Degree of freedom and Gruebler's criterion

Degree of freedom

The degree of freedom (DOF) or mobility of a mechanism is the number of independent input parameters (coordinates) that must be given to fix the position of every link with respect to the frame. A mechanism with $F = 1$ needs one input (for example one motor) to give a definite motion.

Gruebler's criterion

For a planar mechanism, a free link has 3 DOF. Fixing one link as the frame removes 3 DOF. Each lower pair (turning or sliding, 1 DOF) removes 2 DOF and each higher pair removes 1 DOF. So

$$F = 3(n-1) - 2j_1 - j_2$$

where $n$ = total number of links (including the frame), $j_1$ = number of lower pairs (single-DOF), $j_2$ = number of higher pairs (two-DOF).

• $F = 1$: constrained mechanism (definite motion).
• $F \le 0$: structure (no motion).
• $F \ge 2$: needs $F$ inputs, otherwise motion is indefinite.

Numerical (i)

Given $n = 7$, $j_1 = 8$, $j_2 = 1$.

$$F = 3(7-1) - 2(8) - 1 = 18 - 16 - 1 = 1$$

Numerical (ii)

Given $n = 6$, $j_1 = 7$, $j_2 = 0$.

$$F = 3(6-1) - 2(7) = 15 - 14 = 1$$

This is the Watt or Stephenson six-bar chain, which has one DOF.

Answer: (i) $F = 1$, (ii) $F = 1$; both mechanisms have definite motion with one input.