Mechanics of Machines
Dr. Mohammad Kilani
Class 5
CAMS
Introduction to Cams
 Cam-follower systems are frequently
used in many kinds of machines.
Common examples of cam follower
systems are the automobile engine
valves, which are opened by cams.
 Machines used in the manufacture of
many consumer goods are full of
cams. Compared to linkages, cams are
easier to design to give a specific
output function, but they are much
more difficult and expensive to make
than a linkage.
 Cams are a form of degenerate
fourbar linkage in which the coupler
link has been replaced by a half joint.
Cams Terminology
 Cam-follower systems can be classified in several ways:
 by type of follower motion, either translating or
rotating (oscillating);
 by type of cam, radial, cylindrical, threedimensional;
 by type of joint closure, either force- or formclosed;
 by type of follower, curved or flat, rolling or
sliding;
 by type of motion constraints, critical extreme
position (CEP), critical path motion (CPM); by type
of motion program, rise-fall (RF), rise-fall-dwell
(RFD), rise-dwell-fall-dwell (RDFD).
Types of Follower Motion
 Follower motion can be
an oscillation or
translation. An
oscillating follower
rotates around a pivot
point. A rotating
follower moves in
usually rectilinear
translation.
 The choice between
these two forms of the
cam-follower is usually
dictated by the type of
output motion desired.
Type of Joint Closure
 Force closure requires an external force be
applied to the joint in order to keep the two
links, cam and follower, physically in
contact. This force is usually provided by a
spring. This force, defined as positive in a
direction which closes the joint, cannot be
allowed to become negative. If it does, the
links have lost contact because a forceclosed joint can only push, not pull.
 Form closure, closes the joint by geometry.
No external force is required. There are
really two cam surfaces in this arrangement,
one surface on each side of the follower.
Each surface pushes, in its turn, to drive the
follower in both directions.
Type of Follower
 Follower, in this context, refers to that part of
the follower link which contacts the cam.
Three types are available, flat-faced,
mushroom (curved), and roller.
 The roller follower has the advantage of lower
(rolling) friction than the sliding contact of the
other two but can be more expensive.
 Flat-faced followers can package smaller than
roller followers and are often favored for that
reason as well as cost in automotives.
 Roller followers are more frequently used in
production machinery where their ease of
replacement and availability from bearing
manufacturers' stock in any quantities are
advantages.
Type of Cam
 The direction of the follower's
motion relative to the axis of
rotation of the cam determines
whether it is a radial or axial
cam.
 In radial cams, the follower
motion is generally in a radial
direction. Open radial cams are
also called plate cams.
 An axial cam is one whose
follower moves parallel to the
axis of cam rotation. This
arrangement is also called a face
cam if open (force-closed) and a
cylindrical or barrel cam if
grooved or ribbed (form-closed).
Type of Motion Constraints
 There are two general categories of motion constraints
which determine the shape of the cam, critical extreme
position (CEP; also called endpoint specification) and
critical path motion (CPM).
 Critical extreme position refers to the case in which the
design specifications define the start and finish positions
of the follower (i.e., extreme positions) but do not
specify any constraints on the path motion between the
extreme positions. This case is the easier of the two to
design as the designer has great freedom to choose the
cam functions which control the motion between
extremes.
 Critical path motion is a more constrained problem than
CEP because the path motion, and/or one or more of its
derivatives are defined over all or part of the interval of
motion.
Dwells and Type of Motion Program
 A dwell is a period of time for which no change in
output motion appears for a changing input
motion. Dwells are an important feature of camfollower systems because it is very easy to create
exact dwells in these mechanisms.
 The motion programs rise-fall (RF), rise-fall-dwell
(RFD), and rise-dwell-fall-dwell (RDFD) all refer
mainly to the CEP case of motion constraint and
in effect define how many dwells are present in
the full cycle of motion, either none (RF), one
(RFD), or more than one (RDFD)
 The cam-follower is the design type of choice
whenever a dwell is required. Cam-follower
systems tend to be more compact than linkages
for the same output motion.
Dwells and Type of Motion Program
 If your need is for a rise-fall (RF) CEP
motion, with no dwell, then you should
really be considering a crank-rocker linkage
rather than a cam-follower to obtain all the
linkage's advantages over cams of
reliability, ease of construction, and lower
cost.
 If your needs for compactness outweigh
those considerations, then the choice of a
cam-follower in the RF case may be
justified. Also, if you have a CPM design
specification, and the motion or its
derivatives are defined over the interval,
then a cam-follower system is the logical
choice in the RF case
S V A J DIAGRAMS
 The first task faced by
the cam designer is to
select the mathematical
functions to be used to
define the motion of the
follower.
 The easiest approach to
this process is to
"linearize“ the cam, i.e.,
"unwrap it" from its
circular shape and
consider it as a function
plotted on Cartesian
axes.
S V A J DIAGRAMS
 We plot the displacement
functions, its first derivative
velocity v, its second
derivative acceleration a, and
its third derivative jerk}, all
on aligned axes as a function
of camshaft angle.
 Note that we can consider
the independent variable in
these plots to be either time
t or shaft angle θ, as we
know the constant angular
velocity (ω) of the camshaft
and can easily convert from
angle to time and vice versa.
Cam Design Procedure
 A cam design begins with a
definition of the required
cam functions and their svaJ
diagrams.
 Functions for the nondwell
cam segments should be
chosen based on their
velocity, acceleration, and
jerk characteristics and the
relationships at the
interfaces between adjacent
segments including the
dwells..
Double-Dwell Cam Design
Choosing S V A J Functions
 The double-dwell case is quite common design requirement for
cams. A double-dwell cam design specifications are often depicted
on a timing diagram which is a graphical representation of the
specified events in the machine cycle cycle. A machine's cycle is
defined as one revolution of its master driveshaft.
 In a complicated machine. The time relationships among all
subassemblies are defined by their timing diagrams which are all
drawn on a common time axis.
Example 1. Uniform Speed S V A J Diagram
 Consider the following cam design CEP specifications:
dwell
rise
dwell
fall
cam speed (ω)
at zero displacement for 90 degrees of cam rotation (low dwell)
1 in (25 mm) in 90 degrees of cam rotation
at 1 in (25 mm) for 90 degrees of cam rotation (high dwell)
1 in (25 mm) in 90 degrees of cam rotation
1 rev/sec 2π rad/sec
Example 1. Uniform Speed S V A J Diagram
 A uniform velocity cam design merely "connect the dots" on the timing
diagram by straight lines to create the displacement diagram. This approach
ignores the effect on the higher derivatives of the resulting displacement
function.
Example 1. Uniform Speed S V A J Diagram
 The acceleration is zero during the
rise and fall intervals, but becomes
infinite at the boundaries of the
interval, where rise meets low dwell
on one side and high dwell on the
other. Note that the velocity
function is multivalued at these
point, creating discontinuities at
these boundaries.
 The effect of these discontinuities is
to create a portion of the velocity
curve which has infinite slope and
zero duration. This results in the
infinite spikes of acceleration
shown at those points
Example 1. Uniform Speed S V A J Diagram
 Clearly the dynamic forces will
be very large at these
boundaries and will create
high stresses and rapid wear.
This is an unacceptable design.
 In fact, if this cam were built
and run at any significant
speeds, the sharp comers on
the displacement diagram
which are creating these
theoretical infinite
accelerations would be quickly
worn to a smoother contour
by the unsustainable stresses
generated in the materials.
The Fundamental law of Cam Design
 Any cam designed for
operation at other than very
low speeds must be designed
with the following constraints:
 The cam function must be
continuous through the first
and second derivatives of
displacement across the entire
interval (360 degrees).
 Corollary: The jerk function
must be finite across the
entire interval (360 degrees).
The Fundamental law of Cam Design
 In any but the simplest of cams,
the cam motion program cannot
be defined by a single
mathematical expression, but
rather must be defined by several
separate functions, each of which
defines the follower behavior
over one segment, or piece, of
the cam.
 These expressions are sometimes
called piecewise functions, and
must have third-order continuity
at all boundaries. The
displacement, velocity and
acceleration functions must have
no discontinuities.
The Fundamental law of Cam Design
 If any discontinuities exist in the
acceleration function, then there
will be infinite spikes, or Dirac
delta functions, appearing in the
derivative of acceleration, jerk.
 Thus the corollary merely
restates the fundamental law of
cam design. In the uniform
velocity cam design example, the
low-degree (linear) polynomial as
selected for the displacement
function, resulted in
discontinuities in the upper
derivatives.
Simple Harmonic Motion S V A J
Diagrams
 Simple harmonic functions remain continuous
throughout any number of differentiations.
Differentiation of a harmonic functions only
amounts for a phase shift by 90 degrees.
 The equations of a simple harmonic rise of
magnitude h over a cam angle β that starts at a
cam angle θi are as follows:
s
v 
   i  
h
 1  cos 

2


 h
 2

a  

sin 
   i 

 h
   i 
cos 


 2

j   

2
 h
   i 
sin 


 2
3
θi
θi + β
Simple Harmonic Motion S V A J
Diagrams
s
v 
   i  
h
 1  cos 

2


 h
 2

a  

sin 
   i 

 h
   i 
cos 


 2

j   

2
 h
   i 
sin 


 2
3
θi
θi + β
Example 2:
Simple Harmonic Motion S V A J Diagrams
 Consider the cam design CEP specifications of Example 1:
dwell
rise
dwell
fall
cam speed (ω)
at zero displacement for 90 degrees of cam rotation (low dwell)
1 in (25 mm) in 90 degrees of cam rotation
at 1 in (25 mm) for 90 degrees of cam rotation (high dwell)
1 in (25 mm) in 90 degrees of cam rotation
1 rev/sec 2π rad/sec
Plot the S V A J diagrams resulting from simple harmonic motion for the rise
and fall periods.
Example 2:
Simple Harmonic Motion S V A J Diagrams
 Consider the cam design CEP specifications of Example 1:
dwell
rise
dwell
fall
cam speed (ω)
at zero displacement for 90 degrees of cam rotation (low dwell)
1 in (25 mm) in 90 degrees of cam rotation
at 1 in (25 mm) for 90 degrees of cam rotation (high dwell)
1 in (25 mm) in 90 degrees of cam rotation
1 rev/sec 2π rad/sec
Plot the S V A J diagrams resulting from simple harmonic motion for the rise
and fall periods.
rise:
Fall:
    
h
s   1  cos 

h = 1 in., θi = π/2, β = π/2
h = -1 in., θi = 3π/2, β = π/2
i

2
v 
 h
 2

a  

sin 

   i 

 h
   i 
cos 


 2

j   

2
 h
   i 
sin 


 2
3
s  0 . 5  1  cos 2    2 
s  0 . 5  1  cos 2   3 2 
v  sin 2    2 
v  sin 2   3  2 
a  2 cos 2    2 
a  2 cos 2   3  2 
j   4 sin 2    2 
j   4 sin 2   3  2 
Example 2:
Simple Harmonic Motion S V A J Diagrams
0
π/2
π
3π/2
2π
rise:
h = 1 in., θi = π/2, β = π/2
Fall:
h = -1 in., θi = 3π/2, β = π/2
s  0 . 5  1  cos 2    2 
s  0 . 5  1  cos 2   3 2 
v  sin 2    2 
v  sin 2   3  2 
a  2 cos 2    2 
a  2 cos 2   3  2 
j   4 sin 2    2 
j   4 sin 2   3  2 
Simple Harmonic Motion S V A J
Diagrams
 The velocity function resulting from a SHM is continuous but the acceleration
has discontinuities at its starting and ending points. The acceleration is not
zero at these points, and thus do not match the zero acceleration of the
dwell period.
 The simple harmonic motion displacement function, when joint with dwells,
does not satisfy the fundamental law of cam design.
0
π/2
π
3π/2
2π
Simple Harmonic Motion S V A J
Diagrams
 The only case in which the simple
harmonic motion displacement
functions satisfy the fundamental
law of cam design is in which the
rise cam angle βr is equal to the
fall cam angle βf with no dwells in
between.
 A common example is the nonquick return (equal period) risefall (RF), in which the follower
rises in 180° and falls in 180°. βr
= βf = 180° in this case.
0
π
2π
Cycloidal Cam Motion
 The cycloidal motion starts with a full wave
sine wave defined as the acceleration
function for the cam follower through the
cam angle β

  i
a    C sin  2 



 , v ( i )  v  i     0 , s ( i )  0 , s  i     h

v   
 a  d 

s   
 v  d    C
using
s  i   0 , v  i   0 , s     i   h , gives k 1
 C

  i
cos  2 
2



  k 1



  k 1    i   k 2

4
2
2

  i
sin  2 


   i

1
  i
s    h 

sin  2 
 
2



a    2 
h

2

  i
sin  2 


h

, k 2  0 , C  2


h 
  i
 1  cos  2 
  , v   


 




 , j    4 

2

  i
cos  2 



h
3



h


 


2
Example 3:
Cycloidal Motion S V A J Diagrams
 Consider the cam design CEP specifications of Example 1:
dwell
rise
dwell
fall
cam speed (ω)
at zero displacement for 90 degrees of cam rotation (low dwell)
1 in (25 mm) in 90 degrees of cam rotation
at 1 in (25 mm) for 90 degrees of cam rotation (high dwell)
1 in (25 mm) in 90 degrees of cam rotation
1 rev/sec 2π rad/sec
Plot the S V A J diagrams resulting from simple harmonic motion for the rise
and fall periods.
rise:
rise:
h = 1 in., θi = π/2, β = π/2 h = 1 in., θ = π/2, β = π/2
 

1
   
s    h 


i


sin  2 
2

i


h 
  i
 1  cos  2 
v   


 



 



  i
a    2  2 sin  2 






h
j    4 
2

  i
cos  2 
3



h
 


i
s
v 
a 



j
1
2
2

 sin  4  2    2 
1  cos 4
8

32

4
 2
sin 4   2 

cos  4  2 


s
v 
a 

j
1
2
2

 sin  4  2    2 
1  cos 4
8

32

4
 2
sin 4   2 

sin  4  2 



Polynomial Cam Motion
 The displacement equation for general polynomial motion can be written as:
s  C 0  C 1    i   C 2    i     C N    i 
2
N
s 
 C 
k
 i 
N
k
k 0
where s is the follower displacement, θ is the angular position of the cam,
and θi is the initial cam angle at the beginning of the polynomial motion. The
integer N is referred to as the degree of the polynomial
 The velocity, acceleration and kerk are obtained by successive differentiation
as
v 
ds

dt
ds d 
d  dt

  C 1  2 C 2    i     NC N    i 
N
v t     kC k    i 
k 1
k 0
a t  
j t  
dv

dv d 
dt
d  dt
da
da d 
dt

d  dt
N

2
 k k  1 C 
k
 i 
k 2
k 0
N

3
 k k  1 k  1 C 
k
k 0
 i 
k 3
N 1

Example 3:
Polynomial Motion S V A J Diagrams
 Determine the polynomial equation that
will satisfy the following conditions
s  C 0  C 1    i   C 2    i   C 3    i   C 4    i   C 5    i 
2
s = 0 when θ = θi
v = 0 when θ = θi
a = 0 when θ = θi
s = h when θ = θi + β
v = 0 when θ = θi + β
a = 0 when θ = θi + β
1.
2.
3.
4.
5.
6.

 C1  0
2 C 2  0
2
a 
2
2
2 C
2
3
4
 6 C 3    i   12 C 4    i   20 C 5    i 
5
 C 1  2 C 2   3C 3   4 C 4   5 C 5 
2
3
 2 C 2  6 C 3   12 C 4   20 C 5 
2
Solving the equations produces
C0  0
C 3  10 h 
C1  0
C 4   15 h 
C2  0
C3  6h 
5
3
4
5
3
3
2
2
C 0  C1  C 2   C 3   C 4   C 5   h
2
4
v   C 1  2 C 2    i   3C 3    i   4 C 4    i   5 C 5    i 
Substituting the six required conditions ,
we get the following set of equations
C0  0
3
4
 0
 0
3

4

The equations for s, v, a, and j
are therefore
    i  
   i
s  10 h 
  15 h 



 
3
 30 h
v  
 

    i  
60 h

 




2
4

   i
  6 h 

 
   i

 
 60 h     180 h
i
a   2

2
 



   i

 
 60 h 360 h
j  2 
3
 


 360 h
 
3


2
3
   i

 



5

30 h
 


   i

 



4




2
   i

 



3




3

120 h
 
2


   i

 



2




Example 3:
Polynomial Motion S V A J Diagrams
 Determine the polynomial equation that
will satisfy the following conditions
1.
2.
3.
4.
5.
6.
s = 0 when θ = θi
v = 0 when θ = θi
a = 0 when θ = θi
s = h when θ = θi + β
v = 0 when θ = θi + β
a = 0 when θ = θi + β
Because of the form of the displacement
function, this particular follower motion
is referred to as 3-4-5 polynomial
motion.
    i  
   i
s  10 h 
  15 h 



 
3
 30 h
v  
 

    i  
60 h

 




2
4

   i
  6 h 

 
   i

 
 60 h     180 h
i
a   2

2
 



   i

 
 60 h 360 h
j  2 
3
 


 360 h
 
3


2
3
   i

 



5

30 h
 


   i

 



4




2
   i

 



3




3

120 h
 
2


   i

 



2




Polynomial Motion S V A J Diagrams
 There are no jumps in the s,
v or a curves, resulting in
good dynamic
characteristics.
 The curves are similar in
many respected to cycloidal
motion, and the two
motions are comparable.
For the same values of h, ω,
and β , cycloidal motion
produce higher values of
maximum acceleration, but
lower values of maximum
jerk than does the 3-4-5
polynomial motion.
Polynomial Motion S V A J Diagrams
 Using the same
approach, polynomial
motions can be derived
for a wide range of
different boundary
conditions for
displacement and
derivatives.
 Manufacturing precision
requirements, however,
increase as the number
of boundary conditions
increase.
Combined Functions
 Instead of selecting a single function to
represent the rise or fall period of the
follower, one may specify a number of
interconnected segments, each of
which is represented by a different
function.
 As seen earlier, the uniform speed rise
produced infinite acceleration at the
ends of the rise segment. Instead, one
can combine a uniformly increasing
speed rise for a portion of the rise
segment, followed by a uniformly
decreasing speed for the other portion.
The resulting follower motion is
referred to as parabolic motion.
Combined Functions – Parabolic Motion
 Let β be the cam angle turned through
which the follower is given a total rise h.
When the follower displacement is h/2,
the cam has turned by β /2. The constant
acceleration motion specifications are:
a    C , v ( 0 )  0 , s ( 0 )  0
v    C   k 1 , s   
C
2
  k 1  k 2
2
using s 0   0 gives k 2  0
using v 0   0 gives k 1  0
4h
  h
using s    gives C  2

 2  2
 The follower displacement for the first half
of the parabolic motion rise is
s   
2h

2
 , v   
2
4h

2
 , a   
4h

2
Combined Functions – Parabolic Motion
s   
 During the 2nd half of follower motion, the
cam has turned by an additional β /2. The
constant acceleration motion
specifications are:
a     C  
v(

2h
)
, s(

2
4h


2
,
h
)
2
2
v (  )  0, s (  )  h
 Integrating as before, and applying the
boundary conditions
v    
2h

2
  k 3 , s    
using v     
using s     

h

 
2
2
2h

2h

2h
  k 3  0, k 3 
2
  k3  k 4  h
2
  k4  h
 h  2h  k4  h
  k 3  k 4
2
2
2
h

h


2h

2
 , v   
2
4h

2
 , a   
4h

2
Graphical Construction of Radial Cam Profile
Knife-Edge Follower with Uniform Speed Rise and Fall
A cam is to give the following motion to a knife-edged follower :
1. Outstroke during 60° of cam rotation
2. Dwell for the next 30° of cam rotation
3. Return stroke during next 60° of cam rotation
4. Dwell for the remaining 210° of cam rotation.
The stroke is 40 mm and the minimum radius of the cam is 50 mm. The follower
moves with uniform velocity during both the outstroke and return strokes.
Draw the profile of the cam when (a) the axis of the follower passes through the
axis of the cam shaft, and (b) the axis of the follower is offset by 20 mm from the
axis of the cam shaft.
Graphical Construction of Radial Cam Profile
Graphical Construction of Radial Cam Profile
Graphical Construction of Radial Cam Profile
Knife-Edge Follower with Simple Harmonic Rise and Fall
A cam is to be designed for a knife edge follower with the following data :
1. Cam lift = 40 mm during 90° of cam rotation with simple harmonic motion.
2. Dwell for the next 30°.
3. During the next 60° of cam rotation, the follower returns to its original
position with simple harmonic motion.
4. Dwell during the remaining 180°.
Draw the profile of the cam when (a) the line of stroke of the follower passes through
the axis of the cam shaft, and (b) the line of stroke is offset 20 mm from the axis of the
cam shaft. The radius of the base circle of the cam is 40 mm. Determine the maximum
velocity and acceleration of the follower during its ascent and descent, if the cam
rotates at 240 r.p.m.
Graphical Construction of Radial Cam Profile
Knife-Edge Follower with Simple Harmonic Rise and Fall
Graphical Construction of Radial Cam Profile
Knife-Edge Follower with Simple Harmonic Rise and Fall
Graphical Construction of Radial Cam Profile
Knife-Edge Follower with Simple Harmonic Rise and Fall
Graphical Construction of Radial Cam Profile
Roller Follower with Simple Harmonic Rise and Fall
A cam, with a minimum radius of 25 mm, rotating clockwise at a uniform speed is to be
designed to give a roller follower, at the end of a valve rod, motion described below :
1.
2.
3.
4.
To raise the valve through 50 mm during 120° rotation of the cam ;
To keep the valve fully raised through next 30°;
To lower the valve during next 60°; and
To keep the valve closed during rest of the revolution i.e. 150° ;
The diameter of the roller is 20 mm and the diameter of the cam shaft is 25 mm. Draw
the profile of the cam when (a) the line of stroke of the valve rod passes through the
axis of the cam shaft, and (b) the line of the stroke is offset 15 mm from the axis of the
cam shaft.
The displacement of the valve, while being raised and lowered, is to take place with
simple harmonic motion. Determine the maximum acceleration of the valve rod when
the cam shaft rotates at 100 r.p.m.
Draw the displacement, the velocity and the acceleration diagrams for one complete
revolution of the cam.
Graphical Construction of Radial Cam Profile
Roller Follower with Simple Harmonic Rise and Fall
Graphical Construction of Radial Cam Profile
Roller Follower with Simple Harmonic Rise and Fall
Graphical Construction of Radial Cam Profile
Roller Follower with Simple Harmonic Rise and Fall
Graphical Construction of Radial Cam Profile
Flat Follower with Simple Harmonic Rise and Fall
A cam drives a flat reciprocating follower in the following manner :
1. During first 120° rotation of the cam, follower moves outwards through a distance
of 20 mm with simple harmonic motion.
2. The follower dwells during next 30° of cam rotation.
3. During next 120° of cam rotation, the follower moves inwards with simple
harmonic motion.
4. The follower dwells for the next 90° of cam rotation.
The minimum radius of the cam is 25 mm. Draw the profile of the cam.
Graphical Construction of Radial Cam Profile
Flat Follower with Simple Harmonic Rise and Fall