Vehicle Radius

[Mad_DadD]

Can anyone derive an equation to calculate the radius a vehicle will follow given it's wheel base and width between wheels?

[Rule]

Can anyone derive an equation to calculate the radius a vehicle will follow given it's wheel base and width between wheels?

That doesn't make sense...

Oh wait, the answer is r = 42.

[shout]

Can anyone derive an equation to calculate the radius a vehicle will follow given it's wheel base and width between wheels?

That doesn't make sense...

Oh wait, the answer is r = 42.

Clap.

[MyndFyre]

That doesn't make sense...

Oh wait, the answer is r = 42.

It *does* make sense.

Given the ability of an axle to turn its wheels theta degrees, the width of each turning wheel, and the base width of the car, what is the radius of the circle the car would form if the steering wheel was kept in a constant position? 

That's what he was asking.  I'm not sure why you felt the need to quip.

[Rule]

That doesn't make sense...

Oh wait, the answer is r = 42.

It *does* make sense.

Given the ability of an axle to turn its wheels theta degrees, the width of each turning wheel, and the base width of the car, what is the radius of the circle the car would form if the steering wheel was kept in a constant position? 

That's what he was asking.  I'm not sure why you felt the need to quip.

Even as you have phrased it, the question remains ambiguous.  For example, there will be a limit on the radius of the circle based on the length of the car, hence a mathematical model that does not consider this would give inaccurate r values for a range of theta values.

However, I tried to come up with a rough estimate.  I actually feel that it's very possible that I'm wrong.  I would be interested in seeing the correct answer though, so if anyone knows it, please post. 

(This approach is too intuitive for me to be feel reassured)...

Observation:  If the wheels are turned at a constant theta degrees from the front of the car, then the car will travel in a circle of fixed radius a.  This is because the car would be changing direction "at the same rate" throughout its journey.  Or more precisely, the change in the slope of the tangent lines on the path per change in arclength would be a positive constant.

Intuitive guess:
In a given time, the car will move along a certain arclength, S.   The instantaneous change in direction of the car, dPhi, per infinitesimal change in arclength, dS, will be equal to the degrees the wheel is turned.

hold a constant, consider an arc of the circle, S:
S/(2pi*a) = phi/(2pi)            (the ratio of phi to the degrees in a circle is equal to the ratio of the arclength to the circumference of the circle)   
dphi/dS = 1/a
a=r ,  dphi/dS = theta
1/r = Theta
r = 1/(Theta) metres...   ( |theta| < pi ) 

If we consider the wheel having width, the trace of the circle will have width, so the question doesn't make sense.  If we consider the width of the car, there will be two circles, one radius r, the other radius r + wheelbase width, so the question doesn't make sense.

(Problem:  I think that theta = k * dPhi/dS.  I am assuming here that k = 1metre, because there doesn't seem to be any compelling reason it would equal something else.  Also, there is another obvious problem -- the radius should be allowed to get smaller than 1/pi metres).