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Resonance Lineshapes of a Driven Damped Harmonic Oscillator

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 Q of the oscillator, that is, the ratio of the oscillator's resonance frequency to its damping constant γ. 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-Q limit when .

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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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