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Harmonic Oscillation#

Harmonic Oscillation is a special type of periodic motion where the restoring force \(F\) on the moving object is directly proportional to the object's displacement magnitude \(x\) and acts towards the object's equilibrium position. If \(F\) is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude). If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator.

\[F=ma=m{\ddot {x}}=-kx\]

Solving this differential equation, we find that the motion is described by the function

\[x(t)= A \cdot \cos(2\pi f_0 t - \varphi).\]

with the amplitude \(A\), the frequency \(f_0\) and the phase shift \(\varphi\).