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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
Cam Design
 Chapter Objectives:
a- Present methodologies for designing cam-follower systems.
b- Discuss theoretical considerations of common mathematical functions
utilized in generating cam curves.
c- Present methods for derivation of custom polynomial functions which suit
almost any set of boundary conditions.
d- Address major aspects for sizing the cam with considerations of pressure
angle and radius of curvature.
e- Pinpoint cam manufacturing processes and their limitations.
f- Master utilization of DYNACAM software to design cams and illustrate
design concepts and solutions.
 Introduction:
“Cam-follower systems” are commonly utilized in almost all kinds of
machines. For instance, the intake and exhaust valves in an automobile
internal combustion engine are operated by cams. In comparison to
linkages, cams are relatively “easier to design” and they “possess very
specific output functions”. Contrarily, cams are complicated regarding
manufacturing perspective and hence are expensive.
Cams are alternative form of fourbar linkage mechanisms in which the
“coupler link” is “replaced” by “a half joint” as illustrated in Fig. 1.
Figure 1: Cam-follower mechanism with coupler link replaced by half joint simultaneously existing
between cam and follower
Cam Design
Dr. Hany Fayek
2
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
Effectively, a cam-follower system is a fourbar linkage mechanism with
links of “variable lengths”. It is this major conceptual difference that
makes the cam-follower a versatile and useful “function generator”.
Almost “any output function of desire” could be specified and a cam
“curved surface” is generated such that the output function is executed
through the motion of the cam follower. Therefore, cams are not limited
to links of fixed lengths as was previously experienced with linkage
synthesis.
The cam-follower is an extremely useful mechanical device without which
the machine designer’s tasks would be more difficult to accomplish.
Cam Design
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Cam Classification and Terminology:
Cam-follower systems are classified in several ways, more specifically, by:
1- Type of follower motion:
a- Translating follower
b- Rotating (oscillating) follower
Translation follower
Rotating (oscillating) follower
2- Type of cam:
a- Radial cam (planar motion, i.e. 2D motion)
b- Axial cam (planar motion) “The follower axis is parallel to the camshaft
axis”
c- Three dimensional cam
Radial cam
Cam Design
Axial cam
3D cam
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
3- Type of joint closure:
a- Force closure joint
b- Form (geometry) closure joint (engraved track geometry within cam
defines motion of follower)
Force closure
Form (geometry) closure
4- Type of follower:
a- Roller follower (used extensively in production machinery – size is
usually larger than other types of followers – lowest friction due to rolling
over the cam surface)
b- Mushroom follower (custom designed – higher friction due to sliding)
c- Flat-Faced follower (custom designed – higher friction due to sliding –
smaller in packaging – automotive applications)
Roller follower
Cam Design
Mushroom follower
Flat-Faced follower
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
5- Type of motion constraint:
a- Critical Extreme Position (CEP)
“Start” and “end” positions are specified, “but not the path in
between”.
b- Critical Path Motion (CPM)
“Path” or derivative is “defined over all or part of the cam profile”.
6- Type of motion program:
a- Rise-Fall (RF)
b- Rise-Fall-Dwell (RFD)
c- Rise-Dwell-Fall-Dwell (RDFD)
Dwell:
Dwell is defined as: “no input motion of the follower for a specified period
of input continuous motion of the cam.”
Dwells are important features of cam-follower systems because it is very
easy to perform “exact dwells” in these mechanisms.
Dwells could be more precisely classified as:
- High Dwell: HD
- Low Dwell: LD
Cam Design
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 S V A J Diagrams: (Displacement-Velocity-Acceleration-Jerk Diagrams)
The first task faced by the cam designer is to “select” the
“mathematical functions” which will be utilized to “define the motion of
the follower”. The easiest approach is to “linearize” the cam, i.e. “unwrap”
the cam profile from its curved shape and consider it as a function plotted
on a Cartesian axis.
The displacement function “ s ”, its first derivative velocity “ v ”, its second
derivative acceleration “ a ”, and its third derivative jerk “ j ” are all plotted
on aligned axes as a function of the camshaft angle “ ” as shown in Fig. 2.
Figure 2: Plots of cam-follower’s s v a j diagrams
The “independent variable” in these plots could be considered to be either
the time “ t ” or the camshaft angle “ ”.
Since the camshaft is rotating with constant angular velocity “  ”, it is very
simple to convert from angle to time and vice versa through the following
equation:
  t
Cam Design
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 The Fundamental Law of Cam Design:
“The cam function must be continuous through the first and second
derivatives of displacement across the entire interval (i.e. the 360o
revolution of the cam, i.e., the camshaft on which the cam is mounted).”
Corollary: ‫النتيجة‬
“The jerk function must be finite across the entire 3600 interval”
(i.e. it is extremely prohibited to have infinite jerks in your design)
Extremely Important:
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 profile. These are sometimes called “piecewise functions”.
 Family of Functions utilized in SAVJ Rise and Fall Segments:
1- Simple Harmonic Motion (SHM)
2- Cycloidal Displacement
3- Combined Functions
a- Constant Acceleration
b- Trapezoidal Acceleration
c- Modified Trapezoidal Acceleration
d- Modified Sinusoidal Acceleration
4- The SCCA (Sine-Constant-Cosine-Acceleration) Family
5- Polynomial Functions
Cam Design
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Cam Polynomial Functions:
 The class of polynomial functions is perhaps the most versatile functions
that could be utilized for cam design. They can be used and tailored to
almost any design specification.
 They are not limited to “single-” or “double-” dwell applications. They are
adaptable to almost any situation.
 The general form of a polynomial function is:
s  Co  C1x  C2 x 2  C3 x3  C4 x 4  C5 x 5  C6 x 6  ...... +Cn x n
where
s  follower displacement, and
x  the independent variable and in our case will be replaced by either
 
   or time  t  .
 
where
  It is the camshaft angle (global angle) and is measured in radians, for
the entire s-v-a-j plot.
  It is the “local angle” within any segment and measured in radians.
The value of  can be different for each segment.
 
 The value of   must vary from “0” to “1” over any piecewise segment.
 
 
   is a dimensionless ratio.
 
Cam Design
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 The constant coefficients  Cn  are the unknowns to be solved for in favor
of the development of the particular polynomial equation to “suit the
required design specification”.
 The designer must “structure a polynomial cam design problem” by
“deciding how many boundary conditions (BCs) are needed to be
specified on the s-v-a-j diagrams”.
 The number of BC’s then determines the degree of the resulting polynomial.
 If ( k ) represents the number of chosen BCs, the degree of the polynomial
will be ( n  k  1).
Cam Design
Dr. Hany Fayek
10
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Example 9-7: The 3-4-5 Polynomial for the Double Dwell Case
Given:
Dwell
Rise
Dwell
Fall
at zero displacement for 90 degrees (Low Dwell)
1 inch in 90 degrees
at 1 inch in 90 degrees (High Dwell)
1 inch in 90 degrees
cam 
2 radians/s = 1 rev/s
The “Naïve” Cam Designer:
Notice the “discontinuities” at the “velocity boundaries” in Fig. 3.
The effect of these discontinuities forms portions of the velocity curve
which possess “infinite slopes, i.e. the vertical lines” and “zero duration,
i.e. velocity acting over zero time”. This results in the “infinite spikes of
acceleration and thereby jerk” at the boundary points as illustrated in
Fig. 3.
Figure 3: Naïve s v a j diagram
Cam Design
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
Proper Solution: (The 3-4-5 Polynomial)
In order to satisfy the fundamental law of cam design, “the values” of the
“rise” and “fall” functions must match at their boundaries with the dwells
with no discontinuities in “at minimum” the “s, v, and a plots”.
At this stage of the problem, we find that “the dwells are the fully defined
segments at this stage”.
Any “dwell” requires its own segment. Additionally, a “dwell” will always
have “zero velocity” and “zero acceleration”.
For generality, we will let the specified “total rise” be represented by the
variable ( h ).
The “minimum” set of required BC’s for this example is: “6 BC’s”
Cam Design
Dr. Hany Fayek
12
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Rise Segment: (Piecewise – segment function)
For the follower, the “Rise” segment – Fig. 4: (starting from the end of the
low dwell segment)
Figure 4: The s v a j diagram for the “Rise” segment
At (   0o )
then s  0 ,
v  0,
a0
(a1)
For BC’s (a1), the s, v, and a values (magnitudes) at the beginning of the
“Rise” segment should be “zero” in order to match the “end values” of the
“low dwell” segment.
At (   1 )
then s  h ,
v  0,
a0
(a2)
Additionally, the BC’s (a2) at the end of the “Rise” segment should match
the “start values” of the “High Dwell” segment.
Notice: We have “6 BC’s”; hence, we have “5th” degree polynomial
[ k  6 ; therefore, n  k  1  6  1  5 ].
Cam Design
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
s  Co  C1     C2     C3     C4     C5   
2

s  Co  C1 

3
4
5
2 
 3 
4 
 5 

  C2  2   C3  3   C4  4   C5  5 

 
 
 
 
(A)
(A)
 1  2C
3C
5C
ds
4C
 v  C1    22   33  2  44  3  55  4
d



  

1
v  C1  2C2 



2
3
4

 
 
  
  3C3     4C4     5C5    

 
 
  
(B)

6C
20C
dv
1  2C
12C
 a   2  23   3 4  2  4 5  3 
d
 





1 
a  2  2C2  6C3 
 

2

 
 
  12C4     20C5   

 
 
3


(C)
Boundary Conditions: (at end of Low Dwell)
 BCs (a1): At (  0o )

s  0,
v  0,
a0
 Displacement:
 When [   0o  s  0 ]  substitute in Eq. (A)
0  Co  0  0  0  0  0

Co  0
 Velocity:
 When [   0o  v  0 ]  substitute in Eq. (B)
0
1

C1  0  0  0  0
Cam Design

C1  0
Dr. Hany Fayek
14
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Acceleration:
 When [   0o  a  0 ]
0
1
2
 2C2  0  0  0
 substitute in Eq. (C)

 BCs (a2): At (  1 )
C2  0
then s  h ,
v  0,
a0
Boundary Conditions: (at beginning of High Dwell)
 Displacement:
 When [   1  s  h ]  substitute in Eq. (A)
h  C3      C4      C5     
3
4
5
h  C3  C4  C5
(1)
 Velocity:
 When [   1  v  0 ]  substitute in Eq. (B)
 
 
 
1
0  3C3    4C4    5C5  
   
 
 

2
3
4
  3C3  4C4  5C5  0

(2)
 Acceleration:
 When [   1  a  0 ]  substitute in Eq. (C)

1 
0  2 6C3 
 

2
3

 
  
  12C4     20C5      6C3  12C4  20C5  0 (3)

 
  
C3  C4  C5  h
(1)
3C3  4C4  5C5  0
(2)
6C3  12C4  20C5  0
(3)
Cam Design
Dr. Hany Fayek
15
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
In order to solve Eqs. (1), (2), and (3) simultaneously, you can take ( h  1 )
[i.e. set ( h ) equals unity] and calculate ( C3 ), ( C4 ), and ( C5 ). Then the
determined ( C3 ), ( C4 ), and ( C5 ) are multiplied by ( h ).
Solving Eqs. (1), (2), and (3) simultaneously yields:
C3  10h ; C4  15h ; C5  6h
Therefore;
s  10h     15h     6h   
3
4
5
3
4
5
s  h 10     15     6    


[3-4-5 Polynomial]
The expressions for (velocity) and (acceleration) can be obtained by
substituting the determined values of ( C3 ), ( C4 ), and ( C5 ) into Eqs. (B) and
(C) respectively.
Cam Design
Dr. Hany Fayek
16
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Fall Segment: (Piecewise – segment function)
Boundary Conditions: (at end of High Dwell)
For the follower, the “Fall” segment: starting from the end of the High
Dwell segment
Displacement, Velocity, and Acceleration BCs:
When (   0o )
then s  h ,
v  0,
a0
(a3)
When (    2 )
then s  0 ,
v  0,
a0
(a4)
Notice: For BCs (a3), the s, v, and a values (magnitudes) at the beginning of
the “Fall” segment should match the “end values” of the “High Dwell”
segment.
 As a general rule:
- We would like to minimize the number of segments in our polynomial
cam function.
- We would like to minimize the number of BCs specified, because the
degree of the polynomial is tied to the number of BCs.
Cam Design
Dr. Hany Fayek
17
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Important Points:
- As the degree of the function increases, so will the number of its
inflection points and is the number of minima and maxima.
- A high-degree function may have undesirable oscillations between
BCs.
- The polynomial derivation process will guarantee that the function will
pass through all BCs, but “says nothing about the function’s behavior
between the BCs”.
- High velocity magnitudes result in high kinetic energy magnitudes of
the follower. An important objective is to minimize “stored kinetic
energy” especially with large mass follower trains.
- High acceleration magnitudes result in high dynamic force
magnitudes exerted by the follower on the cam resulting in high friction
leading to cam profile wear and ultimately cam fatigue failure.
- High jerk magnitudes result in undesirable vibrations or (vibratory
behavior) during operation of the follower train.
Cam Design
Dr. Hany Fayek
18
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Comparisons between Different Functions for the Position, Velocity, and
Acceleration for the Rise and Fall Segments:
a- Position Comparison:
Figure 5: Position comparison of different functions within the rise segment of the cam
- Figure 5 illustrates that very slight differences occur for the different
functions listed previously in page 7.
- Care should be taken where small position changes can lead to
considerably large acceleration changes.
b- Velocity Comparison:
Figure 6: Velocity comparison of different functions within the rise segment of the cam
- Figure 6 illustrates that the “Modified Sine” function is the best regarding
cam-follower velocity followed by the “3-4-5 polynomial” function.
Cam Design
Dr. Hany Fayek
19
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
- Low magnitudes of velocity denotes “low kinetic energy” especially for
cam-follower systems possessing large masses.
c- Acceleration Comparison:
Figure 7: Acceleration comparison of different functions within the rise segment of the cam
- Figure 7 illustrates that the “Modified trapezoid” function is the best
followed by the “Modified sine” and the “3-4-5 polynomial” functions.
- Low accelerations imply “low forces” on the cam-follower system.
d- Jerk Comparison:
Figure 8: Jerk comparison of different functions within the rise segment of the cam
- The “Cycloidal” function possesses the lowest jerk magnitude followed
by the “4-5-6-7 polynomial” and the “3-4-5 polynomial” functions.
- “Low jerks” imply “low levels of vibrations”.
Cam Design
Dr. Hany Fayek
20
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Sizing the cam, Terminologies:
- Base Circle ( Rb ): the “smallest circle” that can be drawn tangent to the
physical cam surface. (i.e. the smallest circle possessing maximum
circumferential tangency to the physical cam contour or profile)
- Prime Circle ( R p ): the “smallest circle” that can be drawn tangent to
the locus of the centerline of the follower “Pitch curve”. (i.e. the
smallest circle possessing maximum circumferential tangency to the locus
generated by the follower radius R f )
- Pitch Curve: It is the locus of the centerline of the follower path.
Figure 9: Cam geometric terminologies
Cam Design
Dr. Hany Fayek
21
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
- Pressure Angle (  ): the angle between the “direction of motion”
(velocity) of the follower and the direction of the “axis of transmission”
(common normal).
- For “Translating (Rotating) Followers”:   30o
- For “Oscillating Followers”:   35o
- The cam pressure angle represents a measure of the “steepness of the
cam profile”, which if too large can affect the smoothness of the
action.
Figure 10: Cam pressure angle
Cam Design
Dr. Hany Fayek
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NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
- Cam Eccentricity (  ): It is the perpendicular distance between the
“follower’s axis of motion” and the “center of the cam” (i.e. the center
of the prime circle).
- Eccentricity could be beneficial in changing the “pressure angle”.
   0 for “aligned follower”. “Special case of eccentricity”
Figure 11: Cam eccentricity
Cam Design
Dr. Hany Fayek
23
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
- Overturning Moment: for flat-faced follower, the “pressure angle” is
“zero” (   0o ) since the follower axis of motion is parallel to axis of
transmission (common normal).
- There exists a moment on the follower flat face since the force is not
aligned with the direction of follower motion. This condition results in
and is called “overturning moment”.
Figure 12: Overturning moment experienced by flat faced followers
M  0
Cam Design

Fcam d  Fbearing b
Dr. Hany Fayek
24
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
- Radius of Curvature: every point on the cam has an associated radius
of curvature.
Figure 13: Radius of curvature on cam profile
- If the radius of curvature is smaller than the radius of the follower, the
follower “does not move properly”.
- Rule of thumb: min   2  3  R f
- “Negative radius of curvature” on cam profile is “strictly prohibited”
for “Flat-Faced” followers since undercutting will occur, i.e. the
follower will never be able to follow parts or segments of the cam
contour or profile possessing negative radii of curvatures.
Cam Design
Dr. Hany Fayek
25
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
 Cam Manufacturing Considerations:
Unlike linkages, which are very easy to manufacture, cams are very
challenging to properly manufacture.
Cams are usually made from strong, hard materials such as medium to high
carbon steels (case- or through-hardened) or cast ductile iron or grey cast
iron (case-hardened). These materials range from fairly difficult to very
difficult to machine depending on the alloying elements present.
As a minimum requirement, a reasonably accurate milling machine is needed
to manufacture a cam. A computer controlled machining center is far
preferable and is most often the choice for serious cam production.
Cams are typically milled with rotating cutters that in effect “tear” the metal
away leaving a less than perfectly smooth surface at a microscopic level. For
a better finish and better geometric accuracy, the cam can be “ground” after
milling away most of the unneeded material.
Heat treatment is usually necessary to acquire sufficient hardness in order to
prevent rapid wear. Steel cams are typically hardened to about Rockwell Rc
50-55. However, heat treatment introduces some geometric distortion. The
grinding is usually done after heat treatment to correct the contour as well as
to improve the finish. The grinding step nearly doubles the cost of an already
expensive part, so it is often skipped in order to save money. A hardened
but unground cam will have some heat distortion error despite accurate
milling before hardening.
- Geometric Generation:
Geometric generation refers to the continuous “sweeping out” of a surface as
in turning a cylinder on a lathe. This is perhaps the ideal way to make a cam
because it creates a truly continuous surface with an accuracy limited only
by the quality of the machine and tools used. Unfortunately there are very
few types of cams that can be made by this method. The most obvious one is
the eccentric cam which can be turned and ground on a lathe. A cycloid can
also be geometrically generated. Very few other curves can. The presence of
dwells makes it extremely difficult to apply this method. Thus, it is seldom
used for cams. However, when it can be, as in the case of the eccentric cam,
the resulting acceleration, though not perfect, is very close to the theoretical
cosine wave. This eccentric cam was made by turning and grinding on a
high-quality lathe. This is the best that can be obtained in cam manufacture.
Cam Design
Dr. Hany Fayek
26
NU – MENG 311 – Kinematics and Dynamics of Mechanical Systems
Note that the displacement function is virtually perfect. The errors are only
visible in the more sensitive acceleration function measurement.
- Common CAM Manufacturing Methods:
12345-
Manual machining
Numerical control (NC) machining
Continuous Numerical Control (CNC) with Linear Interpolation (LI)
Continuous Numerical Control (CNC) with Circular Interpolation (CI)
Analogue Duplication on a hand-dressed master CAM
 Practical Design Considerations:
The cam designer is often faced with many confusing decisions, especially at
an early stage of the design process. Many early decisions often made
somewhat arbitrarily and without much thought, can have significant and
costly consequences later in the design. The following considerations shed
light on some of the trade-offs that provide the cam designer with some
guidance in making such decisions:
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Translating or oscillating follower?
Force closure or form closure?
Radial or axial cam?
Roller, mushroom or flat-faced follower?
To dwell or not to dwell?
To grind or not to grind?
To lubricate or not to lubricate?
Cam Design
Dr. Hany Fayek