From d106a1b366018c5e8590329a9f9d373829133567 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 9 Sep 2024 13:28:34 -0400 Subject: [PATCH] vault backup: 2024-09-09 13:28:34 --- .obsidian/workspace.json | 7 ++++--- .../2024-09-09 Example.py | 0 .../2024-09-09 Frameworks and Review.md | 19 ++++++++++++++++++- 3 files changed, 22 insertions(+), 4 deletions(-) create mode 100644 300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Example.py diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index f2ffa644..858c69db 100755 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -143,7 +143,8 @@ } } } - ] + ], + "currentTab": 3 }, { "id": "eae71cef5d7338c8", @@ -178,6 +179,8 @@ }, "active": "e7019452a0bd61a5", "lastOpenFiles": [ + "300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Example.py", + "300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Example", "1. Daily Notes/2024/9. September/2024-09-09.md", "300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md", "2024-09-10.md", @@ -203,7 +206,6 @@ "4. Qualifying Exam/2. Writing/2. QE State of the Art.md", "4. Qualifying Exam/2. Writing/1. QE Goals and Outcomes.md", "1. Daily Notes/2024/8. August/2024-08-19.md", - "4. Qualifying Exam/0. Overview/0. QE Overview.md", "4. Qualifying Exam/0. Overview/ME_PhD_Qualifying_Exam_Guideline_Fall2024.pdf", "4. Qualifying Exam/99. Exports/QE Abstract For Dan.pdf", "4. Qualifying Exam/99. Exports", @@ -213,7 +215,6 @@ "4. Qualifying Exam/2. Writing/test.bib", "201. Metadata/My Library/files/4011/contextual.css", "201. Metadata/My Library/files/4011/tabledrag_002.css", - "201. Metadata/My Library/files/4011/ui-dialog.css", "201. Metadata/My Library/files/4011/header-degreefinder-cybersecurity.jpg", "201. Metadata/My Library/files/4011/cathedral-summersmall.jpg", "201. Metadata/My Library/files/4011/johanna-montoya-PRumW--tkc4-unsplash-wind250.jpg", diff --git a/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Example.py b/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Example.py new file mode 100644 index 00000000..e69de29b diff --git a/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md b/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md index 0e867731..9edcaebd 100644 --- a/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md +++ b/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md @@ -41,6 +41,23 @@ The system is at equilibrium where $\frac{dx}{dt} = 0$. It won't move from this Is this system stable? Check the eigenvalues of A. ### Nonlinear Systems **Recall: $\dot{x} = 1-2\cos x$** -#### The Phase Plane +We can qualitatively describe systems using the phase plane: +*Insert graphics from class* +How is this useful to us engineers? +We are going to see systems that are nonlinear, and they can give us ideas about where things could blow up. In our second example, we have generally a pretty safe area below x = 2. Anywhere below there, we know we're going to end up at x = -2, but above x =2, all hell breaks loose. +This is what we care about. We want to know where in our nonlinear system domains things can become dangerous. +## How do we numerically get a time domain response? +Numericaly: +$$ \dot x = f(x) $$ +$$\frac{dx}{dt} = \lim_{\Delta t \rightarrow 0} \frac{f(x(t+\Delta t))-f(x(x))}{\Delta t} $$ +This is the tangent (or the secant while $\Delta t =/ 0$) +>[!note] Eulers Method +>Therefore, for finite $\Delta t$: +> $$ f(x(t+\Delta t)) = \frac{dx}{dt} \Delta t + f(x(t))$$ +> Limitations: innaccurate if time steps are large. +> There are better methods! +> ode45() <- Variable Step Runge-Kutta + +We're going to use a lot of odeint in SciPy \ No newline at end of file