DOF Gruebler’s Criterion Grashof’s four-bar mechanism Kinematic Inversions

2. Summary of previous lecture
• Link / Element
• Kinematic Pairs / Joints
• Classification of Pairs
• Kinematic Chain
• Mechanism
• Degrees of freedom

3. Today’s lecture
• DOF
• Gruebler’s Criterion
• Grashof’s four-bar mechanism
• Kinematic Inversions

4. Degrees of freedom

5. Degrees of freedom
F = 3 (L-1) – 2j – h – Fr
where Fr is the number of redundant degrees of freedom. Now, as the
above mechanism has a cam pair, its degree of freedom must be
found from Gruebler's criterion.
Total number of links = 4
Number of pairs with 1 degree of freedom = 3
Number of pairs with 2 degrees of freedom = 1
F = 3 (L- I) – 2j – 1h – Fr
= 3 (4 - 1) - 2x3 - l x 1- 1 = 1

6. Gruebler’s Criterion
• This criterion is used to find out whether an
assembly of links with 1 d.o.f. lower pairs is a
constrained mechanism or not.
• 3L – 2j – 4 = 0
• L – no. of links j – no.of lower pairs with
one d.o.f
Even Terms

7. F < 0 Pre-loaded structure
Super structure
F = 0 Structure
F = 1 Constrained Mechanism
F > 1 Unconstrained Mechanism

8. Formation of four-bar mechanism
• No. of links – 4, No. of pairs – 4.
• All the pairs are revolute pairs.
Links are :1. Fixed link or Frame
2. Input Link
3. Coupler
4. Output link or Follower
Most of the mechanisms are reversible

9. Assembly Condition
Lengths of links: Longest link - l
Shortest link - s
Intermediate links – p, q

10. Grashof’s four-bar mechanism
• Atleast one link will have full rotation if
S + l ≤ p + q

11. Grashof’s four-bar mechanism
s+l=p+q
But not having pair of equal links
i.e. 2,5,4 and 3
Same
s l p q  

12. Grashof’s four-bar mechanism
s+l=p+q
But having pair of equal links
i.e. 4,4, 3 and 3
s
(1) Parallelogram linkage
Double crank
(2) Deltoid linkage
s
l
l
s
l
l
s
s fixed – Double crank
l fixed – crank- rocker

13. GRASHOF’ S LAW
In a planar four bar revolute pair kinematic chain if the sum of the
lengths of the shortest and the longest links is less than or equal
to the sum of the lengths of the other two intermediate links at
least one link will have full rotation.
Mechanisms obtained from the kinematic chain satisfying these
conditions are known as Grashofian Mechanisms.
Mechanisms obtained from the kinematic chain which are not
obeying these conditions are known as Non-Grashofian
Mechanisms.

14. Kinematic Inversions
• Process of obtaining different mechanisms from
the same kinematic chain, by fixing different
links in turn, is known as kinematic inversion.
• Four inversions are possible from four-bar
kinematic chain.

15. Inversions of four-bar chain

16. Inversions of four bar Mechanisms are named based on
the motions of input link and output link.
Crank - Link with 360 degree rotation
Rocker/Lever – Link with less than 360 degree
rotation

17. Four- bar Inversions
• Crank – Rocker Mechanisms (Two)
• Drag Link / Double Crank Mechanism
• Double – Rocker Mechanism
• Above are Grashofian Inversions
• All four non-Grashofian inversions are
Double – Rocker mechanisms
s l p q  

18. KINEMATIC CHAIN MECHANISM
One Link Fixed
Inversion of the kinematic chain depends
upon which link is fixed.

19. Conditions for Inversions
• POSITION OF F0UR – BAR INVERSION
SHORTEST LINK
• Adjacent to the fixed link Crank – Rocker
• Fixed link itself Drag Link (Double Crank)
• Opposite to fixed link Double Rocker

20. Types of Kinematic Chains
• Four bar chain
• Single slider crank chain
• Double slider crank chain

21. Inversions of Four Bar Chain
• Beam engine (crank and lever mechanism)
• Coupling rod of a locomotive (Double crank mechanism)
• Watt’s indicator mechanism (Double lever mechanism)
The purpose of this mechanism
is to convert rotary motion into
reciprocating motion.

22. Inversions of Four Bar Chain
• Beam engine (crank and lever mechanism)
• Coupling rod of a locomotive (Double crank mechanism)
• Watt’s indicator mechanism (Double lever mechanism)
This mechanism is meant for transmitting rotary motion from
one wheel to the other wheel

23. Inversions of Four Bar Chain
• Beam engine (crank and lever mechanism)
• Coupling rod of a locomotive (Double crank mechanism)
• Watt’s indicator mechanism (Double lever mechanism)
Watt's straight line mechanism

24. (Next Lecture)
Kinematic Inversions
• Inversions of Single Slider Crank Chain
• Inversions of Double Slider Crank Chain

25. Contact details:
Rohit Kumar Singla
Email ID: rohit.kumar@thapar.edu