Displacement, Velocity, and Acceleration Diagrams when the Follower Moves with Uniform Velocity

Displacement, Velocity, and Acceleration Diagrams when the Follower Moves with Uniform Velocity

Displacement, velocity and acceleration diagrams when the knife-edged follower move uniformly velocity are shown in fig (a), (b) and (c). The abscissa shows the time i.e. the number of seconds required for the cam to complete one revolution or it may denote the angular displacement of the cam in degrees. The ordinate represents the displacement, velocity or acceleration of given follower.

As the follower moves with uniform velocity during its rise and return stroke. Therefore slope of displacement curves must constant. In other words, AB₁ and C₁D must be straight lines. A little assumption will show that the follower exists at rest during part of the cam rotation. The periods during which the follower at rest are known to dwell periods as shown by lines B₁C₁ and DE in fig (a). From fig(c). We see that the acceleration or retardation of given follower at the beginning and at the end of each stroke is infinite. This is due to the fact that the follower needs to start from rest and has to gain a velocity within no time. This only occurs if the acceleration or retardation at the beginning and at the end of each stroke is infinite. These conditions are however impracticable.

Fig: Displacement, Velocity, and Acceleration Diagrams when the Follower Moves with Uniform Velocity with modified at right side

In order to have the acceleration and retardation within finite limits, it is required to modify the conditions which govern the motion of the follower. This is done by rounding off the sharp corners of the displacement diagram at the beginning and at the end of each stroke, as shown in fig (a). By doing so, the velocity of the follower increases gradually to its maximum value at beginning of each one stroke and decreases gradually to zero at the end of each stroke as shown in fig (b). The modified displacement, velocity, and acceleration diagrams are shown in fig. The round corners of the displacement diagram are mostly parabolic curves because the parabolic motion results in a very low acceleration of the follower for given strokes and that cam speed.

Displacement, Acceleration, and Velocity Diagrams when the Follower Moves with Simple Harmonic Motion

The displacement, velocity and the acceleration diagrams when the follower moves with simple harmonic motion are shown in fig (a), (b) and (c) respectively. The displacement diagram is drawn as follows

1. Draw a semi-circle on the follower stroke as the diameter.
2. Divide semi-circle into any number of even equal parts i.e. says eight.
3. Divide the angular displacements of the cam at out stroke and return stroke into the same number of equal parts.
4. The displacement diagram is from projecting the points as shown in fig (a).

The velocity and acceleration diagrams are shown in fig (b) and (c) respectively. Since the follower moves with simple harmonic motion. Therefore velocity diagram has a sine curve and the acceleration diagram is a cosine curve. We see from fig (b) that the velocity of the follower is zero at the beginning and at the end of its stroke and increases slowly to a maximum at midstroke. On theother hand, the acceleration of the follower is maximum at the beginning and at ends of that stroke and diminished to zero at midstroke.

Fig:Displacement, velocity and acceleration diagrams when the follower moves with simple harmonic motion

Let,

S=Stroke of the follower,

Θ_o and θ_R = Angular displacement of cam during the out stroke and return stroke of the follower respectively, in radians, and

ω = Angular velocity of the cam in rad/s.

∴Time required for the out stroke of the follower in seconds,

T_o = θ_o /ω

Consider a point P move at a uniform speed ω_p radians per sec round the circumference of a circle with the stroke S as diameter as shown in fig. The point P′ (which is the projection of a point P on the diameter) executes a simple harmonic motion as that point P rotates. The motion of follower is similar to that of point P

′.

Fig: Motion of a point

∴ Peripheral speed of the point P′,

\[{V_p} = \frac{{\pi S}}{2}*\frac{1}{{{t_o}}} = \frac{{\pi S}}{2}*\frac{w}{{{\theta _0}}}\]

And maximum velocity of the follower on the outstroke,

\[{V_o} = {V_p} = \frac{{\pi S}}{2}*\frac{1}{{{t_o}}} = \frac{{\pi S}}{2}*\frac{w}{{{\theta _0}}}\]

We know that the centripetal acceleration of the point P,

\[{a_p} = \frac{{{V_p}^2}}{{OP}} = {\left( {\frac{{\pi wS}}{{2{\theta _o}}}} \right)^2}*\frac{2}{S}\]

∴ Maximum acceleration of the follower on the outstroke,

\[{a_o} = {a_p} = \frac{{{V_p}^2}}{{OP}} = {\left( {\frac{{\pi wS}}{{2{\theta _o}}}} \right)^2}*\frac{2}{S}\]

Similarly, maximum velocity of the follower on the return stroke,

\[{V_R} = \left( {\frac{{\pi wS}}{{2{\theta _R}}}} \right)\]

and maximum acceleration of the follower on the return stroke,

\[{a_R} = {\left( {\frac{{\pi wS}}{{2{\theta _R}}}} \right)^2}*\frac{2}{S}\]

Displacement, Velocity and Acceleration Diagrams when Follower Moves Uniform Acceleration and Retardation

The displacement, velocity and acceleration diagrams when follower moves uniform acceleration and retardation are shown in fig (a), (b) and (c) respectively.

1.Divide the angular displacement of the cam during outstroke (θ_o ) to any even number of equal parts (say eight) and draw vertical lines through these points as shown in fig.

2.Divide the stroke of the follower (S) into the same number of equal even parts.

3. Join Aa to intersect vertical line through the point 1 at B. Similarly, obtain the other points C, D etc. as shown in fig (a). Now join these points to obtain the parabolic curve for the out stroke of the follower.

4. In the similar way, we discussed above the displacement diagram for follower during return stroke may be drawn.

Fig: Displacement, velocity and acceleration diagrams when the follower moves with uniform acceleration and retardation

Since the acceleration and retardation are uniform. Therefore the velocity varies directly with the time. The velocity diagram is shown in fig (b).

Let,

S = Stroke of the follower,

Θ_o and θ_R = Angular displacement of the cam during out stroke and return stroke of follower respectively

ω = Angular velocity of the cam.

We know that time required for the follower during outstroke

t_o = θ_o /ω

Time required for the follower during return stroke

t_R = θ_R/ ω

Mean velocity of follower during outstroke = S/t_o

And mean velocity of follower during return stroke = S/t_R

Since the maximum velocity of the follower is equal to twice the mean velocity. Therefore maximum velocity of the follower during outstroke,

\[{v_{_0}} = \frac{{2S}}{{{t_0}}} = \frac{{2wS}}{{{\theta _0}}}\]

Similarly maximum velocity of the follower during return stroke,

\[{v_R} = \frac{{2wS}}{{{\theta _R}}}\]

We see from the acceleration diagram, as shown in fig (c), that during the first half of outstroke there is uniform acceleration and during the second half of out stroke there is uniform retardation. Thus the maximum velocity of the follower is reached after the time t_o/ 2 and t_R/2.

∴ Maximum acceleration of the follower during outstroke,

\[{a_o} = \frac{{{v_0}}}{{\left( {\frac{{{t_0}}}{2}} \right)}} = \frac{{2*2wS}}{{{t_0}.{\theta _0}}} = \frac{{4{w^2}S}}{{{{\left( {{\theta _0}} \right)}^2}}}\]

Similarly, maximum acceleration of the follower during return stroke,

\[{a_R} = \frac{{4{w^2}S}}{{{{\left( {{\theta _R}} \right)}^2}}}\]

References:
1. H.H. Mabie and C. F. Reinholtz, “Mechanism and Dynamics of Machinery”, Wiley.
2. J.S. Rao & R.V. Dukkipati Mechanisms and Machine Theory, New Age International (P) Limited..
3. J.E. Shigley and J.J. Uicker, Jr., “ Theory of Machines and Mechanisms”, McGraw Hill.
4. B. Paul, “Kinematics and Dynamics of Planar Machinery”, Prentice Hall.
5. C. E. Wilson, J.P. Sadler and W.J. Michels, “Kinematics and Dynamics of Machinery”, Harper Row.