# How to calculate traction force w.r.t. rotational speed of a shaft mounted at a certain angle on a surface?

I am designing a new mobile robot platform and i need a generic solution for the robot in order to establish a proper mathematical modeling. My question is about calculation of the traction force of the robot with respect to rotational speed of the direct contact motor shafts which is mounted as the figure below.

There is a vehicle with 3 contact points with the ground. Two contact points are edges of steel cylindrical shafts with diameter d_c. Other contact is provided by spherical wheel with diameter d_s in order to provide static balance.

Cylindrical shafts are rotating with a high speed, n. The angle between the shafts and the ground is a. Coefficient of static friction is f. Shaft is slipping on the ground because of high speed and low friction. The weight of the vehicle is w. The distances between the contact points and the center line is b.

How to calculate traction force T with respect to rotational speed n for a shaft mounted at a certain angle a on a surface?

Given Parameters:

These parameters are just for example, i need a generic mathematical solution.

d_c = 1 mm

d_s = 10 mm

b = 25 mm

n = 10000 rpm

a = 60 degrees

f = 0.25

w = 0.1 kg

Required:

T(n, a) = ?

Assuming each cylinder take 1/3 of the mass of your robot, it generates a traction of $$\ T=k*m/3$$