Rajalakshmi Engineering College, Thandalam
ERODE SENGUNTHAR ENGINEERING COLLEGE.
Department of Mechanical Engineering
Branch : MECHANICAL ENGINEERING
Sem/Year : III sem. / II year
Subject Name: KINEMATICS OF MACHINERY
Objective:
To study the mechanism, machine and the geometric aspect of
motion.
TEXT BOOKS:
1. Rattan S.S., Theory of Machines, Tata McGraw-Hill Publishing
Company Ltd., New Delhi, 1998.
2. Shigley J.E. and Uicker J.J., Theory of Machines and
Mechanisms, McGraw-Hill, Inc. 1995.
3. Khurmi R.S. and Gupta J.K., Theory of Machines, Eurasia
Publishing House, New Delhi,2006
REFERENCES:
1. Thomas Bevan, Theory of Machines, CBS Publishers and
Distributors, 1984.
2. Ghosh A. and Mallick A.K., Theory of Mechanisms and Machines,
Affiliated East-West Pvt. Ltd., New Delhi, 1988.
3. Rao J.S. and Dukkipati R.V., Mechanism and Machine Theory,
Wiley-Eastern Ltd., New Delhi, 1992.
NOTES OF LESSON
Unit I Basics of Mechanisms
Introduction:
Definitions : Link or Element, Pairing of Elements with degrees
of freedom, Grublers criterion (without derivation), Kinematic
chain, Mechanism, Mobility of Mechanism, Inversions, Machine.
Kinematic Chains and Inversions :
Kinematic chain with three lower pairs, Four bar chain, Single
slider crank chain and Double slider crank chain and
their inversions.
Mechanisms:
i) Quick return motion mechanisms Drag link mechanism, Whitworth
mechanism and Crank and slotted lever mechanism
ii) Straight line motion mechanisms Peaceliers mechanism and
Roberts mechanism.
iii) Intermittent motion mechanisms Geneva mechanism and Ratchet
& Pawl mechanism.
iv) Toggle mechanism, Pantograph, Hookes joint and Ackerman
Steering gear mechanism.
Terminology and Definitions-Degree of Freedom, Mobility
Kinematics: The study of motion (position, velocity,
acceleration). A major goal of understanding kinematics is to
develop the ability to design a system that will satisfy specified
motion requirements. This will be the emphasis of this class.
Kinetics: The effect of forces on moving bodies. Good kinematic
design should produce good kinetics.
Mechanism: A system design to transmit motion. (low forces)
Machine: A system designed to transmit motion and energy.
(forces involved)
Basic Mechanisms: Includes geared systems, cam-follower systems
and linkages (rigid links connected by sliding or rotating joints).
A mechanism has multiple moving parts (for example, a simple hinged
door does not qualify as a mechanism).
Examples of mechanisms: Tin snips, vise grips, car suspension,
backhoe, piston engine, folding chair, windshield wiper drive
system, etc.
Key concepts:
Degrees of freedom: The number of inputs required to completely
control a system. Examples: A simple rotating link. A two link
system. A four-bar linkage. A five-bar linkage.
Types of motion: Mechanisms may produce motions that are pure
rotation, pure translation, or a combination of the two. We reduce
the degrees of freedom of a mechanism by restraining the ability of
the mechanism to move in translation (x-y directions for a 2D
mechanism) or in rotation (about the z-axis for a 2-D
mechanism).
Link: A rigid body with two or more nodes (joints) that are used
to connect to other rigid bodies. (WM examples: binary link,
ternary link (3 joints), quaternary link (4 joints))
Joint: A connection between two links that allows motion between
the links. The motion allowed may be rotational (revolute joint),
translational (sliding or prismatic joint), or a combination of the
two (roll-slide joint).
Kinematic chain: An assembly of links and joints used to
coordinate an output motion with an input motion.
Link or element:
A mechanism is made of a number of resistant bodies out of which
some may have motions relative to the others. A resistant body or a
group of resistant bodies with rigid connections preventing their
relative movement is known as a link.
A link may also be defined as a member or a combination of
members of a mechanism, connecting other members and having motion
relative to them, thus a link may consist of one or more resistant
bodies. A link is also known as Kinematic link or an element.
Links can be classified into 1) Binary, 2) Ternary, 3)
Quarternary, etc.
Kinematic Pair:
A Kinematic Pair or simply a pair is a joint of two links having
relative motion between them
Example:
In the above given Slider crank mechanism, link 2 rotates
relative to link 1 and constitutes a revolute or turning pair.
Similarly, links 2, 3 and 3, 4 constitute turning pairs. Link 4
(Slider) reciprocates relative to link 1 and its a sliding
pair.
Types of Kinematic Pairs:
Kinematic pairs can be classified according to
i) Nature of contact.
ii) Nature of mechanical constraint.
iii) Nature of relative motion.
i) Kinematic pairs according to nature of contact :
a) Lower Pair: A pair of links having surface or area contact
between the members is known as a lower pair. The contact surfaces
of the two links are similar.
Examples: Nut turning on a screw, shaft rotating in a bearing,
all pairs of a slider-crank mechanism, universal joint.
b) Higher Pair: When a pair has a point or line contact between
the links, it is known as a higher pair. The contact surfaces of
the two links are dissimilar.
Examples: Wheel rolling on a surface cam and follower pair,
tooth gears, ball and roller bearings, etc.
ii) Kinematic pairs according to nature of mechanical
constraint.
a) Closed pair: When the elements of a pair are held together
mechanically, it is known as a closed pair. The contact between the
two can only be broken only by the destruction of at least one of
the members. All the lower pairs and some of the higher pairs are
closed pairs.
b) Unclosed pair: When two links of a pair are in contact either
due to force of gravity or some spring action, they constitute an
unclosed pair. In this the links are not held together
mechanically. Ex.: Cam and follower pair.
iii) Kinematic pairs according to nature of relative motion.
a) Sliding pair: If two links have a sliding motion relative to
each other, they form a sliding pair. A rectangular rod in a
rectangular hole in a prism is an example of a sliding pair.
b) Turning Pair: When on link has a turning or revolving motion
relative to the other, they constitute a turning pair or revolving
pair.
c) Rolling pair: When the links of a pair have a rolling motion
relative to each other, they form a rolling pair. A rolling wheel
on a flat surface, ball ad roller bearings, etc. are some of the
examples for a Rolling pair.
d) Screw pair (Helical Pair): if two mating links have a turning
as well as sliding motion between them, they form a screw pair.
This is achieved by cutting matching threads on the two links.
The lead screw and the nut of a lathe is a screw Pair
e) Spherical pair: When one link in the form of a sphere turns
inside a fixed link, it is a spherical pair. The ball and socket
joint is a spherical pair.
Degrees of Freedom:
An unconstrained rigid body moving in space can describe the
following independent motions.
1. Translational Motions along any three mutually perpendicular
axes x, y and z,
2. Rotational motions along these axes.
Thus a rigid body possesses six degrees of freedom. The
connection of a link with another imposes certain constraints on
their relative motion. The number of restraints can never be zero
(joint is disconnected) or six (joint becomes solid).
Degrees of freedom of a pair is defined as the number of
independent relative motions, both translational and rotational, a
pair can have.
Degrees of freedom = 6 no. of restraints.
To find the number of degrees of freedom for a plane mechanism
we have an equation known as Grublers equation and is given by
F = 3 ( n 1 ) 2 j1 j2
F = Mobility or number of degrees of freedom
n = Number of links including frame.
j1 = Joints with single (one) degree of freedom.
J2 = Joints with two degrees of freedom.
If F > 0, results in a mechanism with F degrees of
freedom.
F = 0, results in a statically determinate structure.
F < 0, results in a statically indeterminate structure.
Kinematic Chain:
A Kinematic chain is an assembly of links in which the relative
motions of the links is possible and the motion of each relative to
the others is definite (fig. a, b, and c.)
In case, the motion of a link results in indefinite motions of
other links, it is a non-kinematic chain. However, some authors
prefer to call all chains having relative motions of the links as
kinematic chains.
Linkage, Mechanism and structure:
A linkage is obtained if one of the links of kinematic chain is
fixed to the ground. If motion of each link results in definite
motion of the others, the linkage is known as mechanism. If one of
the links of a redundant chain is fixed, it is known as a
structure.
To obtain constrained or definite motions of some of the links
of a linkage, it is necessary to know how many inputs are needed.
In some mechanisms, only one input is necessary that determines the
motion of other links and are said to have one degree of freedom.
In other mechanisms, two inputs may be necessary to get a
constrained motion of the other links and are said to have two
degrees of freedom and so on.
The degree of freedom of a structure is zero or less. A
structure with negative degrees of freedom is known as a
Superstructure.
Motion and its types:
If the motion between a pair of links is limited to a definite
direction, then it is completely constrained motion. E.g.: Motion
of a shaft
If the motion in a definite direction is not brought about by
itself but by some other means, then it is incompletely constrained
motion.
The three main types of constrained motion in kinematic pair
are,
1.Completely constrained motion : If the motion between a pair
of links is limited to a definite direction, then it is completely
constrained motion. E.g.: Motion of a shaft or rod with collars at
each end in a hole as shown in fig.
2. Incompletely Constrained motion : If the motion between a
pair of links is not confined to a definite direction, then it is
incompletely constrained motion. E.g.: A spherical ball or circular
shaft in a circular hole may either rotate or slide in the hole as
shown in fig.
3. Successfully constrained motion or Partially constrained
motion: If the motion in a definite direction is not brought about
by itself but by some other means, then it is known as successfully
constrained motion. E.g.: Foot step Bearing.
Machine:
It is a combination of resistant bodies with successfully
constrained motion which is used to transmit or transform motion to
do some useful work. E.g.: Lathe, Shaper, Steam Engine, etc.
Kinematic chain with three lower pairs
It is impossible to have a kinematic chain consisting of three
turning pairs only. But it is possible to have a chain which
consists of three sliding pairs or which consists of a turning,
sliding and a screw pair.
The figure shows a kinematic chain with three sliding pairs. It
consists of a frame B, wedge C and a sliding rod A. So the three
sliding pairs are, one between the wedge C and the frame B, second
between wedge C and sliding rod A and the frame B.
This figure shows the mechanism of a fly press. The element B
forms a sliding with A and turning pair with screw rod C which in
turn forms a screw pair with A. When link A is fixed, the required
fly press mechanism is obtained.
Kutzbach criterion, Grashoff's law
Kutzbach criterion:
Fundamental Equation for 2-D Mechanisms: M = 3(L 1) 2J1 J2
Can we intuitively derive Kutzbachs modification of Grublers
equation? Consider a rigid link constrained to move in a plane. How
many degrees of freedom does the link have? (3: translation in x
and y directions, rotation about z-axis)
If you pin one end of the link to the plane, how many degrees of
freedom does it now have?
Add a second link to the picture so that you have one link
pinned to the plane and one free to move in the plane. How many
degrees of freedom exist between the two links? (4 is the correct
answer)
Pin the second link to the free end of the first link. How many
degrees of freedom do you now have?
How many degrees of freedom do you have each time you introduce
a moving link? How many degrees of freedom do you take away when
you add a simple joint? How many degrees of freedom would you take
away by adding a half joint? Do the different terms in equation
make sense in light of this knowledge?
Grashoff's law:
Grashoff 4-bar linkage: A linkage that contains one or more
links capable of undergoing a full rotation. A linkage is Grashoff
if: S + L < P + Q (where: S = shortest link length, L = longest,
P, Q = intermediate length links). Both joints of the shortest link
are capable of 360 degrees of rotation in a Grashoff linkages. This
gives us 4 possible linkages: crank-rocker (input rotates 360),
rocker-crank-rocker (coupler rotates 360), rocker-crank (follower);
double crank (all links rotate 360). Note that these mechanisms are
simply the possible inversions (section 2.11, Figure 2-16) of a
Grashoff mechanism.
Non Grashoff 4 bar: No link can rotate 360 if: S + L > P +
Q
Lets examine why the Grashoff condition works:
Consider a linkage with the shortest and longest sides joined
together. Examine the linkage when the shortest side is parallel to
the longest side (2 positions possible, folded over on the long
side and extended away from the long side). How long do P and Q
have to be to allow the linkage to achieve these positions?
Consider a linkage where the long and short sides are not
joined. Can you figure out the required lengths for P and Q in this
type of mechanism
2. Kinematic Inversions of 4-bar chain and slider crank
chains:
Types of Kinematic Chain: 1) Four bar chain 2) Single slider
chain 3) Double Slider chain
Four bar Chain:
The chain has four links and it looks like a cycle frame and
hence it is also called quadric cycle chain. It is shown in the
figure. In this type of chain all four pairs will be turning
pairs.
Inversions:
By fixing each link at a time we get as many mechanisms as the
number of links, then each mechanism is called Inversion of the
original Kinematic Chain.
Inversions of four bar chain mechanism:
There are three inversions: 1) Beam Engine or Crank and lever
mechanism. 2) Coupling rod of locomotive or double crank mechanism.
3) Watts straight line mechanism or double lever mechanism.
Beam Engine:
When the crank AB rotates about A, the link CE pivoted at D
makes vertical reciprocating motion at end E. This is used to
convert rotary motion to reciprocating motion and vice versa. It is
also known as Crank and lever mechanism. This mechanism is shown in
the figure below.
2. Coupling rod of locomotive: In this mechanism the length of
link AD = length of link C. Also length of link AB = length of link
CD. When AB rotates about A, the crank DC rotates about D. this
mechanism is used for coupling locomotive wheels. Since links AB
and CD work as cranks, this mechanism is also known as double crank
mechanism. This is shown in the figure below.
3. Watts straight line mechanism or Double lever mechanism: In
this mechanism, the links AB & DE act as levers at the ends A
& E of these levers are fixed. The AB & DE are parallel in
the mean position of the mechanism and coupling rod BD is
perpendicular to the levers AB & DE. On any small displacement
of the mechanism the tracing point C traces the shape of number 8,
a portion of which will be approximately straight. Hence this is
also an example for the approximate straight line mechanism. This
mechanism is shown below.
2. Slider crank Chain:
It is a four bar chain having one sliding pair and three turning
pairs. It is shown in the figure below the purpose of this
mechanism is to convert rotary motion to reciprocating motion and
vice versa.
Inversions of a Slider crank chain:
There are four inversions in a single slider chain mechanism.
They are:
1) Reciprocating engine mechanism (1st inversion)
2) Oscillating cylinder engine mechanism (2nd inversion)
3) Crank and slotted lever mechanism (2nd inversion)
4) Whitworth quick return motion mechanism (3rd inversion)
5) Rotary engine mechanism (3rd inversion)
6) Bull engine mechanism (4th inversion)
7) Hand Pump (4th inversion)
1. Reciprocating engine mechanism :
In the first inversion, the link 1 i.e., the cylinder and the
frame is kept fixed. The fig below shows a reciprocating
engine.
A slotted link 1 is fixed. When the crank 2 rotates about O, the
sliding piston 4 reciprocates in the slotted link 1. This mechanism
is used in steam engine, pumps, compressors, I.C. engines, etc.
2. Crank and slotted lever mechanism:
It is an application of second inversion. The crank and slotted
lever mechanism is shown in figure below.
In this mechanism link 3 is fixed. The slider (link 1)
reciprocates in oscillating slotted lever (link 4) and crank (link
2) rotates. Link 5 connects link 4 to the ram (link 6). The ram
with the cutting tool reciprocates perpendicular to the fixed link
3. The ram with the tool reverses its direction of motion when link
2 is perpendicular to link 4. Thus the cutting stroke is executed
during the rotation of the crank through angle and the return
stroke is executed when the crank rotates through angle or 360 .
Therefore, when the crank rotates uniformly, we get,
Time to cutting = =
Time of return 360
This mechanism is used in shaping machines, slotting machines
and in rotary engines.
3. Whitworth quick return motion mechanism:
Third inversion is obtained by fixing the crank i.e. link 2.
Whitworth quick return mechanism is an application of third
inversion. This mechanism is shown in the figure below. The crank
OC is fixed and OQ rotates about O. The slider slides in the
slotted link and generates a circle of radius CP. Link 5 connects
the extension OQ provided on the opposite side of the link 1 to the
ram (link 6). The rotary motion of P is taken to the ram R which
reciprocates. The quick return motion mechanism is used in shapers
and slotting machines. The angle covered during cutting stroke from
P1 to P2 in counter clockwise direction is or 360 -2. During the
return stroke, the angle covered is 2 or .
Therefore,
Time to cutting = 360 -2 = 180
Time of return 2 = = . 360
4. Rotary engine mechanism or Gnome Engine:
Rotary engine mechanism or gnome engine is another application
of third inversion. It is a rotary cylinder V type internal
combustion engine used as an aero engine. But now Gnome engine has
been replaced by Gas turbines. The Gnome engine has generally seven
cylinders in one plane. The crank OA is fixed and all the
connecting rods from the pistons are connected to A. In this
mechanism when the pistons reciprocate in the cylinders, the whole
assembly of cylinders, pistons and connecting rods rotate about the
axis O, where the entire mechanical power developed, is obtained in
the form of rotation of the crank shaft. This mechanism is shown in
the figure below.
Double Slider Crank Chain:
A four bar chain having two turning and two sliding pairs such
that two pairs of the same kind are adjacent is known as double
slider crank chain.
Inversions of Double slider Crank chain:
It consists of two sliding pairs and two turning pairs. They are
three important inversions of double slider crank chain. 1)
Elliptical trammel. 2) Scotch yoke mechanism. 3) Oldhams
Coupling.
1. Elliptical Trammel:
This is an instrument for drawing ellipses. Here the slotted
link is fixed. The sliding block P and Q in vertical and horizontal
slots respectively. The end R generates an ellipse with the
displacement of sliders P and Q.
The co-ordinates of the point R are x and y. From the fig. cos =
x. PR
and Sin = y. QR
Squaring and adding (i) and (ii) we get x2 + y2 = cos2 +
sin2
(PR)2 (QR)2
x2 + y2 = 1
(PR)2 (QR)2
The equation is that of an ellipse, Hence the instrument traces
an ellipse. Path traced by mid-point of PQ is a circle. In this
case, PR = PQ and so x2+y2 =1 (PR)2 (QR)2
It is an equation of circle with PR = QR = radius of a
circle.
2. Scotch yoke mechanism: This mechanism, the slider P is fixed.
When PQ rotates above P, the slider Q reciprocates in the vertical
slot. The mechanism is used to convert rotary to reciprocating
mechanism.
3. Oldhams coupling: The third inversion of obtained by fixing
the link connecting the 2 blocks P & Q. If one block is turning
through an angle, the frame and the other block will also turn
through the same angle. It is shown in the figure below.
An application of the third inversion of the double slider crank
mechanism is Oldhams coupling shown in the figure. This coupling is
used for connecting two parallel shafts when the distance between
the shafts is small. The two shafts to be connected have flanges at
their ends, secured by forging. Slots are cut in the flanges. These
flanges form 1 and 3. An intermediate disc having tongues at right
angles and opposite sides is fitted in between the flanges. The
intermediate piece forms the link 4 which slides or reciprocates in
flanges 1 & 3. The link two is fixed as shown. When flange 1
turns, the intermediate disc 4 must turn through the same angle and
whatever angle 4 turns, the flange 3 must turn through the same
angle. Hence 1, 4 & 3 must have the same angular velocity at
every instant. If the distance between the axis of the shaft is x,
it will be the diameter if the circle traced by the centre of the
intermediate piece. The maximum sliding speed of each tongue along
its slot is given by
v=x where, = angular velocity of each shaft in rad/sec v =
linear velocity in m/sec
3. Mechanical Advantage, Transmission angle:
The mechanical advantage (MA) is defined as the ratio of output
torque to the input torque. (or) ratio of load to output.
Transmission angle.
The extreme values of the transmission angle occur when the
crank lies along the line of frame.
4. Description of common mechanisms-Single, Double and offset
slider mechanisms - Quick return mechanisms:
Quick Return Motion Mechanisms:
Many a times mechanisms are designed to perform repetitive
operations. During these operations for a certain period the
mechanisms will be under load known as working stroke and the
remaining period is known as the return stroke, the mechanism
returns to repeat the operation without load. The ratio of time of
working stroke to that of the return stroke is known a time ratio.
Quick return mechanisms are used in machine tools to give a slow
cutting stroke and a quick return stroke. The various quick return
mechanisms commonly used are i) Whitworth ii) Drag link. iii) Crank
and slotted lever mechanism
1. Whitworth quick return mechanism:
Whitworth quick return mechanism is an application of third
inversion of the single slider crank chain. This mechanism is shown
in the figure below. The crank OC is fixed and OQ rotates about O.
The slider slides in the slotted link and generates a circle of
radius CP. Link 5 connects the extension OQ provided on the
opposite side of the link 1 to the ram (link 6). The rotary motion
of P is taken to the ram R which reciprocates. The quick return
motion mechanism is used in shapers and slotting machines.
The angle covered during cutting stroke from P1 to P2 in counter
clockwise direction is or 360 -2. During the return stroke, the
angle covered is 2 or .
2. Drag link mechanism :
This is four bar mechanism with double crank in which the
shortest link is fixed. If the crank AB rotates at a uniform speed,
the crank CD rotate at a non-uniform speed. This rotation of link
CD is transformed to quick return reciprocatory motion of the ram E
by the link CE as shown in figure. When the crank AB rotates
through an angle in Counter clockwise direction during working
stroke, the link CD rotates through 180. We can observe that /
>/ . Hence time of working stroke is / times more or the return
stroke is / times quicker. Shortest link is always stationary link.
Sum of the shortest and the longest links of the four links 1, 2, 3
and 4 are less than the sum of the other two. It is the necessary
condition for the drag link quick return mechanism.
3. Crank and slotted lever mechanism:
It is an application of second inversion. The crank and slotted
lever mechanism is shown in figure below.
In this mechanism link 3 is fixed. The slider (link 1)
reciprocates in oscillating slotted lever (link 4) and crank (link
2) rotates. Link 5 connects link 4 to the ram (link 6). The ram
with the cutting tool reciprocates perpendicular to the fixed link
3. The ram with the tool reverses its direction of motion when link
2 is perpendicular to link 4. Thus the cutting stroke is executed
during the rotation of the crank through angle and the return
stroke is executed when the crank rotates through angle or 360 .
Therefore, when the crank rotates uniformly, we get,
Time to cutting = =
Time of return 360
This mechanism is used in shaping machines, slotting machines
and in rotary engines.
5. Ratchets and escapements - Indexing Mechanisms - Rocking
Mechanisms:
Intermittent motion mechanism:
1. Ratchet and Pawl mechanism: This mechanism is used in
producing intermittent rotary motion member. A ratchet and Pawl
mechanism consists of a ratchet wheel 2 and a pawl 3 as shown in
the figure. When the lever 4 carrying pawl is raised, the ratchet
wheel rotates in the counter clock wise direction (driven by pawl).
As the pawl lever is lowered the pawl slides over the ratchet
teeth. One more pawl 5 is used to prevent the ratchet from
reversing. Ratchets are used in feed mechanisms, lifting jacks,
clocks, watches and counting devices.
2. Geneva mechanism: Geneva mechanism is an intermittent motion
mechanism. It consists of a driving wheel D carrying a pin P which
engages in a slot of follower F as shown in figure. During one
quarter revolution of the driving plate, the Pin and follower
remain in contact and hence the follower is turned by one quarter
of a turn. During the remaining time of one revolution of the
driver, the follower remains in rest locked in position by the
circular arc.
3. Pantograph: Pantograph is used to copy the curves in reduced
or enlarged scales. Hence this mechanism finds its use in copying
devices such as engraving or profiling machines.
This is a simple figure of a Pantograph. The links are pin
jointed at A, B, C and D. AB is parallel to DC and AD is parallel
to BC. Link BA is extended to fixed pin O. Q is a point on the link
AD. If the motion of Q is to be enlarged then the link BC is
extended to P such that O, Q and P are in a straight line. Then it
can be shown that the points P and Q always move parallel and
similar to each other over any path straight or curved. Their
motions will be proportional to their distance from the fixed
point. Let ABCD be the initial position. Suppose if point Q moves
to Q1 , then all the links and the joints will move to the new
positions (such as A moves to A1 , B moves to Q1, C moves to Q1 , D
moves to D1 and P to P1 ) and the new configuration of the
mechanism is shown by dotted lines. The movement of Q (Q Q1) will
be enlarged to PP1 in a definite ratio.
4. Toggle Mechanism:
In slider crank mechanism as the crank approaches one of its
dead centre position, the slider approaches zero. The ratio of the
crank movement to the slider movement approaching infinity is
proportional to the mechanical advantage. This is the principle
used in toggle mechanism. A toggle mechanism is used when large
forces act through a short distance is required. The figure below
shows a toggle mechanism. Links CD and CE are of same length.
Resolving the forces at C vertically F Sin =P Cos 2
Therefore, F = P . (because Sin /Cos = Tan ) 2 tan Thus for the
given value of P, as the links CD and CE approaches collinear
position (O), the force F rises rapidly.
5. Hookes joint:
Hookes joint used to connect two parallel intersecting shafts as
shown in figure. This can also be used for shaft with angular
misalignment where flexible coupling does not serve the purpose.
Hence Hookes joint is a means of connecting two rotating shafts
whose axes lie in the same plane and their directions making a
small angle with each other. It is commonly known as Universal
joint. In Europe it is called as Cardan joint.
5. Ackermann steering gear mechanism:
This mechanism is made of only turning pairs and is made of only
turning pairs wear and tear of the parts is less and cheaper in
manufacturing. The cross link KL connects two short axles AC and BD
of the front wheels through the short links AK and BL which forms
bell crank levers CAK and DBL respectively as shown in fig, the
longer links AB and KL are parallel and the shorter links AK and BL
are inclined at an angle . When the vehicles steer to the right as
shown in the figure, the short link BL is turned so as to increase
, where as the link LK causes the other short link AK to turn so as
to reduce . The fundamental equation for correct steering is,
CotCos = b / l
In the above arrangement it is clear that the angle through
which AK turns is less than the angle through which the BL turns
and therefore the left front axle turns through a smaller angle
than the right front axle. For different angle of turn , the
corresponding value of and (Cot Cos ) are noted. This is done by
actually drawing the mechanism to a scale or by calculations.
Therefore for different value of the corresponding value of and are
tabulated. Approximate value of b/l for correct steering should be
between 0.4 and 0.5. In an Ackermann steering gear mechanism, the
instantaneous centre I does not lie on the axis of the rear axle
but on a line parallel to the rear axle axis at an approximate
distance of 0.3l above it.
Three correct steering positions will be:
1) When moving straight.2) When moving one correct angle to the
right corresponding to the link ratio AK/AB and angle . 3) Similar
position when moving to the left. In all other positions pure
rolling is not obtainable.
Some Of The Mechanisms Which Are Used In Day To Day Life.
BELL CRANK:
GENEVA STOP:
BELL CRANK: The bell crank was originally used in large house to
operate the servants bell, hence the name. The bell crank is used
to convert the direction of reciprocating movement. By varying the
angle of the crank piece it can be used to change the angle of
movement from 1 degree to 180 degrees.
GENEVA STOP: The Geneva stop is named after the Geneva cross, a
similar shape to the main part of the mechanism. The Geneva stop is
used to provide intermittent motion, the orange wheel turns
continuously, the dark blue pin then turns the blue cross quarter
of a turn for each revolution of the drive wheel. The crescent
shaped cut out in dark orange section lets the points of the cross
past, then locks the wheel in place when it is stationary. The
Geneva stop mechanism is used commonly in film cameras.
ELLIPTICAL TRAMMEL
PISTON ARRANGEMENT
ELLIPTICAL TRAMMEL: This fascinating mechanism converts rotary
motion to reciprocating motion in two axis. Notice that the handle
traces out an ellipse rather than a circle. A similar mechanism is
used in ellipse drawing tools.
PISTON ARRANGEMENT: This mechanism is used to convert between
rotary motion and reciprocating motion, it works either way. Notice
how the speed of the piston changes. The piston starts from one
end, and increases its speed. It reaches maximum speed in the
middle of its travel then gradually slows down until it reaches the
end of its travel.
RACK AND PINION
RATCHET
RACK AND PINION: The rack and pinion is used to convert between
rotary and linear motion. The rack is the flat, toothed part, the
pinion is the gear. Rack and pinion can convert from rotary to
linear of from linear to rotary. The diameter of the gear
determines the speed that the rack moves as the pinion turns. Rack
and pinions are commonly used in the steering system of cars to
convert the rotary motion of the steering wheel to the side to side
motion in the wheels. Rack and pinion gears give a positive motion
especially compared to the friction drive of a wheel in tarmac. In
the rack and pinion railway a central rack between the two rails
engages with a pinion on the engine allowing the train to be pulled
up very steep slopes.
RATCHET: The ratchet can be used to move a toothed wheel one
tooth at a time. The part used to move the ratchet is known as the
pawl. The ratchet can be used as a way of gearing down motion. By
its nature motion created by a ratchet is intermittent. By using
two pawls simultaneously this intermittent effect can be almost,
but not quite, removed. Ratchets are also used to ensure that
motion only occurs in only one direction, useful for winding gear
which must not be allowed to drop. Ratchets are also used in the
freewheel mechanism of a bicycle.
WORM GEAR
WATCH ESCAPEMENT.
WORM GEAR: A worm is used to reduce speed. For each complete
turn of the worm shaft the gear shaft advances only one tooth of
the gear. In this case, with a twelve tooth gear, the speed is
reduced by a factor of twelve. Also, the axis of rotation is turned
by 90 degrees. Unlike ordinary gears, the motion is not reversible,
a worm can drive a gear to reduce speed but a gear cannot drive a
worm to increase it. As the speed is reduced the power to the drive
increases correspondingly. Worm gears are a compact, efficient
means of substantially decreasing speed and increasing power. Ideal
for use with small electric motors.
WATCH ESCAPEMENT: The watch escapement is the centre of the time
piece. It is the escapement which divides the time into equal
segments. The balance wheel, the gold wheel, oscillates backwards
and forwards on a hairspring (not shown) as the balance wheel moves
the lever is moved allowing the escape wheel (green) to rotate by
one tooth. The power comes through the escape wheel which gives a
small 'kick' to the palettes (purple) at each tick.
GEARS
CAM FOLLOWER.
GEARS: Gears are used to change speed in rotational movement. In
the example above the blue gear has eleven teeth and the orange
gear has twenty five. To turn the orange gear one full turn the
blue gear must turn 25/11 or 2.2727r turns. Notice that as the blue
gear turns clockwise the orange gear turns anti-clockwise. In the
above example the number of teeth on the orange gear is not
divisible by the number of teeth on the blue gear. This is
deliberate. If the orange gear had thirty three teeth then every
three turns of the blue gear the same teeth would mesh together
which could cause excessive wear. By using none divisible numbers
the same teeth mesh only every seventeen turns of the blue
gear.
CAMS: Cams are used to convert rotary motion into reciprocating
motion. The motion created can be simple and regular or complex and
irregular. As the cam turns, driven by the circular motion, the cam
follower traces the surface of the cam transmitting its motion to
the required mechanism. Cam follower design is important in the way
the profile of the cam is followed. A fine pointed follower will
more accurately trace the outline of the cam. This more accurate
movement is at the expense of the strength of the cam follower.
STEAM ENGINE.
Steam engines were the backbone of the industrial revolution. In
this common design high pressure steam is pumped alternately into
one side of the piston, then the other forcing it back and forth.
The reciprocating motion of the piston is converted to useful
rotary motion using a crank.
As the large wheel (the fly wheel) turns a small crank or cam is
used to move the small red control valve back and forth controlling
where the steam flows. In this animation the oval crank has been
made transparent so that you can see how the control valve crank is
attached.
6. Straight line generators, Design of Crank-rocker
Mechanisms:
Straight Line Motion Mechanisms:
The easiest way to generate a straight line motion is by using a
sliding pair but in precision machines sliding pairs are not
preferred because of wear and tear. Hence in such cases different
methods are used to generate straight line motion mechanisms:
1. Exact straight line motion mechanism.
a. Peaucellier mechanism, b. Hart mechanism, c. Scott Russell
mechanism
2. Approximate straight line motion mechanisms
a. Watt mechanism, b. Grasshoppers mechanism, c. Roberts
mechanism,
d. Tchebicheffs mechanism
a. Peaucillier mechanism :
The pin Q is constrained to move long the circumference of a
circle by means of the link OQ. The link OQ and the fixed link are
equal in length. The pins P and Q are on opposite corners of a four
bar chain which has all four links QC, CP, PB and BQ of equal
length to the fixed pin A. i.e., link AB = link AC. The product AQ
x AP remain constant as the link OQ rotates may be proved as
follows: Join BC to bisect PQ at F; then, from the right angled
triangles AFB, BFP, we have AB=AF+FB and BP=BF+FP. Subtracting,
AB-BP= AF-FP=(AFFP)(AF+FP) = AQ x AP .
Since AB and BP are links of a constant length, the product AQ x
AP is constant. Therefore the point P traces out a straight path
normal to AR.
b. Roberts mechanism:
This is also a four bar chain. The link PQ and RS are of equal
length and the tracing pint O is rigidly attached to the link QR on
a line which bisects QR at right angles. The best position for O
may be found by making use of the instantaneous centre of QR. The
path of O is clearly approximately horizontal in the Roberts
mechanism.
a. Peaucillier mechanism
b. Hart mechanism
Unit II Kinematics
Velocity and Acceleration analysis of mechanisms (Graphical
Methods):
Velocity and acceleration analysis by vector polygons: Relative
velocity and accelerations of particles in a common link, relative
velocity and accelerations of coincident particles on separate
link, Coriolis component of acceleration.
Velocity and acceleration analysis by complex numbers: Analysis
of single slider crank mechanism and four bar mechanism by loop
closure equations and complex numbers.
7. Displacement, velocity and acceleration analysis in simple
mechanisms:
Important Concepts in Velocity Analysis
1. The absolute velocity of any point on a mechanism is the
velocity of that point with reference to ground.
2. Relative velocity describes how one point on a mechanism
moves relative to another point on the mechanism.
3. The velocity of a point on a moving link relative to the
pivot of the link is given by the equation: V = r, where = angular
velocity of the link and r = distance from pivot.
Acceleration Components
Normal Acceleration: An = 2r. Points toward the center of
rotation
Tangential Acceleration: At = r. In a direction perpendicular to
the link
Coriolis Acceleration: Ac = 2(dr/dt). In a direction
perpendicular to the link
Sliding Acceleration: As = d2r/dt2. In the direction of
sliding.
A rotating link will produce normal and tangential acceleration
components at any point a distance, r, from the rotational pivot of
the link. The total acceleration of that point is the vector sum of
the components.
A slider attached to ground experiences only sliding
acceleration.
A slider attached to a rotating link (such that the slider is
moving in or out along the link as the link rotates) experiences
all 4 components of acceleration. Perhaps the most confusing of
these is the coriolis acceleration, though the concept of coriolis
acceleration is fairly simple. Imagine yourself standing at the
center of a merry-go-round as it spins at a constant speed (). You
begin to walk toward the outer edge of the merry-go-round at a
constant speed (dr/dt). Even though you are walking at a constant
speed and the merry-go-round is spinning at a constant speed, your
total velocity is increasing because you are moving away from the
center of rotation (i.e. the edge of the merry-go-round is moving
faster than the center). This is the coriolis acceleration. In what
direction did your speed increase? This is the direction of the
coriolis acceleration.
The total acceleration of a point is the vector sum of all
applicable acceleration components:
A = An + At + Ac + As
These vectors and the above equation can be broken into x and y
components by applying sines and cosines to the vector diagrams to
determine the x and y components of each vector. In this way, the x
and y components of the total acceleration can be found.
8. Graphical Method, Velocity and Acceleration polygons :
Graphical velocity analysis:
It is a very short step (using basic trigonometry with sines and
cosines) to convert the graphical results into numerical results.
The basic steps are these:
1. Set up a velocity reference plane with a point of zero
velocity designated.
2. Use the equation, V = r, to calculate any known linkage
velocities.
3.Plot your known linkage velocities on the velocity plot. A
linkage that is rotating about ground gives an absolute velocity.
This is a vector that originates at the zero velocity point and
runs perpendicular to the link to show the direction of motion. The
vector, VA, gives the velocity of point A.
4.Plot all other velocity vector directions. A point on a
grounded link (such as point B) will produce an absolute velocity
vector passing through the zero velocity point and perpendicular to
the link. A point on a floating link (such as B relative to point
A) will produce a relative velocity vector. This vector will be
perpendicular to the link AB and pass through the reference point
(A) on the velocity diagram.
5. One should be able to form a closed triangle (for a 4-bar)
that shows the vector equation: VB = VA + VB/A where VB = absolute
velocity of point B, VA = absolute velocity of point A, and VB/A is
the velocity of point B relative to point A.
9. Velocity Analysis of Four Bar Mechanisms:
Problems solving in Four Bar Mechanisms and additional
links.
10. Velocity Analysis of Slider Crank Mechanisms:
Problems solving in Slider Crank Mechanisms and additional
links.
11. Acceleration Analysis of Four Bar Mechanisms:
Problems solving in Four Bar Mechanisms and additional
links.
12. Acceleration Analysis of Slider Crank Mechanisms:
Problems solving in Slider Crank Mechanisms and additional
links.
13. Kinematic analysis by Complex Algebra methods:
Analysis of single slider crank mechanism and four bar mechanism
by loop closure equations and complex numbers.
14. Vector Approach:
Relative velocity and accelerations of particles in a common
link, relative velocity and accelerations of coincident particles
on separate link
15. Computer applications in the kinematic analysis of simple
mechanisms:
Computer programming for simple mechanisms
16. Coincident points, Coriolis Acceleration:
Coriolis Acceleration: Ac = 2(dr/dt). In a direction
perpendicular to the link.
A slider attached to ground experiences only sliding
acceleration.
A slider attached to a rotating link (such that the slider is
moving in or out along the link as the link rotates) experiences
all 4 components of acceleration. Perhaps the most confusing of
these is the coriolis acceleration, though the concept of coriolis
acceleration is fairly simple. Imagine yourself standing at the
center of a merry-go-round as it spins at a constant speed (). You
begin to walk toward the outer edge of the merry-go-round at a
constant speed (dr/dt). Even though you are walking at a constant
speed and the merry-go-round is spinning at a constant speed, your
total velocity is increasing because you are moving away from the
center of rotation (i.e. the edge of the merry-go-round is moving
faster than the center). This is the coriolis acceleration. In what
direction did your speed increase? This is the direction of the
coriolis acceleration.
Example:1
Unit III Kinematics of CAM
Cams:
Type of cams, Type of followers, Displacement, Velocity and
acceleration time curves for cam profiles, Disc cam with
reciprocating follower having knife edge, roller follower, Follower
motions including SHM, Uniform velocity, Uniform acceleration and
retardation and Cycloidal motion.
Cams are used to convert rotary motion into reciprocating
motion. The motion created can be simple and regular or complex and
irregular. As the cam turns, driven by the circular motion, the cam
follower traces the surface of the cam transmitting its motion to
the required mechanism. Cam follower design is important in the way
the profile of the cam is followed. A fine pointed follower will
more accurately trace the outline of the cam. This more accurate
movement is at the expense of the strength of the cam follower.
17. Classifications - Displacement diagrams
Cam Terminology:
Physical components: Cam, follower, spring
Types of cam systems: Oscilllating (rotating), translating
Types of joint closure: Force closed, form closed
Types of followers: Flat-faced, roller, mushroom
Types of cams: radial, axial, plate (a special class of radial
cams).
Types of motion constraints: Critical extreme position the
positions of the follower that are of primary concern are the
extreme positions, with considerable freedom as to design the cam
to move the follower between these positions. This is the motion
constraint type that we will focus upon. Critical path motion The
path by which the follower satisfies a given motion is of interest
in addition to the extreme positions. This is a more difficult (and
less common) design problem.
Types of motion: rise, fall, dwell
Geometric and Kinematic parameters: follower displacement,
velocity, acceleration, and jerk; base circle; prime circle;
follower radius; eccentricity; pressure angle; radius of
curvature.
18. Parabolic, Simple harmonic and Cycloidal motions:
Describing the motion: A cam is designed by considering the
desired motion of the follower. This motion is specified through
the use of SVAJ diagrams (diagrams that describe the desired
displacement-velocity-acceleration and jerk of the follower
motion)
19. Layout of plate cam profiles:
Drawing the displacement diagrams for the different kinds of the
motions and the plate cam profiles for these different motions and
different followers.
SHM, Uniform velocity, Uniform acceleration and retardation and
Cycloidal motions
Knife-edge, Roller, Flat-faced and Mushroom followers.
20. Derivatives of Follower motion:
Velocity and acceleration of the followers for various types of
motions.
Calculation of Velocity and acceleration of the followers for
various types of motions.
21. High speed cams:
High speed cams
22. Circular arc and Tangent cams:
Circular arc
Tangent cam
23. Standard cam motion:
Simple Harmonic Motion
Uniform velocity motion
Uniform acceleration and retardation motion
Cycloidal motion
24. Pressure angle and undercutting:
Pressure angle
Undercutting
UNIT IV
GEARS
SPUR GEARS
Gears are used to transmit power between shafts rotating usually
at different speeds.
Some of the many types of gears are illustrated below :
TERMINOLOGY OF SPUR GEAR
Fig. 1: Terminology of spur gear
A pair of meshing gears is a power transformer, a coupler or
interface which marries the
speed and torque characteristics of a power source and a power
sink (load).
A single pair may be inadequate for certain sources and loads,
in which case more complex combinations such as the above gearbox,
known as gear trains, are
necessary.
In the vast majority of applications such a device acts as a
speed reducer in which the power source drives the device through
the high speed low torque input shaft, while power is fed from the
device to the load through the low speed high torque output
shaft.
Speed reducers are much more common than speed -up drives not so
much because they reduce speed, but rather because they amplify
torque.
Thus gears are used to accelerate a car from rest, not to
provide the initial low speeds
(which could be accomplished by easing up on the accelerator
pedal) but to increase
the torque at the wheels which is necessary to accelerate the
vehicle.
These notes will consider the following aspects of spur gearing
:-
Overall kinetics of a gear pair (for cases only of steady speeds
and loads)
Tooth geometry requirements for a constant velocity ratio (eg.
size and conjugate action)
Detailed geometry of the involute tooth and meshing gears
The consequences of power transfer on the fatigue life of the
components The essentials of gear design.
Some of the main features of spur gear teeth are illustrated.
The teeth extend from the root, or dedendum cylinder (or
colloquially, "circle" ) to the tip, or addendum circle,both these
circles can be measured.
The useful portion of the tooth is the flank (or face), it is
this surface which contacts the mating gear.
The fillet in the root region is kinematically irrelevant since
there is no contact there,but it is important insofar as fatigue is
concerned.
CONDITION FOR CONSTANT VELOCITY RATIO OF TOOTHED WHEELS
(LAW OF GEARING)
Consider the portions of the two teeth, one on the wheel 1 (or
pinion) and the other on the wheel 2, as shown by thick line curves
in Fig. 2.
Let the two teeth come in contact at point Q, and the wheels
rotate in the directions as shown in the figure. Let TT be the
common tangentand MN be the common normal to the curves at the
point of contact Q.
From the centres O1and O2 draw O1M and O2N perpendicular to MN.
A little
consideration will show that the point Q moves in the direction
QC, when considered as a point on wheel 1, and in the direction QD
when considered as a point on wheel
Let v1 and v2 be the velocities of the point Q od the wheels I
and 2 respectively. If the teeth are to remain in contact, then the
components of these ye locities akng the
common normal MN must be equal.
From above, we see that the angular velocity ratio is inversely
proportional to the ratio of the distances of the point P from the
centres O1 and O2, or the common normal to the two surfaces at the
point of contact Q intersects the line of centres at point P which
divides the centre distance inversely as the ratio of angular
velocities
Fig. 2. Law of gearing.
Therefore in order to have a constant angular velocity ratio for
all positions of the
wheels, the point P must be the fixed point (called pitch point)
for the two wheels.
In other words, the common normal at the point of contact
between a pair of teeth must always pass through the pitch point.
This is the fundaniental condition which must be satisfied while
designing the profiles for the teeth of gear wheels. It is also
known as law of gearing.
INTERFERENCE IN INVOLUTE GEARS
Fig. 3 shows a pinion with centre O1 in mesh with wheel or gear
with centre O2.
MN is the common tangent to the base circles and KL is the path
of contact between the two mating teeth.
Fig.3.
Fig.3. Interference in involute gears.
A little consideration will show, that if the radius of the
addendum circle of pinion is increased to O1N the point of contact
L will move from L to N.
When this radius is further increased, the point of contact L
will be on the inside of base circle of wheel and not on the
involute profile of tooth on wheel. The tip of tooth on the pinion
will then undercut the tooth on the wheel at the root and remove
part of the involute profile of tooth on the wheel.
This effect is known as !nterference, and occurs when the teeth
are being cut. In brief, the phenomenon when the tip of tooth
undercuts the root on its mating gear is known as interference.
Similarly, if the radius of the addendum circle of the wht l
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. Obviously,
interference may be avoided if the path of 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.
From the above discussion, we conclude that the interference may
only be avoided, if the point of contact between the two teeth is
always on the involute profiles of both the teeth. In other words,
interference may only be prevented (f the addendu addendum circles
of the two mating gears cut the common tangent to the base circles
between the points of tangency.
Maximum length of path of contact,
MN = MP + PN = r sin +Rsin =(r +R) sin
OVERALL KINETICS OF A GEAR PAIR
Fig : 4 Overall kinetics of a gear pair
Fig : 4 Overall kinetics of a gear pair
Analysis of gears follows along familiar lines in that we
consider kinetics of the overall assembly first, before examining
internal details such as individual gear teeth.
The free body of a typical single stage gearbox is shown.
The power source applies the torque T1 to the input shaft,
driving it at speed 1 in the sense of the torque (clockwise
here).
For a single pair of gears the output shaft rotates at speed 2
in the opposite sense to the input shaft, and the torque T2
supplied by the gearbox drives the load in the sense
The reaction to this latter torque is shown on the free body of
the gearbox apparently the output torque T2 must act on the gearbox
in the same sense as that of the input torque T1.
The gears appear in more detail in Fig 5( i) below. O1 and O2
are th the centres of the pinion and wheel respectively. We may
regard the gears as equivalent pitch
cylinders which roll together without slip - the requirements
for preventing slip due to
the positive drive provided by the meshing teeth is examined
below.
Unlike the addendum and dedendum cylinders, pitch cylinders
cannot be measured directly; they are notional and must be inferred
from other measurements.
Fig: 5 spur gear assembly
One essential for correct meshing of the gears is that the size
of the teeth on the
pinion is the same as the size of teeth on the wheel.
One measure of size is the circular pitch, p, the distance
between adjacent teeth
around the pitch circle Fig 5 ( ii); thus p = D/z where z is the
number of teeth on a
gear of pitch diameter D.
The SI measure of size is the module, m = p/ - which should not
be confused with the SI abbreviation for metre. So the geometry of
pinion 1 and wheel 2 must be such that :
D1 / z1 = D2 / z2 = p / = m
That is the module must be common to both gears. For the rack
illustrated above, both the diameter and tooth number tend to
infinity, but their quotient remains the finite module.
The pitch circles contact one another at the pitch point, P Fig
5( iii), which is also
notional. Since the positive drive precludes slip between the
pitch cylinders, the pinion's pitch line velocity, v, must be
identical to the wheel's pitch line velocity
v = 1 R1 = 2 R2 ; where pitch circle radius R = D/2
Separate free bodies of pinion and wheel appear in Fig 5(iv).Ft
is the tangential
component of action -reaction at the pitch point due to contact
between the gears.
The corresponding radial component plays no part in power
transfer and is therefore not shown on the bodies. Ideal gears only
are considered initially, so the friction force due to sliding
contact is omitted also.
The free bodies show that the magnitude of the shaft reactions
must be Ft, and that for equilibrium :
Ft = T1 / R1 = T2 / R2 in the absence of friction.
The preceding concepts may be combined conveniently into :
1) T2 / T1 = D2 / D1 = z2 / z1 ; D = mz
That is, gears reduce speed and amplify torque in proportion to
their teeth numbers. In practice, rotational speed is described by
N (rev/min or Hz) rather than by (rad/s).
There exists a host of shapes which ensure conjugacy - indeed it
is possible, within certain restrictions, to arbitrarily choose the
shape of one body then determine the shape of the second necessary
for conjugacy.
But by far the most common gear geometry which satisfies
conjugacy is based on the involute, in which case both gears are
similar in form, and the contact point's locus is a simple straight
line - the line of action.
One method of generating an involute is shown in Fig 6. A. A
generating cord, in which there is a knot C, is wrapped around a
fixed cylinder - the base cylinder (idiomatically circle ) of
radius Ro.
When the taut cord is subsequently unwound as shown in this
animation, the knot traces out an involute whose polar coordinates
may be expressed implicitly in terms of the variable generating
angle , reckoned from the radius through the initial knot position,
C'.
The coordinate origin is taken at the circle centre, O, with a
fixed reference direction defined at some constant angle , also
from the initial radius.
The tangent, TC, is normal to the involute at C, and since the
tangent length TC is equal to the arc length TC', the polar
coordinates of C ( r, ) are
2) r = Ro ( 1 + 2 ) = arctan
In order to see how the involute leads to gear teeth and
conjugate action, we place aslightly different interpretation on
the above model. The cord is wrapped around the base cylinder which
in Fig 6.B is now free to rotate about its centre as the cord is
pulled off in a fixed direction.
This fixed cord direction forms the line of action, tangent to
the base cylinder at the fixed point T, and clearly satisfies
conjugacy by cutting the fixed reference at the fixed pitch point P
through which the pitch cylinder passes.
The line of action is inclined to the pitch point tangent at the
pressure angle. The knot C always moves along the line of action,
tracing out an involute with respect to the rotating cylinder.
The relation between the base and pitch circle radii is
evidently :-
3) Ro = R cos
Extending this to two cylinders - representing meshing gears, 1
& 2 Fig C - the taut cord winds off one base cylinder and onto
the other to form the line of action inclined at the pressure
angle
The knot, C, on the mating involutes coincides with the contact
point and moves along the line of action as the gears and base
cylinders rotate. The pitch cylinders extend to the pitch point P
situated at the intersection of the lines of action and of
centres.
Evidently the distance between the cylinders does not affect the
speed ratio since the base cylinder diameters are fixed.
The distance between knots - ie. between tooth flanks along the
line of action is the base pitch, po, given by
po = Do / z = p cos = m cos . . . . . from ( 1 )
For continuous motion transfer, at least two pairs of teeth must
be in contact as one of the pairs comes into or leaves mesh. The
teeth in Fig.6.C are truncated in practice to permit rotation.
Involute generation by knotted cord is all very well
conceptually, but hardly practicable as a basis for
manufacturing.
Only one of the many methods of gear manufacture is considered
here - the rack generation technique is fundamental to the
understanding of gear behaviour.
CONJUGATE TOOTH ACTION
Fig : 7 Conjugate tooth action
We have seen that one essential for correctly meshing gears is
that the size of the teeth ( the module ) must be the same for the
two gears.
We now examine another requirement - the shape of teeth
necessary for the speed ratio to remain constant during an
increment of rotation; this behaviour of the contacting
surfaces (ie. the teeth flanks) is known as conjugate
action.
Consider the two rigid bodies 1 and 2 which rotate about fixed
centres, O, with angular velocities. The bodies touch at the
contact point, C, through which the common tangent and normal are
drawn.
The absolute velocity v of the contact point reckoned as a point
on either body, is
perpendicular to the radius from that body's centre O to the
contact point.
For the bodies to remain in contact, there must be no component
of relative motion along the common normal, so that from the
velocity triangles
v2 cos2 = v1 cos1 where v1 = O1C ; v2 = O2C
Note that the tangential components of velocity are generally
different, so sliding must occur.
For the speed ratio to be constant therefore, from the above and
similar triangles
O1C/v1 . O2C = O1C.cos /O2C.cos= O1C1 /O2C2 = O1 P / O2 P
ie. this ratio also must be constant.
This indicates that, since the centres are fixed, the point P is
fixed too.
In general therefore, whatever the shapes of the bodies, the
contact point C will move along some locus as rotation proceeds;
but if the action is to be conjugate then the body geometry must be
such that the common normal at the contact point passes always
through one unique point lying on the line of centres - this point
is the pitch point referred to above, and the pitch circles' radii
are O1 P and O2 P.
There exists a host of shapes which ensure conjugacy - indeed it
is possible, within certain restrictions, to arbitrarily choose the
shape of one body then determine the shape of the second necessary
for conjugacy.
But by far the most common gear geometry which satisfies
conjugacy is based on the involute, in which case both gears are
similar in form, and the contact point's locus is a simple straight
line - the line of action.
One method of generating an involute is shown in Fig 8 .A. A
generating cord, in which there is a knot C, is wrapped around a
fixed cylinder - the base cylinder (idiomatically circle ) of
radius.Ro.
When the taut cord is subsequently unwound as shown in this
animation, the knot traces out an involute whose polar coordinates
may be expressed implicitly in terms of the variable generating
angle , reckoned from the radius through the initial knot position,
C'.
The coordinate origin is taken at the circle centre, O, with a
fixed reference direction defined at some constant angle , also
from the initial radius.
The tangent, TC, is normal to the involute at C, and since the
tangent length TC is equal to
the arc length TC', the polar coordinates of C ( r, ) are :-
6) r = Ro ( 1 + 2 ) ; = - + arc tan
In order to see how the involute leads to gear teeth and
conjugate action, we place a slightly different interpretation on
the above model.
The cord is wrapped around the base cylinder which in Fig.8.B is
now free to rotate about its centre as the cord is pulled off in a
fixed direction.
This fixed cord direction forms the line of action, tangent to
the base cylinder at the fixed point T, and clearly satisfies
conjugacy by cutting the fixed reference at the fixed pitch point P
through which the pitch cylinder passes.
The line of action is inclined to the pitch point tangent at the
pressure angle,. The knot C always moves along the line of action,
tracing out an involute with respect to the rotating cylinder.
The relation between the base and pitch circle radii is
evidently
Ro = R cos
Extending this to two cylinders - representing meshing gears, 1
& 2 Fig C - the taut cord winds off one base cylinder and onto
the other to form the line of action inclined at the pressure
angle.
The knot, C, on the mating involutes coincides with the contact
point and moves along the line of action as the gears and base
cylinders rotate.
The pitch cylinders extend to the pitch point P situated at the
intersection of the lines of action and of centres. Evidently the
distance between the cylinders does not affect the speed ratio
since the base cylinder diameters are fixed.
A pinion tooth touches a wheel tooth at the contact point C (the
knot) which moves up the line of action and along the teeth faces
as rotation proceeds.
Since contact cannot occur outside the teeth, it takes place
along the line of action only between the points Q2 and Q1 on the
line of action and inside both addendum circles.
The line segment Q2Q1 is named the path of contact.
Fig : 9 The path of contact
The Figure shows clearly :
the contact point marching along the line of action
the path of contact bounded by the two addenda
the orthogonality between line of action and involute tooth
flanks at the contact
point
how load is transferred from one pair of contacting teeth to the
next as rotation
proceeds
relative sliding between the teeth - particularly noticable at
the beginning and end
of contact
guaranteed tooth tip clearance due to the dedendum exceeding the
addendum
a significant gap between the non-drive face of a pinion tooth
and the adjacent
wheel tooth
The gap between the non-drive face of the pinion tooth and the
adjacent wheel tooth is known as backlash. 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.
Shock in a torsionally vibrating drive is exacerbated by
significant backlash, though a small amount of backlash is provided
in all drives to prevent binding due to
manufacturing or mounting inaccuracies and to facilitate
lubrication.
Backlash may be reduced by subtle alterations to tooth profile
or by shortening the centre distance from the extended value,
however we consider gears meshing only at the extended centre
distance.
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 for example, then 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, given by
length of path of contact / distance between teeth along the
line of action
= Q2PQ1 / base pitch, po and for extended centres with for the
20o system :
Gears having a contact ratio below about 1.2 are not normally
recommended as the gears themselves, their shafts and bearings
would all require especial care in design and manufacture to
preserve conjugacy.
A pinion tooth touches a wheel tooth at the contact point C (the
knot) which moves up the line of action and along the teeth faces
as rotation proceeds.
Since contact cannot occur outside the teeth, it takes place
along the line of action only between the points Q2 and Q1 on the
line of action and inside both addendum circles. The line segment
Q2Q1 is named the path of contact.
GEAR TRAIN
SIMPLE GEAR TRAIN:
The only way that the input and output shafts of a gear pair can
be made to rotate in the
same sense is by interposition of an odd number of intermediate
gears as shown in Fig , these do not affect the speed ratio between
input and output shafts.
Such a gear train is called a simple train. If there is no power
flow through the shaft of an intermediate gear then it is an idler
gear.
COMPOUND GEAR TRAIN:
A gear train comprising two or more pairs is termed compound
when the wheel of one stage is mounted on the same shaft as the
pinion of the next stage.
A compound train as in the above gearbox is used when the
desired speed ratio cannot be achieved economically by a single
pair.
Applying ( 1) to each stage in turn, the overall speed ratio for
a compound train is found to be the product of the speed ratios for
the individual stages.
Selecting suitable integral tooth numbers to provide a specified
speed ratio can beawkward if the speed tolerance is tight and the
range of available tooth numbers is limited.
Unlike the above gearbox, the input and output shafts are
coaxial in the train illustrated here; this is rather an unusual
feature, but necessary in certain change speed boxes and the
like.
In the next section we look at a particular gear train
arrangement called an epicyclic
gear train, before focusing on details of gear tooth shape and
manufacture.
EPICYCLIC GEAR TRAINS
An epicyclic train is often suitable when a large torque/speed
ratio is required in a compact envelope. It is made up of a number
of elements which are interconnected to form the train.
Each element consists of the three components illustrated below
:
A central gear ( c) which rotates at angular velocity c about
the fixed axis O-O
of the element, under the action of the torque Tc applied to the
central gear's
integral shaft; this central gear may be either an external gear
(also referred to as
a sun gear) Fig 13.a(1a), or an internal gear, Fig 13.a
(1b).
An arm ( a) which rotates at angular velocity a about the same
O-O axis under
the action of the torque, Ta - an axle A rigidly attached to the
end of the arm
carries
A planet gear ( p) which rotates freely on the axle A at angular
velocity p,
meshing with the central gear at the pitch point P - the torque
Tp acts on the
planet gear itself, not on its axle.
The epicyclic gear photographed here without its arms consists
of two elements. The central gear of one element is an exter+nal
gear; the central gear of the other element is an internal
gear.
The three identical planets of one element are compounded with (
joined to ) those of the second element.
We shall examine first the angular velocities and torques in a
single three-component element as they relate to the tooth numbers
of central and planet gears, zc and zp respectively.
The kinetic relations for a complete epicyclic train consisting
of two or more elements may then be deduced easily by combining
appropriately the relations for the individual elements.
All angular velocities, , are absolute and constant, and the
torques, T, are external to the three-component element; for
convenience all these variables are taken positive in one
particular sense, say anticlockwise as here. Friction is presumed
negligible, ie. the system is ideal.
There are two contacts between the components :
the planet engages with the central gear at the pitch point P
where the action /
reaction due to tooth contact is the tangential force Ft, the
radial component being
irrelevant;
the free rotary contact between planet gear and axle A requires
a radial force
action / reaction; the magnitude of this force at A must also be
Ft as sketched, for
equilibrium of the planet.
UNIT V
FRICTION
INTRODUCTION
When a body slides (rolls) or made to slide (roll) relative to a
second body, with which it is in contact, there is a resistance to
the relative motion. The resistance so
encountered is called friction.
The force resisting relative motion is called force of friction.
Force of friction acts in a
direction opposite to that of relative motion and is tangential
to the contacting surfaces
of the two bodies in contact.
At every joint in a machine, there is a loss of energy owing to
friction. A proper
understanding about friction as a phenomenon enables us to
reduce frictional forces.
In a number of applications, on the other hand, friction is
considered to be quite
useful.
Friction drives like belt and rope drives, friction clutches,
variable speed drives are
some of applications of this type.
TYPES OF FRICTION
Dry Friction
This type of friction exists between two bodies having relative
motion and whose
contacting surfaces are dry and not separated by any
lubricant.
It is further subdivided in two types as Sliding friction and
rolling friction. Sliding friction is the friction in which the
contacting surfaces have a sliding motion relative to each
other.
Rolling friction is the friction between two bodies in contact
when they have a relative
motion of pure rolling.
Skin or Greasy Friction
When contact surfaces of two bodies, in relative motion, are
separated by a film of
lubricant of small thickness, skin or greasy friction is said to
exist between them.
This type of friction is also known as boundary friction.
Film or Viscous Friction
When contacting surfaces of two bodies, in relative motion, are
completely separated
by a relatively thick film of fluid, viscous friction is said to
exist between the two.
Limiting Friction
The maximum value of frictional force, which comes into play
when one body slides or tends to slide over another body, is known
as Limiting Friction.
Laws of Friction
1. Force of friction always acts in a direction in which the
body tends to move.
2. Force of friction is directly proportional to the normal load
between the
surfaces for a given pair of materials.
3. The force of friction depends upon the materials of the
contacting surfaces.
4. The force of friction is independent of the area of contact
surfaces for a given
normal load.
Coefficient of Friction
It is defined as the ratio between the limiting friction (F) and
the normal reaction(R).
F / R
Figure. 1. Angle of friction ()
Let R is the resultant of normal reaction (RN) and the limiting
friction (F). Then the angle
between R and RN is known as the angle of friction.
Tan = F / R. =
Angle of Repose
Consider that a body of weight (W) resting on an inclined plane.
If the angle of
inclination of the plane to the horizontal is such that the body
begins to move down
the plane, then the inclination of the plane is known as angle
of repose.
The angle of repose is equal to angle of friction.
Figure.2. Inclined Plane
Body at Rest
Figure.3. Body at Rest
When a body is at rest on an inclined plane making an angle with
the horizontal, the forces acting on the body are (Figure 3)
Let W = weight of body
RN, = Normal reaction
F' = force resisting the motion of body.
From equilibrium conditions, Wsin= F' a nd Wcos = RN.
If the angle of inclination of plane is increased, the body will
just slide down the plane of its own when
W sin= F' = RN = W cos
Tan = = tan (or) =
This maximum value of angle of inclination of plane with the
horizontal when the body
starts sliding of its own is known as the angle of repose or
limiting angle of friction.
Motion up the Plane
Figure 4: Motion up the plane
Consider a body moving up an inclined plane under the action of
a force F as shown
in Figure 4. Applying conditions of equilibrium and solving the
equations obtained, we get the minimum force required to be
applied, for equilibrium condition as Fmin = W sin (+)
Efficiency:
The efficiency of an inclined plane, when a body slides up the
plane, is defined as the
ratio of the forces required to move the body without
consideration and with consideration of force of friction. From the
analysis the expression for the efficiency is found to be
Motion down the Plane
When the body moves down the plane, the force of friction F' (=
Rn) acts in the
upwards direction and the reaction R, i.e. the combination of Rn
and F' is inclined backwards.
Applying conditions of equilibrium we get the minimum force,
required to be applied as
Fmin = W sin (-)
Efficiency:
Efficiency of the inclined plane when the body slides down the
plane is defined as the
ratio of the forces required to move the body with and without
the consideration of force of friction.
Square Threads
A square threaded screw used as a jack to raise a load W.
Faces of the square threads in the sectional vies are normal to
the axis of the spindle.
Force F acting horizontally is the force at the screw thread
required to slide the load
W up the inclined plane.
The force F required to be applied is given by
Substituting tan= l / d and tan= and simplifying we get
A bar is, usually fixed to the screw head to use as a lever for
the application of force.
Let f = force applied at the end of the bar of length L
Then
f L = F (d/2) = Fr or f = Fr/L = W r / L [tan (+)]
If the weight is lowered, the expressions for F and f are given
by
Screw efficiency is defined as
Therefore the above equation can be obtained in terms of and
as
The efficiency is maximum when d/d= 0 giving the necessary
condition for maximum efficiency as
= 450 - /2
V-THREADS
In this case the faces are inclined to the axis of spindle.
Figure shows a section of Vthread in which 2is the angle between
the faces of the thread.
If RN is the normal reaction, then the axial component of Rn
must be equal to W
i.e. W = Rn cos
This shows that the coefficient of friction (or tan) as used in
relations for the square threads is to be replaced by or /cosor tan
/costo adapt them to V-thread
PIVOTS AND COLLARS
When a rotating shaft is subjected to an axial load, the thrust
(axial force) is taken
either by a pivot or a collar.
Examples are the shaft of a steam turbine, propeller shaft of a
ship etc.
COLLAR BEARING
A collar bearing or simply a collar is provided at any position
along the shaft and
bears the axial load on a mating surface.
The surface of the collar may be plane normal to the shaft or of
conical shape
Figure 5: Collar Bearing
PIVOT BEARING
When the axial load is taken by the end of the shaft, which is
inserted in a recess to
bear the thrust, Figure 6.
Figure 6: Pivot Bearing
It is called a pivot bearing or simply a pivot. It is also known
as footstep bearing.
9
Friction torque of a collar bearing or pivot bearing is
calculated on the basis of following two
assumptions:
Uniform Pressure theory
Uniform Wear theory
Each assumption leads to a different value of torque.
Uniform Pressure theory
In this case the intensity of pressure on the bearing surface is
assumed to be constant
and the intensity of pressure is given by
Where Ro is the outer radius of the collar and Ri is the inner
radius of the collar.
Uniform Wear theory
In this case wearing of the bearing surface is assumed to be
uniform. Under this
assumption
Po ro = Pi ri = P. r = constant for uniform rate of wear
CONE CLUTCH
BELT AND ROPE DRIVE
INTRODUCTION
The velocity of the belt.
The tension under which the belt is placed on the pulleys.
The arc of contact between the belt and the smaller pulley
The conditions under which the belt is used.
It may be noted that
1. The shafts should be properly in line to insure uniform
tension across the belt
section.
2. The pulleys should not be too close together, in order that
the arc of contact on the
smaller pulley may be as large as possible.
3. The pulleys should not be so far apart as to cause the belt
to weigh heavily on the
shafts, thus increasing the friction load on the bearings.
4. A long belt tends to swing from side to side, causing the
belt to run out of the pulleys,
which in turn develops crooked spots in the belt
5. The tight side of the belt should be at the bottom, so that
whatever sag is present on
the loose side will increase the arc of contact at the
pulleys.
6. In order to obtain good results with flat belts, the maximum
distance between the
shafts should not exceed 10 meters and the minimum should not be
less than 3.5
times the diameter of the larger pulley.
Selection of a Belt Drive
Following are the various important factors upon which the
selection of a belt drive depends:
1. Speed of the driving and driven shafts,
2. Speed reduction ratio,
3. Power to be transmitted,
4. Centre distance between the shafts,
5. Positive drive requirements,
6. Shafts layout,
7. Space available, and
8. Service conditions.
Types of Belt Drives
The belt drives are usually classified into the following three
groups:
Light drives:
These are used to transmit small powers at belt speeds up to
about 10 m is, as in
agricultural machines and small machine tools.
Medium drives:
These are used to transmit medium power at belt speeds over 10 m
but up to 22 m as
in machine tools.
Heavy drives:
These are used to transmit large powers at belt speeds above 22
m/s, as in
compressors and generators.
Types of Belts
Fig.7.Types of belts.
Fig.7.Types of belts.
Though there are many types of belts used these days, yet the
following are important
from the subject point of view:
Flat belt:
The flat belt, as shown in Fig. 7 (a), is mostly used in the
factories and workshops,
where a moderate amount of power is to be transmitted, from one
pulley to another when the two pulleys are not more than 8 meters
apart.
V-belt:
The V-belt, as shown in Fig. 7(b), is mostly used in the
factories and workshops,
where a moderate amount of power is to be transmitted, from one
pulley to another, when the two pulleys are very near to each
other.
Circular belt or rope:
The circular belt or rope, as shown in Fig. 7(c), is mostly used
in the factories and
workshops, where a great amount of power is to be transmitted,
from one pulley to
another, when the two pulleys are more than 8 meters apart.
If a huge amount of power is to be transmitted, then a single
belt may not be
sufficient.
In such a case, wide pulleys (for V-belts or circular belts)
with a number of grooves
are used.
Then a belt in each groove is provided to transmit the required
amount of power from
one pulley to another.
ME 2203 - KINEMATICS OF MACHINERY
UNIT I BASICS OF MECHANISMS
1.Define Kinematic link.
It is a resistive body which go to make a part of a machine
having relative motion between them.
2. Define Kinematic pair.
When two links are in contact with each other it is known as a
pair. If the pair makes constrain motion it is known as kinematic
pair.
3. Define Kinematic chain.
When a number of links connected in space make relative motion
of any point on a link with respect to any other point on the other
link follow a definite law it is known as kinematic chain.
4.Write the Grublers criterion for determining the degrees of
freedom of a mechanism having plane motion.
n=3(l-1)-2j
h-Higher pair joint
l-Number of links
j-Lower pair joint
5.Define degree of freedom, what is meant by mobility.
(Ap/May-2008)
The mobility of a mechanism is defined as the number of input
parameters which must be independently controlled in order to bring
the device into a particular position.
6.Write the Kutzbachs relation. (Ap/May-2008)
Kutzbachs criterion for determining the number of degrees of
freedom or movability (n) of a plane mechanism is n=3(l-1)-2j-h
n-Degree of freedom.
l-Number of links.
h-Higher pair joint
j-Lower pair joint.
7.Define Grashoffs law and state its significance?
(Ap/May-2008)
It states that in a planar four bar mechanism, the sum of
shortest link length and longest link length is not greater than
the sum of remaining two links length, if there is to be continuous
relative motion between two members.
Significance:
Grashoffs law specifies the order in which the links are
connected in a kinematic chain.
Grashoffs law specifies which link of the four-bar chain is
fixed.(s+1)=(p+q) should be satisfied, if not, no link will make a
complete revolution relative to another.
s= length of the shorter length
l= length of the longest link
p & q are the lengths of the other two links.
8.Define Inversion of mechanism.
The method of obtaining different mechanism by fixing different
links in a kinematic chain is known as inversion of mechanism.
9.What is meant by Mechanical advantage of mechanism?
It is defined as the ratio of output torque to the input torque
also defined as the ratio of load to effort.
M.A ideal = TB / TA
TB =driven (resisting torque)
TA =driving torque
10.Define Transmission angle.
The acute angle between follower and coupler is known as
transmission angle.
11. Define Toggle position.
If the driver and coupler lie in the same straight line at this
point mechanical advantage is maximum. Under this condition the
mechanism is known as toggle position.
12.List out few types of rocking mechanism?
Pendulum motion is called rocking mechanism.
1. Quick return motion mechanism.
2. Crank and rocker mechanism.
3. Cam and follower mechanism.
13.Define pantograph?
It is device which is used to reproduce a displacement exactly
in a enlarged scale. It is used in drawing offices, for duplicating
the drawing maps, plans, etc. It works on the principle of 4 bar
chain mechanism.
Eg. Oscillating-Oscillating converter mechanism
14.Name the application of crank and slotted lever quick return
motion mechanism?
1. Shaping machines.
2. Slotting mechanism.
3. Rotary internal combustion engine.
15.Define structure?
It is an assemblage of a number of resistant bodies having no
relative motion between them and meant for carrying loads having
straining action.
16.What is simple mechanism?
A mechanism with four links is known as simple mechanism.
17.Define mechanism?
When one of the links of a kinematic chain is fixed, the chain
is known as a mechanism.
18.Define equivalent mechanism; and spatial mechanism?
Equivalent mechanism: The mechanism, that obtained has the same
number of the degree of freedom, as the original mechanism called
equivalent mechanism.
Spatial mechanism: Spatial mechanism have special geometric
characteristics in that all revolute axes are parallel and
perpendicular to the plane of motion and all prism axes lie in the
plane of motion.
19.Define double slider crank chain mechanism?
A kinematic chain which consists of two turning pair and two
sliding pair is known as double slider crank mechanism.
20.Define single slider crank chain mechanism?
A single slider crank chain is a modification of the basic four
bar chain. It consists of one sliding pair and three turning
pair.
Applications:
Rotary or Gnome engines
Crank and slotted lever mechanism
Oscillating cylinder engine
Ball engine
Hand pump
21.Define Sliding pair.
In a sliding pair minimum number of degree of freedom is only
one.
22.Define Turning pair.
In a turning pair also degree of freedom is one. When two l