From d5f4ca3f333680fc456940196f33b130507c53c3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Peter=20Babi=C4=8D?= Date: Sun, 7 Jun 2015 21:18:24 +0200 Subject: [PATCH] damping --- differential-equations/electrical.bib | 7 +++++++ differential-equations/electrical.pdf | Bin 141282 -> 142123 bytes differential-equations/electrical.tex | 10 +++++++++- 3 files changed, 16 insertions(+), 1 deletion(-) diff --git a/differential-equations/electrical.bib b/differential-equations/electrical.bib index 383d3e8..25edab2 100644 --- a/differential-equations/electrical.bib +++ b/differential-equations/electrical.bib @@ -59,3 +59,10 @@ publisher={Springer New York}, pages={240--244} } + +@book{institute1989estimation, + title={Estimation of Nonlinear Damping in Second Order Distributed Parameter Systems}, + author={Institute for Computer Applications in Science and Engineering and Banks, H.T. and Reich, S. and Rosen, I.G.}, + url={https://books.google.es/books?id=OKxCAQAAMAAJ}, + year={1989} +} diff --git 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If $\mu=0$, then the equation reduces to the equation of simple harmonic motion $$y''+y=0$$ +The $\mu$ parameter determines the shape of the limit cycle. As it approaches 0, it gets the shape of a circle. On the other hand, increasing the paramter, involves sharpening of the curves. The Van der Pol equation \eqref{eq:vdp} arises in the study of circuits containing vacuum tubes (triode) and is derived from earlier, Rayleigh equation \cite{nahin2001science}, known also as Rayleigh-Plesset equation - an ordinary differential equation explaining the dynamics of a spherical bubble in an infinite body of liquid. @@ -165,10 +171,12 @@ However, this form is not well-known. Far common form uses the transformation $y \end{align*} which can be plotted onto direction field, as shown on \cref{f:vdp_m}. It is possible to see the stable limit cycle as well as trajectories from both sides attracted towards it. +The Van der Pol oscillator can be forced too, however, this work does not aim to investigate further in this direction. + \begin{figure}[t!] \centering \includegraphics[width=.85\linewidth]{vdp_maxima} - \caption{Van der Pol oscillator plotted on the direction field. Parameter $\mu=1$. The wxMaxima computing software was used for this purpose. } + \caption{Phase portrait of the unforced Van der Pol oscillator, showing a limit cycle and the direction field Parameter $\mu=1$. The wxMaxima computing software was used for this purpose. } \label{f:vdp_m} \end{figure}