This document discusses kinematic chains, mechanisms, inversion, and Grashoff's law. It defines a kinematic chain as a combination of kinematic pairs that connect links, with each link forming part of two pairs. A mechanism is a kinematic chain where one link is fixed. Inversion is obtaining different mechanisms by fixing different links, which changes the absolute motion but keeps relative motion the same. Grashoff's law states the sum of the shortest and longest links must be less than or equal to the sum of the remaining links for continuous relative motion in a 4-bar linkage. Examples of inversions include double crank, crank-rocker, and double rocker mechanisms.

1. (Under the guidance of Prof. Rangaraj M Desai)
Prepared by:
Syed Zabuilla
Chunkyraj Kh
RU1441MDD007
RU1441MDD008
A Presentation on
KINEMATIC CHAIN,
MECHANISM, INVERSION &
GRASHOFF’s LAW

2. Contents
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Kinematic Chain
Mechanism
Inversion
Grashoff’s Law

3. 3
KINEMATIC CHAIN:
 Kinematic chain may be defined as a combination of kinematic
pairs, joined in such a way that each link forms a part of two
pairs and the relative motion between the links or elements is
completely or successfully constrained.
 For example, the crankshaft of an engine forms a kinematic
pair with the bearings which are fixed in a pair, the connecting
rod with the crank forms a second kinematic pair, the piston
with the connecting rod forms a third pair and the piston with
the cylinder forms a fourth pair. The total combination of these
links is a kinematic chain.

4. 4
Kinematic chain may be Open or Closed.
 In a closed kinematic chain, the distal segment is fixed and the end segments are
unite to form a ring or a circuit. When one link moves all the other links will
move in a predictable pattern.
 In an open kinematic chain, the distal segment terminates free in space.
Each segment of an open chain has a characteristic degree of freedom of
motion; the distal possessing a higher degree of freedom than the
proximal ones.
Link 1
Joint 1 Link 2 Joint 2
Joint 3
Link 3
Link 4Open Kinematic Chain
Link 1
Link 2
Link 3
Link 4
Link 5
Link 6
J1
J2
J3
J4
J6
J7
Closed Kinematic Chain

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Types of Kinematic Chains
 Four bar chain or quadric cyclic chain,
 Single slider crank chain, and
 Double slider crank chain.
Fig 1. Four bar chain or quadric cyclic chain,

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Fig 2. Single slider crank chain,
Fig 3. Double slider crank chain.

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MECHANISM:
 When one of the links of a kinematic chain is fixed, the chain is
known as mechanism. It may be used for transmitting or
transforming motion
E.g. Slider-crank mechanism etc.
Robotic arm ( Open type mechanism)
Slider-Crank mechanism
( Closed type mechanism)

8. 8
INVERSION:
 Method of obtaining different mechanism by fixing different links in
a kinematic chain, is Inversion .
 Inversions will have same relative motion between the links but
absolute motion will change.
 No. of inversion = No. of links in kinematic chain.
 Example of Slider-crank Inversion:
1
2
3
4
Inversion #1 :
Slider block translates
1
2
3
4
Inversion #2 :
Slider block has complex motion

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GRASHOFF’s LAW:
 It states that for a Planar 4-bar linkage, Sum of the Shortest and the
longest link lengths must be less than or equal to Sum of the remaining
link lengths, if there is to be a continuous relative motion between 2
members.
 i.e.
1
2
3
4
Inversion #4 :
Link 3 reciprocates
1
2
3
4
Inversion #3 :
Slider block translates
s
l
p
q

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There are 3 inversions of 4-bar mechanisms, which are obtained
by fixing different links of the kinematic chain. They are:
 Double Crank Mechanism
 Crank Rocker Mechanism
 Double Rocker Mechanism
 Double Crank mechanism:
- When the shortest link is fixed, it gives double crank mechanism. A
double crank converts rotary motion from a crank to a second
crank or link in a different plane or axis. It is also known as crank-
crank, drag-crank or rotary-rotary converter. The links p and l
shown rotate through one complete revolution. This is one of the
first inversions of four-bar mechanisms.
- Application: Coupling rod of locomotive
(train wheels) etc. p
l
s
q

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 Crank-Rocker mechanism:
- When any of the link adjacent to the shortest link is fixed, it gives crank rocker
mechanism. A crank-rocker converts rotary motion from a crank to a reciprocating
link in a different plane or axis or vice cersa. It is also known as crank-lever
mechanism or a crank-rocker mechanism or a rotary-oscillating converter . The
links s rotates and q reciprocates.
- Application: Reciprocating engines/IC engines, Compressors etc.
 Double Rocker mechanism:
- When the link opposite to the shortest link is fixed, it gives double rocker
mechanism. A double rocker converts reciprocating motion from one link to another
link in a different plane or axis. The links p and l reciprocates.
- Application:Windshield wipers,
Quick return mechansism etc.

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References:
Theory of Machines by R.S. Khurmi
Kinematic of Machines
Kinematic Fundamentals
Mechanism by Ken Youssefi
www.wikipedia.com

13. THANK
YOU
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