Cam-Follower : Contact Force Analysis

Before you analyse Contact-Force and Contact-Stress, you need to reconfigure the model so that is transfers the forces correctly between the cam and the follower.

See: How to configure the Model for Payload Analysis.


Background

When a load is applied to a rotating bearing, the stresses within its components are extremely difficult to a calculate. Even if the load is constant and it rotates at a constant speed, the stresses in each of the bearing's components [inner race, outer race, and balls or rollers] continually change.

Luckily, suppliers of cam-follower bearings list each cam-follower's capacity as a force rather than a stress.

The cam-follower's force capacity assumes it operates in a 'perfect' environment. Its capacity is, however, strongly influenced by lubrication and contamination. Its capacity also assumes a 90% reliability. To account for these variables, you must apply 'capacity modification factors' to find the cam-follower's actual capacity.

Cam Follower Bearings and 'Load Ratings'

Note: We will assume that, at the load ratings listed below, an internal bearing surface fails rather than the contact surface between the cam and cam-follower. We will consider the life of the contact between the cam and the cam-follower in the next topic.


Cam-followers have three load ratings with which you must compare your application's payload signature.

Basic Static Load Rating, Co
Basic Dynamic Load Rating, C
Fatigue Load Rating, Cu

Each 'rating' might be called the bearing's 'capacity' or 'limit'.  The ratings have units of force [N or lbs] and not stress [N/mm2, or PSI]


Static Load Rating, Co, [Units: N]

A load that is greater than the Static-Load-Rating, Co , deforms the cam-follower by approximately 1×10-4 (0.0001) of its mean diameter (see ISO 76:2006). The load may leave visible marks ['Brinell' marks] on the bearing raceways. Amazingly, this does not have a measurable effect on the life if, afterwards it is used under a much reduced load.

The Static Load Rating is important when the cam-follower:

is stationary, or
rotates very slowly (n < 10RPM), or
performs slow oscillating movements

or

receives impact loads [or 'short duration loads'].

Short duration loads may occur at start-up, emergency-stops, machine jams, or when the active process has a short duration, such as a cutting action. Impact loads may also occur because the machine jams and the product becomes 'rigid'. Of course, unintended machine breakages can give impact loads.

Static-Load Safety Factor, S0

staticsafetyfactor-czero

Czero-basicstaticloadrating

Pzero-basicstaticloadrating

staticbearing-safetyfactor


Static Load Rating and Contact Stress

If you load a cam-follower bearing to its Static Load Rating, then its most stressed component, usually at the contact between the inner race and one of the rolling elements, will have a Hertzian Contact Stress (see Hertz Contact-Stress) of:

~4000MPa for Roller Bearings
~4200MPa for Ball Bearings
~4600MPa for Self-aligning Ball Bearings

This is the limiting stress of the bearing Steel before it plastically deforms.

Dynamic Load Rating, C, [Units: N]

When a bearing is continuously loaded with a contact-force, P, equal to the Dynamic Load Rating, C, the bearing has a life, L10 , of one 'life'.

One life is one million [1 x 106 ] rotations. Life has a probability. The L10  'life' means that 10% of bearings will statistically fail within one million rotations, while 90% will statistically not fail.

The Dynamic Load Rating assumes the load and speed are constant. If the load and speed are not constant then we must calculate an equivalent load.

We can assume the Cam-Follower's payload and rotational speed do change during a machine cycle:

Nearly always, the Payload on the Cam-Follower changes as the cam makes one full rotation. See Payload Signatures
Nearly always, the Rotational Speed of the Cam-Follower changes, since the radius of the cam changes.

The equations are based on the seminal work of G. Lundberg and A. Palmgren's fatigue theory.


Basic Bearing Life [L10]

Bearing Life: Multiples of '1 million' rotations, L10

L10-lifeequation

eqtn-millions cycles

L10 = Life, in millions of rotations [-]

C = Basic Dynamic Load Rating [kN]

P = Basic Dynamic Bearing Load [kN]

p = 3 for bearing with 'balls' [-]

p = 10/3 for bearings with 'rollers' or 'needles' [-]


L10 above assumes that the load and the rotational speed are 'constant'. As stated above, these both vary during a typical machine cycle of a packaging machine. Thus we must find the equivalent load.

EquivalentLoad-Varylingload and speed

EquivalentSpeed

EquivalentLoad-Varylingload and speed-ALT

EquivalentSpeedALT

Bearing Life: Hours, [L10h]

eqtn-hours life

eqtn-lifeinhours

n = rotational speed  of the Cam-Follower [min-1].

[10/3 Needle Rollers in the bearing].

Bearing Life: Hours [Alternative equation]

l10althours1

l10althours2

l10althours3

L10h = Basic Rated Life, hours

fh = Life Factor

fn = Speed Factor

n = Rotational speed RPM

Bearing Life Adjustment Factors

The Dynamic Rating Life, L10, is adjusted by two parameters to compensate for the reality that a bearing does not operate under ideal conditions.

The ISO: 281 standard emerged in 1997, and then refined in 2007 to define the adjustment parameters. Before that time, different manufacturers applied their own factors and thus it was difficult to standardise the life of a cam-follower for a particular operating condition.

Lnm = a1 . aiso . L10

a1 =  Reliability Factor for reliabilities other than 90%

aiso = Integrated Life Modification Factor, accounting for new steel, lubrication, and contamination.

L10 = Basic Life Rating - see above.


Reliability Factor: a1

Reliability against multiples of bearing life.

Reliability against multiples of bearing life.

The occurrence of bearing damage and fatigue failure displays a random character. Thus, even ostensibly identical bearings that are manufactured from the same batch of material, having identical geometrical characteristics, subjected to identical operating conditions (load, speed, lubrication, etc.) will fail after different operating times. Thus, the life of a bearing is found from a statistical evaluation of a large number of bearings operating with similar controlled operating conditions.

The reliability chart, to the left, shows the variation that is characteristic of bearing lives.

Statistically:

1 × L10 : 10% of bearings fail [90% do not fail]
5 × L10 : 70% of bearings fail [30% do not fail]
8 × L10 : 90% of bearings fail [10% do not fail]

Reliability 'Look-up' table.

R %

L10

a1

90

L10

1.00

95

L5

0.64

96

L4

0.55

97

L3

0.47

98

L2

0.37

99

L1

0.25

99.5

L0.5

0.175

99.9

L0.1

0.093

99.95

L0.05

0.077

The Weibull distribution function is commonly used to predict the life of a population of bearings at any given reliability level.

The equation for the life adjustment factor, a1, for reliability is:

eqtn-lifein-A1adjustment

For example: if 90% reliability is substituted for R in the above equation, a1 = 1.

99% reliability: a1 = 0.2484.

Hypothetical reliability of 100% then a1 = 0.05

Life Modification Factor: aiso

The fatigue limit load Cu is defined in accordance with ISO 281 as the load at which the most heavily loaded rolling element reaches the fatigue limit.

Particles that may be in the lubricant, and consequently in the bearing races, can lead to plastic deformations of the raceway. Localised areas of high stress lead to a reduction in the fatigue life. This influence of contaminants in the lubrication gap on the rating life is taken into consideration by the life adjustment factor for contamination eC, see table.

The rating life is reduced by solid particles in the lubrication gap and is dependent on:

the type, size, hardness and quantity of particles
the relative lubricant film thickness
the bearing size.

Due to the complex nature of the interaction between these influencing factors, only an approximate guide value can be attained.

The values in the tables are valid for contamination by solid particles (factor eC). They do not take account of other contamination such as that caused by water or other fluids.

Under severe contamination (eC→ 0), the bearings may fail due to wear. In this case, the operating life is substantially less than the calculated life. There are various inter-dependent. variables that affect the Life Modification Factor.

aiso-systems approach

Life Modification Factor, aiso, is calculated with this equation.

aisofactor

Cu

[N]

Fatigue Limit Load of the bearing steel. This is less than the Basic Dynamic Load Rating. If not available directly in the bearing catalogue

Cu = C0/8.2 for roller and needle bearings

Cu = C0/23 for ball bearings

C0 Static Bearing Rating.

ec

-

Contamination Factor

κ

-

Viscosity Ratio

If κ ≥ 4 then κ =4

If κ ≤ 0.1 then κ = 0.1

P

[N]

Equivalent Load

Chart to find Rated Viscocity, V1

Chart to find Rated Viscocity, V1

Viscosity Ratio, κ

The Viscosity Ratio, κ, rates the quality of the lubricant film formation.

The oil film separate the raceway and rolling elements. The status of the lubricant film is expressed by the viscosity ratio, κ:

viscosityrationk1

We can use the charts to the left and below to find the two parameters:

ν1 : required viscosity' [also called 'reference viscosity'] [mm2/s]

ν : 'operating viscosity at operating temperature' [mm2/s]

Thus, κ, is a function of the bearing diameter, its speed, operating temperature and the selected oil viscosity grade.

Chart to find a recommended ISO grade.

Chart to find a recommended ISO grade.

You can use these equation to calculate ν1

kinematicviscosityeqtn

 

kinematicviscosityeqtn2

AverageRollerDiameter

d = inside diameter of Cam-Follower

D = outside diameter of Cam-Follower

When the operating temperature is known from experience, or from experiments, you can calculate the viscosity at the operating temperature. You need to know the viscosity of the Oil/Grease at 40C and 100C [ν40 and ν100] which is provided with the Oil Specification.


Around 90% of all rolling bearings are lubricated with grease. Grease lubrication presents far fewer sealing problems than oil lubrication and allows much simpler machine designs. With grease-lubricated rolling bearings we differentiate between lifetime lubrication and bearings which require re-lubrication. In general terms lifetime lubrication does not depend on the bearing but on the requirements of the particular application.

 

Example

Bearing Outside Diameter = 60mm ; Bearing Inside Diameter = 20mm ; Operating Speed = 500RPM, Operating Temperature = 70ºC
Mean Bearing Diameter, Dm = (60+20)/2 = 40mm. Thus, Rated or Reference Kinematic Viscosity ν1=40mm2/s

The VG150 oil has a viscosity of ν=40mm2/s at an operating temperature of 70ºC. Thus if we selected the VG 150 oil, then κ=40/40 = 1

However, the VG 320 oil has a viscosity ν=65mm2/s at an operating temperature of 70ºC. Then κ=65/40 = 1.625.

κ= 2 - 3 is about optimal. If it is too high, then friction is increased, heat is generated and the viscosity becomes less than optimal.


Note: A relationship between 'Film Thickness Ratio' and Viscosity Ratio is approximately : κ = λ1.3

Values of Contamination Factor ,ec


ec

Contamination Level

Dm ≤ 100 mm

Dm > 100mm

Extremely high cleanliness:

Particle size same or less than lubricant film thickness
Laboratory-level

1

1

High Cleanliness:

Oil filtered with extremely fine filter.
Filled & sealed greased for life bearings

0.8 to 0.6

0.9 to 0.8

Standard Cleanliness:

Oil filtered with fine filter.
Filled Shielded, greased bearings

0.6 to 0.5

0.8 to 0.6

Minimal / Slight Contamination:

Oil is slightly contaminated.

0.5 to 0.3

0.6 to 0.4

Normal, Typical Contamination:

Bearing contaminated by wear debris from other machine or packaging components

0.3 to 0.1

0.4 to 0.2

High Contamination:

Bearing Environment heavily contaminated
Bearing insufficiently sealed.

0.1 to 0

0.1 to 0

Very high or Extremely high contamination

0

0


Less Detailed Table for Contamination Factor.

Contamination Factor

ec

Very Clean: Debris size similar to lubricant film thickness

1

Clean: Bearings greased for life and sealed

0.8 - 0.9

Normal: Greased for Life and shielded.

0.5 - 0.8

Contaminated: Bearing without seals, particles from surroundings

0.2 - 0.5

Heavily Contaminated: Intruding fluids and particles, extreme conditions.

0.0 - 0.2

Contamination factor: ec

Surfaces internal to the Cam-Follower Bearing:

A solid particle that is caught in the lubricant may indent a raceway and the rolling elements.

If a contaminant particle moves to the inside of bearing, then the rollers [or balls], outer-race and inner-race are prone to 'dent' because of the small internal bearing clearances and the small rolling radii of the rollers [or balls]. The contamination may even prevent the rollers [or balls] rotating.

An indent leads to localised stress, which will decrease the life of the bearing.

The amount of lifetime reduction is a function of the size of the bearing, the lubricant film thickness [viscosity ratio, κ] and on the size, type and hardness of the particle contaminant.

If severe contamination occurs, [ec tends to zero] failure due to wear will probably occur and the lifetime will be much shorter than the L10 lifetime.

External Surface of Cam-Follower and Cam Surface.

If the cam-follower bearings are sealed or shielded, then surfaces of the cam-follower and cam will experience greater contamination than the internal surfaces of the bearings. However, because the radius of the cam and cam-follower are much larger, the effect of a 'dent' is less damaging than the same size 'dent' to a roller or bearing race. The contamination is less likely to prevent the outer race of the cam-follower roller rotating along the cam.

Dm is the Mean Diameter of the Cam-Follower. [See above for calculation].

The factor, eC, tends to be less for smaller bearing because the contamination particles are more likely to be trapped in a smaller bearing than a larger one because the clearance is less, and thus more likely to cause indentations to a bearing surface.

There are various techniques and tools to measure actual contamination.


Note:

Around 90% of all rolling bearings are lubricated with grease. Grease lubrication presents far fewer sealing problems than oil lubrication and allows much simpler machine designs. With grease-lubricated rolling bearings we differentiate between lifetime lubrication and bearings which require re-lubrication. In general terms lifetime lubrication does not depend on the bearing but on the requirements of the particular application.

 

Image ©SKF

Image ©SKF

Life Modification Factor: aiso

When the lubricant is contaminated with solid particles, permanent indentations in the raceway can be generated when these particles are rolled-over. At these indentations, local stress risers are generated, which will lead to reduced life of the rolling bearing. This life reduction due to contamination in the lubricant film is taken into account by the contamination factor eC .

Guide values for the contamination factor can be taken from from the table above, which shows typical levels of contamination for well lubricated bearing.

In the case of severe contamination (eC~0), failure may be caused by wear, and the life of the bearing can be far below a calculated modified rating life.

Area A : very high load and/or severe indentations.

The lubricating conditions in this domain can only marginally improve the expected fatigue life, so the potential improvement to the life depends on what dominates the relationship between the contamination level factor and the load level, Pu/P. To achieve a greater rating life, either the load must be reduced, or the cleanliness must be improved, or both.

Area B : high life modification factors, which is beneficial because a large life modification factor will convert a low basic rating life sufficiently to produce a large rating life.

In this part of the graph, small deviations from estimated load level, cleanliness factor and lubrication conditions will greatly affect the life modification factor. Small changes to lubricating conditions, slightly higher loading and larger indentation severity (for example, from mounting or transport damage) may result in a change from 50 to 5. This would result in a 90% loss of rating life. In cases where the rating life consists of a large life modification factor, aiso and a limited basic rating life L10, the impact of variations in operating conditions should be evaluated in a sensitivity analysis.

Area C : the life modification factor is less sensitive to changes.

Deviations from estimated load level, cleanliness factor and lubrication conditions (for example, from uncertainties in temperature) will not substantially affect the value of aISO, which means the resulting rating life is more robust.

In the load level domain, Area C has the ranges:

oCu ≤ P ≤ 0,5C for ball bearings
oCu ≤ P ≤ 0,33C for roller bearings

EP-additives in the lubricant

ISO 281 STATES 'Exceptionally low rotational speeds [dm x n <10000], the generated film may be less than adequate and unlikely to form elastohydrodynamic lubrication and unlikely to separate the rolling element and raceway contact.' Then you must use EP-additives to improve life.

In accordance with ISO 281, EP-additives can be taken into consideration in the following way:

For operating temperature lower than 80°C (175°F), EP/AW additives in the lubricant may extend bearing service life when κ < 1 and the factor for the contamination level ec > 0.2 and the resulting aiso < 3. Under those conditions, a value of κEP=1 can be applied, in place of the actual κ, in the calculation of aiso for a maximum advantage of up to aiso = 3.
At a viscosity ratio κ<1 and a contamination factor eC≥0.2, a value κ= 1 can be used in calculation in the case of lubricants with EP-additives that have proven effective. Under severe contamination (contamination factor eC ≤0.2), the effectiveness of the additives under these contamination conditions must be proven. The effectiveness of the EP-additives can be demonstrated in the actual application or on a rolling bearing test-rig.
EP/AW additives in the lubricant are used to improve the lubrication condition of the bearing in situations where small κ values are in use, e.g. when κ = 0.5. Furthermore, EP/AW additives are also used to prevent smearing between lightly loaded rollers and raceway, for example, when especially heavy rollers enter a loaded zone at a reduced speed.

Some modern EP/AW additives containing sulphur-phosphorus, which are most commonly used today, can reduce bearing life. Generally, SKF recommends testing chemical reactivity of EP/AW for operating temperatures above 80°C (175°F)

Grease Lubrication

For grease lubrication, the contamination factor, ec , can be determined by means of the diagrams below.

Each diagram represents a level of expected contamination. Using the diameter of the bearing, you can find ec :

Step 1: First consider which diagram to use. Consider the general level of contamination in the environment of the Cam-Follower's application.

For example, is the Cam-Follower running in an open cam-track near to the case magazine in a Case-Packer. In that case, I would use the 'Sever Contamination' diagram.

Step 2: Find the Viscosity Ratio, K, with the diagrams above.

Step 3: Use the plot that relates to the mean diameter, Dm of the Cam-Follower, or interpolate if different to the plots given below.

Step 4: Read the level of Contamination, ec.

Plot of Viscosity Ratio and Bearing Diameter to give Value of Contamination Level, ec,

Operating conditions

EC-HighCleanliness-Grease

High cleanliness

Very clean mounting with careful flushing
Very good sealing
Continuous re-lubrication or short re-lubrication intervals

Bearings with effective sealing

Greased for life

EC-StandardCleanliness-Grease

Standard cleanliness

Clean mounting with flushing

Good sealing
Re-lubrication in accordance with manufacturer's guidelines

Sealed bearings (for example, with sealing washers)

Greased for life
No damage to the seals

EC-SlightTypicalContamination-Grease

Slight to typical contamination

Clean mounting
Moderate sealing
Re-lubrication in accordance with manufacturer's guidelines
Shielded Bearings in area with likely particulate contamination

EC-HeavyContamination-Grease

Heavy contamination

Mounting under workshop conditions
Bearing and application not washed to appropriate standard
Poor sealing
Re-lubrication interval longer than manufacturer's guidelines

Grease Lubrication - ec for different bearing diameters and Viscocity Ratio

Severe contamination

Machine in contaminated environment
Inadequate sealing
Long re-lubrication intervals

When NOT to use the Integrated Life Modification Factor.

It does not apply to the following operating conditions:

Life exceeds 200000 hours, it is indicated "over 200,000".
Very large load imposes to the bearing (More than C0  or more than 50% of C).
Very light load imposed on the bearing.
Very high speed.
Water ingress into lubricant.
Abnormalities such as wear, corrosion, or electric erosion.
Large misalignment between inner and outer rings.
Very large and hard foreign debris intrudes to (Life is shorter than the calculation result).
Viscosity when driving is 10% or less of necessary dynamic viscosity.

Variable Speed and Variable Load

The Dynamic Bearing Life equation assumes that the bearing's speed and load do not change. There are equations you can use to compensate for a variable speed or a variable load and for when speed and load are variable.

Of course, a cam-follower's speed and payload continuously vary when they sit on a cam of varying radius and any sort of motion-law.  If, by chance the speed or load do not change, then the steps below will also apply.

We calculate an equivalent bearing load, Peq. It is a 'semi-graphical' procedure.

 

eqtn-load-varying

where:

Peq = Equivalent Cam-Follower Load

Fi = Cam-follower Payload at machine angle 'i'

ri = Cam-follower rotation at machine step 'i' [revs]

Rc f = Total number of Cam-Follower rotations [revs]

Note: We assume the cam follower has rollers or needle rollers. If it has 'balls' replace 3/10 with 1/3 in all places

eqtn-rollercirc

 

Rcf

Total number of Cam-Follower rotations

Rfollower

Cam-Follower Radius

Cfollower

Circumference of the Cam-Follower

Ccam

'Circumferential Length' of the Cam

Rcam

Cam Radius at each machine step 'i'

eqtn-camcircumference

eqtn-rollerrotations

eqtnloadlifecf

eqtn-hours life-peq

Note: We assume the cam follower is a roller-bearing. If it is a ball-bearing, replace 3/10 with 1/3 in all places

Oscillating Cam-Follower

When the outer race of the cam-follower bearing does not make a complete rotation, but oscillates back and forth, a lower equivalent radial load can be calculated using the formula below:

Equation-OscillatingLoad

where:

Pe = equivalent dynamic radial load
Po = actual oscillating radial load
β = angle of oscillation, in degrees
p = 10/3 Roller Bearings
p = 3.0 Ball Bearings

OscillatingFollowers

Active cam-follower rotations

Active cam-follower rotations are those when the payload between the cam and the cam-follower is positive.

The number of active rotations is dependent on the type of Cam.

Conjugate Cam: active rotations of one cam-follower are when the acceleration is 'positive' during the Rise of the Cam-Follower, and 'negative' acceleration during its 'return', and opposite for the other cam-follower.
Groove Cam: active rotations are throughout the complete machine cycle. However, the roller must reverse its direction when it changes cam-flanks.
Forced Closed Cam: active rotations are throughout a complete machine cycle.

Fatigue Load Rating, Cu, [Units: N]

The fatigue load limit Cu for a bearing is defined as the load level below which metal fatigue will not occur. For this to be valid, the lubricant film must fully separate the rolling elements from the raceways and no indentations, from contaminants or from damage related to handling, may exist on the rolling surfaces - see Lubrication and Surface Finish

For general high-quality materials and bearings with high manufacturing quality, the fatigue stress limit is reached at a contact stress of approximately 1.5GPa between the raceway and rolling elements.

The term fatigue load limit Cu, is defined, in ISO 281:2007, as "bearing load under which the fatigue stress limit is just reached in the most heavily loaded raceway contact" and is affected by factors such as the bearing type, size, and material.

If a catalogue does not list the Fatigue Load Limit, then you should use these approximations, when the bearing diameters is < 150mm:

LifeFatigueLimitBall

LifeFatigueLimitRoller

System Life

Conjugate cams have two or more cam-followers, and often your machine has many cams. All of the cam-followers in a machine are then considered to be a 'system'. For machine design reliability purposes, it is important to know the system life of your machine.  This evaluation process considers the importance of combining the L10 lives of all the bearings so that you can answer the question, 'How long will the machine perform with 90% reliability?'

In simpler terms, the system L10 reliability will be less than the lowest individual L10 rating life. The following formula is used to calculate the 'System Rating Life' of the Cam-Followers.

SystemLife

where:

L10sys = rating life for the system of bearings
L1, L2, Ln = rating life for the individual bearings in the system
m = 9/8   Roller Bearing
m = 10/9 Ball Roller
m = ~1.1

Guideline values for the Static-Load Safety Factor for continuous and/or occasional loads:

The acceptable Static-Load Safety Factor is a function of the certainty of its loading, and whether it has rollers, balls or it is a spherical bearing, and whether a permanent deformation is acceptable.

 

Acceptable Values of So

 

 

Continuous motion

Infrequent motion

 

Permanent deformation acceptance

Permanent deformation acceptance

Certainty of load level

Yes

Some

No

Yes

High certainty:

For example, gravity

0.5 : Ball Elements

1.0 : Roller Elements

1.0 : Ball Elements

1.5 : Roller Elements

2.0 : Ball Elements

3.0 : Roller Elements

0.4 : Ball Elements

0.8 Roller Elements

Low certainty:

For example, vibrational

1.5 :  Ball Elements

2.5 : Roller Elements

1.5 : Ball Elements

3.0 : Roller Elements

2.0 : Ball Elements

4.0 : Roller Elements

1.5 : Ball Elements

2.0 : Roller Elements

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