*Originally posted by rbayer *
**Anyway, there’s a reason I posted in Q&A: none of the info here is anything new to me. What I’m asking is how will increasing our surface area increase our traction/friction/whatever you want to call it. If we double the width of our wheel, for example, does our traction double? Does it go up by root 2? The cube root of 2? Or is it not possible to quantify? **
If your wheel is a smooth surface (i.e., aluminum, two wheel chair wheels), double the area will do nothing to aid your traction at all. It will increase by a factor of 1.
If the material on your wheel is not a smooth surface, in that it interacts with the carpet fibers, your traction will increase. There is no set factor that determines how effective this is.
Whereas the smooth surface of a wheelchair wheel propels the robot due to friction, an irregular surface that meshes with the carpet fibers propels the robot due to both friction (the sliding force between the smooth portions of the wheel and the carpet) and torque (the contact between two surfaces perpendicular to the direction of movement)
Again, think of two spur gears meshed together. Assume that the one on top is your wheel and the one below is the carpet. If each ‘gear’ had no teeth, power transmission occurs because of friction between the two surfaces. When you add teeth to each gear, friction along the surface of the gear is not a factor. Instead, the force is transferred from the face of one tooth on one gear to the face of a tooth on the second gear via torque.
When the motors output a torque that is greater than the force of friction between two smooth surfaces, the surfaces slip. When those same surfaces are meshed, they cannot slip - so power is transferred more efficiently (or, the teeth break ).
Gears are simplified example of the phenomenon that is taking place when irregularly shaped objects interact with the carpet. Because the carpet and belting materials are flexible and irregular, the relationship is similar, but not quite the same.
EDIT: So, to really answer your question; No, the equation you provided isn’t correct. There isn’t really any simple equation to characterize this behavior because the contact surfaces vary at any given movement in myriad ways, including contact surface area and angle, among others. If determining a mathematical relationship between meshed services were an absolute necessity, I would conduct experiments and determine “mu,” realizing that it isn’t friction at all that we’re dealing with. Instead, you’re encapsulating the interaction of the many faces into one system and treating that system like friction. It’s not friction, however. I hope that helps.