Types of Vibrations:

Free vibrations:

This is also called as natural vibration, this is because when a body is vibrating, no extra energy is added to it. Initial energy is provided to start the vibration which slowly and gradually end up with passage of time as energy of the system is liberated or dissipated. As the body starts vibration it tends to move from mean position to extreme position. When it reaches the extreme position, the restoring forces tries to move it to mean position. Such phenomenon is observed in elastic bodies. The forces are proportional to the displacement of the body from the equilibrium position, the acceleration of the body is also proportional to the displacement and is always directed towards the equilibrium position, so that the body moves with SHM.

The number of degrees of freedom of the system determines the different modes of vibration which the body may posses. The motion that all these examples perform is called SIMPLE HARMONIC MOTION (S.H.M.). Consider a simple pendulum which has two degree of freedom. The mass on the spring could be made to swing like a pendulum.

If we will plot the displacement - time graph for above situation, we will get sinusoidal graph. On this contrary I would like to explain that the displacement may be linear or angular. The linear displacement may be distance moved by mass - spring combination and angular displacement may be witnessed by pendulum. 

 Any vibrating body that has a motion that has to be described should must vibrate with S.H.M. and have the same equations for displacement, velocity and acceleration as given below.

Let us consider the Scotch Yoke wheel. The wheel revolves at w radians/sec and the pin forces the yoke to move up and down. The pin slides in the slot and Point P on the yoke oscillates up and down as it is constrained to move only in the vertical direction by the hole through which it slides. The motion of point P is simple harmonic motion. Point P moves up and down so at any moment it has a displacement x, velocity v and an acceleration a.

The pin is located at radius R from the center of the wheel. The vertical displacement of the pin from the horizontal center line at any time is x. This is also the displacement of point P. The yoke reaches a maximum displacement equal to R when the pin is at the top and –R when the pin is at the bottom.

This is the amplitude of the oscillation. If the wheel rotates at w radian/sec then after time t seconds

the angle rotated is q = wt radians. From the right angle triangle we find x = R Sin(wt) and the graph
of x - q 

Velocity is the rate of change of distance with time. The plot is also shown on figure 3a.

v = dx/dt = wR Cos(wt).

The maximum velocity or amplitude is wR and this occurs as the pin passes through the horizontal

position and is plus on the way up and minus on the way down. This makes sense since the
tangential velocity of a point moving in a circle is v = wR and at the horizontal point they are the
same thing.

Acceleration is the rate of change of velocity with time.  a = dv/dt = -w2

R Sin(-w2R)

The amplitude is w2

R and this is positive at the bottom and minus at the top (when the yoke is about

to change direction)

Since R Sin(wR) = x; then substituting x we find a = -w2x

This is the usual definition of S.H.M. The equation tells us that any body that performs sinusoidal 
motion must have an acceleration that is directly proportional to the displacement and is always directed to the point of zero displacement. The constant of proportionality is w2.

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