Resonance Lineshapes of a Driven Damped Harmonic Oscillator

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The plots show (solid lines) the frequency dependence of the amplitude, the phase, the in-phase component, and the quadrature component of a driven damped harmonic oscillator. The variable parameter is the quality factor of the oscillator, that is, the ratio of the oscillator's resonance frequency to its damping constant . The (normalized) lineshapes are presented in dimensionless frequency units, giving the representations a universal character that can be applied to any driven oscillator (mechanics, electronics, optics, etc.). The plots also show (dashed lines) the Lorentzian lineshapes obtained in the high- limit when .

Contributed by: Antoine Weis (University of Fribourg) (March 2011)
Open content licensed under CC BY-NC-SA


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Details

This Demonstration analyzes in which way the (high- limit) Lorentzian lineshapes of a driven damped harmonic oscillator differ from the exact resonance lineshapes.

The equation of motion of a damped harmonic oscillator (with mass , eigenfrequency , and damping constant ) driven by a periodic force is

.

The general solution can be written as

,

where

and .

The solution can thus be parametrized either by the amplitude and phase (||, ) or by the in-phase and quadrature components (. The explicit frequency dependence of those parameters is obtained by inserting the general solution into the equation of motion, yielding

;

The expressions can be rewritten using the dimensionless frequency parameter ξ and the quality factor , defined by , and , to yield

||= ; ;

;.

The four resonance lineshapes are shown in the plots as black solid lines. In order to avoid rescaling during the manipulation of the quality factor , all signals are normalized to their largest value.

The high- limit:

for , that is, for , the expressions can be simplified, yielding

; ;

; ,

or, in dimensionless units,

; ;

; .

The corresponding lineshapes are shown as dashed blue lines.

An alternative parametrization consists in introducing the dimensionless detuning , for which the resonances are given by

; ;

; ,

which shows that the phase and quadrature signals in the high- limit are dispersive and absorptive Lorentzians, respectively.



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