Oscillator undamped

The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control. The natural period of the cycle—the proportional controller contributes no phase shift to alter it—is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band P , which produced the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced using on/off control to cycle the manipulated variable between two limits. The period of the cycle will be close to if the cycle is symmetrical the peak-to-peak amphtude of the controlled variable divided by the difference between the output limits A, is a measure of process gain at that period and is therefore related to for the proportional cycle ... 

According to (2.29), dissipation reduces the spread of the harmonic oscillator making it smaller than the quantum uncertainty of the position of the undamped oscillator (de Broglie wavelength). Within exponential accuracy (2.27) agrees with the Caldeira-Leggett formula (2.26), and similar expressions may be obtained for more realistic potentials. 

The (5-part in (2.50) is responsible for elastic scattering, while the second term, proportional to the Fourier transform of C(t), leads to the gain and loss spectral lines. When the system exercises undamped oscillations with frequency Aq, this leads to two delta peaks in the structure factor. 

Assuming that the pj (t) and Qj (t) can be interpreted as a TS trajectory, which is discussed later, we can conclude as before that ci = ci = 0 if the exponential instability of the reactive mode is to be suppressed. Coordinate and momentum of the TS trajectory in the reactive mode, if they exist, are therefore unique. For the bath modes, however, difficulties arise. The exponentials in Eq. (35b) remain bounded for all times, so that their coefficients q and q cannot be determined from the condition that we impose on the TS trajectory. Consequently, the TS trajectory cannot be unique. The physical cause of the nonuniqueness is the presence of undamped oscillations, which cannot be avoided in a Hamiltonian setting. In a dissipative system, by contrast, all oscillations are typically damped, and the TS trajectory will be unique. 

D. E. F. Harrison, Undamped oscillations of pyridine nucleotide and oxygen tension in chemostat cultures of klebsiella aerogenes,/. Cell Biol. 75, 514—521 (1970). 

For C = 0 (undamped system). The complementary solution is the same as q. (6.93) with the exponential term equal to unity. There is no decay of the sine and cosine terms and therefore the system will oscillate forever. 

Figure 6 shows the results for the more challenging model. Model IVb, comprising three strongly coupled vibrational modes. Overall, the MFT method is seen to give only a qualitatively correct picture of the electronic dynamics. While the oscillations of the adiabatic population are reproduced quite well for short time, the MFT method predicts an incorrect long-time limit for both electronic populations and fails to reproduce the pronounced recurrence in the diabatic population. In contrast to the results for the electronic dynamics, the MFT is capable of describing the almost undamped coherent vibrational motion of the vibrational modes. 

To avoid this handicap, Boersch and coworkers 2) used coupled resonators. The first active laser cavitiy generates the radiation whose absorption is to be measured. The probe is placed in a second cavity, which is coupled to the first one and which is undamped by an active medium just below the threshold for self-oscillation. This arrangement enables changes in the refractive index as small as An 10 ° or absorption coefficients down to a 10 to be detected. 

In the region between to, and to, which for SiC is between about 800 and 1000 cm-1, the reflectance is high not because of large k but because of small n. If n = 0, the normal incidence reflectance is nearly 100% only for the undamped oscillator (y = 0) is the reflectance actually 100%, but solids like SiC approach this rather closely. If the damping constant y in (9.20) is set equal to zero, the real part of the dielectric function becomes... 

A major development reported in 1964 was the first numerical solution of the laser equations by Buley and Cummings [15]. They predicted the possibility of undamped chaotic oscillations far above a gain threshold in lasers. Precisely, they numerically found almost random spikes in systems of equations adopted to a model of a single-mode laser with a bad cavity. Thus optical chaos became a subject soon after the appearance Lorenz paper [2]. 

The time dependence of the perturbations in (3.40) and (3.41) would then be equivalent to an undamped cosine function of frequency cu0, leading to indefinitely sustained oscillations. 

If the uncatalysed reaction rate increases with respect to the rate of catalyst decay, so that ku becomes larger than gk2, there are no real solutions to eqn (3.60). The stationary state can no longer become unstable as /i is varied. Damped oscillatory responses can still be observed when we have a stable focus, but undamped oscillations will not be found. 

We have seen in earlier chapters that kinetic systems with two independent concentrations can show additional complexities of dynamic behaviour beyond those of one-variable systems. Of particular interest are undamped oscillations. The cubic autocatalysis with the additional decay step... 

In the literature it is customary to put y/U (a) = coa, because it is the frequency with which an undamped particle would oscillate in the bottom of the well at a. One also writes / U"(b) = cob, although this is not a real frequency. If one finally puts W for the height of the barrier, one has for the rate of escape... 

Lastly, non-elementary several-stage reactions are considered in Chapters 8 and 9. We start with the Lotka and Lotka-Volterra reactions as simple model systems. An existence of the undamped density oscillations is established here. The complementary reactions treated in Chapter 9 are catalytic surface oxidation of CO and NH3 formation. These reactions also reveal undamped concentration oscillations and kinetic phase transitions. Their adequate treatment need a generalization of the fluctuation-controlled theory for the discrete (lattice) systems in order to take correctly into account the geometry of both lattice and absorbed molecules. As another illustration of the formalism developed by the authors, the kinetics of reactions upon disorded surfaces is considered. 

This type of a pattern of singular points is called a centre - Fig. 2.3. A centre arises in a conservative system indeed, eliminating time from (2.1.28), (2.1.29), one arrives at an equation on the phase plane with separable variables which can be easily integrated. The relevant phase trajectories are closed the model describes the undamped concentration oscillations. Every trajectory has its own period T > 2-k/ujq defined by the initial conditions. It means that the Lotka-Volterra model is able to describe the continuous frequency spectrum oj < u>o, corresponding to the infinite number of periodical trajectories. Unlike the Lotka model (2.1.21), this model is not rough since... 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

Staying within a class of mono- and bimolecular reactions, we thus can apply to them safely the technique of many-point densities developed in Chapter 5. To establish a new criterion insuring the self-organisation, we consider below the autowave processes (if any) occurring in the simplest systems -the Lotka and Lotka-Volterra models [22-24] (Section 2.1.1). It should be reminded only that standard chemical kinetics denies their ability to selforganisation either due to the absence of undamped oscillations (the Lotka model) or since these oscillations are unstable (the Lotka-Volterra model). 

Statement 1. Provided K(t) — K — const, i.e., neglecting change in time of the correlation functions, equations (8.2.12) and (8.2.13) of the concentration dynamics describe undamped concentration oscillations with the frequencies u < = y/aj3, dependent on the initial conditions. The de-... 

For y/3 1, one steady state exists and the regime is globally stable for all values of the Lewis number Lw. For 1 < y/3 < (y/3) and for sufficiently low values of Lewis number the system is again globally stable. Evidently for these conditions only one steady state occurs. For Lw > Lw, undamped oscillations exist. For supercritical values of y/3, y/3 > (y/3), and < a single steady state is stable or unstable according to the value of Lewis number. In the domain l>min < < max>... 

While the multiple steady-state phenomena may be, at least qualitatively, explained in terms of a simple one-step kinetic mechanism and interactions of the intraphase and interparticle heat and mass transfer (thermokinetic model), there is no acceptable explanation for the periodic activity (12). Since the values of the Lewis number are at least by a factor of 10 lower than those necessary to produce undamped oscillations, there is no doubt that the instability cannot be viewed in terms of mutual... 

In the homogeneous systems at low temperatures, oscillation phenomena were also observed for this reaction system. Similar observations were made for the CO and 02 system. In order to explain the oscillation mechanism mathematically, a complicated kinetic scheme was devised which is capable of predicting the sustained oscillations [Yang (26, 27)]. Apparently a similar complicated kinetic mechanism governs the undamped oscillations in the heterogeneous system, however, a pertinent analysis has not been performed in the literature so far. 

The classical problem of multiple solutions and undamped oscillations occurring in a continuous stirred-tank reactor, dealt with in the papers by Aris and Amundson (39), involved a single homogeneous exothermic reaction. Their theoretical analysis was extended in a number of subsequent theoretical papers (40, 41, 42). The present paragraph does not intend to report the theoretical work on multiplicity and oscillatory activity developed from analysis of governing equations, for a detailed review the reader is referred to the excellent text by Schmitz (3). To understand the problem of oscillations and multiplicity in a continuous stirred-tank reactor the necessary and sufficient conditions for existence of these phenomena will be presented. For a detailed development of these conditions the papers by Aris and Amundson (39) and others (40) should be consulted. 

If steam flow is linear with controller output, as it is in Fig. 8-50, undamped oscillations will be produced when the flow decreases by... 

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