Cams (Numerical problem with reference to roller follower and offset condition)

1. Department of Mechanical Engineering
JSS Academy of Technical Education, Bangalore-560060
Kinematics of Machines
(Course Code:18ME44)

2. Kinematics of Machines
Module 4: Numerical problems on Cams
(Type II Problems)

3. Numerical
Q1
Draw the cam profile, given Follower type = roller follower, off set to the right of cam axis by
18mm; lift = 35mm; base circle radius = 50mm; roller radius = 14mm; out stroke with SHM in
0.05sec; dwell for 0.0125sec; return stroke with UARM, during 0.125sec; dwell for the
remaining period. During return stroke, acceleration is 3/5 times retardation. Determine max.
velocity and acceleration during out stroke and return stroke if the cam rotates at 240 rpm.
Cam speed = 240rpm. Therefore, time for one rotation =
240
60
= 4 rps.
To calculate the angle of outstroke, dwell and return stroke, when it is given in terms of seconds.
Angle of outstroke = o = 0.05 x 4 x 360 = 72
Angle of Dwell = d = 0.0125 x 4 x 360 = 18
Angle of return stroke = r = 0.125 x 4 x 360 = 180
Solution

4. Displacement diagram
Step 1: Construct displacement diagram taking lift (S) = 35 mm on vertical axis and angular displacement of
cam on horizontal axis with suitable scale (in this case, scale 15 = 1 cm) as shown in fig.
Step 2: Outstroke is SHM for 72. Divide the outstroke into equal no. of parts (say 6 parts).
Step 3: For SHM, draw a semi circle on the left end of the displacement diagram taking radius = 17.5 mm
with centre as o
Step 4: Divide the semicircle into 6 equal parts.
o

5. Step 5: Since the return stroke takes place with Uniform Acceleration and Retardation motion.
• Construct the displacement diagram for return stroke by adopting the following procedure.
Given that, during return stroke acceleration is 3/5 times retardation.
To calculate the corresponding angular displacement for acceleration and retardation motion.
Acceleration (a) =
3
5
retardation (r)
 a =
3
5
r ; Also,
𝑎
𝑟
=
3
5
i.e. 5a = 3r and total period = 5+3 = 8
Angular rotation of cam during acceleration period =
𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟𝑖𝑜𝑑
𝑡𝑜𝑡𝑎𝑙 𝑝𝑒𝑟𝑖𝑜𝑑
x angular rotation of cam during return stroke.
 Angular rotation for acceleration period =
5
8
x 180 = 112.5
Similarly, angular rotation for retardation period =
3
8
x 180 = 67.5

6. Displacement diagram for return stroke
Now, divide the angular rotation of cam during return stroke into two parts as shown in fig.
i.e. Acceleration period =112.5 and retardation period = 67.5. 112.5 + 67.5 = 180

7. Displacement diagram for return stroke
Step 6: Now divide the return stroke into equal no. of parts (say 6 parts)
• Here the return stroke consists of acceleration period and retardation period.
• Hence divide the acceleration period into 3 equal parts and retardation period into 3 equal parts,
as shown in fig.

8. Step 5 : To calculate the corresponding displacement or lift for acceleration and retardation motion.
Given that, during return stroke acceleration is 3/5 times retardation and maximum lift = 35 mm
We know that, Acceleration (a) =
3
5
retardation (r)
 a =
3
5
r ; Also,
𝑎
𝑟
=
3
5
i.e. 5a = 3r and total period = 5+3 = 8
Lift of the follower during acceleration period =
𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟𝑖𝑜𝑑
𝑡𝑜𝑡𝑎𝑙 𝑝𝑒𝑟𝑖𝑜𝑑
x Lift
 Lift of follower during, acceleration period =
5
8
x 35 = 21.875 mm  21.88 mm
Similarly , angular rotation for retardation period =
3
8
x 35 = 13.125 mm  13.12 mm

9. Displacement diagram for return stroke
Now, divide the Lift / fall i.e. 35 mm of the follower during return stroke into two parts as shown in fig.
i.e. during acceleration lift / fall = 21.88 mm and during retardation lift / fall = 13.12mm.
 21.88 + 13.12 = 35 mm

10. • Now, locate the mid division on the return stroke, here it is the 3rd division.
• Divide the mid (i.e. 3rd ) division into 6 equal parts as shown in fig.

11. • As the mid (i.e. 3rd ) division is divided into 6 equal parts as shown in fig.
• Now join each division as shown in fig. for UARM during return stroke.

12. Now join the corresponding division with a smooth curve as shown in fig. to get the displacement diagram.

13. Construction of cam profile
1. Draw base circle with centre as O, and radius = 50 mm
2. Draw offset circle with centre as O, and radius = 18 mm
3. Draw prime circle with centre as O, and radius = base circle radius + roller radius = 50 + 14 = 64 mm

14. 4. Draw a vertical tangent to the offset circle towards the right side (since the offset is towards right)
5. This tangent cuts the prime circle at the point A
6. With A as center, Draw the roller taking radius of the roller = 14 mm
7. From the center O, join OA.

15. 8. With OA as reference, Mark of the angle of outstroke, dwell and return stroke
i.e. Outstroke =72, Dwell =18 and return stroke = 180, as shown in fig.

16. 9. Since the acceleration is 3/5 times the retardation, divide the return stroke with reference to acceleration and
retardation period as shown in fig.
i.e. acceleration period = 112.5
retardation period = 67.5

17. 10. Now, divide the outstroke into 6 equal parts and mark all the divisions on the prime circle, as shown in fig.
similarly, divide the return stroke into 6 equal parts. i.e. divide acceleration period into 3 equal parts and
retardation period into 3 equal parts and mark all the divisions on the prime circle as shown in fig.
i.e. acceleration period = 112.5 into 3 equal parts
retardation period = 67.5into 3 equal parts

18. 11. From each division that is marked on the prime circle, join the offset circle in tangential manner as shown in fig.
Also, extend these lines beyond the prime circle as shown in fig.

19. 12. From each division on the prime circle, transfer / mark off the corresponding distance from the displacement
diagram for each division. As shown in fig. ( i.e. 1, 2, 3, … 6)

20. 13.Taking the points 1,2,3…6 as reference, draw the roller taking radius = 14 mm as shown in fig.

21. 14.Now, draw a smooth curve that is tangent to the roller, as shown in fig.to get the cam profile.

22. End of Module