Harmonic Oscillator

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An Harmonic Oscillator is a physical system whose equation of motion satisfies a second-order linear differential equation with constant coefficients.

  • Context:
    • It can be defined as the following
[math]\displaystyle{ m\frac{d^2x}{dt}+c\frac{dx}{dt}+kx=F(t) }[/math]
where [math]\displaystyle{ m,\;c }[/math] and [math]\displaystyle{ k }[/math] are constant coefficients, [math]\displaystyle{ x }[/math] is the displacement function and [math]\displaystyle{ F(t) }[/math] is an externally applied force. The solution of this second-order linear differential equation is usually a superposition of periodic functions and a time-dependent amplitude function

References

2015

[math]\displaystyle{ \vec F = -k \vec x \, }[/math]
where k is a positive constant.
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. Depending on the friction coefficient, the system can:
  • Oscillate with a frequency lower than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator).
  • Decay to the equilibrium position, without oscillations (overdamped oscillator).
The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called "critically damped."
If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.

1992

1963

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