structural:expansion, structural:summarization, information:extraction
Driven harmonic oscillators

Driven harmonic oscillators are damped oscillators further affected by an externally applied force F (t).

Newton's second law takes the form F (t) − k x − c d x d t = m d 2 x d t 2. {\displaystyle F(t)-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}}=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}.}

It is usually rewritten into the form d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x = F (t) m. {\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {F(t)}{m}}.}

This equation can be solved exactly for any driving force, using the solutions z (t) that satisfy the unforced equation d 2 z d t 2 + 2 ζ ω 0 d z d t + ω 0 2 z = 0, {\displaystyle {\frac {\mathrm {d} ^{2}z}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} z}{\mathrm {d} t}}+\omega _{0}^{2}z=0,}

and which can be expressed as damped sinusoidal oscillations: z (t) = A e − ζ ω 0 t sin (1 − ζ 2 ω 0 t + φ), {\displaystyle z(t)=Ae^{-\zeta \omega _{0}t}\sin \left({\sqrt {1-\zeta ^{2}}}\omega _{0}t+\varphi \right),} in the case where ζ ≤ 1. The amplitude A and phase φ determine the behavior needed to match the initial conditions.