Exploring Rolling Friction Principles and Practical Applications

Imagine a heavy-duty truck speeding down a highway or a tiny ball bearing silently operating within precision instruments. What force quietly resists their motion? The answer lies in rolling friction. While generally smaller than sliding friction, rolling friction plays a crucial role in numerous engineering applications. This article explores the principles, influencing factors, and real-world applications of this important physical phenomenon.

1. Rolling Friction: The Force That Opposes Motion

Rolling friction, also called rolling resistance, refers to the force that opposes motion when spherical objects, tires, or wheels roll across a surface. While we typically imagine rolling objects as having regular shapes, irregularly shaped objects like rocks or pebbles can also experience rolling friction. A key distinction between rolling and sliding friction is that the former is closely related to deformation of the contacting materials. In ideal conditions where both the rolling object and surface are perfectly rigid, no rolling friction would occur. Furthermore, in reasonably rigid objects, energy loss during rolling is significantly less than during sliding (typically by 2-3 orders of magnitude).

2. Early Research on Rolling Friction

As early as 1781, Coulomb conducted pioneering research on rolling friction and established its first two fundamental laws. Additionally, some laws of sliding friction were extended to rolling scenarios:

• Friction force is proportional to the applied load
• Friction force is inversely proportional to the curvature radius of rolling elements
• Friction is smaller on smooth surfaces than on rough surfaces
• Static friction is typically much greater than kinetic friction
• Kinetic friction shows weak dependence on rolling speed

3. The Complexity of Rolling Friction

Unlike sliding friction, establishing formulas that relate rolling friction to other material properties proves more challenging. This complexity arises because multiple physical mechanisms contribute to rolling friction, including plastic deformation, elastic hysteresis, and adhesion hysteresis.

3.1 Rolling Friction Formulas

Despite these complexities, rolling friction can be empirically expressed as:

Fr = μr × W

Where:

• Fr = rolling friction force (in Newtons, N)
• μr = dimensionless rolling friction coefficient
• W = applied normal load (in Newtons, N)

To account for the radius of rolling elements, rolling resistance can also be expressed as:

Fr = μr1 × W / r

Where:

• μr1 = rolling friction coefficient expressed in millimeters (mm)
• r = radius of the wheel, ball, cylinder, etc.

4. Limitations of Pure Rolling

In most practical situations, pure rolling cannot be achieved because elastic or plastic deformation occurs in the contact area, meaning contact points lie on different planes. Therefore, pure rolling only occurs at a limited number of points, while at other contact points, a combination of rolling, sliding, and slipping is expected. The sliding (or slip) velocity vs is typically less than 5% of the rolling velocity vr. The total rolling friction FR can be expressed as:

FR = (vs / vr) × μk

Where μk is the kinetic coefficient of sliding friction.

5. Energy Loss During Rolling

Several factors contribute to energy loss during rolling:

• Roughness-induced friction from imperfect rolling geometry
• Energy loss from plastic deformation of rough rolling surfaces
• Elastic hysteresis loss caused by stress-release cycles during rolling

These losses typically amount to about 10 ^-4 and should sum to equal the rolling friction force.

6. Specialized Areas of Rolling Friction

6.1 Free Rolling Wheel Friction

When solids are considered rigid, "point" or "line" contact occurs, creating ideal "pure rolling" conditions. However, pure rolling doesn't occur with deformable solids where contact area is finite. "Free rolling" represents the closest approximation to pure rolling. For free rolling solids, energy loss from material cyclic inelastic deformation ("hysteresis") serves as the primary rolling friction mechanism.

6.2 Traction Friction Models in Rolling Contact

The relative rigid-body motion between two solids may involve translation and rotation, each with three components decomposed along normal and tangential directions. In traction applications, the focus typically lies on describing the functional relationship between resultant contact forces (and moments) and corresponding slip or creep ratios, established at constant rolling velocity.

7. Applications of Rolling Friction

Rolling contact appears frequently in industrial applications, particularly in transportation vehicle wheels and tires. By controlling relative wheel or tire motion ("slip"), friction forces of desired magnitude and direction can be generated to control vehicle movement—from minimal forces needed to overcome cruising resistance to larger forces required for acceleration, braking, or turning.

8. Rolling vs. Sliding Friction

Rolling objects offers a significant advantage over sliding, as rolling friction can be two or more orders of magnitude smaller. This principle finds greatest application in ball and roller bearings, where balls or rollers freely move in grooves called races without requiring supporting axles or trunnions.

While small slips between balls and surfaces were once thought responsible for low rolling friction, research shows these contribute minimally to overall resistance. The fundamental mechanism controlling rolling friction involves bulk deformation. When a hard ball rolls on soft metal (Figure 2.7(a)), it creates grooves through plastic metal displacement. The force required equals observed rolling friction, explaining why lubricants have little effect. Similarly, when a hard steel ball rolls on flat rubber (Figure 2.7(b)), work is lost through deformation and imperfect elastic recovery due to internal friction (hysteresis). Highly elastic rubber might recover 95% of deformation energy, while "dead" rubber recovers little.

Ball bearings made from hard steel experience minimal stress within elastic regions, resulting in extremely low rolling resistance (μ ∼ 0.001). Total friction resistance combines adhesion and deformation forces (Equation 2.6):

F = F _adh + F _def

While adhesion forces exist, they're typically small because junctions remain limited and peeling requires less force. Thus, deformation losses dominate rolling friction, particularly influenced by the softer material's hysteresis properties. With lubricant layers present (Figure 2.7(c)), F ≅ F _def .

9. Rolling Friction Coefficients

Rolling friction (resistance) is defined as the force opposing a particle's rolling motion on a surface (Figure 25). Rolling friction coefficients are much smaller than sliding coefficients and can be expressed dimensionlessly (Equation 11) or with length dimensions (Equation 12). Various models exist for specific conditions and applications.

DEM simulations by Fukumoto, Sakaguchi, and Murakami [144] examined how rolling coefficients affect granular material behavior. They found rolling friction influences particle rearrangement during packing and increases fabric anisotropy. Additionally, earth pressure coefficients at rest decrease with higher rolling friction, which also contributes to stress distribution heterogeneity among particles.

Dimensionless rolling friction coefficient:

F = μ _r × W

Rolling friction coefficient (length dimensions):

F = b × W / r, where b = μ _r

Where:

• F = rolling friction force
• μ _r = rolling friction coefficient
• W = normal force (Figure 25), R is reaction force
• b = rolling friction coefficient with length dimension
• r = particle radius or interparticle contact length (coordination number)

Research shows that increasing rolling friction coefficients raises soil's angle of repose [54,57,83,135,136,138,145]. DEM simulations modeling elliptical rice grains demonstrated that ignoring rolling friction—even with sliding coefficients of 1.0—fails to predict repose angles accurately, producing lower values than measurements [146]. Both rolling and sliding friction coefficient increases elevate repose angles due to associated kinetic energy dissipation rate increases [84].

10. Practical Examples of Rolling Friction

Rolling elements (wheels, balls, cylinders) are introduced to reduce sliding friction. Ball bearings in machinery dramatically decrease friction because moving objects roll on balls rather than slide. Wheel friction being substantially lower than sliding friction demonstrates this principle well—theoretically, no relative motion occurs between wheel rims and reaction surfaces during rolling, creating only point contact without sliding friction.

In reality, slight rolling friction persists due to wheel material deformation in contact areas. Deformed portions tend to drag along surfaces, creating friction directly related to deformation magnitude. Surface hardness of rolling elements and reaction surfaces primarily governs this relationship.

11. Additional Factors Influencing Rolling Resistance

Rolling friction or resistance represents the normal force opposing motion when rolling bodies (balls, tires, or wheels) move across surfaces (Figure 9.6). This force relates to elastic and inelastic deformation behavior of rolling materials under applied loads. Not all energy required for rolling motion is recovered after load removal.

Rolling resistance may also stem from slip between wheels and reaction surfaces, dissipating energy through surface plastic deformation and hysteresis losses. Like sliding friction, rolling resistance is often described as the product of rolling friction coefficient and applied normal force. The "Coefficient of Rolling Friction (CRF)" is defined as (Hersey, 1969):

Fr = Cr × N

Where:

• Fr = rolling friction force
• N = normal force (perpendicular to wheel's rolling surface)
• Cr = dimensionless CRF

As noted, rolling friction coefficients are typically much smaller than sliding coefficients (Table 9.1). For a rigid wheel slowly rotating on a smooth, perfectly elastic surface, CRF can be determined geometrically as:

Cr = z / d

Where:

• z = sinkage depth (Figure 9.6)
• d = rigid wheel diameter

An empirical formula for cast iron mine car wheels on steel rails calculates CRF as (Hersey, 1970):

Cr = 0.0048 × (18 / d) ^1/2 × (100 / W) ^1/4

Where:

• d = wheel diameter (inches)
• W = wheel load (pounds)

The driving torque (T) required to overcome rolling friction (Fr) and maintain steady rotation on a plane is:

T = Vs × ω × Cr

Where:

• Vs = linear velocity of rotating body (at axle)
• ω = rotational speed

This friction type proves most significant in cyclic processes like rolling friction and automobile tires. Mechanical damping and delayed recovery cause energy dissipation, making rolling friction and mechanical damping closely related. For hard balls rolling on plastic surfaces, Flom (1961) established:

μ = 0.115 × (G″ / G′) × (W / (G′ × r ² )) ^1/2

Where:

• W = load on rolling ball with radius r
• G′ = storage modulus of polymer surface
• G″ = loss modulus of polymer surface
• G″ / G′ = dissipation factor = tan δ

Equation (25.16) clearly shows that large G″ / G′ = tan δ values produce substantial rolling friction, making transition regions particularly frictional.

Assuming rigid solids creates "point" or "line" contact for ideal "pure rolling." However, pure rolling doesn't occur with deformable solids having finite contact sizes. "Free rolling" along straight paths with deformable wheels best approximates pure rolling. Resistance to constant-velocity straight-path rolling constitutes "rolling friction," primarily caused by energy loss in material cyclic inelastic deformation ("hysteresis").

Free rolling involves minor sliding in small contact pockets, but nearly (antisymmetrically) self-balancing shear tractions contribute relatively small resultant tangential forces. Viscoelastic normal deformation in tire casings, treads, and sidewalls creates significantly asymmetric normal pressure distributions (Figure 8.3.16). The resultant normal reaction shifts forward in the rolling direction, equivalent to a rolling-resistant moment. Free rolling motion is maintained by minimal tangential forces (shown for driven wheels in Figure 8.3.16) or, for driven wheels, minimal applied drive axle torque.

The primary rolling resistance mechanism involves markedly asymmetric normal pressure distributions during rolling, caused by solid nonelastic (viscoelastic) deformation. As Figure 8.3.16 shows, asymmetry mainly appears in pressure p rather than shear traction τ. Rolling resistance is represented by moment Mr. For freely rolling wheels under normal load, linear velocity V and angular velocity ω relate through effective rolling radius re, where:

r ₀ > re > r _h

Here, r ₀ is undeformed wheel radius, while r _h is deformed wheel height above ground.

Automobile and aircraft tire rolling resistance equals about 1% of normal load. Structural materials (primarily rubber) undergo substantial cyclic viscoelastic deformation during rolling. For steel railway wheels rolling on steel rails, contact deformation is much smaller (re ≈ r ₀ ). Small deformation plus steel's relatively low hysteresis produces extremely low rolling resistance—just 0.1%.