Answer to Question #196816 in Mechanical Engineering for muhammed mudhasir

Question

.A cam with minimum radius of 40 mm, rotating clockwise at a uniform speed is to be designed to

give a roller follower at the end of a valve rod, the motions as given below: 1.To rise the valve through

60 mm during120° rotation of the cam; 2.To keep the valve fully raised through next 30°; 3.To lower

the valve during the next 60°; and 4.To keep the valve closed during the rest of the revolution. The

diameter of the roller is 20 mm and diameter of the cam shaft is 25 mm.The displacement of the

follower has to take place with simple harmonic motion. Draw the profile of the when the line of stroke

is along the axis of the cam shaft.

Answer 1

$Given : S = 40 mm = 0.04 m ; \theta_{ O }=120^{\circ}=2 \pi / 3 rad =2.1 rad ; \theta_{ R }=60^{\circ}=θ O ​ =120 ∘ =2π/3rad=2.1rad;θ R ​ =60 ∘ =\pi / 3 radπ/3rad = 1.047 rad ; N = 100 r.p.m.$

Since the valve is being raised and lowered with simple harmonic motion, therefore the displacement diagram, as shown below, is drawn as discussed in the following steps :

1. Draw horizontal line AX = 360^∘ to some suitable scale. On this line, mark AS = 90^{\circ}

90^∘ to represent out stroke; SR = 30∘ to represent dwell; RP = 60^∘ to represent return stroke and PX = 180^o 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 below.

3. Divide the angular displacement during out stroke 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 simple harmonic motion, draw a semicircle with AY as diameter and divide it 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. This is the required displacement diagram.

(a) Profile of the cam when the line of stroke of the valve rod passes through the axis of the camshaft

The profile of the cam is drawn as discussed in the following steps :

1. Draw a base circle with center O and radius equal to the minimum radius of the cam ( i.e., 25 mm ).

2. Draw a prime circle with center O and radius,

$OA= Min. radius of cam + 2 1 ​ Dia. of roller =25+ 2 1 ​ ×20=35mm$

3. Draw angle AOS = 120^∘

 to represent raising or out stroke of the valve, angle SOT = 30^∘

 to represent dwell and angle TOP = 60^∘

 to represent lowering or return stroke of the valve.

4. Divide the angular displacements of the cam during raising and lowering of the valve (i.e., angle AOS and TOP ) into the same number of equal even parts as in the displacement diagram.

5. Join the points 1, 2, 3, etc., with the center O and produce the lines beyond the prime circle as shown in Fig. 20.17.

6. Set off 1B, 2C, 3D, etc., equal to the displacements from the displacement diagram.

7. Join the points A, B, C … N, P, A. The curve drawn through these points is known as the pitch curve.

8. From the points A, B, C … N, P, draw circles of radius equal to the radius of the roller.

9. Join the bottoms of the circles with a smooth curve. This is the required profile of the cam.

(b) Profile of the cam when the line of stroke is offset 15 mm from the axis of the camshaft

The profile of the cam when the line of stroke is offset from the axis of the camshaft, as shown in Fig. 20.18, maybe drawn as discussed in the following steps :

1. Draw a base circle with center O and radius equal to 25 mm.

2. Draw a prime circle with center O and radius OA = 35 mm.

3. Draw an offset circle with center O and radius equal to 15 mm.

4. Join OA. From OA draw the angular displacements of cam, i.e., draw angle AOS = 120^∘

, angle SOT = 30^∘ and angle TOP =60^∘

5. Divide the angular displacements of the cam during raising and lowering the valve into the same number of equal even parts (i.e., six parts ) as in the displacement diagram.

6. From points 1, 2, 3 …. etc. and 0′,1′, 3′, …etc. On the prime circle, draw tangents to the offset circle.

7. Set off 1B, 2C, 3D… etc., equal to displacements as measured from displacement diagram.

8. By joining the points A, B, C … M, N, P, we get a pitch curve with a smooth curve.

9. Now, A, B, C…etc. as the center, draw circles with radius equal to the radius of roller.

10. Join the bottoms of the circles with a smooth curve. This is the required profile of the cam.

Maximum acceleration of the valve rod

We know that angular velocity of the camshaft,

$\omega=\frac{2 \pi N}{60}=\frac{2 \pi \times 100}{60}=10.47 rad / s$

We also know that maximum velocity of the valve rod to raise valve,

$v_{ O }=\frac{\pi \omega . S}{2 \theta_{ O }}=\frac{\pi \times 10.47 \times 0.05}{2 \times 2.1}=0.39 m / s$

and maximum velocity of the valve rod to lower the valve,

$v_{ R }=\frac{\pi \omega . S}{2 \theta_{ R }}=\frac{\pi \times 10.47 \times 0.05}{2 \times 1.047}=0.785 m / s$

The velocity diagram for one complete revolution of the cam . We know that the maximum acceleration of the valve rod to raise the valve.

$a_{ O }=\frac{\pi^{2} \omega^{2} \cdot S}{2\left(\theta_{0}\right)^{2}}=\frac{\pi^{2}(10.47)^{2} 0.05}{2(2.1)^{2}}=6.13 m / s ^{2}$

and maximum acceleration of the valve rod to lower the valve,

$a_{ R }=\frac{\pi^{2} \omega^{2} \cdot S}{2\left(\theta_{ R }\right)^{2}}=\frac{\pi^{2}(10.47)^{2} 0.05}{2(1.047)^{2}}=24.67 m / s ^{2}$

The acceleration diagram for one complete revolution of the cam