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TRANSCRIPT
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Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)
Copyright by Brij Bhooshan @ 2013 Page 1
TThheeoorryy ooff MMeecchhaanniissmm aanndd MMaacchhiinneess
CChhaapptteerr--77 CCaammss
PPrreeppaarreedd BByy
BBrriijj BBhhoooosshhaann
AAsssstt.. PPrrooffeessssoorr
BB.. SS.. AA.. CCoolllleeggee ooff EEnngggg.. AAnndd TTeecchhnnoollooggyy
MMaatthhuurraa,, UUttttaarr PPrraaddeesshh,, ((IInnddiiaa))
SSuuppppoorrtteedd BByy::
PPuurrvvii BBhhoooosshhaann
In This Chapter We Cover the Following Topics
Art. Content Page
7.1 Classification of Followers And Cams 3
7.2 Radial Cam Nomenclature 4
7.3 Description of Follower Movement
Construction of Displacement Diagrams
Uniform Motion
Simple Harmonic Motion
Parabolic or Uniform Acceleration Motion
Cycloidal Motion
5
6
6
7
7
8
7.4 Analysis of Follower Motion
Uniform Motion
Simple Harmonic Motion
Parabolic or Uniform Acceleration Motion
Cycloidal Motion
Advanced Cam Curves
9
9
10
10
11
12
7.5 Determination of Basic Dimensions
Translating Flat-Face Follower
Translating Roller Follower
13
13
14
7.6 Synthesis of Cam Profile
Flat-Face Translating Follower
Translating Roller Follower
Oscillating Flat-Face Follower
Oscillating Roller Follower
16
17
17
18
20
7.7 Cams with Specified Contours
Tangent Cam with Radial-Translating Roller Follower
Circular-Arc Cam with Radial-Translating Flat-Face Follower
21
21
24
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2 Chapter 7: Cams
References:
1. Bevan, T., The Theory of Machines, CBS Publishers and Distributors, 1984.
2. Shigley, J.E., Uicker (Jr.), J.J. and Pennock, G.R. Theory of Machine and Mechanism,
Oxford University Press, New York, 2003.
3. Mallik, A. K., Ghosh, A., Theory of Mechanism and Machines, Affiliated East-West
Press (P) Ltd., New Delhi, 2004.
4. Martin, G.H., Kinematics and Dynamics of Machines, MaGraw-Hill, New York, 1982.
Please welcome for any correction or misprint in the entire manuscript and your
valuable suggestions kindly mail us [email protected].
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3 Theory of Mechanism and Machines By Brij Bhooshan
A cam is a mechanical element used to drive another element, called the follower,
through a specified motion by direct contact. Cam-and-follower mechanisms are simple
and inexpensive, have few moving parts, and occupy a very small space. Furthermore,
follower motions having almost any desired characteristics are not difficult to design.
For these reasons, cam mechanisms are used extensively in modern machinery.
So far, we have studied the mechanisms consisting of only the lower pairs. In this
chapter, we shall discuss the synthesis, analysis, and dynamics of a higher-pair
mechanism. Such a mechanism, known as a cam mechanism, is one of those used most
commonly. In it, the driving member is called the cam, and the driven member is
referred to as the follower. Cam mechanisms can generate complex, coordinated
movements. In general, a cam can be designed in two ways.
(i) the profile of a cam is so designed to give a desired motion to the follower, or
(ii) to choose a suitable profile to ensure a satisfactory performance by follower.
7.1 CLASSIFICATION OF FOLLOWERS AND CAMS
A follower is classified either according to its motion or the nature of its surface in
contact with the cam. The former class has three categories, namely,
(i) the radial-translating follower, where the follower translates along a line passing
through the axis of rotation of the cam,
(ii) the offset-translating follower, where the direction of translation of the follower is
offset from the axis of rotation of the cam in the desired direction, depending on
the direction of rotation of the cam, and
(iii) the oscillating follower, where the follower oscillates about a hinge point as the
cam rotates.
A follower classified according to the nature of its surface in contact with the cam has
four categories. These are:
(i) the knife-edge follower, though simple from the point of view of analysis, is rarely
used because the wear rate is high.
(ii) the flat-face follower, exerts at its bearings a side thrust which is less than that
for the knife-edge and roller followers. This implies reduced friction force and less
chances of jamming in the bearings. This side thrust can be further reduced by
properly offsetting the follower from the axis of rotation of the cam. The sliding
wear in the case of a flat-face translating follower is reduced by offsetting the
follower in a direction perpendicular to the plane of cam rotation so that the
follower rotates about the axis of its translation. The flat-face follower is used in
automobiles, where space is limited.
(iii) the roller follower, is used followers in such situations is restricted by the
minimum size of the pin to be used to connect the roller with the follower. Roller
followers are rather common in larger stationary gas or oil engines. and
(iv) the spherical-face follower.
Cams are classified according to their basic shapes. There are four different types of
cams:
(i) A plate cam, also called a disk cam or a radial cam, are most popular as their
design and manufacture is somewhat simple and many a cam problem can be
solved by using them.
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4 Chapter 7: Cams
(ii) A wedge cam,
(iii) A cylindric cam or barrel cam, profile is not formed on the circumference of a
plate as in the case of disk cams; the profile which acts on the follower is formed
on the surface of a cylinder. The follower moves in a plane parallel tot axis about
which the cam rotates. and
(iv) An end cam or face cam.
In all the cam-follower mechanisms we have stated, the contact between the cam and
the follower is ensured by a spring. But in another type of cams, known as positive-
acting cams, no spring is necessary to bring about the contact.
Diagram 7.1
7.2 RADIAL CAM NOMENCLATURE
Diagram 7.1 shows a radial cam with a radial-translating roller follower. With reference
to this diagram, let us define the various terms we will very frequently use to describe
the geometry of a radial cam.
Base Circle: The base circle is the smallest circle (with its centre at the cam centre) that
can be drawn tangential to the cam profile. The base circle decides the overall size of a
cam and is, therefore, a fundamental feature of the cam.
Trace Point: A trace point is a theoretical point on the follower, its motion describing the
movement of the follower. For a knife-edge follower, the trace point is at the knife-edge
[As only plane motion is being considered, the projection of the cam-follower system on
the plane of motion is sufficient for complete description. So, the projection of the contact
line (i.e., the knife-edge) will be a point.]. For a roller follower, the trace point is at the
roller centre, and for a flat-face follower, it is at the point of contact between the follower
and the cam surface when the contact is along the base circle of the cam. It should be
noted that the trace point is not necessarily the point of contact for all other positions of
the cam.
Pitch Curve: If we apply the principle of inversion, i.e., if we hold the cam fixed and
rotate the follower in a direction opposite to that of the cam, then the curve generated by
the locus of the trace point is called the pitch curve. Obviously, for a knife-edge follower,
the pitch curve and the cam profile are identical.
Pressure Angle: The angle between the direction of the follower movement and the
normal to the pitch curve at any point is referred to as the pressure angle. During a
Pitch curve Cam
rotation
Pressure
angle
Follower motion
Cam profile
Pitch circle
Base circle
Pitch
point
Pitch point
Prime circle
Trace point
O
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5 Theory of Mechanism and Machines By Brij Bhooshan
complete rotation, the pressure angle varies from its maximum to its minimum value.
The greater the pressure angle, the higher will be the side thrust, and consequently the
chances of the translating follower jamming in its guide will increase.
The pressure angle should be as small as possible within the limits of design. In case
of low-speed cam mechanisms with oscillating followers, the highest permissible value of
the pressure angle is 45° whereas it should not exceed 30° in case of cam mechanisms
with translating followers. The pressure angle can be reduced (for a given motion
requirement) by increasing the cam size. However, a bigger cam requires more space
and is more prone to unbalance at high speeds. Another way to control the pressure
angle is by adjusting the offset.
Pitch Point: A pitch point corresponds to the point of maximum pressure angle, and a
circle drawn with its centre at the cam centre, to pass through the pitch point, is known
as the pitch circle.
Prime Circle: The prime circle is the smallest circle that can be drawn (with its centre at
the cam centre) so as to be tangential to the pitch curve. Obviously, for a roller follower,
the radius of the prime circle will be equal to the radius of the base circle plus that of
the roller.
Pitch Circle: It is the circle passing through the pitch point and concentric with the base
circle.
Lift or stroke: It is the maximum travel of the follower from its lowest position to the
topmost position.
Diagram 7.2
7.3 DESCRIPTION OF FOLLOWER MOVEMENT
The cam is assumed to rotate with constant speed and the movement of the follower
during a complete revolution of the cam is described by a displacement diagram, in
which follower displacement y, i.e., the movement of the trace point, is plotted against
the cam rotation θ. Diagram 7.2 shows a typical displacement diagram. The maximum
follower displacement is referred to as the lift L of the follower. It is seen that, in
general, the displacement diagram consists of four parts, namely,
(a) the rise (which is the movement of the follower away from the cam centre),
(b) the dwell (when there is no movement of the follower),
(c) the return (which is now the movement of the follower towards the cam centre),
and
(d) the dwell.
It is always assumed that there is a dwell before and after the rise. The inflexion points
of the displacement diagram (corresponding to the maximum and minimum velocities of
Out stroke
In stroke Dwell
Dwell
C
B
D
A
Rise Dwell Return Dwell
Out stroke In stroke
Cam rotation
Foll
ow
er
dis
pla
cem
en
t
One cycle
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6 Chapter 7: Cams
the follower) correspond to the pitch points. In the case of oscillating followers, the
follower displacement is measured along the arc on which the trace point moves.
Construction of Displacement Diagrams
The rise and return of the follower can take place in many different ways. In this
section, we shall discuss the graphical methods of constructing the displacement
diagrams for the basic follower movements, namely,
(a) uniform motion and its modifications,
(b) simple harmonic motion,
(c) uniform acceleration motion (or parabolic motion) and its modifications, and
(d) cycloidal motion.
The method will be demonstrated for the rise portion of the diagram. A similar
procedure can be adopted for the return movement. Suppose
L = lift of the follower, and θri = angle of cam rotation for the rise phase.
Uniform Motion
By uniform motion, we mean that the velocity of the follower is constant. Since the
follower displacement is from y = 0 to y = L when the cam rotates from θ = 0 to θ = θri, it
will be apparent; that the straight line joining the two points (θ = 0, y = 0) and (θ = θri, y
= L) represents the displacement diagram for uniform motion (Diagram 7.3). The slope
of displacement curve is constant i.e. AB, must be straight line. Similarly if the velocity
is uniform during the return stroke the curve C1D on the displacement diagram must
straight line. As there is an instantaneous change from zero velocity at the beginning of
the rise and a change to zero velocity at the end of the rise, the acceleration of the
follower at these instants will attain a very high value. To avoid this, the straight line of
the displacement diagram is connected tangentially to the dwell at both ends of the rise
by means of smooth curves of any convenient radius, as shown in Diagram 7.3. The bulk
of the displacement takes place at uniform velocity, which is represented by the straight
line in the diagram.
Diagram 7.3
(a) Displacement O B C D E
B1 C1
(b) Velocity
(c) Acceleration
(d) Jerk
(b) Modified
(a) Displacement
O B C D E
B1 C1
(b) Velocity
(c) Acceleration
(d) Jerk
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7 Theory of Mechanism and Machines By Brij Bhooshan
Simple Harmonic Motion
The displacement diagram for simple harmonic motion can be obtained as explained in
Diagram 7.4. The line representing the angle θri is divided into a convenient number of
equal lengths (six divisions are shown in Diagram 7.4). A semicircle of diameter L is
drawn as shown and divided into the same number of circular arcs of equal length.
Horizontal lines are drawn from the points so obtained on the semicircle, to meet the
corresponding vertical lines through the points on the length θri. For simple harmonic
motion, we always have finite velocity, acceleration, jerk, and higher-order derivatives of
displacement.
Diagram 7.4
Parabolic or Uniform Acceleration Motion
With dwell at the beginning and at the end of the rise, when lift of the follower has to
take place in a given time, it is easy to show that the maximum acceleration will be the
least. If the first half of the rise takes place at a constant acceleration and the remaining
displacement is at a constant deceleration (of the same magnitude). This fact makes
parabolic motion very suitable for high-speed cams as it minimizes the maximum inertia
force. The method of constructing the displacement diagram is explained in Diagram
7.5. As in Diagram 7.5, six equal divisions are marked on the line representing the angle
θri. For locating the corresponding six vertical divisions, we make use of the fact that, at
constant acceleration, the displacement is proportional to the square of time (i.e., it is
proportional to the square of the cam rotation as the cam rotates at constant speed) for
the first half. This is also true for the second half of the diagram if the origin is shifted
to the end of the rise.
For cams operating valves of internal-combustion engines, the modified uniform
acceleration motion is used for the follower. It is desired that the valves should open and
close quickly, at the same time maintain the aforementioned advantage of parabolic
(a) Displacement
L
6
5
4
3
2
1
0 1 2 3 4 5 6
(b) Velocity
(c) Acceleration
(d) Jerk
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8 Chapter 7: Cams
motion. In this modified parabolic motion, the acceleration a1 during the first part of the
rise is more than the deceleration a2 during the rest of the rise.
Diagram 7.5
Suppose
Then,
where θa = θri/(1 + K) = angle of cam rotation when the acceleration is a1, and Kθa =
angle of cam rotation with deceleration a2.
The lift L is given by
where L1 = rise with acceleration a1 = L/(l + K), and KL1 = rise with deceleration a2.
The construction of the displacement diagram for this motion is similar to that shown in
Diagram 7.5, and is explained in Diagram 7.5, taking three divisions in each part of the
rise, with an assumed value of K, For uniform acceleration, K = 1. A similar curve is
used for the return phase.
Cycloidal Motion
Cycloidal motion is obtained by rolling a circle of radius L/(2) on the ordinate of the
displacement diagram. A point P on the circle, rolling on the ordinate, describes a
cycloid. A convenient graphical method of constructing the displacement diagram is
shown in Diagram 7.6. A circle of radius L/(2) is drawn with centre at the end A of the
displacement diagram. This circle is divided into the same number of equal divisions (six
divisions, obtained by the radial lines to points 1, 2, ... ,6, are shown in Diagram 7.6) as
the abscissa of the diagram representing the cam rotation θri. The projections of points
1, …, 6 on the circumference are taken on the vertical diameter, represented by points
1',….. 6’. Lines parallel to OA are drawn there from as shown. The displacement
Displacement
L =
K =
2
K =
2
L
Velocity
(a) With modified
Acceleration
(b) Without modified
Jer
k
Displacement
1ʹ 4ʹ 9
4
1
L
Velocity
Acceleration
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diagram is obtained from the intersection of the vertical lines through the points on the
abscissa and the corresponding lines parallel to OA.
Diagram 7.6
7.4 ANALYSIS OF FOLLOWER MOTION
The displacement diagrams, discussed in Section 7.3, give the follower movement y for
any rotation θ of the cam. The displacement y can also be expressed as a function of θ.
Conversely, if the displacement y is given as a function of θ, we can draw the
displacement diagram and thereby obtain the cam profile. Once y is expressed as a
function of θ, the velocity, acceleration, jerk, etc., of the follower can be obtained by
successive differentiation.
The basic follower motions, discussed in Section 7.3, will now be expressed as functions
of θ. The detailed derivation is given here for all motion.
Uniform Motion
Suppose the displacement equation can be represented by
where y represents the displacement corresponding to a cam angle θ and C is constant.
C can be determined from the boundary conditions
At θ = 0; y = 0; and at θ = θri; y = L. Therefore, we have
The velocity and acceleration of the follower can be determined by differentiating Eq.
(7.5), with respect to time.
Now, velocity of the follower is
and acceleration of the follower is
where ω = dθ/dt, the uniform angular velocity of the cam. Also, it is evident that
The velocity and accelerations are shown in Diagram 7.3(a). It can be seen, that very
large inertia forces will act on the cam due to infinite accelerations at the beginning and
Jerk
Acceleration
Displacement
Velocity
(a) Displacement diagram of cycloid motion
3
0ʹ,3ʹ,6ʹ
4ʹ,5ʹ
5
0,6
1
1ʹ,2ʹ
2
4
Cycloid
P
P P
0 1 2 3 4 5 6
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10 Chapter 7: Cams
completion of the rise, even for very small angular velocities. Hence, this motion is not
very practicable.
To avoid such difficulties, sometimes the uniform motion is modified. The simplest
modification used generally, is to use a simple arc at the beginning and completion of
the motion as shown in Diagram 7.3(b). This curve can be used for moderate speeds of
the cam.
Simple Harmonic Motion
The equation of displacement for this simple harmonic motion can be
The velocity, acceleration expressions can be obtained in the usual manner as
Now, velocity of the follower is
and acceleration of the follower is
The rate of change of acceleration, i.e., da/dt, is called jerk and is a useful index of the
quality of the motion.
There is no transition point in this case. Simple harmonic motion is also called cosine
acceleration motion since the acceleration is a cosine curve as given by Eq. (7.11). The
acceleration, is finite, and hence inertia forces can be limited. However there are two
infinite jerks at the beginning and end of the motion, because the acceleration is
suddenly brought to zero from a finite value. This motion is used only for moderate
speeds.
Parabolic Motion
The displacement y can be written, for the first half of the motion i.e. during the
accelerating period of parabolic motion by
where C is a constant. Since y = L/2 when θ = θri/2, we get
Then,
So, the velocity of the follower is
The maximum velocity of the follower (at θ = θri/2) is
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11 Theory of Mechanism and Machines By Brij Bhooshan
and the acceleration of the follower is
During the retardation period of the rise, let
To obtain the three constants C1, C2, and C3, we make use of three conditions, namely
θ = θri, y = L,
θ = θri, = L, (assuming a dwell at the end of the rise),
θ = θri/2, = 2ωL/θri [from (7.16), maintaining the velocity continuity at θ = θri/2].
Thus,
When
So, the velocity of the follower is
and the acceleration of the follower is
The rate of change of acceleration, i.e., da/dt, is called jerk and is a useful index of the
quality of the motion. Assuming dwell at the beginning and the end of the rise, it can be
noted that
The parabolic motion no doubt gives finite acceleration, which means finite inertia
forces, it however introduces three infinite jerks in one motion as indicated. The jerks
will give rise to shock loads and may give rise to undue vibrations and stresses. This
motion can be used where the cam speeds are low and moderate.
Cycloidal Motion
The equation of displacement for this cycloidal motion can be
The velocity, acceleration and jerk expressions can be obtained in the usual manner as
Now, velocity of the follower is
and acceleration of the follower is
and jerk of the follower is
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12 Chapter 7: Cams
As can be seen from Eq. (7.24), this produces sine acceleration, and hence sometimes,
the cycloidal motion is called sine acceleration motion. In this case, the acceleration as
well as jerk are limited to finite values and hence this is much better suited for high
speed cams, than other motions so far discussed.
Advanced Cam Curves
In most situations, the basic follower motions discussed so far are inadequate for smooth
operation. This is particularly so for high-speed operation. One method of rectifying this
deficiency is to combine several portions of these basic curves to obtain the displacement
diagram. While doing this, the velocities and accelerations should be matched at the
junction points. Another method is to represent the basic follower motions by polynomial
curves. Such curves are called advanced cam curves and can be used to approximately
satisfy any requirement, but this involves many computational difficulties. For our
further discussion, the displacement y will be taken to be a polynomial in θ, that is,
The number of terms to be taken is equal to the to satisfy the six boundary conditions,
namely,
we can use terms up to C5. Thus, we get
Using these boundary conditions, we get six linear simultaneous equations, from which
the six constants C0, C1, …., C5 may be obtained,
and finally we get, the displacement equation
This carve is referred to as a 3-4-5 curve since terms of only these orders finally appear
in the polynomial.
Now, velocity of the follower is
and acceleration of the follower is
and jerk of the follower is
These equations show that the acceleration and jerk are finite and hence suitable for
high-speed operation.
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7.5 DETERMINATION OF BASIC DIMENSIONS
To determine the shape of the cam profile for generating a prescribed motion of the
follower, it is necessary to have information about some basic dimensions, viz., the base
circle radius and the offsets. However, in most cases, these parameters are not specified
by the customer and have to be found out by the designer based on certain requirements
on the quality of motion transmission. Certain conditions have also to be satisfied by the
cam surface curvature in order to limit the contact stresses. This section presents the
procedures followed for determining the basic dimensions in a few commonly-used cam-
follower configurations.
Diagram 7.7
Translating Flat-face Follower
A typical cam mechanism with an offset-translating flat-face follower is shown in
Diagram 7.7. The base circle, with a radius rb, touches the cam profile at the point S
and, so, the follower starts lifting when it touches the cam at S. At the instant shown in
the diagram, the cam has rotated by an angle θ in the CCW direction from the position
corresponding to the beginning of the rise. The flat face touches the cam at A where the
profile has a radius of curvature ρ with the centre at C. The offset is e towards the right.
The force exerted (in the direction of follower motion) by the cam on the follower acts at
A and the eccentricity of this driving effort is given by e as indicated in the diagram. As
explained in previous, the velocity and acceleration of the follower at the instant will
remain unchanged if the actual cam is replaced by a cam having a circular profile with a
radius of curvature ρ and the centre at C. It should be further noted that with this cam
(in place of the actual cam at the instant) both ρ and OC remain invariant. Since both
the points S and C are fixed on the body of the cam,
θ + φ = constant
From the diagram,
where y(θ) represents the rise of the follower from its lowermost position (i.e., when it
touches the base circle). Differentiating both sides of (7.32), we obtain
where prime (´) denotes the differentiation with respect to θ.
C
A
S
O
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14 Chapter 7: Cams
Using (7.31) in this equation, we have
Differentiating (7.33) once more with respect to time and again using (7.31), we get
Now, using this equation in (7.32), the expression for the radius of curvature of the cam
profile becomes
If ρmin be the minimum permitted radius of curvature from the contact stress point of
view, then the minimum permissible base circle radius becomes
The minimum value of [y(θ) + y"(θ)] can be found out from the prescribed follower
motion. It be further noted that at no point can the cam profile be concave when a flat-
face follower is used.
The eccentricity of the driving effort, ε, cannot be also allowed to exceed some limiting
value beyond which the follower rod may get jammed in the prismatic guide. From
Diagram 7.7,
Using (7.33) in this relation, we get
If εmax be the maximum permissible eccentricity of the driving effort, then the minimum
required offset
There is no driving effort during return (i.e., y'(0) < θ) and the spring force ensures the
desired movement of the follower. The minimum required width of the follower face can
be determined from (7.37) as
Translating Roller Follower
A cam-follower mechanism with an offset-translating roller follower is shown in
Diagram 7.8. In this case, the force acting on the follower always passes through the
roller centre P (which is also the trace point). Neglecting the effect of friction on the
roller, the force will be normal to the cam profile at the point of contact and, hence, will
pass through the centre of curvature C. The tendency of jamming of the follower in its
guide, in this case, depends on the angle of the driving force with the direction of
follower motion, termed as the pressure angle. This angle ( in the diagram) should not
exceed a permissible limit for a proper functioning of the system. Furthermore, to limit
the contact stress, the radius of curvature of the cam profile should never be less than a
minimum value. It is also obvious that at no point can the cam profile be permitted to be
concave with a radius of curvature less than that of the roller.
The point of intersection, P0, of the path of the trace point and the prime circle indicates
the lowest position of the follower. The rising motion of the follower starts when the
point S coincides with P0. The radius of curvature of the pitch curve PC, at the instant
considered, is denoted by ρp as shown in the figure. So, if ρ be the corresponding radius
of curvature of the cam profile, then
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Diagram 7.8
where rR is the roller radius. If ON be the normal dropped on the line PC from O, then,
from Diagram 7.8,
where θ is the cam rotation from the position when the rising motion of the follower
starts. Since the component of the velocity of the point P along the normal CP has to be
equal [This is because the points P and N are connected by two rigid bodies whose
common normal at the point of contact passes through P and N] to the velocity of the
point N (which is along CP and is equal to ω.ON),
or
Substituting the expression for ON from (7.41) in this equation and rearranging, we get
the relation
The equation (7.43) can be representing graphically in Diagram 7.9. The radius of
curvature of the cam profile can be also found out in the following manner. Referring to
Diagram 7.10,
where ρp = ∠AOC, and
Differentiating this equation with respect to θ (and again considering the actual cam to
be instantaneously replaced by a circular cam with C as the centre and CP as the
radius), we get
But since the lines OS and OC are rigidly attached to the cam, dθ = ‒d and this
equation become;
Roller of radius rR
Pitch curve
P
N
C
A
S
O
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16 Chapter 7: Cams
or
Substituting OC in (7.44) by the RHS of this equation, we get
or
Next, differentiating both sides of (7.43) with respect to θ and rearranging the terms
after some manipulations, we get
Again,
Using (7.43), (7.47), and (7.48), (7.46) yields
Once ρp is determined, (7.40) yields the radius of curvature of the cam profile as
Diagram 7.9 Diagram 7.10
Once the required offset, e, is determined, ρ can be evaluated. It should be checked
whether at any point ρ becomes less than the minimum permissible value based on the
contact stress. Even if no restriction is imposed on ρ from the point of view of the contact
stress, cusp formation should be avoided. Cusp is formed when at a point ρp = rR as
indicated in Diagram 7.10a. Diagram 7.10b shows what happens if ρp < rR.
7.6 SYNTHESIS OF CAM PROFILE
If the displacement of the follower y is expressed as a function of the cam rotation θ, i.e.,
y = f (θ), then the cam profile can be analytically obtained and expressed in the polar
coordinates (rc, θc) in the form of parametric equations, where θc is measured from a
Cam profile
Pitch curve
O
P R
Q
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17 Theory of Mechanism and Machines By Brij Bhooshan
reference line on the cam and rc is the corresponding radial distance of the point on the
cam from the origin. In this section, a generalized approach, applied to different
configurations, is presented.
Flat-face Translating Follower
A cam mechanism with a flat-face translation follower as shown in Diagram 7.11 in
which the axis of follower translation is offset from the cam centre O. since the line OS
is a fixed line on the cam, this can be taken as a reference for measuring θc. So, the
values of θc and the length of the corresponding radius vector rc (OA) define the cam
profile. The simplest way to get the cam profile is to express θc and rc as functions of the
cam rotation θ. From Diagram 7.11,
Diagram 7.11
Using (7.37), this relation gives
Again, from Diagram 7.11,
or
The pair of equations (7.50a) and (7.50b) represents the parametric equations of the cam
profile. By changing θ from 0° to 360°, the polar coordinates of the points on the cam
profile can be found out.
Translating Roller Follower
Diagram 7.12 shows a cam mechanism with an offset-translating follower. In this case,
the roller centre P is the trace point. The lowest position of P, represented by P0, is at
the intersection of the line of translation and the prime circle. The corresponding point
on the pitch curve is S. The point U on the prime circle (corresponding to the start of the
rise period) rotates to V as the cam rotates through an angle θ and the follower rises by
y(θ), as shown in the diagram. So, ∠UOV = θ = ∠ P0OS. The line OV is taken as the
reference and the parametric equations of the pitch curve are given by r (= OP) and θP (=
∠VOP) expressed as functions of θ. Thus,
B
C
A
S
O
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18 Chapter 7: Cams
Diagram 7.12
or
Again,
or
The corresponding point on the cam profile is the point A, where the roller touches the
cam. The parametric equation of the cam profile can be expressed as
where is determined from (7.43).
Oscillating Flat-face Follower
The kinematic view of a cam mechanism with an oscillating flat-face follower is shown
in Diagram 7.13. The flat face of the oscillating follower is at a distance e from the
follower hinge Q whose x- and y-coordinate are a and b, respectively. The cam centre is
chosen as the origin of the coordinate system. The point of tangency of the flat face with
the cam profile is at A at the instant shown in the diagram. At this instant, the follower
face makes an angle with the reference line which is chosen to be parallel to the x-axis.
The position of the follower face at the beginning of the rise period is indicated by the
dashed line making an angle b with the reference line. The point of tangency at this
instant is A0 as shown, which moves to S when the cam rotates through an angle θ.
Correspondingly, the point U (intersection of the cam profile and the y-axis at the
beginning of the rise period) moves to V as indicated. Thus, when the cam rotates
through an angle θ, the oscillating follower rotates through an angle δ(θ) from its lowest
position and
Prime circle
Base circle
Cam profile
U
V
B
Roller
Pitch curve
P
M
C
A
S
O
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19 Theory of Mechanism and Machines By Brij Bhooshan
Diagram 7.13
where δ(θ) is the displacement function of an oscillating follower. It is simple to show
that
Since the components of the velocities of two coincident points at A (one on the cam and
the other being on the follower) along the common normal at A are the same, we can
write
where OM is the perpendicular dropped on the common normal. From (7.53), = δ'(θ).ω
and using this in (7.55), we get
From Diagram 7.13,
and using this in (7.56), we get
The coordinates of the point A can be expressed as
Finally, the polar equation of the cam profile in parametric form can be expressed as
The centre of curvature of the cam corresponding to the point A is at C on the common
normal AM at a distance ρ from A. Taking OC = h, from the diagram, we can write
where ∠UOC = χ. Differentiating (7.60b) with respect to θ and considering ρ and h as
constants (assuming the real cam to be replaced by an instantaneously equivalent
circular cam), we get
because χ' = ‒1. Therefore,
Q
V
S
U
O
M
C
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20 Chapter 7: Cams
Again, differentiating (7.57) with respect to θ, we get (after some algebraic
manipulations)
Substituting l from (7.57), h sin χ from (7.61), and l’ from (7.62) in (7.60a), we can get an
expression for the radius of curvature of the cam profile in the form (after some
manipulations)
Oscillating Roller Follower
The approach followed to determine the cam profile of a mechanism with an oscillating
roller follower is similar to that used in the previous case. Diagram 7.14 shows the
kinematic features of a cam-follower mechanism with an oscillating roller follower. The
lowest position of the follower, making an angle b with the reference line, corresponds
to the situation in which the trace point (the roller centre) lies on the prime circle at P0.
In the position shown, the cam has rotated through an angle θ from the beginning of the
rise period. As in the previous case,
The x-and y-coordinate of the follower hinge are a and b, respectively, and the
coordinates of the trace point P at the instant shown are
Diagram 7.14
Since the components of the velocities of the point along the common normal are the
same.
or
Again, from the diagram,
Roller
Pitch curve
Prime circle
P
A
Q
V
S U
O
M
C
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21 Theory of Mechanism and Machines By Brij Bhooshan
Using this in (7.65), we get
Rearranging the terms in this relation, we finally get
Once is determined, the coordinates of A can be found out as
The polar equations of the cam profile in parametric form become
The basic principle for determining the radius of curvature is the same as that followed
in the previous cases. From Diagram 7.14, we get
where ρp = CP. We also get
Differentiating (7.69b) with respect to B and using χ' = ‒1, we obtain
Substituting h sin χ from this equation into (7.69a), we get
Rearranging the terms in this equation, we get
The radius of curvature of the cam profile
7.7 CAMS WITH SPECIFIED CONTOURS
For the mass production of cams, an iterative design approach is taken rather than the
synthesis approach. A trial cam is designed with a combination of simple curves such as
straight lines, circular arcs, and involutes. These curves are simple from the
manufacturing point of view. The follower movement is analyzed with this trial cam and
modifications are introduced in the cam surface till satisfactory follower movement is
obtained. Once this is achieved, the master cam thus produced is copied for mass
production.
The motion of the follower on two such cams with specified contours is discussed in this
section.
Tangent Cam with Radial-translating Roller Follower
The cam profile shown in Diagram 7.15a consists of two straight lines AB and EF (say,
of length l) which are tangential to the base circle. The portions BU and EV are circular
arcs, each of radius r2, with centres at G and D, respectively. The portion UV is also a
circular arc with the cam centre as its centre. Thus, when the contact is along UV, the
follower will have dwell. The follower movement will be symmetric as the cam profile is
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22 Chapter 7: Cams
symmetric. We will consider the motion of the follower during the rise in two parts,
namely,
(i) when the contact is along the straight flank AB, i.e., 0 ≤ θ ≤ , θ being the
rotation of the cam, and
(ii) when the contact is along the nose BU, i.e., ≤ θ ≤ .
Diagram 7.15
The pitch curve is shown by the dashed line in Diagram 7.15a. The prime circle radius
where rb = base circle radius, and rR = radius of the roller follower. From Diagram 7.15,
we see that
The angles and can be obtained from these equations
Displacement Equation for 0 ≤ θ ≤ :
The rise of the follower is
The velocity of the follower is
where ω = angular velocity of the cam
The acceleration of the follower is
The jerk, of the follower can be similarly obtained.
Displacement Equation for ≤ θ ≤ .:
The distance of the roller centre C' from the point G remains constant during contact
along the nose. So, the motion is equivalent to a slider-crank mechanism with OG as the
crank and GC' as the connecting rod (Diagram 7.15b).
Direction of follower movement
Cam profile
Pitch curve
F
O
R
E V D
G U
B
K H C
A
G
O
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23 Theory of Mechanism and Machines By Brij Bhooshan
The rise of the follower is
where
The lift of the follower is obtained by putting γ = 0 and θ = in (7.75), that is,
The velocity of the follower is obtained by using (7.75), we get
Differentiating (7.76) with respect to θ, we get
Again, using (7.76) to replace cos γ in this equation, we get
Now, from (7.78) and this equation, we have
where OG = (rb ‒ r2) sec and GC' = r2 + rR.
The acceleration of the follower is
Diagram 7.16
D
O Q
G
B
O
D
E
P
G
Q
C A
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24 Chapter 7: Cams
Circular-arc Cam with Radial-translating Flat-face Follower
The cam profile shown in Diagram 7.16a consists of circular arcs of radii rb (base circle
radius), r1, and r2 (nose radius), is the angle of cam rotation during the rise. Here also,
the displacement equation will be derived in two parts, namely,
(i) when the contact is along the circular arc of radius r1 (centre at P), i.e., when
0 ≤ θ ≤ , and
(ii) when the contact is along the circular arc of radius r2 (centre at Q), i.e., when
≤ θ ≤ .
Displacement Equation for 0 ≤ θ ≤ :
From Diagram 7.16a, we see that the rise of the follower is
The velocity of the follower is
The acceleration of the follower is
The jerk, of the follower can be similarly obtained.
Displacement Equation for ≤ θ ≤ α:
From Diagram 7.16b, the rise of the follower is
The lift will be given by
The velocity of the follower is
The acceleration of the follower is
The jerk, of the follower can be similarly obtained.