Tire data for suspension design

[Greg Locock]

Here's a rough way of doing it assuming zero real toe compliance and zero roll steer and zero weight transfer and linear range (and we approximate the cow to a sphere).

Rear axle load = Wr. At 1g the lateral laod is also Wr. This is provided by two tires with cornering stiffness k, N/deg. So rear axle, and hence body, slip angle is SAr= Wr/2/k

The same holds for the front axle, for it to turn at 1g the front axle must be operating at a slip angle of SAf=Wf/2/k. The difference between those 2 angles is the average steer angle of the front wheels. at 1g, ignoring thrust effects.

Everything else you do is a modification of this process.

[koushikatti]

Thanks Greg and Jersey!

I think i found out an approach to go about the slip angles.
1. Assume a yaw rate r and a slid slip angle beeta at the C.M. of the car.
2. Find out the vertical loads, toe angle and the camber angle of each tire in steady state cornering situation .
3. Find the slip angles of each tires in terms of beeta (vehicle side slip angle ) and r (yaw rate), this includes the effects of toe angle , track width and the distance of the rear and the front axle from the C.M.
4. Find the required lateral forces required in the front and the rear axle by force and torque balance about the CM.
5. iterate for various valued of beeta and r to converge to a solution
One additional thing that i assumed is that the lateral force in each tires will be in a direction perpendicular to the tire vertical plane , so the component of this force for each tire in a direction perpendicular to the vehicle velocity would be the force that would balance the inertia force (centripital force), the other component along the tire will be the drag force which the engine has to overcome.

[Jersey Tom]

If you are in a steady state condition and you know both the vehicle velocity and lateral acceleration (or path curvature or radius) then you can already determine the yaw rate exactly. There's no need to iterate on it.

[koushikatti]

What I thought was that yaw rate depende on the velocity (magnitude and the direction) of the four wheels and the path curvature. In this case the body side slip angle is not known and also the slip angles of each tire are not known, so how can the yaw rate be determined exactly.

[Lycoming]

That sounds really complicated, but if you're just wondering how yaw rate can be determined from vehicle velocity and radius, that's a simple algebraic formula, if you assume no over/under steer. If you make that assumption, then:

2*pi*r/v is the time it would take the car to traverse 360 degrees. Since the car must then also yaw 360 degrees in the same time frame, yaw rate is just 360/(2*pi*rv).

Assuming steady state of course.

You don't need to know body/tire slip angle for that, though they will probably be some non-trivial value.

[koushikatti]

That sounds really complicated, but if you're just wondering how yaw rate can be determined from vehicle velocity and radius, that's a simple algebraic formula, if you assume no over/under steer. If you make that assumption, then:

2*pi*r/v is the time it would take the car to traverse 360 degrees. Since the car must then also yaw 360 degrees in the same time frame, yaw rate is just 360/(2*pi*rv).

Assuming steady state of course.

You don't need to know body/tire slip angle for that, though they will probably be some non-trivial value.

@lycoming the assumption that the car has no under/over steer may lead to conclusions distant from what actually it is, and may lead to wrong values of slip angles, thats what i am worried about.


@shayanjameel08 i could not understand what you said can you just elaborate ?

[Tim.Wright]

That sounds really complicated, but if you're just wondering how yaw rate can be determined from vehicle velocity and radius, that's a simple algebraic formula, if you assume no over/under steer. If you make that assumption, then:

2*pi*r/v is the time it would take the car to traverse 360 degrees. Since the car must then also yaw 360 degrees in the same time frame, yaw rate is just 360/(2*pi*rv).

Assuming steady state of course.

You don't need to know body/tire slip angle for that, though they will probably be some non-trivial value.

@lycoming the assumption that the car has no under/over steer may lead to conclusions distant from what actually it is, and may lead to wrong values of slip angles, thats what i am worried about.


@shayanjameel08 i could not understand what you said can you just elaborate ?

For steady state conditions: LatAcc = Vel x YawRate
regardless of if there is understeer or oversteer.

[Greg Locock]

Note the method i described makes no mention of under or oversteer, that is an outcome from the calculation.

[koushikatti]

Hello,

I have been able to derive equations for a four wheel model (bicycle model with front and rear tracks included) to determine the slip angles and steerig wheel angle for a given pair of velocity and path curvature (radius). I iterated through various combinations of vehicle sideslip angle and steering angle to the point when the solution converged to give the required lateral force for steady state cornering. Now the next thing I am confused about it to determine the understeer gradient, all the articles that I read concerning this topic is based on the two wheel bicycle model. I am trying to extend the approach for a four wheel model. For the bicycle model the understeer gradient is given as

K = (Wf/Cf)-(Wr/Cr) = alpha_front - alpha_rear

To determine the understeer gradient for the two track model using this formula, can I use the mean (average) of the front left and the right slip angle and consider it as alpha_front (for using it in the formula) and similarly for the rear?

Regards!

[Jersey Tom]

K = (Wf/Cf)-(Wr/Cr) = alpha_front - alpha_rear

To determine the understeer gradient for the two track model using this formula, can I use the mean (average) of the front left and the right slip angle and consider it as alpha_front (for using it in the formula) and similarly for the rear?

Regards!

What you have above is not correct... understeer gradient is well, a gradient - a rate of change. It is expressed in degrees per G. Whereas what you have written in the final expression on the far right is the difference between two angles.

Barring that however, you would indeed replace "Cf" by "Clf + Crf" simple as that.

[Greg Locock]

You are on the right track, at a vehicle level understeer is (d SWA/dlataccg)/steering ratio. So the average front axle steer is used.

[koushikatti]

What you have above is not correct... understeer gradient is well, a gradient - a rate of change. It is expressed in degrees per G. Whereas what you have written in the final expression on the far right is the difference between two angles.

Barring that however, you would indeed replace "Cf" by "Clf + Crf" simple as that.

@ Jersey tom ..The expression the extreme right shoud be (alpha_front-alpha_rear)/lateral accleration , I forgot to write the latter term while posting the previous post. Thanks for the correction.

@Greg locock .. thanks for elaborating the expression and clarifing my doubt.

Well coming back to yaw rate. This is something that has been confusing me for very long time and somehow even after reading many articles I am unable to convince myself. In steady-state yaw rate is given by Velocity times the path curvature. This is what is confusing me. Considering a case in which the vehicle goes in a straight line (infinite radius turn). Now if a constant wind blows in a direction perpendicular to the vehicle heading direction, then the front and rear tires will react to the lateral force due to the wind and some slip angle will be induced. The front and rear slip angle may or may not be same , depending on the front and rear axle weight distribution and the tire stiffness. Consider a situation in which the front axle developes more slip angle as compares to the rear, so in this case the vehicle will start to move in the curve away from the wind (understeer), so this will induce a side-slip angle (due to the tire slip angles) and a yaw rate (depending on the relative front and rear slip angles , since the vehicle moves in a curved path) at the center of mass. So this case shows that yaw rate is dependant on the slip angles, then why is it not true for steady state cornering ? In that case the force due to wind will be replaced by the cornering force (mv^2 / R) . PLease correct me if I am going wrong somewhere.


Regards!

[Tim.Wright]

Since yawrate is the time rate of change of heading angle (units deg/sec), it is therefore not the slip angles which create the yawrate in your example, but the change of slip angles with time. More specifically its the sideslip velocity (typically at the CG) which is imparting an extra component of yawrate.

So in general your yawrate is:
Yawrate = Ay/vel + Betap

With the first term (Ay/vel) coming from the trajectory kinematics, and the second term (Betap = sideslip velocity) coming from the rate of change of slip angles which is superimposed on the trajectory yawrate.

In steady state, your sideslip angles at the tyres, and hence the CG are constant, so therefore the yawrate is only Ay/vel.

Its a neat relationship which allows you to obtain the slip angle reasonably well, without a slip sensor, under some carefully controlled conditions, such as a constant sinusoisal steering input on a flat road at constant speed.

[koushikatti]

@ Tim Wright .. Thanks for the reply , this would certainly help.

Currently I have equations for the slip angles of the four tires in terms of the vehicle yaw rate, body side slip angle and the steering angle of the front wheels. I have iterated through various combinations of steering wheel angle and body sideslip angle to converge to a solution. So here is what I plant to do next :

Understeer Gradient = ((average steering angle for the front wheels obtained through iterations ) -(akerman steering angle )) / lateral acceleration

akerman steering angle (in deg) = 57.3 L/R
lateral acceletarion = V^2 /gR

I hope I am going the right way!

Regards!

[Tim.Wright]

do you have Milliken?

The equations and their derivation for understeer gradient (lateral acceleration response to steer) are in there.