Vehicle Dynamics (Milliken)

[BROW]

Hi,

I was hoping to get an answer here, as I having been going through Milliken's Race Car Vehicle Dynamics recently but just struggle to understand some of the theory even after reading a couple times.

1. Page 131: It is stated that "the vehicle slip angle is obtained by rotating the entire vehicle by the amount alpha(r).

Don't really get how this works. Does this suggest that there is no body slip with no rear slip angles? I presume this only applies for a constant radius test.

2. What is the order of events? Is it steer angle (input) --> front slip angle --> front lateral force --> body slip --> rear slip angle --> rear lateral force? Is the body slip angle what actually induces the rear tire slip angle?

3. I noticed that the on page 148 the front tire slip angle is on the right side of the wheel plane (clockwise from it), whereas the page 142 shows it being to the left of it, on the inside, which surely changes up the equation for calculating the slip angle?

4.. Finally, something that probably sounds idiotic, but how does the input at the steering wheel, the steer angle, induce slip angle at the front tire.Is it because steering it put the entire wheel (and upper tire) at the steer angle, but then the actual contact patch lags behind it a little?

I've been trying to put things together but sometimes one thought makes sense for a certain point, but contradicts the other, and because of this I lack any good foundation to base everything off.

[Greg Locock]

1) Correct. In order to accelerate the rear of the car sideways you must generate a lateral force at the tire, which means it must have a slip angle.

2)Yes

3) Not in my copy it isn't.

4) Not idiotic. yes.

I'd say you've got a good grasp of what's going on.

[Tim.Wright]

1. The order of transformations described in that paragraph are regarding the comparison between figure 5.3a (low speed kinematic/ackermann cornering) and 5.5 (high speed cornering with slip angles). It's not the chronologial order in which a vehicle responds to a steer input.

He's saying that if you take the car in figure 5.3a and rigidly roate the whole car about the CG by the angle alpha_rear then it will give you the complete cornering state for the neautral steer car (i.e. radius, front+rear slip angles, CG slip angle, yaw rate and steer angle). It's a bit abstract and honestly not so important to grasp in my opinion.

2 The chronologial order will become clearer as you read further but it's more or less like this (assuming initial condition is straight line constant speed):
1. Steer angle given while the car is going straight causes a front slip angle (equation 5.4 with ß and r = 0)
2. Front slip angle gives a lateral tyre force
3. Front lateral force creates a yaw and lateral acceleration
4. Combination of lateral acc & yawrate causes a sideslip velocity at the CG (eqation 5.2)
5. Sideslip angle grows and creates a rear slip angle (eq. 5.3) as well as changing the front slip angle (eq. 5.4)
6. Rear slip angle creates a rear lateral tyre force
7. Rear lateral force adds to the lateral acceleration and reduces the yaw acceleration

If the steering is held constant at some value, then eventually the rear axle force will increase to an amount which balances the front axle yaw moment about the CG and the car will be in steady state cornering (eq 5.10)

3. The disgram on page 148 is showing just showing positive sign conventions, not the actual cornering condition. Using the SAE convention that Milliken use in the book, the slip angle is always the opposite sign to the lateral force (i.e. cornering stiffness is negative).

4. The answer is above in (2.1). Steering input creates a front slip angle.

Milliken is a good fundamentals book but it took me about 3 or 4 attempts at each chapter to understand everything cause I'm a bit thick. Investing time to understand this book will give you a better schooling than any seminar will - I can tell you from experience. You will only get anything out of it if you have an engineering background though.

[BROW]

Thanks guys, most of it makes sense!

One things I found in a book was the following diagram of the progression of forces and yaw velocity/displacement due to step steer input. One thing I don't understand is why at 0.3s would you have a positive force on the rear? The body angle is steeper than the travel path (straight line), so surely the force should be acting the other direction.

https://imgur.com/3iJcDPK

[WilO]

I think the longitudinal line that appears to be attached to the CoM might be throwing you off(?). Remember that at this point the vehicle has both lateral and yaw velocity, as indicated by the clockwise rotational vector at the CoM (The vehicle may not be rotating about the CoM just yet, so I think the clarity of this image could be improved).
So, if the instantaneous point of rotation lies within the wheelbase of the car, say a little in front of the rear axle, the rear axle is developing a tangential velocity to the left, which is giving rise to a slip angle at the rear axle. All this places the car on a curved path, if you will, and gives rise to a small centripetal acceleration.
Conversely, imagine the car rotating about a point that lies outside the wheelbase (yaw dynamic index greater than 1), just behind the rear axle. Now the rear axle will initially go in the other direction (the 'wrong' way), before reversing direction and finally developing path curvature and centripetal acceleration.
Another great book from the Millikens' is Chassis Design - Principles and Analysis. There is a section in that book that does a good job of describing the cornering sequence for vehicles having different yaw dynamics indices. Tim's post is pure gold, as usual. Having some background engineering knowledge is a necessary (but insufficient) prerequisite to getting the most out of the Milliken text. I think a search for 'center of percussion' might be helpful, if you haven't already done so.