Search results for "Harmonic balance"
showing 9 items of 9 documents
Analysis of oscillations in discontinuous Lurie systems via LPRS method
2019
We discuss advantages and limitations of the harmonic balance method and the locus of a perturbed relay system (LPRS) method in the problem of finding periodic oscillations. In this paper we present the results of using harmonic balance method and LPRS method while investigating a 3rd order dynamic system in Lurie form. In this system a symmetric periodic oscillation is found, while other two asymmetric periodic motions are not found using both methods. peerReviewed
The Egan problem on the pull-in range of type 2 PLLs
2021
In 1981, famous engineer William F. Egan conjectured that a higher-order type 2 PLL with an infinite hold-in range also has an infinite pull-in range, and supported his conjecture with some third-order PLL implementations. Although it is known that for the second-order type 2 PLLs the hold-in range and the pull-in range are both infinite, the present paper shows that the Egan conjecture may be not valid in general. We provide an implementation of the third-order type 2 PLL, which has an infinite hold-in range and experiences stable oscillations. This implementation and the Egan conjecture naturally pose a problem, which we will call the Egan problem: to determine a class of type 2 PLLs for …
Hidden oscillations in nonlinear control systems
2011
Abstract The method of harmonic linearization, numerical methods, and the applied bifurcation theory together discover new opportunities for analysis of hidden oscillations of control systems. In the present paper new analytical-numerical algorithm for hidden oscillation localization is discussed. Counterexamples construction to Aizerman's conjecture and Kalman's conjecture on absolute stability of control systems are considered.
Harmonic balance analysis of pull-in range and oscillatory behavior of third-order type 2 analog PLLs
2020
The most important design parameters of each phase-locked loop (PLL) are the local and global stability properties, and the pull-in range. To extend the pull-in range, engineers often use type 2 PLLs. However, the engineering design relies on approximations which prevent a full exploitation of the benefits of type 2 PLLs. Using an exact mathematical model and relying on a rigorous mathematical thinking this problem is revisited here and the stability and pull-in properties of the third-order type 2 analog PLLs are determined. Both the local and global stability conditions are derived. As a new idea, the harmonic balance method is used to derive the global stability conditions. That approach…
Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems
2011
The method of harmonic linearization, numerical methods, and the applied bifurcation the- ory together discover new opportunities for analysis of oscillations of control systems. In the present survey analytical-numerical algorithms for hidden oscillation localization are discussed. Examples of hidden attrac- tor localization in Chua's circuit and counterexamples construction to Aizerman's conjecture and Kalman's conjecture are considered.
Response Power Spectrum of Multi-Degree-of-Freedom Nonlinear Systems by a Galerkin Technique
2003
This paper deals with the estimation of spectral properties of randomly excited multi-degree-of-freedom (MDOF) nonlinear vibrating systems. Each component of the vector of the stationary system response is expanded into a trigonometric Fourier series over an adequately long interval T. The unknown Fourier coefficients of individual samples of the response process are treated by harmonic balance, which leads to a set of nonlinear equations that are solved by Newton’s method. For polynomial nonlinearities of cubic order, exact solutions are developed to compute the Fourier coefficients of the nonlinear terms, including those involved in the Jacobian matrix associated with the implementation o…
Analysis of oscillations in discontinuous Lurie systems via LPRS method
2019
We discuss advantages and limitations of the harmonic balance method and the locus of a perturbed relay system (LPRS) method in the problem of finding periodic oscillations. In this paper we present the results of using harmonic balance method and LPRS method while investigating a 3rd order dynamic system in Lurie form. In this system a symmetric periodic oscillation is found, while other two asymmetric periodic motions are not found using both methods.
Relay feedback systems - established approaches and new perspectives for application
2019
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Harmonic Balance Method and Stability of Discontinuous Systems
2019
The development of the theory of discontinuous dynamical systems and differential inclusions was not only due to research in the field of abstract mathematics but also a result of studies of particular problems in mechanics. One of the first methods, used for the analysis of dynamics in discontinuous mechanical systems, was the harmonic balance method developed in the thirties of the twentieth century. In our work, the results of analysis obtained by the method of harmonic balance, which is an approximate method, are compared with the results obtained by rigorous mathematical methods and numerical simulation.