Abstract
A non-linear seales method is presented for the analysis of strongly non-linear oseillators of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbdiqb-Hha4zaadaGaey4kaSIa% am4zaiaacIcacqWF4baEcaGGPaGae8xpa0JaeqyTduMaamOzaiaacI% cacqWF4baEcqWFSaalcuWF4baEgaGaaiaabMcaaaa!4FEC!\[\ddot x + g(x) = \varepsilon f(x,\dot x{\text{)}}\], where g(x) is an arbitrary non-linear function of the displacement x. We assumed that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbdiab-Hha4jaacIcacqWF0baD% cqWFSaalcqaH1oqzcaGGPaGaeyypa0Jae8hEaG3aaSbaaSqaaiaaic% daaeqaaOGaaiikaiabe67a4jaacYcacqaH3oaAcaGGPaGaey4kaSYa% aabmaeaacqaH1oqzdaahaaWcbeqaaiaad6gaaaaabaGaamOBaiabg2% da9iaaigdaaeaacaWGTbGaeyOeI0IaaGymaaqdcqGHris5aOGae8hE% aG3aaSbaaSqaaiab-5gaUbqabaGccaGGOaGaeqOVdGNaaiykaiabgU% caRiaad+eacaGGOaGaeqyTdu2aaWbaaSqabeaacaWGTbaaaOGaaiyk% aaaa!67B9!\[x(t,\varepsilon ) = x_0 (\xi ,\eta ) + \sum\nolimits_{n = 1}^{m - 1} {\varepsilon ^n } x_n (\xi ) + O(\varepsilon ^m )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabsgacqaH+oaEcaGGVaGaaeizaiaadshacqGH9aqpdaaeWaqa% aiabew7aLnaaCaaaleqabaGaamOBaaaaaeaacaWGUbGaeyypa0JaaG% ymaaqaaiaad2gaa0GaeyyeIuoakiaadkfadaWgaaWcbaGaamOBaaqa% baGccaGGOaGaeqOVdGNaaiykaaaa!4FFC!\[{\text{d}}\xi /{\text{d}}t = \sum\nolimits_{n = 1}^m {\varepsilon ^n } R_n (\xi )\], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabsgacqaH3oaAcaGGVaGaaeizaiaadshacqGH9aqpdaaeWaqa% aiabew7aLnaaCaaaleqabaGaamOBaaaaaeaacaWGUbGaeyypa0JaaG% imaaqaaiaad2gaa0GaeyyeIuoakiaadofadaWgaaWcbaGaamOBaaqa% baGccaGGOaGaeqOVdGNaaiilaiabeE7aOjaacMcaaaa!5241!\[{\text{d}}\eta /{\text{d}}t = \sum\nolimits_{n = 0}^m {\varepsilon ^n } S_n (\xi ,\eta )\], and R n,S nare to be determined in the course of the analysis. This method is suitable for the systems with even non-linearities as well as with odd non-linearities. It can be viewed as a generalization of the two-variable expansion procedure. Using the present method we obtained a modified Krylov-Bogoliubov method. Four numerical examples are presented which served to demonstrate the effectiveness of the present method.
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Xu, Z., Cheung, Y.K. A non-linear seales method for strongly non-linear oscillators. Nonlinear Dyn 7, 285–299 (1995). https://doi.org/10.1007/BF00046304
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DOI: https://doi.org/10.1007/BF00046304