# On a Roll: Roll rates in a Vehicle

This could be treated as part 2 of the spring rates post.

When the vehicle goes through the corner, it experience the roll motion when the roll center of the vehicle doesn’t coincide with the vehicle center of mass(which usually is the case). Vehicle roll motion affects the wheel loading which in turn affects the available tire lateral/longitudinal forces and hence the performance of the vehicle.

This body roll during the corner is opposed by a roll stiffness. Roll stiffness can be defined as the roll resisting moment of the body (sprung mass) about the roll axis. It is usually expressed in (torque) N-m/degree of roll.

It can be provided by various components in the vehicle: springs, ARBs , tires and compliances with the primary stiffness coming from springs and Anti-Roll Bars. The vehicle level properties like location of center of mass, roll center location and axle track width also plays a big part. Roll stiffness of the complete vehicle is the sum of the separate roll stiffness rates of all vehicle elements.

**Roll Gradient**

The stiffness in roll is also conveniently expressed in normalized form for the entire vehicle as degrees of roll per unit lateral acceleration. (deg/g) This is called roll gradient. Some typical roll gradient values are :

Large Passenger Cars: 7.0 degrees/g

Small Passenger Cars: 5.0 Degrees/g

Sports cars: 4.0 Degrees/g

These are indicative values only. Usually if the springs alone are not able to provide the roll stiffness required, Anti roll bars or ARBs are deployed. You usually have some roll gain target in mind for the entire vehicle and try to match that number using different components. The spring rates/ride rates come from the ride frequencies/ wheel travel considerations and then the remaining roll stiffness (if applicable) from ARBs (as ARB and springs are the two main contributors to roll stiffness).

**Anti Roll Bars**

An anti-roll bar (roll bar, anti-sway bar, sway bar, stabilizer bar) is a part of many automobile suspensions that helps reduce the body roll of a vehicle during fast cornering or over road irregularities. It connects opposite (left/right) wheels together through short lever arms linked by a torsion spring. A sway bar increases the suspension’s roll stiffness — its resistance to roll in turns, independent of its spring rate in the vertical direction — Wikipedia

Anti-roll bar acts as a second spring to add spring stiffness at the tire but only in cornering.

The are many mathematical formulations available to estimate the component level ARB stiffness with varied level of accuracy. One of them is from Fred Puhn’s book How to Make Your Car Handle

Please note that the ARB stiffness from the formula above would be in lbf/in which needs to be converted to SI units. Whatever formula you use, the stiffness of an anti-roll bar is proportional to the stiffness of the material, the fourth power of its radius, and the inverse of the length of the lever arm. These formulas can predict the ARB roll stiffness to a reasonable accuracy. However if you want something very accurate, you would need to do FEA.

## Single wheel bump Vs Roll

During single wheel bump only one of the wheel moves and hence the bar twists over its entire length while in a roll, both wheel moves. As a result the ARB wheel rate in roll is double than in single bump.

## Effect of ARB bushings

D block bushing stiffness as well as there spacings affect the roll stiffness. They can be treated as just one more stiffness element which would be in series with bar’s own roll resistance.

# Methodology

How to get to ARB sizing from a system level Roll Gradient target ?

Brace yourself !! This is going to be a lot of calculations. Most of these are taken from the book Race Car Vehicle Dynamics by Milliken. There might be multiple approaches to arrive at the same result.

To get to the basics- suppose there is a suspension with equal spring rates left and right . When the Vehicle rolls by an angle phi.

Then the roll stiffness would be equal to :

Where

Kφ = Roll Stiffness

T = Torque

K = wheel rate

t = distance between the springs or the wheels ( so trackwidth)

This signifies the roll stiffness from the springs. The same formula can be used for different components.

To get to the actual ARB size, following steps can be followed:

1**- Finding sprung mass Roll moment Lever Arm using sprung mass CG and RCHs of front and rear**

Roll center heights are different for front/rear axle and are dependent on suspension geometries. Roll moment lever arm is a vertical distance between Sprung mass CG and roll axis connecting front and rear RCHs

2- **Finding Roll Moment per g Lat acc using RMLA**

Now that we know the lever arm we can calculate

Proper units should be used. Here we assume lever arm in m and we also converted the g acceleration.

3- For a given desired Roll gain (deg/g), the total roll rate is calculated which is equal to **Total Roll Stiffness Rate per (desired) Roll Gradient (N-m/deg)**

4- **Front and Rear roll stiffness from the springs alone** is then calculated which is different for independent and dependent suspensions. Here I assume that Front suspension is independent and rear is dependent.

5- **Total roll rate = front +rear** which is then subtracted from total roll rate to determine the roll rate to be provided by ARBs

6- To calculate the requirements of the front and rear anti-roll bars, it is important to know the total lateral load transfer distribution per g of acceleration. Also for proper initial understeer of the vehicle, the **Front Lateral Load Transfer needs to be 5% above the total front weight distribution**. which can be calculated from :

7- Using this front load transfer, **front roll stiffness** required is calculated

8- Front Arb stiffness = Front total stiffness- front spring stiffness

9- Rear ARB stiffness = Total — front ARB- rear spring stiffness

where:

Now this can be translated to wheel center and then the motion ratios can be applied to get the actual ARB rate. Overall installation ratio in terms of ARBs is degrees of bar twist per degree of chassis roll to convert the contribution of anti roll bar rate to total roll rate of the car. This is due to following:

- ARB lever length between centerline to point of suspension attachment
- installation ratio between that suspension attachment and wheel center.

These equations can be used to arrive at the correct ARB diameter required in order to match the system level roll gradient.

That’s all for now.

*Written while listening to Ummeed with Zakir Khan*