Tom Rauji said:
So I treat the axle wrap as a couple and it just adds into other forces as a vector? Can you recommend a good book that deals with this? Not necessarily for cars, but the physics or mechanics involved?
The basics for it is statics analysis. I found
this website which seems to give good tutorials about it. The chapters about moments and trusses should relate closely to a suspension analysis.
Tom Rauji said:
The issue I have to solve is I plant the tires really well but once I apply over a certain threshold of power the car gets unstable. It does a nice small lift in the front over a wide range of power up to a threshold, but once beyond that it will go right on the back bumper. Something rapidly changes in the geometry at a certain front end lift or rear extension down out of the body, and up it goes.
Now we're discussing a specific issue. From the OP, it seemed like you were asking for some general knowledge, but you really have a problem to solve. Give us some specific about that problem. What do «the car gets unstable» means?
You are right about the suspension changing and that is really important to understand. I'm not sure what you already understand and what are your references for your info about suspensions so let me try explaining what I think you need to know.
Here is a simple trailing axle:
The motion of this axle - with respect to the frame - is easy to analyze: The whole axle can only rotate about the joint IC.
The trailing axle has many default as a suspension, thus most suspension designs have multiples links:
The motion of this axle is neither pure translation nor rotation as for the trailing axle. Instead, it will travel along some curvy path.
But a curvy path is only a succession of arcs with a given location for their centers. So, for every position of the suspension links, you can locate that center (Instantaneous Center - or IC - since it is always changing). There are solutions already found for most typical suspension designs. For example, for our four-link suspension design:
So when the suspension is in that precise position, we know about what point the axle is in
pure rotation (no translation).
Why is that important?
If the axle is in pure rotation, so is any point on that axle, like the tire-road contact point where the reaction forces are applied. So we could «re-position» our arm this way and still get the same result:
On the previous figure, there is also the thrust ##T##, the normal force ##N## and the reaction forces at the IC, which are equal and opposite.
If you just look at ##T## and its horizontal counterpart at the IC, you can intuitively see that this couple will try to rotate the axle counterclockwise.
If you just look at ##N## and its vertical counterpart at the IC, you can intuitively see that this couple will try to rotate the axle clockwise.
The moments for each couple can be calculated easily: ##-Th## and ##NL## (I assumed the clockwise direction is positive).
If both moments have equal magnitude, then they will compensate each other and the virtual link will be in pure compression and it will not try to rotate in either direction. So the ratio ##\frac{h}{L}## becomes significant, which can be further simplified to only one value, i.e. the angle of our virtual link since ##\tan \theta = \frac{h}{L}##.
Now imagine ##\theta = 90°## for an instant. Then the normal force of the car would be entirely taken in compression by the virtual link, meaning the suspension spring would be useless. On the other hand, if ##\theta = 0°## (IC is at the ground level), then the full thrust of the tire is taken by the virtual link. Remember that even if this link is virtual, it still means that the actual suspension links are taking the loads in compression or tension somehow.
Anti-squat
When you accelerate (because of the thrust force ##T##), there is a weight transfer from the front to the rear. This means ##N## will increase. That normal force increase will be visible by a compression of the rear springs (and similarly by a decompression of the front springs). We can easily calculate that increase ##\Delta N## this way:
$$\Delta N = T\frac{h_{cg}}{L_w}$$
Where ##h_{cg}## is the center of gravity height and ##L_w## is the wheelbase length.
For now, note that the higher the ##cg##, the larger the weight transfer toward the rear axle, something that is desirable when accelerating.
Back to the suspension design, you can position the IC such that there will be a counterclockwise (negative) moment that will compensate for the additional weight transfer. If you perfectly achieve that, the rear springs will not compress because the axle will try to lift the frame while using the ground as a pivoting point. So:
$$Th = \Delta N L$$
But we already know the relationship between ##\Delta N## and ##T##, so:
$$\frac{h}{L} = \frac{h_{cg}}{L_w}$$
With this simple geometric relationship, no matter the thrust, no matter the weight, we know the rear suspension will not squat under acceleration. We call that achieving 100% anti-squat. By moving the IC location, we can achieve partial anti-squat or we could even create a moment so high that the rear end would lift under acceleration.
Real Life
I'm getting closer to your situation. Say you position your IC such that you want to achieve 100% anti-squat and you expect the rear end stable, right? Not quite.
Once you will accelerate, your front end will lift (although you could technically put anti-lift in the front suspension too if it was an AWD, but you most likely won't do that). If the front end lifts, 2 things happen:
- The ##cg## height will increase, thus increasing the weight transfer;
- With the front end lifting, it means the car frame rotates about the rear tires' contact patch, thus the IC location moves, thus ##\theta## increases.
So your whole geometric relationship ##\left(\frac{h}{L} = \frac{h_{cg}}{L_w}\right)## can change as well. If you don't have 100 % anti-squat, then ##\theta## will change even more. If the change is noticeable quickly (when you have a short swing arm length, for example), then unexpected things can happen rapidly. So it's not only where your IC is that matters, but also how fixed it remains as the suspension and frame move all around.
If that is not enough trouble, if the suspension is too rigid (high ##\theta##, i.e. a lot of anti-squat), the suspension doesn't do its job efficiently as it cannot absorb vertical wheel motion very wheel (rigid links instead of springs). So it is possible you will see wheel hop appearing.
In your case, when «it goes right on the back bumper», it is because you have enough thrust to transfer all the weight from the front end to the rear end - according to your (effective) ##cg## height and wheelbase. Once you go past that thrust threshold, then the car body is in pure rotational acceleration and it will lift up until it flips over. (And remember that the ##cg## height is still increasing as the front end lifts up!) You can control that by adjusting the thrust (like what bikers do when they ride on a wheelie).
Long story short, your ##cg## height is too high at some point during the initial acceleration. This can be adjusted by making sure your front and/or rear suspension will not lift as much as they do right now.
