Oscillation with computed period

Figure 9. Experimental period of oscillation in comparison with computer period...

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

Both the geometrical contribution and the Rayleigh contribution can thus be expressed in terms of material constants and the geometry of the lens, and therefore be directly compared. The complex summation of (7.39) and (7.42) enables V(z) to be computed and leads immediately to oscillations with period... 

The direct measurement of the interaction force between two mica surfaces1 indicated a large repulsion at relatively short distances, which could not be accounted for by the DLVO theory. This force was associated with the structuring of water in the vicinity of the surface.2 Theoretical work and computer simulations8-5 indicated that, in the vicinity of a planar surface, the density of the liquid oscillates with the distance, with a periodicity of the order of molecular size. This reveals that, near the surface, the liquid is ordered in quasi-discrete layers. When two plan ar surfaces approach each other at sufficiently short distances, the molecules of the liquid reorder in discrete layers, generating an oscillatory force.6... 

We also realized viscosity oscillations of a polymer solution based on different mechanisms [26]. It is known that a terpyridine ligand binds or dissociates with a Ru metal ion depending on the redox states of the Ru metal ion [29]. Generally, when the Ru metal ion is in the reduced Ru(II) state, the Ru(II) metal ion forms bis-complexes with terpyridine (Ru(terpy)2). However, when the Ru metal ion is in the oxidized Ru(III) state, the Ru(in) metal ion forms a monocomplex with terpyridine (Ru(terpy)) (Fig. 11.7). Therefore, supramolecular block copolymers have been made by using Ru(terpy)2 as a junction point [29], If the Ru-terpyridine complex acts as a catalyst of the BZ reaction, the redox oscillation may cause periodical binding/dissociation of the Ru-terpyridine complex. Recently, a theoretical computational simulation in the case that the Ru-terpyridine complex acts as a reversible crosslinking point of polymer network during the BZ reaction has been reported by Balazs et al. [30]. The... 

More complex oscillations have been found when the full TWC microkinetic model (Eqs. 1-31 in Table 1) has been used in the computations, cf. Fig. 4. The complex spatiotemporal pattern of oxidation intermediate C2H2 (Fig. 4, right) illustrates that the oscillations result from the composition of two periodic processes with different time constants. For another set of parameters the coexistence of doubly periodic oscillations with stable and apparently unstable steady states has been found (cf. Fig. 5). Even if LSODE stiff integrator (Hindmarsh, 1983) has been succesfully employed in the solution of approx. 10 ODEs, in some cases the unstable steady state has been stabilised by the implicit integrator, particularly when the default value for maximum time-step (/imax) has been used (cf. Fig. 5 right and Fig. 3 bottom). Hence it is necessary to give care to the control of the step size used, otherwise false conclusions on the stability of steady states can be reached. 

Statistical thermodynamics and computer simulations showed that the density profiles of hard-sphere and Lennard-J ones fluids normal to a planar interface oscillate about the bulk density with a periodicity of roughly one molecular diameter [1079-1086], The oscillations decay exponentially and extend over a few molecular diameters. In this range, the molecules are ordered in layers. The amplitude and range of density fluctuations depend on the specific boundary condition at the wall and on the size and interaction between the molecules. A steep repulsive wall-fluid... 

Typical values for the amplitude Vq are 10 -10 Nm. The decay length Xh ranges from 0.2 to 1.0 nm. In some cases, oscillatory forces with a period of 0.2-0.3 nm have also been observed [303,1146,1147,1157,1158]. Computer simulations confirm that oscillations are expected in water between surfaces such as mica [1102,1159,1160]. 

Another remarkable feature of thin film rheology to be discussed here is the quantized" property of molecularly thin films. It has been reported [8,24] that measured normal forces between two mica surfaces across molecularly thin films exhibit oscillations between attraction and repulsion with an amplitude in exponential growth and a periodicity approximately equal to the dimension of the confined molecules. Thus, the normal force is quantized, depending on the thickness of the confined films. The quantized property in normal force results from an ordering structure of the confined liquid, known as the layering, that molecules are packed in thin films layer by layer, as revealed by computer simulations (see Fig. 12 in Section 3.4). The quantized property appears also in friction measurements. Friction forces between smooth mica surfaces separated by three layers of the liquid octamethylcyclotetrasiloxane (OMCTS), for example, were measured as a function of time [24]. Results show that friction increased to higher values in a quantized way when the number of layers falls from n = 3 to n = 2 and then to M = 1. 

Simulate the three-CSTR system on a digital computer with an on-off feedback controller. Assume the manipulated variable is limited to +1 mol of A/lt around the stcadystate value. Find the period of oscillation and the average value of for values of the load variable of 0.6 and 1. 

When the same kind of electrode is introduced in a solution with a high pH (i.e., pH= 10) and a lower substrate concentration (first order kinetics), an oscillation in time of the measured pH inside the membrane spontaneously occurs. This enzyme, which has been extensively studied, does not give oscillation for any conditions of pH and substrate concentration. The period of oscillation is around one-half minute, and the oscillation is abolished by introducing an enzyme inhibitor. The phenomenon can be explained by the autocatalytic effect and by a feedback action of OH- diffusion in from the outside solution. The diffusion of this ion is quicker than the diffusion of the substrate. There is a qualitative agreement between the computer simulation and the experimental results. 

Repeating such computations for other values of p within the range of instability, we can see how the waveform, amplitude, and period vary with the reactant concentration. Figure 2.6(a) plots the maximum and minimum values of the concentration of A as a function of p. This representation emphasizes how the system oscillates about the unstable stationary state... 

The size and period of the oscillations, or of the corresponding limit cycle, varies with the dimensionless reactant concentration pi. We may determine this dependence in a similar way to that used in 2.5. Close to the Hopf bifurcation points we can in fact determine the growth analytically, but in general we must employ numerical computation. For now we will merely present the basic result for the present model. The qualitative pattern of response is the same for all values of ku < g. 

Fig. 4.10 Instantaneous nondimensional velocity profiles in a circular duct with an oscillating pressure gradient. The mean velocity, averaged over one full period, shows that the parabolic velocity profile or the Hagen-Poiseuille flow. These solutions were computed in a spreadsheet with an explicit finite-volume method using 16 equally spaced radial nodes and 200 time steps per period. The plotted solution is that obtained after 10 periods of oscillation.

In the present paper we study common features of the responses of chemical reactor models to periodic forcing, and we consider accurate methods that can be used in this task. In particular, we describe an algorithm for the numerical computation and stability analysis of invariant tori. We shall consider phenomena that appear in a broad class of forced systems and illustrate them through several chemical reactor models, with emphasis on the forcing of spontaneously oscillating systems. 

Fig. 7.3 Deterministic (a) and noisy (b) computer simulations of the time course of affective disorders showing the intervals between successive disease episodes (interval duration) as a function of a disease variable S and examples of episode generation from different disease states (figure modified after [2]). In deterministic simulations (a), there is a progression from steady state (S = 18) to subthreshold oscillations (S = 22) with immediate onset of periodic event generation at a certain value of S (slightly below S = 60). With further increase of S, the intervals between successive episodes are continuously...

The effect of anion ordering on the stability of FISDW phases has been confirmed quantitatively by calculation of the spin susceptibility following the standard approach and including an additional periodic potential with wave vector Q = (0, 5, 0) [135]. After this numerical computation even phases (N = 0, 2) are suppressed, whereas odd phases N = 1,3 are not. The same model also explains the normal-phase reentrance above 17 T. However, the predicted oscillation for Tc versus the magnetic field in the N = 0 phase is still lacking in the experimental data. 

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