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This is where physics gets stupid.

Lets start with the fundamentals. If an object follows a circular path, it must experience a centripetal force to keep it on this path. The same applies to cars, there must be a centripetal force to pull the car into the turn. The only way a car will experience a lateral force is via the tire contact patches. There aren’t any other areas where an external force is applied to the car, so there must be a lateral (or sideways) force being applied to the tires to pull the car into the turn.

Image Added

The picture above shows how the tire deforms when going through a turn.

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The amount of lateral force generated is based on the slip angle of the wheel, as seen in the above photo.

This is fairly complicated to explain. But the lateral force scales proportional to the weight on the wheel as well as the slip angle. However these relation on non-linear, we can’t throw more weight on the wheel and expect it to produce a directly proportional increase in lateral force. This characterization is typically discretized, meaning that for a certain selection of weights, the characterization of slip angle to lateral force is given. On Dynamics - MSXV there are the data sheets for the tires where you can find these characterizations, but below is a good example.

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Notice that the lateral force produced starts to diminish past ~10 degrees. It’s fairly normal to see the maximum lateral force be produced somewhere around 7 degrees.

Now how do we determine the slip angle on each wheel? We cry.

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There is a bicycle model of steering which can approximate the angle, which I will actually write something about this laterdidn’t do because frankly it didn’t make much sense. Now I have a better ideal of how to determine this step by step. This bicycle model simplifies a 4- or 3-wheeled vehicle into 2 wheels to simplify the math that goes on here during steering. Y’know, cause they wanted to mitigate complexity. To reiterate, this is a simplification that isn’t 100% accurate, but it’s good enough.

First, move the steering system to the maximum possible turn. Now let’s drive the car at a slow speed. At low speeds, less lateral force is required to produce the smallest possible turning radius due to a lower requirement of centripetal acceleration, which is dependent on speed. This means we will get a small turning radius. By fixing the steering angle, we determine the slip angle based on our target speed, which gives us the lateral force required to keep the car in the turn. We determine the target speed based on the ideal path for the car and the time requirement. But notice how the path and the lateral force required are linked, since the centripetal force required is dependent on the centripetal acceleration of the car. I’m not going to do the math for this, but you should get the point.

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The part where the bicycle model comes in is that both the front and rear must experience slip to generate lateral force. The relative values between the front and rear must balance to reach the appropriate rotational velocity for the entire car. That’s right the car rotates.

Think of a 2D coordinate system, there are 3 degrees of freedom; horizontal, vertical, and rotational movement. The rotation is based around an axis that is perpendicular to the 2D plane. When a car takes a turn at a constant radius rotates around it’s yaw axis at the same rate at which it rotates around the circle. This kind of relates to the no-slip condition for a rolling object.

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title What is yaw?
In 3D there are 3 axes that an object can rotate around. Generally this is thought of as X, Y, and Z. However, when it comes to vehicles, the cartesian coordinates are aligned and generalized to the vehicle position. Image Added Yaw describes the rotation around the vertical axis of the car.

Now the rotation velocity of the car around its yaw axis is precisely that, a velocity. Neglecting the transition from straight line driving to driving a curved path, under a constant speed in a curved path, the angular acceleration of the car must be zero. This means (neglecting air resistance and other small losses) that the lateral forces produced at each tire in the bicycle model must balance out around the CG. This is a just a sum of moments produced around the center of gravity of the car must equal zero.

There’s something to be said about oversteer vs understeer. Just know that if we get to choose if our front or rear wheels lose grip in a steering situation, we prefer to lose the front wheels, also known as understeer.

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BE WARNED

This has largely been a qualitative assessment of a steering linkage because putting numbers to this is disgusting.

Yes, models were made to help validate performance, but they were thrown out. Why? Because programming in all the factors to consider is an absolute pain in the ass. Yes, I have created multiple models, each with their own improvements, but it still cannot factor in everything. Especially fits into the chassis.

This is meant to give the reader a run down on some fundamentals to understand what’s going on in steering. The details for the calculation are to be completed by the reader, but at least the general process has been outlined.