Підвищення ефективності розрахунку стаціонарних періодичних режимів електронних кіл на основі спектрального аналізу сигналів

Спектральні характеристики сигналів лежать в основі вибору кроку їх дискретизації у часі, якщо для обчислення цих сигналів застосовується метод аналізу стаціонарних періодичних режимів нелінійних елект-ронних кіл на основі ряду Котельникова-Шеннона. Оскільки отримання самих сигналів і є метою застосування цього методу, виникає замкнене коло: щоб отримати сигнали, необхідно визначити крок дискретизації у часі, а щоб визначити крок дискретизації, необхідно знати спектральні властивості сигналу, а саме верхню граничну частоту, яка обмежує його частотний спектр. У роботі запропонований метод визначення кроку дискретизації сигналу на основі обчислення часткової реакції схеми на пробний сигнал у вигляді функції Хевісайда. Реакція визначається будь-яким чисельним методом, придатним для розв'язування систем нелінійних диференціальних рівнянь першого порядку. За спектральною густиною енергії реакції визначається верхня гранична частота і крок дискретизації сигналу у часі, який визначає необхідну кількість відліків. Наведені приклад застосування запропонованого методу та його порівняльна ефективність.A key problem in the periodic steady-state analysis of electronic circuits is that the duration of transient pro-cesses in a circuit might be much larger than the period of a steady-state response. Thus, application of traditional transient methods becomes ineffective due to a huge amount of redundant computations, and special periodic-steady state methods should be used. The method for periodic steady-state analysis using the Kotelnikov-Shannon series is a time-domain method that has proved to be effective for a such type of circuits. In this method the unknown signals are expanded in the Kotelnikov-Shannon series and the derivatives of these signals are calculated as the derivatives of the series. A matrix form of the derivatives approximation leads to simple matrix expressions in a mathematical model. When using the method for periodic steady-state analysis of non-linear circuits using the Kotelnikov-Shannon series to find the steady-state response of a circuit, a time discretization step is chosen based on the spectral characteristics of the signals. As far as the goal of the method is to calculate the unknown signals in a circuit, a vicious circle occurs: to calculate the signals, the time discretization step has to be chosen, and to choose the time discretization step, the spectral characteris-tics of the signals have to be known, namely the upper frequency in these characteristics. In order to choose the time discretization step, we propose to calculate a partial transient response of a circuit for an input signal of the form of the Heaviside step function, which is usually used to obtain a step response of a linear circuit. The response is calculated with any method, suitable for solving a system of non-linear ordinary differential equations, which usually represents the mathematical model of a circuit. The upper frequency in the spectral characteristic of the partial transient response depends on the duration of the computational domain. The upper frequency versus the dura-tion of the computational domain dependency can be approximated with a hyperbolic function. Thus, calculating few values of the upper frequency at different durations of the computational domain, the value of the upper frequency when the duration of the computational domain is equal to the period of a steady-state response can be forecasted using the hyperbolic approximation

An Investigation into Dynamic Stability of Waterborne Aircraft on Take-off and Landing

This research contributes to the knowledge of dynamic stability of waterborne aircraft and ground effect phenomenon. Hereto an analytical and computational study has been performed during which the motion of waterborne aircraft in take-off and landing is predicted. An analytical tool that can be used to predict the nonlinear heaving and pitching motions of seaplanes is presented. First, the heaving and pitching equations of motion are presented in their general Lagrangian form. Then, the equations are simplified to a form of nonlinear equations known as the forced Duffing equations with cubic nonlinearity. The system of motion is assumed to be driven by a sinusoidal head sea wave. The equations are then solved using the Poincare-Lindstedt perturbation method. The analytical solution is verified with CFD simulations performed on Ansys Fluent and AQWA. The solution is used to extend Savitsky’s method to predict porpoising which is a form of dynamic instability found in high-speed boats and seaplanes. The results of the analytical tool are in very good agreement with the results obtained from Fluent and AQWA. However, as the motion is assumed to be 2D in Fluent, heaving amplitude is slightly over predicted. Moreover, the frequency of oscillations of the 2D simulations is found to be unsteady. The unsteadiness in frequency increases with the increase of the length of the hull. Nevertheless, the amplitude of the pitch motion is slightly less than the amplitude predicted analytically. The discrepancy in the results is due to the characteristics of the 2D simulations that assumes that sea water will only pass underneath the hull which will make the buoyancy force greater as less damping is experienced. This is also a consequence of the fact that parameters within the analytical model of heave and pitch are calculated using a strip theory which considers only hydrodynamic effects, while Fluent also incorporate aerodynamic contributions. Similarly, AQWA is a 3D platform that only takes in consideration hydrodynamic effects. Hence, the results of AQWA are slightly less in amplitude than that predicted analytically. In addition, it was found that the frequency of oscillations obtained using AQWA increases with time while in the analytical approach, the frequency of oscillations can only be assumed to be constant for the whole period of motion. The increment in the oscillations indicates that porpoising is taking place. Nevertheless, it was found that heaving terms control the amplitude of motion and pitching terms control frequency of oscillations. The pitching nonlinear term has an effect on the amplitude of motion but not significant. Finally, the analytical method of Savitsky that is used to predict the porpoising stability limit is extended to find the porpoising limit for a wider range of pitch angles. In addition, the porpoising limit is predicted for a planing hull that is moving under the effect of head sea waves. When the seaplane is moving through head sea waves at a fixed pitch angle, porpoising takes place at a lower speed than what Savitsky has predicted

Nonlinear Dynamics

This volume covers a diverse collection of topics dealing with some of the fundamental concepts and applications embodied in the study of nonlinear dynamics. Each of the 15 chapters contained in this compendium generally fit into one of five topical areas: physics applications, nonlinear oscillators, electrical and mechanical systems, biological and behavioral applications or random processes. The authors of these chapters have contributed a stimulating cross section of new results, which provide a fertile spectrum of ideas that will inspire both seasoned researches and students

Нелінійна динаміка — 2013

The book of Proceedings includes extended abstracts of presentations on the Fourth International conference on nonlinear dynamics

14th Conference on Dynamical Systems Theory and Applications DSTA 2017 ABSTRACTS

From Preface: This is the fourteen time when the conference “Dynamical Systems – Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and the Ministry of Science and Higher Education. It is a great pleasure that our invitation has been accepted by so many people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcome nearly 250 persons from 38 countries all over the world. They decided to share the results of their research and many years experiences in the discipline of dynamical systems by submitting many very interesting papers. This booklet contains a collection of 375 abstracts, which have gained the acceptance of referees and have been qualified for publication in the conference proceedings [...]

Нелінійна динаміка — 2013

The book of Proceedings includes extended abstracts of presentations on the Fourth International conference on nonlinear dynamics

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding