The Art of the Helicopter John Watkinson - Karatunov.net

48 The Art of the Helicopter
Fig. 2.29 Fourier analysis of a square wave into fundamental and harmonics. A, amplitude; δ, phase of
fundamental wave in degrees; 1, first harmonic (fundamental); 2 odd harmonics 3–15; 3, sum of harmonics 1–15;
4, ideal square wave.
To overcome this problem the Fourier analysis searches for each frequency using both
a sine wave and a cosine wave basis function. Thus at each frequency two coefficients
will be obtained. The ratio of the two coefficients can be used to determine the phase
of the frequency component concerned.
The full frequency accuracy of the Fourier transform is only obtained if the averaging
process is performed over a very long, ideally infinite, time. In practice this may cause
difficulties, not least with the amount of computation required, and the averaging
time may need to be reduced. If a short-term average is used, the same result will be
obtained whether the frequency is zero or very low, because a low frequency doesn’t
change very much during the averaging process. Thus the short-term Fourier transform
(STFT) allows quicker analysis at the expense of frequency accuracy. For best frequency
accuracy, the signal has to be analysed for a long time and so the exact time at which
a particular event occurs would be lost. On the other hand if the time when an event
occurs has to be known, the analysis must be over a short time only and so the frequency
analysis will be poor. In the language of transforms, this is known as the Heisenberg
inequality. Werner Heisenberg explained the wave-particle duality of light. When light
is analysed over a short time, its frequency cannot be known, but its location can.
Light can then be regarded as a particle called a photon. When light is analysed over
a long time its frequency can be established but its location is then unknown and so it
is regarded as a wave motion.
In helicopters the frequencies of interest will generally be known from the rotor speed
and usually only the first few harmonics contribute to the aerodynamic result although
higher harmonics may result in vibration. These harmonics are normally sufficiently
far apart in frequency that the time span of the analysis is not critical.
One peculiarity arises with Fourier analysis of helicopter rotors. The behaviour
of a typical rotor is such that the coefficients of the harmonics are always negative.