APPLICATION OF MODIFIED CONVERSION METHOD TO A NONLINEAR DYNAMICAL SYSTEM
Read the full article
For citation: Melnikov G.I., Ivanov S.E., Melnikov V. G., Malykh K.S. Application of modified conversion method to a nonlinear dynamical system. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 1, pp. 149–154 (in Russian)
The paper deals with a mathematical model of dynamical system with single degree of freedom, presented in the form of ordinary differential equations with nonlinear parts in the form of polynomials with constant and periodic coefficients. A modified method for the study of self-oscillations of nonlinear mechanical systems is presented. A refined method of transformation and integration of the equation, based on Poincare-Dulac normalization method has been developed. Refinement of the method lies in consideration of higher order nonlinear terms by Chebyshev economization technique that improves the accuracy of the calculations. Approximation of the higher order remainder terms by homogeneous forms of lower orders is performed; in the present case, it is done by cubic forms. An application of the modified method for the Van-der-Pol equation is considered as an example; the expressions for the amplitude and the phase of the oscillations are obtained in an analytical form. The comparison of the solution of the Van-der-Pol equation obtained by the developed method and the exact solution is performed. The error of the solution obtained by the modified method equals to 1%, which shows applicability of the developed method for analysis of self-oscillations of nonlinear dynamic systems with constant and periodic parameters.
1. Blanchard P., Devaney R.L., Hall G.R. Differential Equations. 2nd ed. Brooks Cole, 2011, 864 p.
2. Warminski J., Lenci S., Cartmell M.P., Rega G., Wiercigroch M. Nonlinear Dynamic Phenomena in Mechanics. Berlin, Springer-Verlag, 2012, 276 p.
3. Teschl G. Ordinary Differential Equations and Dynamical Systems. American Mathematical Society, 2012, vol. 140, 356 p.
4. Demidovich B.P., Maron I.A., Shuvalova E.Z. Chislennye Metody Analiza. Priblizhenie Funktsii, Differentsial'nye i Integral'nye Uravneniya [Numerical Methods of Analysis. Functions Approximation, Differential and Integral Equations]. St. Petersburg, Lan' Publ., 2010, 400 p.
5. Gesztesy F., Holden H., Michor J., Gerald T. Soliton Equations and their Algebro-Geometric Solutions. Cambridge University Press, 2008, vol. 2, 438 p.
6. Melnikov G.I. Dinamika Nelineinykh Mekhanicheskikh i Elektromekhanicheskikh Sistem [Nonlinear Dynamics of Mechanical and Electromechanical Systems]. Leningrad, Mashinostroenie Publ., 1975, 198 p.
7. Ivanov S.E., Melnikov G.I. Avtonomizatsiya nelineinykh dinamicheskikh system [Off-line interaction of the nonlinear dynamic systems]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2012, no. 1 (89), pp. 151–156.
8. Bibikov Yu.N. Kurs Obyknovennykh Differentsial'nykh Uravnenii [Course of Ordinary Differential Equations]. St. Petersburg, Lan' Publ., 2011, 304 p.
9. Melnikov V.G. Chebyshev economization in transformations of nonlinear systems with polynomial structure. Proc. 14th WSEAS Int. Conf. on Systems. Corfu Island, Greece, 2010, vol. 1, pp. 301–303.
10. Melnikov V.G. Mnogochlennye preobrazovaniya nelineinykh sistem upravleniya [Polynomial transformations of non-linear control systems]. Izv. vuzov. Priborostroenie, 2007, vol. 50, no. 5, pp. 20–25.
11. Melnikov V.G., Melnikov G.I., Ivanov S.E. Komp'yuternye Tekhnologii v Mekhanike Pribornykh System [Computer Technologies in Mechanics of Instrument Systems]. St. Petersburg, SPbSU ITMO Publ., 2006, 127 p.
12. Ivanov S.E. Algoritmicheskaya realizatsiya metoda issledovaniya nelineinykh dinamicheskikh sistem [Algorithmic realization for research method of the nonlinear dynamic systems] Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2012, no. 4 (80), pp. 90–92.
13. Melnikov V.G. Preobrazovanie dinamicheskikh mnogochlennykh sistem s primeneniem approksimatsii Chebysheva [Transformation of dynamic polynomial systems by Chebyshev approximation]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2012, no. 4 (80), pp. 85–90.
14. Melnikov V.G. Chebyshev economization in Poincare-Dulac transformations of nonlinear systems. Nonlinear Analysis, Theory, Methods and Applications, 2005, vol. 63, no. 5–7, pp. e1351–e1355. doi: 10.1016/j.na.2005.01.080