Energy analysis of Duffing oscillators with quadratic damping: exact solutions

Oscillators with a Duffing-type restoring force and quadratic damping are dealt with in this paper. Four characteristic cases of this restoring force are analysed: hardening, softening, bistable and a pure cubic one. Their energy-displacement relationships are considered, and the corresponding closed-form exact solutions are obtained in terms of the incomplete Gamma function, which represent new results. Such results provide insight into damped dynamics of the class of system, including finding the phase trajectories as well as the comparison between these cases from the viewpoint of the energy loss per cycle.

S-, P- and R-striations as attractors for electron phase trajectories in spatially periodic resonance fields

A new point of view on the appearance of S-, P- and R-striations in a positive column of inert gases is proposed, based on a dynamic analysis of the resonance properties of electron phase trajectories in spatially periodic fields. The positive column may be considered as a resonator containing a set of resonant modes. Like a tuning fork, being disturbed, it responds with one of the modes, in particular with of S-, P-, or R-modes or striations, depending on the discharge conditions. The dynamic approach eliminates the difficulties of the kinetic theory associated with the long length of the solution of Boltzmann equation, which is much greater than the length of the positive column.

A Study On Methods for Determining Phase Space Reconstruction Parameters

Several pairs of algorithms were used to determine the phase space reconstruction parameters to analyze the dynamic characteristics of chaotic time series. The reconstructed phase trajectories were compared with the original phase trajectories of the Lorenz attractor, Rössler attractor, and Chens attractor to obtain the optimum method for determining the phase space reconstruction parameters with high precision and efficiency. The research results show that the false nearest neighbor method and the complex auto-correlation method provided the best results. The saturated embedding dimension method based on the saturated correlation dimension method is proposed to calculate the time delay. Different time delays are obtained by changing the embedding dimension parameters of the complex auto-correlation method. The optimum time delay occurs at the point where the time delay is stable. The validity of the method is verified through combing the application of correlation dimension, showing that the proposed method is suitable for the effective determination of the phase space reconstruction parameters.

Analysis and Synthesis of a Fuzzy Controller by the Phase Plane Method

The article is devoted to solving the problem of analysis and synthesis of a control system with a fuzzy controller by the phase plane method. The nonlinear transformation, built according to the Sugeno fuzzy model, is approximated by a piecewise linear characteristic consisting of three sections: two piecewise linear and one piecewise constant. This approach allows us to restrict ourselves to three sheets of phase trajectories, each of which is constructed on the basis of a second-order differential equation. Taking this feature into account, the technique of "stitching" of three sheets of phase trajectories is considered and an analytical base is obtained that allows one to determine the conditions for "stitching" of phase trajectories for various variants of piecewise-linear approximation of the characteristics of a fuzzy controller. In view of the specificity of the approximated model of the fuzzy controller used, useful analytical relations are given, with the help of which it is possible to calculate the time of motion of the representing point for each section with the involvement of the numerical optimization apparatus. For a variant of the approximation of three sections, a technique for synthesizing a fuzzy controller is proposed, according to which the range of parameters and the range of input signals are determined, at which an aperiodic process and a given control time are provided. On the model of the automatic control system of the drive level of the mechatronic module, it is shown that the study of a fuzzy system by such an approximated characteristic of a fuzzy controller gives quite reliable results. The conducted studies of the influence of the degree of approximation on the quality of control show that the approximated characteristic of a fuzzy controller gives a slight deterioration in quality in comparison with the smooth characteristic of a fuzzy controller. Since the capabilities of the phase plane method are limited to the 2nd order of the linear part of the automatic control system, the influence of the third order on the dynamics of the system is considered using the example of a mechatronic module drive. It is shown that taking into account the electric time constant leads to overshoot within 5-10 %. Such overshoot can be eliminated due to the proposed recommendations for correcting the static characteristic of the fuzzy controller.

Analysis of multi-stage running-in process of Sn-11Sb-6Cu alloy and AISI 1045 with phase trajectory plot

Running-in of the main bearings of diesel engine is a crucial process before service. Multi-stage running-in is a better way to enhance running-in quality and efficiency. In order to reveal the evolution of phase trajectory and compare the running-in quality, the running-in tests were performed with the material of bearing bush (Sn-11Sb-6Cu) and shaft (AISI 1045). The running-in quality was comprehensively evaluated via friction coefficient, phase trajectory and surface topography. Results indicate that the phase trajectories show a trend of stage-by-stage convergence. The multi-stage running-in can achieve a more stable attractor, lower friction coefficient and smoother surface, that is, a better running-in quality than the constant running-in scheme. This study provides a reference for formulating running-in specifications for sliding bearings.

Fuzzy Phase Trajectories in Hemispherical Resonator Gyroscopes

The paper considers elementary fuzzy oscillator models represented by hard and fuzzy second-order differential equations with hard and fuzzy initial conditions. Linear models describe wave processes in ring resonators of hemispherical resonator gyroscopes.We show that in the case 1 (a hard model with fuzzy initial conditions), when there is no internal friction (model 1), phase trajectories appear as a fuzzy centre shaped as an elliptical ring. When internal friction is present (model 2), phase trajectories appear as a fuzzy focus shaped as a circular logarithmic spiral. In the case 2, for a fuzzy hemispherical resonator gyroscope model with hard initial conditions, when there is no internal friction (model 1), a representative point of a fuzzy phase trajectory does not stop or increase its oscillations with time, meaning that the system is asymptotically unstable, while for the model 2 the origin singularity is a fuzzy stable focus. In the case 3, for a fuzzy hemispherical resonator gyroscope model with fuzzy initial conditions, when there is no internal friction (model 1), there is a fuzzy asymptotic instability in the model 1 of a hemispherical resonator gyroscope, while in the presence of internal friction (model 2), the phase trajectory is also a function of time and controls the asymptotic stability of the fuzzy model 2 of a hemispherical resonator gyroscope. Asymptotic stability is determined for all cases and models

Применение анализа совместных рекуррентностей к оценке фазовой синхронизации физиологических сигналов

Based on the analysis of joint recurrences, differences in phase synchronization between rhythmic photostimulation and brain responses were revealed in individuals with atrial fibrillation of paroxysmal and persistent types. As a measure of phase synchronization between two signals, the cross-correlation coefficient between the probabilities of recurrences of the corresponding phase trajectories is considered. With a lengthening of the lifetime of atrial fibrillation and an increase in the degree of decline in cognitive functions, the value of this coefficient increases for brain responses to theta-range frequencies.

Application of the dynamic Lorentz model to modeling the motion of a rigid body

A qualitative research of the field of phase trajectories of the system of dynamic equations of an absolutely rigid body was carried out, moving around the selected pole under the influence of gyroscopic, dissipative forces and Coriolis inertia forces. The equations of body motion are reduced to a dynamical system generating a Lorentz attractor. Under parametric constraints imposed on the equations of a dynamical system, the structure of its phase trajectories is described depending on the values of the system parameters.