Driven Harmonic Oscillator
A Damped Harmonic Oscillator is an Harmonic Oscillator that is damped. It is a physical system whose equation of motion satisfies a inhomogeneous second-order linear differential equation with constant coefficients.
- AKA: Forced Vibration.
- Context:
- It can be defined as the simple case of the second-order linear differential equation that describes Harmonic Oscillator motion when an external force [math]\displaystyle{ F(t)=F_0\;cos\;\omega\;t }[/math] is applied:
[math]\displaystyle{ m\frac{d^2x}{dt}+c\frac{dx}{dt}+k\;x=F(t) }[/math]
- Its general solution is of the form
- [math]\displaystyle{ x(t)=\phi(t)+\frac{F_0cos(\omega t-\delta)}{\sqrt{(k-m\omega^2+c^2\omega^2}} }[/math]
- The function [math]\displaystyle{ \phi(t) }[/math] as solution of homogeneous second-order linear differential equation which describes the Damped Harmonic Oscillator
- When the damping factor is removed from the system the solution can be expressed as a sum of periodic functions of different periods:
[math]\displaystyle{ x(t)=a\;cos(\omega_0t)+b\;sin(\omega_0t)+\frac{F_0}{m(\omega_0^2-\omega^2)} }[/math]
- where [math]\displaystyle{ \omega_0=\sqrt{k/m} }[/math] is called the angular frequency. When the angular frequency [math]\displaystyle{ \omega }[/math] of the external force equals the natural angular frequency of the system this corresponds to the resonance case. Consequently, the solution bacames:
[math]\displaystyle{ x(t)=a\;cos(\omega_0t)+b\;sin(\omega_0t)+\frac{F_0}{2m\omega_0}\; sin(\omega_0t) }[/math]
- where the third term represents an oscillation with an increasing amplitude. Thus, when the external force is in resonance with the natural frequency of the system, it will cause unbounded oscillations.
- Example(s):
- Counter-Example(s):
- See: Harmonic Oscillator, Second-Order Linear Differential Equation.
References
2015
- (Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/Harmonic_oscillator#Driven_harmonic_oscillators
- QUOTE: Driven harmonic oscillators are damped oscillators further affected by an externally applied force F(t).
- Newton's second law takes the form
- [math]\displaystyle{ F(t)-kx-c\frac{\mathrm{d}x}{\mathrm{d}t}=m\frac{\mathrm{d}^2x}{\mathrm{d}t^2}. }[/math]
- It is usually rewritten into the form
- [math]\displaystyle{ \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{F(t)}{m}. }[/math]
- This equation can be solved exactly for any driving force, using the solutions z(t) which satisfy the unforced equation:
- [math]\displaystyle{ \frac{\mathrm{d}^2z}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}z}{\mathrm{d}t} + \omega_0^2 z = 0, }[/math]
- and which can be expressed as damped sinusoidal oscillations,
- [math]\displaystyle{ z(t) = A \mathrm{e}^{-\zeta \omega_0 t} \ \sin \left( \sqrt{1-\zeta^2} \ \omega_0 t + \phi \right), }[/math]
- in the case where ζ ≤ 1. The amplitude A and phase φ determine the behavior needed to match the initial conditions.
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
- See: 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/
- See: Chapter 21, Harmonic Oscillator
⇒http://www.feynmanlectures.caltech.edu/I_21.html