Damped Harmonic Oscillator

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.

• AKA: Damped Free Vibration.
• Context:
  • It can be defined as the second-order linear differential equation that describes Harmonic Oscillator motion when the frictional force [math]\displaystyle{ F_v(t)=-c\frac{dx}{dt} }[/math] is included:

[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.

• Example(s):
  • Homogeneous Second-Order Linear Differential Equation.
• Counter-Example(s):
  • Simple Harmonic Oscillator.
  • Driven Harmonic Oscillator.
• See: Harmonic Oscillator, Second-Order Linear Differential Equation.

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References

2015

• (Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/Harmonic_oscillator#Damped_harmonic_oscillator
  • The value of the damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be:

• 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]

1992

• (Martin Braun, 1992)) ⇒ Martin Braun (1974, 1977 1982, 1992) "Differential Equations and their Applications", Spring-Verlag New York, Inc. ⇒ http://www.springer.com/us/book/9780387978949
  • Section 2.6, Mechanical Vibrations, pages 165-174

1963

• (Feynman et al., 1963) ⇒ Richard P. Feynman, Robert B. Leighton and Matthew Sands (1963, 1977, 2006, 2010, 2013) "The Feynman Lectures on Physics": New Millennium Edition is now available online by the California Institute of Technology, Michael A. Gottlieb, and Rudolf Pfeiffer ⇒ http://www.feynmanlectures.caltech.edu/
  • Chapter 21, Harmonic Oscillator ⇒http://www.feynmanlectures.caltech.edu/I_21.html

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