Explain the behaviour of the oscillator when the driving frequency is far from the natural frequency in small damped oscillations.

(N/A) For small damping,the amplitude $A$ of a forced oscillator is given by $A = \frac{F_0}{\sqrt{m^2(\omega^2 - \omega_d^2)^2 + (\omega_d b)^2}}$.
When the driving frequency $\omega_d$ is far from the natural frequency $\omega$,such that $\omega_d b << m|\omega^2 - \omega_d^2|$,the term $(\omega_d b)^2$ can be neglected in the denominator.
Thus,the amplitude simplifies to $A \approx \frac{F_0}{m|\omega^2 - \omega_d^2|}$.
In this regime,the amplitude is primarily determined by the stiffness and inertia of the system rather than the damping constant $b$. As $\omega_d$ moves further away from $\omega$,the amplitude decreases significantly.
When $\omega_d = \omega$,the amplitude is limited only by the damping term,$A = \frac{F_0}{\omega_d b}$. If $b = 0$,the amplitude becomes infinite at resonance. As damping $b$ increases,the peak amplitude decreases and shifts slightly.