Periodic Oscillations In Nonlinear Systems Research Articles

Інтелектуальний регулятор запасу газодинамічної стійкості компресора авіаційного ГТД

The gas-dynamic stability of compressors of aircraft gas turbine engines is one of the most important conditions that determine their reliability and level of flight safety. Unstable operation of the compressor in the engine system (surge) leads to loss of thrust accompanied by an increase in gas temperature in front of the turbine and increased vibration because of large amplitudes of pressure pulsations and mass flow through the engine path. To improve the parameters of ACS aviation gas turbine engines are increasingly using regulators built using fuzzy logic algorithms. The implementation of fuzzy control algorithms differs from classical algorithms, which are based on the concept of feedback and reproduce a given functional dependence or differential equation. These functions are related to the qualitative assessment of system behavior, analysis of the current changing situation, and the selection of the most appropriate for the situation supervision of the gas turbine engine. This concept is called advanced management. ACS gas turbine engines with fuzzy regulators are nonlinear systems in which stable self-oscillations are possible. Approximate methods are used to solve the problems of analysis of periodic oscillations in nonlinear systems. Among them, the most developed in theoretical and methodological aspects is the method of harmonic linearization. The scientific problem is solved in the work – methods of synthesis of intelligent control system with the fuzzy regulator as a separate subsystem based on the method of harmonic linearization and design on its basis of fuzzy ACS reserve of gas-dynamic stability of aviation gas turbine engine. Based on the analysis of the principles of construction of fuzzy control systems, it is shown that the use of fuzzy logic provides a new approach to the design of control systems for aviation gas turbine engines in contrast to traditional control methods. It is shown that the fuzzy controller, as the only essentially nonlinear element when using numerical integration methods, can be harmonically linearized. Harmonic linearization allows using the oscillation index to assess the quality in the separate channels of fuzzy ACS aviation gas turbine engines. A fuzzy expert system has been developed for optimal adjustment of the functions of belonging of typical fuzzy regulators according to quality criteria to transients.

Forced Oscillations in Systems with Arg min Type Operators

The properties of the i>Argmin operator and periodic oscillations in nonlinear systems described by ordinary differential equations and equations with a delayed argument whose right sides are i>Argmin operators are investigated.

An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems

In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.

On verification of stability of periodic oscillations in nonlinear systems

A method for verification of stability of periodic oscillations in nonlinear systems is proposed. The main oscillation is represented in the exponential form and is substituted into the initial differential equation. Then the approximating differential equation with respect to the same exponent is taken. The terms representing higher than the first-order derivatives are omitted in the corresponding operator equation. The real and imaginary parts of this simplified operator equation are separated and give two nonlinear differential equation of the first order for the amplitudes of the main oscillation. These equations can be used for calculation of transients. The differential equations of variations obtained from the equations of the amplitudes are used for verification of stability of the main oscillation.

Stability of periodic oscillations of nonlinear systems

Stability of periodic oscillations of nonlinear systems

The describing function method and free vibrations in nonlinear systems

The most popular method of investigation of periodic oscillations in nonlinear systems is the describing function method. Two theorems on the existence of a free vibration in a given neighbourhood of the approximate vibration obtained by this method are presented in the paper. Moreover, for a typical transfer function K(jω) of the linear part of the system and a typical describing function of the non-linear element, the theorem on the estimation of frequency of the free vibration is given.