follower motion with uniform acceleration and retardation (parabolic motion)

FOLLOWER MOTION WITH UNIFORM ACCELERATION AND RETARDATION (PARABOLIC MOTION)

• In uniform acceleration and retardation motion (UARM), the first half of outward stroke is completed with uniform acceleration and the remaining half of the stroke is with uniform retardation. Similarly, the first half of return stroke is completed with uniform acceleration and the remaining half of the stroke is with uniform retardation.

• The uniform acceleration and retardation motion is also known as parabolic motion.

• The parabolic motion, like SHM, is recommended for follower when cam rotates at low or moderate speeds.

1. Construction of Displacement Diagram

The displacement diagram when the follower moves with UARM is shown in Fig.3.18.

The displacement diagram can be constructed as follows:

Step 1: Divide he angular displacement of the cam during outstroke into any even number of equal parts (say 6) and draw vertical lines through these points.

Step 2: Divide the follower lift into same number of equal parts (say 6) at the centre vertical life and name the intersection points as a, b, c, d, e, and f.

Step 3: Join 0a meeting the first point at A. Similarly 0b, 0c, 0d, 0e and 0f meeting the 2nd 3rd 4th 5th and 6th vertical lines at B, C, D, E and F as shown.

Step 4: Join these points to obtain the parabolic displacement curve for the outstroke of the follower.

Step 5: Follow the same procedure to draw the displacement curve for the return stroke of the follower.

2. Determination of Displacement, Velocity, Acceleration and Jerk of Follower having Parabolic Motion

(i) Displacement of the Follower

At any instant, the displacement equation for the follower having parabolic motion is given by

(ii) Velocity of the Follower

We know that,

Velocity = Rate of change of displacement with respect to time

v = dy/dt

Velocity of follower during outstroke is given by

Maximum velocity of follower during outward and return strokes:

(iii) Acceleration of the Follower

We know that, Acceleration = Rate of change of velocity with respect to time

a = dv/dt

Acceleration of follower during outstroke is given by

Maximum acceleration of follower during outward and return strokes

Since the acceleration is uniform (and is not a function of θ), therefore the value of acceleration is same during both first half and second half of each stroke, but the direction of acceleration in second half is opposite to that of first half acceleration.

(iv) Jerk of the Follower

• Fig.3.19 illustrates the displacement, velocity, acceleration and jerk diagrams when the follower moves with parabolic motion.

• From Fig.3.18, the following points may be observed.

■ The velocity of the follower increases from zero to maximum during uniform acceleration and decreases to zero during retardation.

■ The acceleration of the follower remains constant during both the first and second half of each stroke, but the direction of acceleration in second half is opposite to that of first half acceleration.

■ Because of zero jerk value during both outward and return strokes, the parabolic, motion may appear to be the best cam curve (but actually it is not). But the value of jerk is infinite at the beginning and end of each stroke. This feature of parabolic motion is undesirable and objectionable.

■ The infinite jerk produces abruptly changing contact stresses (at the bearings and on the cam surface) that leads to noise, surface wear, and eventual failure.

■ Due to this reason, the cams with parabolic motion for followers are recommended only for low or moderate speeds.