Free and Forced Oscillations

We all have enjoyed swinging on a swing in our childhood.

And swinging a swing executes an oscillatory motion.

Oscillatory motion is the motion in which the objects do the motion like back and forth motion.

It is the type of periodic motion, in which object repeats their motion after a regualar interval of time.

This oscillation is mainly of two types.

One is free oscillation and another is forced oscillations.

Let's discuss the free and forced oscillations.

Free oscillation is the type of oscillation in which oscillations occur due to its natural frequency.

Means, no external force is acting during the oscillations and oscillations stop after some time.

Suppose a person pushes his friend on a swing and leave, then, the swing stops after some time.

Hence, this is the free oscillation.

Forced oscillation is the type of oscillation in which an external force continuously acts on the body.

And in this case, the body performs oscillations continuously, never stops.

Suppose, a person is continuously pushing his friend on the swing, then the swing will never stop.

Hence, this is the forced/driven oscillations.

Suppose, an external force $F(t)$ of amplitude $F_0$ is applied to a damped oscillator.

And it varies periodically with time $t$.

Then, the forced oscillations can be represented as,

Now, when the motion of a particle is under the combined action of linear restoring force, damping force, and a time-dependent driving force.

Then, the equation can be written as,

And after rearranging, it can be written as,

Hence, this is the equation of the oscillator.

The oscillator initially oscillates with its natural frequency of $\omega$.

But, when we apply external periodic force on the oscillator, then it oscillates with natural frequency ${\omega }_d$.

And its natural frequency will be,

The amplitude is the function of the natural frequency and the driving frequency.

It is given as,

And the phase angle is given by,

Now, there may be two possibilities, when we apply a periodic force on the oscillator.

The first factor is that there may be much difference between the driving frequency and the natural frequency.

And there may be a very less difference between the driving frequency and the natural frequency.

Let's consider, the first case when the driving frequency is far from natural frequency and damping is small.

Then, the term $m({\omega }^2-{\omega }_d^2)$ will be much higher than the term ${\omega }_{d}b$.

Hence, the term ${\omega }_{d}b$ can be neglected.

Hence, it will be as,

Now, we take the second case, when the driving frequency is very close to the natural frequency.

Then, the term $m({\omega }^2-{\omega }_d^2)$ will be much smaller than the term ${\omega }_{d}b$.

Hence, the term $m({\omega }^2-{\omega }_d^2)$ can be neglected.

Hence, we get amplitude as,

Now, the term ${\omega }_{d}b$ is very small. Hence, the amplitude will be very large.

Hence, the phenomenon of increase in amplitude, when the driving frequency is very close to the natural frequency is called resonance.

Therefore, this is all about free and forced oscillations.

Revision

Free oscillation is the type of oscillation in which oscillations occur due to its natural frequency.

Forced oscillation is the type of oscillation in which an external force continuously acts on the body.

The amplitude is the function of the natural frequency and the driving frequency.

When the driving frequency is far from natural frequency and damping is small.

And when the driving frequency is very close to the natural frequency.

The End