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

Explanation

Amplitude of forced oscillation `A= (F_0)/([m^(2)(omega^(2)-omega_(0)^(2))^(2)+(omega_(d)^(2)b^(2))]^(1/2))`
If `omega_(d)` is very close to `omega`, then `m(omega^(2)-omega_(d)^(2))` is much less than `omega_(d)b` so it is neglected `A= (F_0)/(omega b)`.
Hence, the maximum amplitude for a given driving frequency is governed by the driving frequency and the damping and is never infinity.
Resonance : ..The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is called resonance...