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 ; 3. 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