Introduction

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.

Kinematic chain and inversion

A kinematic chain is an assembly of links connected by kinematic pairs so that relative motion of the links is definite (one DOF). Fixing different links of the same chain one at a time gives different mechanisms. Each of these is called an inversion of the chain.

Single slider crank chain

It has four links: a crank, a connecting rod, a slider (sliding pair) and the frame, joined by three turning pairs and one sliding pair.

   crank 2      rod 3     slider 4
   O ---------- A ------- B ==> (guide, link 1)

A chain of four links has four inversions. The first (fixing the cylinder) is the basic engine mechanism.

Inversion 1: fixed link = cylinder (frame)

The basic reciprocating engine or compressor mechanism. The crank rotates, the slider reciprocates.

• Application: reciprocating steam/IC engine, reciprocating pump.

Inversion 2: fixed link = crank

The connecting rod rotates about the fixed crank pin and the slider oscillates in a rotating slotted frame.

• Application: Whitworth quick-return motion (shaper) and the rotary engine (Gnome engine).

Inversion 3: fixed link = connecting rod

The crank becomes a rocker and the slider moves in a slot that oscillates about the fixed rod pivot.

• Application: Crank and slotted-lever quick-return mechanism (shaper), oscillating cylinder engine.

Inversion 4: fixed link = slider

The slider is fixed, the connecting rod oscillates, and the crank end moves in a sliding fashion.

• Application: hand pump (the plunger link is fixed) and pendulum pump / Bull engine.