3.9: Waveform Analysis
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Harmonic decomposition
As described in appendix 19.9, when superposition applies, then a Fourier series decomposition of the form ??? can be made of any periodic function where
F(t)=N∑n=1αncos(nω0t+ϕn)
or the more general Fourier Transform can be made for an aperiodic function where
F(t)=∫α(ω)cos(ωt+ϕ(ω))dt
Any linear system that is subject to the forcing function F(t) 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 F(t) can be decomposed into a sum of harmonic terms. As a consequence two equivalent representations can be used to describe signals and waves; the first 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.
For example, the superposition of two equal intensity harmonic oscillators in the time domain is given by
y(t)=Acos(ω1t)+Acos(ω2t)=2Acos[(ω1+ω22)t]cos[(ω1−ω22)t]
The free linearly-damped linear oscillator
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 damped waveform for the underdamped case, shown in figure (3.5.1) is given by equation (3.5.12), that is
f(t)=Ae−Γ2tcos(ω1t−δ)t≥0f(t)=0t<0
where ω21=ω20−(Γ2)2 and where ω0 is the angular frequency of the underdamped system. The Fourier transform is given by
G(ω)=ω0(ω2−ω21)2+(Γω)2[(ω2−ω21)−iΓω]
which is complex and has the famous Lorentz form.
The intensity of the wave gives
|f(t)|2=A2e−Γtcos2(ω1t−δ)
|G(ω)|2=ω20(ω2−ω21)2+(Γω)2
Note that since the average over 2π of cos2=12, then the average over the cos2(ω1t−δ) term gives the intensity I(t)=A22e−Γt which has a mean lifetime for the decay of τ=1Γ. The |G(ω)|2 distribution has the classic Lorentzian shape, shown in Figure 3.9.2, which has a full width at half-maximum, FWHM, equal to Γ. Note that G(ω) is complex and thus one also can determine the phase shift δ which is given by the ratio of the imaginary to real parts of Equation 3.9.5, i.e. tanδ=Γω(ω2−ω21).
The mean lifetime of the exponential decay of the intensity can be determined either by measuring τ from the time dependence, or measuring the FWHM Γ=1τ of the Fourier transform |G(ω)|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 τ usually can be measured when τ≳10−12s whereas for shorter lifetimes the radiation width Γ becomes sufficiently large to be measured. Thus the two experimental approaches are complementary.
Damped linear oscillator subject to an arbitrary periodic force
Fourier’s theorem states that any arbitrary forcing function F(t) can be decomposed into a sum of harmonic terms. Consider the response of a damped linear oscillator to an arbitrary periodic force.
F(t)=N∑n=0αnF0(ωn)cos(ωnt+δn)
For each harmonic term ωn the response of a linearly-damped linear oscillator to the forcing function F(t)=F0(ω)cos(ωnt) is given by equation (3.6.18-3.6.20) to be
x(t)Total=x(t)T+x(t)S=F0(ωn)m[e−Γ2tcos(ω1t−δn)+1√(ω20−ω2n)2+(Γωn)2cos(ωnt−δn)]
The amplitude is obtained by substituting into Equation 3.9.11 the derived values F0(ωn)m from the Fourier analysis.
Example 3.9.1: Vibration isolation
Frequently it is desired to isolate instrumentation from the influence of horizontal and vertical external vibrations that exist in its environment. One arrangement to achieve this isolation is to mount a heavy base of mass m on weak springs of spring constant k plus weak damping. The response of this system is given by Equation ??? which exhibits a resonance at the angular frequency ω2R=ω20−2(Γ2)2 associated with each resonant frequency ω0 of the system. For each resonant frequency the system amplifies the vibrational amplitude for angular frequencies close to resonance that is, below √2ω0, while it attenuates the vibration roughly by a factor of (ω0ω)2 at higher frequencies. To avoid the amplification near the resonance it is necessary to make ω0 very much smaller than the frequency range of the vibrational spectrum and have a moderately high Q value. This is achieved by use a very heavy base and weak spring constant so that ω0 is very small. A typical table may have the resonance frequency at 0.5 Hz which is well below typical perturbing vibrational frequencies, and thus the table attenuates the vibration by 99% at 5 Hz and even more attenuation for higher frequency perturbations. This principle is used extensively in design of vibration-isolation tables for optics or microbalance equipment.