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
,
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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