Equivalent mass, $m$
Equivalent stiffness, $k$
Equivalent damping coefficient, $c$
Natural angular frequency, $\omega_n=\sqrt{k/m}$
Damping ratio: this value can be found by a number of methods, as shown below.
Determined by equivalent mass, stiffness, and damping coefficient.
$$ \zeta= \frac{c}{2\sqrt{km}}= \frac{c}{2m\omega_n}= \frac{c\omega_n}{2k} $$
Determined by the log decrement of a free vibration response
$$ \zeta=\frac{\delta}{\sqrt{4\pi^2+\delta^2}},\>\delta=\frac{1}{n-1}\ln\frac{x_1}{x_n} $$
Determined by the half power method from the frequency response
$$ \zeta=\frac{\omega_2-\omega_1}{2\omega_n} $$
Damped angular frequency, $\omega_d=\omega_n\sqrt{1-\zeta^2}$
Frequency ratio, $r=\omega/\omega_n$
<aside> <img src="/icons/cellular_yellow.svg" alt="/icons/cellular_yellow.svg" width="40px" /> ODE for a damped oscillator
$$ m\ddot x+c\dot x+kx=f(t) $$
$$ \ddot x+\frac cm\dot x+\frac kmx=\frac{f(t)}m $$
$$ \ddot x+(2\zeta\omega_n)\dot x+(\omega_n^2)x=u(t) $$
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The homogenous solution depends on whether the damping is sufficient to inhibit the oscillatory component.
<aside> <img src="/icons/cellular_yellow.svg" alt="/icons/cellular_yellow.svg" width="40px" /> Underdamped System, $(\zeta<1)$
$$ x_h=e^{-\zeta\omega_nt}(C_1\cos(\omega_dt)+C_2\sin(\omega_dt)) $$
Critically Damped System, $(\zeta=1)$
$$ x_h=e^{-\zeta\omega_nt}(C_1+C_2t)) $$
Overdamped System, $(\zeta>1)$
$$ x_h=e^{-\zeta\omega_nt}(C_1\cosh(\omega_*t)+C_2\sinh(\omega_t))\>,\>\>\omega_=\omega_n\sqrt{\zeta^2-1} $$
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In this case, a mass normalized force given by $u(t)$ is applied to the mass, having a quadratic profile with respect to time.
$$ u(t)=at^2+bt+c $$
$$ x_p=\frac c{\omega_n^2}+\frac b{\omega_n^2}(t-\frac{2\zeta}{\omega_n})+\frac a{\omega_n^2}(t^2-\frac{4\zeta}{\omega_n}t-\frac 2{\omega_n^2}(1-4\zeta^2)) $$
The particular solution for the special case of a constant force, $mg$, is given as follows.
$$ x_p=\frac{mg}{\omega_n^2} $$