Damped oscillator

The damped oscillations with a period of about 1 nm corresponded well to the size of the OMCST molecules and extended to about 5 nm, or about five solvent layers. An example of these forces for the same system from Christenson and Blom [68] is shown in Fig. VI-7. 

Kerner E H 1958 Note on the forced and damped oscillator In quantum mechanics Can. J. Phys. 36 371-7... 

Equation (2.41) describes either damped oscillations (at tls < 2do) or exponential relaxation OItls > 2do). Since tls grows with increasing temperature, there may be a cross-over between these two regimes at such that 2h QoJ Ao)coth P hAo) = 2Aq. If the friction coefficient... 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

Nevertheless, Leggett et al. [1987] have argued, with some provisos, that [with the initial condition (O) = 1] and C(t) may be practically taken the same. If C(t) then obeys the damped oscillator equation (2.41), then the inelastic part of the structure factor has the Lorentzian form with the peaks at [Pg.25]

In the real space the correlation function (6) exhibits exponentially damped oscillations, and the structure is characterized by two lengths the period of the oscillations A, related to the size of oil and water domains, and the correlation length In the microemulsion > A and the water-rich and oil-rich domains are correlated, hence the water-water structure factor assumes a maximum for k = k 7 0. When the concentration of surfac-... 

In the microemulsion the role of A is played by the period of damped oscillations of the correlation functions (Eq. (7)). The surface-averaged Gaussian curvature Ky, = 2t x/ S is the topological invariant per unit surface area. Therefore the comparison between Ryyi = Kyy / in the disordered microemulsion and in the ordered periodic phases is justified. We calculate here R= Since K differs for diffused films from cor-... 

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

At x = 0, B(0) is equal to the uniform density of electrons. The first term of the right hand side makes a bulk peak around x = 0. It sharply damps outside, because the k-integration over the occupied states is similar in structure to the following damping oscillation function ... 

It should be noted that in the case of a damped oscillator, the condition given by Eq. (60) yields a resonant frequency that does not correspond to its natural frequency, as... 

As a somewhat more complicated example, consider the electrical circuit of the damped oscillator shown in Fig. 5-3. The charge q(t) is determined by Eq. (5-45), namely,... 

If V is localized, say, near the origin, then for locations far from the origin, this behaves like j 2kFr)/r2, which means as cos(2kFr)/ r3. These damped oscillations of frequency 2kF are the Friedel oscillations, which always arise when an electron gas is perturbed the frequency of oscillation comes from the kink in the dielectric function at 2kF. We see the Friedel oscillations (in planar rather than in spherical geometry) for the electron gas at a hard wall [Eq. (12) et seq.] and for the electron density at the surface of a metal. 

Figure 5 Schematic diagram of a torsion pendulum and a typical damped oscillation curve. Modified from L. E. Nielsen,...

Although, strictly speaking, total compensation cannot be achieved, partial compensation may well lead to a negligible residual ohmic drop, although the presence of damped oscillations does not yet prevent detection of the Faradaic current. Such a situation is typically reached for ARu = 417 in the system shown in Figure 1.8. This figure illustrates how the positive feedback ohmic drop compensation should be carried out in practice. The procedure may be summarized as follows ... 

With real cells, the resistance Ru is measured by augmenting positive feedback until sustained oscillations are observed. Then R = Re and the value of Ru is obtained by a simple reading of Re. The amount of positive feedback is then decreased back to a new value of Re so as to obtain damped oscillations compatible with the measure of the Faradaic component of the current, as in Figure 1.8 for ARu =412. The remaining resistance is thus obtained as ARu = R — Re. In a number of cases, the residual ohmic drop is negligible. If not, it may be taken into account in a simulation of the voltammograms, as depicted below. 

Figure 24. The nullclines (upper panels, gray lines) and time courses (lower panels) for oscillatory solutions of the minimal model of glycolysis. Left panels Damped oscillations. The...

M. R. Roussel, Slowly reverting enzyme inactivation A mechanism for generating long lived damped oscillations. J. Theor. Biol. 195, 233 244 (1998). 

A particular question of interest is whether the DNA torsional motions observed on the nanosecond time scale are overdamped, as predicted by simple Langevin theory, and as observed for Brownian motions on longer time scales, or instead are underdamped, so that damped oscillations appear in the observed correlation functions. A related question is whether the solvent water around the DNA exhibits a normal constant viscosity on the nanosecond time scale, or instead begins to exhibit viscoelastic behavior with a time-, or frequency-, dependent complex viscosity. In brief, are the predictions for... 

The feedstock to the TCC reactor varies in its coke-forming properties. This is an important source of disturbance in the kiln operation. Figure 23 shows the effect of a 20% change in coke. The temperature above the air inlet responds with a damped oscillation of the temperature before reaching a steady state 40°F (22°C) above the previous steady state. The temperature at the bottom of the lower zone rises monotonically 75°F (42°C) to the new steady-state value. This rise in catalyst temperature will influence the performance of the reactor, since the catalyst is returned to the top of the reactor after some heat losses from cooling coils in the bottom of the kiln and in the air lift and the separator. The change of temperature... 

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