follower motion with simple harmonic motion

FOLLOWER MOTION WITH SIMPLE HARMONIC MOTION

• The simple harmonic motion is recommended for follower motion when cam rotates at low or moderate speeds.

What is meant by SHM?

• When a body rotates on a circular path with uniform angular velocity, its projection on the diameter will have simple harmonic motion (SHM).

• The velocity of the projection will be maximum at the centre and ze at the ends of the diameter.

• In case of acceleration and retardation, the values will be zero at the centre and maximum at the ends of diameter.

• Accordingly, when a follower moves with SHM during its outward stroke and return stroke, its motion will have the same characteristics.

1. Construction of Displacement Diagram

The displacement diagram when the follower moves with SHM is shown in Fig.3.16.

The displacement diagram can be constructed as follows:

Step 1: Draw a semi-circle with lift (L) of the follower as diameter. This circle is also known as harmonic circle.

Step 2: Divide the semi-circle into any number of even equal parts (say 6 or 8).

Step 3: Divide the outstroke angle (0 ̧) and return stroke angle (8,) into the same number of equal parts.

Step 4: Project intercepts on the semi-circle to corresponding divisions on the cam displacement interval. Mark the points of intersection (A, B, C, .....) so obtained.

Step 5: Join the intersection points by free hand with a curve. The diagram so obtained is the displacement curve for SHM.

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

(i) Displacement of the Follower

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

(ii) Velocity of the Follower

We know that,

Velocity, v = Rate of change of displacement with respect to time

v = dy / dt

Velocity of follower during outstroke is given by

Similarly, the velocity of follower during return stroke is given by

Maximum velocity of follower during outward and return strokes:

Since the variation in velocity is a sine curve, therefore the maximum velocity of follower during outward stroke occurs at 

Similarly, the maximum velocity of follower during return stroke occurs at  and is given by

(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 outward stroke is given by

Similarly, acceleration of follower during return stroke is given by

Maximum acceleration of follower during outward and return strokes

Since the variation in acceleration is a cosine curve, therefore the maximum acceleration of follower during outward stroke occurs at θ = 0 and θ = θ。.

Therefore, substituting θ = 0 and θ = θ。in equation (3.21), we get

(iv) Jerk of the Follower

We know that,

Jerk = Rate of change of acceleration with respect to time

Jerk, j = da/dt

Jerk of follower during outward stroke is given by

Similarly, jerk of follower during return stroke is given by

Maximum jerk of follower during outward and return strokes

Since the variation in jerk is a sine curve, therefore the maximum jerk of follower during outward 'stroke occurs at 

• Fig.3.17 illustrates the displacement, velocity, acceleration and jerk diagrams when the follower moves with SHM.

• From Fig.3.17, the following points may be observed:

■ The velocity of the follower is zero at the beginning and at the end of its strokes and increases gradually to a maximum at mid-stroke.

■ On the other hand, the acceleration of the follower is a maximum at the ends of the stroke and decreases to zero at mid-stroke.

■ The jerk of the follower is zero at the beginning and at the end of its stroke and increases gradually to a maximum at mid-stroke. However, there are two infinite jerks at the beginning and end of stroke because a finite value of acceleration is to be generated in no time.

• Because of the presence of infinite jerks, the cam is subjected to large dynamic loads which is not desirable. Hence the cams with SHM for followers are recommended only. for low or moderate speeds.