Number Synthesis:

From Grumbler's criterion, the degrees of freedom for a kinematic chain may be described by the following expression:

 F = 3 (n - 1) - 2 j      . . . (7.1)

Here n is the number of links & j is the number of simple hinges.

 Assume,         

n₂ be number of binary links,

n₃ be number of ternary links, and                  

n₄ be number of quarternary links & so on

n = n₂ + n₃ + n₄ +.. . + n_i                         . . . (7.2)

 If a mechanism involves higher order hinges also, the equivalent number of simple hinges may be finding out by the following equation (Unit 2).

j =  j₁  + 2 j₂  + 3 j₃  + ... + ij_i

here j_i is the number of hinges connecting (i + 1) links.

A mechanism might contain higher pairs along with lower pairs. The degrees of freedom for such a mechanism might be expressed as follows (Unit 2) :

F = 3 (n - 1) - 2 j - h

Where h is the number of higher pairs.

To determine, the number of degrees of freedom of a mechanism, the following also can be considered:

 (a) Sometimes one or more links of a mechanism might have redundant degrees of freedom. If a link may be moved without causing any movement in the rest of the mechanism as illustrated in the following Figure (a), then the link is said to contain a redundant degree of freedom (Fr). In this mechanism, roller may rotate without causing any movement in the rest of the mechanism.

 (b) A mechanism might possess one or more links that do not introduce any additional constraint. Such links are redundant. A mechanism illustrated in Figure (b) contains links AB & CD that are identical and thus, every links leads to similar constraint. Therefore, while determining the degrees of freedom the redundant links (nr) must not be taken into account. Likewise redundant joints (jr) must also not be taken into account.

Therefore, at last, if a mechanism contains redundant links (nr), redundant kinematic pairs (jr), & redundant degrees of freedom (Fr), the degrees of freedom may be resolute by the following expression:

 F = 3 (n - n_r - 1) - 2 (j - j_r ) - h - F_r                           . . . (7.2(a))