Light-induced drift in a selectively excited photochemical reaction

It is shown that the velocity-selective excitation of a photochemical reaction is accompanied by drift of the components relative to each other. This new type of drift can be used to control the reaction and, in particular, to separate the reaction products and the reactants. The effect is not associated with the change in the transport cross sections upon excitation of the molecule, which in most cases is quite small.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 105–108, March–April, 1990.     

Nonlinear nonplanar dynamics of a parametrically excited inextensional elastic beam

The nonlinear dynamics of a clamped-clamped/sliding inextensional elastic beam subject to a harmonic axial load is investigated. The Galerkin method is used on the coupled bending-bending-torsional nonlinear equations with inertial and geometric nonlinearities and the resulting two second order ordinary differential equations are studied by the method of multiple time seales and by direct numerical integration. The amplitude equations are analyzed for steady and Hopf bifurcations. Depending on the amplitude of excitation, the damping and the ratio of principal flexural rigidities, various qualitatively distinct frequency response diagrams are uncovered and limit cycles and chaotic motions are found. In the truncated two-degree-of-freedom system the transition from periodic to chaotic amplitude-modulated motions is via the process of torus doubling and subsequent destruction of the torus.     

Analysis of global dynamics in a parametrically excited thin plate

The global bifurcations and chaos of a simply supported rectangular thin plate with parametric excitation are analyzed. The formulas of the thin plate are derived by von Karman type equation and Galerkin's approach. The method of multiple scales is used to obtain the averaged equations. Based on the averaged equations, the theory of the normal form is used to give the explicit expressions of the normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple program. On the basis of the normal form, a global bifurcation analysis of the parametrically excited rectangular thin plate is given by the global perturbation method developed by Kovacic and Wiggins. The chaotic motion of thin plate is also found by numerical simulation. The project supported by the National Natural Science Foundation of China (10072004) and by the Natural Science Foundation of Beijing (3992004)     

Technical stability of parametrically excited panels in a gas glow

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 6, pp. 73–81, June, 1989.     

Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The locked pendulum mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.     

Nonlinear dynamics of a resonant gas sensor

We develop a mathematical model for a resonant gas sensor made up of an microplate electrostatically actuated and attached to the end of a cantilever microbeam. The model considers the microbeam as a continuous medium, the plate as a rigid body, and the electrostatic force as a nonlinear function of the displacement and the voltage applied underneath the microplate. We derive closed-form solutions to the static and eigenvalue problems associated with the microsystem. The Galerkin method is used to discretize the distributed-parameter model and, thus, approximate it by a set of nonlinear ordinary-differential equations that describe the microsystem dynamics. By comparing the exact solution to that associated with the reduced-order model, we show that using the first mode shape alone is sufficient to approximate the static behavior. We employ the Finite Difference Method (FDM) to discretize the orbits of motion and solve the resulting nonlinear algebraic equations for the limit cycles. The stability of these cycles is determined by combining the FDM discretization with Floquet theory. We investigate the basin of attraction of bounded motion for two cases: unforced and damped, and forced and damped systems. In order to detect the lower limit of the forcing at which homoclinic points appear, we conduct a Melnikov analysis. We show the presence of a homoclinic point for a loading case and hence entanglement of the stable and unstable manifolds and non-smoothness of the boundary of the basin of attraction of bounded motion.     

Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink

Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass,a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra,and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed.The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.     

Global dynamics of a harmonically excited oscillator with a play: Numerical studies

In this paper a harmonically excited linear oscillator with a play is investigated. Direct numerical simulation and numerical continuation techniques were employed to study the system behaviour. To conduct the numerical analysis, the system differential equations were transformed into the autonomous form and were then solved using our newly developed in-house Matlab-based computational suite ABESPOL [1]. The results are presented in form of trajectories and Poincaré maps on the phase plane, bifurcation diagrams and basins of attraction. The bifurcation analysis was supported by a path following method. The influence of each system parameter (except gap) on the system dynamics was studied in detail. The bifurcations known as interior crisis and boundary crisis were observed and discussed in this work. Notably, the parameter regions where various types of grazing induced bifurcations occurred were detected and investigated.     

Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam

The sub- and super-critical dynamics of an axially moving beam subjected to a transverse harmonic excitation force is examined for the cases where the system is tuned to a three-to-one internal resonance as well as for the case where it is not. The governing equation of motion of this gyroscopic system is discretized by employing Galerkin’s technique which yields a set of coupled nonlinear differential equations. For the system in the sub-critical speed regime, the periodic solutions are studied using the pseudo-arclength continuation method, while the global dynamics is investigated numerically. In the latter case, bifurcation diagrams of Poincaré maps are obtained via direct time integration. Moreover, for a selected set of system parameters, the dynamics of the system is presented in the form of time histories, phase-plane portraits, and Poincaré maps. Finally, the effects of different system parameters on the amplitude-frequency responses as well as bifurcation diagrams are presented.     

Non-linear dynamics of a thermomechanical pseudoelastic oscillator excited by non-ideal energy sources

This paper discusses the non-linear dynamical response of a shape-memory non-ideal oscillator. The non-ideal excitation originates from a DC electric motor with limited power supply driving an unbalanced rotating mass. The restoring force provided by the shape-memory device is described by a thermomechanical model capable of accounting for the hysteretic behavior via the evolution of a suitable internal variable. The non-linear dynamic response of the system is investigated with the voltage as control parameter. Numerical simulations show the occurrence of regular and quasi-periodic motions, which are investigated via bifurcation diagrams and phase plane portraits. The 0–1 test is used for quantitative characterization of chaotic responses. The computation of basins of attraction points out the strong dependence of the response on small changes of initial conditions, along with meaningful modifications of competing basins with variations of the control parameter. Finally, variations of the mechanical and thermal parameters of the pseudoelastic oscillator are considered, with the aim to evaluating the effects produced by the non-ideal excitation source on the non-linear dynamics of the shape memory device.     

Global transient dynamics of nonlinear parametrically excited systems

In this paper we examine the response of a typical nonlinear system that is subjected to parametric excitation. Particular attention is paid to how basins of attraction evolve such that the global transient stability of the system may be assessed. We show that at a forcing level that is considerably smaller than that at which the steady-state attractor loses its stability, there may exist a rapid erosion and stratification of the basin, signifying a global loss of engineering integrity of the system.We also show, for a system near its equilibrium state, that the boundaries in parameter space can become fractal. The significance of such an analysis is not only that it corresponds to a failure locus for a system subjected to a sudden pulse of excitation, but since the phase-space basin is often eroded throughout its central region, the determination of basin boundaries in control space can often reflect the characteristics of the phase-space basin structure, and hence on the macroscopic level they provide information regarding the global transient stability of the system.     

On the dynamics of randomly excited nonlinear systems

Summary The problem of characterising the dynamics of randomly excited systems is examined. It is shown that the probability approach, though conceptually more rigorous, is difficult to apply to statistics other than normal ones. The direct method of Axelby is recalled and applied to a nonlinear system with random excitation characterised by a statistic of great interest in real physical systems. The application is developed parametrically with reference to a second order system for which the calculations are developed and the quantitative results discussed.

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Sommario Nella disamina del problema di caratterizzare il comportamento dinamico di sistemi eccitati da segnali casuali, si mette in rilievo come l'approccio probabilistico, sebbene concettualmente più rigoroso, sia di difficile applicazione ai casi statistici oltre che a quelli normali. Si richiama quindi il metodo diretto di Axelby che viene applicato ad un sistema non lineare eccitato da un segnale casuale. Lo sviluppo dell'applicazione, in termini parametrici, fa riferimento ad un sistema di secondo grado e i relativi risultati numerici vengono discussi nel loro significato quantitativo.

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Solution of the equations of motion for a selectively radiating gas in a shock layer

Numerical solutions are obtained for the system of integro-differential equations describing the flow of a viscous, heat-conducting, selectively radiating gas in the region between the shock wave and a blunt body. The calculations are made for bodies of radius from 0.1 to 3 m with stagnation temperature from 6000° to 15 000° K. As a result of the calculations the convective and radiative thermal fluxes in the vicinity of the stagnation point are obtained. The effect of injection on convective and radiative heat transfer is studied.The first calculations of radiative thermal fluxes in air were made about 10 years ago in [1,2]. However, the results did not take account of the effects of emission and reabsorption, nor the interaction of the convective and radiative heating processes. These effects have been studied primarily with the use of simplified models of a radiating gas. Most often the approximation used is that of a gray gas with absorption coefficient which is independent of wavelength ([3–6] and others).The appearance in the literature of quite detailed data on the selective spectral absorption coefficients of air over a wide temperature range [7,8] has made it possible to solve the direct problem of calculating the flow field of a selectively radiating gas behind a shock wave with account for all the effects mentioned above.     

Effect of electromagnetic actuations on the dynamics of a harmonically excited cantilever beam

The influence of electromagnetic actuators (EMAs) on the frequency response of a harmonically excited cantilever beam is investigated analytically, numerically and experimentally in this paper. Specifically, the intensity of the current generating the EMAs force is varied and its effect on the dynamic behavior of the system is analyzed. Analytical treatment based on perturbation analysis is performed on a simplified equation modeling the one mode vibration of the cantilever beam. Results indicated that EMAs produce a softening behavior in the system. Further, it is shown that as the current intensity of EMAs increases, the resonance curve shifts toward smaller values of frequency and the non-linear characteristic of the system becomes softer. The analytical predictions have been verified numerically and confirmed experimentally using a test rig.