TYPES OF CONSTRAINED MOTIONS-

1)      Completely Constrained Motion- when the motion between a pair is limited to a definite direction irrespective of the direction of force applied. Example- the motion of a square bar in a square hole.

2)      Incompletely Constrained Motion- when the motion between a pair can take place in more than one direction. Example- a circular bar in a circular hole where the bar can either rotate or slide in the hole.

3)      Successfully constrained motion- when the motion between a pair is such that the constrained motion is not completed by itself but by some other means. Example- piston reciprocating inside an engine cylinder.

KINEMATIC CHAIN-

It is defined as a combination of kinematic pairs joined in such a way that each link forms a part of two pairs and relative motion between the links is completely or successfully constrained. When any one link is fixed then the kinematic chain becomes a mechanism.

 Relation between no. of pairs (p) and no. of links (l)

l= 2p – 4

 Relation between no. of links (l) and no. of joints (j)

j = ³/₂ l – 2

 klein’s formula,

j + ^h/₂ = ³/₂ l – 2

 When L.H.S > R.H.S, then the chain is locked.

When L.H.S < R.H.S, then the chain is unconstrained chain.

When L.H.S = R.H.S, then the chain is a kinematic chain.

MECHANISM-

When one of the links of a kinematic chain is fixed then the chain is called a mechanism. It transmits and modifies motion. The process of obtaining different mechanisms by fixing different links in a kinematic chain is known as inversion of the mechanism.

 KINEMATIC CHAINS AND THEIR MECHANISMS-

• Four bar chain or Quadric cycle chain- It consists of four links and each of them forms a turning pair. The inversions of four bar chain mechanism,

o   Crank and lever mechanism (oscillatory motion) – Here the crank is able to rotate completely but the follower can only give an oscillatory motion. Example- beam engine.

o   Double crank mechanism- Here in this inversion there is a complete rotation of the crank and the follower. Example- coupling rod of a locomotive.

o   Double lever mechanism- example- watt’s indicator mechanism, pantograph. 

• Slider crank mechanism- It consists of three turning pairs and one sliding pair. It converts rotary motion into reciprocating motion and is found in reciprocating steam engines. The inversions of slider crank mechanism,

o   1^st inversion- Example- Internal combustion engine, pump, reciprocating steam engine.

o   2^nd inversion- It is obtained by keeping the connecting rod fixed. It is called as crank and slotted lever mechanism. Example- oscillating cylinder engine.

o   3^rd inversion- It is obtained by keeping the crank fixed. Example- Whitworth quick return motion mechanism. This mechanism is used in shapers.

• Double slider crank chain mechanism- It consists of two sliding pairs and two turning pairs. The inversion of double slider crank chain mechanism, 

·         Elliptical trammels

·         Scotch yoke mechanism

·         Oldham’s coupling