Roll Rate Calculator

Roll stiffness, more precisely body roll stiffness, describes the ability of a car's suspension to reduce the vehicle's rolling during cornering. When entering a curve, the body leans toward the outside of the wheels. Springs and stabilizers counteract this lean. The higher the stiffness of an axle is, the more stable the vehicle is and how it reacts differently to steering movements.

This calculator will calculate the roll stiffness of an axle by adding together the roll stiffness created by the springs and the roll stiffness contributed by the stabilizer bar. By leaving fields blank the tool can back-calculate values so that you can not only read results but also achieve a desired roll stiffness.

What is roll stiffness?

Roll stiffness is the torque required to roll the body through a given angle and is usually quoted in newton metres per radian or newton metres per degree. The roll stiffness of an axle consists of two components:

Springs act through a combination of wheel stiffness and the square of track width. A larger track width gives a longer lever arm for the spring to work on, providing more contribution to stability. Stabilizers only act when the wheels move in opposite directions on opposite sides, which is required when cornering. So stabilizers can increase handling ability while cornering without affecting ride comfort from road roughness such as both wheels hitting a bump at the same time.

How to use this calculator.

Three values are needed. The wheel stiffness is the actual spring constant experienced by the wheel and not the original spring constant of the coil spring. If the suspension has a gearing ratio, please open the tool for calculating wheel stiffness. The spring constant can be converted.

Track width is the distance between the centerlines of the tires on an axle left and right. Enter the value for stabilizer stiffness of the stabilizer. If this axle has no stabilizer enter zero. If you fill in three out of four fields, the tool will complete the fourth one.

Calculation equation:

The roll stiffness is the sum of the contributions of the spring and stabilizer.

Here k-phi is the axle roll stiffness, k-w is the wheel stiffness, t is the track width and k-arb is the stabilizer bar stiffness. The factor 1/2 and the square are direct derivations from the geometric relationships where the body rotates about two springs spaced apart by the track width.

Let's consider a simple example. Let's say the wheel stiffness of the front axle is 20 newtons per millimeter, the track width is 1.6 meters and the stabilizer bar stiffness is 5,000 newton-meters per radian. The spring contribution is 25,600 newton-meters per radian. If you add in the contribution of the stabilizer bar, the result is 30,600 newton-meters per radian.

The table below shows the numerical values of each symbol in this calculation example.

Why track gauge matters so much:

Track width affects the value of the squared term in the formula and therefore changes the spring rate contribution proportionally to the square. Even small changes can have unexpected effects. In this example, increasing track width from 1.6 meters to 1.8 meters (an increase of just 12.5 percent) increases the spring rate contribution from 25,600 to 32,400 newton-meters per radian (a 26 percent improvement). This is one reason why race cars with wider bodies can use relatively soft springs and still be stable in corners.

From roll stiffness to body deflection:

The roll stiffness itself is just an abstract number. What the driver actually feels is the deflection of the body. When you open the layer for the curve analysis and add the weight above the spring as well as the lateral acceleration g, the tool converts the roll stiffness into the actual deflection angle of the body.

The body rolls further until the rolling moment caused by the rotational movement is in balance with the roll stiffness.

W is the weight on the springs, a-y is the lateral acceleration in g and h is the height of the center of gravity above the roll axis. By dividing the roll moment by the total roll stiffness of both axles you get the roll angle. This angle is calculated per g and expressed as the roll gradient, a figure that chassis engineers attach great importance to. Comfort-oriented production cars typically have a roll gradient of four to seven degrees per g. Racing cars usually have less than two degrees per g.

Front and rear roll balance

One of the most valuable aspects in using roll stiffness is being able to compare a vehicle's front and rear behavior. The distribution of roll stiffness between the front and rear axles determines the handling balance.

If the stiffness of the front wheels is increased compared to that of the rear axle, the vehicle will push outwards. This is understeer. If the stiffness of the rear wheels is increased, the tail turns more freely and leads to oversteer. So if a driver complains about handling, race engineering experts usually resort first to adjustable stabilizers rather than working on springs or dampers. By entering a second axle in the cornering analysis area, the tool will show the current contribution of that axle.

Wheel stiffness and spring stiffness

A common mistake is to confuse wheel stiffness with spring stiffness. Spring stiffness is a value that is measured directly on the spring itself. Wheel stiffness is the stiffness experienced by the wheel after the action of the suspension mechanism. If there is a translation between the two, they have different values.

MR is the motion ratio, i.e. the value resulting from wheel travel divided by suspension travel. As this ratio enters into the equation in a squared form, only about half of the wheel stiffness remains with a motion ratio of 0.7. An integrated tool for calculating wheel stiffness performs this conversion.

Tips for accurate values:

If the roll ratio is relevant make sure you enter the wheel stiffness and not the original spring rate. Measure the tire footprint width at contact area and do not measure the circumference of the body. For values for stabilizers you must use the roll stiffness in units like Newton per radian or degrees instead of normal torque. When comparing front to rear, calculate the values for each axle separately then check the distribution ratio. A value of "Front 60/40" means that the front wheels take a share of 60 percent of the roll resistance which typically indicates understeer.