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

School of Mechanical Engineering

The University of Birmingham

These notes are regularly interspersed with questions to illustrate the points made and to

understanding. These are highlighted in italics for clarity.


the follower, through a prescribed motion programme by direct contact. The cam mechanism consists of

the cam, the follower and the frame.


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



These notes are regularly interspersed with questions to illustrate the points made and to increase


The cam mechanism consists of


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

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间歇

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反转

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点缀

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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



(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



output motion

Slight manufacturing inaccuracy has little effect on

ed analysis involved in

critical and quieter in operation

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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



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.


different followers? What should be considered when selecting a follower? See TMM pp 203



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



configuration, the cam


TMM pp 203-204, make



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.

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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.



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


nd the requirements of the follower motion and will seek an optimum cam


strated with examples largely featuring disc cams and reciprocating


angle of rotation rather than a linear displacement, and to other cam types where the input may be



studies with gears, it is convenient initially to separate the study of kinematics of the


nd the requirements of the follower motion and will seek an optimum cam


strated with examples largely featuring disc cams and reciprocating


re the input may be

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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



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.


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



) 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


ollower curve. Follower responses fall into three

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Cam mechanisms

Dr K D Dearn

Figure 4 Follower Displacement Diagrams





Cam mechanisms School of Mechanical Engineering

Dr K D Dearn The University of Birmingham

Issue 1.1 of 28 Mechanical Design B

Figure 5 Linear Displacement Function

θ (rad)

θ (rad)

θ (rad)

y (min)

y’ (min/rad)

y’’ (min/rad2)




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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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.

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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:



Parabolic (or constant acceleration) motion of a cam follower



teristics of SHM can apply to cam followers. Figure 7 shows displacement, velocity and


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





teristics of SHM can apply to cam followers. Figure 7 shows displacement, velocity and


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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.




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


Figure 8 A cycloidal curve




, on the circumference of a cylinder rolling on a flat plane, as


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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.

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Cam mechanisms

Dr K D Dearn

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.



Figure 9 Cycloidal follower motion


rise. However, parabolic motion has the advantage of comparatively low accelerations –





in fact for a given


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Cam mechanisms

Dr K D Dearn

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




the acceleration low. One of the mot common means of achieving this is called


illustrated in figure 10, further details are given in the recommended texts.



differences in the displacement curve.

What does this imply about cam manufacture and follower design?

Figure 10 The modified-trapezoid motion curve




the acceleration low. One of the mot common means of achieving this is called




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



Figure 11 Comparison of basic curves

With the advent of computer methods for design and analysis, the use of high order polynomials is


order greater than 2 allows for the higher kinematic derivatives to be continuous and smooth, and


Typically, polynomial expressions of order 5 to 11 are used.

The following pages give illustrated instructions for the graphical construction of cam profiles.


line roller follower graphical construction procedure



sign and analysis, the use of high order polynomials is


order greater than 2 allows for the higher kinematic derivatives to be continuous and smooth, and the


graphical construction of cam profiles.


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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



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



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



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




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


Cam mechanisms

Dr K D Dearn

Figure 14 In

Figure 15 Sw



In-line flat faced follower graphical construction

Swing arm roller follower graphical construction






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.




a smaller cam will have lower sliding speeds and can show reduced vibration. However


In addition there are practical considerations such as cam shaft size, hub size, bearing size and follower

design which will also impose design constraints.


inal return from 330° to 360°. You should find that the common tangent is difficult if not


described a loop, and the cam is said to be undercut – clearly a manufacturing impossibility. Undercutting


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


be a problem. Consider the system shown in figure 17. C, marks the centre of


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.




a smaller cam will have lower sliding speeds and can show reduced vibration. However


ft size, hub size, bearing size and follower


inal return from 330° to 360°. You should find that the common tangent is difficult if not


clearly a manufacturing impossibility. Undercutting


by the application,


marks the centre of


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

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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



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



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



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


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



Width = y’max – y’min



sheet of A4 graph paper. Try sticking two sheets together).



r follower mounted in a frame.

Figure 18 Forces exerted on a roller follower



must always be positive and, in practice, we generally want to ensure that the radius of

We can thus use the inequality:


. 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







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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

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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



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


affected by the roller diameter. Figure 21a shows two rollers following the same pitch curve. Note how the



For a given y(θ), the

locity is positive (the rise) and negative (the

to ensure that the pressure


rve. Note how the

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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




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


Again the end result is complex and in practice would be solved with computational methods, or the use of


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




a feature that may well

ne to exaggerated wear. A similar argument to that covered in the previous section can be used to


be solved with computational methods, or the use of


nomogram to choose a minimum cam radius for the cam in question 5a. Then use the charts in

imum pressure angle for

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Cam mechanisms

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Figure 21 Examples of undercutting and roller size limitations



Figure 21 Examples of undercutting and roller size limitations



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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