6.3 Rolling resistance

Rolling Resistance

Rolling resistance is the force that opposes the motion of a wheel or cylinder as it rolls across a surface. Understanding it matters in statics and strength of materials because it directly affects force equilibrium in systems with rolling contact, and it shows up in problems involving wheels, bearings, and conveyor systems.

Definition and Causes

When a rigid or semi-rigid object rolls on a surface, neither the object nor the surface is perfectly rigid. The contact zone deforms slightly, creating a small indentation ahead of the rolling object. This deformation shifts the normal force's line of action forward of the center of contact, producing an asymmetric pressure distribution. That asymmetry is what generates a net force opposing the direction of rolling.

Several factors contribute to the magnitude of rolling resistance:

• Deformation of materials at the contact zone (this is the dominant factor)
• Surface roughness of both the rolling object and the surface
• Material properties like elasticity and hardness (softer materials deform more, increasing resistance)
• Contaminants or lubricants on the contact surface

The key distinction from sliding friction: rolling resistance is typically much smaller. This is exactly why wheels are far more efficient than sleds for moving loads across a surface.

The Rolling Resistance Model

In a statics context, rolling resistance is often modeled using the concept of a rolling resistance arm (sometimes called the coefficient of rolling resistance with units of length). Here's the idea:

The normal force $N$ shifts forward by a small distance $a$ from directly beneath the center of the wheel. This offset distance $a$ is the rolling resistance arm. For a wheel of radius $r$, the moment about the contact point gives the relationship:

$F_r = \frac{a \cdot N}{r}$

where:

• $F_r$ is the rolling resistance force
• $a$ is the rolling resistance arm (units of length, typically mm)
• $N$ is the normal force
• $r$ is the radius of the wheel

This is the physically accurate model you'll encounter in statics courses. The offset $a$ depends on the materials and surface conditions, and it's typically very small (fractions of a millimeter for hard surfaces).

Simplified Force Equation

A common simplified approach treats rolling resistance similarly to sliding friction:

$F_r = C_{rr} \times N$

where $C_{rr}$ is a dimensionless coefficient of rolling resistance. Note that $C_{rr}$ and the rolling resistance arm $a$ are related by $C_{rr} = a / r$, so the coefficient depends on wheel radius.

Typical values of $C_{rr}$:

Normal Force on Horizontal and Inclined Surfaces

For a horizontal surface, the normal force equals the weight of the object:

$N = m \times g$

For a surface inclined at angle $\theta$ from horizontal, the component of weight perpendicular to the surface is reduced:

$N = m \times g \times \cos(\theta)$

A 1000 kg cart on a flat surface has $N = 9810 \text{ N}$. On a 15° incline, $N = 9810 \times \cos(15°) = 9475 \text{ N}$. The rolling resistance force decreases on the incline, but now you also have a gravity component pulling the cart downhill, which usually dominates.

Solving Rolling Resistance Problems

When you encounter a rolling resistance problem in statics, follow these steps:

1. Draw a free body diagram of the rolling object, showing the weight, normal force, applied forces, and the rolling resistance force opposing the direction of motion (or impending motion).
2. Identify the contact geometry. Note the wheel radius $r$ and whether you're given the rolling resistance arm $a$ or the dimensionless coefficient $C_{rr}$.
3. Calculate the normal force using equilibrium in the direction perpendicular to the surface.
4. Calculate the rolling resistance force using $F_r = a \cdot N / r$ or $F_r = C_{rr} \times N$.
5. Apply equilibrium equations (sum of forces, sum of moments) to solve for unknowns like the force needed to maintain constant velocity or the force needed to start rolling.

Power to Overcome Rolling Resistance

Once an object is moving at velocity $v$, the power required to overcome rolling resistance is:

$P_r = F_r \times v$

For example, a vehicle with $F_r = 200 \text{ N}$ traveling at $25 \text{ m/s}$ (about 90 km/h) requires $P_r = 200 \times 25 = 5000 \text{ W}$, or 5 kW, just to overcome rolling resistance. At highway speeds, air resistance typically dominates, but at low speeds rolling resistance is the larger factor.

Applications in Engineering

Bearings: Ball and roller bearings replace sliding contact with rolling contact, dramatically reducing resistance. Material hardness and surface finish are selected to minimize the rolling resistance arm $a$.

Vehicle design: Rolling resistance, air resistance, and grade resistance are the three main forces opposing vehicle motion. Tire manufacturers balance tread pattern, rubber compound, and recommended inflation pressure to minimize $C_{rr}$ without sacrificing traction or durability. Under-inflated tires increase the contact patch deformation and raise rolling resistance noticeably.

Conveyor systems and industrial rollers: Selecting harder roller materials and smoother surfaces keeps rolling resistance low, reducing the motor power needed to drive the system. Proper lubrication of bearing surfaces further reduces energy losses at the roller supports.