Damped Harmonic Oscillator

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A Damped Harmonic Oscillator is an Harmonic Oscillator that is damped. It is a physical system whose equation of motion satisfies a homogeneous second-order linear differential equation with constant coefficients and includes the frictional force.

[math]\displaystyle{ m\frac{d^2x}{dt}+c\frac{dx}{dt}+k\;x=0\quad\iff\quad \frac{d^2x}{dt}+2\zeta\omega_0\frac{dx}{dt}+\omega_0^{2}k\;x=0 }[/math]

where [math]\displaystyle{ \omega_0=\sqrt{k/m} }[/math] is called the undamped angular frequency and [math]\displaystyle{ \zeta = \frac{c}{2 \sqrt{mk}} }[/math] is damping ratio
  • There are three types of solution
[math]\displaystyle{ \zeta\lt 1 \Rightarrow x(t)=ae^{r_1t}+ae^{r_2t} }[/math]
In this case, the system is overdamped. The system will return to a steady state without oscillating.
[math]\displaystyle{ \zeta=1 \Rightarrow x(t)=(a+b)e^{-ct/2m} }[/math]
In this case, the system is critically damped. The system will return to a steady state quickly.
[math]\displaystyle{ \zeta\gt 1 \Rightarrow x(t)=Re^{-c\;t/2m}cos(\omega_1t-\delta) }[/math]
In this case, the system is underdamped. Any initial disturbance of the system is dissipated by the damping present, it oscillates between [math]\displaystyle{ Re^{-c\;t/2m} }[/math] where the amplitude gradually decreasing to zero.


References

2015

  • Overdamped (ζ > 1): The system returns (exponentially decays) to steady state without oscillating. Larger values of the damping ratio ζ return to equilibrium slower.
  • Critically damped (ζ = 1): The system returns to steady state as quickly as possible without oscillating (although overshoot can occur). This is often desired for the damping of systems such as doors.
  • Underdamped (ζ < 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. The angular frequency of the underdamped harmonic oscillator is given by [math]\displaystyle{ \omega_1 = \omega_0\sqrt{1 - \zeta^2}, }[/math] the exponential decay of the underdamped harmonic oscillator is given by [math]\displaystyle{ \lambda = \omega_0\zeta. }[/math]
The Q factor of a damped oscillator is defined as
[math]\displaystyle{ Q = 2\pi \times \frac{\text{Energy stored}}{\text{Energy lost per cycle}}. }[/math]
Q is related to the damping ratio by the equation [math]\displaystyle{ Q = \frac{1}{2\zeta}. }[/math]

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1963