Laplace frequency

This self-consistent equation has a simple message the relevant friction for the reaction is determined by the (Laplace) frequency component of the time dependent friction at the reactive frequency X. This frequency sets the basic time scale for the microscopic events affecting k. 

Due to the isotropic nature of the liquid, the linearized hydrodynamic equations are easily solved when written in the Fourier (wave number) plane. Thus, the basic equations in fluid mechanics in the wavenumber and Laplace frequency (z) plane are written as... 

The expression of the transverse current autocorrelation function can also be derived from the linearized hydrodynamic equations. Because it is decoupled from all the longitudinal modes, the derivation is simple and the final expression in wavenumber and Laplace frequency plane can be written as... 

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,... 

Note that here z is not the usual Laplace frequency (used in rest of the review), but it is a Fourier frequency. Equation (226) can be written in the Fourier-Laplace space (z) as... 

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... 

The total frequency-dependent friction calculated from the MCT, (z), is plotted against the Laplace frequency (z) in Fig. 14. In the same figure the Enskog friction (e and the binary contribution (z) are also shown. Note here that in the high-frequency regime the frequency-dependent total friction is much less than the Enskog friction and is dominated entirely by the binary... 

Figure 14. The frequency-dependent total friction Ctoui(z) (solid line) and the binary friction (B(z) (dashed-dot line) plotted as a function of Laplace frequency (z). For comparison, the calculated Bnskog friction is also shown (dashed line). The calculation has been performed for p = 0.85 and T = 0.85. The frequency-dependent friction, the Enskog friction, and the frequency are scaled by xsc l. This figure has been taken from Ref. 170.

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. 

This is the Laplace -frequency dependent rate matrix. The time-domain counterpart of eqn (13.51) is ... 

Equation (13.55) with the initial conditions of i (0) = S o reads in the Laplace frequency domain as ... 

Acceleration-Velocity-Displacement The simplest kind of conversion is integration and differentiation to obtain different time derivatives of ground motion. For example, to convert an acceleration to a velocity or from velocity to displacement at a given frequency /, one simply divides by the ampUtude by the angular frequency (0 = 2nf. This follows straightforwardly from the equivalence of differentiatiOTi in the time domain to multipUcation in the frequency domain by the Laplace frequency s = jm. Thus for sinusoidal ampUtudes of acceleration a. ... 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

We first illustrate the idea of frequency response using inverse Laplace transform. Consider our good old familiar first order model equation, 1... 

Note In the text, we emphasize the importance of relating pole positions of a transfer function to the actual time-domain response. We should get into the habit of finding what the poles are. The time response plots are teaching tools that reaffirm our confidence in doing analysis in the Laplace-domain. So, we should find the roots of the denominator. We can also use the damp () function to find the damping ratio and natural frequency. 

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. 

I have chosen the five languages listed above simply because I have had some exposure to all of them over the years. Let me assure you that no politieal or nationalistic motives are involved. If you would prefer French, Spanish, Italian, Japanese, and Swahili, please feel free to make the appropriate substitutions My purpose in using the language metaphor is to try to break some of the psychological barriers that students have to such things as Laplace transforms and frequency response. It is a pedagogical gimmick that I have used for over two decades and have found it to be very effective with students. 

There are a variety of feedback controller tuning methods. Probably 80 percent of all loops are tuned experimentally by an instrument mechanic, and 75 percent of the time the mechanic can guess approximately what the settings will be by drawing on experience with similar loops. We will discuss a few of the time-domain methods below. In subsequent chapters we will present other techniques for tinding controller constants in the Laplace and frequency domains. 

There are several methods for testing for stability in the Laplace domain. Some of the most useful are discussed below. Frequency-domain methods will be discussed in Chap. 13. 

In Chap. 12 we will show that we can convert from the Laplace domain (Russian) into the frequency domain (Chinese) by merely substituting ia for s in the transfer function of the process. This is similar to the direct substitution method, but keep in mind that these two operations are different. In one we use the transfer function. In the other we use the characteristic equation. 

This very remarkable result [Eqs. (12.7) and (12.8)] permits us to go from the Laplace domain to the frequency domain with ease. 

The design of feedback controllers in the frequency domain is the subject of this chapter. The Chinese language that we learned in Chap. 12 is now put to use to tune controllers. Frequency-domain methods are widely used because they have the significant advantage of being easier to use for high-order systems than the time- and Laplace-domain methods. 

Thus the system is closedioop stable if K, > 1/Kp. This is exactly the conclusion we reached using root locus methods. So the Chinese frequency-domain conclusions are the same as the Russian Laplace-domain conclusions. 

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... 

Third-order process in the time, Laplace, and frequency domains. 

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