Laplace notation

Gearbox, lead-screw and machine-table. With reference to Figure 2.9 (free-body diagram of a gearbox), the motor-shaft will have zero viscous friction Cm, hence equation (2.22), using Laplace notation, becomes... 

Expressing these in Laplace notation (see Appendix I) gives two simultaneous equations (Equations 19.2 and 19.3)/... 

Take Laplace transforms with zero initial conditions and using lower-case notation... 

There are many acceptable notations of Laplace transform. We choose to use a capitalized letter, and where confusion may arise, we further add (s) explicitly to the notation. 

We will use the same notation <1> for the time function and its Laplace transform, and only add the tors dependence when it is not clear in which domain the notation is used. 

Be careful with the notation. Upper case C is for concentration in the Laplace domain. The boldface upper case C is the output matrix. 

In Fig. 5.1, we use the actual variables because they are what we measure. Regardless of the notations in a schematic diagram, the block diagram in Fig. 5.2 is based on deviation variables and their Laplace transform. 

The first term is the double-layer charging response, while the second is a measure of the overlap between double-layer charging and Faradaic reaction, which eventually tends toward the Faradaic response that would have been obtained if double-layer charging were absent. As to the expression of the characteristic functions f(s) and f(t) in the Laplace and original spaces, respectively, with the same notations as in Section 6.1.4,... 

I a this section we will study the time-dependent behavior of some chemical. engineering systems, both openloop (without control) and closedloop (with controllers included). Systems will be described by diflerential equations, and solutions will be in terms of time-dependent functions. Thus, our language for this part of the book will be English. In the next part we will learn a little Russian in order to work in the Laplace domain where the notation is more simple than in English. Then in Part V we will study some Chinese because of its ability to easily handle much more complex systems. 

The use of Laplace transfonnations yields some very useful simplifications in notation and computation. Laplace-transforming the linear ordinary differential equations describing our processes in terms of the independent variable t converts them into algebraic equations in the Laplace transform variable s. This provides a very convenient representation of system dynamics. 

After transforming equations into the Laplace domain and solving for output variables as functions of s, we sometimes want to transform back into the time domain. This operation is called mission or inverse Laplace tran ormation. We are translating from Russian into English. We will use the notation. 

Defining in this way permits us to use transfer functions in the z domain [Eq. (18.57)] just as we use transfer functions in the Laplace domain. G,, is the z transform of the impulse-sampled response of the process to a unit impulse function <5( . In z-transforming functions, we used the notation =... 

From now on we will use the shorthand, Laplace-domain notation, but keep in mind what is implied in its use. 

In order to facilitate the way of notation, it is useful to represent the virtual voltage sources by SF = [SF] cos (2coLf) and Sc = [Sc] cos (2col ). Again, the Laplace transform method is most useful for solving... 

We have seen already that the general solution of Fick s second law in the Laplace domain can be formulated in various convenient ways. Here, we choose the notation of eqn. (147)... 

With appropriate adaptations, the other monomolecular reaction schemes can be treated by similar procedures. It may suffice, therefore, to give below the resulting Laplace transforms of the interfacial concentrations in terms of parameters, the meanings of which are given in Table 8. Where possible, references to the literature are given. However, notations and formulations are often quite different due to the personal preferences of the authors. 

In the case of reaction kinetics described by eqn. (3) and eqn. (3) of Chap. 4, the problem under study, as in refs. 16 and 17, can be solved by using the inverse Laplace transform. Actually, differentiating both sides of eqn. (3) of Chap. 4 with respect to t and using the notation x = ve exp ( 2Rjae) we have... 

All the system response curves in frequency and time domains were calculated numerically from equations that are much too involved to reproduce in detail here. Transfer functions in Laplace transform notation are easily defined for the potentiostat and cell of Figure 7.1. Appropriate combinations of these functions then yield system transfer functions that may be cast into time- or frequency-dependent equations by inverse Laplace transformation or by using complex number manipulation techniques. These methods have become rather common in electrochemical literature and are not described here. The interested reader will find several citations in the bibliography to be helpful in clarifying details. 

We were also able to obtain a continuous linewidth distribution curve 0(ryv,K) from the regularization approach by inversion of equation (15). The result obtained from the Laplace inversion procedure is a set of delta functions G. K) which approximate the continuous G(ryv,K) curve sampled at equal intervals. The subscript w shall henceforth be omitted in the interest of clarity in notation. In order to obtain a size distribution from G(IL,K) we need to convert both the ordinate and the abscissa (IL) in the following way. 

You will notice that I am using a mathematician s favorite notation for this operator, which is not the same as the standard physics text the symbol A. This is the Laplace operator, or the Laplacian. With this notation in place, Schrodinger s equation is ... 

Use dummy variables in the Laplace domain (ua, ub, etc) for brevity of notation. 

For many purposes, we will find that antiplane shear problems in which there is only one nonzero component of the displacement field are the most mathematically transparent. In the context of dislocations, this leads us to first undertake an analysis of the straight screw dislocation in which the slip direction is parallel to the dislocation line itself. In particular, we consider a dislocation along the X3-direction (i.e. = (001)) characterized by a displacement field Usixi, X2). The Burgers vector is of the form b = (0, 0, b). Our present aim is to deduce the equilibrium fields associated with such a dislocation which we seek by recourse to the Navier equations. For the situation of interest here, the Navier equations given in eqn (2.55) simplify to the Laplace equation (V ms = 0) in the unknown three-component of displacement. Our statement of equilibrium is supplemented by the boundary condition that for xi > 0, the jump in the displacement field be equal to the Burgers vector (i.e. Usixi, O" ") — M3(xi, 0 ) = b). Our notation usixi, 0+) means that the field M3 is to be evaluated just above the slip plane (i.e. X2 = e). 

Since many equations are involved in a control system analysis, it is desirable that each equation be written as simply as possible. Operational calculus provides a useful notation, and in particular the Laplace transformation permits a very simple treatment if the differential equations are linear. A further simplification results if the same types of initial conditions are taken for all problems, or if only steady-state sinusoidal behavior is considered. Churchill (C6) and Carslaw and Jaeger (C2)... 

A necessary and sufficient condition for identifia-bility is the concept that with p-estimable parameters at least p-solvable relations can be generated. Traditionally, for linear compartment models this involves using Laplace transforms. For example, going back to the 1-compartment model after first-order absorption with complete bioavailability, the model can be written in state-space notation as... 

We use the same notation as with the Fourier transform, denoting a Laplace transform by a capital letter and the function by a lowercase letter. You will have to tell from the context whether we are discussing a Fourier transform or a Laplace transform. We use the letter t for the independent variable of the function, since Laplace transforms are commonly applied to functions of the time. The letter x could also have been used. 

The time dependence is significant, because the Laplace transformation of (6.2.2) cannot be obtained as it could in deriving (5.4.13), and the mathematics for sweep experiments are greatly complicated as a consequence. The problem was first considered by Randles (1) and Sevcik (2) the treatment and notation here follow the later work of Nicholson and Shain (3). The boundary condition (6.2.2) can be written... 

Using an operator notation for and taking the Laplace transform, we may also write... 

The Laplace transformation of a function of time /(,) consists of operating on the function by multiplying it by and integrating with respect to time t from 0 to infinity. The operation of Laplace transforming is indicated by the notation... 

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