Oscillations summary
Oscillations are characterised by an amplitude $$(x_{0})$$ and a period $$(T)$$ (or frequency $$(f)$$).
Angular frequency: $$\omega=2\pi f=2\pi/T$$
Simple harmonic motion (SHM) : periodic and $$a\propto x$$
Equations for SHM

in terms of position :
$$$v=\pm\omega\sqrt{x_{0}^{2}x^{2}} \quad \quad a=\omega^{2}x $$$ 
in terms of energy:
$$$E_{\text{T}}=\frac{1}{2}m\omega^{2}x_{0}^{2}$$$$$$E_{k}=\frac{1}{2}m\omega^{2}(x_{0}^{2}x^{2}) \quad \quad E_{\text{p}}=E_{\text{T}} E_{\text{k}}=\frac{1}{2}m\omega^{2}x^{2}$$$
Damped oscillations:

Light damping: amplitude of oscillation decreases gradually and period is constant.

Critical damping: no oscillation and shortest time to reach the state of rest.

Heavy damping: no oscillation and long time to reach the state of rest.
Forced oscillation: damped oscillation maintained by a driver force.
Resonance : driver frequency=natural frequency $$\Rightarrow$$ peak in the amplitude.