$$ \text{Damping forces} \; \Rightarrow \; \text{oppose motion} \; \Rightarrow \; \text{dissipate energy} \; \Rightarrow \; \text{oscillations stop} $$
$$ \ddot{x}+\gamma\dot{x}+\omega_0^2x=0 $$
where $\gamma=\frac{b}{m}$(s$^{-1}$) and $\omega_0^2=\frac{k}{m}$. $\omega_0$(rad s$^{-1}$) is the undamped angular frequency (natural frequency)
<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/feather_1fab6.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/feather_1fab6.png" width="40px" /> Light damping: It is when the $\omega_0^2>\frac{\gamma^2}{4}$, its solution is:
$$ x(t) = A \exp\left( - \frac{\gamma}{2} t\right) \cos ( \omega t + \Phi) $$
Where $A,\Phi$ are constants and $\omega^2=\omega_0^2-\frac{\gamma^2}{4}$ </aside>
$\omega$ is real
$\omega<\omega_0$
if $\gamma=0 \Rightarrow$ undamped case recovered
$A=1, \Phi=0, \omega=1,\gamma=0.2$
<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/man-lifting-weights_1f3cb-fe0f-200d-2642-fe0f.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/man-lifting-weights_1f3cb-fe0f-200d-2642-fe0f.png" width="40px" /> Heavy damping: It is when $\omega_0^2<\frac{\gamma^2}{4}$, its solution is:
$$ x(t) = \exp\left( - \frac{\gamma}{2} t\right) [A e^{\alpha t} + B e^{-\alpha t}] $$
$A=B=0.5,\omega_0=1,\gamma=2.5$
<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/rock_1faa8.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/rock_1faa8.png" width="40px" /> Critical damping: It is when $\omega_0^2=\frac{\gamma^2}{4}$, its solution is:
$$ x(t) = A \exp\left( - \frac{\gamma}{2} t\right) + Bt \exp\left( - \frac{\gamma}{2} t\right), $$
where $A,B$ are constant </aside>
This is where the system returns to equilibrium in the shortest time, without oscillations
$A=1,B=1,\omega_0=1,\gamma=2$
