Laplace frequency dynamics

The expression for the dynamic structure factor has been presented in the previous section, and that in the Laplace frequency plane is given by Eq. (210). When written in the time plane, Eq. (210) provides the following equation of motion for the dynamic structure factor,... 

Equation (228) is the normalized density correlation function in the Fourier frequency plane and has the same structure as Eq. (210), which is the density correlation function in the Laplace frequency plane. i/ (z) in Eq. (228) is the memory function in the Fourier frequency plane and can be identified as the dynamical longitudinal frequency. Equation (228) provides the expression of the density correlation function in terms of the longitudinal viscosity. On the other hand, t]l itself is dependent on the density correlation function [Eq. (229)]. Thus the density correlation function should be calculated self-consistently. To make the analysis simpler, the frequency and the time are scaled by (cu2)1 2 and (cu2)-1 2, respectively. As the initial guess for rjh the coupling constant X is considered to be weak. Thus rjt in zeroth order is... 

Many packages are available for steady-state simulation, as discussed in Chapter 4. To manipulate the linearized models in the Laplace, frequency, and time domains, MATLAB and SIMULINK are used commonly, and example scripts are introduced in Section 21.6. The most recent commercial packages permit steady-state and dynamic simulations. These include HYSYS.Plant, CHEMCAD, and ASPEN DYNAMICS, with the former used in this section and in Section 21.5. 

As you will see, several different approaches are used in this book to analyze the dynamics of systems. Direct solution of the differential equations to give functions of time is a time domain teehnique. Use of Laplace transforms to characterize the dynamics of systems is a Laplace domain technique. Frequency response methods provide another approaeh to the problem. 

At this point it might be useful to pull together some of the concepts that you have waded through in the last several chapters. We now know how to look at and think about dynamics in three languages time (English), Laplace (Russian) and frequency (Chinese). For example, a third-order, underdamped system would have the time-domain step responses sketched in Fig. 14.10 for two different values of the real TOOt. In the Laplace domain, the system is represented by a transfer function or by plotting the poles of the transfer function (the roots of the system s characteristic equation) in the s plane, as shown in Fig. 14.10. In the frequency domain, the system could be represented by a Bode plot of... 

We could go through the Laplace domain by approximating and then inverting. However, there is a direct conversion V. V. Solodovnilcov, Introduction to Statistical Dynamics of Autoinatic Control, Dover, 1960). Suppose we want to find the impulse response of a stable system (defined as g,), given the system s frequency response. Since the Laplace transformation of the impulse input is unity,... 

In the next chapter we take a quantitative look at the dynamics of these CSTR systems using primarily rigorous nonlinear dynamic simulations (time-domain analysis). However, some of the powerful linear Laplace and frequency-domain techniques will be used to gain insight into the dynamics of these systems. 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

Thus, application of delta function is equivalent to exciting all the circular frequencies with equal emphasis. This is the basis of finding the natural frequency of any oscillator via impulse response. When the oscillator is subjected to an impulse, all frequencies are equally excited and the system dynamics picks out the natural frequency of vibration, leaving others to decay in due course of time. It is noted that this result also applies to Laplace transform and we are going to use it often by replacing time by space and circular frequency by wave numbers. 

The complex variable z (Im z < 0) is homogenetic to a frequency. The resolvent l/(z — L) is the Fourier-Laplace transform of the evolution operator (see Appendix A). Expression (93) shows that the dynamics is reduced to the determination of the matrix element of the resolvent between two observables. Therefore only a reduced dynamics has to be investigated. For that purpose we shall define more precisely the observables and the operators of interest. The theory is formulated in the framework of the Liouville space of the operators and based on hierarchies of effective Liouvillians which are especially convenient to study reduced dynamics at various macroscopic and microscopic timescales (see Appendix B). 

In Eq. [7], the frequency-dependent friction is the Laplace transform of the time-dependent friction The presence of the Laplace transform means that the time-dependence of the friction must be known in order to determine the Laplace transform. This friction can be readily determined from molecular dynamics simulations in the approximation where the motion along the reaction coordinate is fixed at x = 0. (A discussion of some subtle, but important, aspects of this approximation is given by Carter et al. ) In that case, the random force R(t) can be calculated from equilibrium dynamics in the presence of this one constraint. From R(t), the time-dependent friction (t) can be calculated and the implicit Eq. [7] solved. The result gives the Grote—Hynes value of the transmission coefficient for that system. 

The Relationship Between Time and Frequency Domain Laplace and Fourier Transform As long as this is true, Fourier transform (F(V(t)), F(I(t)) - settled situation, a static case) or Laplace transform (L(V(t)), L(I(t)) - transition situation, a dynamic case) can be used for the conversion between time and frequency domain, where z(t) is the impulse function characterizing the complex impedance Z(j(o) in time domain. 

This type of excitation was introduced by the laboratory LAPLACE (Laboratoire PLAsma et Conversion d Energie - Plasma and Energy Conversion Lab) in Toulouse [TUR 08]. The shape of the excitation current is exactly the same as that shown in Figure 2.33. The only change, in fact, is the frequency domain. Indeed, we shall see frequencies which dynamically excitate the various phenomena. This may lead to an increase in the frequency up to ten Hz. 

The characteristic function h (t) contains complete information of the dynamic behavior of the system. Extracting this information from Eq. (7-13) is a rather complex mathematical procedure in the time domain. Instead, it is more advantageous to transform the problem into the frequency domain (i). This is performed by applying Laplace or Fourier transformation... 

In the above discussion of the frequency dependent permittivity, the analysis has been based on either the single particle rotational diffusion model of Debye, or empirical extensions of this model. A more general approach can be developed in terms of time correlation functions [6], which in turn have to be interpreted in terms of a suitable molecular model. While using the correlation function approach does not simplify the analysis, it is useful, since experimental correlation functions can be compared with those deduced from approximate theories, and perhaps more usefully with the results of molecular dynamics simulations. Since the use of correlation functions will be mentioned in the context of liquid crystals, they will be briefly introduced here. The dipole-dipole time correlation function C(t) is related to the frequency dependent permittivity through a Laplace transform such that ... 

In previous chapters, Laplace transform techniques were used to calculate transient responses from transfer functions. This chapter focuses on an alternative way to analyze dynamic systems by using frequency response analysis. Frequency response concepts and techniques play an important role in stability analysis, control system design, and robustness analysis. Historically, frequency response techniques provided the conceptual framework for early control theory and important applications in the field of communications (MacFarlane, 1979). We introduce a simplified procedure to calculate the frequency response characteristics from the transfer function of any linear process. Two concepts, the Bode and Nyquist stability criteria, are generally applicable for feedback control systems and stability analysis. Next we introduce two useful metrics for relative stability, namely gain and phase margins. These metrics indicate how close to instability a control system is. A related issue is robustness, which addresses the sensitivity of... 

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