Kinematics of Machines

TE2030E
Class code: TE2030E - 157243
Instructor: Dr. Minh Bau Luong
bau.luongminh@hust.edu.vn
Kinematics of Machines TE2030E
⚫ Class hour
▪ 12:30 – 15:05, Periods 1-3, Thursday, Room D8-106
⚫ Instructor: Minh Bau Luong. Ph.D
Computational Engineering and Energy Lab (CEELab)
https://sites.google.com/hust.org.vn/ceelab
Department of Vehicle and Energy Conversion Engineering
School of Mechanical Engineering (SME), HUST
Office: Room 810M – C7 Building, HUST
Mobile: +84 9636 14452 Email: bau.luongminh@hust.edu.vn

⚫ Main textbooks:
▪ George Henry Martin, Kinematics and Dynamics of Machines, 1982, McGraw-Hill, Inc. IS
BN: 0-07-040657-X
⚫ References:
▪ Norton, Design of Machinery: An Introduction To The Synthesis and Analysis of Mechanisms and Mac
hines, 6th ed., 2019, McGraw Hill, New York, NY,
https://designofmachinery.com/books/design-of-machinery
▪ John J. Uicker Jr, Gordon R. Pennock, Joseph E. Shigley, Theory of Machines and Mechanisms, 5 th e
d., 2017, McGraw-Hill Inc., ISBN 9780190264482
▪ Khurmi, R., Theory of Machines, 14th ed., 2005, S. Chand & Co. Ltd., ISBN 9788121925242
▪ S S Rattan., Theory of Machines, 3rd ed., 2009, Tata McGraw-Hill, ISBN 13: 978-0-07-014477-4

▪ Search titles and download ebooks at https://www.libgen.is

Kinematics of Machines TE2030E
Volume: 3(3-0-1-6)
- Theory: 45 hours
- Exercise/ Project: … hour
- Experiment: 15 hours
Prerequisite courses: None
Courses learned first: Math Calculus I, II; Math Linear Algebra, Physic 1
Parallel courses: Analytical Dynamics and Tutorial
Course description
This subject provides the knowledge on kinematics of basic mechanisms
commonly found in a system or machine. By the end of the course students
will be able to analyze the position, velocity, acceleration, and forces of the
elements of single and multiple degree-of-freedom linkages; understand and
have ability to design some common mechanisms in machines.
Videos:
Mechanical mechanism & applications https://youtu.be/yB-fM7uAwaE?si=wwsSDN6tojmpi_lm
200 Mechanical Principles Basic https://youtu.be/m4M4PClovd8?si=rFkzSxipi3hqgJ7u
How to learn effectively
 Chapter 1
FUNDAMENTAL CONCEPTS

1.1 Kinematics, kinetics and dynamics

1.2 Machine and Mechanism
1.3 Inversion
1.4 Pairing
1.5 Degrees of Freedom
1.6 Types of Motion
 Chapter 1. FUNDAMENTAL CONCEPTS

1.1 Kinematics, kinetics and

dynamics

- What is mechanics/dynamics/statics/
kinematics/kinetics?
- What is the difference between
kinematics and kinetics?
Chapter 1. FUNDAMENTAL CONCEPTS

Mechanics: concerned with the

state of rest or motion of bodies that are
subjected to the action of forces

Dynamics: treats with the forces acting

on the parts of a machine and the
motions resulting from these forces; deals
with the systems that change with time
Statics: deals with the analysis of
stationary systems which are not
changing with time
Kinematics: Study of the relative motion of
machine parts without regard to forces
(Study the displacement, velocities, and
accelerations)

Kinetics: Study of forces on systems in

motion (the study of forces causing or
resulting from motion)
 Chapter 1. FUNDAMENTAL CONCEPTS
Kinematics
Displacement Kinetics
Pkt
Velocity
Pj
N
Acceleration
P1 Ptt
A TDC
§ CT
x Displacement
B 
Pk

 Z T
BDC
§ CD 
l + Ptt
Kinematics: Study of the relative motion of
D C machine parts without regard to forces
 (Study the displacement, velocities, and
O R accelerations)
Kinetics: Study of forces on systems in
motion (the study of forces causing or
resulting from motion)
Kinematic variables Kinetic Variables
• Time • Force
• Displacement, position, distance • Pressure
• •
Are there any inverse kinematics equations for the
Velocity OpenManipulator Link? · Issue #96 ·
ROBOTIS-GIT/open_manipulator · GitHub
Torque
• Acceleration • Work
• Power
• Momentum …
1.2 Machine and Mechanism

Mechanism Machine
A device which transforms input Typically contains mechanisms,
motion to some desirable output designed to provide significant
motion/pattern and typically forces and transmits significant
develops very low forces and power (to do useful work)
transmits little power

Sewing machine Internal

combustion engine

Adjustable
desk lamp Cam and follower mechanism
 There is a direct analogy between the terms structure, mechanism and
machine and the three branches of mechanics.
A structure is also a combination of resistant (rigid) bodies connected by
joints, but its purpose is not to do work or to transform motion.
A structure (such as a truss) is intended to be rigid. It can perhaps be
moved from place to place and is movable in this sense of the word;
however, it has no internal mobility, no relative motions between its
various members, whereas both machines and mechanisms do.

Understanding and Analysing Trusses

https://youtu.be/Hn_iozUo9m4?si=KgGqwKBgNZXojtA9
 Classification of Mechanisms:
Based on the relative motion of the rigid bodies
Planar mechanism Spherical mechanisms Spatial mechanisms
Each link has some points that
All particles describe No restrictions on the
remains stationary as the linkage
plane curves in space relative motions of the
moves, the stationary points of all
and all these curves lie particles
links lie at a common location with
in parallel planes. the locus of each point is a curve
contained in a spherical surface,
defined by several arbitrarily chosen
points that are all concentric.

u re o utio r i r roc er ou e roc er P r eo r i e

oth i s
co ti uous otio co ti uous otio o co ti uous otio co ti uous otio

https://commons.wikimedia.org/wiki/File:Linkage_four_bar.svg
 Classification of Mechanisms:
Based on the degree of freedom (D.O.F) of output only
Constrained mechanism
• One independent output motion. Output member is
constrained to move in a particular manner only.
• Example: slider crank mechanism, four-bar
mechanism, five-bar mechanism with two inputs

Unconstrained mechanism
Slider crank mechanism, 1 D.O.F
• Output motion has more than one D.O.F.
• Example: Automobile differential during
turning the vehicle on a curve, five-bar
mechanism with one input

https://testbook.com/question-answer/a-five-bar-mechanism-is-
shown-in-the-figure-what--63c7bb6903a0945eccd6eb87
Five-bar mechanism (2 D.O.F)
 Terminology: Linkages, links, joints, and kinematic chains
Linkages are the basic building blocks of all mechanisms, made up of
links and joints.
A link is an (assumed) rigid body that possesses at least two nodes that
are points for attachment to other links.

A joint is a connection between two

or more links (at their nodes), which
allows some motion, or potential
motion, between the connected links.
Joints (also called kinematic pairs)
can be classified in several ways.
A kinematic chain (chuỗi động học) is a system of links, that is, rigid
bodies, which are either joined together or are in contact with one
another in a manner that permits them to move relative to one another.
The connections are the joints between the links, called kinematic pairs
(khớp động); thus a link can be defined as the rigid connection between
two or more elements of different kinematic pairs.

For example, in an IC engine:

1. The crankshaft with the bearings
forms a kinematic pair.
2. The connecting rod with the crank
forms a second kinematic pair.
3. The piston with the connecting rod
forms a third pair
4. The piston with the cylinder forms a
fourth pair.
→ The combination of these links is a
kinematic chain.
Open and closed kinematic chains
Open kinematic chain: there is one or more link which is connected to one
other link. The distal segment possesses a higher degree of freedom than
the proximal ones.

Closed kinematic chain: every link is connected to at least two other links,
the chain forms one or more closed loops.
When one link moves all the other links will move in a predictable pattern

Open vs. Closed mechanism chain

 1.3 Inversion
Obtaining a different mechanism by making a different link in a
kinematic chain the fixed member
Inversions will have same relative motion between the links but absolute
motion will change.

Example of Slider-crank Inversion

 1.4 Pairing
Two bodies in contact constitute a pair.
The connections, joints between the links, are called kinematic pairs.

Classification of pairs
y 
x
Based on the nature of contact between links:
Lower pairs: surface contact between the
pair elements, such as the pin joint

Higher pairs: line or point contact between

the elemental surfaces, such as the
connection between a cam and its follower,
ball and roller bearings, gear contacts.
 Higher pairs

The cam-follower mechanism converts rotational motion (crankshaft) into linear

motion (follower) as used in IC engines, reciprocating pumps, and compressors.
 If two links have no motion relative to each other, they are
considered as a single link (e.g: bearing and cylinder wall).

The link of a mechanism/the part of a machine which is

stationary, and which supports the moving members is called the
frame and is designated link 1.

If a link is not completely rigid, e.g bell or chain, that can be

called a flexible link. However, in some cases, that can be replaced
by a rigid link.
 A flexible link
A rigid link: deformations are so
small that they can be
neglected in determining the
motions of the various other
links in a machine.
if it is always
in tension

the fluid in the

hydraulic press

Assuming an
incompressible fluid

d2/d1 = A1/A2
The six lower pairs

Turning/revolute pair or Screw pair or helical pair

Prismatic pair
pin joint ( hớp ả ề) ( hớp vít)
( hớp trượt)

Cylindric pair Globular or Spheric Flat pair or planar pair

( hớp trụ) pair ( hớp cầu) ( hớp phẳ )
Note: Spheric pair with pin that is different from spheric pair due to
having 2 degrees of freedom, a rotation about two coordinate axes

Pin 3
Slot 4

All other joint types are called higher pairs.

E.g: mating gear teeth, a wheel rolling on a rail, a cam contacting
its follower… An unlimited variety of higher pairs exist

If a mechanism has only lower pairs, it is called a linkage

Based on the type of mechanical constraint/How the contact is
maintained:

Self closed pairs: the links in the

pair have direct mechanical
contact, even without the
application of external force (no
external force).

Force closed pair: the links in the

pair are kept in contact by the
application of external forces.
Based on the degree of freedom (D.O.F)

1. Type I / Class I – One D.O.F

2. Type II / Class II – Two D.O.F
3. Type III / Class III – Three D.O.F
4. Type IV / Class IV – Four D.O.F
5. Type V / Class V – Five D.O.F
 1.5 Degrees of freedom (DOF)
The system's DOF: number of independent parameters (number of independent
coordinates) needed to uniquely define its position in space at any instant of time.

A link has 3 DOF in a 2D space

- Two linear coordinates (x, y) to define the
position of anyone point on the pencil 
- One angular coordinate () to define the
angle of the pencil with respect to the axes.

6 DOF in spatial 3D space motion

A planar prismatic pair (P-pair)
Joints remove degrees of freedom
 Degree of freedom in planar 2D mechanisms
Gruebler's equation: M = 3L – 2J – 3G
M = degree of freedom or mobility
L = number of links
J = number of joints
G = number of grounded links, G = 1
In any real mechanism, even if more than one link of the kinematic chain is
grounded, there can be only one ground plane; thus G is always one, G= 1, and :

Gruebler's equation becomes: M = 3(L – 1) – 2J

In the above equations, the value of J must reflect the value of all joints in the
mechanism, that means half joints count as 0.5 because they remove only 1 DOF.
Kutzbach’s modification of Gruebler's equation:
Kutzbach’s equation: M = 3(L – 1) – 2J1 – J2
M = degree of freedom or mobility
L = number of links
J1 = number of 1 DOF full joints
J2 = number of 2 DOF half joints

Notation for kinematic diagrams

 Examples: Determine the number degree of freedom

2 3

L = 3; J1= 3; J2 = 0
M = 3x(3 – 1)– 2x3 –1x0
L = 4; J1= 4; J2= 0
=6–6–0=0
M = 3x(4 – 1) – 2x4 – 1x0
M  0: structure =9–8–0=1
L = 5; J1= 5; J2= 0
M =3x(5 – 1) – 2x5 – 1x0
M = 1: constrained mechanism = 12 – 10 – 0 = 2

M > 1: Unconstrained mechanism

L = 2; J 1= 2; J2 = 0 Nếu 2 thông số đầu vào được

M = 3 x(2 – 1) – 2 x2 – 1x 0 điều khiển sẽ xác định 1 vị trí đầu
= 3 – 4 – 0 = -1 ra -> constrained mechanism
M < 0: preload/super structure)
 M = 3(L – 1) – 2J1 – J2

Note: Ground is usually considered as link 1

M = 3(L – 1) – 2J1 – J2
Homework: Determine the DOF of the planar mechanism below
a. An automobile hood hinge mechanism
b. An automobile hatchback lift mechanism
c. An electric can opener
d. A folding ironing board
Cần 3 thông số đầu vào (vị trí của 3 piston thủy lực) mới
xác định được 1 thông số đầu ra (vị trí của gầu)
It is important to realize that the Kutzbach criterion can give an incorrect
result. In the development of the Kutzbach criterion, no consideration was
given to the lengths of the links or other dimensional properties. Therefore, it
should not be surprising that exceptions to the criterion are found for
particular cases with equal link lengths, parallel links, or other special
geometric features.
For example, Fig. a) represents a structure and that the criterion properly predicts m
= 0. However, if link 5 is arranged as in Fig. b), the result is a double-parallelogram
linkage with a mobility of m = 1.
The actual mobility of m = 1 results only if the parallelogram geometry is achieved.

M=0 M= 1
Examples:

Determine the mobility of the planar mechanism below

Examples:

Determine the mobility of the planar mechanism below

Degree of freedom in spatial mechanisms

Where the subscript refers to the number of freedoms of the joint.

1.6 Types of Motion
Plane Motion (Planar Motion)

A body has plane motion if all its points move in planes which are parallel to
some reference planes, which called the plane of motion.
Plane motion can be one of three types: translation, rotation, or a combination of
translation and rotation.
Summarized table
 Pure translation
A body has translation if it moves so that all straight lines in the body
move to parallel positions.
Rectilinear translation is a motion wherein all points of the body
move in straight-line paths.

The piston has rectilinear translation A translation in which points in the

body move along curved paths is
called curvilinear translation.

The connecting link 3 has curvilinear translation

Pure rotation
All points in a body remain at fixed distances from a line which is
perpendicular to the plane of motion. This line is the axis of
rotation, and points in the body describe circular paths about it.

The crank has a motion of rotation if

the frame of the engine is fixed
Spherical Motion
A point has spherical motion if it moves in three-dimensional space and
remains at a fixed distance from some fixed point.
A body has spherical motion if each point in the body has spherical motion.

In the ball-and-socket joint in the Figure below, if either the socket or rod
is held fixed , the other will move with spherical motion.
Helical Motion
A point which rotates about an axis at a fixed distance and at the
same time moves parallel to the axis describes a helix. A body has
helical motion if each point in the body describes a helix.

The motion of a nut along a

screw is a common example.
Cycle, Period and Phase of Motion
A mechanism completes a cycle of motion when it moves through all its
possible configurations and returns to its starting position.
E.g: The slider-crank mechanism (in the figure below) completes a cycle of
motion as the crank makes one revolution.

The time required for one cycle is the period.

The relative positions of the links at any instant during the cycle of motion for the
mechanism constitute a phase.
E.g: When the crank is in position 1, the mechanism is in one phase of its motion.
When the crank is in position 2, the mechanism is in another phase.
 Chapter 2
Properties of Motion, Relative Motion,
Methods of Motion Transmission

2.1 Introduction
2.2 Properties of Motion
2.3 Relative Motion
2.4 Methods of Motion Transmission
 Chapter 2. Properties of Motion, Relative Motion,
Methods Of Motion Transmission
2.1 Introduction

- The motion of a rigid body can be defined in terms of the motion of

one or more of its points.

- Path of motion: the locus of its successive positions

- Distance: the length of its path of motion, distance is a scalar quantity
since it has magnitude only
Path
A

u re o utio r i r roc er ou e roc

oth i s
co ti uous otio co ti uous otio o co ti uous
 2.2 Properties of Motion
Linear displacement of a point is the change of its position and is
a vector quantity

In term of vector: s = x + y
In term of magnitude:

Direction with respect to the x axis:

The motion of a point at any instant is

in a direction tangent to its path
Linear velocity: the time rate of change of linear displacement.

Point P moves from position B to position C in time t:

The instantaneous linear velocity of

the point, when it is at position B:

V is directed tangent to the path.

 Angular displacement and velocity
The body rotate about the fixed axis O.
As P moves to P’, the angular displacement
of line OP or the body is , occurred in t.

Average angular velocity

Instantaneous angular velocity

Velocity of point P

or:

Let’s s y: then
n: revolutions per minute (RPM)
Velocity at a point
It can be seen that: since the radii of rotation for all points in a rotating
body have the same angular velocity , the magnitudes of their linear
velocities are directly proportional to their radii.
Linear acceleration: the time rate of change of linear velocity
Considering the case of a point having rectilinear motion
If the initial velocity be V0 and the velocity after a time interval t be V

Average acceleration

Instantaneous acceleration
 Angular acceleration: the time rate of change of angular velocity

For a body having uniform angular acceleration,  is constant.

Normal and tangential acceleration: for the point having curvilinear motion

Normal acceleration resulting from a change in direction of its linear velocity

Tangential acceleration resulting from a change in magnitude of its linear velocity

The tangential acceleration At of the point at position B

The normal acceleration An of the point at position B

The direction of An is always toward the center of curvature. A point

having curvilinear motion always has a normal component of
acceleration.
A point having rectilinear motion has no normal component of
acceleration since:

The total linear acceleration A of the moving point is the vector sum
of An and At. Its magnitude:

And its direction:

2.3 Simple harmonic motion, absolute and relative motion

Simple harmonic motion

Acceleration is proportional to the displacement of the particle from

a fixed point and is of opposite sign

It is often convenient to represent simple harmonic motion by the

projection upon a diameter of a point moving on a circle
Line OP rotate with constant angular velocity .
Let B be the projection of point P on the x axis.
Displacement, velocity and acceleration of point
B from point O, are:

Scotch-yoke mechanism, if
link 2 rotates with constant
angular velocity, link 4 has
simple harmonic motion.
Exercise 1:

A 6-in-diameter steel cylinder is to be machined in a lathe.

The cutting speed is to be 100 ft/min.
Determine the speed of rotation in revolutions per minute.
(1 inch = 0.0254 m, 1 foot = 0.304 m)

Exercise 2:
An automobile engine has a bore (cylinder diameter) of 95.3 mm. The
stroke (distance the piston travels from one extreme position to the other)
is 88.9 mm. The car is running at 96.5 km/h and the outside diameter of the
tires is 686 mm. If the number of revolutions per minute of the engine is
four times that of the wheels, find (a) the revolutions per minute of the
wheels, (b) the revolutions per minute of the engine, (c) crankpin velocity
(in meters per second), (d) angular velocity of crank (in radians per
minute), (e) average piston velocity (in meters per second), and (f) distance
piston travels per kilometer of car travel.
Exercise 3:
Exercise 4:
Absolute motion vs. Relative motion
Absolute motion is the motion of a body relative to another body at rest.
• The motion of a point relative to fixed coordinate axes is absolute motion.
• Concept of relativity: Since no body in the universe is truly at absolute rest,
motion is always described in relation to another body.
• The term “absolute” is commonly omitted (e.g., a car moving at 100 km/h
means relative to the ground).
 Relative motion
A body has relative motion to another body if
their absolute motions differ.
The displacement, velocity and acceleration
(x,v,a) of a body A relative to a body B are the
absolute (x,v,a) of A minus the absolute those of B.

vector quantities

If a body 2 & body 3 have motion in a plane or parallel planes, then their relative
angular motions are defined as the difference in their absolute angular motions.
 2.4 Methods of Transmitting motion
All mechanisms transmit motion, classified based on how the motion is
transferred

Link 2 is the driver; Link 4 is driven,

called the follower.
Link 3 in Fig. a is a rigid connector and
is called the coupler.

a. Rigid linkage (Coupler)

Band 3 in Fig. b represents a flexible connector (belt

or chain drive).
Motion is transmitted from the driver to the follower
along the line of transmission (Line P2P4).

b. Flexible connectors
 In Fig. c, motion is transmitted by direct contact
between the driver and follower.
The motion depends on the shape of the outlines on
links 2 and 4, and on the relative position of the links.
Example: Cam and follower, gear pairs.

c. Direct-contact mechanism

In the direct-contact mechanism in Fig.

c, Cam (driver) transmit motion to the
driven link only if the driver has motion
in the direction of the common normal.
Here the line of transmission is the
common normal.
 Angular-velocity ratio Radii of rotation: O2P2 and O4P4
Velocity of P2: P2E ⊥ O2P2.
The components of P2E along the normal and
tangent: P2S and P2L.

Velocity of P4: P4F ⊥ O4P4.

The components of P4F along the normal and
tangent: P4S and P4M.
Note that the components of velocities P2E and
P4F along the normal direction must be equal
Angular velocities of links 2 and 4 for continuous contact.

∆O2GP2 ∼ ∆P2SE
∆O4HP4 ∼ ∆P4SF

∆O2GQ ∼ ∆O4HQ Finally,

 The mechanisms in Figs a and b have identical velocities, S, along the line of
transmission P2P4.

The angular-velocity ratio of these driver-follower types: mechanisms of any is

inversely proportional to the perpendicular distances from their centers of
rotation to the transmission line.

For a belt drive, the transmission ratio is based on the

radii of the pulleys:
O2G and O4H are R2 and R4, the radii of the pulleys
Constant angular-velocity ratio: require
the line of transmission intersected the line
of centers at a fixed point.
line of centers
The belt drive can provide a constant
angular-velocity ratio.

In gear mechanisms, mating gear-tooth profiles must be designed to

maintain a constant angular-velocity ratio.

For the four-bar linkage, the angular-velocity ratio is constant only if

cranks 2 and 4 are equal in length, and the coupler length P2P4 is equal
to O2O4. Then 2/4 = 1.
 Types of contact in direct-contact mechanism
Sliding contact: Sliding occurs in a direct-contact
mechanisms whenever the bodies have relative motion
along the tangent through their point of contact.

Velocity Components:
• Normal components o e ocities P₂E d P₄F re
equal → Represe ted y P₂S.
• Tangential components o e ocities P₂L d P₄M re
different, causing relative motion between bodies 2 and 4
→ Sliding velocity Vs

Direction & Magnitude:

• Body 2 s ides o ody 4 in the tangential direction toward L.
• Magnitude of sliding velocity is represented by length ML.
 Rolling contact: rolling contact exists only if there is no sliding at their point of contact.
Conditions for rolling contact:
• Tangential velocities at the contact point (P₂E & P₄F) must be equal in magnitude and direction.
• Normal components of velocities (P₂E and P₄F) must also be equal.
• It occurs only if the radii (O₂P₂ and O₄P₄) are along a common line → the line of centers (O₂O₄)

Rolling vs. Sliding:

• A point of contact on the line of centers
is necessary for rolling, but not sufficient.
• Even with identical tangential velocities,
sliding can still occur if P₂E and P₄F are
not identical

Summary
• For ro i co t ct, linear velocities at the point of
contact must be identical.
• The angular velocity ratio of the driver and follower
is inversely proportional to the contact radii.
.
Positive drive: exists in direct-contact mechanisms when the motion of the driver compels the follower to move
Conditions for positive drive:
• I link 2 (driver) rotates counterclockwise, it exerts a force
on link 4 (follower) along the common normal at point P.
• This orce cre tes torque arm about O₄, making link 4
rotate clockwise.
• Si i r y, i link 4 is the driver and rotates
counterclockwise, link 2 will be forced to rotate clockwise
due to torque around O₂.

Dead center condition (No positive drive):

• I the common normal passes through a center of
rotation (O₂ or O₄), motion of the driver does not
compel the follower to move → called dead center,
meaning there is no positive drive.

Friction drive:
• Whe bodies 2 and 4 are circular disks, motion is
not transmitted unless there is sufficient friction.
• This type o dri e is ow s friction drive.
Homework: exercise 2.19 in the textbook
Homework: Exercise 2.22 in the textbook
 Chapter 3 Linkages

3.1 Four-Bar Linkage

3.2 Other mechanisms
 Chapter 3. LINKAGES
3.1 Four-bar linkage
Four-bar linkage is one of the most
Rocker
useful and most common mechanisms.
Components:
• Li 1 → Frame (fixed). Frame (fixed).
• Li s 2 d 4 → Crank (rotating)/Rocker.
• Li 3 → Coupler (connecting rod).
Application:
• M y ech is s c e analyzed or replaced
using a four-bar linkage or a combination of
multiple four-bar linkages.

The link that is connected to the power source

or prime mover is called the input link. The
output link connects the moving pivot C to
ground pivot O4, CO4

Animation
1. https://www.geogebra.org/m/BueCG9ch
2. https://www.geogebra.org/classic/zZ5Bt9hX
3. https://youtube.com/playlist?list=PLrqEfTEEzLU5CTY__gh-
CtEutUwddf9Tj&si=s9NYQaXOGHaBZwqa
Examples of four-bar linkage

Windshield wiper

Level-luffing crane

One or more links may have infinity in

length, e.g slider-crank mechanism with a
slider replacing an infinity long output link.

Slider-crank mechanism
 To ensure a complete revolution of the input crank, Grashof’s law states that:
“For a planar four-bar linkage, the sum of the shortest and longest link lengths
cannot be greater than the sum of the remaining two link lengths if there is to
be continuous relative rotation between two links.”
If the longest link has length l, the shortest link has
length s, and the other two links have lengths p and q
s+lp+q

Depending on the location of link s relative

to the fixed link, the different variations of
the four-bar linkage are created.
https://www.geogebra.org/m/BueCG9ch

u re o utio r i r roc er ou e roc er P r eo r i e

oth i s
co ti uous otio co ti uous otio o co ti uous otio co ti uous otio
Reuleaux’s approach:
If links are labeled: crank s, lever p, coupler l (need not be the longest link), frame q
s + l + p ≥ q, if not, the links cannot be connected (a)
s + l − p ≤ q, if not, s is incapable of rotation (b)
s + q + p ≥ l, if not, the links cannot be connected (c)
s + q − p ≤ l, if not, is incapable of rotation (d)

s+l+p <q s+l−p > q

s+q−p > l
s+q+p < l
Example:
 Types of four-bar linkage
a. Parallel-crank four-bar linkage
Cranks 2 and 4 are of equal length
Coupler 3 is equal in length to the line of
centers O2O4,
Cranks 2 and 4 always have the same
angular velocity.

Dead point/dead center: at which the follower could begin to rotate in a direction
opposite to that of the driver (as link 4 is colinear with link 3).
Usually inertia, springs, or gravity prevent the undesired reversal at the dead point.

b. Non-parallel-crank four-bar linkage

Cranks 2 and 4 are of equal length
Coupler 3 is equal to the line of centers O2O4,
The cranks are nonparallel and rotate in opposite directions.
If crank 2 turns with constant angular velocity, crank 4 will
have a varying angular velocity.
 c. Crank and rocker A Path
The mechanism transforms motion of
rotation into oscillating motion.
B
Crank 2 rotates completely about
pivot O2, then coupler 3 causes
crank 4 to uoscillate
re o utioabout O 4r. i r roc er ou e roc er
Following oth i s
conditions must exist
co ti uous otio co ti uous otio o co ti uous otio
Reuleaux’s approach
O2B + BC + O4 ≥ O2O4
O2B + O2O4 + O4 ≥ B
O2B + BC - O4C  O2O4
O2O4 + O2B - O4C  BC

Either 2 or 4 can be the driving crank.

• If link 2 drives, the mechanism will always operate.
• If 4 is the driver, a flywheel or some other aid will be required to carry the mechanism
beyond the dead points B’ and B’’. The dead points exist when BC is in line with O2B.
d. Drag link
A drag-link four-bar mechanism: the shortest link is fixed.
Animation https://www.geogebra.org/classic/zZ5Bt9hX
Both 2 and 4 make complete rotations.
If one crank rotates at constant speed, the other
crank will rotate in the same direction at a
varying speed.

The relations between the links must be as

follows:
O2O4 + O4C - O2B  BC
O4C - O2O4 + O2B ≥ BC
These relations can be derived from triangles
O2B'C' and O2B"C".

This is applied for quick-return mechanisms.

e. Slider crank mechanism
It is a special case of the four-bar linkage
when crank 4 is made infinite in length,
then point C has rectilinear motion and
crank 4 is replaced by a slider.
Examples of application:
+ In Gasoline and diesel engines, the combustion gas
force acts on the piston, link 4. Motion is transmitted A § TDC
CT
through the connecting rod to crank 2. There are two piston x Displacement
B
dead-center positions during the cycle, one required to
carry the crank beyond these positions. 
BDC
§ CD
+ Air compressors, where an electric motor or gasoline
engine drives the crank and in turn the piston compresses l

the air. D C

O R
 f. Scotch-yoke mechanism

A variation of the slider-crank mechanism.

The Scotch yoke is the equivalent of a

slider crank having an infinitely long

connecting rod and the slider has simple

harmonic motion.

This mechanism is used in testing

machines to simulate vibrations

having simple harmonic motion.

Scotch-Yoke Compressor
 3.2 Other mechanisms
Quick-return mechanisms
Used in machine tools (e.g., shapers, power-
driven saws) to provide a slow cutting/advance
stroke and a faster return stroke.
• Function: The driving crank rotates at a
constant angular velocity, causing an
asymmetric motion cycle.
• Time Ratio of the time for the cutting
stroke to the return stroke is greater than 1.

time of cutting/advance stroke

Time ratio = >1
time of return stroke

Some types of quick-return mechanisms:

Crank-Shaper, Whitworth, Drag Link,
Offset Slider Crank…

Animations of quick-return mechanisms https://www.geogebra.org/m/qxsTdpCv

 Crank-Shaper (shaper linkage)
A six-bar linkage: link 2 rotates completely and link 4 oscillates.
If link 2 rotates counterclockwise at constant velocity, slider 6 will have a slow
stroke to the left and a fast return stroke to the right.
Time ratio = 1/ 2 > 1
Applied in shaper (shaping machine).
Whitworth mechanism https://www.geogebra.org/m/qxsTdpCv
Obtained by making the distance O2O4 in crank-shaper mechanism less than the
crank length O2B, O2O4 < O2B
Both links 2 and 4 rotate completely.
If the driver 2 rotates counterclockwise with constant angular velocity, slider 6 will
move from D' to D" with a slow motion while 2 rotates through angle 1.
As 2 rotates through the smaller angle 2, slider 6 will have a quick-return motion
from D" to D'. Time ratio = 1/ 2 > 1
Drag link
Links 1, 2, 3 and 4 comprise a drag-link mechanism.
If link 2, the driver, rotates counterclockwise with constant angular velocity, then
slider 6 makes a slow stroke to the left and returns with a quick stroke to the right.
Time ratio = 1/ 2 > 1
Offset Slider Crank
A slider-crank mechanism with an offset y so that the path of the slider does not
intersect the crank axis.
It has kinematic characteristics that differ from the in-line (or on-center) slider-
crank, it is a quick-return mechanism.
Time ration = 1/ 2 only a little larger than 1.
If the length of connecting rod 3 is long compared to the length of crank 2, then
the resulting motion is nearly harmonic (exact harmonic motion can be obtained
from the Scotch-yoke linkage). https://www.geogebra.org/m/qxsTdpCv
Example:
Determine the advance-to-return ratio for the slider-crank linkage with the offset e.
Also, determine in which direction the crank should rotate to provide quick return.
 Straight line mechanisms
Linkages have a point that moves along a straight line, or nearly along a
straight line , without being guided by a plane surface.
Most of these mechanisms were designed in early days before plane surfaces
to be used as guides could be machined.

a. Watt's mechanism
Point P traces a figure-eight-shaped path, a
considerable portion of which is approximately a
straight line.
The lengths must be proportioned:
b. Roberts’ linkage
That produces approximate straight-line
motion. Point P moves very nearly along line
AB.
Length AC = CP = PD = DB and CD = AB/2
The accuracy of the motion can be increased by
increasing the ratio of the height of mechanism
to its width.

+ Scott-Russell mechanism

That gives exact straight-line motion of

point P (note that it employs a slider).
Length AC = BC = CP
c. Tchebysheff’s mechanism

That gives approximate straight-line motion.

Point P, the midpoint of CB, moves very
nearly along line CB.
Length AB = CD = 1.25AD, and AD = 2CB.

+ Peaucellier's mechanism
That produces exact straight-line motion
for point P.
The following relationships must hold: AB
= AE, BC = BD, and PC = PD = CE = DE.
It is proved that point O, the projection of
P on a line through AB, is fixed, and point
P moves along PO, which is a straight line
perpendicular to AB.
Parallel mechanisms
They give parallel motion, and are used to enlarge or
reduce movements.
Links 2, 3, 4, and 5 form a parallelogram.
Link 3 is extended and contains point D.
F is the point of intersection of lines AD and CE.

Another application of a parallel

mechanism.
Toggle mechanisms
These mechanisms are used whenever a large force
acting through a short distance is required.
Links 4 and 5 are the same length.
Let P be the vertical component of the force which
link 3 exerts on the pin at C. When the angle
between BC and O2C becomes small, a force
analysis gives: 𝑃 P
𝐹=
2 𝑡𝑎𝑛𝑔

Thus for a given value of P, as links 4 and 5

approach a colinear position,  is very small,
force F rises rapidly (much higher than P).

Toggle mechanisms are used in toggle

clamps, riveting machines, punch presses,
and rock crushers.
Oldham coupling

This coupling is a mechanism for connecting two shafts having parallel

misalignment.
Disk 3 has a tongue on each side. These are at 90° to one another and
slide in grooves in members 2 and 4.
The shafts are coupled in such a way that if one shaft rotates, the other
shaft also rotates at the same speed due to no relative rotation between
bodies 2, 3, and 4.
Link 3 can slide or reciprocate in the slots in the flanges (links 2 and 4).

Link 3
 Universal joints
Connecting intersecting shafts
Allowing one or more rotating shafts to be linked
together, to transmit torque and/or rotary motion
between two points that are not in line with each
other, to withstand heavy loads.
The most common type is the Hooke or Cardan joint
(e.g: a driveshaft in rear-wheel-drive vehicles)
It is possible to give the relation of angular velocity
and acceleration below:

When 3 is constant

If  become large enough, the variation in speed of

the other shaft is considerable, and accelerations
can then cause vibrations which are intolerable.
 Using two universal joints so that the second joint compensates for the variation in
speed produced by the first. Condition for 2/4 = 1 at all times:
Angle 1 must equal angle 2,
Yoke 1 must be made to lie in the plane of 2 and 3
Yoke 2 lies in the plane of 3 and 4.

Several universal joints have

been invented which give a
constant-velocity ratio, such as
Bendix-Weiss joint.
Intermittent-motion mechanism (Indexing Mechanisms)
Converting continuous motion into intermittent motion.
Commonly used on machine tools for indexing a shaft.
+ Geneva Wheel
Link 2 (driver) contains a pin which engages slots in the
driven link 3.
The slots in link 3 are positioned so that the pin enters and
leaves them tangentially.
The driven member makes one-fourth of a revolution for each
revolution of the driver. However, velocity ratios other than
4: 1 may be used.
The locking plate prevents the driven member from rotating
except during the indexing period.

+ Indexer uses standard gear teeth

+ Ratchets
Used to transform motion of rotation or
translation into intermittent rotation or translation.
Member 2: ratchet wheel
Member 3: pawl.
Member 4: pawl lever, when 4 oscillates, the
ratchet will rotate counterclockwise with an
intermittent motion.
Member 5: holding pawl, prevent the ratchet from
reversing.

A ratchet-drive mechanism in which the throw

of the driving crank, link 6, is adjustable.
Crank 6 rotates (in either direction), link 4
oscillates and member 2 rotates counterclock -
wise with intermittent motion.
If the pawl 3 is placed in the dotted position,
the ratchet will rotate clockwise.
Some silent ratchets which have no teeth on the ratchets, and the
device depends upon the wedging together of smooth or flat surfaces
 - Elliptic trammel
It is an instrument for drawing ellipses.
Link 3 is pivoted to sliders 2 and 4, which slide in link 1, and point P describes an
ellipse.

a: half the major axis

b: half the minor axis

If P is placed at point C, which is midway

between A and B, then:
x2 + y2 = a2
which is the equation of a circle of radius a.