# How to calculate optimal bend angle for moving monowheel vehicle?

Imagine you are driving a monowheel vehicle (https://www.google.com/search?q=monowheel) at some speed S (km/h) and you need to take a curve, so you steer by angle A in the respective direction, BUT to not fall over and crash you actually need to bend the whole thing (it's same for motorcycles and bikes) for some B angle.

So given speed S, steer angle A, tire size (radius R) and position of mass center C, can I calculate the optimum bend angle B that makes the vehicle successfully make the curve (not fall over)? And the threshold where no bending (not matter how much) would help to not to crash?

Assuming an ideal curve, (an arc of a circle), then the horizontal force (static friction from the road must supply the centripetal acceleration, and the normal force must support the weight. For stability, the resultant should be directed through the center of mass.

• Makes sense. I am thinking How do I put this ideas in united formula? Commented Jul 9, 2021 at 18:05

you can use those equations:

$$\tan(\alpha)=\frac{m\,v^2/r}{m\,g}=\frac{v^2}{r\,g}$$

with a "single track model"

$$\frac{1}{r}\approx\frac{\delta}{L}$$

thus $$\tan(\alpha)=\frac{v^2}{L\,g}\,\delta$$

where

• $$v~$$ vehicle speed
• $$r~$$ curve radius
• $$\delta~$$ steer angle
• $$m~$$ vehicle mass
• $$~L$$ wheel base
• $$~\alpha$$ bend angle
• And what is g ? L is space between wheels? In my case it's 0? :) Commented Jul 10, 2021 at 15:49
• I guess by now g is gravity constant? Commented Jul 10, 2021 at 15:55
• this is correct, but what is your wheel base, because, you just have one wheel?
– Eli
Commented Jul 10, 2021 at 16:00
• yes monowheel vehicle has one wheel. So I have no idea how to represent the wheel base. Commented Jul 10, 2021 at 16:02
• you can take some “fictive” wheel base
– Eli
Commented Jul 10, 2021 at 16:04