How To Calculate Roll Rates

The rolling resistance coefficient (Crr) is a key factor in how much energy is lost when a wheel or other object rolls over a surface. This calculator helps you determine the force of rolling resistance and the coefficient of rolling resistance.

Intermediate Values

Calculated Rolling Resistance Force (Fr): — N

Coefficient of Rolling Resistance (Crr) Formula: Fr / Fn

Coefficient of Rolling Resistance (Crr) from Contact Patch: —

Note: The first result uses the directly measured force (Fr). The third intermediate result provides an alternative calculation of Crr using the contact patch length, which can be useful in different scenarios.

What is Roll Rate? Understanding Rolling Resistance

The term "roll rate" most commonly refers to the Coefficient of Rolling Resistance (Crr). It's a dimensionless number that quantifies the resistance a surface exerts on an object that is rolling over it. This resistance arises from the deformation of both the rolling object (like a tire) and the surface it's rolling on. Energy is lost as these materials flex and return to their original shape, generating a force that opposes motion. Understanding how to calculate roll rates is crucial in fields ranging from automotive engineering and cycling performance to materials science and robotics.

A lower Crr means less energy is wasted, resulting in greater efficiency. For example, cyclists strive for low rolling resistance to travel further with less effort, while automakers focus on it to improve fuel economy. Conversely, some applications might require higher rolling resistance for stability or braking. This calculator helps you quantify this important physical property.

The Roll Rate (Crr) Formula and Explanation

The fundamental way to calculate the Coefficient of Rolling Resistance (Crr) involves understanding the forces at play. There are a couple of common approaches, depending on the data available.

Method 1: Using Measured Rolling Resistance Force

This is the most direct method when you can measure the force opposing the motion.

Formula: Crr = Fr / Fn

Where:

Variables and Units for Crr Calculation

Variable	Meaning	Unit	Typical Range
Crr	Coefficient of Rolling Resistance	Unitless	0.001 – 1.0+ (depending on surfaces and materials)
Fr	Rolling Resistance Force	Newtons (N)	Varies widely; typically a fraction of Fn
Fn	Normal Force	Newtons (N)	Weight of the object + any additional downward force

Method 2: Using Contact Patch Length (for deformation-based models)

In some engineering contexts, particularly when analyzing deformation, Crr can be approximated using the length of the contact patch (b) and the rolling radius (r):

Formula Approximation: Crr ≈ b / (2r)

This formula is a simplification and assumes a specific model of deformation. It's often used as a conceptual tool or in specific material studies. The calculator provides this as a third intermediate value for comparison.

Practical Examples of Calculating Roll Rates

Let's see how these formulas work in practice.

Example 1: A Car Tire on Asphalt

Consider a car tire with a normal force (weight on that tire) of 5000 N. If the measured rolling resistance force is 100 N:

• Inputs:
• Normal Force (Fn) = 5000 N
• Rolling Resistance Force (Fr) = 100 N
• Rolling Radius (r) = 0.35 m (approximate for many car tires)
• Contact Patch Length (b) = 0.05 m (typical for a car tire)

Calculation: Crr = Fr / Fn = 100 N / 5000 N = 0.02

The Coefficient of Rolling Resistance is 0.02.

Using the contact patch approximation: Crr ≈ b / (2r) = 0.05 m / (2 * 0.35 m) = 0.05 / 0.7 = 0.071

Note the significant difference, highlighting that the approximation is highly dependent on the material properties and deformation model. The direct measurement method (Fr/Fn) is generally more representative of actual performance.

Example 2: A Bicycle Tire on Pavement

A cyclist weighs 75 kg (approx. 735 N). Assume the load distribution means 60% is on the rear tire, so Fn = 735 N * 0.6 = 441 N. If the measured rolling resistance force is 4.41 N:

• Inputs:
• Normal Force (Fn) = 441 N
• Rolling Resistance Force (Fr) = 4.41 N
• Rolling Radius (r) = 0.35 m
• Contact Patch Length (b) = 0.015 m (smaller for a bicycle tire)

Calculation: Crr = Fr / Fn = 4.41 N / 441 N = 0.01

The Coefficient of Rolling Resistance is 0.01, indicating very low resistance, which is typical for good bicycle tires on smooth pavement.

Using the contact patch approximation: Crr ≈ b / (2r) = 0.015 m / (2 * 0.35 m) = 0.015 / 0.7 = 0.021

Again, the approximation differs from the measured value. For practical efficiency calculations, using the directly measured rolling resistance force (Fr) is preferred.

How to Use This Roll Rate Calculator

1. Identify Your Inputs: Determine the Normal Force (Fn) acting on the object, the measured Rolling Resistance Force (Fr) opposing its motion, the object's Rolling Radius (r), and the Contact Patch Length (b). Units must be consistent (Newtons for forces, meters for lengths).
2. Enter Values: Input these values into the corresponding fields in the calculator. The "Normal Force" is typically the weight of the object plus any downward external forces. The "Rolling Resistance Force" is the force you measure that directly opposes the rolling motion.
3. Select Units (if applicable): This calculator primarily uses metric units (Newtons and meters) for consistency in physics calculations. Ensure your input values are in these units.
4. Click Calculate: Press the "Calculate" button.
5. Interpret Results: The calculator will display the primary result: the Coefficient of Rolling Resistance (Crr). It will also show intermediate values, including the calculated rolling resistance force based on your inputs, and alternative Crr calculations.
6. Copy Results: Use the "Copy Results" button to easily transfer the calculated values and their units for use in reports or further analysis.
7. Reset: If you need to start over or test new values, click the "Reset" button to clear the fields and restore default placeholders.

Key Factors That Affect Roll Rate (Crr)

1. Surface Type and Condition: Rougher, softer, or more deformable surfaces (like sand, gravel, or soft soil) significantly increase Crr compared to hard, smooth surfaces (like asphalt or polished concrete).
2. Tire/Object Material and Construction: The rubber compound, tread design, tire pressure (for pneumatic tires), and internal construction all influence how much deformation occurs, thus affecting Crr. Softer, more pliable materials generally lead to higher Crr.
3. Tire Pressure (for pneumatic tires): Lower tire pressure increases the contact patch area and the degree of tire deformation, leading to higher Crr. Maintaining optimal pressure is key for efficiency.
4. Load (Normal Force): A heavier load (higher Fn) generally increases the rolling resistance force (Fr), although the Crr itself may not change linearly. Increased load deforms the tire and surface more.
5. Tire/Object Radius and Width: Larger radii and wider objects can sometimes lead to lower Crr under similar loads, as the pressure distribution might change, and the deformation mechanics differ. However, this is complex and depends on specific geometry.
6. Speed: At very low speeds, Crr is relatively constant. However, at higher speeds, Crr can increase due to factors like aerodynamic effects and increased flexing of the tire material.
7. Temperature: Material properties change with temperature. For rubber compounds, colder temperatures can increase stiffness and slightly increase Crr, while very high temperatures might cause softening and also increase Crr.

FAQ about Roll Rate Calculation

What is the difference between Rolling Resistance Force (Fr) and Coefficient of Rolling Resistance (Crr)?

Fr is the actual force in Newtons (N) that opposes the motion of a rolling object due to deformation. Crr is a dimensionless ratio (Fr / Fn) that quantifies *how much* resistance occurs relative to the load. Crr is independent of the absolute load, while Fr is directly dependent on it.

Are the units important when calculating roll rates?

Yes, extremely important. For the formula Crr = Fr / Fn, both Fr and Fn must be in the same units of force (e.g., Newtons). For formulas involving dimensions like contact patch length (b) and rolling radius (r), they must be in consistent units of length (e.g., meters). This calculator uses Newtons and meters.

Can Crr be negative?

No, the Coefficient of Rolling Resistance (Crr) is physically a non-negative value. It represents energy loss, so the resistance force (Fr) always opposes motion and acts in the opposite direction to the normal force (Fn) in the ratio calculation, resulting in a positive Crr.

What is a "good" Crr value?

"Good" depends on the application. For efficiency (vehicles, bicycles), a low Crr is desirable (e.g., 0.005 to 0.02). For applications needing traction or stability, a higher Crr might be acceptable or even beneficial. Typical values range from 0.001 for specialized low-resistance tires on smooth surfaces to over 0.5 for wide tires on soft sand.

How does the contact patch length (b) relate to Crr?

The contact patch length (b) is a measure of how long the area of deformation is where the rolling object meets the surface. In simplified models, Crr is sometimes approximated as b / (2r). A longer contact patch relative to the radius often implies greater deformation and thus potentially higher rolling resistance. However, this is an approximation and doesn't account for all factors.

Does speed affect the roll rate?

Yes, while often considered constant at moderate speeds, Crr can increase at very high speeds due to increased material hysteresis and flex. At very low speeds, it's generally stable.

Can I use this calculator for non-tire applications?

Yes, the principles apply to any object rolling over a deformable surface, such as bearings, spools, or even sleds, provided you can accurately determine the normal force, the resistance force, and the relevant dimensions (like rolling radius and contact patch). The accuracy will depend on how well the simplified models match the real-world physics.

How is the rolling resistance force (Fr) typically measured?

It can be measured using a force gauge connected between the rolling object and the towing mechanism, or indirectly by measuring the power required to maintain a constant speed and calculating the force from power and velocity (Force = Power / Velocity). Alternatively, it can be calculated if Crr is known: Fr = Crr * Fn.

Related Tools and Internal Resources

• Friction Calculator: Explore the concept of static and kinetic friction, another important force related to motion.
• Aerodynamic Drag Calculator: Understand the resistance caused by air, crucial for high-speed applications.
• Power and Energy Calculator: Calculate the energy expenditure required to overcome rolling resistance and other forces.
• Material Deformation Analysis: Learn about the physics behind why surfaces and objects deform under load.
• Vehicle Efficiency Estimator: See how factors like rolling resistance impact fuel or energy consumption.
• Surface Roughness Measurement Guide: Understand how surface texture affects friction and rolling resistance.