kinematics of machines-15me42
TRANSCRIPT
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 1
KINEMATICS OF MACHINES
10ME44/15ME42
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 2
UNIT 1 SIMPLE MECHANISMS
CONTENTS
1.1 Introduction
Objectives
1.2 Kinematics of Machines
1.3 Kinematic Link or an Element
1.4 Classification of Links
1.5 Degree of Freedom
1.6 Kinematic Pairs
1.7 Different Pairs
1.7.1 Types of Lower Pair
1.7.2 Higher Pair
1.7.3 Wrapping Pair
1.8 Kinematic Chains
1.9 Inversions of Kinematic Chain
1.10 Machine
1.11 Other Mechanisms
1.11.1 Pantograph
1.11.2 Straight Line Motion Mechanisms
1.11.3 Automobile Steering Gear
1.11.4 Hooks Joint or Universal Coupling
1.12 Key Words
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 3
Objectives:
1. Study about kinematic chain, machine & mechanisms
2. Get knowledge about various inversions of kinematic chains
1.1 INTRODUCTION
In our daily life, we come across a wide array of machines. It can be a sewing machine, a
cycle or a motor car. Power is produced by the engine which makes use of a mechanism
called slider crank mechanism. It converts reciprocating motion of a piston into rotary
motion of the crank. The power of the engine is transmitted to the wheels with the help
of different mechanisms. If you visit LPG gas filling plant or a bottling plant almost all
the functions are done by making use of mechanisms. These are only few examples.
Generally, manual handling in industries has been reduced to the minimum. In
engineering, mechanisms and machines are two very common and frequently used terms.
We shall start with simple definition of these terms.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 4
Theory of Machines In this unit, you will also study about link, mechanism, machine, kinematic quantities,
different types of motion and planar mechanism. You will study about degree of
freedom, kinematic pairs and classification of links in this unit.
A moving body has to be assigned coordinates according to the axes assigned. The
motion of the bodies is constrained according to the requirement in a mechanism. The
links which are the basic elements of the mechanism are connected to form kinematic
pairs which are of different types. The links may further be connected to several links in
order to impart motion and they are classified accordingly.
In this unit, you will be explained how to get different mechanisms by using four bar
chain which is a basic kinematic chain. The four bar chain has four links which are
connected with each other with the help of four lower kinematic pairs. This chain
provides different mechanisms of common usage. For example, one mechanism,
provided by this, is used in petrol engine, diesel engine, steam engine, compressors, etc.
One mechanism makes possible to complete idle stroke in machine tool in lesser time
than cutting stroke which reduces machining time. This mechanism being termed as
quick return mechanism. Similarly, there are some mechanisms which can provide
rocking motion which can be used in materials handling. You will be explained
terminology and classification of cams and followers also.
Objectives After studying this unit, you should be able to
determine degrees of freedom for a link and kinematic pair,
describe kinematic pair and determine motion,
distinguish and categorise different type of links,
know inversions of different kinematic chains,
understand utility of various mechanisms of four bar kinematic chain,
make kinematic design of a mechanism,
know special purpose mechanisms,
know terminology of cams, and
know classification of followers and cams.
1.2 KINEMATICS OF MACHINES
The kinematics of machines deals with analysis and synthesis of mechanisms. Before proceeding to this, you are introduced to the kinematics.
Kinematics implies displacement, velocity and acceleration of a point of interest at a
particular time or with passage of time. A point or a particle may be displaced from its
initial position in any direction. The motion of a particle confined to move in a plane can
be defined by x, y or r, or some other pair of independent coordinates. The motion of a
particle constrained to move along a straight line can be defined by any one coordinate.
The concerned coordinate shall describe its location at any instant.
1.2.1 Displacement The distance of the position of the point from a fixed reference point is called displacement. In rectilinear motion the displacement, is along one axis say x-axis, therefore,
Displacement ‘s’ = x In a general plane motion,
Displacement ‘s’ = x + i y
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 5
1.2.2 Velocity The velocity of a particle is defined as the rate of change of displacement, therefore, the velocity
V S 2 S1 s
t t
t 2
1
where, s S 2 S1
and t t 2 t1
s is the distance traveled in time t. The direction of velocity shall be tangent to the
path of motion. Y
S2
S1
X
Figure 1.1 : Plane Motion
1.2.3 Acceleration The acceleration of a particle is defined as the rate of change of velocity, therefore,
Acceleration ‘a’ V2 V1 V
t t
t 2
1
where V V2 V1
and t t 2 t1
V is the change in velocity in time t.
1.3 KINEMATIC LINK OR AN ELEMENT
Machines consist of several material bodies, each one of them being called link or
kinematic link or an element. It is a resistant body or an assembly of resistant bodies.
The deformation, if any, due to application of forces is negligible. If a link is made of
several resistant bodies, they should form one unit with no relative motion of parts with
respect to each other.
For example, piston, piston rod and cross head in steam engine consist of different parts
but after joining together they do not have relative motion with respect to each other and
they form one link. Similarly, ropes, belts, fluid in hydraulic press, etc. undergo small
amount of deformation which, if neglected, will work as resistant bodies and, thereby,
can be called links.
SAQ 1
(a) What is a resistant body?
(b) Define link.
Simple Mechanisms
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 6
Theory of Machines
1.4 CLASSIFICATION OF LINKS
A resistant body or group of resistant bodies with rigid connections preventing their relative movement is known as a link. The links are classified depending on number of joints.
Singular Link
A link which is connected to only one other link is called a singular link (Figure 1.2).
Figure 1.2 : Singular Link
Binary Link
A link which is connected to two other links is called a binary link (Figure 1.3).
2 3
1
Figure 1.3 : Binary Link
Ternary Link
A link which is connected to three other links is called a ternary link (Figure 1.4).
Figure 1.4 : Ternary Link Quarternary Link
A link which is connected to four other links is called quarternary link (Figure 1.5).
Figure 1.5 : Quarternary Link
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 7
1.5 DEGREE OF FREEDOM
The degree of freedom of a body is equal to the number of independent coordinates
required to specify the movement. For a cricket ball when it is in air, six independent
coordinates are required to define its motion. Three independent displacement
coordinates along the three axes (x, y, z) and three independent coordinates for rotations
about these axes are required to describe its motion in space. Therefore, degrees of
freedom for this ball is equal to six. If this cricket ball moves on the ground, this
movement can be described by two axes in the plane.
When the body has a plane surface to slide on a plane, the rotation about x and y-axes
shall be eliminated but it can rotate about an axis perpendicular to the plane, i.e. z-axis.
At the same time, while executing plane motion, this body undergoes displacement
which can be resolved along x and y axis. The rotation about z-axis and components of
displacement along x and y axes are independent of each other. Therefore, a sliding body
on a plane surface has three degrees of freedom.
These were the examples of unconstrained or partially constrained bodies. If a cylinder
rolls without sliding along a straight guided path, the degree of freedom is equal to one
only because rotation in case of pure rolling is dependent on linear motion. This is a case
of completely constrained motion.
The angle of rotation rx
where, r is radius of cylinder and x is linear displacement. z z
Z y
x x
Y
x o x
O
Y Figure 1.6 : Degree of Freedom
o
x
Figure 1.7 : Completely Constrained Motion
1.6 KINEMATIC PAIRS
In a mechanism, bodies or links are connected such that each of them moves with respect
to another. The behaviour of the mechanism depends on the nature of the connections of
the links and the type of relative motion they permit. These connections are known as
Simple Mechanisms
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 8
kinematic pairs. Hence kinematic pair is defined as a joint of two links having relative
motion between them.
Broadly, kinematic pairs can be classified as :
(a) Lower pair,
(b) Higher pair, and
(c) Wrapping pair.
1.7 DIFFERENT PAIRS
When connection between two elements is through the area of contact, i.e. there is
surface contact between the two links, the pair is called lower pair. Examples are motion
of slider in the cylinder, motion between crank pin and connecting rod at big end.
1.7.1 Types of Lower Pairs
There are six types of lower pairs as given below :
(a) Revolute or Turning Pair (Hinged Joint)
(b) Prismatic of Sliding Pair
(c) Screw Pair
(d) Cylindrical Pair
(e) Spherical Pair
(f) Planar Pair
Revolute or Turning Pair (Hinged Joint)
A revolute pair is shown in Figure 1.8. It is seen that this pair allows only one
relative rotation between elements 1 and 2, which can be expressed by a single
coordinate ‘’. Thus, a revolute pair has a single degree of freedom.
1
2
Figure 1.8 : Revolute or Turning Pair
Prismatic or Sliding Pair
As shown in Figure 1.9, a prismatic pair allows only a relative translation between elements 1 and 2, which can be expressed by a single coordinate ‘s’, and it has one degree of freedom.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 9
S
1
2
Figure 1.9 : Prismatic or Sliding Pair
Screw Pair
As shown in Figure 1.10, a screw pair allows rotation as well as translation but these two movements are
related to each other. Therefore, screw pair has one degree of freedom because the relative movement
between 1 and 2 can be expressed by a single coordinate ‘’ or ‘s’. These two coordinates are related by the
following relation :
2L
s
where, L is lead of the screw.
2
S
1
Figure 1.10 : Screw Pair Cylindrical Pair
As shown in Figure 1.11, a cylindrical pair allows both rotation and translation parallel to the axis of
rotation between elements 1 and 2. These relative movements can be expressed by two independent
coordinates ‘’ or ‘s’ because they are not related with each other. Degrees of freedom in this case are equal
to two.
S
1
2
Figure 1.11 : Cylindrical Pair Spherical Pair
A ball and socket joint, as shown in Figure 1.12, forms a spherical pair. Any rotation of element 2 relative
to 1 can be resolved in the three components. Therefore, the complete description of motion requires three
independent coordinates. Two of these coordinates ‘’ and ‘’ are required to specify the position of axis
OA and the third coordinate ‘’ describes the rotation about the axis of OA. This pair has three degrees of
freedom.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 10
Theory of Machines A
A
2
o 1
o
Figure 1.12 : Spherical Pair
Planar Pair
A planar pair is shown in Figure 1.13. The relative motion between 1 and 2 can be described by x and y coordinates in x-y plane. The x and y coordinates describe
relative translation and describes relative rotation about z-axis. This pair has three degrees of freedom.
Z
Y
1
X 2
Figure 1.13 : Planar Pair
1.7.2 Higher Pair
A higher pair is a kinematic pair in which connection between two elements is only a
point or line contact. The cam and follower arrangement shown in Figure 1.14 is an
example of this pair. The contact between cam and flat faced follower is through a line.
Other examples are ball bearings, roller bearings, gears, etc. A cylinder rolling on a flat
surface has a line contact while a spherical ball moving on a flat surface has a point
contact.
Follower
B A Cam
Rotating
with Shaft
Figure 1.14 : Higher Pair
1.7.3 Wrapping Pair
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 11
Wrapping pairs comprise belts, chains and such other devices. Belt comes from one side of the pulley and moves over to other side through another pulley as shown in Figure 1.15.
Figure 1.15 : Wrapping Pair
1.8 KINEMATIC CHAINS
In a kinematic chain, four links are required which are connected with each other with the help of lower pairs.
These pairs can be revolute pairs or prismatic pairs. A prismatic pair can be thought of as the limiting case of a
revolute pair. Before going into the general theory of mechanisms it may be observed that to form a simple closed chain we need
at least three links with three kinematic pairs. If any one of these three links is fixed, there cannot be relative
movement and, therefore, it does not form a mechanism but it becomes a structure which is completely rigid.
Thus, a simplest mechanism consists of four links, each connected by a kinematic lower pair (revolute etc.), and it
is known as four bar mechanism.
3
3 2
4
2
1
A
1
2
3
B
Q
1
Figure 1.16 : Planar Mechanism
For example, reciprocating engine mechanism is a planner mechanism in which link 1 is fixed, link 2 rotates and
link 4 reciprocates. In internal combustion engines, it converts reciprocating motion of piston into rotating motion
of crank. This mechanism is also used in reciprocating compressors in which it converts rotating motion of crank
into reciprocating motion of piston. This was a very common practical example and there are many other examples
like this. More about planar mechanisms shall follow in following sections. Let us consider the two mechanisms shown in Figure 1.17. The curved slider in figure acts similar to the revolute
pair. If radius of curvature ‘’ of the curved slider becomes infinite, the angular motion of the slider changes into
linear displacement and the revolute pair R4 transforms to a prismatic pair. Depending on different type of
kinematic pairs, four bar kinematic chain can be classified into three categories :
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 12
4R-kinematic chain which has all the four kinematic pairs as revolute pairs. (a) 3R-1P kinematic chain which has three revolute pairs and one prismatic pair. This is also called as single
slider crank chain. (b) 2R-2P kinematic chain which has two revolute pairs and two prismatic pairs. This is also called as
double slider crank chain.
3 B
R3
B
3 R3 4
R2
A
R2
2 4
2
R1 R4 O4
O2 R1 R4 O4
O2
1 1
1 1
Figure 1.17 : Kinematic Chain
SAQ 2
Form a kinematic chain using three revolute pairs and one prismatic pair.
1.9 INVERSIONS OF KINEMATIC CHAIN
If in a four bar kinematic chain all links are free, motion will be unconstrained. When
one link of a kinematic chain is fixed, it works as a mechanism. From a four link
kinematic chain, four different mechanisms can be obtained by fixing each of the four
links turn by turn. All these mechanisms are called inversions of the parent kinematic
chain. By this principle of inversions of a four link chain, several useful mechanisms can
be obtained.
3 3 3
2 4 2 4 2 4
1 1 1
3 3
2 4
2 4
1
1
Figure 1.18 : Inversion of Kinematic Chain
1.9.1 Inversions of 4R-Kinematic Chain
Kinematically speaking, all four inversions of 4R-kinematic chain are identical.
However, by suitably altering the proportions of lengths of links 1, 2, 3 and 4
respectively several mechanisms are obtained. Three different forms are illustrated here.
In Figure 1.19, links have been shown by blocks and lines connecting them represent
pairs.
1 T1
2 T2
3 T3
4
T4
Figure 1.19 : First Inversion
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 13
Crank-lever Mechanism or Crank-rocker Mechanism Simple Mechanisms
This mechanism is shown in Figure 1.20. In this case for every complete rotation of link 2 (called a crank), the link 4 (called a lever or rocker), makes oscillation
between extreme positions O4B1 and O4B2. B
B1 B2
A 3
4
2
A2
A1 O2 O4
Figure 1.20 : Crank-rocker Mechanism
The position of O4B1 is obtained when point A is A1 whereas position O4B2 is
obtained when A is at A2. It may be observed that crank angles for the two strokes
(forward and backward) of oscillating link O4B are not same. It may also be noted
that the length of the crank is very short. If l1, l2, l3 and l4 are lengths of links 1, 2, 3 and 4 respectively, the proportions of the link may be as follows :
(l1 l2 ) (l3 l4 )
(l2 l3 ) (l1 l4 )
Double-leaver Mechanism or Rocker-Rocker Mechanism
In this mechanism, both the links 2 and 4 can only oscillate. This is shown in Figure 1.21.
A1 A I3 B
I4
B2
B1
2
I2 A2 O4
O2
Figure 1.21 : Double-lever Mechanism
Link O2A oscillates between positions O2A1 and O2A2 whereas O4B oscillates
between positions O4B1 and O4B2. Position O4B2 is obtained when O2A and AB
are along straight line and position O2A1 is obtained when AB and O4B are along straight line. This mechanism must satisfy the following relations.
(l3 l4 ) (l1 l2 )
(l2 l3 ) (l1 l4 )
It may be observed that link AB has shorter length as compared to other links.
If links 2 and 4 are of equal lengths and l1 > l3, this mechanism forms automobile
steering gear.
Double Crank Mechanism
The links 2 and 4 of the double crank mechanism make complete revolutions.
There are two forms of this mechanism.
Parallel Crank Mechanism
In this mechanism, lengths of links 2 and 4 are equal. Lengths of links 1 and
3 are also equal. It is shown in Figure 1.22.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 14
Theory of Machines
A 3 B
2
4
O2 O4
1 1
Figure 1.22 : Double Crank Mechanism
The familiar example is coupling of the locomotive wheels where wheels
act as cranks of equal length and length of the coupling rod is equal to
centre distance between the two coupled wheels.
Drag Link Mechanism
In this mechanism also links 2 and 4 make full rotation. As the link 2 and 4
rotate sometimes link 4 rotate faster and sometimes it becomes slow in
rotation.
A1 B1
A B
B2 A4
O O B4
A2
A3 B3
Figure 1.23 : Drag Link Mechanism
The proportions of this mechanism are
l3 l1 ; l4 l2 l3
(l1 l4 l2 )
and l3 (l2 l4 l1)
It may be observed from Figure 1.23 that length of link 1 is smaller as compared to other links.
SAQ 3
Why 4R-kinematic chain does not provide four different mechanisms?
SAQ 4
In this mechanism, if length of link 2 is equal to that of link 4 and link 4 has lengths equal to that of link 2 which mechanism will result and analyse motion.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 15
1.9.2 Inversions of 3R-1P Kinematic Chain or Inversions of
Slider Crank Chain In this four bar kinematic chain, four links shown by blocks are connected through three
revolute pairs T1, T2 and T3 and one prismatic pair.
T1
T2
T3
1 2 3 4
S
Figure 1.24 ; Inversion of Slider Crank Chain
First Inversion
In this mechanism, link 1 is fixed, link 2 works as crank, link 4 works as a slider
and link 3 connects link 2 with 4. It is called connecting rod. Between links 1 and
4 sliding pair has been provided.
1 T1
2 T2
3 T3
4
S
2 3
4
1
Figure 1.25 : First Inversion
This mechanism is also known as slider crank chain or reciprocating engine
mechanism because it is used in internal combustion engines. It is also used in
reciprocating pumps as it converts rotatory motion into reciprocating motion and
vice-versa.
Second Inversion
In this case link 2 is fixed and link 3 works as crank. Link 1 is a slotted link which facilitates movement of link 4 which is a slider. This arrangement gives quick return motion mechanism. The motion of link 1 can be taped through a link and provided to ram of shaper machine. Figure 1.26 shows this mechanism and it is called Whitworth Quick Return Motion Mechanism. The forward stroke starts
when link 3 occupies position O4Q. At that time, point A is at A1. The forward
stroke ends when link 3 occupies position O4P and point A occupies position A2.
The return stroke takes place when link 3 moves from position O4P to O4Q. The stroke length is distance between A1 and A2 along line of stroke. If
acute angle < PO4 Q = and crank rotates at constant speed .
Quick Return Ratio = Time taken in forward stroke
Time taken in return stroke
(2 )
2
Simple Mechanisms
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 16
Theory of Machines
1 T1
2 T2
3 T3
4
3 4
S
2
1
4
B
3
O4
1
2 Q Ram
P
6
A1 O2 A2
5
A
Figure 1.26 : Second Inversion
Third Inversion
This inversion is obtained by fixing link 3. Some applications of this inversion are
oscillating cylinder engine and crank and slotted lever quick return motion
mechanism of a shaper machine. Link 1 works as a slider which slides in slotted or
cylindrical link 4. Link 2 works as a crank. The oscillating cylinder engine is
shown in Figure 1.27(a).
1
T1 2
T2 3
T3 4
2
4 1
3
S
3
(a) Oscillating Cylinder Engine
6 Ram
5
1
A
2
O2
A1 4
A2
3
O4
(b) Crank and Slotted Lever Mechanism
Figure 1.27 : Third Inversion
The motion of link 4 in crank and slotted lever quick return motion mechanism
can be taped through link 5 and can be transferred to ram. O2A1 and O2A2 are two positions of crank when link 4 will be tangential to the crank circle and corresponding to which ram will have extreme positions. When crank travels from
position O2A1 and O2A2 forward stroke takes place. When crank moves further
from position O2A2 to O2A1 return stroke takes place. Therefore, for constant
angular velocity for crank ‘’.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 17
(2)
2
Time for forward stroke
Quick Return Ratio =
Time for return stroke
Fourth Inversion – Pendulum Pump
Simple Mechanisms
It is obtained by fixing link 4 which is slider. Application of this inversion is
limited. The pendulum pump and hand pump are examples of this inversion. In
pendulum pump, link 3 oscillates like a pendulum and link 1 has translatory
motion which can be used for a pump.
1 4
3 1
2
Figure 1.28(a) : Pendulum Pump
2
1
3
1 T1
2 T2
3 T3
4
S
4
Figure 1.28(b) : Hand Pump
1.9.3 Inversions of 2R-2P Kinematic Chain or Double
Slider Crank Chain
This four bar kinematic chain has two revolute or turning pairs – T1 and T2 and two
prismatic or sliding pairs – S1 and S2. This chain provides three different mechanisms.
T1 T2 S1 1 2 3 4
S2
Figure 1.29 : Inversion of 2R-2P Kinematic Chain
First Inversion
The first inversion is obtained by fixing link 1. By doing so a mechanism called
Scotch Yoke is obtained. The link 1 is a slider similar to link 3. Link 2 works as a
crank. Link 4 is a slotted link. When link 2 rotates, link 4 has simple harmonic
motion for angle ‘’ of link 2, the displacement of link 4 is given by
x OA cos
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 18
Theory of Machines
1 T1
2 T2
3 S1
4
S2
A T2
3 2
s2
S T1
4
O
1 1
Figure 1.30 : Scotch Yoke Mechanism Second Inversion
In this case, link 2 is fixed and a mechanism called Oldham’s coupling is obtained.
This coupling is used to connect two shafts which have eccentricity ‘’. The axes
of the two shafts are parallel but displaced by distance . The link 4 slides in the
two slots provided in links 3 and 1. The centre of this link will move on a circle
with diameter equal to eccentricity.
Fixed Link
1 T1
2 T2
3 S1
4
S2
S1 1 2
3 T1
2 4 S2
T2
Figure 1.31 : Oldham’s Coupling Third Inversion
This inversion is obtained by fixing link 4. The mechanism so obtained is called
elliptical trammel which is shown in Figure 1.32. This mechanism is used to draw
ellipse. The link 1, which is slider, moves in a horizontal slot of fixed link 4. The
link 3 is also a slider moves in vertical slot. The point D on the extended portion
of link 2 traces ellipse with the system of axes shown in the figure, the position
coordinates of point D are as follows :
x AD sin or sin xD
D AD
y D CD cos or cos
yD
CD
Since, sin 2 cos
2 1
Substituting for sin and cos in this equation, the following equation of ellipse is
obtained.
x D2
y1
D 1
AD 2 CD
2
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 19
The semi-major axis of the ellipse is AD and semi-minor axis is CD. Simple Mechanisms
T1
T2 Y
1 2 3 S1
4
S2 4 T1
1 D
O X
4
S2
3
2
B C
A
T2 S1
Figure 1.32 : Elliptical Trammel
SAQ 5
(a) Explain why only three different mechanisms are available from 2R-2P kinematic chain.
(b) If length of crank in the reciprocating mechanism is 15 cm, find stroke
length of the slider.
(c) If length of fixed link and crank in crank and slotted lever quick return mechanism are 30 cm and 15 cm respectively, determine quick return ratio.
(d) If an ellipse of semi-major axis 30 cm and semi-minor axis 20 cm is to be
drawn, what should be the length of link ACD in elliptical trammel.
1.10 MACHINE
A machine is a mechanism or collection of several mechanisms which transmits force
from power source to the resistance to be overcome and, thereby, it performs useful
mechanical work. A common type of example is the commonly used internal-combustion
engine. The burning of petrol or diesel in cylinder results in a force on the piston which is
transmitted to the crank to result in driving torque. This driving torque overcomes the
resistance due to any external agency or friction, etc. at the crankshaft and thereby doing
useful work.
1.10.1 Difference between Machine and Mechanism A system can be defined as a mechanism or a machine on the basis of primary objective.
Sl. No. Machine Mechanism
1 If the system is used with the If the objective is to transfer or objective of transforming transform motion without
mechanical energy, then it is considering forces involved, the
called a machine system is said to be a mechanism
2 Every machine has to transmit It is concerned with transfer of motion because mechanical motion only
work is associated with the
motion, and thus makes use of
mechanisms
3 A machine can use one or It is not the case with mechanisms. more than one mechanism to A mechanism is a single system to
perform the desired function, transfer or transform motion
e.g. sewing machine has
several mechanisms
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 20
Theory of Machines
1.11 OTHER MECHANISMS
Geometry of motion of well known lower pair mechanisms will be examined and their
actual working and application will be dealt with. On the strength of analytical study of
these mechanisms, new mechanisms can be developed for specific requirements, for
modern plants and machinery.
1.11.1 Pantograph Pantograph is a geometrical instrument used in drawing offices for reproducing given
geometrical figures or plane areas of any shape, on an enlarged or reduced scale. It is
also used for guiding cutting tools. Its mechanism is utilised as an indicator rig for
reproducing the displacement of cross-head of a reciprocating engine which, in effect,
gives the position of displacement. There could be a number of forms of a pantograph. One such form is shown in Figure 1.33. It comprises of four links : AB, BC, CD, DA, pin-jointed at A, B, C and D.
Link BA is extended to a fixed pin O. Suppose Q is a point on the link AD of which the
motion is to be enlarged, then the link BC is extended to P such that O, Q, P are in a
straight line. It may be pointed out that link BC is parallel to link AD and that AB is
parallel to CD as shown. Thus, ABCD is a parallelogram.
P1 P
C D
C1 D1
Q
Q1
B A
O
A1
B1 Figure 1.33 : Pantograph Mechanism
Suppose a point Q on the link AD moves to position Q1 by rotating the link OAB
downward. Now all the links and the joints will move to the new positions : A to A1, B to B1, C to C1, D to D1 and P to P1 and the new configuration of the mechanism will
be as shown by dotted lines. The movement of Q (QQ1) will stand enlarged to PP1 in a definite ratio and in the same form as proved below : Triangles OAQ and OBP are similar. Therefore,
OBOA OQ
OP
In the dotted position of the mechanism when Q has moved to position Q1 and
correspondingly P to P1, triangles OA1Q1 and OB1P1 are also similar since length of
the links remain unchanged.
OA1
OQ1
OB
OP
1 1
But OB1 OB
OA1 OA
OA
OQ1
OB OP
1
OQ
QQ1
OP OP
1
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 21
As such triangle OQQ1 and OPP1 are similar, and PP1 and QQ1 are parallel and further, Simple Mechanisms
PP QQ
1
1
OP OQ
PP OQ QQ1
1 1 OQ
QQ1 OB
OA
Therefore, PP1 is a copied curve at enlarged scale.
1.11.2 Straight Line Motion Mechanisms A mechanism built in such a manner that a particular point in it is constrained to trace a
straight line path within the possible limits of motion, is known as a straight line motion
mechanism.
The Scott Russel Mechanism
This mechanism is shown in Figure 1.34. It consists of a crank OC, connecting rod
CP, and a slider block P which is constrained to move in a horizontal straight line
passing through O. The connecting rod PC is extended to Q such that
PC CQ CO
It will be proved that for all horizontal movements of the slider P, the locus of point Q will be a straight line perpendicular to the line OP.
Q
Locus o
f Q
O
Extension
Rod CD
Crank C
P
Figure 1.34 : Scott Russel Mechanism
Draw a circle of diameter PQ as shown. It is will known that diameter of a circle
always subtends a right angle or any point on the circle. Thus, at point O, the
angle QOP is a right angle. For any position of P, the line connecting O with P
will always be horizontal. Therefore, line joining the corresponding position of Q
with O will always a straight line perpendicular to OP. Thus, the locus of point Q
will be straight line perpendicular to OP. Thus, a horizontal straight line motion of
slider block P will enable point Q to generate a vertical straight line, both passing
through O.
Generated Straight Line Motion Mechanisms
Principle
The principle of working of an accurate straight line mechanism is based
upon the simple geometric property that the inverse of a circle with
reference to a pole on the circle is a straight line. Thus, referring to
Figure 1.35, if straight line OAB always passes through a fixed pole O and
the points A and B move in such a manner that : OA OB = constant, then
the end B is said to trace an inverse line to the locus of A moving on the
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 22
Theory of Machines
circle of diameter OC. Stated otherwise if O be a point on the circumference
of a circle diameter OP, OA by any chord, and B is a point on OA produced,
such that OA OB = a constant, then the locus of a point B will be a straight
line perpendicular to the diameter OP. All this is proves as follows :
B
A
O Pole O1 C D
Figure 1.35 : Straight Line Motion Mechanism
Draw a horizontal line from O. From A draw a line perpendicular to OA
cutting the horizontal at C. OC is the diameter of the circle on which the
point A will move about O such that OA OB remains constant.
Now, s OAC and ODB are similar.
Therefore, OA
OD
OC
OB
OD OA OB
OC
But OC is constant and so that if the product OA OB is constant, OD will
be constant, or the position of the perpendicular from B to OC produced is
fixed. This is possible only if the point B moves along a straight path BD
which is perpendicular to OC produced.
A number of mechanisms have been innovated to connect O, B and A in
such a way as to satisfy the above condition. Two of these are given as
follows :
The Peaucellier Mechanism
The mechanism consists of isosceles four bar chain OKBM (Figure 1.36). Additional links AK and AM from, a rhombus AKBM. A is constrained to move on a circular path by the radius bar CA which is equal to the length of the fixed link OC.
CA K
B
L
A
M
O C D
Figure 1.36 : The Peaucellier Mechanism
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 23
From the geometry of the figure, it follows that Simple Mechanisms
OA OB (OL AL) (OL LB ) OL2 AL
2 [ AL LB]
(OK 2 KL
2 ) ( AK
2 KL
2 ) OK
2 AK
2 constant
Hence, OA OB is constant for a given configuration and B
traces a straight path perpendicular to OC produced.
The Hart’s Mechanism
This is also known as crossed parallelogram mechanism. It is
an application of four-bar chain. PSQR is a four-bar chain in
which
SP = QR
and SQ = PR (Figure 1.37)
On three links SP, SQ and PQ, then it can be proved that for
any configuration of the mechanism :
OA OB = Constant
Q
a
x b
B R
P M
y
A
a
n
d
T
h
e
n
B
u
t
Hence,
S
Q
P
R
b
P
Q
x
SR y
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 24
N
OA OB OS
a x OP
a y xy constant
x SM NM [QK SR]
y SM NM [QN || PS]
xy ( SM 2 NM
2 )
(b 2 QM
2 ) ( a
2 QM
2 ) (b
2 a
2 )
OA OB xy constant = constant
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 25
It is therefore, concluded if the mechanism is pivoted at Q as a
fixed point and the point A is constrained to move on a circle
through O the point B will trace a straight line perpendicular to
the radius OC produced.
Approximate Straight-line Mechanisms
With the four-bar chain a large number of mechanisms can be devised which give a path which is approximately straight line. These are given as follows :
The Watt Mechanism
OABO is the mechanism used for Watt for obtaining approximate straight
line motion (Figure 1.38). It consists of three links : OA pivoted at O. OB
pivoted at O and both connected by link AB. A point P can be found on the
link AB which will have an approximate straight line motion over a limited
range of the mechanism. Suppose in the mean position link OA and OB are
in the horizontal position an OA and OB are the lower limits of movement
of these two links such that the configuration is OABO. Let I be the
instantaneous centre of the coupler link AB, which is obtained by
producing OA and BO to meet at I. From I draw a horizontal line to meet
AB at P. This point P, at the instant, will move vertically.
O A
a
A1
P
P1
B O
1
b
B1
Figure 1.38 : The Watts Mechanism
Considering angles and being exceedingly small, as an approximation,
AP
AA
BB
b
BP a b a
Where a and b are the lengths of OA and OB respectively. Since, both OA
and OB are horizontal in the mean or mid-position ever point in the
mechanism then moves vertically.
Hence, if P divides AB in the ratio
AP : BP = b : a
then P will trace a straight line path for a small range of movement on either side of the mean position of AB.
The Grass-hopper Mechanism
It is shown in Figure 1.39. It is a modification of Scott-Russel mechanism. It
consists of crank OC pivoted at O, link OP pivoted at O and a link PCR as
shown. It is, in fact, a laid out four-bar mechanism. Line joining O and P is
horizontal in middle position of the mechanism. The lengths of the link are
so fixed such that :
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 26
OC (CP)
2 Simple Mechanisms
CR
If this condition is satisfied, it is found that for a small angular displacement
of the link OP, the point R on the link PCR will trace approximately a straight line path, perpendicular to line PQ.
R
D
R1 C
C1 P P1
O P2
C2
R2
O1
Figure 1.39 : Grass-hopper Mechanism
In Figure 1.39, the positions both of P and R have been shown for three
different configuration of links. It may be noted that the pin at C is slidable
along with link RP such that at each position the above equation is satisfied. Robert’s Mechanism
This is also a four-bar chain ABCD in which links AD = BC (Figure 1.40).
The tracing point P is obtained by intersection of the right bisector of the
couple CD and a perpendicular on the horizontal from the instantaneous
centre I. Thus, an additional link E is connected to the coupler link BC and
the path of point P is approximately horizontal in this Robert’s mechanism.
I
E
D
C
A B
P
Figure 1.40 : Robert’s Mechanism
Tchebicheff’s Mechanism
This consists of four-bar chain in which two links AB and CD of equal
length cross each other; the tracing point P lies in the middle of the
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 27
Theory of Machines
connecting link BC (Figure 1.41). The proportions of the links are usually
such that P is directly above A or D in the extreme position of the
mechanism, i.e. when CB lies along AB or when CB lies along CD. It can be
shown that in these circumstances the tracing point P will lies on a straight
line parallel to AD in the two extreme positions and in the mid position if
BC : AD : AB : : 1 : 2 : 2.5
B2 C1
C
P2 P
P1
B
C2 B1
A D
Figure 1.41 : Tchebicheff’s Mechanism
SAQ 6
Which mechanisms are used for
(a) exact straight line, and
(b) approximate straight line.
1.11.3 Automobile Steering Gear
In an automobile vehicle the relative motion between its wheels and the road surface
should be one of pure rolling. To satisfy this condition, the steering gear should be so
designed that when the vehicle is moving along a curved path, the paths of points of
contact of each wheel with the road surface should be concentric circular arcs. The
steering or turning of a vehicle to one side or the other is accomplished by turning the
axis of rotation of the front two wheels. Each front wheel has a separate short axle of
rotation, known as stub axle such as AB and CD. These sub axles are pivoted to the
chassis of the vehicle. To satisfy the condition of pure rolling during turning, the design
of the steering gear should be such that at any instant while turning the axes of rotation
of the front and the rear wheels must intersect at one point which is known as
instantaneous centre denoted by I. The whole vehicle is assumed to be revolving about
this point at the instant considered. In Figure 1.42, AB and CD are the short axles of the
front wheel and EF for the rear wheels.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 28
Outer Front
Right turn Simple Mechanisms
Wheel
B
C
Inner Front
A D Wheel
a
E
F
I
Rear Wheels
Figure 1.42 : Automobile Steering Gear
While turning to the right side, axes of the front and the rear wheels meet at I.
Suppose = The angle by which the inner wheel is turned;
= The angle by which the outer wheel is turned;
A = Distance between the points of the front axles; and
l = Wheel base AE.
As may be seen from the geometry of the Figure 1.42, the angle of turn of the inner
front wheel is always more than the angle of turn of the outer front wheel. From Figure 1.42,
a AC EF EI FI l (cot cot )
a l (cot cot )
This is the fundamental equation of steering. If this equation is satisfied in a vehicle,
it is assured that the vehicle while taking a turn of any angle would not slip but would
have pure rolling motion between its wheels and the road surface.
Types of Steering Gears
There are mainly two types of steering gear mechanisms :
(a) Davis steering gear,
(b) Ackerman’s steering gear,
Both these mechanisms are described separately as follows :
Davis Steering Gear
This steering gear mechanism is shown in Figure 1.43(a). It consists of the
main axle AC having a parallel bar MN at a distance h. The steering is
accomplished by sliding bar MN within the guides (shown) either to left or
to the right hand side. KAB and LCD are two bell-crank levers pivoted with
the main axle at A and C respectively such that BAK and DCL remain
always constant. Arms AK and CL have been provided with slots and these
house die-blocks M and N. With the movement of bar MN at the fixed
height, it is the slotted arms AK and CL which side relative to the die-blocks
M and N.
In Figure 1.43(a), the vehicle has been shown as moving in a straight path
and both the slotted arms are inclined at an angle as shown.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 29
Theory of Machines Now suppose, for giving a turn to the right hand side, the base MN is moved
to the right side by distance x. The bell-crank levers will change to the
positions shown by dotted lines in Figure 1.43(b). The angle turned by the
inner wheel and the outer wheels are and respectively. The arms BA and
CD when produced will meet say at I, which will be the instantaneous
centre.
K L
M h N
B D
A C
(a)
b b
x x
B K
L
h
A
C
D
Suppose
Now,
or
(b) I
Figure 1.43 : Davis Steering Gear
2b = Difference between AC and MN, and
= Angle AK and CL make with verticals in normal position.
tan b
. . . (1.1(a))
h
tan () (b x)
[considering point A] . . . (1.2(b))
h
tan () (b x)
[considering point C] . . . (1.2(c))
h
tan () tan tan
1 tan tan
b
b x
tan
h
h
1
b
tan
h
b h tan (b x) (h
b
tan
)
h
hb h 2 tan (b x ) ( h b tan )
bh hx b 2 tan xb tan
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 30
h 2 tan b
2 tan xb tan bh hx bh hx
tan ( h 2 b
2 xb) hx
tan hx
. . . (1.2(d))
(h 2
b 2 xb)
Similarly, tan () tan tan
b x
1 tan tan h
Studying for tan and simplifying :
Simple Mechanisms
tan hx
(h 2
b 2 xb)
After obtaining the expressions for tan and
fundamental equation of steering :
cot cot h
2
b
2
xb
hx
cot cot 2h
b 2 tan
But for correct steering,
. . . (1.2(e))
tan , let us not take up the
( h 2 b
2 xb)
xh
. . . (1.2(f))
cot cot a
l
2 tan a
l
tan a
. . . (1.2(g))
2l
The ratio al varies from 0.4 to 0.5 and correspondingly to 14.1
o.
The demerits of the Davis gear are that due to number of sliding pairs,
friction is high and this causes wear and tear at contact surfaces rapidly,
resulting in in-accuracy of its working.
Ackermann Steering Gear
The mechanism is shown in Figure 1.44(a). This is simpler than that of the
Davis steering gear system. It is based upon four-bar chain. The two
opposite links AC and MN are unequal; AC being longer than MN. The other
two opposite links AM and CN are equal in length. When the vehicle is
moving on a straight path link AC and MN are parallel to each other. The
shorter links AM and CN are inclined at angle to the longitudinal axis of
the vehicle as shown. AB and CD are stub axles but integral part of AM and
CN such that BAM and DCN are bell-crank levers pivoted at A and C. Link
AM and CN are known as track arms and the link MN as track rod. The track
rod is moved towards left or right hand sides for steering. For steering a
vehicle on right hand side, link NM is moved towards left hand side with the
result that the link CN turns clockwise. Thus, the angle is increased and
that on the other side, it is decreased. From the arrangement of the links it is
clear that the link CN or the inner wheel will turn by an angle which is
more than the angle of turn of the outer wheel or the link AM.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 31
Theory of Machines
a
A C
B B
M N
(a)
A C O
B
D
N
M
a
0.3t
I
(b)
Figure 1.44 : Ackermann Steering Gear
To satisfy the basic equation of steering :
cot cot a
l ,
the links AM and MN are suitably proportioned and the angle is suitable
selected. In a given automobile, with known dimensions of the four-bar
links, angle is known. For different angle of turn , the corresponding
value of are noted. This is done by actually drawing the mechanism to a
scale. Thus, for different values of , the corresponding value of and
(cot cot ) are tabulated. As given above, for correct steering,
cot cot a
l
For approximately correct steering, value of al should be between 0.4
and 0.5. Generally, it is 0.455. In fact, there are three values of which give correct
steering; one when = 0, second and third for corresponding turning to the
right and the left hand.
Now there are two values of corresponding to given values of . The
value actually determined graphically by drawing the mechanism and
tabulating corresponding to different values of is known as actual
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 32
or a. But the value of obtained from the fundamental equation
cot cot a
l corresponding to different values of is known as
correct or c. On making comparison between the two values it is found
that a is higher than c for small values of , a are lower than c. The
difference is negligible for small value of but for large values of , it is substantial. This would reduce the life of the tyres due to greater wear on
account of slipping but then for large value of , the vehicle takes a sharper turn as such its speed is reduced and accordingly the wear is also reduced.
Thus, the greater difference between a and c for large value of will not
matter much.
As a matter of fact the position for correct steering is only an arbitrary
condition. In Ackermann steering, for keeping the value of angle on the
lower side, the instantaneous centre of the front wheels does not lie on the
line of axis of the rear wheel as shown in Figure 1.44(b).
1.11.4 Hook’s Joint or Universal Coupling It is shown in Figure 1.45, it is also known as universal joint. It is used for connecting
two shafts whose axes are non-parallel but intersecting as shown in Figure 1.45. Both the
shafts, driving and the driven, are forked at their ends. Each fork provides for two
bearings for the respective arms of the cross. The cross has two mutually perpendicular
arms. In fact, the cross acts as an intermediate link between the two shafts. In the figure,
the driven shaft has been shown as inclined at an angle with the driving shaft.
Driven Shaft Forks
Driving Shaft
Cross
Figure 1.45 : Hook’s Joint
The Hook’s joint is generally found being used for transmission of motion from the gear
box to the back axle of automobile and in the transmission of drive to the spindles in a
multi-spindle drilling machines. There are host of other applications of the Hook’s joint
where motion is required to be transmitted in non-parallel shafts with their axes
intersecting.
Figure 1.46(a) gives the end of the driving shafts. AB and CD are the mutually
perpendicular arms of the cross in the initial position. Arm AB is of the driving shaft and
CD for the driven shaft. The plan of rotation of the driving shaft and its arm AB will be
represented in the plane of the paper in elevation. In Figure 1.46(b), i.e. in the plan the direction of driving and driven shaft and that of the
cross arms are given. The driven shaft is inclined at an angle with the axis of the
driving shaft. PP gives the direction of the arm connected to the driving shaft and QQ gives the
direction of the arm connected to the driven shaft. In fact, the traces PP and QQ give the
plane of rotation of the arms of the cross, as seen in the plan view. Now, suppose, the driving shaft turns by angle . The arm AB will also turn by and
will take the position A1B1 as shown in the elevation. Suppose, correspondingly the
driven shaft and its arm CD are rotated by . The new position of CD is C2O. With the
Simple Mechanisms
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 33
Theory of Machines
rotation of AB by , it is the projection C1D1 of CD which will rotate through angle .
OC1 is the projection of OC and its sure length is given by OC2 and accordingly the
angle of rotation of the arm CD of the driven shaft, is known.
C
C1 C2
A1
A M N
O B
B1
D1
D
(a) End Elevation
Driving Shaft
O
M N
P
P
O
N1
Drivin
g
Shaft
O
(b) Plan
Figure 1.46
Ratio between and
As given above,
= The angle through which the driving shaft is rotated, and
= The corresponding angle through which the driven shaft is rotated.
Refer Figures 1.46(a) and (b).
tan OM
OM
. . . (1.3)
OC NC
1 2
tan
ON
NC
2
tan OM NC2 OM . . . (1.4)
NC
ON ON
tan
2
But from Figure 1.46(b)
OM cos
ON
tan 1 . . . (1.5)
cos
tan
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 34
Ratio between Speed of Driven and Driving Shafts Simple Mechanisms
= Angular speed of the driving shaft, and
1 = Angular speed of the driven shaft.
d
dt
d
1 dt
By Eq. (1.4)
tan
1
tan
cos
tan cos tan
Differentiating both sides,
sec2 d cos sec
2 d
dt
dt
sec 2 cos sec
2
1
cos sec2
cos cos2 sec
2 . . . (1.4(a))
1
sec
2
But sec2 1 tan
2
From Eq. (1.4),
tan
tan
cos
tan
2
tan2
cos
sec2
1 tan
2
1 sin
2
cos2
cos2 cos
2
cos
2 cos
2 sin
2
cos2 cos
2
cos
2 (1 sin
2 ) 1 cos
2
cos2 cos
2
cos
2 cos
2 sin
2 1 cos
2
cos2 cos
2
Hence, sec2
1 cos
2 sin
2
cos
2 cos
2
But as per Eq. (1.4(a)),
cos cos 2 sec
2
1
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 35
Theory of Machines Substituting for sec
2
(1 cos2
sin2 ) cos
2 sec
2
1
sec2 cos
2
1 cos2 sin
2
cos
Speed of driven Speed of driver
cos
1 1 cos 2 cos
2
Condition for Maximum and Minimum Speed Ratio
For a given value of :
will be a maximum
1
2 7
c o s
6
3
/ cos
. . . (1.5)
Speed of
Driver Speed of
1 Driven
5
4
Figure 1.47 : Polar Velocity Diagram
when in the equation :
cos ;
1 1 cos2 sin
2
The denominator (1 cos 2 sin
2 ) is minimum, i.e.
cos 2 1 or when cos 1
or when = 0 or 180o, corresponding to points 5 and 6 in Figure 1.47 and the
expression for the maximum speed ratio would be
cos 1 . . . (1.6)
1 sin2
1 cos
or [Represented by points 5 and 6 in Figure 1.47] . . . (1.7)
1 cos
Similarly for a given angle ,
will be minimum when in the equation.
1
cos
1 1 cos2 sin
2
the denominator is maximum.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 36
It will be so when = 90o or 270
o.
In that case cos . . . (1.8)
1
For maximum speed ratio, 1 = cos . . . (1.9)
Represented by points 7 and 8 in Figure 1.47.
Condition for the Same Speed
Simple Mechanisms
cos
1 1 sin
2 cos
2
For
to be unity
1
1
cos
1 sin2 cos
2
cos (1 sin 2 cos
2 )
cos2
1
cos
(1
cos
)
sin2 (1 cos
2 )
For the same speed,
cos
1
1 cos
1
1 cos
. . . (1.10)
Condition for Maximum Variation of Driven Speed
Maximum variation of speed of driven shaft
(
1 max
1 min
)
of driven shaft
mean
where mean =
cos
(1 cos 2 )
Maximum variation cos
cos
sin2
tan sin
. . . (1.11)
cos
Maximum variation = tan sin
If is small, tan = as well as sin =
Maximum variation 2 if is very small . . . (1.12)
Generally, the speed of the driving shaft is constant. As such it can be represented
by a circle of radius . In that case the maximum and minimum speed of the
driven shaft will be
and cos respectively, represented by an ellipse of
cos
major axis
and minor axis cos . This is shown in Figure 1.47 which is
cos
known as polar velocity diagram.
37
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 37
Angular Acceleration of the Driven Shaft
That angular acceleration of the driven shaft is given by
d 1 d 1 d d1
1 dt d dt d
But by Eq. (1.4),
1 cos2 sin
2
cos
1
1
cos
1 cos2 sin
2
d cos sin2
sin 2
1
d
(1 cos2 sin
2 )
2
By Eq. (1.13), 1 d1
d
Angular acceleration of driven shaft :
2 cos sin
2 sin 2
1 (1 cos
2 sin
2 )
2
For determining conditions for maximum acceleration, differentiate 1, w.r.t. and equate it to zero. The
resulting expression is, however, very complicated, and it will be found that the following expression
which is derived from the exact expression by a simple approximation, gives results which are sufficiently
close for most practical purpose.
For maximum 1, cos 2 2sin2 . . . (1.15)
sin2
2
This equation gives the value of almost accurate upto a maximum value of as 30o. It should be noted
that the angular acceleration of the driven shaft is a maximum when is approximate 45o, 135
o, etc., i.e.
when the arms of the cross are inclined at 45o to the plane contacting the axes of the two shafts.
MECHAICAL ADVANTAGE
The mechanical advantage of a mechanical is the ratio of the output force or torque at any instant to the input force or torque.
If friction and inertia forces are ignored.
Power input = Power output
If T2 be input torque,
2 be input angular velocity,
T4 be output torque, and
4 be output angular velocity.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 38
T2 2 T4 4
or T4
2
T
2 4
Mechanical Advantage T4
2
T2
4
Thus, mechanical advantage is the reciprocal of the velocity ratio.
1.12 KEY WORDS Machine : A machine is a mechanism or a collection of
mechanisms which transmits force from the
source of power to the resistance to be
overcome, and thus performs useful
mechanical work.
Mechanism : A mechanism is a combination of rigid or
restraining bodies so shaped and connected that
they move upon each other with definite
relative motion. slider-crank mechanism used
in internal combustion engine or reciprocating
air compressor is the simplest example.
Kinematic Pair : A pair is a joint of two elements that permits
relative motion. The relative motion between
the elements of links that form a pair is
required to be completely constrained or
successfully constrained.
Kinematic Link or Element : Kinematic link is a resistant body or an
assembly of resistant bodies which go to make
a part or parts of a machine connecting other
parts which have motion relative to it.
Resistant Body : A body is said to be a resistant body if it is
capable of transmitting the required forces with
negligible deformation.
Completely Constrained Motion : When the motion between a pair is limited to a
definite direction, irrespective of direction of
force applied and only one independent
variable is required to define motion. Such a
motion is said to be a completely constrained
motion.
Incompletely Constrained Motion : When a connection between the elements,
forming a pair, is such that the constrained
motion is not completed by itself, but by some
other means. Such a motion is said to be a
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 39
successfully constrained motion.
Kinematic Chain : A kinematic chain is 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 constrained.
Kinematic Inversions : In a mechanism, one of the links in a kinematic
chain is fixed. One may obtain different
mechanisms by fixing in different links in a
kinematic chain. This method of obtaining
different mechanisms is known as kinematic
inversions.
Crank : The link which makes complete rotation.
Rocker : The link oscillates between two extreme
positions.
Coupler : It transfers motion from input link to output
link or vice-versa.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 40
OUT COMES
1) Students understand link, Mechanism and various mechanisms
2) Students able to solve mobility of mechanisms by applying Grubler’s criterion
3) Demonstrate various inversions of kinematic chain and their applications
Exercise
1. Define link, Inversion, Kinematic pair, Kinematic chain, Machine 2. List inversions of a four bar chain and explain whit worth quick return motion
mechanism 3. Explain Pantograph 4. Explain Ackerman steering gear mechanism with a neat sketch 5. Explain ratchet and pawl mechanism
FURTHER READING
1. "Theory of Machines”, Rattan S.S, Tata McGraw-Hill Publishing Company Ltd., New
Delhi, and
3rd edition -2009.
2. "Theory of Machines”, Sadhu Singh, Pearson Education (Singapore) Pvt. Ltd, Indian
Branch New
Delhi, 2nd Edi. 2006
3. “Theory of Machines & Mechanisms", J.J. Uicker, , G.R. Pennock, J.E. Shigley.
OXFORD 3rd
Ed. 2009.
4. Mechanism and Machine theory, Ambekar, PHI, 2007
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 41
UNIT 4 Velocity Analysis by Instantaneous Center
Method
CONTENTS
4.1 Definition, Kennedy's Theorem,
4.2 Determination of linear and angular velocity using instantaneous center method
4.3 Klein's Construction: Analysis of velocity and acceleration of single slidercrank
mechanism.
OBJECTIVES:
Learn Instantaneous centre method to do velocity analysis of a kinematic
chain
Study Klein’s construction method to do velocity and acceleration analysis
of slider crank mechanisms
4.1 Definition, Kennedy's Theorem
II Method Instantaneous
Method
To explain instantaneous centre let us consider a plane body P having a nonlinear
motion relative to another body q consider two points A and B on body P having velocities
as Va and Vb respectively in the direction shown.
A V a
B V b
I P
q
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 42
If a line is drawn r to Va, at A the body can be imagined to rotate about some point on
the line. Thirdly, centre of rotation of the body also lies on a line r to the direction of Vb at B.
If the intersection of the two lines is at I, the body P will be rotating about I at that
instant. The point I is known as the instantaneous centre of rotation for the body P. The
position of instantaneous centre changes with the motion of the body.
A V a P
B V b
I q
Fig. 2
In case of the r
lines drawn from A and B meet outside the body P as shown in Fig 2.
A V a
B V b
I at
Fig. 3
If the direction of Va and Vb are parallel to the r at A and B met at . This is the case when the
body has linear motion.
• Number of Instantaneous Centers
The number of instantaneous centers in a mechanism depends upon number of
links. If N is the number of instantaneous centers and n is the number of links. • Types of Instantaneous Centers
There are three types of instantaneous centers namely fixed, permanent and neither
fixed nor permanent.
Example: Four bar mechanism. n = 4. n n 1
4 4 1
N = = 6
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 43
Fixed instantaneous center I12, I14
Permanent instantaneous center I23, I34
Neither fixed nor permanent instantaneous center I13, I24 • Arnold Kennedy theorem of three centers: Statement: If three bodies have motion relative to each other, their instantaneous
centers should lie in a straight line. Proof:
1 V A3 V A2
I 12
I 13
2 3
I 23 A
Consider a three link mechanism with link 1 being fixed link 2 rotating about
I12 and link 3 rotating about I13. Hence, I12 and I13 are the instantaneous centers for link 2
and link 3. Let us assume that instantaneous center of link 2 and 3 be at point A i.e. I23.
Point A
is a coincident point on link 2 and link 3.
Considering A on link 2, velocity of A with respect to I12 will be a vector VA2
link A I12. Similarly for point A on link 3, velocity of A with respect to I13 will be I13. It is seen that velocity vector of VA2 and VA3 are in different directions which is impossible. Hence, the instantaneous center of the two links cannot be at the
assumed position.
It can be seen that when I23 lies on the line joining I12 and I13 the VA2 and VA3
will be same in magnitude and direction. Hence, for the three links to be in relative
motion all the three centers should lie in a same straight line. Hence, the proof.
4.2 Determination of linear and angular velocity using instantaneous
center method
r to
r to A
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 44
Steps to locate instantaneous centers:
Step 1: Draw the configuration
diagram. Step 2: Identify the number of instantaneous centers by using the relation n 1 n N = .
2 Step 3: Identify the instantaneous centers by circle diagram.
Step 4: Locate all the instantaneous centers by making use of Kennedy’s theorem.
To illustrate the procedure let us consider an example.
A slider crank mechanism has lengths of crank and connecting rod equal to 200 mm
and 200 mm respectively locate all the instantaneous centers of the mechanism for the
position of the crank when it has turned through 30o from IOC. Also find velocity of
slider and angular velocity of connecting rod if crank rotates at 40 rad/sec. Step 1: Draw configuration diagram to a suitable scale.
Step 2: Determine the number of links in the mechanism and find number of instantaneous
centers. n 1 n N =
2
4 4 1
n = 4 linksN = = 6
2
I 13
I 24
A
3
2
200 I 23 800 B
I 12
30o
4
O 1 1 I 12
I 14 to I 14 to
Step 3: Identify instantaneous centers. o Suit it is a 4-bar link the
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 45
resulting figure will be a square.
1 I12 2
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 46
I 24
1 2 3 4
I 12 I 23
I 34
I 13 I 24
I 13 I 14
I41
I23 OR
4 I34 3
o Locate fixed and permanent instantaneous centers. To locate neither fixed nor
permanent instantaneous centers use Kennedy’s three centers theorem.
Step 4: Velocity of different points. Va = 2 AI12 = 40 x 0.2 = 8 m/s
also Va = 2 x A13
3 = Va
AI13
Vb = 3 x BI13 = Velocity of slider.
OUT COME
Analyze the given the simple mechanisms for calculating velocity and acceleration
at various points
Exercise
1. Explain Kennedy’s theorem with proof.
2. A reciprocating engine mechanism has connecting rod 200mm long and
crank 50mm long. By KLEIN’s construction, determine the velocity and acceleration of piston,
and angular acceleration of connecting rod, when the crank has turned through 450 from IDC
clockwise and is rotating at 240rpm.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 47
UNIT 6 SPUR GEARS
CONTENTS
6.1 Introduction
6.2 Gear terminology,
6.3 Law of gearing,
6.4 Path of contact, arc of contact, contact ratio of spur gear.
6.5 Interference in involute gears, methods of avoiding interference,
6.6 Back lash, condition for minimum number of teeth to avoid
interference,
6.7 Expressions for arc of contact and path of contact
OBJECTIVES
1. Overview the various types gears and terminologies used
2. Comprehend the theorems and expressions used for design of gears
6.1 Introduction
Spur gears: Spur gears are the most common type of gears. They have straight teeth, and
are mounted on parallel shafts. Sometimes, many spur gears are used at once to create
very large gear reductions. Each time a gear tooth engages a tooth on the other gear,
the teeth collide, and this impact makes a noise. It also increases the stress on the gear
teeth. To reduce the noise and stress in the gears, most of the gears in your car are
helical.
Spur gears (Emerson Power Transmission Corp)
External contact Internal contact
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 48
Spur gears are the most commonly used gear type. They are characterized by teeth,
which are perpendicular to the face of the gear. Spur gears are most commonly available,
and are generally the least expensive. • Limitations: Spur gears generally cannot be used when a direction change between
the two shafts is required. • Advantages: Spur gears are easy to find, inexpensive, and efficient.
2. Parallel helical gears: The teeth on helical gears are cut at an angle to the face of the
gear. When two teeth on a helical gear system engage, the contact starts at one end of
the tooth and gradually spreads as the gears rotate, until the two teeth are in full
engagement.
Helical gears
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 49
(Emerson Power Transmission Corp) Herringbone gears
This gradual engagement makes helical gears operate much more smoothly and quietly than
spur gears. For this reason, helical gears are used in almost all car transmission. Because of the angle of the teeth on helical gears, they create a thrust load on the gear when
they mesh. Devices that use helical gears have bearings that can support this thrust load. One interesting thing about helical gears is that if the angles of the gear teeth are correct,
they can be mounted on perpendicular shafts, adjusting the rotation angle by 90 degrees.
Helical gears to have the following differences from spur gears of the same size:
o Tooth strength is greater because the teeth are longer,
o Greater surface contact on the teeth allows a helical gear to carry more load than a spur
gear o The longer surface of contact reduces the efficiency of a helical gear relative to a spur
gear Rack and pinion (The rack is like a gear whose axis is at
infinity.): Racks are straight gears that are used to convert
rotational motion to translational motion by means of a
gear mesh. (They are in theory a gear with an infinite pitch
diameter). In theory, the torque and angular velocity of the pinion gear are related to the Force and the velocity of the rack by the radius of the
pinion gear, as is shown. Perhaps the most well-known application of a rack is the rack and pinion steering system
used on many cars in the past Gears for connecting intersecting shafts: Bevel gears are useful when the direction of
a shaft's rotation needs to be changed. They are usually mounted on shafts that are 90
degrees apart, but can be designed to work at other angles as well. The teeth on bevel gears can be straight, spiral or hypoid. Straight bevel gear teeth actually
have the same problem as straight spur gear teeth, as each tooth engages; it impacts the
corresponding tooth all at once. Just like with spur gears, the solution to this problem is to curve the gear teeth. These spiral
teeth engage just like helical teeth: the contact starts at one end of the gear and
progressively spreads across the whole tooth.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 50
Straight bevel gears Spiral bevel gears
On straight and spiral bevel gears, the shafts must be perpendicular to each other, but they
must also be in the same plane. The hypoid gear, can engage with the axes in different
planes. This feature is used in many car differentials. The ring gear of the differential and the
input pinion gear are both hypoid. This allows the input pinion to be mounted lower than
the axis of the ring gear. Figure shows the input pinion engaging the ring gear of the
differential. Since the driveshaft of the car is connected to the input pinion, this also
lowers the driveshaft. This means that the driveshaft
Hypoid gears (Emerson Power Transmission Corp)
doesn't pass into the passenger compartment of the car as much, making more room for people
and cargo. Neither parallel nor intersecting shafts: Helical gears may be used to mesh two shafts that
are not parallel, although they are still primarily use in parallel shaft applications. A special
application in which helical gears are used is a crossed gear mesh, in which the two shafts are
perpendicular to each other.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 51
Crossed-helical gears
Worm and worm gear: Worm gears are used when large gear reductions are
needed. It is common for worm gears to have reductions of 20:1, and even up
to 300:1 or greater. Many worm gears have an interesting property that no other gear set has: the
worm can easily turn the gear, but the gear cannot turn the worm. This is
because the angle on the worm is so shallow that when the gear tries to spin
it, the friction between the gear and the worm holds the worm in place. This feature is useful for machines such as conveyor systems, in which the
locking feature can act as a brake for the conveyor when the motor is not
turning. One other very interesting usage of worm gears is in the Torsen
differential, which is used on some high-performance cars and trucks.
6.2 Gear terminology,
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 52
Addendum: The radial distance between the Pitch Circle and the top of the teeth.
Arc of Action: Is the arc of the Pitch Circle between the beginning and the end of
the engagement of a given pair of teeth.
Arc of Approach: Is the arc of the Pitch Circle between the first point of contact of the gear
teeth and the Pitch Point.
Arc of Recession: That arc of the Pitch Circle between the Pitch Point and the last point of
contact of the gear teeth.
Backlash: Play between mating teeth.
Base Circle: The circle from which is generated the involute curve upon which the tooth profile
is based.
Center Distance: The distance between centers of two gears.
Chordal Addendum: The distance between a chord, passing through the points where the Pitch
Circle crosses the tooth profile, and the tooth top.
Chordal Thickness: The thickness of the tooth measured along a chord passing through the
points where the Pitch Circle crosses the tooth profile.
Circular Pitch: Millimeter of Pitch Circle circumference per tooth.
Circular Thickness: The thickness of the tooth measured along an arc following the Pitch Circle
Clearance: The distance between the top of a tooth and the bottom of the space into which it fits
on the meshing gear.
Contact Ratio: The ratio of the length of the Arc of Action to the Circular Pitch.
Dedendum: The radial distance between the bottom of the tooth to pitch circle.
Diametral Pitch: Teeth per mm of diameter.
Face: The working surface of a gear tooth, located between the pitch diameter and the top of the
tooth.
Face Width: The width of the tooth measured parallel to the gear axis.
Flank: The working surface of a gear tooth, located between the pitch diameter and the bottom
of the teeth
Gear: The larger of two meshed gears. If both gears are the same size, they are both called
"gears".
Land: The top surface of the tooth.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 53
Line of Action: That line along which the point of contact between gear teeth travels, between
the first point of contact and the last.
Module: Millimeter of Pitch Diameter to Teeth.
Pinion: The smaller of two meshed gears.
Pitch Circle: The circle, the radius of which is equal to the distance from the center of the gear
to the pitch point.
Diametral pitch: Teeth per millimeter of pitch diameter.
Pitch Point: The point of tangency of the pitch circles of two meshing gears, where the Line of
Centers crosses the pitch circles.
Pressure Angle: Angle between the Line of Action and a line perpendicular to the Line of
Centers.
Profile Shift: An increase in the Outer Diameter and Root Diameter of a gear, introduced to
lower the practical tooth number or acheive a non-standard Center Distance.
Ratio: Ratio of the numbers of teeth on mating gears.
Root Circle: The circle that passes through the bottom of the tooth spaces.
Root Diameter: The diameter of the Root Circle.
Working Depth: The depth to which a
tooth extends into the space between teeth
on the mating gear.
6.3 Law of gearing
Gear-Tooth Action
Figure 5.2 shows two mating gear teeth, in
which • Tooth profile 1 drives tooth profile 2 by acting
at the instantaneous contact point K. • N1N2 is the common normal of the two profiles.
• N1 is the foot of the perpendicular from O1 to
N1N2
• N2 is the foot of the perpendicular from O2 to
N1N2.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 54
Although the two profiles have different velocities V1 and V2 at point K, their velocities along N1N2 are
equal in both magnitude and direction. Otherwise the two tooth profiles would separate from each other.
Constant Velocity Ratio For a constant velocity ratio, the position of P should remain unchanged. In this case, the motion
transmission between two gears is equivalent to the motion transmission between two imagined slip-less
cylinders with radius R1 and R2 or diameter D1 and D2. We can get two circles whose centers are at O1 and O2, and through pitch point P. These two circles are termed pitch circles. The velocity ratio is equal to the
inverse ratio of the diameters of pitch circles. This is the fundamental law of gear-tooth action. The fundamental law of gear-tooth action may now also be stated as follow (for gears with fixed center
distance) A common normal (the line of action) to the tooth profiles at their point of contact must, in all positions
of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch point Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate curves,
and the relative rotation speed of the gears will be constant(constant velocity ratio). Conjugate Profiles
To obtain the expected velocity ratio of two tooth profiles, the normal line of their profiles must pass
through the corresponding pitch point, which is decided by the velocity ratio. The two profiles which
satisfy this requirement are called conjugate profiles. Sometimes, we simply termed the tooth profiles
which satisfy the fundamental law of gear-tooth action the conjugate profiles. Although many tooth shapes are possible for which a mating tooth could be designed to satisfy the
fundamental law, only two are in general use: the cycloidal andinvolute profiles. The involute has
important advantages; it is easy to manufacture and the center distance between a pair of involute gears
can be varied without changing the velocity ratio. Thus close tolerances between shaft locations are not
required when using the involute profile. The most commonly used conjugate tooth curve is the involute
curve. (Erdman & Sandor). conjugate action : It is essential for correctly meshing gears, the size of the teeth ( the module ) must be
the same for both the gears. Another requirement - the shape of teeth necessary for the speed ratio to remain constant during an
increment of rotation; this behavior of the contacting surfaces (ie. the teeth flanks) is known as conjugate action.
Involute Curve
The following examples are involute spur gears. We use the word involute because the contour of gear
teeth curves inward. Gears have many terminologies, parameters and principles. One of the important
concepts is the velocity ratio, which is the ratio of the rotary velocity of the driver gear to that of the
driven gears.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 55
Generation of the Involute Curve
The curve most commonly used for gear-tooth
profiles is the involute of a circle. This involute
curve is the path traced by a point on a line as the line
rolls without slipping on the circumference of a
circle. It may also be defined as a path traced by the
end of a string, which is originally wrapped on a
circle when the string is unwrapped from the
circle. The circle from which the involute is derived is called the base circle.
Figure 4.3 Involute curve
Properties of Involute Curves
1. The line rolls without slipping on the circle.
2. For any instant, the instantaneous center of the motion of the line is its point of tangent with the circle.
Note: We have not defined the term instantaneous center previously. The instantaneous center or
instant center is defined in two ways.
1. When two bodies have planar relative motion, the instant center is a point on one body about which
the other rotates at the instant considered.
2. When two bodies have planar relative motion, the instant center is the point at which the bodies are
relatively at rest at the instant considered.
3. The normal at any point of an involute is tangent to the base circle. Because of the property (2) of the
involute curve, the motion of the point that is tracing the involute is perpendicular to the line at any
instant, and hence the curve traced will also be perpendicular to the line at any instant.
There is no involute curve within the base circle.
Cycloidal profile:
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 56
Epicycliodal Profile:
Hypocycliodal Profile:
The involute profile of gears has important advantages;
• It is easy to manufacture and the center distance between a pair of involute gears can be varied
without changing the velocity ratio. Thus close tolerances between shaft locations are not
required. The most commonly used conjugate tooth curve is the involute curve. (Erdman &
Sandor). 2. In involute gears, the pressure angle, remains constant between the point of tooth engagement
and disengagement. It is necessary for smooth running and less wear of gears.
But in cycloidal gears, the pressure angle is maximum at the beginning of engagement, reduces to zero
at pitch point, starts increasing and again becomes maximum at the end of engagement. This results in
less smooth running of gears. 3. The face and flank of involute teeth are generated by a single curve where as in cycloidalgears, double
curves (i.e. epi-cycloid and hypo-cycloid) are required for the face and flank respectively. Thus the
involute teeth are easy to manufacture than cycloidal teeth.
In involute system, the basic rack has straight teeth and the same can be cut with simple tools.
Advantages of Cycloidal gear teeth:
1. Since the cycloidal teeth have wider flanks, therefore the cycloidal gears are stronger than the involute
gears, for the same pitch. Due to this reason, the cycloidal teeth are preferred specially for cast teeth.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 57
2. In cycloidal gears, the contact takes place between a convex flank and a concave surface, where as in
involute gears the convex surfaces are in contact. This condition results in less wear in cycloidal gears
as compared to involute gears. However the difference in wear is negligible 3. In cycloidal gears, the interference does not occur at all. Though there are advantages of cycloidal gears
but they are outweighed by the greater simplicity and flexibility of the involute gears. Properties of involute teeth:
1. A normal drawn to an involute at pitch point is a tangent to the base circle. 2. Pressure angle remains constant during the mesh of an involute gears. 3. The involute tooth form of gears is insensitive to the centre distance and depends only on the
dimensions of the base circle. 4. The radius of curvature of an involute is equal to the length of tangent to the base circle. 5. Basic rack for involute tooth profile has straight line form. 6. The common tangent drawn from the pitch point to the base circle of the two involutes is the line of
action and also the path of contact of the involutes. 7. When two involutes gears are in mesh and rotating, they exhibit constant angular velocity ratio and is
inversely proportional to the size of base circles. (Law of Gearing or conjugate action) 8. Manufacturing of gears is easy due to single curvature of profile.
System of Gear Teeth
The following four systems of gear teeth are commonly used in practice:
1. 14 ½O
Composite system 2. 14 ½
O Full depth involute system
3. 20
O Full depth involute system
4. 20
O Stub involute system
The 14½O
composite system is used for general purpose gears. It is stronger but has no interchangeability. The tooth profile of this system has cycloidal curves at the top
and bottom and involute curve at the middle portion. The teeth are produced by formed milling cutters or hobs.
The tooth profile of the 14½O
full depth involute system was developed using gear hobs for spur and
helical gears.
The tooth profile of the 20o full depth involute system may be cut by hobs.
The increase of the pressure angle from 14½o to 20
o results in a stronger tooth, because the tooth acting
as a beam is wider at the base.
The 20o stub involute system has a strong tooth to take heavy loads.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 58
Constant Velocity Ratio
For a constant velocity ratio, the position of P should remain unchanged. In this case, the motion
transmission between two gears is equivalent to the motion transmission between two imagined
slip-less cylinders with radius R1 and R2 or diameter D1 and D2. We can get two circles whose
centers are at O1 and O2, and through pitch point P. These two circles are termed pitch circles.
The velocity ratio is equal to the inverse ratio of the diameters of pitch circles. This is the
fundamental law of gear-tooth action. The fundamental law of gear-tooth action may now also be stated as follow (for gears with
fixed center distance) A common normal (the line of action) to the tooth profiles at their point of contact must, in all
positions of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch
point Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate
curves, and the relative rotation speed of the gears will be constant(constant velocity ratio). Conjugate Profiles
To obtain the expected velocity ratio of two tooth profiles, the normal line of their profiles must
pass through the corresponding pitch point, which is decided by the velocity ratio. The two
profiles which satisfy this requirement are called conjugate profiles. Sometimes, we simply
termed the tooth profiles which satisfy the fundamental law of gear-tooth action the conjugate
profiles. Although many tooth shapes are possible for which a mating tooth could be designed to satisfy
the fundamental law, only two are in general use: the cycloidal andinvolute profiles. The involute
has important advantages; it is easy to manufacture and the center distance between a pair of
involute gears can be varied without changing the velocity ratio. Thus close tolerances between
shaft locations are not required when using the involute profile. The most commonly used
conjugate tooth curve is the involute curve. (Erdman & Sandor). conjugate action : It is essential for correctly meshing gears, the size of the teeth ( the module )
must be the same for both the gears. Another requirement - the shape of teeth necessary for the speed ratio to remain constant during
an increment of rotation; this behavior of the contacting surfaces (ie. the teeth flanks) is known
as conjugate action.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 59
Involute Curve
The following examples are involute spur gears. We use the word involute because the contour
of gear teeth curves inward. Gears have many terminologies, parameters and principles. One of
the important concepts is the velocity ratio, which is the ratio of the rotary velocity of the
driver gear to that of the driven gears.
Generation of the Involute Curve
The curve most commonly used for gear-tooth
profiles is the involute of a circle. This
involute curve is the path traced by a point on a
line as the line rolls without slipping on the
circumference of a circle. It may also be
defined as a path traced by the end of a string,
which is originally wrapped on a circle when the
string is unwrapped from the circle. The circle from which the involute is derived is called
the base circle. Properties of Involute Curves 3. The line rolls without slipping on the circle. 4. For any instant, the instantaneous center of the motion of the line is its point of tangent with
the circle.
Note: We have not defined the term instantaneous center previously. The instantaneous center
or instant center is defined in two ways.
4. When two bodies have planar relative motion, the instant center is a point on one body about
which the other rotates at the instant considered. 5. When two bodies have planar relative motion, the instant center is the point at which the
bodies are relatively at rest at the instant considered.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 60
6. The normal at any point of an involute is tangent to the base circle. Because of the property
(2) of the involute curve, the motion of the point that is tracing the involute is perpendicular to
the line at any instant, and hence the curve traced will also be perpendicular to the line at any
instant.
There is no involute curve within the base circle.
Cycloidal profile:
Epicycliodal Profile:
Hypocycliodal Profile:
The involute profile of gears has important advantages;
• It is easy to manufacture and the center distance between a pair of involute gears can be
varied without changing the velocity ratio. Thus close tolerances between shaft locations
are not required. The most commonly used conjugate tooth curve is the involute curve.
(Erdman & Sandor).
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 61
4. In involute gears, the pressure angle, remains constant between the point of tooth
engagement and disengagement. It is necessary for smooth running and less wear of gears.
But in cycloidal gears, the pressure angle is maximum at the beginning of engagement,
reduces to zero at pitch point, starts increasing and again becomes maximum at the end of engagement. This results in less smooth running of gears.
5. The face and flank of involute teeth are generated by a single curve where as in
cycloidalgears, double curves (i.e. epi-cycloid and hypo-cycloid) are required for the face
and flank respectively. Thus the involute teeth are easy to manufacture than cycloidal teeth.
In involute system, the basic rack has straight teeth and the same can be cut with simple tools.
Advantages of Cycloidal gear teeth:
9. Since the cycloidal teeth have wider flanks, therefore the cycloidal gears are stronger than
theinvolute gears, for the same pitch. Due to this reason, the cycloidal teeth are preferred
specially for cast teeth. 10. In cycloidal gears, the contact takes place between a convex flank and a concave surface,
where as in involute gears the convex surfaces are in contact. This condition results in less
wear in cycloidal gears as compared to involute gears. However the difference in wear is
negligible 11. In cycloidal gears, the interference does not occur at all. Though there are advantages of
cycloidal gears but they are outweighed by the greater simplicity and flexibility of the involute
gears. Properties of involute teeth:
1. A normal drawn to an involute at pitch point is a tangent to the base circle. 2. Pressure angle remains constant during the mesh of an involute gears. 3. The involute tooth form of gears is insensitive to the centre distance and depends only on
thedimensions of the base circle. 12. The radius of curvature of an involute is equal to the length of tangent to the base circle. 13. Basic rack for involute tooth profile has straight line form. 14. The common tangent drawn from the pitch point to the base circle of the two involutes is
theline of action and also the path of contact of the involutes. 15. When two involutes gears are in mesh and rotating, they exhibit constant angular velocity
ratioand is inversely proportional to the size of base circles. (Law of Gearing or conjugate
action)
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 62
16. Manufacturing of gears is easy due to single curvature of profile.
System of Gear Teeth
The following four systems of gear teeth are commonly used in practice:
5. 14 ½O
Composite system
6. 14 ½O
Full depth involute system
7. 20O
Full depth involute system
8. 20O
Stub involute system
The 14½O
composite system is used for general purpose gears.
It is stronger but has no interchangeability. The tooth profile of this system has cycloidal
curves at the top and bottom and involute curve at the middle portion. The teeth are produced by formed milling cutters or hobs.
The tooth profile of the 14½O
full depth involute system was developed using gear hobs for
spur and helical gears.
The tooth profile of the 20o full depth involute system may be cut by hobs.
The increase of the pressure angle from 14½o to 20
o results in a stronger tooth, because the
tooth acting as a beam is wider at the base.
The 20o stub involute system has a strong tooth to take heavy loads.
6.5 Interference in involute gears, methods of avoiding interference,
The study of the geometry of the involute profile for gear teeth is called involumetry. Consider
an involute of base circle radius ra and two points B and C on the involute as shown in figure.
Draw normal to the involute from the points B and C. The normal BE and CF are tangents to
the Base circle.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 63
Let ra= base circle radius of gear rb= radius of point B on the involute rc= radius of point C on the
involute and From the properties of the Involute:
Arc AE = Length BE and
Arc AF = Length CF
2rc
Using this equation and knowing tooth thickness at any point on the tooth, it is possible
to calculate the thickness of the tooth at any point Number of Pairs of Teeth in Contact
Continuous motion transfer requires two pairs of teeth in contact at the ends of the path
of contact, though there is only one pair in contact in the middle of the path, as in Figure.
R
r
The average number of teeth in contact is an important parameter - if it is too low due to the use
of inappropriate profile shifts or to an excessive centre distance.The manufacturing
inaccuracies may lead to loss of kinematic continuity - that is to impact, vibration and noise. The average number of teeth in contact is also a guide to load sharing between teeth; it is
termed the contact ratio
6.4 Path of contact, arc of contact, contact ratio of spur gear.
Length of path of contact for Rack and Pinion:
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 64
Let
r = Pitch circle radius of the pinion = O1P
= Pressure angle ra. = Addendu m
radius of the pinion a = Addendum of
rack
EF = Length of path of contact
EF = Path of approach EP + Path of recess PF
From AP a
sin
triang EP EP
(1)
leO1N
a
P :Path of approach EP
NP O1Psin
rsin
sin (2)
O1N O1Pcos Path of recess PF NF NP (3)
rcos
From triangle O1NF:
NF O1F2 O1N2 12 ra2 r2 cos2 12
Substituting NP and NF values in theequation (3)
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 65
Path of racess PF ra2
r2 cos
2 1
2 r sin
Path of
length
of
contact EF EP PF a
ra2
r2 cos
2
12 r sin sin
Exercise problems refer presentation slides
6.5 Interference in involute gears, methods of avoiding interference,
Pitch
Pitch
Circle
Figure shows a pinion and a gear in mesh with their center as O1andO2 respectively. MN is the
common tangent to the basic circles and KL is the path of contact between the two mating teeth.
Consider, the radius of the addendum circle of pinion is increased to O1N, the point of contact L
will moves from L to N. If this radius is further increased, the point of contact L will be inside
of base circle of wheel and not on the involute profile of the pinion. The tooth tip of the pinion will then undercut the
tooth on the wheel at the root and damages Wheel
part
of the involute profile. This effect is known as
interference, and occurs when the teeth are
being cut and weakens the tooth at its root.
In general, the phenomenon, when the tip of
Undercut Pinion
tooth undercuts the root on its mating gear is
known as interference.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 66
Similarly, if the radius of the addendum circles of the wheel increases beyond O2M, then the tip
of tooth on wheel will cause interference with the tooth on pinion. The points M and N are
called interference points. Interference may be avoided if the path of the contact does not extend beyond interference
points. The limiting value of the radius of the addendum circle of the pinion is O1N and of the
wheel is O2M. The interference may only be prevented, if the point of contact between the two teeth is
always on the involute profiles and if the addendum circles of the two mating gears cut the
common tangent to the base circles at the points of tangency. When interference is just prevented, the maximum length of path of contact is MN.
Methods to avoid Interference
1. Height of the teeth may be reduced. 2. Under cut of the radial flank of the pinion. 3. Centre distance may be increased. It leads to increase in pressure angle. 4. By tooth correction, the pressure angle, centre distance and base circles remain unchanged,
buttooth thickness of gear will be greater than the pinion tooth thickness. Minimum
number of teeth on the pinion avoid Interference
The pinion turns clockwise and drives the gear as shown in Figure.
Points M and N are called interference points. i.e., if the contact takes place beyond M and
N, interference will occur. The limiting value of addendum circle radius of pinion is O1N and the limiting value of
addendum circle radius of gear is O2M. Considering the critical addendum circle radius of
gear, the limiting number of teeth on gear can be calculated. Let
Ф = pressure angle
R = pitch circle radius of gear = ½mT r =
pitch circle radius of pinion = ½mt T & t =
number of teeth on gear & pinion
m = module
aw = Addendum constant of gear (or) wheel ap = Addendum
constant of pinion aw. m = Addendum of gear ap. m = Addendum
of pinion
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 67
6.6 Back lash, condition for minimum number of teeth to avoid interference,
Minimum number of teeth on the wheel avoid Interference
From triangle O2MP, applying cosine rule and simplifying, The limiting radius of wheel
addendum circle: Minimum number of teeth on the pinion for involute rack to avoid Interference
The rack is part of toothed wheel of infinite
diameter. The base circle diameter and profile of the involute teeth are straight lines.
PITCH LINE
RACK
Let t= Minimum number of teeth on the pinion r=
Pitch circle radius of the pinion = ½ mt
= Pressure angle
AR.m= Addendum of rack
The straight profiles of the rack are tangential to the pinion profiles at the point of contact
and perpendicular to the tangent PM. Point L is the limit of interference.
Backlash:
The gap between the non-drive face of the pinion tooth and the adjacent wheel tooth is known
as backlash.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 68
If the rotational sense of the pinion were to reverse, then a period of unrestrained pinion
motion would take place until the backlash gap closed and contact with the wheel tooth re-
established impulsively. Backlash is the error in motion that occurs when gears change direction. The term "backlash"
can also be used to refer to the size of the gap, not just the phenomenon it causes; thus, one could
speak of a pair of gears as having, for example, "0.1 mm of backlash." A pair of gears could be designed to have zero backlash, but this would presuppose perfection
in manufacturing, uniform thermal expansion characteristics throughout the system, and no
lubricant. Therefore, gear pairs are designed to have some backlash. It is usually provided by reducing
the tooth thickness of each gear by half the desired gap distance. In the case of a large gear and a small pinion, however, the backlash is usually taken entirely
off the gear and the pinion is given full sized teeth. Backlash can also be provided by moving the gears farther apart. For situations, such as
instrumentation and control, where precision is important, backlash can be minimised
through one of several techniques.
Let r = standard pitch circle radius of pinion
R = standard pitch circle radius of wheel c =
standard centre distance = r +R r’ =
operating pitch circle radius of pinion
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 69
R’ = operating pitch circle radius of wheel
c’ = operating centre distance = r’ + R’
Ф = Standard pressure angle
Ф’ = operating pressure angle h = tooth thickness of pinion on
standard pitch circle= p/2 h’ = tooth thickness of pinion on
operating pitch circle Let
H = tooth thickness of gear on standard pitch circle
H1 = tooth thickness of gear on operating pitch circle p =
standard circular pitch = 2п r/ t = 2пR/T
p’ = operating circular pitch = 2п r1/t = 2пR1/T
∆C = change in centre distance B
= Backlash t = number of teeth
on pinion
T = number of teeth on gear.
Involute gears have the invaluable ability of providing conjugate action when the gears' centre
distance is varied either deliberately or involuntarily due to manufacturing and/or mounting
errors. There is an infinite number of possible centre distances for a given pair of profile shifted gears,
however we consider only the particular case known as the extended centre distance. Non Standard Gears:
The important reason for using non standard gears are to eliminate undercutting, to prevent
interference and to maintain a reasonable contact ratio.
The two main non- standard gear systems:
(1) Long and short Addendum system and
(2) Extended centre distance system.
Long and Short Addendum System: The addendum of the wheel and the addendum of the pinion are generally made of equal lengths.
Here the profile/rack cutter is advanced to a certain increment towards the gear blank and the same quantity of increment will be withdrawn from the pinion blank.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 70
Therefore an increased addendum for the pinion and a decreased addendum for the gear
is obtained. The amount of increase in the addendum of the pinion should be exactly
equal to the addendum of the wheel is reduced. The effect is to move the contact region from the pinion centre towards the gear centre,
thus reducing approach length and increasing the recess length. In this method there is
no change in pressure angle and the centre distance remains standard. Extended centre distance system:
Reduction in interference with constant contact ratio can be obtained by increasing the
centre distance. The effect of changing the centre distance is simply in increasing the
pressure angle. In this method when the pinion is being cut, the profile cutter is withdrawn a certain
amount from the centre of the pinion so the addendum line of the cutter passes through
the interference point of pinion. The result is increase in tooth thickness and decrease in
tooth space. Now If the pinion is meshed with the gear, it will be found that the centre distance has
been increased because of the decreased tooth space. Increased centre distance will have
two undesirable effects.
OUT COME
Students able to Analyse & Calculate the design specifications of
simple involute gears with given inputs.
Exercise
1. With a neat sketch derive the Length of arc of contact Derivation
2. Two gears in mesh have a module of 8mm and a pressure angle of 200. The larger
gear has 57 teeth while the pinion has 23 teeth. If the addendum on pinion and gear
wheel are equal to one module, find
a) The number of pairs of teeth in contact
b) The angle of action of the pinion and the gear wheel
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 71
UNIT 7 Gear Trains
CONTENTS
7.1 Simple gear trains,
7.2 Compound gear trains for large speed. reduction,
7.3 Epicyclic gear trains,
7.4 Algebraic and tabular methods of finding velocity ratio of epicyclic gear trains.
7.5 Torque calculations in epicyclic gear trains
OBJECTIVES:
Outline the procedure for analyzing gear train to calculate speeds and
velocities of gears
7.1 Simple gear trains
A gear train is two or more gear working together by meshing their teeth and
turning each other in a system to generate power and speed. It reduces speed and
increases torque. To create large gear ratio, gears are connected together to form
gear trains. They often consist of multiple gears in the train. The most common of the gear train is the gear pair connecting parallel shafts. The
teeth of this type can be spur, helical or herringbone. The angular velocity is
simply the reverse of the tooth ratio. Any combination of gear wheels employed to transmit motion
from one shaft to the other is called a gear train. The meshing of
two gears may be idealized as two smooth discs with their edges
touching and no slip between them. This ideal diameter is called
the Pitch Circle Diameter (PCD) of the gear.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 72
Simple Gear Trains
The typical spur gears as shown in diagram. The direction of rotation is reversed
from one gear to another. It has no affect on the gear ratio. The teeth on the gears
must all be the same size so if gear A advances one tooth, so does B and C. t = number of teeth on the gear,
D = Pitch circle diameter, N = speed in rpm
D
m = module =
t
and module must be the
same for all gears
otherwise they would not
mesh.
DA
DB
D m = = = C
tA tB tC
DA = m tA; DB = m tB and DC = m tC
= angular velocity.
GEAR 'A' GEAR 'B' GEAR 'C'
D
v = linear velocity on the circle. v = = r (Idler gear)
2
Application:
a) to connect gears where a large center distance is required b) to obtain desired direction of motion of the driven gear ( CW or CCW) c) to obtain high speed ratio
Torque & Efficiency
The power transmitted by a torque T N-m applied to a shaft rotating at N rev/min is
given by:
P 2 NT
60
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 73
In an ideal gear box, the input and output powers are the same so;
P2 N1T1 2 N2 T2
60 60
T2 N1 GR
N1T1 N2 T2
T1 N2
It follows that if the speed is reduced, the torque is increased and vice versa. In a real
gear box, power is lost through friction and the power output is smaller than the power
input. The efficiency is defined as:
Power out 2 N2T2 60 N 2T2
Power In 2 N1T1 60 N1T1
Because the torque in and out is different, a gear box has to be
clamped in order to stop the case or body rotating. A holding
torque T3 must be applied to the body through the clamps. The total torque must add up to zero.
T1 + T2 + T3 = 0
If we use a convention that anti-clockwise is positive and clockwise is negative we
can determine the holding torque. The direction of rotation of the output shaft
depends on the design of the gear box.
7.2 Compound gear trains for large speed. reduction, Compound Gear train
Compound gears are simply a chain of simple gear trains with the input of the second
being the output of the first. A chain of two pairs is shown below. Gear B is the output
of the first pair and gear C is the input of the second pair. Gears B and C are locked to
the same shaft and revolve at the same speed. For large velocities ratios, compound gear
train arrangement is preferred.
The velocity of each tooth on A and B are the same
GEAR 'D' so: A tA = B tB -as they are simple gears.
Likewise for C and D, C tC = D tD.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 74
Reverted Gear train
The driver and driven axes lies on the same line. These
are used in speed reducers, clocks and machine tools.
GR NA tB tD
ND tA tC
If R and T=Pitch circle radius & number of teeth of the gear
RA + RB = RC + RD and tA + tB = tC + tD
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 75
7.3 Epicyclic gear trains
Observe point p and you will see that gear B also revolves once on its own axis. Any object
orbiting around a center must rotate once. Now consider that B is free to rotate on its shaft
and meshes with C.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 76
Suppose the arm is held stationary and gear C is rotated once. B spins about its own center
and the Now consider that C is unable to rotate and the arm A is revolved once. Gear B will revolve
because of the orbit. It is this extra rotation that causes confusion. One way to get round this
is to imagine that the whole system is revolved once. Then identify the gear that is fixed and
revolve it back one revolution. Work out the revolutions of the other gears and add them up.
The following tabular method makes it easy. Suppose gear C is fixed and the arm A makes one revolution. Determine how
many revolutions the planet gear B makes. Step 1 is to revolve everything once about the center.
Step 2 identify that C should be fixed and rotate it backwards one revolution keeping the
arm fixed as it should only do one revolution in total. Work out the revolutions of B. Step 3 is simply add them up and we find the total revs of C is zero and for the arm is 1.
Opposite
Step Action A B C
1 Revolve all once 1 1 1
2
Revolve C by –1 revolution, 0
-1
keeping the arm fixed tC tB
3
Add
1 1
0
tC tB
tC
The number of revolutions made by B is 1 tB Note that if C revolves -1, then
the direction of B is
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 77
7.4 Algebraic and tabular methods of finding velocity ratio of epicyclic
gear trains.
Example: A simple epicyclic gear has a fixed sun gear with 100 teeth and a planet gear with
50 teeth. If the arm is revolved once, how many times does the planet gear revolve? Solution:
Step Action A B C
1 Revolve all once 1 1 1
2 Revolve C by –1 revolution,
0
-1
keeping the arm fixed
3 Add 1 3 0
Gear B makes 3 revolutions for every one of the arm.
The design so far considered has no identifiable input and output. We need a design that
puts an input and output shaft on the same axis. This can be done several ways. Problem 1: In an ecicyclic gear train shown in figure, the arm A is fixed to the shaft S. The
wheel B having 100 teeth rotates freely on the shaft S. The wheel F having 150 teeth driven
separately. If the arm rotates at 200 rpm and wheel F at 100 rpm in the same direction; find
(a) number of teeth on the gear C and (b) speed of wheel B.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 78
Solution:
TB=100; TF=150; NA=200rpm; NF=100rpm:
Since the module is same for all gears : the number
of teeth on the gearsis proportional to the pitch cirlce: The gear B and gear F rotates in the opposite directions:
The Gear B rotates at 350 rpm in the same direction of gears F and Arm A.
Problem 2: In a compound epicyclic gear train as shown in the figure, has gears A and an
annular gears D & E free to rotate on the axis P. B and C is a compound gear rotate about
axis Q. Gear A rotates at 90 rpm CCW and gear D rotates at 450 rpm CW. Find the speed
and direction of rotation of arm F and gear E. Gears A,B and C are having 18, 45 and 21
teeth respectively. All gears having same module and pitch.
Solution:
TA=18 ; TB=45; TC=21; NA= -90rpm; ND=10rpm:
Since the module and pitch are same for all gears:
the number of teeth on the gearsis proportional to the pitch cirlce:
rD rA rB rC
TD TA TB TC
TD 18 45 21 84 teeth on gear D
Gears A and D rotates in the opposite directions:
TB TD NA NF
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 79
Problem 3: In an epicyclic gear of sun and planet type shown in figure 3, the pitch circle
diameter of the annular wheel A is to be nearly 216mm and module 4mm. When the annular
ring is stationary, the spider that carries three planet wheels P of equal size to make one
revolution for every five revolution of the driving spindle carrying the sun wheel. Determine the number of teeth for all the wheels and the exact pitch circle diameter of the
annular wheel. If an input torque of 20 N-m is applied to the spindle carrying the sun wheel, determine the fixed torque on the annular wheel.
Solution: Module being the same for all the meshing gears:
Operation Spider Sun Wheel S Planet wheel P Annular wheel A
arm L TS
TP
TA = 54
Arm L is fixed & TS TP
Sun wheel S is 0 +1
TS TP TS
given +1
TP TA TA
revolution
Multiply by m
0 m
TS m TS m
(S rotates through TP TA
m revolution)
Add n revolutions
n m+n
n TS m n TS m
to all elements
TP
TA
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 80
Problem 4: The gear train shown in figure 4 is used in
an indexing mechanism of a milling machine. The
drive is from gear wheels A and B to the bevel gear
wheel D through the gear train. The following table
gives the number of teeth on each gear. How many revolutions does D makes for one Figure 4
revolution of A under the following situations:
a. If A and B are having the same speed
and same direction
b. If A and B are having the same speed and opposite direction
c. If A is making 72 rpm and B is at rest
d. If A is making 72 rpm and B 36 rpm in the same direction
Solution:
Gear D is external to the epicyclic train and thus C and D constitute an ordinary train.
Operation
Arm E (28) F (24)
A (72) B (72) G (28) H (24)
C (60)
Arm or C is fixed 28 7 28 7
24
6
& wheel A is
0 -1
+1 -1 +1
24 6
given
+1 revolution
Multiply by m
(A rotates through 0 -m m
+m -m +m m
m revolution)
Add n revolutions n n - m
n + m n - m n + m
to all elements n m
nm
(i) For one revolution of A: n + m = 1 (1)
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 81
For A and B for same speed and direction: n + m = n – m (2)
From (1) and (2): n = 1 and m = 0
If C or arm makes one revolution, then revolution made by D is given by:
ND TC 60
2
NC TD 30
ND 2NC
(ii)A and B same speed, opposite direction: (n + m) = - (n – m)(3) n = 0; m
= 1
When C is fixed and A makes one revolution, D does not make any revolution.
(iii) A is making 72 rpm: (n + m) = 72
B at rest (n – m) = 0 n = m = 36 rpm
C makes 36 rpm and D makes 36 72 rpm
(iv) A is making 72 rpm and B making 36 rpm
(n + m) = 72 rpm and (n – m) = 36
rpm
(n + (n – m)) = 72; n = 54
D makes 54 108 rpm
Problem 5: Figure 5 shows a compound
epicyclic gear train, gears S1 and S2 being
rigidly attached to the shaft Q. If the shaft P
rotates at 1000 rpm clockwise, while the
Q
annular A2 is driven in counter clockwise
direction at 500 rpm, determine the speed and
direction of rotation of shaft Q. The number of
teeth in the wheels are S1 = 24; S2 = 40; A1 = 100; A2 = 120.
Solution: Consider the gear train P A1 S1:
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 82
Operation Arm A1
S1 (24)
P (100)
100 P1
Arm P is fixed &
P1 24
wheel A1 is given 0 +1
+1 revolution
Multiply by m
(A1 rotates through 0 +m m
m revolution)
Add n revolutions n n+ m
to all elements n
m
If A1 is fixed: n+ m; gives n = - m
Now consider whole gear train:
OR Operation
Arm A1 S1 (24)
P (100)
Arm P is fixed
A1
P1
& wheel A1 is
0 -1
P1 S1
given -1
A1
revolution
S1
100 25
0 -1
24 6
Add +1
revolutions to +1 0 1
all elements
Operation A1 A2 S1 (24), S2 (40)
(100)
(120)
and Q Arm P
A1 is fixed & 120 P2
3
wheel A2 is given 0
+1 P2 40
+1 revolution 3
Multiply by m
(A1 rotates through 0
+m
3m m
m revolution)
Add n revolutions n
n+ m
n 3m
to all elements
n m
When P makes 1000 rpm: n m = 1000
(1) and A2 makes – 500 rpm: n+ m = -500 (2)
from (1) and (2):
500 m m 1000
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 83
31 1000 500 31 49m
m 949rpm
and n 949 500 449rpm
NQ = n – 3 m = 449 – (3 -949) = 3296 rpm
OUT COME
Students able to Calculate the velocities and speeds of various gears in gear trains .
Exercise
1. An epicyclic gear train is shown in fig 3. The wheel A is fixed and the input
at the arm R is 3kw at 600 rpm. Find the speed of wheel D and the torque on
it and the torque required to hold the wheel ‘A’ neglect frictional losses. 2. Explain torque in an epicyclic gear train?
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 84
UNIT 8 CAMS
CONTENTS
8.1 Types of cams, types of followers
8.2 displacement, velocity and acceleration curves for uniform velocity, Simple Harmonic
Motion, Uniform Acceleration Retradation,
8.3 . Cam profiles: disc cam with reciprocating / oscillating follower having knife-edge,
roller and flat-face follower inline and offset.
OBJECTIVES:
1. Outline the classification various types of cams and followers and their
Arrangements.
2. Comprehend the procedure of developing Cam profiles.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 85
8.1 Types of cams, types of followers
Definition A cam may be defined as a rotating, reciprocating or oscillating machine part, designed to impart
reciprocating and oscillating motion to another mechanical part, called a follower. A cam and follower have, usually, a line contact between them and as such they constitute a higher
pair. The contact between them is maintained by an external force which is generally, provided by a
spring or sometimes by the sufficient weight of the follower itself.
Classification of Cams
Cams are classified according to :
(a) Shape
(b) Follower movement
(c) Type of constraint of the follower
According to Shape
Wedge and Flat Cams
It is shown in Figures 1.48(a), (b), (c) and (d).
In Figure 1.48(a), on imparting horizontal translatory motion to wedge, the follower also
translates but vertically in Figure 1.48(b), the wedge has curved surface at its top. The
follower gets a oscillatory motion when a horizontal translatory motion is given to the
wedge.
In Figure 1.48(c), the wedge is stationary, the guide is imparted translatory motion within the
constraint provided. This results in translatory motion of the follower in Figure 1.48(d),
instead of a wedge, a rectangular block or a flat plate with a groove is provided. When
horizontal translatory motion is imparted to the block, the follower is constrained to have a
vertical translatory motion.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 86
Follower Follower Oscillates
Guide
(a) Wedge Cam (b) Wedge Cam
Follower Follower
Constraint
Guide
Guide
Block / Flat
Plat
Fixed Groove
Wedge Wedge
(c)
Fixed
Wedge
Cam (d) Flate Wedge Cam
F
i
g
u
r
e
1
.
4
8
Guide
Oscillating Foll
ower
(a) Follower Reciprocating (b) Follower Oscillating
Guide Offset
Cam
Offset
(c) Offset Follower Reciprocating (d) Offset Follower Oscillating
(e)
Figure 1.49 : Radial or Disc Cams
Wedge Wedge
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 87
It is pointed out that the radial cams are very popular due to their simplicity and compactness,
Cylindrical Cams
Cylindrical cams have been shown in Figures 1.50(a) and (b). In
Figure 1.50(a) the follower reciprocates whereas in Figure 1.50(b) the
follower oscillates. Cylindrical cams are also known as barrel or drum
cams.
Follower
Follower
(a) Follower Reciprocates (b) Follower Oscillates
Figure 1.50 : Cylindrical Cams
Spiral Cams
It is shown in Figure 1.51. The cam comprises of a plate on the face of
which a groove of the form of a spiral is cut. The spiral groove is provided
with teeth which mesh with pin gear follower.
This cam has a limited use because it has to reverse its direction to reset the position of the follower. This cam has found its use in computers.
Follower
Guide
Spiral groove
Cam (plate)
Figure 1.51 : Spiral Cam
Conjugate Cams
As the name implies, the cam comprises of two discs, keyed together and
remain in constant touch with two rollers of a follower as shown in
Figure 1.52.
Cam disc Roller
Follower
Roller
Figure 1.52 : Conjugate Cam
Simple Mechanisms
41
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 88
Theory of Machines This cam is used where the requirement is of high dynamic load, low wear,
low noise, high speed and better control of follower. Globoidal Cams
This cam has two types of surfaces : convex and concave. A helical contour
is cut on the circumference of the surface of rotation of the cam as shown in
Figures 1.53(a) and (b). The end of the follower is constrained to move
along the contour and then oscillatory motion is obtained. In this cam, a
large angle of oscillation of the follower is obtained.
Concave Follower
Follower Surface
Convex surface Support
Support
(a) (b)
Figure 1.53 : Globoidal Cams Spherical Cams
In this cam, as shown in Figure 1.54, the cam is of the shape of a sphere on
the peripheral of which a helical groove is cut. The roller provided at the
end of the follower rolls in the groove causing oscillatory motion to the
follower in an axis perpendicular to the axis of rotation of the cam.
Cam Bearing
Follower
Figure 1.54 : Spherical Cam
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 89
8.2 displacement, velocity and acceleration curves for uniform velocity, Simple Harmonic Motion,
Uniform Acceleration Retradation,
Rise-return-rise (RRR)
In this type of cam, its profile or contour is such that the cam rises, returns
without rest or dwell, and without any dwell or rest, it again rises. Follower
displacement and cam angle diagram for this type of cam is shown in
Figure 1.55(a).
Dwell, Rise-return Dwell (DRRD)
In this type of cam after dwell, there is rise of the follower, then it returns to
its original position and dwells for sometimes before again rising.
Generally, this type of cam is commonly used. Its displacement cam angle
diagram is shown in Figure 1.55(b).
Dwell-rise-dwell-return
It is the most widely used type of cam. In this, dwell is followed by rise.
Then the follower remains stationary in the dwell provided and then returns
to its original position [Figure 1.55(c)].
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 90
Dwell-rise-dwell
As may be seen in the follower-displacement verses cam angle diagram,
shown in Figure 1.55(d) in this cam, the fall is sudden which necessities
enormous amount of force for this to take place.
Y Y
Fo
llow
erdi
spla
cem
ent
Follo
wer
disp
lace
men
t Return
Rise
Rise
X
O X Dwell
Cam angle, 360o
(a) [R – R – R]
(b) [D – R – R – D]
Y Dwell
Rise
Fa
ll
Fol
low
erdi
spl
acem
ent
F o l l o w e r d i s p l a c e m e n t
Rise Return
X
Dwell X Dwell
Cam angle, 360o
(c) [D – R – D – R – D]
(d) [D – R – D]
Figure 1.55 : Dwell-rise-dwell
According to Type of Constraint of the Follower
Pre-loaded Spring Cam
For its proper working there should be contact between the cam and the
follower throughout its working, and it is achieved by means of a pre-loaded
spring as shown in Figures 1.48(a) and (b), etc.
Positive Drive Cam
In this case, the contact between the cam and the follower is maintained by
providing a roller at the operating end of the follower. This roller operates
in the groove provided in the cam. The follower cannot come out of the
groove, as shown in Figures 1.52 to 1.54.
Gravity Drive Cam
In this type of cam, the lift or rise of the follower is achieved by the rising
surface of the cam (Figure 1.48(c)) and the follower returns or falls due to
force of gravity of the follower. Such type of cams cannot be relied upon
due to their uncertain characteristics.
Classification of Followers
Followers may be classified in three different ways :
(a) Depending upon the type of motion, i.e. reciprocating or oscillating.
(b) Depending upon the axis of the motion, i.e. radial or offset.
(c) Depending upon the shape of their contacting end with the cam.
Those of followers falling under classification (a) and (b) have already been dealt with as indicated above. Followers of type (c) will be taken up now.
Simple Mechanisms
43
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 91
Theory of Machines Depending upon the Shape of their Contacting End with the Cam
Under this classification followers may be divided into three types :
(a) Knife-edge Follower (Figure 1.55(a))
(b) Roller Follower (Figure 1.55(b))
(c) Flat or Mushroom Follower (Figure 1.56(c))
Knife-edge Follower
Knife-edge followers are generally, not used because of obvious high rate of
wear at the knife edge. However, cam of any shape can be worked with it.
During working, considerable side thrust exist between the follower and the
guide.
Roller Follower
In place of a knife edge, a roller is provided at the contacting end of the
follower, hence, the name roller follower. Instead of sliding motion between
the contacting surface of the follower and the cam, rolling motion takes
place, with the result that rate of wear is greatly reduced. In roller followers
also, as in knife edge follower, side thrust is exerted on the follower guide.
Roller followers are extensively used in stationary gas and oil engines. They
are also used in aircraft engines due to their limited wear at high cam
velocity.
While working on concave surface of a cam the radius of the surface must
be at least equal to radius of the roller.
Roller
(a) Knife-edge Follower (b) Roller Follower
Offset
Flat
(c) Flat or Mushroom Follower (i)
Spherical (d) : Spherical Follower
44 Figure 1.56 : Classification of Followers
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 92
Flat or Mushroom Follower
At the name implies the contacting end of the follower is flat as shown. In
mushroom followers there is no side thrust on the guide except that due to
friction at the contact of the cam and the follower. No doubt that there will
be sliding motion between the contacting surface of the follower and the
cam but the wear can be considerably reduced by off-setting the axis of the
followers as shown in Figure 1.56(c)(i). The off-setting provided causes the
follower to rotate about its own axis when the cam rotates.
Flat face follower is used where the space is limited. That is why it is used
to operate valves of automobile engines. Where sufficient space is available
as in stationary gas and oil engines, roller follower is used as mentioned
above. The flat faced follower is generally preferred to the roller follower
because of the compulsion of having to use small diameter of the pin in the
roller of the roller follower.
In flat followers, high surface stresses are produced in the flat contacting
surface. To minimise these stresses, spherical shape is given to the flat end,
as shown in Figure 1.56(d). The curved faced or spherical faced followers
are used in automobile engines.
With flat followers, it is obviously, essential that the working surface of the cam should be convex everywhere.
8.3 Terminology of Cam and Follower
The Cam Profile
The working contour of a cam which comes into contact with the follower to
operate it, is known as the cam profile. In Figure 1.57, A-B-C-D-A is the cam
profile or the working contour.
Simple Mechanisms
Roller
Prime circle Base circle
-A)
4
Dw
ell(
D
D
Pitch Curve
Tracing point for roller
Tracing point for knife edge
Cam profile Normal
Out stroke (A-B)
A
1
B’
2 B
Dw
ell
C’
C)-(B
3 C
(C-D) Instroke
Pressure Angle
Tangent
Figure 1.57 : Cam Profile
The Base Circle
The smallest circle, drawn from the centre of rotation of a cam, which forms part
of the cam profile, is known as the base circle and its radius is called the least
radius of the cam. A circle with centre O and of radius OA forms the base circle.
Size of a cam depends upon the size of the base circle.
The Tracing Point
The point of the follower from which the profile of a cam is determined is called
the tracing point. In case of a knife-edge follower, the knife edge itself is the
tracing point. In roller follower, the centre of roller is the tracing point.
45
Theory of Machines The Pitch Curve
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 93
The locus or path of the tracing point is known as the pitch curve. In knife-edge
follower, the pitch curve itself will be the cam profile. In roller follower, the
cam profile will be determined by subtracting the radius of the roller radially
throughout the pitch curve.
The Prime Circle
The smallest circle drawn to the pitch curve from the centre of rotation of the
cam is called as the prime circle. In knife-edge follower, the base circle and the
prime circle are the same. In roller follower, the radius of the prime circle is the
base circle radius plus the radius of the roller.
The Lift or Stroke
It is the maximum displacement of the follower from the base circle of the cam.
It is also called as the throw of the cam. In Figure 1.57, distance BB and CC is
the lift, for the roller follower.
The Angles of Ascent, Dwell, Descent and Action
Refer Figure 1.57, the angle covered by a cam for the follower to rise from its lowest position to the highest position is called the angle of ascent denoted as
1.
The angle covered by the cam during which the follower remains at rest at
its highest position is called the angle of dwell, denoted by 2.
The angle covered by the cam, for the follower to fall from its highest position
to the lowest position is called the angle of descent denoted as 3.
The total angle moved by the cam for the follower to return to its lowest position after the period of ascent, dwell and descent is called the angle of
action. It is the sum of 1, 2 and 3.
The Pressure Angle
The angle included between the normal to the pitch curve at any point and the
line of motion of the follower at the point, is known as the pressure angle. This
angle represents the steepness of the cam profile and as such it is very important
in cam design.
The Pitch Point
The point on the pitch curve having the maximum pressure angle is known as the pitch point.
The Cam AngleIt is the angle of rotation of the cam for a certain displacement of the follower.
KINEMATICS OF MACHINES-15ME42
Depart of Mechanical Engineering , ATMECE MYSORE Page 94
OUT COME
Students able to Develop cam profiles for given input parameters
Exercise
1. With a neat sketch list the classification of followers.
2. A cam rotating clockwise at uniform speed of 300 rpm operates a reciprocating follower through a
roller 1.5cm diameter. The follower motion is defined as below
a) Outward during 1500 with UARM
b) Dwell for next 300
c) Return during next 1200 with SHM
d) Dwell for the remaining period
Stroke of the follower is 3cm.minimum radius of the cam is 3cm.Draw the cam profile the
follower axis passes through the cam.
FURTHER READING
1. "Theory of Machines”, Rattan S.S, Tata McGraw-Hill Publishing Company Ltd., New Delhi, and
3rd edition -2009.
2. "Theory of Machines”, Sadhu Singh, Pearson Education (Singapore) Pvt. Ltd, Indian Branch New
Delhi, 2nd Edi. 2006
3. “Theory of Machines & Mechanisms", J.J. Uicker, , G.R. Pennock, J.E. Shigley. OXFORD 3rd
Ed. 2009.
4. Mechanism and Machine theory, Ambekar, PHI, 2007