Inversion of four bar chain

A. Beam engine (crank and lever mechanism).

A part of the mechanism of a beam engine (also known as cranks and lever mechanism) which consists
of four links is shown in Fig. In this mechanism, when the crank rotates about the fixed centre A, the
lever oscillates about a fixed centre D. The end E of the lever CDE is connected to a piston rod which
reciprocates due to the

rotation of the crank. In other words, the purpose of this mechanism is to convert rotary motion into
reciprocating motion.

Fig- Beam engine

B. Coupling rod of a locomotive (Double crank mechanism).

The mechanism of a coupling rod of a locomotive (also known as double crank mechanism) which
consists of four links, is shown in Fig. In this mechanism, the links AD and BC (having equal length) act as
cranks and are connected to the respective wheels. The link CD acts as a coupling rod and the link AB is
fixed in order to maintain a constant centre to centre distance between them. This mechanism is meant
for transmitting rotary motion from one wheel to the other wheel.

Fig- Coupling rod of a locomotive

C. Watt’s indicator mechanism (Double lever mechanism).

Watt’s indicator mechanism (also known as Watt's straight line mechanism or double lever mechanism)
which consists of four links, is shown in fig. The four links are : fixed link at A, link AC, link CE and link
BFD. It may be noted that BF and FD form one link because these two parts have no relative motion
between them. The links CE and BFD act as levers. The displacement of the link BFD is directly
proportional to the pressure of gas or steam which acts on the indicator plunger. On any small
displacement of the mechanism, the tracing point E at the end of the link CE traces out approximately a
straight line.

Fig- Watt indicator mechanism

The initial position of the mechanism is shown in Fig by full lines whereas the dotted lines show the
position of the mechanism when the gas or steam pressure acts on the indicator plunger.

2. Single slider crank chain:-

Fig:- Single slider crank chain

A single slider crank chain is a modification of the basic four bar chain. It consists of one sliding pair and
three turning pairs. It is, usually, found in reciprocating steam engine mechanism. This type of
mechanism converts rotary motion into reciprocating motion and vice versa. In a single slider crank
chain, as shown in Fig., the links 1 and 2, links 2 and 3, and links 3 and 4 form three turning pairs while
the links 4 and 1 form a sliding pair.

The link 1 corresponds to the frame of the engine, which is fixed. The link 2 corresponds to the crank ;
link 3 corresponds to the connecting rod and link 4 corresponds to cross-head. As the crank rotates, the
cross-head reciprocates in the guides and thus the piston reciprocates in the cylinder.

Inversions of Single Slider Crank Chain:-

We have seen that a single slider crank chain is a four-link mechanism. We know that by fixing, in turn,
different links in a kinematic chain, an inversion is obtained and we can obtain as many mechanisms as
the links in a kinematic chain. It is thus obvious, that four inversions of a single slider crank chain are
possible. These inversions are found in the following mechanisms.

A. Pendulum pump or Bull engine.-

In this mechanism, the inversion is obtained by fixing the cylinder or link 4 (i.e. sliding pair), as shown in
Fig. In this case, when the crank (link 2) rotates, the connecting rod (link 3) oscillates about a pin pivoted
to the fixed link 4 at A and the piston attached to the piston rod (link 1) reciprocates. The duplex pump
which is used to supply feed water to boilers have two pistons attached to link 1, as shown in Fig.
Fig- Pendulum pump or Bull engine

B. Oscillating cylinder engine. –

The arrangement of oscillating cylinder engine mechanism, as shown in Fig. It is used to convert
reciprocating motion into rotary motion. In this mechanism, the link 3 forming the turning pair is fixed.
The link 3 corresponds to the connecting rod of a reciprocating steam engine mechanism. When the
crank (link 2) rotates, the piston attached to piston rod (link 1) reciprocates and the cylinder (link 4)
oscillates about a pin pivoted to

The fixed link at A.

Fig:- Oscillating cylinder engine

C. Rotary internal combustion engine. –

Sometimes back, rotary internal combustion engines were used in aviation. But now-a-days gas turbines
are used in its place. It consists of seven cylinders in one plane and all revolves about fixed centre D, as
shown in Fig., while the crank (link 2) is fixed. In this mechanism, when the connecting rod (link 4)
rotates, the piston (link 3) reciprocates inside the cylinders forming link 1.

Fig:- Rotary internal combustion engine

 D. Crank and slotted lever quick return motion mechanism.

This mechanism is mostly used in shaping machines, slotting machines and in rotary internal combustion
engines.

Fig:- Crank and slotted lever quick return motion mechanism

In this mechanism, the link AC (i.e. link 3) forming the turning pair is fixed, as shown in Fig. The link 3
corresponds to the connecting rod of a reciprocating steam engine. The driving crank CB revolves with
uniform angular speed about the fixed centre C. A sliding block attached to the crank pin at B slides
along the slotted bar AP and thus causes AP to oscillate about the pivoted point A. A short link PR
transmits the motion from AP to the ram which carries the tool and reciprocates along the line of stroke
R1R2. The line of stroke of the ram (i.e. R1R2) is perpendicular to AC produced. In the extreme positions,
AP1 and AP2 are tangential to the circle and the cutting tool is at the end of the stroke. The forward or
cutting stroke occurs when the crank rotates from the position CB1 to CB2 (or through an angle) in the
clockwise direction. The return stroke occurs when the crank rotates from the position CB2 to CB1 (or
through angle) in the clockwise direction.

E. Whitworth quick return motion mechanism.-

 This mechanism is mostly used in shaping and slotting machines. In this mechanism, the link CD (link 2)
forming the turning pair is fixed, as shown in Fig. The link 2 corresponds to a crank in a reciprocating
steam engine. The driving crank CA (link 3) rotates at a uniform angular speed. The slider (link 4)
attached to the crank pin at A slides along the slotted bar PA (link 1) which oscillates at a pivoted point
D. The connecting rod PR carries the ram at R to which a cutting tool is fixed. The motion of the tool is
constrained along the line RD produced, i.e. along a line passing through D and perpendicular to CD.
When the driving crank CA moves from the position CA1 to CA2 (or the link DP from the position DP1 to
DP2) through an angle in the clockwise direction, the tool moves from the left hand end of its stroke to
the right hand end through a distance 2 PD.

Now when the driving crank moves from the position CA2 to CA1 (or the link DP from DP2 to DP1 )
through an angle in the clockwise direction, the tool moves back from right hand end of its stroke to
the left hand end.

A little consideration will show that the time taken during the left to right movement of the ram (i.e.
during forward or cutting stroke) will be equal to the time taken by the driving crank to move from CA1
to CA2. Similarly, the time taken during the right to left movement of the ram (or during the idle or
return stroke) will be equal to the time taken by the driving crank to move from CA2 to CA1. Since the
crank link CA rotates at uniform angular velocity therefore time taken during the cutting stroke (or
forward stroke) is more than the time taken during the return stroke. In other words, the mean speed of
the ram during cutting stroke is less than the mean speed during the return stroke.
Fig- Whitworth quick return motion mechanism

Inversions of Double Slider Crank Chain

A. Elliptical trammels.-

It is an instrument used for drawing ellipses. This inversion is obtained by fixing the slotted plate (link 4),
as shown in Fig. 5.34. The fixed plate or link 4 has two straight grooves cut in it, at right angles to each
other. The link 1 and link 3, are known as sliders and form sliding pairs with link 4. The link AB (link 2) is a
bar which forms turning pair with links 1 and 3. When the links 1 and 3 slide along their respective
grooves, any point on the link 2 such as P traces out an ellipse on the surface of link 4, as shown in Fig. A
little consideration will show that AP and BP are the semi-major axis and semi-minor axis of the ellipse
respectively.

Fig- Elliptical trammels

 B. Scotch yoke mechanism. –

This mechanism is used for converting rotary motion into a reciprocating motion. The inversion is
obtained by fixing either the link 1 or link 3. In Fig., link 1 is fixed. In this mechanism, when the link 2
(which corresponds to crank) rotates about B as centre, the link 4 (which corresponds to a frame)
reciprocates. The fixed link 1 guides the frame.

Fig.- Scotch yoke mechanism

C. Oldham’s coupling.-

An oldham’s coupling is used for connecting two parallel shafts whose axes are at a small distance
apart. The shafts are coupled in such a way that if one shaft rotates, the other shaft also rotates at the
same speed. This inversion is obtained by fixing the link 2, as shown in Fig. The shafts to be connected
have two flanges (link 1 and link 3) rigidly fastened at their ends by forging. The link 1 and link 3 form
turning pairs with link 2. These flanges have diametrical slots cut in their inner faces, as shown in Fig.
The intermediate piece (link 4) which is a circular disc, have two tongues (i.e. diametrical projections) T1
and T2 on each face at right angles to each other, as shown in Fig. 5.36 ©. The tongues on the link 4
closely fit into the slots in the two flanges (link 1 and link 3). The link 4 can slide or reciprocate in the
slots in the flanges. When the driving shaft A is rotated, the flange C (link 1) causes the intermediate
piece (link 4) to rotate at the same angle through which the flange has rotated, and it further rotates the
flange D (link 3) at the same angle and thus the shaft B rotates. Hence links 1, 3 and 4 have the same
angular velocity at every instant. A little consideration will show that there is a sliding motion between
the link 4 and each of the other links 1 and 3. If the distance between the axes of the shafts is constant,
the centre of intermediate piece will describe a circle of radius equal to the distance between the axes
of the two shafts. Therefore, the maximum sliding speed of each tongue along its slot is equal to the
peripheral velocity of the centre of the disc along its circular path.