Frequency-domain analysis

Control system design in the frequeney domain ean be undertaken using a purely tlieoretieal approaeli, or alternatively, using measurements taken from the eompon-ents in the eontrol loop. The teehnique allows transfer funetions of both the system elements and the eomplete system to be estimated, and a suitable eontroller/eompen-sator to be designed. 

Frequeney domain analysis is eoneerned with the ealeulation or measurement of the steady-state system output when responding to a eonstant amplitude, variable frequeney sinusoidal input. Steady-state errors, in terms of amplitude and phase relate direetly to the dynamie eharaeteristies, i.e. the transfer funetion, of the system. 

The steady-state response of a linear system will be 

For a given value of w, a set of algebraic equations is solved using any numerical scheme. The displacement of a mass can be written as 

From displacements, accelerations, velocities, strains and stresses can be computed. The amplification function (AF) for each frequency xijx may be derived. Repeated solutions of Eq. (3.92) are necessary for a proper definition of this function. If the fast Fourier transform is used then AF must be tabulated at each frequency interval. 

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The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

With frequency-domain analysis, the average spectmm for a machine-train signature can be obtained. Recurring peaks can be normalized to present an accurate representation of the machine-train condition. Figure 43.20 illustrates a simplified relationship between the two methods (i.e., time-domain and frequency-domain). 

The real advantage of frequency-domain analysis is the ability to normalize each vibration component so that a complex machine-train spectrum can be divided into discrete components. This ability simplifies isolation and analysis of mechanical degradation within the machine-train. 

In addition, it should be noted that frequency-domain analysis can be used to determine the phase relationships for harmonic vibration components in a typical machine-train spectrum. Frequency-domain normalizes any or all running speeds, where time-domain analysis is limited to true running speed. 

Reciprocating Limitations of the frequency-domain analysis prevent total analysis of reciprocating compressors. It is limited to the evaluation of the rotary forces generated by the main crankshaft. Therefore, time-domain and phase analysis are required for complete diagnostics. 

Redford, G. I. and Clegg, R. M. (2005). Polar plot representation for frequency-domain analysis of fluorescence lifetimes. J. Fluoresc. 15, 805-15. 

Forde, T. and Hanley, Q. S. (2006). Spectrally resolved frequency domain analysis of multi-fluorophore systems undergoing energy transfer. Appl. Spectrosc. 60, 1442-52. 

Processing of time domain data may cause artefacts in the frequency domain. One example for these distortions are truncations at the beginning or at the end of the FID which could lead to severe baseline artefacts which can be reduced by an appropriate filter. Undesired resonances leading to broad lines in the final spectra can be more easily eliminated in time domain by truncating the first few data points. Furthermore, the model functions in time domain are mathematically simpler to handle than the frequency domain analogues, which leads to a reduction of computation time. The advantage of the frequency domain analysis is that the quantification process can be directly interpreted visually. 

Another important insight gained from the frequency-domain analysis is the optimal shape of the modulation that reduces the decoherence rate. Given a specific coupling spectrum, one should choose a modulation so as to decrease the overlap between the modulation and coupling spectra [29]. This optimal... 

Callaghan, P. T., and Stepisnik, J. (1995a). Frequency-domain analysis of spin motion using modulated gradient NMR. J. Magn. Reson. A 117, 53-61. 

P. C. Searson, D. D. Macdonald andL. M. Peter, Frequency domain analysis of photoprocesses at illuminated semiconductor electrodes by transient transformation, J. Electrochem. Soc. 139, 2538, 1992. 

In general, understanding how a system behaves with respect to the frequency components of an applied stimulus is called frequency domain analysis. ... 

In traditional ac analysis in the complex plane, the voltages and currents were complex numbers. But the frequencies were always real. However now, in an effort to include virtually arbitrary waveforms into our analysis, we have in effect created a complex frequency plane too, (a + jco). This is called s-plane, where s = a + jco. Analysis in this plane is just a more generalized form of frequency domain analysis. 

The existence of this eddy shedding is also reported by the authors numerical experiment of a separating flow downstream of a rearward-facing step using a vortex code. The eddy shedding and pairing frequencies are determined using frequency domain analysis of local time traces to obtain the local dominant Strouhal... 

Now let us apply the capacity-based approach. Positive and negative 10 percent disturbances are made in the fresh feed flow rate Fg and in the fresh feed composition Zq. Dynamic simulations (confirmed by frequency-domain analysis, to be discussed in Chapter 10) show that the variability in product quality xb is decreased by increasing reactor volume or by decreasing the number of trays in the stripper. 

CHAHiiR I I Frequency-Domain Analysis of Closedloop Systems 381 Im (.G Gc)... 

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