Peak acceleration of a harmonic oscillator with a sinusoidal driving force

At the natural frequency, for a lightly damped system (\(\zeta \ll 1\)), the peak acceleration is given by the expression

\[
a_{peak} = \frac{F_{0}}{2 \zeta m}
\]

where

\[
\zeta = \frac{c}{2 \sqrt{mk}}
\]

Derivation

The system in Figure 1 is described by the equation

\[\begin{align}
\sum F &= ma \\
F_{0} \sin \omega t - c \frac{dx}{dt} - kx &= m \frac{d^{2}x}{dt^{2}} \\
\frac{d^{2}x}{dt^{2}} + \frac{c}{m}\frac{dx}{dt} + \frac{k}{m} x &= \frac{F_{0}}{m} \sin \omega t
\end{align}\]

which can be expressed as

\[
\frac{d^{2}x}{dt^{2}} + 2 \zeta \omega_{n} \frac{dx}{dt} + w_{n}^{2} x = \frac{F_{0}}{m}\sin \omega t
\]

We note that \(\ref{c}\) is at a maximum where \(\omega = \omega_{n} (1 - 2 \zeta^{2})^{-0.5}\) (see derivation), which we will approximate as \(\omega \approx \omega_{n}\) for small \(\zeta\) using a binomial approximation, which we will substitute into our expression for \(Z_{m}\) in \(\ref{zm}\) to give: