﻿ General Design Information > Cam Mechanisms > Design Checks with the Cam-Data FB > Force Analysis: Cam-Follower: Contact-Force & Life

# Force Analysis: Cam-Follower: Contact-Force & Life

## Cam-Follower : Contact Force Analysis

 Before you analyze Contact-Force and Contact-Stress, you need to configure the model to transfer power to and from the cam and cam-follower.

### Background

When a load is applied to a ball or roller bearing, the stresses within its elements (inner race, outer race, balls or rollers) are difficult and complex to calculate, even if the load and speed are constant.

Luckily, suppliers of cam-follower bearings have done the hard work for engineers. They publish the capacity of a cam-follower as a force/load rather than as a stress. The load, as a force, is much easier to calculate.

However, the capacity assumes it operates in a perfect environment. Its capacity is actually strongly influenced by ambient lubrication, contamination, and temperature, as well as the required reliability.

To account for these, we must use capacity modification factors .

ISO 281-2007 is a standard to find the modification factors for a bearing, design, and application.

## Cam-Roller Bearings and 'Load Ratings'

Note: We will assume that the ratings that are listed below apply to the internal surfaces of the bearing rather than the contact-surface between the cam and cam-follower. We will consider the contact between the cam and the cam-follower in the next topic, when we review the cam's capacity.

Note: Here, rating, capacity, load-limit are equivalent terms.  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. The load may leave visible marks (Brinell marks) on the bearing's raceways. Amazingly, this does not have a measurable effect on the life, if,  it is used under a much reduced load after.

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

is stationary, or

rotates very slowly (n < 10RPM), or

has slow oscillating movements

or

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 rapidly almost rigid. Of course, unintended machine breakages may lead to impact loads.

Static Load Rating and Contact Stress

According to the ISO 281 standard, a static load to a cam-follower bearing at 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 stress at which the bearing deforms permanently by 0.0001 of its mean diameter.

### 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 × 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 survive.

If the load and speed are not constant then we must calculate an equivalent load.

Nearly always, the load on the Cam-Follower continually changes as the cam makes one full rotation.

Nearly always, the rotational speed of the Cam-Follower continually changes, since the radius of the cam continually changes.

Basic Bearing Life (L10)

 L10 = Life, in millions of rotations (-) C = Basic Dynamic Load Rating (kN) P = Basic Dynamic Bearing Load (kN) p = 3 : ball bearing (-) p = 10/3 : roller (or needles) bearings (-) As stated above, the speed and load both vary within a machine cycle of a packaging machine, even if the machine is operating at a constant speed. We must find the load and speed that are equivalent to varying load and speed. Bearing Life: Hours, (L10h) n = rotational speed  of the Cam-Follower (min-1) 10/3 : Needle Rollers in the bearing. 3       : Ball Bearings Bearing Life: Hours - Alternative formulation  (L10h) L10h = Basic Rated Life, hours fh = Life Factor fn = Speed Factor n = Rotational speed RPM

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

The ISO 281 standard makes sure that different manufacturers do not apply their own factors.

Lnm = a1 . aiso . L10

a1 =  Reliability Factor for reliabilities other than 90%

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

L10 = Basic Life Rating - see above.

Reliability Factor: a1

Reliability against multiples of bearing life.

The occurrence of bearing damage and fatigue failure has a random character. Thus, even seemingly identical bearings, from the same batch of steel, with 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 reliability of bearing.

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:

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 Cu is defined 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, rather than fatigue. In this case, the bearing's life is much less than the calculated life. There are various inter-dependent factors and variable that affect the Life Modification Factor.

Life Modification Factor, aiso:

 Cu (N) Fatigue Limit Load Cu Cu = C0/8.2 for roller and needle bearings (dm<100mm) = C0/22 for ball bearings (dm<100mm) 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 Reference Viscocity, V1

Viscosity Ratio, κ

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

The lubricant-film separates the raceway and rolling-elements. It's status is expressed as:

ν1 : Reference viscosity (mm2/s) - function of the Bearing's Mean Diameter and its Rotating Speed.

ν : Viscosity at operating temperature (mm2/s) - function of oil viscosity grade and temperature.

We can use the charts to the left and below to find the two parameters. We can also use various equations.

Chart to find a recommended ISO grade.

You can use these equation to calculate ν1

 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 / Grease 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, 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 VG 150 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 localized 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, is less for smaller bearing.

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 tools to measure actual contamination.

It is possible that eC < 0 when the Bearings < 50mm and there is Severe 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.

Note: Grease Lifetime - a function of Temperature and n × dm (RPM × mean bearing diameter)

SKF Graphs are:

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

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. It can also be determined using

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 Severe 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.

Note: for bearing with a Mean-Diameter less than 50, that ec becomes undefined. If ISO 281 is used, then ec actually becomes less than 0!. should be very low.

 Plots of Viscosity Ratio, K and Bearing Diameter to give Value of Contamination Level, ec, Operating conditions 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 Standard Cleanliness •Clean mounting with flushing•Good sealing•Re-lubrication in accordance with manufacturer's guidelinesSealed bearings (for example, with sealing washers): •Greased for life•No damage to the seals Slight to typical contamination •Clean mounting•Moderate sealing•Re-lubrication in accordance with manufacturer's guidelines•Shielded Bearings in area with likely particulate contamination Heavy contamination •Mounting under workshop conditions•Bearing and application not washed to appropriate standard•Poor sealing•Re-lubrication interval longer than manufacturer's guidelines 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.

Grease Life Estimates

a function of Temperature and n × dm (RPM × mean bearing diameter)

SKF Graphs

#### 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, a variable load, and for when speed and load are variable.

Of course, the speed and load on a cam-follower 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.

where:

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

 Rcf Total number of Cam-Follower rotations for each Cam rotation. Rfollower Radius of the Cam-Follower Radius Cfollower Circumference of the Cam-Follower Ccam Circumferential Length of the Cam Rcam Cam Radius at each machine step 'i'

#### 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: 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

#### 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 after it changes cam-flanks. It is not loaded as the roller traverses backlash between cam-flanks.•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 catalog does not list the Fatigue Load Limit, then you should use these approximations, when the mean bearing diameter is < 100mm:

#### 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. 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

#### 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, impacts, shocks,  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