- Simple Harmonic Motion
- Single Spring
Simple Harmonic Motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated SHM) is a special type of periodic motion where the restoring force on the moving object is directly proportional to the magnitude of the object's displacement and acts towards the object's equilibrium position. It results in an oscillation which continues indefinitely, if uninhibited by friction or any other dissipation of energy.
Single Spring
\[F=-kx\]
\[ma=-kx\]
\[a=-\frac{k}{m}x\]
\[a=\frac{d^2x}{dt^2}\]
\[\frac{d^2x}{dt^2}=-\frac{k}{m}x\]
\[x(t)=Acos(\omega t)+B sin(\omega t)\]
\[\dot x (t) =\omega A sin(\omega t)+\omega B cos(\omega t)\]
\[\omega ^2 = \frac{k}{m}\]
\[T=\frac{2\pi}{\omega}\]
\[f=\frac{1}{T}=\frac{\omega}{2\pi}\]
\[\omega=\frac{2\pi}{T}=2\pi f\]
Let \(x(t=0)=x_0\) and \(\dot x(t=0)=\dot x _0 = v_0\)
\[x_0=A, v_0=\omega B\]
\[\omega ={\sqrt {\frac {k}{m}}}\]
\[\frac{d^2x}{dt^2}=-\frac{k}{m}x-\frac{c}{m}v\]
\[\frac{d^2x}{dt^2}+\frac{c}{m}\frac{dx}{dt}+\frac{k}{m}x\]
Tuning
\[v'=v-(\frac{c}{m}v'+\frac{k}{m}(x+v'h))h\]
\[v'+\frac{c}{m}v'h=v+\frac{k}{m}x'h\]
\[v'=v-(\frac{c}{m}v'+\frac{k}{m}(x+v'h))h
\\=v-\frac{c}{m}v'h-\frac{k}{m}xh-\frac{k}{m}v'h^2\]
\[v'=\frac{v-\frac{k}{m}xh}{1+\frac{c}{m}h+\frac{k}{m}h^2}
\]
Useful References
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