Explain the behaviour of the oscillator when the driving frequency is close to the natural frequency in small damped oscillation and define resonance.

(N/A) The amplitude $A$ of a forced oscillator is given by the expression:
$A = \frac{F_{0}}{\left[m^{2}(\omega_{0}^{2} - \omega^{2})^{2} + (\omega b)^{2}\right]^{1/2}}$
where $F_{0}$ is the driving force amplitude,$m$ is the mass,$\omega_{0}$ is the natural frequency,$\omega$ is the driving frequency,and $b$ is the damping constant.
When the driving frequency $\omega$ is very close to the natural frequency $\omega_{0}$,the term $m^{2}(\omega_{0}^{2} - \omega^{2})^{2}$ becomes very small compared to $(\omega b)^{2}$.
In this condition,the amplitude is approximately $A \approx \frac{F_{0}}{\omega b}$.
Since the damping constant $b$ is non-zero,the amplitude remains finite and does not reach infinity.
Resonance: The phenomenon where the amplitude of oscillation increases significantly when the driving frequency of an external force is close to the natural frequency of the oscillator is called resonance.