Lecture - 10 ( four bar mechanism and grashof's law)

Inversions of mechanisms 

Different mechanisms can be obtained by fixing different-different links of the same kinematic chain. These are called as inversions of the mechanism. 
By changing the fixed link, the number of mechanisms which can be obtained is equal to the number of links. The inversion of a mechanism does not change the motion of its links relative to each other.
Remember if number of links are 'L' than the number of inversions will be less than or equal to 'L' .

FOUR BAR MECHANISM 

This mechanism consist of 4 links and 4 turning pairs. One of the most useful and most common mechanisms is the four-bar linkage. In this mechanism, the link which can make complete rotation is known as crank. The link which oscillates is known as rocker or lever. The link connecting these two is known as coupler and the link which is stationary is known as fixed link. Four bar mechanism is also called as Quadratic cycle mechanism.
Example is shown in below figure

Fixed position is the best position of mechanism because fixed link govern both input and output.
Coupler is the worst position of mechanism because it is just a transmitting body, coupler has no effect on input and output.

INVERSIONS OF FOUR BAR MECHANISMS ARE LISTED BELOW - 
  • DOUBLE CRANK MECHANISM - when 2 links of mechanism are under complete rotation.
  • CRANK-ROCKER MECHANISM - when 1 link is under complete rotation and 1 link is under oscillation motion.
  • DOUBLE ROCKER MECHANISM - when 2 links of mechanism are under oscillation.

GRASHOF'S LAW 

For the continuous relative motion between the number of links in mechanism, the summation of lengths of shortest and longest links should not be greater than the summation of lengths of other 2 links.
'S' represents the shortest link of mechanism
'L' represents the longest link of mechanism
'P' & 'Q' represents other two links
    Grashof's law---   S + L ≤ P + Q 
for continuous relative motion above equation must be satisfied. Here continuous relative motion represents that atleast one link must have continuous motion or rotational motion.
Rotational motion is a continuous motion and oscillation motion is not a continuous motion.
 # If (S+L) < (P+Q)
In this condition grashof's law is satisfied so we can get continuous relative motion.


  • If smallest link (S) of mechanism is fixed than we get double crank mechanism.



  • If smallest link (S) of mechanism is adjacent to fixed link than we get crank-rocker mechanism.
  • If smallest link (S) of mechanism is coupler link than we don't get continuous relative motion hence we get double rocker mechanism
# IF (S+L) = (P+Q)
In this condition also our mechanism satisfies grashof's law, hence we can get continuous relative motion. But here 2 cases will generate as follows-
CASE 1 - When lengths of each link is different (example- S=2 ; L=5 ; P=3 ; Q=4)

  • When 'S' is fixed we get double crank mechanism
  • When 'S' is adjacent to fixed link then we get crank-rocker mechanism
  • When 'S' is coupler link then we get double rocker mechanism
CASE 2 - When we have 2 pairs of equal length (example - S=P ; L=Q or S=Q ; L=P)

  • We get parallelogram linkage or golden linkage. Here whether we fix 'S' link or we fix 'L' link we always get double crank mechanism.

  • We get deltoid linkage. Here if 'S' is fixed then we get double crank mechanism and if 'L' is fixed then we get crank-rocker mechanism.
# IF (S+L) > (P+Q) 
In this condition of mechanism grashof's law is not satisfied, therefore there is no possibility of continuous relative motion at all. Whether we fix any link we always get double rocker mechanism.

PRACTICAL EXAMPLES OF FOUR BAR MECHANISM 

  • Beam engine - in this mechanism rotation is converted into oscillation, it is a crank-rocker mechanism. This mechanism is used in car wipers and sewing Machines
  • Coupling rod of locomotives - in this mechanism both input and output are in rotation. It is a double crank mechanism. This mechanism is also called as parallelogram linkage.

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