How does friction applied to a rolling sphere in 3 dimensional space

Question

I am doing a rigid body physics simulation program. This simple program can simulate the behaviour of a rigid sphere rolling/sliding on some surface. I need help with computing the friction force on the 3-dimensional space.

Basically the ball have an initial angular velocity vector $\vec{\omega}$ and translation vector $\vec{v}$ at the same time. I understand that if their magnitude $\omega$ and $v$ fulfill $v=\omega r$, then the ball is in pure rolling, and there will be no sliding friction forces, but what if $\vec{\omega}$ and $\vec{v}$ are not perpendicular. In such situations, sliding and rolling will create friction because they are not relatively still to the contact surface. How can I compute the sum friction force in this situation?

Similarly, what if $\vec{\omega}$ is perpendicular to $\vec{v}$ but not perpendicular to the contact normal? We can observe the ball is rolling in an arc, but how does the friction contribute to this situation?

Answer 1

The plane is described with u , v parameter and the trajectory on the plane with s parameter . Thus the position vector to the contact point is

$$ \vec R=[x(u(s),v(s)), y (u(s),v(s)),z (u(s),v(s))]^T$$

From here you can obtain the tangent vector $~\hat t(s)~$ and the normal vector $~\hat n(s)~$

The friction force is $~F_\mu=-\mu\,N\,\hat t~$ where N is the normal force $~N=m\,g\,\vec e_z\cdot \hat n~$

The rolling condition is $~\omega\,r=v=\dot s~$

Notice

the rotation axis of the ball is $~\hat t\times \hat n~$