• Mechanisms • Degrees of Freedom (DOF) for Plane Mechanisms• Kutzbach Criterion • Grubler’s Criterion• Inversion of Mechanisms• Types of Kinematic Chains

Lecture Outline

Mechanisms Mechanism: When one of the links of kinematic chains is fixed, the chain is known as mechanisms.

Used for transmitting or transforming motion. For example Engine indicators. Typewriters.

Simple Mechanisms: A mechanisms with four links is known as simple mechanisms.

Compound Mechanisms: A mechanisms with more than four links is known as Compound mechanisms.

Machine:When a mechanism is required to transmit power or to do some particular type of work, it then becomes a machine.

Mechanisms Degree of Freedom: DOF (Also called as movability) is defined as the number of input parameters (usually pair variables) which must be independently controlled in order to bring the mechanism into useful engineering purpose.

Four bar chain, as shown in figure (a) shows that only

one variable θ is needed to define the relative positions of all links.

Five bar chain, as shown in figure (b) shows that two variables θ1 and θ2 are needed to define completely the relative positions of all the links.

Kutzbach CriterionKutzbach equation to calculate the movability of a mechanism having plane motion is

n = 3(l-1) - 2j – h

If there are no two degree freedom pairs (i.e higher pairs), then h = 0.

n = 3(l-1) – 2j - h

Kutzbach CriterionFor binary joint n = 0

For four bar mechanism n = 1For five bar mechanism n = 2For five bar mechanism n = 0

For six bar mechanism n = -1 (redundant constraints forming statically indeterminate structure)

Kutzbach CriterionFor higher pair or two degree of freedom joints is shown as below.

n = 3 (3-1) – 2 * 2 – 1 = 1

n = 3 (4-1) – 2 * 3 – 1 = 2

Grubler’s Condition:

Inversion of MechanismsAs discussed in a kinematic chain one of a fixed link is called as mechanism. We can obtain a number of mechanisms by changing the fixed link.

Mechanism Inversion:Method of obtaining different mechanism by fixing different links in a kinematic chain is known as inversion of mechanisms.

Driver:The part of mechanism which initially moves with respect to frame or fixed link is known as driver.

Follower:The part of mechanism to which the motion is transmitted is called as follower.

Note: Most of mechanisms are reversible. For example, in a reciprocating engine the piston is driver and flywheel is follower while in a reciprocating air compressor, the flywheel is a driver.

Types of Kinematic ChainsFour lower pair kinematic chain system is the most important in which each pair being a sliding or a turning pair.

The kinematic chain with four lower pairs are as follow:

1. Four bar chain or quadric cyclic chain.2. Single slider crank chain.3. Double slider crank chain.

Four Bar Chain or Quadric Cyclic Chain

Four bar chain or quadric cyclic chain is the simplest of kinematic chains. It contains four links each forms a turning pair at A,B,C and D.

According to Grashof’s law :

Sum of smallest and largest link lengths < sum of the remaining two linkage lengths

Important consideration in designing a mechanism is to ensure that the input crank makes a complete revolution relative to other links.

Four Bar Chain or Quadric Cyclic Chain

From figure the shortest link will make a complete revolution relative to the other three links, satisfying Grashof’s law.

1. Shortest link is known as crank or driver.

2. Link 2 BC is known as lever or rocker or follower as it makes a partiall

rotation or oscillates only.

3. Link 3 CD which connects the crank and lever is known as connecting rod

or coupler.

4. The fixed link AB is known as frame of the mechanism.

Inversion of Four Bar Chain

Beam Engine (crank and lever mechanism)

In this when crank rotates about fixed centre A, the lever oscillates about a fixed centre D.The end E of the lever CDE is connected to piston rod which reciprocates due to rotation of crank.

This mechanism converts rotary motion into reciprocating motion.

Inversion of Four Bar Chain

Coupling rod of a locomotive (Double crank mechanism)

Consists of four links. The links AD and BC (equal length) act as cranks and are connected to respective wheel.

CD acts as a coupling rod and AB is fixed to maintain constant centre to centre distance between them.

Used to transmit rotary motion from one wheel to other wheel.

Inversion of Four Bar Chain

Watt’s indicator mechanism (Double lever mechanism)

Also known as watt’s straight line mechanism or double lever mechanism consists of four links.

Four links are : fixed link at A, link AC, link CE and link BFD.

Link BF and FD makes one single link as there is no relative movement between them.

CE and BFD acts as levers. Displacement of link BFD is proportional

to pressure of gas or steam which acts on indicator plunger.

E traces out as a straight line. Initial and final position of the

mechanism is shown by dotted lines.

Single Slider Crank Chain

Slider crank chain is a modification of four bar chain. It consist of one sliding and three turning pairs.

Usually found in reciprocating steam engine. This mechanism converts rotary motion into reciprocating motion.

In single slider crank chain, links 1 and 2, links 2 and 3, and links 3 and 4 form three turning pairs while links 4 and 1 from a sliding pair.

Inversions of Single Slider Crank Chain

Pendulum pump or bull engine:

In this mechanism the inversion is obtained by fixing the cylinder or link 4(i.e sliding pair)

when the crank (link 2) rotates, the connecting rod (link 3) oscillates about a pin pivoted connected at A.

The piston attached to (link 1) reciprocates along point A.

The duplex pump which is used to supply feed water to boilers have two pistons attached to link 1.

Inversions of Single Slider Crank Chain

Oscillatory cylinder :

To convert reciprocating motion to rotary motion.

Link 3 forming a turning pair is fixed. When crank (link 2) rotates, piston attached to

piston rod (link 1) reciprocates and cylinder (link 4) oscillates a pin pivoted point A.

Rotary internal Combustion Engine:

Previously used in aviation. Consist seven cylinders in one plane and

revolves about fixed point D. When connecting rod (link 4) rotates, the

piston (link 3) reciprocates inside cylinder forming link 1.

Inversions of Single Slider Crank Chain

Crank and Slotted lever quick return mechanism:

The link AC (link 3) forming the turning pair is fixed.

link 3 corresponds to connecting rod of a reciprocating steam engine.

Driving crank CB revolves with uniform angular speed about fixed centre C.

A sliding block attached to crank pin at B slides along the slotted bar AP and thus causes AP to oscillate about A.

Small link PR transmits motion from AP to ram which carries the tool and reciprocate along line of stroke R1R2.

Line of stroke is perpendicular to AC.

Inversions of Single Slider Crank Chain

In extreme positions AP1 and AP2 are tangential to circle and cutting tool is at the end of stroke.

Forward stroke occurs when crank rotates from position CB1 to CB2 through angle β in clockwise direction.

The return stroke occurs when crank rotates from CB2 to CB1 through angle α in clockwise direction.

Inversions of Single Slider Crank Chain

Since tool travels a distance of R1R2 during cutting and return stroke. So length of stroke is

End of Lecture 2Thank You!