Equating real parts

In his paper On Governors , Maxwell (1868) developed the differential equations for a governor, linearized about an equilibrium point, and demonstrated that stability of the system depended upon the roots of a eharaeteristie equation having negative real parts. The problem of identifying stability eriteria for linear systems was studied by Hurwitz (1875) and Routh (1905). This was extended to eonsider the stability of nonlinear systems by a Russian mathematieian Lyapunov (1893). The essential mathematieal framework for theoretieal analysis was developed by Laplaee (1749-1827) and Fourier (1758-1830). 

The oniy difference between the roots given in equation (5.9) and those in equation (5.i0) is the sign of the reai part. If the real part cr is negative then the system is stabie, but if it is positive, the system wiii be unstabie. This iioids true for systems of any order, so in generai it can be stated If any of the roots of the characteristic equation have positive reai parts, then the system wiii be unstabie . 

Use the Routh-Hurwitz eriterion to determine the number of roots with positive real parts in the following eharaeteristie equations... 

If all roots of the characteristic equation have negative real parts, the point of equilibrium, = 0 is asymptotically stable whatever are the terms Xt. 

If the characteristic equation does not have any roots with positive real parts, but has some roots with zero real parts, the terms in Xt may influence stability. This case belongs to the so-called critical case that requires special investigation. 

Si is the laminar flame velocity, the function Z(co) is the heat response function Equation 5.1.16, whose real part is plotted in Figure 5.1.10. The function f(r, giJ is a dimensionless acoustic structure factor that depends only on the resonant frequency, a , the relative position, r, of the flame, and the density ratio Pb/Po-... 

With x t) = q t) + ip t), the corresponding cosine iterative equation (including absorption) for the real part of x, following identical arguments that led to Eq. (11), is... 

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

Equation (1.8) represents a plane wave exp[i(A x — mt)] with wave number k, angular frequency m, and phase velocity m/A, but with its amplitude modulated by the function 2 cos[(AA x — Amt)/2]. The real part of the wave (1.8) at some fixed time to is shown in Figure 1.2(a). The solid curve is the plane wave with wavelength X = In jk and the dashed curve shows the profile of the amplitude of the plane wave. The profile is also a harmonic wave with wavelength... 

Figure 1.5 shows the real part of the plane wave exp[i(A ox — coot)] with its amplitude modulated by B(x, t) of equation (1.20). The plane wave moves in the positive x-direction with phase velocity Uph equal to o)o/ko. The maximum amplitude occurs at x = v t and propagates in the positive x-direction with group velocity Ug equal to (dm/dA )o. 

From the imaginary part equation, the ultimate frequency is u = Vl 1. Substituting this value in the real part equation leads to the ultimate gain Kc u = 60, which is consistent with the result of the Routh criterion. 

Thus we have either = 0 or - 2 + (1 + Kc) = 0. Substitution of the real part equation into the nontrivial imaginary part equation leads to... 

The solution of this equation is the ultimate frequency cou = 0.895, and from the real part equation, the corresponding ultimate proportional gain is Kc u = 5.73. Thus the more accurate range of Kc that provides system stability is 0 < Kc < 5.73. 

We can confirm the answer by substituting the values into our analytical equations. We should find that the real part of the closed-loop pole agrees with what we have derived in Example 7.5, and the value of the proportional gain agrees with the expression that we derived in this example. 

By the very definition of the GF, the real parts of the poles of its frequency Fourier component correspond to natural frequencies of the system (see, for example, Eqs. (A1.23) or (A1.55)). Consequently, the spectrum of natural frequencies of the perturbed system, cop, should fit the equation... 

The real part of the general solution (1-22) to the one-dimensional wave equation is... 

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