FORCED OSCILLATIONS AND RESONANCE -

FREE OSCILLATIONS

A system is said to execute free oscillation if on being disturbed from its position of equilibrium, it oscillates itself without outside interference.

Its frequency is called natural frequency and is denoted by $\fn_cm \large \nu _0$

$\fn_cm \large i.e \left [ \nu_0=\frac{1}{2\pi}\sqrt{\frac{k}{m}} \right ]$

FORCED OSCILLATIONS

All free oscillations eventually die out because of the ever-present damping forces. However, an external agency can maintain its oscillations. These are called force or driven oscillations.

“When a body is maintained in a state of oscillations by an external periodic force of frequency other than the natural frequency of the body, the oscillations are called forced oscillations.”

We consider the case when the external force is itself periodic with a frequency $\fn_cm \large \omega_d$ called the driven frequency.

A most important fact of forced periodic oscillations is that the system oscillates not with its natural frequency ω, but at frequency $\fn_cm \large \omega_d$ of the external agency because the free oscillations die due to damping.

Let an external periodic force $\fn_cm \large F=F_0 cos\omega_d t$ be applied to the body. Then the total force acting on the oscillator is given by

$\fn_cm \large F=kx-bv+F_0 cos\omega_dt$

$\fn_cm \large \Rightarrow m\frac{d^2 x}{dt^2}=-kx-b\frac{dx}{dt}+F_0 cos\omega_d t$

$\fn_cm \large \left [ \frac{d^2x}{dt^2}+\left ( \frac{b}{m} \right )\frac{dx}{dt}+\left ( \frac{k}{m} \right )x=\left ( \frac{F_0}{m} \right ) cos\omega_d t\right ]$

Which is the differential equation of the oscillator.

The oscillator initially oscillates with its natural frequency ω, when we apply the external periodic force, the oscillations with the natural frequency die out and then the body oscillates with the frequency of the external periodic force.

The solution of the equation is given by

$\fn_cm \large X=X_m cos(\omega_d t+\phi)$

where $\fn_cm \large X_m$ is the amplitude of the forced oscillator and is given by

$\fn_cm \large X_m=\frac{F_0}{\sqrt{m^2 (\omega^2-\omega_d ^2)^2+b^2 \omega_d ^2}} \;\;and\;\; tan\phi=\frac{-v_0}{\omega_d x_0}$

where $\fn_cm \large m$ is the mass of the particle, $\fn_cm \large v_0$ and $\fn_cm \large x_0$ are the velocity and the displacement of the particle at t=0

If damping is small then $\fn_cm \large b^2$ can be neglected

$\fn_cm \large i.e\;\;\;X_m=\frac{F_0}{m(\omega^2-\omega_d^2)}$

If we go changing the driving frequency ( $\fn_cm \large \omega_d$ ), the amplitude tends to ∞ when it is equal to the natural frequency ( $\fn_cm \large \omega$ ). i.e $\fn_cm \large \omega$ = $\fn_cm \large \omega_d$

This phenomenon is known as Resonance.

But this is the ideal case of zero dampings, a case that never arises in a real system as the damping is never perfectly zero.

(The curves in this fig. shows that smaller the damping, the taller and narrower is the resonance peak)