Frequency higher harmonic Fourier transforms

Generally, the FR parameters can be derived for the equivalent fundamental sine-wave perturbations from the first harmonic Fourier transformation of the input and the pressure response signals. The higher harmonics can, also, be used to extend the experimental frequency range by a factor of n (n > 1, where n is an odd number) when high quality response data are available [9,37,39]. 

If a metal has more than one extremal area of the FS the usual way to de-convolute the different dHvA frequencies is done by a Fourier transformation with respect to l/B. This is, in addition, the easiest way to resolve higher harmonics r of the dHvA signal. 

The combustor is naturally unstable under certain operating conditions. Figure 16.7 shows combustor pressure oscillations and the Fast Fourier Transform (FFT) spectrum under atypical, unstable operating condition. The fundamental mode at 39 Hz and its higher harmonics were observed. The fundamental-mode frequency corresponds to the inlet quarter-wave mode of acoustic oscillations. During stable operation as shown in Fig. 16.8, the amplitude of pressure oscillations is much less. Also, no significant peak was observed in the pressure spectrum. 

Figure 11.23 A shows the real and imaginary parts of the time-dependent overlap integral (Ji(O)LV(t)) as given by Eq. (11.54) for a single harmonic vibrational mode with frequency v, A = 2.0, and a relaxation time constant Tc of 2t. Panel B shows the normalized Fourier transform. As expected, the calculated spectrum has a vibronic line at the 0-0 transition frequency and a ladder of higher-frequency lines at intervals of o. The vibrational lines are approximately Lorentzian, and although the figure does not demonstrate this, their widths depend inversely on the relaxation time constant The overall width of the spectrum (dashed line in...

Fig. 2 Schematic depiction of the procedure for conducting LAOS experiments and analysis using FT-rheology. a Measurement of the oscillatory shear strain and shear stress response in the time domain, b Normalized frequency spectra after the Fourier transformation of the shear stress exhibit the fundamental peak at the angular frequency (d. Higher harmonics / /i with n being a positive odd integer are detected for a periodic nonlinear shear stress, c By variation of yo the transition from linear to nonlinear mechanical behavior can be observed in the increase of/ /i...

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