Formula Undamped Harmonic Oscillator Acceleration Amplitude Time Angular frequency Phase
$$a(t) ~=~ -\omega^2 \, A \, \cos(\omega \, t + \varphi)$$ $$a(t) ~=~ -\omega^2 \, A \, \cos(\omega \, t + \varphi)$$ $$A ~=~\frac{ -a(t) }{ \omega^2 \, \cos(\omega \, t + \varphi) }$$ $$t ~=~ \frac{1}{\omega} \, \left[ \arccos\left( -\frac{a(t)}{\omega^2 \, A} \right) ~-~ \varphi \right]$$ $$\varphi ~=~ \arccos\left( -\frac{a(t)}{\omega^2 \, A} \right) ~-~ \omega \, t$$
Acceleration
$$ a(t) $$ Unit $$ \frac{\mathrm{m}}{\mathrm{s}^2} $$ Acceleration of the harmonic oscillator at the time \(t\). This can be, for example, the acceleration of the oscillating mass hanging on a spring.
Amplitude
$$ A $$ Unit $$ \mathrm{m} $$ Maximum deflection of a harmonic oscillator. In the case of a spring pendulum, it is the maximum distance of the mass from the rest position of the spring (undeflected mass).
Time
$$ t $$ Unit $$ \mathrm{s} $$ Time at which the acceleration is \(a(t)\).
Angular frequency
$$ \omega $$ Unit $$ \frac{\mathrm{rad}}{\mathrm s} $$ Angular frequency \(\omega = 2\pi\,f\) describes how fast the harmonic oscillator oscillates.
Phase
$$ \varphi $$ Unit $$ \mathrm{rad} $$ Phase is an angular quantity that determines what acceleration \(a(0)\) the harmonic oscillator had at time \(t=0\):$$ a(0) ~=~ -\omega^2\,A\,\cos(\varphi) $$