How to calculate effect of elevation change on vehicle normal force?

Question

My name is Ricky and I run a site called Race Optimal.

http://www.raceoptimal.com/about/physics

I'm working on converting the 2D physics model to 3D. I have an algorithm to calculate a continuous piecewise polynomial for the x, y, and z coordinates. As the z coordinate changes, more and less normal force is experienced depending on the acceleration relative to the z-axis. Typically this would be solved by calculating the radius of curvature. However, since the path is 3D, I'm not sure how to calculate curvature relative to the z axis. Any suggestions?

What if you just took a derivative of the polynomial to get the acceleration in the Z direction, then used $F = ma$? — user363165, Sep 07 '16 at 00:36
I think the problem with that approach is that acceleration depends on velocity, and achievable velocity depends on how much the normal force is affected. If I know the effective radius of curvature I can calculated max velocity explicitly. — , Sep 07 '16 at 01:49
Look at radius of curvature for 3D curvature and curvature of space curve — John Alexiou, Sep 07 '16 at 02:15
Cool question, but it is more about math at this stage. Later you should ask about physics here. — John Alexiou, Sep 07 '16 at 02:21
For some reason I missed the section of the Wikipedia article on space curves when I was researching. Thank you! — , Sep 07 '16 at 04:17

score 1 · Answer 1 · answered Sep 07 '16 at 04:11

Wikipedia to the rescue!

https://en.wikipedia.org/wiki/Curvature#Curvature_of_space_curves

Basically, when the 3D curve is in parameterized form, the first derivative with respect to t gives the tangent vector to the path. The second derivative gives the normal vector pointing in the direction of acceleration due to curvature. Radius of curvature is then given by

$$ R=\frac{(x'^{2}+y'^{2}+z'^{2})^{3/2}}{\sqrt{(z''y'-y''z')^{2}+(x''z'-z''x')^{2}+(y''x'-x''y')^{2}}} $$

The radius of curvature relative to the vertical can be found by dividing this total radius by the z component of the unit normal. The curvature relative to the x and y directions, i.e. for cornering, can be found by dividing by the magnitude of the x and y components combined. Thus, the effective radius of curvature in for cornering or vertical acceleration with be equal to or larger than the 3D radius of curvature.

How to calculate effect of elevation change on vehicle normal force?

1 Answers1