Object: Resonance frequency, amplitude response and phase are calculated.
A vibrating system with one degree of freedom consists of mass, spring constant and damping coefficient.
The system and responses at an excitation frequency are analysed.
First, the natural frequency is calculated.

\(\displaystyle \omega_0 = \sqrt{ \cfrac{k}{m} } \)

\(\displaystyle c_c = 2\sqrt{ mk } \)

\(\displaystyle \zeta = \cfrac{c}{c_c} \)

\( \omega_0 \): Natural angular frequency
\( c_c \): Critical damping
\( \zeta \): Damping ratio
\( m \): Mass
\( c \): Damping coefficient
\( k \): Spring constant

In this function, the excitation is conducted by force.
Thus, rasonance angular frequency, ampitude response and phase are shown as follows;

\(\displaystyle \omega_r = \omega_0 \sqrt{ 1-2\zeta^2} \)

\(\displaystyle A = \cfrac{1}{ \sqrt{ (1- (\cfrac{\omega}{\omega_0})^2 )^2 + 4\zeta^2(\cfrac{\omega}{\omega_0})^2 ) } } \)

\(\displaystyle \phi = \arctan( \cfrac{ 2\zeta\cfrac{\omega}{\omega_0} }{1-(\cfrac{\omega}{\omega_0})^2} ) \)

\( \omega_r \): Resonance angular frequency
\( A \): Amplitude response
\( \phi \): Phase
\( \omega \): Excitation frequency

The damping ratio controls vibrations.
\(\displaystyle \zeta \ge \cfrac{1}{\sqrt{2}} \): Resonance frequency does not exist.

\(\displaystyle \zeta \lt \cfrac{1}{\sqrt{2}} \): Resonance frequency exists.