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 a/differential-equations/electrical.pdf b/differential-equations/electrical.pdf index 2952344..dad0ca0 100644 Binary files a/differential-equations/electrical.pdf and b/differential-equations/electrical.pdf differ diff --git a/differential-equations/electrical.tex b/differential-equations/electrical.tex index 3379cc3..d4f3a7a 100755 --- a/differential-equations/electrical.tex +++ b/differential-equations/electrical.tex @@ -122,6 +122,11 @@ If the system is perturbed slightly, it always returns to the standard cycle. Li \label{f:lc_hst} \end{figure} +\subsection{Damping} +Mentioning damping is important mainly because, in a real world, oscillations eventually stop, due to Newton's law of Thermodynamics (the frictional force). In electronics, there is no ideal oscillator, too - small amount of energy is lost every cycle, due to electric resistance. + +Generally, the damping is linear either linear or non-linear. As a rule of thumb, the linear one is easily modeled mathematically, obeying known rules, while the non-linear one is not \cite{institute1989estimation}. There are some use cases, where non-linear damping is advantageous, but the research is still ongoing about this topic. + \subsection{Li\'{e}nard Equation} A non-linear second-order ordinary differential equation \begin{equation} @@ -146,6 +151,7 @@ y''-\mu\left(1-y^2\right)y'+y=0 \end{equation} where $y$ is the dynamical variable and $\mu>0$ is a parameter. 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}