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Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)

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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



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


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




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].




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.




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





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




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






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


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




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





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




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.


The equation of displacement for this simple harmonic motion can be

The velocity, acceleration expressions can be obtained in the usual manner as



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





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



and jerk of the follower is




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.



and jerk of the follower is

These equations show that the acceleration and jerk are finite and hence suitable for

high-speed operation.





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




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


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





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




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





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.


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




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





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




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





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




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





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.


Diagram 7.16

D

O Q

G

B

O

D

E

P

G

Q

C A




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




Displacement Equation for ≤ θ ≤ α:

From Diagram 7.16b, the rise of the follower is

The lift will be given by




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