-
Notifications
You must be signed in to change notification settings - Fork 1
Chapter 20
-
A cam is a rotating machine element which gives -
reciprocating or oscillating motion to another element known
-
as follower.
-
Terms Used in Radial Cams -
erms -
Fig. 20.3 shows a radial cam with reciprocating roller follower. 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 centre of the roller represents the trace point.
-
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.
-
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 centre 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 centre 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 -
topmost position.
-
Motion of the Follower -
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.
-
∴ Time required for the out stroke of the follower in seconds, -
tO = θO / ω -
Consider a point P moving at a uniform speed ωP radians per sec round the circumference -
of a circle with the stroke S as diameter, as shown in Fig. 20.7.
-
The point P′ (which is the projection of a point P on the diam-
-
eter) 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′, -
πS 1 πS ω -
vP = × = × -
2 θO -
2 tO -
and maximum velocity of the follower on the outstroke,
-
π S ω πω. S -
vO = vP = × = -
2 θO 2 θO -
We know that the centripetal acceleration of the point P, -
2 -
(vP )2 π ω.S 2 π2 ω2 .S -
aP = = × = -
2 θO S 2 ( θO ) -
2 -
OP -
∴ Maximum acceleration of the follower on the outstroke, -
π 2 ω2 .S -
aO = aP = -
2 ( θO ) 2 -
Similarly, maximum velocity of the follower on the return stroke, -
πω.S -
vR = -
2 θR -
and maximum acceleration of the follower on the return stroke,
-
π2 ω2 .S -
aR = -
2 (θ R ) 2 -
In order to draw the cam profile for a radial cam, first of all the displacement diagram for the -
given motion of the follower is drawn. Then by constructing the follower in its proper position at
-
each angular position, the profile of the working surface of the cam is drawn.
-
In constructing the cam profile, the principle of kinematic inversion is used, i.e. the cam is -
imagined to be stationary and the follower is allowed to rotate in the opposite direction to the cam
-
rotation.
-
The construction of cam profiles for different types of follower with different types of -
motions are discussed in the following examples.
-
Example 20.1. A cam is to give the following motion to a knife-edged follower : -
1. Outstroke during 60° of cam rotation ; 2. Dwell for the next 30° of cam rotation ; -
- Return stroke during next 60° of cam rotation, and 4. Dwell for the remaining 210° of cam
-
rotation.
-
The stroke of the follower is 40 mm and the minimum radius of the cam is 50 mm. The -
follower moves with uniform velocity during both the outstroke and return strokes. Draw the pro-
-
file of the cam when (a) the axis of the follower passes through the axis of the cam shaft, and
-
(b) the axis of the follower is offset by 20 mm from the axis of the cam shaft.
-
785 -
l -
Chapter 20 : Cams -
Construction -
Fig. 20.10 -
First of all, the displacement diagram, as shown in Fig. 20.10, is drawn as discussed in the -
following steps :
-
1. Draw a horizontal line AX = 360° to some suitable scale. On this line, mark AS = 60° to -
represent outstroke of the follower, ST = 30° to represent dwell, TP = 60° to represent -
return stroke and PX = 210° to represent dwell. -
2. Draw vertical line AY equal to the stroke of the follower (i.e. 40 mm) and complete the -
rectangle as shown in Fig. 20.10. -
3. Divide the angular displacement during outstroke and return stroke into any equal number -
of even parts (say six) and draw vertical lines through each point. -
4. Since the follower moves with uniform velocity during outstroke and return stroke, there- -
fore the displacement diagram consists of straight lines. Join AG and HP. -
5. The complete displacement diagram is shown by AGHPX in Fig. 20.10. -
(a) Profile of the cam when the axis of follower passes through the axis of cam shaft
-
The profile of the cam when the axis of the follower passes through the axis of the cam shaft, -
as shown in Fig. 20.11, is drawn as discussed in the following steps :
-
Fig. 20.11 -
786 l Theory of Machines
-
1. Draw a base circle with radius equal to the minimum radius of the cam (i.e. 50 mm) with -
O as centre. -
2. Since the axis of the follower passes through the axis of the cam shaft, therefore mark -
trace point A, as shown in Fig. 20.11. -
3. From OA, mark angle AOS = 60° to represent outstroke, angle SOT = 30° to represent -
dwell and angle TOP = 60° to represent return stroke. -
4. Divide the angular displacements during outstroke and return stroke (i.e. angle AOS and -
angle TOP) into the same number of equal even parts as in displacement diagram. -
5. Join the points 1, 2, 3 ...etc. and 0 ′ , 1′ , 2′ , 3′ , ... etc. with centre O and produce beyond -
the base circle as shown in Fig. 20.11. -
6. Now set off 1B, 2C, 3D ... etc. and 0′ H, 1′ J ... etc. from the displacement diagram. -
7. Join the points A, B, C,... M, N, P with a smooth curve. The curve AGHPA is the complete -
profile of the cam. -
Notes : The points B, C, D .... L, M, N may also be obtained as follows :
-
1. Mark AY = 40 mm on the axis of the follower, and set of Ab, Ac, Ad... etc. equal to the distances 1B, -
2C, 3D... etc. as in displacement diagram.
-
2. From the centre of the cam O, draw arcs with radii Ob, Oc, Od etc. The arcs intersect the produced -
lines O1, O2... etc. at B, C, D ... L, M, N.
-
(b) Profile of the cam when the axis of the follower is offset by 20 mm from the axis of the cam
-
shaft -
The profile of the cam when the axis of the follower is offset from the axis of the cam shaft, -
as shown in Fig. 20.12, is drawn as discussed in the following steps :
-
Fig. 20.12 -
1. Draw a base circle with radius equal to the minimum radius of the cam (i.e. 50 mm) with -
O as centre. -
2. Draw the axis of the follower at a distance of 20 mm from the axis of the cam, which -
intersects the base circle at A. -
3. Join AO and draw an offset circle of radius 20 mm with centre O. -
4. From OA, mark angle AOS = 60° to represent outstroke, angle SOT = 30° to represent -
dwell and angle TOP = 60° to represent return stroke. -
787 -
l -
Chapter 20 : Cams -
5. Divide the angular displacement during outstroke and return stroke (i.e. angle AOS and -
angle TOP) into the same number of equal even parts as in displacement diagram. -
6. Now from the points 1, 2, 3 ... etc. and 0′,1′, 2′, 3′ ... etc. on the base circle, draw tangents -
to the offset circle and produce these tangents beyond the base circle as shown in Fig. -
20.12. -
7. Now set off 1B, 2C, 3D ... etc. and 0 ′ H, 1′ J ... etc. from the displacement diagram. -
8. Join the points A, B, C ...M, N, P with a smooth curve. The curve AGHPA is the complete -
profile of the cam. -
Example 20.2. A cam is to be designed for a knife edge follower with the following data : -
1. Cam lift = 40 mm during 90° of cam rotation with simple harmonic motion. -
2. Dwell for the next 30°. -
3. During the next 60° of cam rotation, the follower returns to its original position with -
simple harmonic motion. -
4. Dwell during the remaining 180°. -
Draw the profile of the cam when -
(a) the line of stroke of the follower passes through the axis of the cam shaft, and -
(b) the line of stroke is offset 20 mm from the axis of the cam shaft. -
The radius of the base circle of the cam is 40 mm. Determine the maximum velocity and -
acceleration of the follower during its ascent and descent, if the cam rotates at 240 r.p.m.
-
Solution. Given : S = 40 mm = 0.04 m; θO = 90° = π /2 rad = 1.571 rad ; θR = 60° = -
π /3 rad = 1.047 rad ; N = 240 r.p.m.
-
Fig. 20.13 -
First of all, the displacement diagram, as shown in Fig 20.13, is drawn as discussed in the -
following steps :
-
1. Draw horizontal line AX = 360° to some suitable scale. On this line, mark AS = 90° to -
represent out stroke ; SR = 30° to represent dwell ; RP = 60° to represent return stroke -
and PX = 180° to represent dwell. -
2. Draw vertical line AY = 40 mm to represent the cam lift or stroke of the follower and -
complete the rectangle as shown in Fig. 20.13. -
3. Divide the angular displacement during out stroke and return stroke into any equal num- -
ber of even parts (say six) and draw vertical lines through each point. -
4. Since the follower moves with simple harmonic motion, therefore draw a semicircle with -
AY as diameter and divide into six equal parts. -
5. From points a, b, c ... etc. draw horizontal lines intersecting the vertical lines drawn through -
1, 2, 3 ... etc. and 0 ′ , 1′ , 2′ ...etc. at B, C, D ... M, N, P. -
6. Join the points A, B, C ... etc. with a smooth curve as shown in Fig. 20.13. This is the -
required displacement diagram. -
788 l Theory of Machines
-
(a) Profile of the cam when the line of stroke of the follower passes through the axis of the cam
-
shaft -
The profile of the cam when the line of stroke of the follower passes through the axis of the -
cam shaft, as shown in Fig. 20.14, is drawn in the similar way as is discussed in Example 20.1.
-
Fig. 20.14 -
(b) Profile of the cam when the line of stroke of the follower is offset 20 mm from the axis
-
of the cam shaft -
The profile of the cam when the line of stroke of the follower is offset 20 mm from the axis -
of the cam shaft, as shown in Fig. 20.15, is drawn in the similar way as discussed in Example 20.1.
-
Fig. 20.15 -
789 -
l -
Chapter 20 : Cams -
Maximum velocity of the follower during its ascent and descent
-
We know that angular velocity of the cam, -
2π N 2π × 240 -
ω= = = 25.14 rad/s -
60 60 -
We also know that the maximum velocity of the -
follower during its ascent,
-
πω.S π× 25.14 × 0.04 -
vO = = = 1 m/s Ans. -
2θO 2 ×1.571 -
and maximum velocity of the follower during its
-
descent,
-
πω.S π× 25.14 × 0.04 -
vR = = = 1.51 m/s Ans. -
2θR 2 ×1.047 -
Maximum acceleration of the follower during its
-
ascent and descent
-
Role of cams in piston movement. -
We know that the maximum acceleration of the -
follower during its ascent,
-
π2 ω2 .S π2 (25.14) 2 0.04 -
aO = = = 50.6 m/s2 Ans. -
2 ( θO ) 2 2 (1.571) 2 -
and maximum acceleration of the follower during its descent,
-
π 2 ω2 .S π 2 (25.14) 2 0.04 -
aR = = = 113.8 m/s2 Ans. -
2 ( θR ) 2 2 (1.047) 2 -
S = Stroke of the follower,dia of the circle -
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. -
tO = θO / ω