Cam design, cam profile

1. Mechanisms of Machinery
(MEng 3071)
Chapter 5: Cams
University of Gondar
Institute of Technology
School of Mechanical & Industrial Engineering
Prepared by: Biniam T.
8 January 2018
1

2. Outline
▀ Classification of followers
▀ Classification of Cams
▀ Type of joint closure
▀ Nomenclature
▀ Type of follower motion
▀ Displacement diagram
▀ Graphical design of cams curves
▀ Analytical cam design
▀ Tangent cam with reciprocating roller
follower 2

3. INTRODUCTION
 A cam is an irregular shaped mechanical member for transmitting a
desired motion to another element known as the follower, by direct
contact.
 A cam may remain stationary, translate or rotate while the follower
may translate or oscillate. They have a line contact and constitute a
higher pair.
 The cams are usually rotated at uniform speed by a shaft, but the
follower motion is predetermined and will be according to the shape
of the cam.
 The cams are widely used for operating
• The inlet and exhaust valves of internal combustion engines,
• Automatic attachment of machineries,
• Paper cutting machines,
• Spinning and weaving textile machineries,
• Feed mechanism of automatic lathes etc. 3

4. Application
 The inlet and exhaust valves of IC engines,
4

5. Classification of follower
 Cams are classified based on:
1) Depending upon the shape of their contacting end with the cam
a) A Knife-edge follower: A sharp knife edge is in contact with the cam. Such followers
produce excessive cam wear, so they are of little practical use.
b) A Roller follower: A cylindrical roller, held by a pin to the follower assembly, is in
contact with the cam. At low speed pure rolling contact is possible but at high speeds
some sliding can also occurs. These type of followers reduce wear of the cam surface at
high peripheral speeds.
c) A flat-faced follower: these types of followers cause high surface stress and to reduce
this stress, the flat face is modified to a spherical surface with a large radius. 5

6. Continue
2) Depending upon the type of motion
3) Depending upon the axis of motion
(a) Offset (b) Radial
(b) Translatory follower(a) Oscillatory follower
6

7. Classification of Cam
 Cams are classified based on:
1. Cam shape as disc, translation, cylindrical cams
2. Follower motion: dwell-rise-dwell-return, dwell-rise-return-dwell
3. Cam constraints as spring or pre-loaded cams and positive return cams
 Some types of cam are
a) Radial cam and flat faced, offset translating follower
b) Radial cam and spherical-faced, oscillating follower
c) Radial (heart) cam and translating, knife edge follower
d) Radial two-lobe frog cam and translating, offset, roller follower
e) Wedge cam and translating roller follower 7

8. Continue
f) Cylindrical cam and oscillating roller follower
g) End or face cam and translating roller follower
h) Yoke cam and translating roller follower
 Cylindrical cam and follower motion
8

9. Type of Joint Closure
 Force Closed Cams: requires a external force to keep the cam in contact with
the follower. A spring usually supplies this force.
 Form Closed Cams: are closed by joint geometry.
(a) Force Closed (b) Form Closed 9

10. Nomenclature
 The following terms are important in order to draw the cam profile.
1) Base circle: It is the smallest circle that can be drawn to the cam profile.
2) Trace point: It is a reference point on the follower and is used to generate the
pitch curve. In case of knife edge follower, the knife edge represents the trace
point and the pitch curve corresponds to the cam profile. In a roller follower, the
center of the roller represents the trace point.
10

11. Nomenclature
3) Pressure angle: It is the angle between the direction of the follower motion
and a normal to the pitch curve. This angle is very important in designing a cam
profile. If the pressure angle is too large, a reciprocating follower will jam in its
bearings. It is an important factor determining the size of the cam.
4) Pitch point: It is a point on the pitch curve having the maximum pressure
angle.
5) Pitch circle: It is a circle drawn from the center of the cam through the pitch
points.
6) Pitch curve: It is the curve generated by the trace point as the follower moves
relative to the cam. For a knife edge follower, the pitch curve and the cam
profile are same whereas for a roller follower, they are separated by the radius
of the roller.
7) Prime circle: It is the smallest circle that can be drawn from the center of the
cam and tangent to the pitch curve. For a knife edge and a flat face follower,
the prime circle and the base circle are identical. For a roller follower, the prime
circle is larger than the base circle by the radius of the roller.
8) Lift or stroke: It is the maximum travel of the follower from its lowest position
to the top most position. 11

12. Type of follower motion
 The follower, during its travel, may have one of the following motions.
1) Uniform velocity,
2) Simple harmonic motion,
3) Uniform acceleration and retardation, and
4) Cycloidal motion.
 The x-axis represents the time (i.e. the number of seconds required for the cam
to complete one revolution) or it may represent the angular displacement of the
cam in degrees.
 The y-axis represents the displacement, or velocity or acceleration of the
follower.
 Since the follower moves with uniform velocity during its rise and return stroke,
therefore the slope of the displacement curves must be constant. In other words,
𝐴𝐵1 and 𝐶1 𝐷 must be straight lines.
 A little consideration will show that the follower remains at rest during part of the
cam rotation. The periods during which the follower remains at rest are known
as dwell periods, as shown by lines 𝐵1 𝐶1 and 𝐷𝐸 in Fig. 1 (a).
12
Displacement, Velocity and Acceleration Diagrams (knife-edged follower)
with Uniform Velocity

13. Type of follower motion
 From Fig. 1 (c), we see that the acceleration or retardation of the follower at
the beginning and at the end of each stroke is infinite. This is due to the fact
that the follower is required to start from rest and has to gain a velocity within
no time.
 This is only possible if the acceleration or retardation at the beginning and at
the end of each stroke is infinite. These conditions are however, impracticable.
 In order to have the acceleration and retardation within the finite limits, it is
necessary to modify the conditions which govern the motion of the follower.
 This may be done by rounding off the sharp corners of the displacement
diagram at the beginning and at the end of each stroke, as shown in Fig. 2 (a).
 By doing so, the velocity of the follower increases gradually to its maximum
value at the beginning of each stroke and decreases gradually to zero at the
end of each stroke as shown in Fig. 2 (b).
 The modified displacement, velocity and acceleration diagrams are shown in
Fig. 2. The round corners of the displacement diagram are usually parabolic
curves because the parabolic motion results in a very low acceleration of the
follower for a given stroke and cam speed.
13

14. Type of follower motion
 Uniform motion can be represented by a simple equation like
𝑦 = 𝑐𝜃
Where:
y= follower displacement corresponding to the cam angle θ
c= constant to be determined form boundary conditions
if
d=the total distance through which the follower is to rise, and 𝛽 =
𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑐𝑎𝑚 𝑖𝑠 𝑡𝑜 𝑟𝑜𝑡𝑎𝑡𝑒 𝑡𝑜 𝑝𝑟𝑜𝑑𝑢𝑐𝑒 𝑡ℎ𝑒 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑟𝑖𝑠𝑒
then
𝑑 = 𝑐𝛽 → 𝑐 = 𝑑/𝛽 therefore the equation that represent the follower displacement is:
𝑦 =
𝑑
𝛽
𝜃 14

15. Type of follower motion
15
 The velocity and acceleration of the follower can be obtained by differentiation.
𝑣 =
𝑑𝑦
𝑑𝑡
=
𝑑
𝛽
𝑑𝜃
𝑑𝑡
=
𝑑
𝛽
𝜔 𝑎𝑛𝑑 𝑎 =
𝑑2 𝑦
𝑑𝑡2 =
𝑑
𝛽
𝑑𝜔
𝑑𝑡
= 0

16. Type of follower motion
 The displacement diagram is drawn as follows :
1) Draw a semi-circle on the follower stroke as diameter.
2) Divide the semi-circle into any number of even equal parts (say eight).
3) Divide the angular displacements of the cam during out stroke and return stroke into the
same number of equal parts.
4) The displacement diagram is obtained by projecting the points as shown in Fig. 3 (a)
 Since the follower moves with a simple harmonic motion, therefore velocity
diagram consists of a sine curve and the acceleration diagram is a cosine curve.
 We see from Fig. 3 (b) that the velocity of the follower is zero at the beginning
and at the end of its stroke and increases gradually to a maximum at mid-stroke.
 On the other hand, the acceleration of the follower is maximum at the beginning
and at the ends of the stroke and diminishes to zero at mid-stroke.
Let s= stroke of the follower,
θo and θR = angular displacement of the cam during out stroke and return
stroke of the follower respectively, in radians, and
ω = angular velocity of the cam in rad/s 16
Displacement, Velocity and Acceleration Diagrams with Simple
Harmonic Motion

17. Type of follower motion
17

18. Type of follower motion
 Time required for the out stroke of the follower in seconds,
𝑡 𝑜 = 𝜃 𝑜 𝜔
 Consider a point P moving at a uniform speed 𝜔 𝑃 radians per sec round the circumference of a
circle with the stroke S as diameter. The point 𝑃′
(w/c is the projection of a point p on the
diameter) executes a simple harmonic motion as the point P rotates. The motion of the follower is
similar to that of point 𝑃′
.
 Peripheral speed of the point 𝑃′
𝑣 𝑃 =
𝜋𝑆
2𝑡 𝑜
=
𝜋𝑆𝜔
2𝜃 𝑜
 And maximum velocity of the follower on the outstroke
𝑣 𝑜 = 𝑣 𝑃 =
𝜋𝑆𝜔
2𝜃 𝑜
 We know that the centripetal acceleration of the point P,
𝑎 𝑃 =
𝑣 𝑃
2
𝑂𝑃
=
𝜋𝑆𝜔
2𝜃 𝑜
2
×
2
𝑆
 Maximum acceleration of the follower on the outstroke
𝑎 𝑜 = 𝑎 𝑃 =
𝜋𝑆𝜔
2𝜃 𝑜
2
×
2
𝑆
18

19. Type of follower motion
 Similarly, maximum velocity of the follower on the return stroke
𝑣 𝑅 =
𝜋𝑆𝜔
2𝜃 𝑅
 Maximum acceleration of the follower on the return stroke
𝑎 𝑅 =
𝜋𝑆𝜔
2𝜃 𝑅
2
×
2
𝑆
Equation: a simple harmonic motion of the follower
(refer your note book)
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20. Type of follower motion
 The displacement diagram consists of a parabolic curve and may be drawn as
discussed below:
1) Divide the angular displacement of the cam during outstroke (𝜃 𝑜) into any even number
of equal parts (say eight) and draw vertical lines through these points as shown in Fig. 4
(a).
2) Divide the stroke of the follower (𝑆) into the same number of equal even parts.
3) Join 𝐴𝑎 to intersect the vertical line through point 1 at 𝐵. Similarly, obtain the other
points 𝐶, 𝐷 etc. as shown in Fig. 4 (a). Now join these points to obtain the parabolic
curve for the out stroke of the follower.
4) In the similar way as discussed above, the displacement diagram for the follower during
return stroke may be drawn.
 Since the acceleration and retardation are uniform, therefore the velocity varies directly with the
time. The velocity diagram is shown in Fig. 4 (b).
Let s= stroke of the follower,
θo and θR = angular displacement of the cam during out stroke and return
stroke of the follower respectively, in radians, and
ω = angular velocity of the cam in rad/s 20
Displacement, Velocity and Acceleration Diagrams with Uniform Acceleration
(Parabolic)

21. Type of follower motion
 We know that time required for the follower during outstroke,
𝑡 𝑜 = 𝜃 𝑜 𝜔
 and time required for the follower during return stroke,
𝑡 𝑅 = 𝜃 𝑅 𝜔
 Mean velocity of the follower during outstroke
= 𝑆 𝑡 𝑜
 and mean velocity of the follower during return stroke
= 𝑆 𝑡 𝑅
 Since the maximum velocity of follower is equal to twice the mean velocity, therefore maximum
velocity of the follower during outstroke,
𝑣 𝑂 =
2𝑆
𝑡 𝑜
=
2𝑆𝜔
𝜃 𝑜
 Similarly, maximum velocity of the follower during return stroke,
𝑣 𝑅 =
2𝑆𝜔
𝜃 𝑅
 We see from the acceleration diagram, as shown in fig. 4(c), that during first half of the out stroke
there is uniform acceleration and during the second half of the out stroke there is uniform
retardation. Thus, the maximum velocity of the follower is reached after the time 𝑡 𝑜 2 (during out
stroke) and 𝑡 𝑅 2 (during return stroke). 21

22. Type of follower motion
22

23. Type of follower motion
 Maximum acceleration of the follower during outstroke,
𝑎 𝑜 =
𝑣 𝑜
𝑡 𝑜 2
=
2 × 2𝜔𝑆
𝑡 𝑜 𝜃 𝑜
=
4𝜔2
𝑆
𝜃 𝑜
2
 Similarly, maximum acceleration of the follower during return stroke,
𝑎 𝑅 =
4𝜔2
𝑆
𝜃 𝑅
2
Equation: a Uniform Acceleration motion of the follower
(refer your note book)
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