3.9.1
Harmonic decomposition
Figure 3.11: The time and frequency rep- resentations of a system exhibiting beats. As described in appendix , when superposition applies, then a
Fourier series decomposition of the form3101 can be made of any periodic function where
() =
X
=1
cos(0+) (3.101)
A more general Fourier Transform can be made for an aperiodic function where
() =
Z
() cos(+()) (3.102) Any linear system that is subject to the forcing function() has an output that can be expressed as a linear superposition of the solutions of the individual harmonic components of the forcing function. Fourier analysis of periodic waveforms in terms of harmonic trigonometric functions plays a key role in describing oscillatory motion in classical mechanics and signal processing for linear systems. Fourier’s theorem states that any arbitrary forcing function()can be decomposed into a sum of harmonic
terms. As a consequence two equivalent representations can be used to describe signals and waves; thefirst is in the time domain which describes the time dependence of the signal. The second is in the frequency domain which describes the frequency decomposition of the signal. Fourier analysis relates these equivalent representations.
Figure 3.12: The intensity ()2 and Fourier transform |()|2 of the free linearly-underdamped harmonic oscillator with0= 10and dampingΓ= 1. For example, the superposition of two equal intensity har-
monic oscillators in the time domain is given by () = cos (1) +cos (2) = 2cos ∙µ 1+2 2 ¶ ¸ cos ∙µ 1−2 2 ¶ ¸ (3.103) which leads to the phenomenon of beats as illustrated for both the time domain and frequency domain infigure 311
3.9.2
The free linearly-damped linear oscilla-
tor
The response of the free, linearly-damped, linear oscillator is one of the most frequently encountered waveforms in science and thus it is useful to investigate the Fourier transform of this waveform. The waveform amplitude for the underdamped case, shown in
figure 35is given by equation(335), that is
() = −Γ2cos (1−) ≥0 (3.104)
() = 0 0 (3.105)
where21=20−¡Γ2¢2 and where0is the angular frequency of the undamped system. The Fourier transform is given by
() = 0 (2−2 1) 2 + (Γ)2 £¡ 2−21¢−Γ¤ (3.106) which is complex and has the famous Lorentz form.
The intensity of the wave gives |()|2 = 2−Γcos2(1−) (3.107) |()|2 = 2 0 (2−2 1) 2 + (Γ)2 (3.108)
Note that since the average over 2 of cos2 = 1
2 then the average over the cos2(1−) term gives the intensity() = 22−Γwhich has a mean lifetime for the decay of= 1
ΓThe|()| 2
distribution has the classic Lorentzian shape, shown infigure312, which has a full width at half-maximum, FWHM, equal toΓ. Note that()is complex and thus one also can determine the phase shiftwhich is given by the ratio of the imaginary to real parts of equation3105i.e. tan= Γ
(2−2 1)
.
The mean lifetime of the exponential decay of the intensity can be determined either by measuring from the time dependence, or measuring the FWHM Γ = 1 of the Fourier transform|()|2. In nuclear and atomic physics excited levels decay by photon emission with the wave form of the free linearly-damped, linear oscillator. Typically the mean lifetime usually can be measured when & 10−12 whereas for shorter lifetimes the radiation widthΓbecomes sufficiently large to be measured. Thus the two experimental approaches are complementary.
3.9.3
Damped linear oscillator subject to an arbitrary periodic force
Fourier’s theorem states that any arbitrary forcing function()can be decomposed into a sum of harmonic terms. Consider the response of a damped linear oscillator to an arbitrary periodic force.
() =
X
=0
0() cos (+) (3.109)
For each harmonic term the response of a linearly-damped linear oscillator to the forcing function
() =0() cos()is given by equation(365−67)to be
() = ()+() = 0() ⎡ ⎣−Γ2cos (1− ) + 1 q (2 0−2) 2 + (Γ)2 cos (−) ⎤ ⎦ (3.110)
The amplitude is obtained by substituting into(3110) the derived values 0() from the Fourier analysis.
3.2
Example: Vibration isolation
Seismic isolation of an optical bench. Frequently it is desired to isolate instrumentation from the
influence of horizontal and vertical external vibrations that exist in the environment. One arrangement to achieve this isolation is to mount a heavy base of mass on weak springs of spring constant plus weak damping. The response of this system is given by equation 3109 which exhibits a resonance at the angu- lar frequency 2 = 20−2
¡Γ 2
¢2
associated with each resonant frequency0of the system. For each resonant frequency the sys- tem amplifies the vibrational amplitude for angular frequencies close to resonance that is, below √20 while it attenuates the vibration roughly by a factor of ¡0
¢2
at higher frequencies. To
avoid the amplification near the resonance it is necessary to make 0 very much smaller than the frequency range of the vibrational spectrum and have a moderately high value. This is achieved by use a very heavy base and weak spring constant so that 0 is very small. A typical table may have the resonance frequency at 05 which is well below typical perturbing vibrational frequencies, and thus the table attenuates the vibration by 99% at 5 and even more attenuation for higher frequency perturbations. This principle is used extensively in design of vibration-isolation tables for optics or microbalance equipment.