Oscillations characteristic equation

Aris and Amundson (1958) solved the coupled, time-dependent material and energy balances, linearizing the equations about the operating point by a Taylor series expansion. This made the solution possible by the method of characteristic equations. The solution yielded two equations, one the slope condition and the other recognized by Gilles and Hofmann (1961) as the condition that sets the limits to avoid rate oscillation. This is called the... 

When the test temperature is raised, the rate of Brownian motion increases by a certain factor, denoted Ox. and it would therefore be necessary to raise the frequency of oscillation by the same factor flx to obtain the same physical response, as shown in Figure 1.6. The dependence of Uj upon the temperature difference T—Tg follows a characteristic equation, given by Williams, Landel, and Ferry (WLF) [11] ... 

To analyze stability of the linearized model, we have to examine the eigenvalues that are solutions of the characteristic equation of A. Usually the eigenvalue is a complex number ( = fi+iui. If yti = Re < 0, then the solution is a decaying oscillating function of time, so we have a stable situation. If fi = Re ( > 0 on the other hand, then the solution diverges in an oscillatory fashion and the solution is unstable. The boundary between these two situations, where fi = Re ( = 0, defines a Hopf bifurcation in which an eigenvalue crosses from the left-hand to the right-hand complex plane. 

A simple result is obtained for this model if the nonreactive mode is fast relative to the reactive mode. [From Eq. (6.76) the time scale associated with X2 is given by the roots of its characteristic equation, l/2[y2 (yi — 4( 2) ] ] In this case the nonreactive oscillator may be considered as part of the heat... 

We will now neglect the higher modes V n because they oscillate fast whereas ao(a ) varies slowly in space, and approximate the sum in eq. (5.35) by the dominant first term with Aq > 0. We obtain the characteristic equation for the eigenvalue A ... 

For a supercritical value of K = 1 /12 the roots of the characteristic equation, corresponding to Eq. (11.55), are imaginary. The solution (11.58) describes a damped oscillation. Once the particle has lost velocity, the opposite motion of liquid brings it back to the surface. Therefore, when the distance of the particle from the surface is maximum at time its velocity approaches zero. Based on (11.58), we obtain... 

We now solve the equation of motion for the harmonic oscillator, Eq. (8.12). We begin by finding the characteristic equation. 

I EXERCISE 8.1 Show that the characteristic equation for Eq. (8.12) for the harmonic oscillator is... 

For the sake of simplicity, let us now set up the case of a second-order bandpass filter and a comparator with saturation levels This closed-loop system verifies the required premises the system is autonomous, the nonlinearity is both separable and frequency-independent, and the linear transfer function contains enough low-pass filtering to neglect the higher harmonics at the comparator output. Choosing adequately the band-pass filter, it can be forced that the first-order characteristic equation for the closed-loop system of Fig. 4 has an oscillation solution being and the oscillation frequency and... 

The general case, of which the Krebs cycle with eight reactions in circle is a particular case can be obtained by extension of the expressions derived for three reactions in circle and four reactions in circle. Aliter to this would be the method of eigenvalues and eigenvectors. The cases when A, eigenvalues are imaginary represent situations in the system, when the concentration of the species will exhibit subcritical damped oscillations. These are given by the characteristic equation ... 

The theoretical aspects of the technique are described by Verschaffelt (1915) and Kestin et al. (1957a, 1958). The characteristic equation which relates the motion of an oscillating body to that of the fluid is (Kestin and Newell, 1957b)... 

Equation (14.9) is called the secular determinant. When it is expanded, a 3N order characteristic equation for A will be obtained which can be solved for the 3N characteristic values for A in terms of the/and m values. Each value of A can be then put back into Eq. (14.8) to calculate the corresponding values for Aj. Actually only the ratios of the Aj values, that is, the ratio of the amplitudes for each A, can be determined but this is sufficient to describe the vibration. The result indicates that each atom is oscillating about its equilibrium position with amplitude Aj, generally different for each coordinate, but with the same frequency v = A /(2 r) and phase constant a which means all the atoms go through their equilibrium positions simultaneously. Such a motion is called a normal mode of vibration. 

Thus, the imposed initial conditions for the characteristics are equivalent to the requirement that the semiclassical solution exp(-Wo(q)/ i) at the potential minimum coincides with the wave function of the ground state of harmonic oscillator. Inserting Equation (6.90) into the Hamilton-Jacobi Equation (6.86), we come to the relation... 

One characteristic property of dyes is their colour due to absorption from the ground electronic state Sq to the first excited singlet state Sj lying in the visible region. Also typical of a dye is a high absorbing power characterized by a value of the oscillator strength/ (see Equation 2.18) close to 1, and also a value of the fluorescence quantum yield (see Equation 7.135) close to 1. 

Recall that the calculation of 4>m relies on equation 13.4. It is important to emphasize that the validity of this equation rests on two assumptions. The first is a direct consequence of the way the microphone responds to the rate of the process, as it originates the photoacoustic wave. All the fast processes yield a measured waveform with the same time profile all the slow processes produce virtually no signal. Between these two extremes an intermediate regime exists, in which the time profile of the measured waveform varies to reflect the rate of the process. The fast and slow process rates are defined in relation to the intrinsic response of the microphone, determined by its characteristic oscillation frequency v. 

---