In 2D machines, why does higher pair joints deduct 1 D.O.F.?

Question

I have been taught that higher pair joints (e.g. Gears, cams, rollers) deduct 1 d.o.f. Due to the fact that they still allow 2 motions

translation along the tangent surface
rotation around the instantaneous Contact point

However, from my reasonings, the higher pair does constrain 2 motions

as it is non-slipping (like gears and rollers) it gives a relationship between the angle and the location (such as x=theta * r)
as it rotates on a surface it gives a relationship between x and y that it household be on the surface (such as x = y+2)

John Alexiou · Answer

You are right. A non-slip roller or gear has two constraints, one along the contact normal (no interpenetration) and the second one no relative slip velocity. So the total degrees of freedom added with such a joint is 1.

A slipping roller though only has once constraint (no interpenetration) and thus provides a total of two degrees of freedom to the system.

+-----------+------------+------------+
|   JOINT   |  #MOTIONS  | #REACTIONS |
+-----------+------------+------------+
|   FIXED   |     0      |      3     |
|   PIN     |     1      |      2     |
|   SLIDER  |     1      |      2     |
|   GEAR    |     1      |      2     |
|   ROLLER  |     2      |      1     |
|   NO-ROT  |     2      |      1     |
|   FREE    |     3      |      0     |
+-----------+------------+------------+

In 2D machines, why does higher pair joints deduct 1 D.O.F.?

One Answer

Add your own answers!

Ask a Question