Editing Harmonic oscillator (section)

==== Amplitude part ====
[[Image:Harmonic oscillator gain.svg|thumb|[[Bode plot]] of the frequency response of an ideal harmonic oscillator]]
Squaring both equations and adding them together gives
<math display="block">\left. \begin{aligned}
 A^2 (1-\omega^2)^2 &= \cos^2\varphi \\
 (2 \zeta \omega A)^2 &= \sin^2\varphi
\end{aligned} \right\}
\Rightarrow A^2[(1 - \omega^2)^2 + (2 \zeta \omega)^2] = 1.</math>

Therefore,
<math display="block">A = A(\zeta, \omega) = \sgn \left( \frac{-\sin\varphi}{2 \zeta \omega} \right) \frac{1}{\sqrt{(1 - \omega^2)^2 + (2 \zeta \omega)^2}}.</math>

Compare this result with the theory section on [[resonance]], as well as the "magnitude part" of the [[RLC circuit]]. This amplitude function is particularly important in the analysis and understanding of the [[frequency response]] of second-order systems.