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Terminology
BITS Pilani
Pilani Campus

1. Structure, mechanism and machine

2. Kinematic chain
3. Links
4. Kinematic pairs/joints
5. Four link planar mechanism/Slider-crank mechanism
6. Mobility and Degree of freedom : Kutzbach equation
7. Range of motion : Grashof’s law

Introduction to Kinematics  Crank - crank

 Crank - rocker
 Rocker - rocker
Lecture – 2 8 Jan, 2025
Dr J S Rathore 2
BITS Pilani, Pilani Campus

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Watt’s mechanism Rigid body assumption

Dynamics
 Watt's linkage is an example of four-link mechanism.
 Invented by James Watt in which, a point of the linkage is constrained to travel  Kinematics : Study of motion (without system inertia/ forces involved)
on an approximation to a straight line.  Kinetics : Study of forces (due to motion)
 Described in Watt's patent specification of 1784 for the Watt steam engine.
1. Separation of dynamics into kinematics and kinetics :
Rigid body assumption (Euler)
Watt’s mechanism is a simple four link mechanism of double-rocker type
2. Actual shape does not matter

For Flexible bodies Shape and motion?

Study of motion and forces take place simultaneously

https://en.wikipedia.org/wiki/Watt's_linkage
Watt’s approximate straight line mechanism
Punch Mechanism
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Kinematics and Dynamics of machines

Kinematic chain
Structure Machine  Interconnected system of links, in which, not a single link is fixed.

F  Closed chain: If every link is connected to at least 2 other links

 Open chain
2 3 4-link mechanism

 Combination of mechanisms A
 Forces/energy transmitted are OPEN CHAIN
CLOSED CHAIN
significant
 Mechanism When one of the links in chain is fixed. B
Combination of rigid bodies (Constrained kinematic chain)
with definite relative motion
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Link Kinematic pairs/joints

A rigid body OR part of a chain, which moves relative to some other  Connection between links
part  Can be classified in several ways
1. Based on type of contact between links (surface, line or point)
Binary link : If a link is connected to two other links OR a link with two nodes. 2. Based on the number of degrees of freedom allowed at the joint

Ternary link Nodes

Quaternary link and so on… Lower pair : Surface or area contact between links
Kinematic pairs
(by Reuleaux in 1870)
Binary link Ternary link Quaternary link
Higher pair : line or point contact

 Are assumed to be rigid

 Can be any shape (not just those shown)
 Have nodes for attachment
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Lower pairs/joints Higher pairs/joints

Sliding (prismatic) pair = P

Turning (revolute) pair = R

Ball and socket pair

(Spherical pair)

Cylindrical pair

Screw pair

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Four-link nomenclature Mobility and DOF

B Degrees of freedom (DOF)
Turning (revolute) pair
 The number of independent coordinates required to specify mechanism’s
A Coupler Four link mechanism (4R) configuration
Follower  Or the number of inputs that need to be provided in order to create a predictable
Crank output
∆y2 ∆y
Link 1 ∆y1
Ground Link ∆x2 ∆x
Pivot 02 Pivot 04
∆x1

∆θ1
∆θ1 ∆θ2
 Ground Link : Fixed wrt reference frame 3R1P ∆θ2
 Links pivoted to ground:  By connecting two previously disconnected links by a revolute joint, two
− Crank : Makes complete revolutions DOF are eliminated.
− Follower : crank/rocker (Oscillatory motion)
 Coupler : complex motion, not attached to ground
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Mobility and DOF Mobility and DOF

Degrees of freedom (DOF) ∆y2
∆y1
 The number of independent coordinates required to specify mechanism’s ∆x2
configuration
∆x1 3(n)
 Or the number of inputs that need to be provided in order to create a predictable
output
∆y
∆θ1 3(n-1)
∆y2 ∆θ2
∆y1
∆x2 ∆x
∆x1 3(n-1)-2P1
∆θ1 P1= Number of pairs with 1 DOF
∆θ1 ∆y
∆θ2 ∆θ2
∆x P2= Number of pairs with 2 DOF
1. By connecting two previously disconnected links by a revolute/sliding joint,
two DOF are eliminated.
2. Since the revolute and prismatic joints make up all lower-pair joints in planar ∆θ1 F = 3(n-1) - 2P1 -1P2
mechanisms, the above results can be expressed as a rule: a lower-pair joint ∆θ2
reduces the mobility of a planar mechanism by two DOF. 13 14
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Degree of Freedom Exercise : DOF ?

F = 3(n-1) - 2P1 -1P2
Kutzbach equation
P1= Number of pairs with 1 DOF
F = 3(n-1) - 2P1 -1P2
P2= Number of pairs with 2 DOF
E
F

D G
For spatial mechanism
A
F = 6(n-1) - 5P1 - 4P2 - 3P3 - 2P4 - 1P5
C B

Wiper mechanism
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Exercise : DOF ? Degree of Freedom

F = 3(n-1) - 2P1 -1P2
Kutzbach equation (for planar mechanism)

 P1: number of pairs with 1 DOF

F = 3(n-1) - 2P1 -1P2
R  P2: number of pairs with 2 DOF
R
P

R P F 0 Mechanism
R
F 0 Structure (Statically Determinate)

F 0 Structure (Statically Indeterminate)

R

Quick-return mechanism
(Shaper)
Gruebler criterion 1 = 3(n-1) - 2P1  3n - 2P1 = 4
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Three or more links at a joint Exercise : Structure or a mechanism ?

Multiple joints

3
2
4

3 F = 3(n-1) - 2P1 -1P2

2
4

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Contents
BITS Pilani
Pilani Campus

1. Review of few key points from last class

2. Degree of freedom : Kutzbach equation / Gruebler criterion

3. Approximate straight line mechanisms : Applications

4. Range of motion : Grashof’s law

Introduction to Kinematics
Lecture – 3 9 Jan, 25
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BITS Pilani, Pilani Campus

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Update* Kinematic pairs/joints

 This will be mailed to all the students  Connection between links
 Can be classified in several ways
1. Based on type of contact between links (surface, line or point)
2. Based on the number of degrees of freedom allowed at the joint

1. Evaluative tutorials before midsem exam will be closed book and

open book post midsem. Lower pair : Surface or area contact between links
Kinematic pairs
(by Reuleaux in 1870)
2. In open book tutorials only hand written class notes are allowed. Higher pair : line or point contact
No text book in open book tutorials.

3. For OPEN BOOK in Compre exam students are allowed to carry

text books also along with the class notes.
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Lower pairs/joints Higher pairs/joints

 
Sliding (prismatic) pair = P
Turning (revolute) pair = R

Ball and socket pair

Cylindrical pair (Spherical pair)

Screw pair
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Mobility and DOF Mobility and DOF

Degrees of freedom (DOF) ∆y2
∆y1
 The number of independent coordinates required to specify mechanism’s ∆x2
configuration
∆x1 3(n)
 Or the number of inputs that need to be provided in order to create a predictable
output
∆y
∆θ1 3(n-1)
∆y2 ∆θ2
∆y1
∆x2 ∆x
∆x1 3(n-1)-2P1
∆θ1 P1= Number of pairs with 1 DOF
∆θ1 ∆y
∆θ2 ∆θ2
∆x P2= Number of pairs with 2 DOF
1. By connecting two previously disconnected links by a revolute/sliding joint,
two DOF are eliminated.

2. Since the revolute and prismatic joints make up all lower-pair joints in planar ∆θ1 F = 3(n-1) - 2P1 -1P2
mechanisms, the above results can be expressed as a rule: a lower-pair joint ∆θ2
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reduces the mobility of a planar mechanism by two DOF. BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

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Degree of Freedom Three or more links at a joint

Kutzbach equation (for planar mechanism) Multiple joints

 P1: number of pairs with 1 DOF 3

F = 3(n-1) - 2P1 -1P2 2
 P2: number of pairs with 2 DOF
4

F 0 Mechanism
3
2
F 0 Structure (Statically Determinate) 4

F 0 Structure (Statically Indeterminate)

Gruebler criterion 1 = 3(n-1) - 2P1  3n - 2P1 = 4

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Exercise : DOF ? Exercise : DOF ?

R R
R
F = 3(n-1) - 2P1 -1P2 F = 3(n-1) - 2P1 -1P2
R

R
R R

Drum - brake mechanism

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Mechanisms to explore : Watt’s mechanism

Home assignment
1. Watt’s mechanism
2. Roberts’ mechanism Approximate Straight line mechanisms
3. Chebychev mechanism
4. Peaucillier mechanism
 Watt's engine, 1784, the first engine
to produce power directly on a shaft
5. Theo Janson walking mechanism without the intervention of a water
https://en.wikipedia.org/wiki/Jansen%27s_linkage fed by a reciprocating pumping
engine.
The Strandbeest by Theo Janson  The coupler point M describes the
figure-eight shaped coupler point
curve possessing two nearly straight
line segments
 The vertical segment was used for
piston rod guidance
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Watt’s mechanism Approximate straight line mechanisms

 Watt's linkage is used to prevent axle movement in the longitudinal direction
of the train.
 Watt's linkage is used in the rear axle of some car suspensions as an
improvement over the Panhard rod
 Prevent relative sideways motion between the axle and body of the car.

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Watt's linkage in a 1998 Ford RangerBITS
EVPilani,
suspension
Pilani Campus BITS Pilani, Pilani Campus

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Approximate straight line mechanisms Hoeken’s straight line mechanisms

AC = L
AB = 2L
BD = 2.5L
CD = DM = 2.5 L

http://mechanical-design-handbook.blogspot.com/2011/02/hoekens-straight-line-mechanism.html
Hoeken Mechanism application
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Approximate straight line mechanisms Exact straight line mechanism

 invented in 1864, was the first true

planar straight line mechanism
 the first planar linkage capable of
transforming rotary motion into perfect
Post-hole boring mechanism, 1956, German design, straight-line motion, and vice versa
capable of boring a vertical hole 6 ft deep  named after Charles-Nicolas Peaucellier,
and Yom Tov Lipman Lipkin
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Dump truck mechanisms Approximate straight line mechanisms

Transport mechanism for material handling

Skelton of dump truck mechanism
 Point C on the coupler of four-bar linkage
OAABOB describes path c as crank rotates
through 360˚
 This motion is transferred to the transport
member 5 by means of parallelogram
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linkage BITS Pilani, Pilani Campus

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Exercise : Excavator

A simple 4-link mechanism which we all know

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Exercise : DOF ?
Midsem problem for 5 marks : DOF ?
The Fig is a schematic of the entire linkage for a large power shovel used in mining.
Determine the number of degrees of freedom associated with this mechanism.

F = 3(n-1) - 2P1 -1P2 F = 3(n-1) - 2P1 -1P2

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Exercise : DOF ?
BITS Pilani
Pilani Campus

(3)

(4) F = 3(n-1) - 2P1 -1P2

(2)
(P2)

(5) (1)

Introduction to Kinematics
Lecture – 4 10 Jan, 25
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BITS Pilani, Pilani Campus

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Outline
Limitation : Kutzbach equation
1. Limitation Kutzbach equation
F = 3(n-1) - 2P1 -1P2
2. Range of motion : Grashof’s law
 Four bar mechanism with 4 Revolute (Turning) pairs (4R) F = 3(5-1) – 2(6) = 0
− Crank – crank
− Crank – rocker
− Rocker – rocker
3. Kinematic Inversions of
 Four bar mechanism with 3 Revolute + 1 Prismatic pairs (3R-1P)
 Four bar mechanism with 2 Revolute + 2 Prismatic pairs (2R-2P)
 No consideration given to lengths of links or
 Home assignment specific geometry.
 Kutzbach criterion can be violated due to non-
4. World of Mechanisms
uniqueness of geometry for a given connectivity
5. Velocity analysis: Vector polygon method of links.
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Kinematic Inversion
Limitation : Kutzbach equation F = 3(n-1) - 2P1 -1P2
Different mechanisms behavior can be obtained by fixing different links of
Due to over-constrained portion the same kinematic chain.
The process of choosing different links in the chain as frames known as
Kinematic inversion.

F = 3(6-1) – 2(8) = -1

Replace the portion of the  Kutzbach equation

mechanism with a single rigid  Grashof’s law on range of motion
F = 3(9-1) – 2(12) = 0 body

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Kinematic Inversion Grashof conditions Or Grashof’s rules

Different mechanisms behavior can be obtained by fixing different links of  Grashof condition predicts behavior of linkage based only on length of links
the same kinematic chain.  Identify the four bar linkages that have continuously rotatable joints
The process of choosing different links in the chain as frames known as s = length of shortest link
Kinematic inversion. l = length of longest link
p, q = length of two remaining links
coupler
Follower
crank

 Inversions result from grounding different links in the chain. Grashof’s law states that for a planar four-link mechanism, the sum of the
 So, there are as many inversions as links. shortest and longest link lengths cannot be greater than the sum of the
 Not all inversions will have unique kinds of motion. remaining two link lengths if there is to be continuous relative motion
 A Four-bar (4R) has only 3 distinct inversions, 2 crank-rockers, 1 double-crank, between two members.
and 1 double-rocker.
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Grashof conditions Or Grashof’s rules Grashof conditions Or Grashof’s rules

 Grashof condition predicts behavior of linkage based only on length of links  Grashof condition predicts behavior of linkage based only on length of links
 Identify the four bar linkages that have continuously rotatable joints  Identify the four bar linkages that have continuously rotatable joints
s = length of shortest link s = length of shortest link
l = length of longest link l = length of longest link
p, q = length of two remaining links p, q = length of two remaining links

 If s + l ≤ p + q the linkage is Grashof : At least one link is capable of making a

complete revolution
(class I kinematic chain)

 Otherwise the linkage is non-Grashof : No link will make a complete revolution

relative to another
(class II kinematic chain)
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For s + l < p + q
For s + l = p + q (Grashof neutral linkage)
 Crank-rocker if either link adjacent to shortest is grounded  Four inversions result in mechanisms similar to those obtained when s + l < p + q

p p
q q  The situation is also true when a linkage has two pairs of equal lengths.
s
l s
l This results in 2 special mechanisms:
s ̶ l ̶ s ̶ l
Case 1: Parallelogram form
1. Equal links are not adjacent
s 2. All 4 inversions lead to double-crank
mechanisms
q
s

s l
At least one link is s
Double crank capable of making a
(if shortest link is grounded) complete revolution Double rocker if link opposite
Grashof’s inversions to shortest is grounded
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For s + l = p + q For s + l > p + q

Case 2: Equal links are adjacent s ̶ s ̶ l ̶ l  All inversions will be double rockers
 No link can fully rotate
1. Any longer links (l) is fixed two crank –rocker mechanisms
2. If any of the shorter link (s) is fixed two double crank mechanisms

In this situation, two revolutions of shorter

link will cause one revolution of longer link
l
Patented by Elijah Galloway in 1829 s l

Inversions of non-Grashof’s law

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Grashof condition 4-link mechanism : 3R1P

Check Grashof’s condition and identify the mechanism.

Grashof If s +l < p + q
Special Grashof If s + l = p + q
Non- Grashof Otherwise

Prismatic pair is a limiting

case of revolute pair with
center at infinity

Center of rotation at a very large distance

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4-link mechanism : 3R1P 4-link mechanism : 3R1P

4-link kinematic chain (3R1P) lead to 4-link kinematic chain (3R1P) lead to
four different types of mechanism four different types of mechanism

Slider-crank mechanism Hand pump mechanism

Slider-crank mechanism

Offset Slider-crank mechanism

Oscillating cylinder mechanism
Quick-return mechanism
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4-bar mechanism : 2 R- 2 P Home assignment

Elliptical trammel Oldham’s coupling
Scotch-yoke mechanism
 Trammels were used for curve drawing The Oldham Coupler is named for Irish
 converting the linear motion of a slider into rotational motion, or vice versa
in earlier days engineer John Oldham who invented the
 Used to draw /cut or machine ellipses three disc coupler in 1821, to resolve a
on wood or other sheet materials paddle placement problem in a paddle
steamer design

Why is slider-crank preferable than scotch-

yoke mechanism in Internal Combustion
Engines ?

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World of mechanisms:

Computational Design of Mechanical Characters

by
Disney Research

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