Power spectrum density

Fig. 6 shows the FFT spectrum for calculated bed pressure drop fluctuations at various centrifugal accelerations. The excess gas velocity, defined by (Uo-U ,, was set at 0.5 m/s. Here, 1 G means numerical result of particle fluidization behavior in a conventional fluidized bed. In Fig. 6, the power spectrum density function has typical peak in each centrifugal acceleration. However, as centrifugal acceleration increased, typical peak shifted to high frequency region. Therefore, it is considered that periods of bubble generation and eruption are shorter, and bubble velocity is faster at hi er centrifugal acceleration. 

In order to analyze both systems, some techniques from nonlinear science are burrowed. Firstly, a phase portrait is constructed from delay coordinates, a Poincare map is also computed, FFT is exploited to derive a Power Spectrum Density (PSD) Maximum Lyapunov Exponents (MLE) are also calculated from time series. Although we cannot claim chaos, the evidence in this chapter shows the possible chaotic behavior but, mostly important, it exhibits that the oscillatory behavior is intrinsically linked to the controlled systems. The procedures are briefly described before discuss each study case. 

A spectrum is the distribution of physical characteristics in a system. In this sense, the Power Spectrum Density (PSD) provides information about fundamental frequencies (and their harmonics) in dynamical systems with oscillatory behavior. PSD can be used to study periodic-quasiperiodic-chaotic routes [27]. The filtered temperature measurements y t) were obtained as discrete-time functions, then PSD s were computed from Fast Fourier Transform (FFT) in order to compute the fundamental frequencies. 

Fig. 7. Power spectrum density. The measured time series comprise several fundamental frequencies. Since frequencies have low-order (< 0.03 Hz) noise effect can be neglected. Note that if the values reference outlet substrate and the control gains decrease (experiment E.ld), then the number of fundamental frequencies in PSD decreases. This leads us to belief that there is a suitable values such that system displays hmit cycle. However, this behavior was not experimentally found.

The power spectrum density (PSD) is a widely used tool to find fundamental frequencies (and their harmonics) in d mamical systems with oscillatory... 

The perturbation strength for which the Nekhoroshev s theorem holds is also so small that it cannot be applied to realistic physical and chemical situations. Indeed it was shown that the range of perturbation strength is much smaller than the situation where the power spectrum density of observables exhibits a continuous one [24]. This means that, in its rigorous sense, the Nekhoroshev s theorem can only be applied to sufficiently weak perturbed systems. For the same reason as mentioned above, Nekhoroshev s theorem is nevertheless a key guiding principle to sticky or stagnant motions in nearly integrable Hamiltonian systems. 

As mentioned in the introduction, the power spectrum density is used to probe the long-time correlation decay. Appearance of l//v-type spectra is an indication that there are, in principle, infinitely many time scales in the relaxation process. Geisel et al. [37] gave an example of mixed Hamiltonian systems... 

Figure 4. (a) The power spectrum density of total system potential in liquid water. The... 

Figure 6. The power spectrum density of the lowest mode energy for the FPU p model. The initial value of each mode energy is fixed at E = 0.5. The system size is given as N = 8,16,32,64,128 from left to right, respectively. The line F(f) =A/ 1 + (f/fs) 1 is obtained by fitting the power spectrum density in the frequency between two J. marks. The shoulder at/ =fs is indicated by a " mark. [Reprinted with permission from Jpn. J. Appl. Phys. 35 2387-2393 (1996). Copyright 1996 by The Institute of Pure and Applied Physics.]...

As in the study of water dynamics, the power spectrum density is useful to detect the long-time correlation or to detect decay slow energy transfers between optical and acoustic modes in the model given in Eq. (11). The relaxation inside optical modes is much faster than that in acoustic modes, since the frequency spectrum in optical modes is sharply localized and almost resonant while the spectrum is broadly spread in acoustic modes. 

Figure 7. The power spectrum density of energy fluctuation of (a) the acoustic subsystem h and (b) an individual mode in optical and acoustic subsystem. In the simulation, the ratio of perturbation strength is taken as X = /1 / K — 5.0.

Figure 10. The power spectrum density of the total dipole moment fluctuation of liquid water (solid line) and water cluster (H2O)108 (dashed line). The simulation of liquid water was performed for 216 water molecules under the periodic boundary condition.

Figure 13. (a) The power spectrum density for total (upper) and individual (lower) translational energies. The number of molecules is 216 and SPC potential is used for simulations. The temperature is 305 K. (b) The power spectrum density for total (upper) and individual (lower) rotational energies. 

Figure 16. The power spectrum density for molecules for which rigid rotators are replaced by point particles, keeping the radial distribution function of liquid water unchanged.

Figure 17. The power spectrum density for total rotational (upper) and total translational (lower) energies in the case of (a) alcohol and (b) oxygen molecules. In both figures, the spectra for translational energies are shifted below for clarity. The number of molecules is 216 in each case.

FIGURE 4-23 Typical power spectrum density for fast fluidized bed (Xia et al., 1992). 

The MEDUSA monitors vibration levels (r.m.s. and peak levels), auto power spectrum densities (APSDs), waveforms and probability density functions at every hour automatically, then the r.m.s. levels and the APSDs are compared with acceptable vibration levels to judge whether the machine s condition is normal. 

The local values of the drag reduction, mean values of the velocities and turbulence intensities are presented along with the power spectrum density functions. It is shown that extremely high local values of drag reduction can be obtained by this method with very low average concentration of polymer solution. 

The power spectrum density functions were obtained by applying the Fourier transform to the original set of data (time series of velocity). The Fast Fourier Transform algorithm described by Singleton [17] was applied. 

In view of a quantitative treatment, these fluctuations have been recorded and whence analyzed in terms of power spectrum densities (psd). It is known that, in the framework of a linear theory and assuming the system as ergodic, the mass flux psd W V and the velocities psd W% can be linked through the following relationship [9]. 

Figure 23 Ni dots electrodeposited throughout 45-nm thick PS-PVP nanotemplate, lateral scale 1x1 pm (a) topography image, z scale 30 nm, (b) power spectrum density, the main peak (24 nm) corresponds to the SMA periodicity (inset FFT image of (a) showing perfect hexagonal ordering of Ni dots). Occasional lacunas appear because of the inhomogeneity of electrodeposition. (Reproduced from Ref. 57. American Chemical Society, 2003.)...

Fig. 2.5. ACROBAT tail buffet alleviation experiments a single-input singleoutput (SISO) control law design for active rudder and piezoelectric wafers excitation, b power spectrum density (PSD) peak values for the root bending moment at the first bending resonance [11]...

Figure 4.11 shows a typical sample of the cutting force and noise from the micro-milling experiment. As can be seen, the SNR is very low and the absolute value of the noise is nearly one-third that of the force. The noise statistics are computed as mean = -0.1093, variance = 0.2631, skew = 0.0004, kurtosis = 1.2621. It is super-Gaussian, with a longer tail than Gaussian distribution and is well fitted with Laplacian distribution (see Figure 4.11b). The power spectrum density (PSD) and autocorrelation...

Figure 36 Power spectrum density of pressure fluctuations in a bubbling fluidized bed of sand (van den Schaaf et al., 1999a). Dt. 0.80 m uq. 0.44 m/s dp 390 fim settled bed height = 2.19m. (With permission of American Society of Mechanical Engineers)...

We should calculate the power spectrum density for selected electrodes above sensory-motor and parietal areas in both hemispheres (usually C3, C4, P3, and P4) which can be done using different methods such as combination of fast Eourier transform combined with Hanning cosine window for selecting the EEG signals. There are several methods of processing and classifications of signals which should be studied in order to achieve the optimum method [10], Based on the studied patterns the proper feedback commands are... 

Figure 11.7 Power spectrum densities for hot-leg piping design.

The equivalence of these pressure fluctuations is further demonstrated by the power spectrum density functions, normalized with respect to the relevant frequency range of 0-10 Hz. These are reported in Figure 16.9,... 

Figure 16.9 Power spectrum density fxmctions (PSDF) for the two matched systems.

The energy spectral density function (or power spectrum) P f) is given by the absolute square of P f) ... 

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