From 8591c8375ec80432f0f4064a6e45e90613789926 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 23 Sep 2024 13:26:13 -0400 Subject: [PATCH] vault backup: 2024-09-23 13:26:12 --- .../2024-09-23 Temporary Title.md | 20 +++++++++++++++++++ 1 file changed, 20 insertions(+) diff --git a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-23 Temporary Title.md b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-23 Temporary Title.md index 8ae9ef91..aba4dbfd 100644 --- a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-23 Temporary Title.md +++ b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-23 Temporary Title.md @@ -50,3 +50,23 @@ These things relate directly to the eigenvectors and eigenvectors of the system. ## Eigenvalues and Eigenvectors $z = \beta^2 - 4 \omega^2$ Solving for lambda ($\det[A - I\lambda]$) will lead us towards the same expression for z. This is what's under the square root. +# Nonlinear Things (Finally!) +## Nonlinear Pendulum +### Undamped +$$ \ddot{\theta} = -\frac{g}{l} \sin(\theta) $$ +$\dot \theta = \zeta = P(\theta, \zeta)$ +$$ {\bf J} = +\left[ \matrix{ \frac{\partial P}{\partial \theta} & \frac{\partial P}{\partial \zeta} \\ \frac{\partial Q}{\partial \theta} & \frac{\partial Q}{\partial \zeta}} \right] = + +\left[ \matrix{ 0 & 1\\ -\omega^2 \sin(\theta) & 0 } \right] + $$ + What ar the equilibrium points? +$$\left[ \matrix{ \dot \theta \\ \dot \zeta} \right ] = \left[ \matrix{ \zeta \\ -\omega^2 \sin(\theta) } \right] $$ +This system is in equilibrium when $\dot \theta = \dot \zeta = 0$, which is when $\zeta = \sin(\theta) = 0$ . +For $\bf J$: +- $\tau = 0$ +- $\Delta = \omega^2 \cos(\theta)$ +Then: +- $\theta$ is 0, $\Delta = \omega^2 >0$, center, marginally stable +- $\theta = n \pi$, $\Delta = - \omega^2 <0$, saddle. Unstable +What does the phase plane look like? \ No newline at end of file