Forced Periodic Oscillations

A related oscillatory phenomenon is that in which the concentration of one or more reactants, fed to a flow reactor, is varied in time. Such forced periodic feed oscillations during oxidation reactions have now been studied by a number of authors. It is found that not only can conversion be increased but the selectivity of certain parallel reactions can be improved, which may be of value in industrial applications. Cutlip and Abdul-Karem and Jain  

Morton and M. G. Goodman, Proceedings of Workshop on Modelling of Chem. React. Systems, Heidelberg, 1/5 Sept., 1980, Modelling of Chem. React. Systems, Springer Series in Chemical Physics, 1981, Vol. 18. 

Goodman, C. N. Kenney, W. Morton, and M. B. Cutlip, 2nd World Congress of Chem. Eng., 9th Interamerican Congress of Chem. Eng., 31st Canadian Eng. Conf., Montreal, Canada, 1981. 

Renken, M. Miiller, and C. Wandrey, Proceedings of 4th Interntional Conference on Chemical Reaction Engineering, Heidelberg, 1976, p. 107. 

Another advantage of forced periodic feed experiments, which has not been fully exploited so far, is that the technique could be used for kinetic model discrimination, a technique in which large deviations could be induced into calculated reponses between rival models under consideration. Hawkins has carried out experiments on oxidation of CO for discriminating between several Hougen and Watson rival models. Cutlip et al have compared experimental forced periodic feed CO oxidation experimental transients with simulations using an elementary step model and compared theory with experiment in studies of the variation of the conversion as a function of time period of the forced oscillation. 

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There are several other comparable rheological experimental methods involving linear viscoelastic behavior. Among them are creep tests (constant stress), dynamic mechanical fatigue tests (forced periodic oscillation), and torsion pendulum tests (free oscillation). Viscoelastic data obtained from any of these techniques must be consistent data from the others. 

Cutlip, M. B., Hawkins, C. J., Mukesh, D., Morton, W. and Kenney, C. N., 1983, Modelling of forced periodic oscillations of carbon monoxide over platinum catalyst. Chem. Eng. Commun. 22, 329-344. 

The modification of the surface force apparatus (see Fig. VI-4) to measure viscosities between crossed mica cylinders has alleviated concerns about surface roughness. In dynamic mode, a slow, small-amplitude periodic oscillation was imposed on one of the cylinders such that the separation x varied by approximately 10% or less. In the limit of low shear rates, a simple equation defines the viscosity as a function of separation... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Surface force apparatus has been applied successfully over the past years for measuring normal surface forces as a function of surface gap or film thickness. The results reveal, for example, that the normal forces acting on confined liquid composed of linear-chain molecules exhibit a periodic oscillation between the attractive and repulsive interactions as one surface continuously approaches to another, which is schematically shown in Fig. 19. The period of the oscillation corresponds precisely to the thickness of a molecular chain, and the oscillation amplitude increases exponentially as the film thickness decreases. This oscillatory solvation force originates from the formation of the layering structure in thin liquid films and the change of the ordered structure with the film thickness. The result provides a convincing example that the SFA can be an effective experimental tool to detect fundamental interactions between the surfaces when the gap decreases to nanometre scale. 

K. Tomita. Periodically forced nonlinear oscillators. A.V. Holden, Princenton Univ. Press. 

Fig. 13.9. The forced Takoudis-Schmidt-Aris model with a forcing frequency twice that of the natural oscillation (a) zero-amplitude forcing (autonomous oscillation and limit cycle) (b) r, = 0.002 (c) r, = 0.003 (d) r, = 0.004 (e) r, = 0.005 (f) rr = 0.006 (g) rf = 0.007 (h) rf = 0.01. Traces show the time series 0p(t) over 10 natural periods (or 20 forcing periods) and the associated limit cycle in the 0 -6, plane.

In the previous subsection, the forcing frequency was exactly twice the natural oscillatory frequency. Thus the motion around one oscillation gives exactly two circuits of the forcing cycle for one revolution of the natural limit cycle. The full oscillation of the forced system has the same period as the autonomous cycle and twice the forcing period. The concentrations 0p and 6r return to exactly the same point at the top of the cycle, and subsequent oscillatory cycles follow the same close path across the toroidal surface. This is known as phase locking or resonance. We can expect such locking, with a closed loop on the torus, whenever the ratio of the natural and forcing... 

V-shaped regions, but in general close up again at sufficiently large amplitude. The number marked in each of the tongues indicates how many forcing periods are required for one full oscillation or, equivalently, how many points appear in the stroboscopic map. 

For sufficiently large forcing amplitudes the oscillation becomes completely entrained, with a period exactly equal to one forcing period, whatever that value of a>/a>0. The entrainment may arise from a phase-locked response—as seen previously in Fig. 13.9—or from a quasi-periodic pattern. The boundary for full entrainment appears as an almost straight line with positive slope of oj/oj0 > 1 and negative slope for oj/oj0 < 1. 

Glass, L., Guevara, M., Belair, J. and Shrier, A., 1984, Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A 29,1348-1357. 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

A schematic view of the nanomechanical GMR device to be considered is presented in Fig. 1. Two fully spin-polarized magnets with fully spin-polarized electrons serve as source and drain electrodes in a tunneling device. In this paper we will consider the situation when the electrodes have exactly opposite polarization. A mechanically movable quantum dot (described by a time-dependent displacement x(t)), where a single energy level is available for electrons, performs forced harmonic oscillations with period T = 2-k/uj between the leads. The external magnetic field is perpendicular to the orientation of the magnetization in both leads. 

A very rich scenario was, on the other hand, observed with Pt(110) if perturbed under conditions for which regular autonomous oscillations with well-defined frequencies existed (91-93). Typically, these autonomous oscillations were established at fixed external parameters and then one of the partial pressures was periodically modulated by use of a feedback-regulated gas inlet system with frequencies up to 0.5 s l and relative amplitudes around 1% (31, 33). Following the pioneering mathematical treatment of forced nonlinear oscillations by Kai and Tomita (SO), the results can be rationalized in terms of a dynamic phase diagram characterizing the response of the system as a function of the amplitude A and of the period of the pressure modulation Tcx with respect to that of the... 

Some processes may have forces operating far away from equilibrium where the linear phenomenological equations are no longer applicable. Such a domain of irreversible phenomena, such as some chemical reactions, periodic oscillations, and bifurcation, is examined by extended nonequilibrium thermodynamics. Extending the methods of thermodynamics to treat the linear and nonlinear phenomena, and such dissipative structures are attracting scientists from various disciplines. 

Chaotic behavior and synchronization in heterogeneous catalysis are closely related. Partial synchronization can lead to a complex time series, generated by superposition of several periodic oscillators, and can in some cases result in deterministically chaotic behavior. In addition to the fact that macroscopically observable oscillations exist (which demonstrates that synchronization occurs in these systems), a number of experiments show the influence of a synchronizing force on all the hierarchical levels mentioned earlier. Sheintuch (294) analyzed on a general level the problem of communication between two cells. He concluded that if the gas-phase concentration is the autocatalytic variable, then synchronization is attained in all cases. However, if the gas-phase concentration were the nonautocatalytic variable, then this would lead to symmetry breaking and the formation of spatial structures. When surface variables are the model variables, the existence of synchrony is dependent upon the size scale. Only two-variable models were analyzed, and no such strict analysis has been provided for models with two or more surface concentrations, mass balances, or heat balances. There are, however, several studies that focused on a certain system and a certain synchronization mechanism. 

The time average of W denoted as (W) is found by integrating the sinusoidal factors over one period of the field E and then using standard results from the theory of forced damped oscillations. (W) can be calculated as... 

In physics such oscillatory objects are denoted as self-sustained oscillators. Mathematically, such an oscillator is described by an autonomous (i.e., without an explicit time dependence) nonlinear dynamical system. It differs both from linear oscillators (which, if a damping is present, can oscillate only due to external forcing) and from nonlinear energy conserving systems, whose dynamics essentially depends on initial state. Dynamics of oscillators is typically described in the phase (state) space. Periodic oscillations, like those of the clock, correspond to a closed attractive curve in the phase space, called the limit cycle. The limit cycle is a simple attractor, in contrast to a strange (chaotic) attractor. The latter is a geometrical image of chaotic self-sustained oscillations. 

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