Simple Harmonic Oscillator

A Simple Harmonic Oscillator is an Harmonic Oscillator that is neither driven nor damped. It is a physical system whose equation of motion satisfies a homogeneous second-order linear differential equation with constant coefficients.

• AKA: Free Vibration.
• Context:
  • It can be defined as the simple case of the second-order linear differential equation that describes Harmonic Oscillator motion:

[math]\displaystyle{ m\frac{d^2x}{dt}+k\;x=0 }[/math]

• • Its general solution is

[math]\displaystyle{ x(t)=a\;cos\;\omega_0+b\;sin\;\omega_0=A\;cos(\omega_0\;t-\delta) }[/math]

where [math]\displaystyle{ \omega_0=\sqrt{k/m} }[/math] is called the angular frequency (a.k.a natural frequency), [math]\displaystyle{ A=\sqrt{a^2+b^2} }[/math] is called the amplitude and [math]\displaystyle{ \delta=tan^{-1}(b/a) }[/math] is called the phase (a.k.a phase shift, phase angle). This describes periodic motion which repeats over a time interval of length [math]\displaystyle{ T_0=2\pi/\omega_0 }[/math] (this called the period) with a constant amplitude (i.e. [math]\displaystyle{ x(t) }[/math] always lies between [math]\displaystyle{ -A }[/math] and [math]\displaystyle{ +A }[/math])

• Example(s):
  • Simple Pendulum.
• Counter-Example(s):
  • Damped Harmonic oscillator.
  • Driven Harmonic Oscillator.
• See: Harmonic Oscillator, second-order linear differential equation.

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