Friction in Machine Elements

Introduction

Friction is the tangential resistance that develops when two surfaces have relative motion or a tendency to move relative to one another. It is indispensable in some machine elements—belts transmit power, clutches engage shafts, and brakes absorb energy—yet harmful in others because it causes energy loss, heat generation, wear, noise, and possible seizure.

In mechanical engineering, friction is not eliminated completely; it is controlled. Design therefore requires a realistic friction model, suitable materials, adequate lubrication, acceptable temperature rise, and reliable safety margins. This chapter develops models for sliding and rolling contacts, screws, pivots, collars, bearings, belts and ropes, followed by the analysis of clutches and brakes.

Learning Objectives

After studying this chapter, you should be able to:

• distinguish static, kinetic, sliding, rolling, boundary and fluid friction;
• calculate friction forces, torque, power loss and efficiency in common machine elements;
• analyse square-thread screws, pivots, collars, journal bearings, belts and ropes;
• determine torque capacity and heat dissipation requirements for clutches and brakes;
• select materials, lubrication and safety factors using engineering judgment;
• recognize the limitations of idealized Coulomb-friction models.

1. Fundamentals of Friction

1.1 Physical origin and terminology

At a microscopic level, real surfaces touch at asperities rather than over their apparent geometric area. Friction arises from adhesion at these junctions, deformation of asperities, ploughing by hard particles, and mechanical interlocking. Consequently, the coefficient of friction depends on material pair, surface finish, lubricant, temperature, speed, contamination and contact pressure.

The principal terms are:

• Normal reaction, N: force perpendicular to the contacting surfaces.
• Friction force, F: tangential resistance to relative motion or impending motion.
• Coefficient of friction, μ: dimensionless ratio characterizing friction under specified conditions.
• Angle of friction, φ: angle satisfying tan φ = μ.
• Static friction: friction before gross relative motion begins.
• Kinetic friction: friction during sliding.
• Limiting friction: maximum static friction immediately before motion.

For dry sliding under the ideal Coulomb model,

Here μ_s and μ_k are static and kinetic coefficients. The model assumes a rigid-body normal load, approximately constant coefficient, and a contact condition for which adhesion and pressure effects do not change substantially. It is not reliable for lubricated contacts, elastohydrodynamic contacts, very high temperatures or strongly speed-dependent friction.

The frictional power loss during sliding is

where P_f is power in watts, F is friction force in newtons, and v is sliding speed in metres per second. For a rotating contact,

where T_f is friction torque in newton-metres and ω is angular speed in radians per second.

1.2 Static equilibrium and friction angle

For a body on an inclined plane at angle θ, impending downward motion occurs when the downslope component of weight equals limiting friction:

Therefore,

The angle at impending sliding is the angle of repose. This result applies to a uniform contact with a single effective coefficient and neglects vibration and dynamic effects.

1.3 Sliding and rolling friction

Sliding friction generally produces more heat and wear than rolling friction. Rolling resistance is caused mainly by elastic hysteresis, local deformation, micro-slip and surface irregularity. A simple rolling-resistance model represents the resisting force as

where b is the rolling-resistance coefficient with units of length, and N is normal load. Alternatively, a dimensionless rolling coefficient C_r may be used:

The chosen definition must be stated because manufacturers and texts use both conventions. Rolling bearings also experience lubricant drag, seal friction and raceway losses, so their total torque cannot be predicted from pure rolling resistance alone.

2. Friction in Screws and Screw Mechanisms

Power screws convert rotary motion into linear motion and include screw jacks, presses, vises and lifting mechanisms. Their threads behave like an inclined plane wrapped around a cylinder.

For a square-thread screw, let W be the axial load, d_m the mean thread diameter, l the lead, and α the helix angle. For a single-start screw, lead equals pitch. The helix angle is

Let μ be the thread coefficient of friction and φ the friction angle, where tan φ = μ. The torque required to raise the load is

The torque required to lower the load, when controlled lowering occurs, is

These equations assume square threads, uniform friction, negligible collar friction and a steady axial load. For V-threads, the normal force is increased by the thread half-angle. If β is the included thread angle, the effective coefficient is approximately

and μ' replaces μ in the friction-angle relation.

A screw is self-locking when the load cannot rotate the screw backward:

The efficiency of a square-thread screw while raising a load is

If collar friction is present, its torque must be added to thread torque. For a uniform-wear collar with inner and outer radii r_i and r_o,

where μ_c is the collar coefficient. The total raising torque is T_R + T_c.

Worked Example 1: Screw-jack torque and efficiency

A square-thread screw lifts W = 20 kN. Its mean diameter is d_m = 40 mm, lead is l = 8 mm, and thread coefficient is μ = 0.10. Neglect collar friction. Determine raising torque, self-locking condition and efficiency.

First,

tan α = 0.008/(π × 0.040) = 0.06366, so α ≈ 3.64°. Also φ = tan^-1(0.10) ≈ 5.71°.

Thus,

T_R = 20,000(0.040/2)tan(5.71° + 3.64°) ≈ 65.8 N·m.

Since φ > α, the screw is self-locking. The efficiency is

η_s = tan 3.64°/tan 9.35° ≈ 0.345,

or approximately 34.5%. The low efficiency is a deliberate trade-off for self-locking.

3. Pivots, Collars and Journal Bearings

3.1 Flat pivot and collar friction

A pivot supports an axial load through a plane surface. The friction torque depends on the pressure distribution. Two limiting assumptions are commonly used.

For a uniform-pressure collar with inner radius r_i, outer radius r_o and total axial load W,

For a uniform-wear collar, pressure varies inversely with radius and

For a solid circular pivot of radius R, the uniform-pressure torque is

These formulas assume dry or boundary friction with a constant effective coefficient. Uniform pressure is more appropriate for a new or lightly worn surface; uniform wear is often used after running-in.

3.2 Journal bearings

A journal bearing supports a rotating shaft in a sleeve. Its friction is influenced by viscosity, radial clearance, speed, load and bearing geometry. In full-film hydrodynamic lubrication, the shaft is separated from the bearing by a pressurized lubricant film generated by relative motion. Coulomb friction alone is then inadequate.

The bearing pressure is often estimated by

where L is bearing length and d is journal diameter. The bearing surface speed is

when d is in metres and n is rotational speed in revolutions per minute.

A useful screening parameter is the Sommerfeld number,

where μ_o is lubricant dynamic viscosity, n is rotational speed in revolutions per second, r is journal radius and c is radial clearance. Detailed bearing performance requires solution of the Reynolds equation and consultation of bearing charts or computational models. The simple pressure equation must not be interpreted as a complete bearing design method.

3.3 Lubrication regimes

The Stribeck concept relates friction to viscosity, speed and load. As a contact moves from boundary lubrication to mixed lubrication and then full-film lubrication, friction may initially decrease because asperity contact is reduced. At very high viscosity or speed, viscous drag increases.

• Boundary lubrication: surface films carry much of the load; additives and material chemistry are crucial.
• Mixed lubrication: both asperity contact and fluid-film pressure carry load.
• Hydrodynamic lubrication: a full fluid film separates surfaces.
• Elastohydrodynamic lubrication: elastic deformation and pressure-dependent viscosity are important, especially in rolling contacts.

Lubricant selection must consider viscosity, temperature, oxidation stability, compatibility with seals, contamination and environmental requirements.

4. Belts and Ropes

Belts and ropes transmit power through friction between the flexible element and a pulley. Let T₁ be tight-side tension, T₂ slack-side tension, μ coefficient of friction, and θ angle of contact in radians. For a flat belt,

This capstan equation assumes a flexible belt, uniform friction, negligible bending stiffness, no centrifugal tension and impending slip. The transmitted torque on a pulley of radius r is

and power is

where v is belt speed.

For a V-belt with groove angle 2β, the wedging effect increases friction. Under the idealized model,

where β is the groove half-angle. This relation neglects belt stiffness, groove deformation and centrifugal effects. V-belts must be selected using manufacturer ratings rather than this equation alone.

At high speed, centrifugal tension T_c reduces the tension available for frictional transmission. If T is total tight-side tension and T₀ is the effective tension excluding centrifugal tension, then approximately

The maximum power condition for a simple belt model occurs near T_c = T_max/3, where T_max is allowable total tension. Real belt drives also require allowance for start-up loads, fatigue, misalignment, temperature and wear.

Worked Example 2: Flat-belt power transmission

A flat belt runs at v = 10 m/s, with coefficient μ = 0.30 and wrap angle θ = π rad. If the slack-side tension is T₂ = 500 N, determine the tight-side tension and power transmitted at impending slip.

T₁ = 500e^0.30π ≈ 1,283 N. Therefore,

P = (1,283 − 500)10 ≈ 7,830 W,

or approximately 7.83 kW. This is an ideal slip-limit value; the rated operating power should be lower because of service and safety factors.

5. Clutches

A clutch connects a driving shaft to a driven shaft and transmits torque through friction. A friction clutch must provide sufficient torque without excessive heat, judder or wear.

For a single-disc clutch with two friction surfaces, axial force W, coefficient μ, outer radius r_o and inner radius r_i, the torque is derived by integrating elemental friction torque over the annulus.

Under the uniform-pressure assumption,

For n friction surfaces, multiply the result by n. Under the uniform-wear assumption,

Uniform wear is widely used for a run-in clutch; uniform pressure is useful for a new clutch or engagement analysis. The mean radius under uniform wear is (r_o + r_i)/2, while under uniform pressure it is

The clutch must also dissipate engagement energy. If the relative angular speed before engagement is ω_r and the equivalent rotational inertia referred to the relative motion is J_eq, the ideal kinetic energy to be dissipated is

Repeated engagements require a thermal analysis because the average heat rate is approximately energy per engagement multiplied by engagement frequency. Thermal capacity, convection, conduction, allowable lining temperature and fade characteristics must be considered.

6. Brakes

A brake absorbs mechanical energy and converts it primarily into heat. Its design objectives are adequate stopping torque, controllable actuation, acceptable temperature, stable friction and fail-safe behavior.

For a disc brake with friction radius R_m, clamp force W, coefficient μ and n active friction interfaces,

The effective radius is selected according to the pressure model. For a narrow annular pad under uniform wear, it is approximately the mean of inner and outer radii. For a broad pad, integration over the actual contact area is preferable.

For a simple drum brake with shoe normal reaction N and effective friction radius r,

The relation between applied force and drum reaction depends on whether the shoe is leading or trailing. A leading shoe is self-energizing: friction assists actuation. This reduces required input force but can cause grabbing and instability if poorly controlled.

For a vehicle of mass m stopped from speed v₀ on a level surface, neglecting aerodynamic drag,

If constant deceleration a is assumed, stopping distance is

These equations describe ideal energy and kinematics. Actual braking includes tire-road adhesion, brake distribution, slope, aerodynamic drag, rolling resistance and thermal fade.

7. Heat, Wear and Safety

Frictional heat generation is approximately the product of friction force and sliding speed. For a contact with torque T_f rotating at speed ω, heat generation is T_fω. Temperature rise depends on heat capacity, conduction paths, convection, radiation, contact duration and thermal cycling.

Wear may be classified as adhesive, abrasive, fatigue, corrosive or erosive. A preliminary estimate for sliding wear uses Archard’s relation:

where V is wear volume, k is a dimensionless wear coefficient, W is normal load, L is sliding distance and H is hardness of the softer material. Archard’s law is empirical and should be calibrated for the relevant materials, lubricant and environment; it is not a universal prediction of service life.

Good practice includes:

1. define the duty cycle, loads, speed and environment;
2. identify whether sliding, rolling or mixed contact dominates;
3. select a friction model appropriate to lubrication and geometry;
4. calculate torque, power loss, heat and wear;
5. apply service factors for shock, uncertainty and degradation;
6. provide inspection, adjustment, guarding and emergency stopping;
7. validate the design through testing and condition monitoring.

Safety is especially important in brakes, hoists, presses and screw lifting devices. A self-locking screw may prevent back-driving, but it does not replace a brake or positive load-retention device when failure could injure personnel. Guards must prevent contact with belts and rotating parts, and brake systems should be designed for loss of power, lining wear and thermal fade.

8. Illustrative Case: Overheating of a Conveyor Drum Brake

Background

An industrial conveyor uses a drum brake to stop a loaded belt during emergency shutdown. Operators report smoke and reduced braking effectiveness after repeated stops.

Problem

The original design was checked only for static braking torque. It did not assess the energy absorbed per stop, stop frequency, drum temperature or friction variation with temperature.

Analysis

The engineering team measured stopping speed, rotating inertia, brake torque and ambient conditions. The energy per stop was estimated from E = Jω²/2. The average heat load was then calculated from energy per stop and the number of stops per hour. Infrared measurements showed that the lining temperature exceeded the supplier’s continuous operating limit. The coefficient of friction decreased at high temperature, explaining the observed fade.

Approach and solution

The brake was redesigned with a larger heat-dissipating drum, a lining rated for the measured temperature range and improved airflow. The control system limited repeated emergency resets and provided a temperature warning. A redundant holding brake was retained so that the dynamic stopping brake was not relied upon to hold the stationary load. The design was verified by a repeated-stop test at maximum rated load.

Outcome

Brake temperature remained below the specified limit, stopping torque became more consistent, and lining inspection intervals were established. The case demonstrates that torque capacity alone is insufficient: friction components must be designed for energy, temperature, wear and failure consequences.

9. Contemporary Engineering Practice

Modern friction design increasingly combines analytical models with experimental and numerical methods. Finite-element analysis can estimate contact pressure and thermal gradients, while tribometers measure friction and wear for candidate material pairs. Sensors can monitor vibration, temperature, torque and acoustic emissions to detect lubrication loss or bearing damage.

Data-driven and AI-assisted systems can identify abnormal friction behavior from condition-monitoring data, but they do not replace physical models, calibration or safety certification. Training data may not represent rare overloads or contamination events. Engineering decisions must therefore retain traceable assumptions, verified limits and human responsibility.

Environmentally responsible design favors long-life components, low-toxicity lubricants, remanufacturing and reduced energy loss. However, a lubricant or lining should not be selected solely for environmental labeling; compatibility, fire safety, disposal requirements and actual life-cycle performance must also be assessed.

Chapter Summary

Friction is both a source of useful force transmission and a cause of energy loss, heat and wear. Coulomb friction provides a first model for dry sliding, but lubricated bearings and rolling contacts require more appropriate tribological descriptions. Screw mechanisms are analysed using the friction angle and helix angle; self-locking occurs when the friction angle exceeds the helix angle. Pivot and collar torque depends on pressure assumptions. Belts and ropes follow the capstan relation, while centrifugal tension and service conditions limit practical capacity. Clutches transmit torque through friction interfaces and must also absorb engagement energy. Brakes convert kinetic energy into heat and must be designed for torque, thermal capacity, fade, wear and safe failure behavior. Sound design combines equations with material selection, lubrication, testing, monitoring and appropriate safety factors.

Key Terms

Friction: Tangential resistance between contacting bodies.

Static friction: Friction acting before gross relative motion begins.

Kinetic friction: Friction during sliding motion.

Coefficient of friction: Ratio characterizing friction relative to normal load.

Angle of friction: Angle whose tangent equals the coefficient of friction.

Rolling resistance: Resistance associated with rolling deformation and micro-slip.

Self-locking screw: Screw that cannot be back-driven by its axial load.

Helix angle: Inclination of a screw thread’s mean helix to a plane normal to its axis.

Collar: Annular surface supporting an axial load.

Journal bearing: Bearing in which a cylindrical journal rotates in a sleeve.

Boundary lubrication: Lubrication regime with substantial asperity interaction.

Hydrodynamic lubrication: Full-film lubrication produced by relative motion.

Capstan equation: Relation between tight- and slack-side belt tensions.

Clutch: Device that connects and transmits torque between shafts.

Brake: Device that absorbs energy and resists motion.

Uniform pressure: Assumption of constant contact pressure over an area.

Uniform wear: Assumption that wear rate is constant across the contact radius.

Friction fade: Reduction of friction performance, commonly due to overheating.

Archard’s law: Empirical relation for estimating sliding wear volume.

Tribology: Study of friction, lubrication and wear.

Review Questions

Short Answer Questions

1. Define static friction, kinetic friction and angle of friction.
2. State the assumptions behind the capstan equation.
3. What condition makes a square-thread screw self-locking?
4. Distinguish boundary, mixed and hydrodynamic lubrication.
5. Why must a brake be checked for thermal capacity as well as torque?

Long Answer Questions

1. Derive the torque equation for a uniform-wear collar and explain its assumptions.
2. Explain the operation of a square-thread screw and derive its raising-torque and efficiency equations.
3. Discuss belt power transmission, including tension ratio, centrifugal tension and practical limitations.
4. Compare uniform-pressure and uniform-wear theories for friction clutches.
5. Explain the sources of friction, heat and wear in journal bearings and describe suitable design controls.

Analytical / Application Questions

1. A screw jack has a 30 kN load, mean diameter 50 mm, lead 10 mm and thread coefficient 0.12. Determine the raising torque and state whether it is self-locking, neglecting collar friction.
2. A belt has coefficient of friction 0.25, wrap angle 210° and slack tension 400 N. Determine the ideal tight-side tension and power transmitted at 8 m/s.
3. A clutch must absorb repeated high-speed engagements. Propose a design-analysis procedure that checks torque capacity, engagement energy, temperature, wear, lining selection and safety redundancy.

Multiple Choice Questions with Answers

Question 1

For dry sliding at impending motion, the friction force is equal to:

A. μ_k/N

B. μ_s N

C. N/μ_s

D. μ_s + N

Correct Answer: B. μ_s N

Question 2

A square-thread screw is self-locking when:

A. α > φ

B. α = 90°

C. φ > α

D. μ = 0

Correct Answer: C. φ > α

Question 3

A belt runs at 10 m/s with tight tension 1200 N and slack tension 400 N. Its ideal transmitted power is:

A. 0.8 kW

B. 4 kW

C. 8 kW

D. 16 kW

Correct Answer: C. 8 kW

Question 4

The capstan equation primarily relates:

A. Bearing pressure and viscosity

B. Tight-side and slack-side belt tensions

C. Brake energy and stopping distance

D. Wear volume and hardness

Correct Answer: B. Tight-side and slack-side belt tensions

Question 5

For a uniform-wear collar, the mean friction radius is:

A. r_o − r_i

B. r_o r_i

C. (r_o + r_i)/2

D. (r_o^2 + r_i^2)/2

Correct Answer: C. (r_o + r_i)/2

Question 6

Hydrodynamic lubrication is characterized by:

A. Complete absence of lubricant

B. A full fluid film separating the surfaces

C. Only dry asperity contact

D. A coefficient of friction exactly equal to one

Correct Answer: B. A full fluid film separating the surfaces

Question 7

If friction torque is 20 N·m and angular speed is 100 rad/s, frictional power loss is:

A. 0.2 W

B. 2 W

C. 200 W

D. 2,000 W

Correct Answer: D. 2,000 W

Question 8

For a friction clutch, increasing the number of active friction surfaces generally:

A. Reduces torque capacity proportionally

B. Increases torque capacity proportionally in the ideal model

C. Eliminates heat generation

D. Makes the coefficient of friction zero

Correct Answer: B. Increases torque capacity proportionally in the ideal model

Question 9

Archard’s wear relation includes which quantity as a denominator?

A. Sliding distance

B. Normal load

C. Hardness

D. Wear volume

Correct Answer: C. Hardness

Question 10

The kinetic energy that a brake must absorb from a rotating inertia is proportional to:

A. Angular speed

B. The square of angular speed

C. The inverse of angular speed

D. The cube of angular speed

Correct Answer: B. The square of angular speed