Lecture - 9 ( Grubler's approach and simple mechanism)

GRUBLER'S EQUATION

Grubler explain that to convert any linkage combination in a Kinematic chain or constrained motion mechanism, the following equation must be satisfy
    3L - 2J - 4 = 0  
This explanation is given for those mechanism whose degrees of freedom is one (F=1) and having no higher pairs (H=0).

If we apply Grubler's conditions in KUTZBACH equation we get Grubler's equation.
F = 3(L-1) - 2J - H ; (KUTZBACH equation)
now apply Grubler's conditions (F=1, H=0)
1 = 3(L-1) - 2J - 0
0 = 3L - 2J - 4 ; (GRUBLER'S EQUATION)

Grubler's equation is derived from KUTZBACH equation after applying certain conditions, therefore Grubler's equation is also called as modified Kutzbach equation.

CONCLUSION OF GRUBLER'S APPROACH

As we know for making a mechanism, Grubler's equation must be satisfied.We know the term of grubler's equation '-2J - 4' will always be even for each value of 'J'. To satisfy the equation, the term '3L' must also be even and '3L' will be even when value of 'L' will be a even number. 
Now conclusion we get is that to make a mechanism, number of links 'L' will always be an even number. 
So our first even number is '2' , but 2 links can never form a mechanism. So we move to next even number that is '4'. Therefore according to Grubler's approach where no higher pair is present, minimum number of links required to make a mechanism is '4'. According to GRUBLER'S equation if L=4 then value of J should be 4 to satisfy the equation.
It forms our first mechanism without any higher pair which is called as Simple mechanism. 
Simple mechanism is the mechanism which is formed with 4 links and 4 lower pairs or binary joints.
Example is 

TYPES OF SIMPLE MECHANISM 

  • FOUR BAR MECHANISM
  • SINGLE SLIDER CRANK MECHANISM
  • DOUBLE SLIDERS CRANK MECHANISM

Points to remember - 
  • Minimum number of links requirement to make a mechanism with lower pairs only or without higher pair are 4.
  • But the minimum number of links required to make a mechanism with both lower and higher pair are 3. Example is shown below - 

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