Friction, Contact and Flexible Elements

Introduction

Friction and contact forces are present wherever bodies touch or transmit motion through surfaces, ropes, belts, cables, bearings, brakes, clutches, wedges, and screws. Unlike ideal smooth contact, a frictional contact can support a tangential force, but only up to a limiting value. This makes friction both useful—for example, in braking and power transmission—and potentially harmful through wear, heat generation, energy loss, and uncertain motion.

This chapter develops engineering models for dry friction and flexible elements. The central model is Coulomb friction, supplemented by specialized relations for wedges, screws, journal bearings, rolling resistance, belts, brakes, and clutches. These models are idealizations; sound engineering judgment requires recognizing when their assumptions are not adequate.

Learning Objectives

After studying this chapter, you should be able to:

1. Construct free-body diagrams involving normal reaction and friction.
2. Distinguish static friction, impending motion, and kinetic friction.
3. Analyze wedges and power screws using friction angles.
4. Determine torque, power loss, and resisting forces in bearings and rolling contacts.
5. Analyze belt tensions, belt drives, brakes, and clutches.
6. Apply equilibrium and compatibility principles to cables and pulleys.
7. Evaluate the assumptions and limitations of ideal friction and contact models.

1. Coulomb Friction and Contact

For two dry, contacting surfaces, the normal reaction acts perpendicular to the contact plane. The friction force acts tangentially and opposes relative or impending relative motion.

For static contact,

where F is the friction-force magnitude in newtons, N is the normal reaction in newtons, and μ_s is the coefficient of static friction. The inequality is important: static friction is not automatically equal to μ_sN. It takes whatever value is needed for equilibrium until the limiting value is reached.

At impending sliding,

For sliding motion,

where μ_k is the coefficient of kinetic friction. Usually μ_k < μ_s, although actual behavior depends on materials, lubrication, speed, temperature, and surface condition.

The friction angle φ is defined by

For impending motion, μ = μ_s and φ = φ_s. For sliding motion, μ = μ_k and φ = φ_k. The resultant contact force R combines N and F:

At impending motion, F = μ_sN, so R makes angle φ_s with the normal.

Contact modeling procedure

A reliable solution normally follows these steps:

1. Isolate the body or bodies of interest.
2. Draw all applied loads, weights, support reactions, cable tensions, and contact forces.
3. Assume a possible direction of impending motion.
4. Draw friction opposite that motion.
5. Apply ∑F_x = 0, ∑F_y = 0, and ∑M = 0 for a body in equilibrium.
6. Check whether the calculated friction satisfies F ≤ μ_sN. If not, the assumed sticking condition is impossible.
7. For actual sliding, use F = μ_kN and verify the direction.

A negative friction value is not an error; it indicates that the actual direction is opposite to the assumed direction.

Inclined-plane example

A 200 N block rests on a 30° incline with μ_s = 0.40. The component of weight parallel to the plane is

The normal reaction is

The maximum available static friction is

Because 100 N > 69.3 N, the block cannot remain at rest and slides downward. If μ_k = 0.30, the kinetic friction is 52.0 N upward along the plane, giving a net downward force of 48.0 N.

2. Wedges

A wedge converts a relatively small applied force into a large separating force. Wedges are used in lifting, positioning, clamping, and adjustment mechanisms. Friction can make a wedge self-locking, but it also increases the force required for insertion or removal.

For a wedge, draw separate free-body diagrams for the wedge and the body being lifted. At each rough contact, the normal reaction is perpendicular to the contact surface and the friction force is tangent to it. At impending motion, the resultant contact force is inclined by the friction angle toward the direction opposing motion.

For a simple wedge lifting a load W, the required force depends on wedge angle α and the friction coefficients at the interfaces. A general force balance is safer than memorizing a single formula because formulas differ according to whether the wedge is pushed, pulled, or removed and whether both surfaces have identical friction.

For a symmetric wedge with identical friction at two interfaces, the force needed to raise the load is commonly obtained by resolving the two contact-resultant forces. In a simplified idealization,

where P is the horizontal force, α is the wedge half-angle or effective wedge angle as defined by the geometry, and φ is the friction angle. The exact factor multiplying φ depends on the stated wedge geometry; therefore, the free-body diagram must define α unambiguously.

A wedge is self-locking when it will not move backward under the load after the applied force is removed. In practical designs, self-locking requires the resisting friction angle to exceed the relevant wedge angle. Vibration, lubrication, wear, and elastic deformation can destroy self-locking, so a positive mechanical lock is preferable for safety-critical lifting.

3. Screws and Power Screws

A screw thread can be unwrapped into an inclined plane. This analogy permits analysis of screw jacks, vises, presses, and lead screws.

For a single-start screw with mean diameter d_m and pitch p, the lead angle λ is

For a multi-start thread, lead equals the pitch multiplied by the number of starts. If friction acts on the thread with coefficient μ, the friction angle is φ = tan⁻¹μ.

For raising an axial load W, the torque required at the thread, neglecting collar friction, is

For lowering the load,

The lowering expression assumes controlled impending motion and φ > λ for self-locking. If λ ≥ φ, the screw may overhaul or back-drive under the load.

A collar or thrust bearing adds torque. For a uniform-wear collar with mean radius r_m,

Thus total input torque is

The screw efficiency during raising is

where p is the lead, measured in metres per revolution. The expression assumes one revolution raises the load by one lead and neglects other losses.

Worked example. A screw jack raises W = 10 kN using a screw with d_m = 30 mm, lead p = 6 mm, thread coefficient μ = 0.10, and collar coefficient μ_c = 0.08. Let r_m = 25 mm.

First,

so λ ≈ 3.64°. Also φ = tan⁻¹(0.10) ≈ 5.71°. The thread torque is

The collar torque is

Therefore T_in ≈ 44.7 N·m. The efficiency is

or approximately 21.4%. The low efficiency is expected for a self-locking screw with substantial friction.

4. Bearings and Journal Friction

A bearing supports load while permitting relative motion. Bearing friction depends on lubrication, viscosity, speed, load, alignment, surface finish, temperature, and bearing geometry.

For a lightly loaded journal bearing modeled by a friction coefficient μ_b, the resisting torque is approximately

where W is the radial load and r is journal radius. The corresponding power loss at angular speed ω is

with P_loss in watts when T_f is in newton-metres and ω in radians per second.

This simple model is useful for preliminary estimates, but journal bearings often operate in hydrodynamic or mixed lubrication. In hydrodynamic lubrication, a pressure-generating oil film supports the load and direct metal-to-metal contact is reduced. The coefficient of friction is then not a fixed material constant. Viscosity, clearance, speed, and load must be considered.

Rolling-element bearings replace sliding contact with rolling contact, although rolling resistance and internal sliding remain. Manufacturer catalogs should be used for life, static capacity, lubrication, and speed limits rather than relying only on a Coulomb coefficient.

5. Rolling Resistance

Rolling resistance arises from elastic hysteresis, local deformation, micro-slip, and surface irregularity. It is not identical to sliding friction.

A common engineering model represents rolling resistance by a resisting moment M_r or by an equivalent horizontal force F_r:

where r is the rolling radius, N is the normal load, and c_r is a rolling-resistance length with units of metres. Alternatively, engineers may define a dimensionless rolling coefficient C_rr by F_r = C_rrN.

The resisting force acts opposite the direction of rolling. On a level surface, the force required to maintain constant speed is approximately F_r, neglecting aerodynamic drag and bearing losses. Rolling resistance is affected strongly by tire pressure, material, temperature, load, speed, and surface texture.

6. Belts and Flexible Power Transmission

Belts transmit power between pulleys through friction. Let T_1 be the tight-side tension and T_2 the slack-side tension, with T_1 > T_2. The transmitted torque on a pulley of radius r is

and power transmitted at belt speed v is

For a flat belt on a pulley, the capstan relation at impending slip is

where μ is the coefficient of friction and θ is the angle of wrap in radians. This relation assumes a flexible belt, uniform friction, static equilibrium of a differential belt element, negligible belt mass and bending stiffness, and impending sliding. For actual operation without slip, the tension ratio must be below the limiting value.

For a V-belt with groove half-angle β, the wedging action increases friction. An idealized relation is

provided β is defined as the half-angle between the belt side and the groove centerline and the contact model is appropriate. V-belts must be selected using manufacturer data because groove geometry, belt construction, centrifugal tension, and temperature influence capacity.

The total belt tension includes the difference-producing tension and pretension. Excessive pretension increases bearing loads, while insufficient pretension causes slip. Centrifugal tension becomes important at high speed and changes the effective belt-force distribution.

Belt example. A flat belt has μ = 0.30 and wraps 180° around a pulley. Since θ = π rad,

If T_2 = 400 N, the limiting tight-side tension is approximately 1,026 N. The maximum transmitted force difference is 626 N. At belt speed 8 m/s, the corresponding limiting power is approximately 5.01 kW, neglecting centrifugal and other losses.

7. Brakes and Clutches

A brake absorbs mechanical energy and converts it mainly into heat. A clutch transmits torque between rotating members. Both rely on friction, but a clutch generally aims to transmit torque repeatedly with controlled engagement, whereas a brake aims to reduce or stop speed.

For a single-disc clutch with two friction surfaces, uniform-pressure theory gives

where n is the number of friction surfaces, W is the axial clamping force, and R_m is the mean friction radius. Under uniform pressure,

where r_o and r_i are outer and inner friction radii. Under uniform wear, pressure varies approximately as p ∝ 1/r and

The correct theory depends on whether the surfaces have reached steady wear. The transmitted power is P = Tω, and the heat generated during engagement must be checked against the thermal capacity and allowable temperature.

For a band brake, the capstan relation also applies:

where T_t and T_s are tight- and slack-side tensions. If the brake drum radius is r, braking torque is approximately

The direction of self-energizing depends on the arrangement of the actuating lever and the direction of drum rotation. A self-energizing brake reduces actuation force, but excessive self-energizing can cause grabbing and loss of controllability.

Brake design must consider stopping energy, heat dissipation, fade, wear, lining compatibility, emergency behavior, guarding, and applicable safety standards. A friction calculation alone does not establish safe performance.

8. Cables, Ropes, and Pulleys

An ideal cable is flexible, inextensible, and capable of carrying tension but not compression or bending moment. In a massless cable passing over a frictionless pulley, tension is the same throughout a continuous segment. A real cable has mass, elastic stretch, bending stiffness, internal friction, and possible wear.

For a cable segment with distributed load w per unit horizontal distance, the shape may be parabolic for small sag. For a uniformly distributed load w per unit cable length, the exact shape is a catenary. The catenary equation for a cable with horizontal tension component H is

where a = H/w and x and y are measured from the lowest point, with w in force per unit horizontal or cable-length coordinate as defined by the model. The distinction between parabolic and catenary models must be stated.

For a cable carrying a point load, equilibrium of each cable segment gives constant horizontal tension component H. If the cable tangent makes angle θ with the horizontal,

and the vertical component is H tan θ. As sag decreases, θ becomes larger and the support tension can become very high. A nearly level cable is therefore not automatically safe; it may impose large anchor forces.

A pulley with bearing friction has different tensions on its two sides. If the rope wraps through angle θ and the pulley friction coefficient is μ_p, an approximate capstan relation is

This relation should not be confused with the ideal frictionless-pulley assumption.

9. Contact, Wear, and Model Limitations

Coulomb friction is a useful first-order model, not a universal law. Important limitations include:

• The coefficient μ may vary with speed, temperature, pressure, humidity, lubrication, contamination, and surface history.
• Real contact occurs at microscopic asperities, so apparent contact area does not directly determine friction.
• Friction may be anisotropic or direction-dependent.
• Static friction can show time dependence, presliding displacement, and stick-slip behavior.
• Contact pressure may be highly nonuniform; local stresses can cause yielding, pitting, or fatigue.
• Adhesion, wear particles, oxidation, and lubricant starvation can change the mechanism of friction.
• Flexible members stretch and have mass; dynamic tension waves can be important.
• Belts may creep even without gross slip, and centrifugal effects matter at high speed.
• Brakes and clutches are thermal systems as well as force systems.

For high-consequence designs, engineers use measured friction data, contact-stress analysis, fatigue calculations, thermal models, dynamic simulation, and standards-based testing. Safety factors should account for uncertainty rather than merely compensating for careless modeling.

Applied Case: Conveyor Belt Drive with a Friction Brake

Background

Consider an illustrative conveyor driven by a motor through a flat belt. A friction brake is mounted on the driven shaft to stop the conveyor during an emergency. The design must transmit 4 kW continuously and provide 250 N·m of braking torque.

Problem

The belt speed is 10 m/s, the wrap angle is 170°, μ = 0.28, and the driven pulley radius is 0.20 m. Determine the minimum ideal tension ratio and the required tension difference for power transmission. Then identify design issues not captured by the ideal calculation.

Analysis

Convert the wrap angle to radians:

The limiting tension ratio is

The required tension difference for 4 kW is

Let T_1 = 2.296T_2. Then

so T_2 ≈ 309 N and T_1 ≈ 709 N. These are ideal limiting values; operating tension should be lower in ratio than the slip limit and should include appropriate pretension.

The brake torque requirement corresponds to a tangential force at a 0.20 m drum radius of

The brake lining, lever, and drum must generate this force while limiting temperature and wear.

Approach and outcome

A responsible design would next check belt fatigue, pretension, pulley bearing loads, pulley alignment, transient acceleration, emergency stopping time, brake thermal capacity, lining wear, guarding, and failure modes. A spring-applied, power-released brake may be selected where loss of electrical power must produce a safe stop. The final design requires manufacturer data and applicable machinery-safety requirements.

Learning points

1. The capstan equation determines a slip limit, not a guaranteed operating capacity.
2. Power transmission determines tension difference, whereas pretension determines contact and bearing loading.
3. Emergency braking requires energy and thermal analysis, not torque analysis alone.
4. Safety-related friction systems require controlled failure behavior and inspection provisions.

Design Practice and Professional Relevance

Engineers should document friction coefficients with their source, test condition, temperature, surface condition, and uncertainty. Free-body diagrams should distinguish applied force from contact reaction and should identify whether motion is impending, actual, or absent. For flexible elements, specify whether mass, elasticity, pulley friction, and bending stiffness are neglected.

Computer-aided design and finite-element tools can estimate contact pressure and deformation, while multibody simulation can model belts, cables, and pulleys. However, software does not remove the need to select a valid friction law, verify units, test sensitivity to uncertain parameters, and compare results with physical evidence. Sensors, digital twins, and condition-monitoring systems can detect belt slip, bearing temperature, brake wear, and abnormal vibration, but they should support—not replace—safe mechanical design.

Chapter Summary

Coulomb friction describes dry contact through a normal reaction and a tangential force limited by μ_sN during sticking and equal approximately to μ_kN during sliding. Friction angle methods simplify analysis of wedges and screws. Screw-jack torque includes thread and collar friction, while self-locking depends on the relationship between friction angle and lead angle. Bearing friction produces resisting torque and power loss, but lubrication often makes a constant-coefficient model inadequate. Rolling resistance is associated with deformation and hysteresis rather than simple sliding.

Belts transmit power through a tension difference, with the capstan equation defining the limiting tension ratio. Brakes and clutches use friction to absorb or transmit energy and must be checked thermally. Cables carry tension only; ideal pulleys equalize tension, whereas real pulleys create tension differences. All models depend on assumptions about motion, geometry, material, lubrication, temperature, and surface condition. Professional analysis therefore combines equilibrium with material, thermal, dynamic, safety, and maintenance considerations.

Key Terms

Coulomb friction: An idealized model in which friction is limited by a coefficient multiplied by normal force.

Static friction: Tangential contact force acting without relative sliding.

Kinetic friction: Friction acting while surfaces slide relative to one another.

Impending motion: The limiting state immediately before sliding begins.

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

Normal reaction: Contact force component perpendicular to a surface.

Wedge: Tapered machine element that produces separation or lifting force.

Self-locking: Condition in which a mechanism does not back-drive under load.

Lead angle: Angle associated with the helix of a screw thread.

Journal bearing: Sliding bearing supporting a rotating shaft.

Rolling resistance: Resistance caused mainly by deformation and hysteresis during rolling.

Capstan equation: Relation between belt or rope tensions and wrap angle.

Pretension: Initial tension applied to a belt or cable before service loading.

Clutch: Friction device that transmits torque between rotating members.

Brake: Device that absorbs mechanical energy to reduce or stop motion.

Uniform-pressure theory: Clutch model assuming constant contact pressure.

Uniform-wear theory: Clutch model assuming wear intensity is constant across the friction surface.

Catenary: Exact ideal shape of a flexible cable under uniform self-weight.

Stick-slip: Alternating sticking and sliding caused by unstable frictional behavior.

Review Questions

Short Answer Questions

1. Why is static friction represented by an inequality rather than always by F = μ_sN?
2. Define friction angle and explain its use in contact analysis.
3. State two assumptions behind the capstan equation for a flat belt.
4. Distinguish rolling resistance from sliding friction.
5. Why can a nearly horizontal cable impose large anchor forces?

Long Answer Questions

1. Explain a systematic free-body-diagram procedure for a block on a rough inclined plane, including checks for sticking and sliding.
2. Derive the screw-jack raising-torque expression by treating the thread as an inclined plane.
3. Compare uniform-pressure and uniform-wear theories for a single-disc clutch.
4. Explain how pretension, friction, and belt speed affect the performance of a belt drive.
5. Discuss the limitations of using a constant coefficient of friction in industrial design.

Analytical / Application Questions

1. A 500 N block rests on a 25° incline with μ_s = 0.35 and μ_k = 0.25. Determine whether it remains at rest and, if it slides, calculate its acceleration.
2. A belt transmits 6 kW at 12 m/s. If μ = 0.25 and the wrap angle is 210°, determine the limiting tension ratio and the tension difference required for the stated power.
3. Design a preliminary friction-brake calculation for a rotating drum. Identify the torque, stopping-energy, thermal, wear, and safety information required before selecting the brake.

Multiple Choice Questions with Answers

Question 1

For a body at rest on a rough surface, static friction is

A. Always equal to μ_sN

B. Always zero

C. Equal to the value required for equilibrium, up to μ_sN

D. Equal to μ_kN

Correct Answer: C. Equal to the value required for equilibrium, up to μ_sN

Question 2

At impending sliding, the friction angle φ satisfies

A. sin φ = μ

B. tan φ = μ

C. cos φ = μ

D. φ = μN

Correct Answer: B. tan φ = μ

Question 3

A 100 N block has a normal reaction of 80 N and μ_s = 0.25. The maximum static friction is

A. 20 N

B. 25 N

C. 80 N

D. 320 N

Correct Answer: A. 20 N

Question 4

The ideal belt relation for impending slip is

A. T_1 + T_2 = μθ

B. T_1/T_2 = e^(μθ)

C. T_1 − T_2 = e^(μθ)

D. T_1/T_2 = μ/θ

Correct Answer: B. T_1/T_2 = e^(μθ)

Question 5

A belt transmits power at speed v. The transmitted power is

A. (T_1 + T_2)v

B. (T_1 − T_2)v

C. T_1T_2v

D. (T_1/T_2)v

Correct Answer: B. (T_1 − T_2)v

Question 6

For a screw with mean diameter d_m and lead p, the lead angle is defined by

A. tan λ = πd_m/p

B. tan λ = p/(πd_m)

C. sin λ = p/d_m

D. λ = μd_m

Correct Answer: B. tan λ = p/(πd_m)

Question 7

The approximate resisting torque of a journal bearing using a constant coefficient model is

A. W/(μr)

B. μWr

C. μW/r

D. Wr/μ

Correct Answer: B. μWr

Question 8

Under uniform wear in a disc clutch, the mean friction radius is

A. (r_o − r_i)/2

B. (r_o + r_i)/2

C. (2/3)(r_o³ − r_i³)

D. r_or_i

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

Question 9

An ideal flexible cable can carry

A. Tension only

B. Compression only

C. Bending moment only

D. Equal tension and compression

Correct Answer: A. Tension only

Question 10

A brake differs fundamentally from a clutch because a brake primarily

A. Stores elastic energy

B. Absorbs mechanical energy to reduce motion

C. Eliminates all friction

D. Increases shaft speed

Correct Answer: B. Absorbs mechanical energy to reduce motion