearing contact points– Calculate the loads– THEN

• Estimate and sketch the indirect (unintentional) loads applied to the component

• Calculate the loads– THEN

• Calculate L10 or PV life/load limits


Load/Life: PV for Sliding Contact Bearings

• The PV value is the product of the force and the velocity• Sliding contact bearings have a maximum allowable pressure and a maximum PV

value`– The allowable product of pressure (F/(D*L)) and velocity to have acceptable wear rates– For a typical Delrin bearing used as a bushing (Nylon has ½ these values)

– http://www.dupont.com/enggpolymers/americas/products/deldata.html• The maximum tensile stress we should be putting on the material is 10ksi• The maximum contact pressure for a material is typically sqrt(3) times the yield

strength (Von Mises criteria)• Perform calculations and if in doubt, do a Bench Level Experiment

Maximum Pressure (N/mm^2, psi) 140 19,895PV continuous (N/mm^2-m/s, psi-ips) 1.8 9,791PV short periods (N/mm^2-m/s, psi-ips) 3.5 19,581Compressive Modulus (GPa, psi) 4 579,710


X

P=xdP

M

Load/Life: Moment Loading

• Pitch occurs about the center of stiffness– Each bearing surface has a linearly increasing contact pressure from the center of

stiffness to the edge– For both top and bottom pressure triangles:

– Integrating from 0 to L and solving for dP:

– The maximum pressure is just LdP

dPdxxdFdM x2 22 ==

LM

dP 32

3=

Bearing_Pitch_PVBy Alex Slocum 3/8/98

Dimensions in inches and poundsEnter numbers in boldTotal slider length 1.5Slider contact width 0.25Pivot point height above center of stiffness 1Force 150Max PV (psi-ips) 4000Moment 150Maximum contact pressure 1600Speed (inchs/sec) 1PV (psi-inch/sec) 1600

To determine bearing contact pressure in a slider loaded by a moment


Load/Life: Pinned Joints

• Select a bearing that can handle the stresses:• Ffulcrum/Abearings’ projected area < allowable bearing surface pressure

• Be very careful to make the bearings robust enough to handle side loads (prevent wobble)!– Make sure the actuator is mounted using clevis’ so bending moments are

not transmitted to the actuator!– Sloppy pin joints are often sufficient to function as clevises

• Also check the PV value: Load*Velocity<Bearing Max value


Load/Life: L10

• Rolling element bearings have an L10 life– The allowable load for a given number of revolutions where only 10% of the

bearings will fail• La: millions of revolutions.

– a1: 1.0 for a 10% probability of failure– a2: a materials factor, which is typically 3.0 for steel bearings .– a3: lubrication factor, which typically is 1.0 for oil mist.– C: basic dynamic load rating from a table of available bearings.– Fe: the applied equivalent radial load, determined by bearing type.– γ: 3 for balls and 10/3 for rollers.– Precision life is about 90% of the L10 life

• For high speed applications, the operating speed must be taken into account when calculating the equivalent radial load Fe

• Kω: rotation factor = 1 for rotating inner ring and 2 for a rotating outer ring• Kr: radial load factor = 1 (almost always)• KA: axial load factor

F e = K ω K r F r + K A F A L a = a 1 a 2 a 3 (C/ F e ) γ


Load/Life: Ball Bearing Speed Limits

• Bearing speed is limited by its DN value, where D is the diameter in mm and N is the allowable rpm

• Shear power is the product of velocity (Rw) and force (RωµA/h) or (Rω)2µA/h• Centrifugal load is proportional to rω2

Single row, non-filling slot type Single row filling slot type Radial and angular -contact single row Angular-contact single and double row Single row angular-contact

Bearing Type

Molded nylon PRB pressed-steel Molded nylon PRB pressed-steel Molded nylon PRC composite CR (ring piloted) Molded nylon PRB pressed-steel Metallic (ring-piloted)

Type of Cage

250,000 300,000 - - 400,000 - - -

(selected)Grease

ABEC-1 ABEC-3 ABEC-7

200,000 250,000 200,000 200,000 300,000 200,000 200,000 250,000

(1) Grease

250,000 300,000 200,000 250,000 350,000 250,000 250,000 300,000

(2) Oil

(1) Grease filled to 30 to 50% of capacity. Type of grease must be carefully chosen to achieve the above speed values. Consult Fafnir for complete recommendations. (2) For oil bath lubrication, oil level should be maintained between 1/3 to 1/2 from the bottom of the lowest ball.

200,000 250,000 - - 300,000 - - -

(1) Grease

250,000 300,000 - - 400,000 - - -

(2) Oil

250,000 350,000 - - 600,000 - - -

oil Circulating

250,000 400,000 - - 750,000 - - -

Oil mist


Lubrication• Lubrication

– Separates the structural materials, and prevents chemical bonding.– Allows for viscous shear of a fluid thereby reducing material wear– Tribology is the study of lubrication and wear

• Most bearings require lubrication– Oil is the most common lubricant– Grease is just a soap that holds oil and releases it as it gets hot– LESS IS BETTER!

• Lubricants attract dirt, and thus cleanliness (via seals) it of extreme importance!• Some bearings are impregnated with solid lubricants which are released as they get hot

– E.g., porous bronze bushings• Some bearings are inherently lubricious and can function “dry”

– E.g., Teflon, Rulon, Delrin….• In general, these bearings do even better when also lubcricated• Measure coefficient of friction by an incline plane: µ=tanθ

– Sliding contact (e.g., plastic on metal) with modest friction (µ=0.1-0.05)– Rolling elements (e.g., ball bearings) with very low friction (µ=0.01-.005)


Lubrication: Surface Finish

• Surfaces with positive (left) and negative (right) skewness– Both surfaces have the same average roughness Ra value:

• Sliding contact bearings tend to average out surface finish errors and wear less when the skewness is negative

– Negative skewness holds the lubricant• As the system heats up, the lubricant flows and the friction drops

– The larger the positive skewness, the greater the wear-in period

R a = 1 L

y(x)dx0

L

ValleyPeak


Lubrication: Regions• There are three regions of lubrication as shown by the Stribeck curve

– Boundary, or sliding contact, where there is sliding contact between bodies, or the lubricant layer is very thin and the relative velocity is very low

• Coefficients of friction are on the order of 0.05-0.15 for bearing materials• Surface speeds are less than 3 m/min

– Mixed, where there is partial separation of the sliding surfaces by hydrodynamic action of the lubricant film, but there is still some solid rubbing

• Coefficients of friction are on the order of 0.02-0.1 for bearing materials– Full Film (hydrodynamic), where there is separation of the sliding surfaces by

hydrodynamic action of the lubricant film• Coefficients of friction are on the order of 0.001-0.005• Friction also depends on velocity and lubricant viscosity• Surface speeds are greater than 5 m/min

– Stiction: when static µ is greater than dynamic µ, cause stick-slip which causes position errors

• Static friction never equals dynamic friction.

Velocity

Fric

tion

Bou

ndar

y

Mix

ed Full film

(Hydrodynamic)

Osborne Reynolds (1842 - 1912) Claude Louis Marie Henri Navier (1785-1836)

George Gabriel Stokes (1819-1903)


Lubrication: Rolling Contact

• Hertz contact creates a contact footprint– Body is rolling with one center and multiple diameters– There has to be some slip

• Still, static friction approximately equals dynamic friction at low speeds, so stick slip is often minimized

• For heavily loaded tables, static friction is still significantly greater than dynamic friction

• Effect of stick-slip in a machine used to cut a hole using circular interpolation:– Errors will appear at velocity crossovers:

• 10-20 microns for sliding contact bearings• 5 microns for rolling element bearings


Lubrication: Hertz Effects• Rolling contacts have friction because the elements deform under load and

cause rolling across different effective diameters (slip)

– The rolling bearing pulls in lubricant, whose viscosity increases with pressure, to form an elastohydrodynamic lubrication layer between the ball and the race

• The EHD layer accommodates the differential slip, but generates heat via viscous shear

• The geometry of the rolling contact interface also plays a significant role


Error Motions• Bearings are not perfect, and when they move, errors in their motion can affect

system performance– Accuracy standards are known as (ABEC) classes as set by the Annular Bearing

Engineers Committee of the Anti-Friction Bearing Manufacturers Association, Inc. (AFBMA)

• ABEC 3 rotary motion ball bearings are common and low cost• ABEC 9 rotary ball bearings are used in the highest precision machines

• Remember Abbe errors!

• Manufacturers often provide repeatability data


Error Motions: Rotary Bearings

• Disc drives exist because of accurate repeatable rotary motion bearings– Radial, Axial, and Tilt error motions are of concern

• Precision Machine Designers measure error motions and use FFTs to determine what is causing the errors…

250200150100500

0

100

200

300

400

500

Frequency (Hz)

Dis

plac

emen

t (n

anom

eter

s) Displacement due to machine deformation

Average Error Motion Fundamental Error Motion

Total Error Motion

PC center

MRS center

Inner motion

MRS center error motion value PC center error motion value

Outer motion


Error Motions: Linear Bearings

• Semiconductor circuits are created on machines called steppers, whose linear motion accuracy must be 10x more accurate than the VLSI line width!

– Horizontal, vertical, roll, pitch, and yaw errors are of concern– Once again, the FFT is your friend!

0.00E+00

5.00E-08

1.00E-07

1.50E-07

2.00E-07

2.50E-07

1.00E-03 1.00E-02 1.00E-01

Wavelength (m)

Am

plitu

de (

m)

Overall bow in railSurface finish effects

Peaks likely due to rolling elements (ball and cam roller surface errors)


Error Motions: Horizontal Straightness

• Assume all bearings move horizontally


Error Motions: Vertical Straightness

• Assume all bearings move vertically


Error Motions: Roll

• Assume all bearings on each rail move vertically in an opposite direction


Error Motions: Pitch

• Assume front and rear bearing pairs move in opposite vertical directions


Error Motions: Yaw

• Assume front and rear bearing pairs move in opposite horizontal directions


Conclusions: Fundamentals Rule

• All the fundamental principles discussed in Topic 3 come to bear when designing bearing systems

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Types of Bearings

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