Kinematic Chain

Kinematic chain is define as the combination of kinematic pairs in which each link forms a part of two kinematic pairs and relative motion between the link is either completely constrained or successfully constrained.

Kinematic Chain

A kinematic pair is joint of two elements that permits relative motion. The relative motion between the elements of links that form a pair is required to be completely or successfully constrained. {alertInfo}

It can be said that in a kinematic chain, links must have motion in one direction either by itself or by external means. If number of links are l and number of lower pairs are p then to form a kinematic chain

l=2p-4                     …(i)

In equation (i), we have not taken higher pairs.

In the case of a higher pair.

1 higher pair= 2 lower pairs

For constrained motion other criteria is

\( j+\frac{H}{2}=\frac{3}{2}l-2 \)       …(ii)

Where, j= Number of joints

H= Number of higher pair

l=Number of links

If a higher pair exists then no need to add one additional link in Equation (ii).

It is necessary that is kinematic chain satisfies the equation (i) and (ii). {alertInfo}

A W Klein specified a rule that in equation (i) and (ii), if

LHS>RHS, the chain is locked or structure

LHS=RHS, chain is constrained

LHS<RHS, chain is unconstrained


Determine the motion of kinematic chain.

Solution: In this Chain there are

l=6, p=7, h=0




Hence, constrained motion.

Let see a flow chart from link to machine

flow chart from link to machine

Inversion of a Kinematic Chain

Assume that there are n number of links in a mechanism then by fixing each link we can get n number of mechanisms. Thus, the process of choosing different links of a chain for the frame is called kinematic inversion.

Now, let see inversion of some kinematic chain.

Double slider-crank kinematic chain

A kinematic chain consists of two pairs and two sliding pairs is called a double slider crank chain. Links 3 and 4 reciprocate, link 2 rotates and link 1 is fixed. Two pairs of the same kind are adjacent.

Double slider-crank kinematic chain
Double slider-crank kinematic chain

First Inversion (Elliptical Trammel)

It is a device to draw ellipses. Fig below shows an elliptical trammel in which two grooves are cut at right angles in a plate that is fixed. The plate forms the fixed link 4. Two sliding blocks are fitted into the grooves. The slides forms two sliding links 1 and 3. The link joining slides form the link 2. Any point on link 2 or on its extension traces out an ellipse on the fixed plate, when relative motion occurs.

Elliptical Trammel
Elliptical Trammel

X= BC cosӨ
or \( \frac{x}{BC} \)= cosӨ
y=AC sinӨ
or \( \frac{y}{AC} \)= sinӨ
Squaring and adding, we get
\( \frac{x²}{BC²} + \frac{y²}{AC²}=1 \)
Which is the equation of an ellipse.

Second Inversion (Scotch Yoke)

If any of the slide-blocks of the first inversion is fixed, the second inversion of the double-slider-crank chain is obtained. This mechanism gives SHM. Its early application was on steam pumps, but it is now used as a mechanism on a test machine to produce vibrations. It is also used as a sine-cosine generator for computing elements Fig below shows a sketch of scotch yoke mechanism.

Scotch Yoke
Scotch Yoke

x = r - rcosӨ
x = r(1- cosӨ)
x = r(1- cos ωt)
v = \( \frac{dx}{dt} \) = rω sin ωt
a =\( \frac{d²x}{dt²} \) = rω² cos ωt

A scotch-yoke mechanism is used to convert the rotary motion into a sliding motion. As the crank 3 rotates, the horizontal portion of the link 1 slider as reciprocates in the fixed link 4.

Third Inversion (Oldham's coupling)

The Oldham's coupling shown in Fig Below is used to connect two parallel shafts, the distance between whose axes is small and variable. The shafts connected by the coupling rotates at the same speed. The shafts have flanges at the ends, in which slots are cut. These form links 1 and 3. An intermediate piece circular in shape and having tongues at right angles on opposite sides, is fitted between the flanges of the two shafts in such a way that the tongues of the intermediate piece get fitted in the slots of the flanges. The intermediate piece forms link 4, which slides or reciprocates in links 1 and 3. The link 2 is fixed.

Oldham's coupling
Oldham's coupling

Maximum sliding speed of each tongue along at slot = distance between the axes of the shaft X angular velocity of each shaft

Four Bar Kinematic Chain

This mechanism consists of four turning pairs

four bar kinematic chain

Similarly, by fixing links 3 and 4, we get 3rd and 4th inversions respectively.

four bar kinematic chain

Watt mechanism is an inversion of the four bar chain mechanism.

Single Slider Kinematic Chain

It is also a four bar chain mechanism consisting of three turning pairs and one sliding pair.

First inversion

Reciprocating engine mechanism is its example.

single slider kinematic Chain first Inversion

Second inversion

This mechanism is used in a shaper machine known as Whitworth quick return mechanism. This mechanism is shown in figure as

single slider kinematic Chain second inversion

Movement of driving crank <=> tool movement 

(i) ac' ➡ ac" through an angle Ө, f' ➡ f"

(ii) ac" ➡ ac' through an angle Ф, f" ➡ f'

$$ \frac{Time of cutting stroke}{Time of return  stroke}=\frac{\theta/\omega  }{\phi /\omega } $$

$$=\frac{\theta}{360°-\ theta} $$

(Because crank moves with constant angular velocity ω)

Third inversion

Third inversion of this mechanism is used in shaper and slotter machines and known as crank and slotted lever type quick return mechanism.

single slider kinematic Chain third inversion

Movement of driving crank⇔ tool movement

(i) b′ ➡ b″ through angle Ө,  f' ➡ f”

(ii) b″ ➡ b′ through angle Ф,  f" ➡ f'

As driving crank rotates with uniform angular velocity.

$$ \frac{Time of cutting stroke}{Time of return stroke}=\frac{\theta/\omega }{\phi /\omega } =\frac{\theta}{360°-\ theta} $$

Fourth inversion

Fourth inversion provides the mechanism of hand pump

single slider kinematic Chain fourth inversion

In both mechanisms, quick-worth and crank and slotted lever, cutting stroke is corresponding to high angle turns. {alertInfo}


[1] Theory of Machines and Mechanisms(John J.Uicker, Gordon R. Pennock, Joseph E Shingley )

[2] Theory of Machines & Vibration(S.S.Rattan)