Rudimentary Set-Up

Imagine the following kindergarten-inspired drawing as a six-bar steering linkage.

• Green-ish circles are the rotation axes for the upright in the front suspension*.

• Yellow lines are the steering arms.

• Blue lines are the tie rods.

• Red line in the middle is the center bar.

*This is a simplification so it doesn’t get too difficult too illustrate, but remember that the axes are not perfectly vertical, but at an angle in the front suspension to account for caster and scrub radius.

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Dimension A: Horizontal Tie Rod Length

• This is a constant based on the control arm lengths from FSU.

• In our case it needs to be ~162.5 mm

• This is HORIZONTAL length, not overall length of the tie rod. This is critical for zero bump steer geometry.

What is bump steer geometry?

 

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Dimension B: Steering Arm Length

Think of the tie rod (blue line) moving in straight line.

Moving from position 1 to position 2 is a constant linear distance. This would be representational of the constant displacement the rack and pinion has.

The two dotted circles represent the path the steering arm follows with different lengths. The steering arm can be thought of as a radius of a circle. The path the tie rod follows is simplified to the chord of a circle.

With a constant chord length, and a varying diameter for the circle in which the chord is defined, the arc it isolates is reduced.

The yellow and purple dashed lines show the “angular displacement” for a constant cord. There is a larger angle between the yellow dashed lines than the purple dashed lines. This means for a constant displacement from the rack and pinion, a shorter steering arm will increase the angular displacement, ie greater steering angle for the wheel.

Dimension C: Steering Arm Angle

The angle of the steering arm is a bit trickier.

The green triangles are equal in width, but not equal in height (obviously).

Think of the solid lines as the initial position, and the dashed as the final for each colour.

For an equivalent horizontal movement to the right, the amount of angular displacement is dependent on the original angle (or dimension C) of the steering arm.

This is what can create Ackerman or Anti-Ackerman geometry.

In the most recent linkage system I created, Dimension C is >90 deg. This is what creates Anti-Ackerman geometry. This is only because the linkage is placed in front of the “axle.” Yes, I’m aware that the car does not currently have, or likely ever will have an axle, but think of an imaginary line that connects the two spindles.

If the current linkage was placed behind the axle, there would be Ackerman Geometry. These symmetries can be summarized in a table.

Dimension D: Tie Rod Angle

There isn’t an elegant description of how this impacts the rest of the linkage system, since it is dependent on how the steering arm is set up. But generally this should be adjusted for two reasons, to make the linkage system solve, or to get the linkage system to behave more optimally.

There will be some “inefficiency” in the horizontal displacement of the center bar because the tie rod must move vertically as well to stay connected to the steering arm. The path that the steering arm travels is a circle.

Remember! The angle doesn’t matter, the HORIZONTAL length does. So long at the HORIZONTAL length is the correct value, then there’s nothing to worry about.

I will also quickly note here that this angle will change as the linkage is actuated. Meaning that zero bump steer geometry is only met when the car is driving straight. This might be of concern to some, but not me.

Linkage Geometry Concerns

Creating linkage systems that are impossible to operate is pretty easy. It is incredibly difficult to describe a failure mode to a single word or phrase. And these will not be all of them, I encourage readers to add failure modes they’ve encountered to this section.

Minimum Tie Rod Length

Lets say we’re looking at the left steering arm in this example.

But we don’t know where the steering arm is supposed to be. We know how long it is, that’s what the dotted black circle represents; all possible positions of the steering arm.

Let’s move the center bar as far to the right as possible, creating the maximum possible distance between the point where tie rod connects to the center bar and the upright’s rotation axis.

If we draw a line from the wheel rotation axis to where the tie rod would connect to the center bar, we see the minimum length of the tie rod (the green line).

Since this is the furthest distance possible from the upright rotation axis, if the tie rod were any shorter, the linkage would not be able to fully utilize the range of motion from the rack and pinion.

Tie Rod Position Limit

Related to the previous concern, the tie rod cannot be the minimum length.

Suppose that the steering system does make use of the minimum length position described in the previous section. When it comes time to rotate the steering system to the straight position, the steering linkage can in theory move in either direction, following the purple arrow or the green arrow. This is physically possible if the joint is designed correctly.

Instead of adding more complexity to the mechanical design, just don’t do this.

Overlap

This is a catch all term, but there can be mathematically correct solutions for the linkage system. These solutions don’t account for the fact that only one part can occupy a given volume in space.

Swivel Joint Limitations

Determine the angular displacement of both tie rod joints (to the center bar and to the steering arm). Often times the linkage will require more displacement that is available in standard rod ends.

Though the section is called “Swivel Joint Limitations” avoid using spherical bearings, the custom bearing housing isn’t worth it. The tie rod is a two-force member, so it will not experience bending. Just use rod ends.

Asymmetric Swivel Requirements

Cool, the total angular displacement for the joint is in less than the rod end specification. There might be a case where steering to one side requires more than half the swivel specified for the rod end. This has work arounds, but depending on the steering arm geometry, the tie rod might not be in the shape of a rod anymore.

This is what I did in my most recent revision for MSXV, but I would advise against adjusting the tie rod, but make alterations to the steering arm such that it accommodates the asymmetric angular displacements.

The lack of rotational symmetry can have the tie rod clash and potentially get caught on other components

Deflection

We have a valid nodal of the linkage system that won’t break the rods ends. YAY! But in the mechanical design of everything we can’t allow excessive deflection of our parts under load.

How we determine the load is a nightmare we won’t talk about here. In MSXV, we calculated that at worst we see about 500 N. So we rounded up to 1 kN to be safe and showed that the parts deflected less than 0.5 mm or 1.0 mm (I can’t remember anymore, but it was small).

This means that designs must be simulated to validate this.

Interference

The bane of steering’s existence…

As MSXV learned, run some collision checks in CAD before committing to a design. One of the reasons I designed the tie rod in MSXV as I did was that I wanted to have a lot of clearance from the chassis through movement of the linkage system.

Center Bar

One might think that the center bar is just the rack of the rack and pinion controlling the motion. This is severely limiting.

In MSXV, the design I create for the center bar was to change two things: width and position.

Width is adjusted to make a viable linkage system (as are most parameters). We cannot buy a rack for any center bar width we want. This means we have to take an off-the shelf-component and modify it or create a structure around it to meet our width requirements.

The position is fairly straight forward. There’s two priorities, the center bar must be located such that the linkage functions as designed, and the steering column is as direct as possible to the driver. In MSXV, there was only one way to get the steering column to be in a reasonable position without using a u-joint.

How much turn?

*snorts line of cocaine*

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.

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

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.

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.

There is a bicycle model of steering which can approximate the angle, which I didn’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.

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.

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.

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.