Download - Cam Mechanisms Pack
SCHOOL OF MECHANICAL ENGINEERING
Mechanical Design B
Cam mechanisms
Learning pack - Version 1.1
Recommended text:
You particularly are directed to:
“Theory of Machines and Mechanisms” by J E Shigley and J J Uicker, McGraw-Hill, 2nd edition, 1995.
This text book covers all the main points on the topic and provides a considerable amount of detail (also
good for linkages and gears). Throughout these notes cross references will be made to sections or pages of
this book, and it will be referred to as TMM. If designing a cam system, complete design guidance is
available in the ESDU Mechanisms notes volumes 3a to 3d. These cover all the course materials plus
detailed considerations covering comprehensive design. Many of the figures are taken from Fundamentals
of Applied Kinematics by D C Tao, Adison-Wesley, 1967.
You may find this and other standard theory of machines texts useful.
Cam mechanisms
Dr K D Dearn
These notes are regularly interspersed with questions to illustrate the points made and to
understanding. These are highlighted in
INTRODUCTION
The cam is a mechanical component of a machine that is used to transmit motion to another component,
the follower, through a prescribed motion programme by direct contact.
three elements – the cam, the follower and the frame.
They are versatile and can produce any type of motion in the follower. In addition they can convert rotary
motion to linear and vice-versa. The follower as the driven m
- Non-uniform motion
- Intermittent motion
- Reversing motion
Uses of the cam mechanism include:
- Valve timing in internal combustion (IC) engines
- Textile and sewing machines
- Computers
- Printers
- Paper handling devices (photo
- Machine tools
Figure 1 shows four different configurations of cams used for IC engine valve timing.
Try and answer the following: What kind of cams are these? What sort of follower is employed? How is
contact maintained between the cam and follower? How would you expect the contact force to vary as the
engine goes through its cycle? What are the main advantages of cams in this application?
Figure
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These notes are regularly interspersed with questions to illustrate the points made and to
understanding. These are highlighted in italics for clarity.
The cam is a mechanical component of a machine that is used to transmit motion to another component,
the follower, through a prescribed motion programme by direct contact. The cam mechanism consists of
the cam, the follower and the frame.
They are versatile and can produce any type of motion in the follower. In addition they can convert rotary
versa. The follower as the driven member may respond through:
Valve timing in internal combustion (IC) engines
Textile and sewing machines
Paper handling devices (photo-copiers and automatic telling machines)
Figure 1 shows four different configurations of cams used for IC engine valve timing.
What kind of cams are these? What sort of follower is employed? How is
between the cam and follower? How would you expect the contact force to vary as the
engine goes through its cycle? What are the main advantages of cams in this application?
Figure 1 Cam shaft configurations in IC engines
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These notes are regularly interspersed with questions to illustrate the points made and to increase
The cam is a mechanical component of a machine that is used to transmit motion to another component,
The cam mechanism consists of
They are versatile and can produce any type of motion in the follower. In addition they can convert rotary
ember may respond through:
What kind of cams are these? What sort of follower is employed? How is
between the cam and follower? How would you expect the contact force to vary as the
Cam mechanisms
Dr K D Dearn
NOMENCLATURE
Cams may be categorised by:
(1) The shape of the cam
(2) The shape of the follower
(3) The motion of the follower
(4) The position of the follower relative to the cam
(5) The means by which the follower is held in contact with the cam.
Table
Cams
Easily designed to coordinate large number of
input-output motion requirements
Can be made small and compact
Dynamic response is sensitive to the manufacturing
accuracy of the cam contour
Expensive to produce
Easy to obtain dynamic balance
Subject to surface wear
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(4) The position of the follower relative to the cam
(5) The means by which the follower is held in contact with the cam.
1 Comparison between cams and linkages
Linkages
Easily designed to coordinate large number of Satisfy limited number of input-output motion
requirements
Occupy more space
Dynamic response is sensitive to the manufacturing Slight manufacturing inaccuracy has little effect on
output response
Less expensive
Difficult and complicated analysis involved in
dynamic balancing
Joint wear is non-critical and quieter in operation
Figure 2 Cam nomenclature
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output motion
Slight manufacturing inaccuracy has little effect on
ed analysis involved in
critical and quieter in operation
Cam mechanisms
Dr K D Dearn
Figure 2 shows the layout of the most common type of cam
rotates on a fixed centre and the follower bears on the edge.
What follower configurations exist (at least 5 types)? What are the advantages and disadvantages of the
different followers? What should be considered when selecting a f
your own notes. In each case indicate how contact is maintained between follower and cam, (in some cases
external means must be used, in others contact is maintained by the geometry). Indicate on the diagram:
conjugate cams, yoke cams, translating cams and cylindrical cams. What is a face cam?
Figure 3 shows a number of different cam and follower types. Note that the follower may be in
the centre of rotation (i.e. radial) or offset from it, the follower may als
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Figure 2 shows the layout of the most common type of cam – a disc cam. In this configuration, the cam
rotates on a fixed centre and the follower bears on the edge.
What follower configurations exist (at least 5 types)? What are the advantages and disadvantages of the
different followers? What should be considered when selecting a follower? See TMM pp 203
your own notes. In each case indicate how contact is maintained between follower and cam, (in some cases
external means must be used, in others contact is maintained by the geometry). Indicate on the diagram:
ams, yoke cams, translating cams and cylindrical cams. What is a face cam?
Figure 3 shows a number of different cam and follower types. Note that the follower may be in
the centre of rotation (i.e. radial) or offset from it, the follower may also oscillate or be translating.
In-line cam followers
Offset cam followers
Pivoted arm followers
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configuration, the cam
What follower configurations exist (at least 5 types)? What are the advantages and disadvantages of the
TMM pp 203-204, make
your own notes. In each case indicate how contact is maintained between follower and cam, (in some cases
external means must be used, in others contact is maintained by the geometry). Indicate on the diagram:
Figure 3 shows a number of different cam and follower types. Note that the follower may be in-line with
o oscillate or be translating.
Cam mechanisms
Dr K D Dearn
Figure
CAM KINEMATICS
As with previous studies with gears, it is convenient initially to separate the study of kinematics of the
system from its dynamics.
Generally, the output from a cam system is the motion of the follower. When specifying a cam system, the
design engineer will have in mind the requirements of the follower motion and will seek an optimum cam
configuration and profile to achieve this. It is logical, therefore, that a study of cam kinematics should start
with a study of follower motion.
The following discussion will be illustrated with examples largely featuring disc cams and reciprocating
followers. However, the principles may be readily extended to oscillating followers, where the output is an
angle of rotation rather than a linear displacement, and to other cam types whe
rotation or a translation.
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Cylindrical cam
Translating cam
Positive-acting cams
Figure 3 Common cam configurations
studies with gears, it is convenient initially to separate the study of kinematics of the
Generally, the output from a cam system is the motion of the follower. When specifying a cam system, the
nd the requirements of the follower motion and will seek an optimum cam
configuration and profile to achieve this. It is logical, therefore, that a study of cam kinematics should start
strated with examples largely featuring disc cams and reciprocating
followers. However, the principles may be readily extended to oscillating followers, where the output is an
angle of rotation rather than a linear displacement, and to other cam types where the input may be
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studies with gears, it is convenient initially to separate the study of kinematics of the
Generally, the output from a cam system is the motion of the follower. When specifying a cam system, the
nd the requirements of the follower motion and will seek an optimum cam
configuration and profile to achieve this. It is logical, therefore, that a study of cam kinematics should start
strated with examples largely featuring disc cams and reciprocating
followers. However, the principles may be readily extended to oscillating followers, where the output is an
re the input may be
Cam mechanisms
Dr K D Dearn
The input to the system is the cam movement (usually an angle of rotation,
follower movement (usually a displacement or oscillation angle,
Consider a disc cam. During 1 revolution a follower is said to move in three possible ways:
Rise – i.e. move away from the cam centre
Return – i.e. move toward the cam centre
Dwell – i.e. maintain a constant distance from the cam centre
What shape of disc cam would result in a permanent
made to dwell is one of their main advantages.
Commonly the relationship between cam rotation and follower response is sketched on a displacement
diagram as shown in figure 4. This figure shows a typical f
common categories (although there is an unlimited number of more complicated
RRR – Rise Return Rise
DRR – Dwell Rise Return
DRD – Dwell Rise Dwell
Indicate below 4b figure the category of response show
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The input to the system is the cam movement (usually an angle of rotation, θ) and the output is the
follower movement (usually a displacement or oscillation angle, y)
ion a follower is said to move in three possible ways:
i.e. move away from the cam centre
i.e. move toward the cam centre
i.e. maintain a constant distance from the cam centre
What shape of disc cam would result in a permanent dwell? The simple way that cam followers may be
made to dwell is one of their main advantages.
Commonly the relationship between cam rotation and follower response is sketched on a displacement
diagram as shown in figure 4. This figure shows a typical follower curve. Follower responses fall into three
common categories (although there is an unlimited number of more complicated
Indicate below 4b figure the category of response shown, and then sketch the other two
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) and the output is the
ion a follower is said to move in three possible ways:
dwell? The simple way that cam followers may be
Commonly the relationship between cam rotation and follower response is sketched on a displacement
ollower curve. Follower responses fall into three
Cam mechanisms
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Figure 4 Follower Displacement Diagrams
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Issue 1.1 Page 8 of 28 Mechanical Design B
Figure 5 Linear Displacement Function
θ (rad)
θ (rad)
θ (rad)
y (min)
y’ (min/rad)
y’’ (min/rad2)
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Issue 1.1 Page 9 of 28 Mechanical Design B
STANDARD DISPLACEMENT CURVES
Assuming the basic follower response is known (periods of dwell, the lift, the time for rise and return); the
next step in cam design is to decide on the exact relationship between input and output. Consider the
common case of the response DRD illustrated in figure 5. Here, the simplest possible relationship between
input and output for the rise and return is shown – i.e. the follower displacement is directly proportional to
cam rotation. In practice this relationship is never used. To understand why, it is necessary to consider the
velocities and accelerations that would be imposed on the follower.
Consider a cam that rotates at constant speed, then the cam angle is directly proportional to time. The
derivative of displacement with respect to the angle is proportional to the velocity, and the second
derivative is proportional to the acceleration.
In other words:
t∝θ , dtdy
ddy ∝
θ and 2
2
2
2
dt
yd
d
yd ∝θ
Sketch, in the space below figure 5, the first and second derivatives of the displacement curve
Note that y’ (used to denote ��/��) takes constant values for the rise and return, and that �” (used to
denote ���/���) is zero except at the instantaneous transition between rise and dwell, where is becomes
infinite. These transitions are manifest on the cam as sharp discontinuities and would result in unsteady
motion, large contact forces and rapid wear. What is required is a smooth rise without sharp variations in
acceleration and hence contact force. Standard solutions to this problem follow.
PARABOLIC MOTION
The simplest approach to solving discontinuity posed by a simple linear relationship is to employ a second
order parabolic relationship between input and output. The resulting curve is a blend of two parabolas,
usually with a point of inflexion at the half way point.
In the following analysis, the total lift during a parabolic rise is denoted by h and the lift takes place during a
cam rotation angle of β.
Let:
� � � �� � ���
Hence:
y’ = C1 + 2C2θ
y = 2C2
Consider the first half of the rise for 0< θ < β/2:
Applying initial conditions when, θ = 0, y = 0, y’ = 0
Then
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Issue 1.1 Page 10 of 28 Mechanical Design B
C1 = 0
C0 = 0
Applying mid-point conditions when, θ = β/2, y = h/2
C2 = 2
2
β
h
Thus for: 0< θ < β/2:
2
2
2θ
β
hy =
Now derive the relationship between y and θ for the interval β/2 < θ < β (Remember by definition: when θ =
β, y = h and y’ = 0)
(Also remember the curve should be continuous at the halfway point, so the first derivative of the function
must be the same as the first derivative of the previous function for θ = β/2)
For the period: β/2 < θ < β
2
2
24θ
βθ
β
hhhy −+−=
Try question 1 on the tutorial sheet to simplify this expression TMM pp 213-215
Given in figure 6 is the displacement, first and second derivatives of cam motion. Note that the second
derivative (‘acceleration’) is first constant during the first half rise, then changes sign at the point of
inflexion and is constant again during the second half rise. When the inflexion point is at the half way
point, the acceleration is the same magnitude as the deceleration, therefore parabolic motion is sometimes
referred to as constant acceleration.
Question 2 of the tutorial deals with the case of unequal magnitudes of acceleration and deceleration
Sketch on figure 6(c) a graph of the third derivative, y’’’, a quantity proportional to jerk for constant speed
cams. You should note that the jerk becomes infinite at transition points and that the curve is only
continuous for the first derivative.
The figure also demonstrates a method of graphical construction for the displacement diagram. This will be
covered in the tutorial session
Tutorial question 4a covers this technique
Discontinuity in the kinematic derivatives can affect the smooth running of machinery. The remaining
functions described in these notes have been specially chosen to relieve some of the problems whilst
retaining low values of acceleration.
Cam mechanisms
Dr K D Dearn
Figure 6 Parabolic (or constant acceleration) motion of a cam follo
SIMPLE HARMONIC MOTION
You will be familiar with simple harmonic motion, SHM, (consider a pendulum bob that has zero velocity
and maximum acceleration at the two extremes of its travel and zero acceleration at the midpoint of its
travel). The characteristics of SHM can apply to cam followers. Figure 7 shows displacement, velocity and
acceleration for this case, and shows how to construct an SHM curve graphically. Note that the
acceleration curve is continuous at inflexion.
To practice this, try the first part of tutorial question 5a
The follower displacement may be described by the relationship:
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Parabolic (or constant acceleration) motion of a cam follower
You will be familiar with simple harmonic motion, SHM, (consider a pendulum bob that has zero velocity
and maximum acceleration at the two extremes of its travel and zero acceleration at the midpoint of its
teristics of SHM can apply to cam followers. Figure 7 shows displacement, velocity and
acceleration for this case, and shows how to construct an SHM curve graphically. Note that the
acceleration curve is continuous at inflexion.
first part of tutorial question 5a
The follower displacement may be described by the relationship:
−=
β
πθcos1
2
hy
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You will be familiar with simple harmonic motion, SHM, (consider a pendulum bob that has zero velocity
and maximum acceleration at the two extremes of its travel and zero acceleration at the midpoint of its
teristics of SHM can apply to cam followers. Figure 7 shows displacement, velocity and
acceleration for this case, and shows how to construct an SHM curve graphically. Note that the
Cam mechanisms
Dr K D Dearn
In the space below, write expressions for y’ and y’’. Check these against the amplitudes of velocity and
acceleration shown in the figure. Also see
Figure 7
CYCLOIDAL MOTION
A cycloid is the shape traced by a point,
shown in figure 8. As the cylinder rotates about its centre, so the central axis translates relative to the
plane.
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In the space below, write expressions for y’ and y’’. Check these against the amplitudes of velocity and
igure. Also see TMM.
='y
=''y
7 Simple harmonic motion of a cam follower
A cycloid is the shape traced by a point, P, on the circumference of a cylinder rolling on a
shown in figure 8. As the cylinder rotates about its centre, so the central axis translates relative to the
Figure 8 A cycloidal curve
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In the space below, write expressions for y’ and y’’. Check these against the amplitudes of velocity and
, on the circumference of a cylinder rolling on a flat plane, as
shown in figure 8. As the cylinder rotates about its centre, so the central axis translates relative to the
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Issue 1.1 Page 13 of 28 Mechanical Design B
If the cylinder has a radius, r, then the horizontal distance moved by P for an angle of rotation (0 ≤ φ ≤ 2π),
is given by OM – NM (the line ON ). Since the cylinder rolls without slip,
OM = rφ
Whilst by considering the triangle PO’Q,
NM = r sin φ
Hence,
y = r φ – r sin φ
If the rise takes place over a cam rotation of β, then when θ = 0, φ = 0 and when θ = β,
φ = 2π, or:
π
φ
β
θ
2=
If the cylinder is of radius, r, and h, is the horizontal distance covered by point P in one complete cycle,
then:
h = 2πr
Therefore, y in terms of θ is obtained combining these expressions, thus:
−=
β
πθ
πβ
θ 2sin
2
1hy
Write the expressions for y’ and y’’ in the space below and check against figure 9 (also see TMM).
='y
=''y
The displacement, velocity and acceleration profiles for cycloidal motion are shown in figure 9. Note that
all the curves are continuous and the acceleration curve has zero values at the start and end of the motion.
Also illustrated is instruction on how to construct a cycloid graphically.
Practice this technique in tutorial question 5. Slightly modify the procedure because you are sketching a
return rather than rise. Make sure that you draw the circle used for construction to the right scale – to work
correctly you must draw a circle of radius h/2π.
Cam mechanisms
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MODIFIED TRAPEZOID MOTION
We have seen that SHM and cycloidal motions have the advantages of continuous derivatives during the
rise. However, parabolic motion has the advantage of comparatively low accelerations
rise parabolic motion will yield the lowest possible acceleration and hence the lowest possible contact
forces.
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Figure 9 Cycloidal follower motion
We have seen that SHM and cycloidal motions have the advantages of continuous derivatives during the
rise. However, parabolic motion has the advantage of comparatively low accelerations –
rise parabolic motion will yield the lowest possible acceleration and hence the lowest possible contact
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We have seen that SHM and cycloidal motions have the advantages of continuous derivatives during the
in fact for a given
rise parabolic motion will yield the lowest possible acceleration and hence the lowest possible contact
Cam mechanisms
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It would be useful therefore, to modify the parabolic function such that the derivatives are made
continuous while keeping the acceleration low. One of the mot common means of achieving this is called
‘modified trapezoidal’ motion (after the shape of the acceleration curve) The displacement curves are
illustrated in figure 10, further details are given in the recommended tex
Figure 11 is a comparison of the displacement, velocity and acceleration functions for SHM, cycloidal and
trapezoidal motions. Note that very large differences in acceleration are manifested as relatively small
differences in the displacement curve.
What does this imply about cam manufacture and follower design?
Figure
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It would be useful therefore, to modify the parabolic function such that the derivatives are made
the acceleration low. One of the mot common means of achieving this is called
‘modified trapezoidal’ motion (after the shape of the acceleration curve) The displacement curves are
illustrated in figure 10, further details are given in the recommended texts.
Figure 11 is a comparison of the displacement, velocity and acceleration functions for SHM, cycloidal and
trapezoidal motions. Note that very large differences in acceleration are manifested as relatively small
differences in the displacement curve.
What does this imply about cam manufacture and follower design?
Figure 10 The modified-trapezoid motion curve
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It would be useful therefore, to modify the parabolic function such that the derivatives are made
the acceleration low. One of the mot common means of achieving this is called
‘modified trapezoidal’ motion (after the shape of the acceleration curve) The displacement curves are
Figure 11 is a comparison of the displacement, velocity and acceleration functions for SHM, cycloidal and
trapezoidal motions. Note that very large differences in acceleration are manifested as relatively small
Cam mechanisms
Dr K D Dearn
Figure
POLYNOMIAL
With the advent of computer methods for de
becoming the most common and convenient way to describe a follower’s motion. Using polynomials of an
order greater than 2 allows for the higher kinematic derivatives to be continuous and smooth, and
values of these derivatives can be fixed at zero at the extremes of motion or matched with other functions.
Typically, polynomial expressions of order 5 to 11 are used.
CAM PROFILE SKETCHING
The following pages give illustrated instructions for the
Use these techniques to answer tutorial questions 4, 5 and 6. If required, you will find more detailed
instructions in the supporting texts.
In-line roller follower graphical construction procedure
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Figure 11 Comparison of basic curves
With the advent of computer methods for design and analysis, the use of high order polynomials is
becoming the most common and convenient way to describe a follower’s motion. Using polynomials of an
order greater than 2 allows for the higher kinematic derivatives to be continuous and smooth, and
values of these derivatives can be fixed at zero at the extremes of motion or matched with other functions.
Typically, polynomial expressions of order 5 to 11 are used.
The following pages give illustrated instructions for the graphical construction of cam profiles.
Use these techniques to answer tutorial questions 4, 5 and 6. If required, you will find more detailed
line roller follower graphical construction procedure
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sign and analysis, the use of high order polynomials is
becoming the most common and convenient way to describe a follower’s motion. Using polynomials of an
order greater than 2 allows for the higher kinematic derivatives to be continuous and smooth, and the
values of these derivatives can be fixed at zero at the extremes of motion or matched with other functions.
graphical construction of cam profiles.
Use these techniques to answer tutorial questions 4, 5 and 6. If required, you will find more detailed
Cam mechanisms
Dr K D Dearn
Figure 12
1. Draw the base circle. This is the circle of minimum cam radius
2. Draw the cam follower at its lowest position, tangent to the base circle
3. Draw the reference circle passing through the fo
4. Draw radial lines from the centre of the cam, spaced at equal angular intervals (those used on the
follower displacement diagram)
5. Measure the follower displacements corresponding to each angular interval and transfer them to
the appropriate radial line (this gives the location of the follower centres).
6. Draw the circles representing the follower at each location
7. Draw a SMOOTH curve tangent to these circles
Remember that for periods of dwell, the cam profile is that of an arc.
Offset roller follower graphical construction procedure
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12 In-line roller follower graphical construction
Draw the base circle. This is the circle of minimum cam radius
Draw the cam follower at its lowest position, tangent to the base circle
Draw the reference circle passing through the follower centre.
Draw radial lines from the centre of the cam, spaced at equal angular intervals (those used on the
follower displacement diagram)
Measure the follower displacements corresponding to each angular interval and transfer them to
radial line (this gives the location of the follower centres).
Draw the circles representing the follower at each location
curve tangent to these circles
Remember that for periods of dwell, the cam profile is that of an arc.
ollower graphical construction procedure
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Draw radial lines from the centre of the cam, spaced at equal angular intervals (those used on the
Measure the follower displacements corresponding to each angular interval and transfer them to
Cam mechanisms
Dr K D Dearn
Figure 13
1. Draw the base circle
2. Draw the follower at its lowest position, tangent to the base circle
3. Draw the reference circle through the c
4. Draw a circle with its centre at the centre of the cam’s rotation and tangent to the follower axis
5. Divide this circle into a number of divisions on the follower displacement diagram
6. Draw tangents to the circle at each of these divi
7. Lay off the displacements along the appropriate tangent line, measuring from the reference circle
8. Draw circles representing the follower at these positions
9. Draw a SMOOTH curve tangent to the followers’ circles.
Flat faced follower and swinging arm follower
Figures 14 and 15 show similar procedure for the construction of cam profiles with flat faced and swinging
followers
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13 Offset roller follower graphical construction
Draw the follower at its lowest position, tangent to the base circle
Draw the reference circle through the centre of the follower
Draw a circle with its centre at the centre of the cam’s rotation and tangent to the follower axis
Divide this circle into a number of divisions on the follower displacement diagram
Draw tangents to the circle at each of these divisions
Lay off the displacements along the appropriate tangent line, measuring from the reference circle
Draw circles representing the follower at these positions
curve tangent to the followers’ circles.
follower
Figures 14 and 15 show similar procedure for the construction of cam profiles with flat faced and swinging
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Draw a circle with its centre at the centre of the cam’s rotation and tangent to the follower axis
Divide this circle into a number of divisions on the follower displacement diagram
Lay off the displacements along the appropriate tangent line, measuring from the reference circle
Figures 14 and 15 show similar procedure for the construction of cam profiles with flat faced and swinging
Cam mechanisms
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Figure 14 In
Figure 15 Sw
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In-line flat faced follower graphical construction
Swing arm roller follower graphical construction
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Issue 1.1 Page 20 of 28 Mechanical Design B
NOTE ON THE RELATIONSHIP BETWEEN / � AND / �
Derivatives of follower displacement with respect to the cam rotation angle (y’, y’’ etc.) are referred to as
kinematic derivatives. As mentioned previously, they are closely related to the derivative with respect to
time – i.e. velocity, acceleration and jerk (dy/dt, d2y/dt
2, d
3y/dt
3)
Given that � � � ��� and that � � � ���
dt
d
d
dy
dt
dy θ
θ⋅=
And
2
22
2
2
2
2
θ
θθ
θ d
yd
dt
d
dt
d
d
dy
dt
yd⋅
+⋅=
Let
ωθ
=dt
dand α
θ=
2
2
dt
d
Then
ω⋅= 'ydt
dy
And
2
2
2
''' ωα ⋅+⋅= yydt
yd
In many machines the cam is designed to run at constant speed and therefore α = 0. In this case the
acceleration term simplifies to:
2
2
2
'' ω⋅= ydt
yd
Beware: During the start-up of all machinery, this condition does not apply and care must be taken to
calculate the correct acceleration to ensure adequate estimation of forces and proper design for strength
and stiffness.
Cam mechanisms
Dr K D Dearn
SIZE LIMITATIONS
One of the principle advantages of cam systems is their relative compactness. Generally, the smaller the
cam the better – a smaller cam will have lower sliding speeds and can show reduced vibration. However
there are limitations and any application will have a minimum size dedicated by kinematic considerations.
In addition there are practical considerations such as cam sha
design which will also impose design constraints.
Flat faced follower considerations
Try to construct the cam profile in tutorial question 6 with a minimum cam radius of 75mm. To save time,
only construct the final return from 330° to 360°. You should find that the common tangent is difficult if not
impossible to draw.
Figure 16 shows a similar graphical construction. The common tangent to the follower’s positions
described a loop, and the cam is said to be un
may be avoided by increasing the cam radius, increasing the angle over which the rise takes place, or
decreasing the lift. In many cases, the lift and the associated angle of rotation are fixed
and the only recourse is to increase the base circle radius.
Figure 16 Undercutting of a flat
The key to avoiding undercutting is to consider the curvature of the cam. So long as the cam remains
convex there will not be a problem. Consider the system shown in figure 17.
curvature of the cam profile at the point of contact. Clearly, the centre of curvature must lie somewhere
on a line normal to the follower. The cam is shown at some arbitrary
has started to raise, such that the cam has rotated through an angle
position y above the base circle, radius
horizontal distance from the centre of rotation to the point of contract is denoted by
curvature of the cam at the instant shown is denoted by
shown at a radius r, and angle α in a coordinate system that rotates with the cam.
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One of the principle advantages of cam systems is their relative compactness. Generally, the smaller the
a smaller cam will have lower sliding speeds and can show reduced vibration. However
there are limitations and any application will have a minimum size dedicated by kinematic considerations.
In addition there are practical considerations such as cam shaft size, hub size, bearing size and follower
design which will also impose design constraints.
Try to construct the cam profile in tutorial question 6 with a minimum cam radius of 75mm. To save time,
inal return from 330° to 360°. You should find that the common tangent is difficult if not
Figure 16 shows a similar graphical construction. The common tangent to the follower’s positions
described a loop, and the cam is said to be undercut – clearly a manufacturing impossibility. Undercutting
may be avoided by increasing the cam radius, increasing the angle over which the rise takes place, or
decreasing the lift. In many cases, the lift and the associated angle of rotation are fixed by the application,
and the only recourse is to increase the base circle radius.
Figure 16 Undercutting of a flat-faced follower
The key to avoiding undercutting is to consider the curvature of the cam. So long as the cam remains
be a problem. Consider the system shown in figure 17. C, marks the centre of
curvature of the cam profile at the point of contact. Clearly, the centre of curvature must lie somewhere
on a line normal to the follower. The cam is shown at some arbitrary position sometime after the follower
has started to raise, such that the cam has rotated through an angle θ and the follower has risen to the
above the base circle, radius Ro. The follower is mounted with an eccentricity ε, whilst the
horizontal distance from the centre of rotation to the point of contract is denoted by
f the cam at the instant shown is denoted by ρ and the position of C, the centre of curvature is
in a coordinate system that rotates with the cam.
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One of the principle advantages of cam systems is their relative compactness. Generally, the smaller the
a smaller cam will have lower sliding speeds and can show reduced vibration. However
there are limitations and any application will have a minimum size dedicated by kinematic considerations.
ft size, hub size, bearing size and follower
Try to construct the cam profile in tutorial question 6 with a minimum cam radius of 75mm. To save time,
inal return from 330° to 360°. You should find that the common tangent is difficult if not
Figure 16 shows a similar graphical construction. The common tangent to the follower’s positions
clearly a manufacturing impossibility. Undercutting
may be avoided by increasing the cam radius, increasing the angle over which the rise takes place, or
by the application,
The key to avoiding undercutting is to consider the curvature of the cam. So long as the cam remains
marks the centre of
curvature of the cam profile at the point of contact. Clearly, the centre of curvature must lie somewhere
position sometime after the follower
and the follower has risen to the
. The follower is mounted with an eccentricity ε, whilst the
horizontal distance from the centre of rotation to the point of contract is denoted by s, the radius of
, the centre of curvature is
Cam mechanisms
Dr K D Dearn
Figure 17 Flat-faced follower to determine radius of curvature
Show that:
Differentiating (1) yields:
(Making the approximation the dr/dθ,
Similarly differentiate (2) to get:
Differentiate again to show
Combining (2) and (5)
Or
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faced follower to determine radius of curvature
( ) sr =+ αθcos (1)
and
( ) yRor +=++ ραθcos (2)
( )θ
αθd
dsr =+− sin (3)
, dα/dθ and dρ/dθ are zero over a small range of cam rotation)
( ) 'cos yr =+− αθ (4)
( ) ''sin yr =+− αθ (5)
yRoy +=+− ρ''
''yyRo −−= ρ
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are zero over a small range of cam rotation)
Cam mechanisms
Dr K D Dearn
We know that ρ must always be positive and, in practice, we generally want to ensure that the radius of
curvature is larger than some specified minimum value
Although this is a differential equation, in practical cases it is not necessary to solve it rigorously. What is
required is the largest value of the inequality to assign a
always positive. The largest value will, therefore, occur at an angle where
value. In other words you must identify the angle at which the largest
calculate the value of displacement and pseudo
Note also from equations (1) and (4) that:
In other words, the horizontal distance of the point of contact from the centre of the cam is gi
direct function of rotation angle. For a given cam it is possible to predict the maximum (positive) and
minimum (negative) values of this function and thus the face width of the cam is given by:
Now attempt tutorial question 6. You should find that Ro min must be greater than 90 mm to give a
minimum radius of curvature of 2.5 mm 0r 87.5 mm to just avoid undercutting (Your sketch will not fit on a
sheet of A4 graph paper. Try sticking two sheets
Translating roller follower
The design of systems with roller followers requires similar considerations. In this case, not only must
undercutting be avoided, but the question of pressure angle must be addressed. Figure 18 shows a cam
and roller follower mounted in a frame.
Figure 18 Forces exerted on a roller follower
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must always be positive and, in practice, we generally want to ensure that the radius of
ture is larger than some specified minimum value ρmin. We can thus use the inequality:
''min
yyRo −−⟩ ρ
Although this is a differential equation, in practical cases it is not necessary to solve it rigorously. What is
f the inequality to assign a minimum value to Ro. Note that
always positive. The largest value will, therefore, occur at an angle where y’’ takes its largest negative
value. In other words you must identify the angle at which the largest follower deceleration occurs, and
calculate the value of displacement and pseudo-acceleration at this angle.
Note also from equations (1) and (4) that: s = y’
In other words, the horizontal distance of the point of contact from the centre of the cam is gi
direct function of rotation angle. For a given cam it is possible to predict the maximum (positive) and
minimum (negative) values of this function and thus the face width of the cam is given by:
Width = y’max – y’min
Now attempt tutorial question 6. You should find that Ro min must be greater than 90 mm to give a
minimum radius of curvature of 2.5 mm 0r 87.5 mm to just avoid undercutting (Your sketch will not fit on a
sheet of A4 graph paper. Try sticking two sheets together).
The design of systems with roller followers requires similar considerations. In this case, not only must
undercutting be avoided, but the question of pressure angle must be addressed. Figure 18 shows a cam
r follower mounted in a frame.
Figure 18 Forces exerted on a roller follower
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must always be positive and, in practice, we generally want to ensure that the radius of
We can thus use the inequality:
Although this is a differential equation, in practical cases it is not necessary to solve it rigorously. What is
. Note that ρmin and y are
takes its largest negative
follower deceleration occurs, and
In other words, the horizontal distance of the point of contact from the centre of the cam is given by a
direct function of rotation angle. For a given cam it is possible to predict the maximum (positive) and
minimum (negative) values of this function and thus the face width of the cam is given by:
Now attempt tutorial question 6. You should find that Ro min must be greater than 90 mm to give a
minimum radius of curvature of 2.5 mm 0r 87.5 mm to just avoid undercutting (Your sketch will not fit on a
The design of systems with roller followers requires similar considerations. In this case, not only must
undercutting be avoided, but the question of pressure angle must be addressed. Figure 18 shows a cam
Cam mechanisms School of Mechanical Engineering
Dr K D Dearn The University of Birmingham
Issue 1.1 Page 24 of 28 Mechanical Design B
N is the normal force exerted by the cam on the follower, R1 and R2 are the reactions between the follower
stem and the frame, F is the sum of the external loads on the follower (comprising spring loads, inertia
loads and external machine loads). The sleeve in which the follower is supported is given a length, b, and
the follower is shown in a position with a follower overhang length, a. The follower stem has a diameter d.
Consider the external forces in the x-direction and the y-direction.
∑ =−−= 0sin21
αNRRFx
∑ =−++= 0cos)(21
αµ NRRFFy
Taking moments about point A (shown in figure 19)
∑ =−−++= 0cos2
sin2
22 bRNd
aNdRFd
M A ααµ
Eliminate R1 and R2 (and show that)
( ) αµµµα sin2cos2dbab
b
F
N
−+−=
Thus N and F are related purely by the geometry of the system and the pressure angle, which is itself purely
a function of the size and displacement characteristics of the cam. Notice that it is possible for the
denominator to take a zero of negative valve. Should it become zero, N/F = ∞, and the follower will jam.
Thus to avoid jamming,
( ) 0sin2cos2 ⟩−+− αµµµα dbab
And since μ2d is small,
( )ba
b
+=
2tan
µα
Generally this requires α to be less than about 30°
Relationship between pressure angle and displacement curve
Figure 19 shows the pitch circle of a roller follower in contact with a disc cam. The follower has an offset ε
from the centre of rotation of the cam and at the instant shown the pressure angle is α. The instantaneous
centre of velocity of the cam and follower is shown at point P. If the cam rotates at ω, then the velocity of
the cam centre relative to P is Rω. As this is the instantaneous centre of velocity of the follower to point P
must be the same. Thus:
��/�� � ��
Dividing both sides by ω gives this expression in terms of θ
Rd
dydtdy
==θω
Cam mechanisms
Dr K D Dearn
Figure 19 Construction to derive pressure angle for roller follower
By inspection of the figure
Similarly,
Thus, it can be shown that
(See TMM p 237 for a full development)
The point of this proof is to show that pressure angle depends on
designer has control over Ro and ε.
What is the effect of increasing ε? Is it the same when the ve
return)? What happens if you increase Ro?
In practice a designer would use a computational method to select
angle remains in bounds. This may also be done using the nomo
Undercutting and minimum roller diameter
As with flat faced followers the minimum cam radius is a function of follower motion. However it is also
affected by the roller diameter. Figure 21a shows two rollers following the same pitch cu
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Figure 19 Construction to derive pressure angle for roller follower
αε tan)( ayR ++=
( )22 ε−= Roa
( )22
'tan
ε
εα
−+
−=
Roy
y
p 237 for a full development)
The point of this proof is to show that pressure angle depends on y(θ), y’(θ), Ro and ε. For a given
What is the effect of increasing ε? Is it the same when the velocity is positive (the rise) and negative (the
return)? What happens if you increase Ro?
In practice a designer would use a computational method to select Ro and ε to ensure that the pressure
angle remains in bounds. This may also be done using the nomogram in figure 20.
Undercutting and minimum roller diameter
As with flat faced followers the minimum cam radius is a function of follower motion. However it is also
affected by the roller diameter. Figure 21a shows two rollers following the same pitch curve. Note how the
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For a given y(θ), the
locity is positive (the rise) and negative (the
to ensure that the pressure
As with flat faced followers the minimum cam radius is a function of follower motion. However it is also
rve. Note how the
Cam mechanisms
Dr K D Dearn
smaller roller describes an acceptable cam shape, but the large describes a loop. Figure 21b describes a
limiting case, where the roller path would require a cam profile with a sharp point
be prone to exaggerated wear. A similar argument to that covered in the previous section can be used to
produce a relationship between the geometric parameters and the follower motion to avoid undercutting.
Again the end result is complex and in practice would
design charts such as those shown in figure 22. This shows families of curves relating minimum cam radius,
radius of curvature, lift, angle and roller radius for a simple harmonic motion rise.
Use the nomogram to choose a minimum cam radius for the cam in question 5a. Then use the charts in
figure 22 to select suitable roller diameter for the first part of the motion
Figure 20 Nomogram for selection of a suitable minimum cam radius for a given max
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smaller roller describes an acceptable cam shape, but the large describes a loop. Figure 21b describes a
limiting case, where the roller path would require a cam profile with a sharp point – a feature that may well
ne to exaggerated wear. A similar argument to that covered in the previous section can be used to
produce a relationship between the geometric parameters and the follower motion to avoid undercutting.
Again the end result is complex and in practice would be solved with computational methods, or the use of
design charts such as those shown in figure 22. This shows families of curves relating minimum cam radius,
radius of curvature, lift, angle and roller radius for a simple harmonic motion rise.
nomogram to choose a minimum cam radius for the cam in question 5a. Then use the charts in
figure 22 to select suitable roller diameter for the first part of the motion.
Figure 20 Nomogram for selection of a suitable minimum cam radius for a given maximum pressure angle for
various displacement functions
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smaller roller describes an acceptable cam shape, but the large describes a loop. Figure 21b describes a
a feature that may well
ne to exaggerated wear. A similar argument to that covered in the previous section can be used to
produce a relationship between the geometric parameters and the follower motion to avoid undercutting.
be solved with computational methods, or the use of
design charts such as those shown in figure 22. This shows families of curves relating minimum cam radius,
nomogram to choose a minimum cam radius for the cam in question 5a. Then use the charts in
imum pressure angle for
Cam mechanisms
Dr K D Dearn
Figure 21 Examples of undercutting and roller size limitations
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Figure 21 Examples of undercutting and roller size limitations
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Cam mechanisms
Dr K D Dearn
Figure 22 Charts for the sizing of disc cams and radial roller followers where the follower moves with simple
harmonic motion
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Figure 22 Charts for the sizing of disc cams and radial roller followers where the follower moves with simple
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Figure 22 Charts for the sizing of disc cams and radial roller followers where the follower moves with simple