follower motion with uniform velocity (or uniform motion)

FOLLOWER MOTION WITH UNIFORM VELOCITY (OR UNIFORM MOTION)

• The uniform velocity motion, also known as uniform motion, is considered not suitable from practical point of view. However, the discussion about uniform motion is presented here for its academic importance.

1. Construction of Displacement Diagram

• The displacement diagram, when a follower moves with uniform velocity is shown in Fig.3.13.

• Since the velocity of the follower is uniform, therefore the slope of the displacement curve must be constant.

• The constant slope is represented by straight lines BC and DE during the rise and return strokes respectively. CD and EF represent the dwell periods.

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

(i) Displacement of Follower

At any instant, the velocity of the follower having uniform velocity motion is given by

v = constant            .... (3.1)

or

dy/dt = C

By integrating, we get

y = Ct + A

where A is constant of integration.

At t = 0, y = 0.     ⸫ A = 0.

The equations (3.3) and (3.4) imply that follower displacement is directly proportional to cam rotation, during outstroke and return stroke. Since the slope is constant, the displacement curve will be a straight line.

(ii) Velocity of Follower

We know that velocity is the rate of change of displacement with respect to time.

The equations (3.5) and (3.6) imply that the velocity of the follower remains constant during outstroke and return stroke.

Maximum velocity of follower during outward and return strokes

Since the velocity of follower remains constant, the maximum velocity of follower during outward and return strokes are given as

(iii) Acceleration of Follower

• We know that acceleration is the rate of change of velocity with respect to time.

• Thus the acceleration of the follower during outstroke and return stroke is zero.

Maximum acceleration of follower

At the beginning of outstroke, a finite velocity is to be obtained within no time and this requires infinite (+∞) acceleration. Similarly, at the end of the outstroke, a finite velocity is to be brought down to zero within no time and this requires infinite retardation (-∞). In the same way, at the beginning of the return stroke, infinite retardation (-∞) is required, whereas at the end of the return stroke infinite acceleration (+∞) is required.

(iv) Jerk of Follower

We know that,

Jerk = Rate of change of acceleration with respect to time

Since the acceleration is zero during both outward and return strokes, therefore the jerk is also zero during both strokes.

Maximum jerk of follower

Since the acceleration is infinite at the beginning and at the end of the outward and return strokes, therefore the jerk is also infinite at the beginning and at the end of both strokes.

• Fig.3.14 illustrates the displacement, velocity, acceleration and jerk diagrams when a follower moves with uniform velocity.

• The infinite acceleration or retardation at the beginning and at the end of each stroke results in infinite inertia forces (= ma, where a = ∞) that are exerted at the follower bearings and also at the contact point on the cam surface.

• Due to the above drawback, the constant velocity motion is not suitable from practical point of view and hence it is rarely used.

3. Modified Uniform Motion

• In order to achieve the finite values of acceleration and retardation at the beginning and end of each stroke, the uniform motion conditions need to be modified. This can be done by rounding off the sharp corners of the displacement diagram.

• Now the changes in velocity are gradual and hence finite values of acceleration and retardation are obtained at the beginning and end of each stroke.

• The displacement, velocity, and acceleration diagrams for the modified uniform motion of the follower are shown in Fig.3.15.

• Even this modified uniform motion is considered practically not suitable, because the values of acceleration and retardation at the beginning and end of each stroke are still very high.