Forced oscillators

Gyroscopic moments Drive torque Cyclic forces Oscillating torques... 

M. Iwamatsu. A molecular theory of solvation force oscillations in nonpolar Uquids. J Colloid Interface Sci 204 374-388, 1998. 

There is less information available in the scientific literature on the influence of forced oscillations in the control variables in polymerization reactions. A decade ago two independent theoretical studies appeared which considered the effect of periodic operation on a free radically initiated chain reaction in a well mixed isothermal reactor. Ray (11) examined a reaction mechanism with and without chain transfer to monomer. 

Combustion-generated noise is a problem in itself. However, if an acoustic wave can interact with the combustion zone, so that the heat release rate is a function of the acoustic pressure, q = f p ), then Equation 5.1.14 describes a forced oscillator, whose amplitude can potentially reach a high value. The condition for positive feedback was first stated by Rayleigh [23] ... 

The Maxwell theory of X-ray scattering by stable systems, both solids and liquids, is described in many textbooks. A simple and compact presentation is given in Chapter 15 of Electrodynamics of Continuous Media [20]. The incident electric and magnetic X-ray helds are plane waves Ex(r, f) = Exo exp[i(q r — fixO] H(r, t) = H o exp[/(q r — fixO] with a spatially and temporally constant amplitude. The electric field Ex(r, t) induces a forced oscillation of the electrons in the body. They then act as elementary antennas emitting the scattered X-ray radiation. For many purposes, the electrons may be considered to be free. One then finds that the intensity /x(q) of the X-ray radiation scattered along the wavevector q is... 

How relevant are these phenomena First, many oscillating reactions exist and play an important role in living matter. Biochemical oscillations and also the inorganic oscillatory Belousov-Zhabotinsky system are very complex reaction networks. Oscillating surface reactions though are much simpler and so offer convenient model systems to investigate the realm of non-equilibrium reactions on a fundamental level. Secondly, as mentioned above, the conditions under which nonlinear effects such as those caused by autocatalytic steps lead to uncontrollable situations, which should be avoided in practice. Hence, some knowledge about the subject is desired. Finally, the application of forced oscillations in some reactions may lead to better performance in favorable situations for example, when a catalytic system alternates between conditions where the catalyst deactivates due to carbon deposition and conditions where this deposit is reacted away. 

An important example in mechanical and electrical systems is that of forced oscillations of a vibrational system. If ah external force f(t) is imposed on the mechanical oscillator considered above, Eq. (40) becomes... 

Bi, H. T., Grace, J. R., and Zhu. J., Propagation of Pressure Waves and Forced Oscillations in Gas-solid Fluidized Beds and their Influence on Diagnostics of Local Hydrodynamics, Powder Technol., 82 239 (1995)... 

In a general way, when the radiation from two laser beams with frequencies (oi and (02 are superimposed in a medium with a sufficiently large nonlinear part of the susceptibility, each atom is induced to undergo forced oscillations and generates... 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

G.R. Meira. Forced oscillations in continuous polymerization reactors and molecular weight distribution control. A survey. J. Macromol. Sci.- Rev. Macro-mol. Chem., 20(2) 207-241, 1981. 

Sivasegaram, S., R-F. Tsai, and J. H. Whitelaw. 1995. Control of combustion oscillations by forced oscillation of part of the fuel. Combustion Science Technology 105 67-83. 

Tangent of the phase angle difference (< between stress and strain during forced oscillations. 

Ratio of the amplitude of the stress in phase with the strain (cxo cos d) to the amplitude of the strain (yo) in the forced oscillation of a material... 

Ratio of complex stress ( ct ) to complex strain (y ) in the forced oscillation of material M =... 

Note 1 See Definition 5.10 for the definition of a forced oscillation in which... 

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