Formula Sinusoidally Forced Damped Oscillator Amplitude Force Excitation frequency Natural frequency Mass Damping constant
$$A_0 ~=~ \frac{F_0}{ m \sqrt{ (\omega^2 - {\omega_0}^2)^2 ~+~ \frac{b^2\,\omega^2}{m^2}}}$$ $$A_0 ~=~ \frac{F_0}{ m \sqrt{ (\omega^2 - {\omega_0}^2)^2 ~+~ \frac{b^2\,\omega^2}{m^2}}}$$ $$F_0 ~=~ m \, A_0 \, \sqrt{ (\omega^2 - {\omega_0}^2)^2 ~+~ \frac{b^2\,\omega^2}{m^2}}$$ $$m ~=~ \sqrt{ \frac{ (F_0/A_0)^2 ~-~ (b\,\omega)^2}{ (\omega^2 - {\omega_0}^2)^2 } }$$ $$b ~=~ \frac{1}{\omega} \, \sqrt{ \left(\frac{F_0}{A_0}\right)^2 ~-~ m^2 \, (\omega^2 - {\omega_0}^2)^2 }$$
Amplitude
$$ A $$ Unit $$ \mathrm{m} $$ The maximum displacement of a forced damped oscillation.
Force
$$ F_0 $$ Unit $$ \mathrm{N} $$ Amplitude of an external sinusoidal force with which the oscillator is excited:$$ F_{\text{ext}} ~=~ F_0 \, \cos(\omega\,t) $$
Here \(t\) is the time.
Excitation frequency
$$ \omega $$ Unit $$ \frac{\mathrm{rad}}{\mathrm s} $$ Excitation frequency of an external sinusoidal force:$$ F_{\text{ext}} ~=~ F_0 \, \cos(\omega\,t) $$
Natural frequency
$$ \omega_0 $$ Unit $$ \frac{\mathrm{rad}}{\mathrm s} $$ Frequency at which the oscillator oscillates when it is made to oscillate once.
Mass
$$ m $$ Unit $$ \mathrm{kg} $$ Mass of the damped oscillator. For a spring pendulum, it is in good approximation the mass hanging on the spring. In the case of a spring whose mass cannot be neglected, a part of the mass of the spring must also be taken into account in \(m\) because it also oscillates.
Damping constant
$$ b $$ Unit $$ \frac{\mathrm{kg}}{\mathrm s} $$ The damping constant is a measure of how fast the oscillations decay. Depending on the value of the damping constant, we get a underdamped, critically damped or overdamped oscillator.