Modellieren dynamischer Systeme

Damit technisch-physikalische Systeme möglichst adäquat dimensioniert und geregelt werden können, ist es unerlässlich, das dynamische Verhalten derartiger Systeme gut vorherzusagen. Dieses Modul zeigt auf, wie dies mit exakten und numerischen Methoden gelingt.

Modellierung

Klassifikation von Differenzialgleichungen

Graphische Lösung gewöhnlicher Differenzialgleichungen 1. Ordnung

Numerische Lösung gewöhnlicher Differenzialgleichungen

Analytische Lösung gewöhnlicher Differenzialgleichungen

Approximative Methoden

Die obigen Inhalte werden dabei an ausgewählten Problemstellungen illustriert (Ausflussprobleme, mechanischen Feder-Masse Systemen, elektrischen RCL-Systemen oder einfachen thermischen Systemen.

Modeling: Students can use selected laws of physics as input and, with further assumptions, derive the corresponding differential equation.

Classification of Differential Equations: Students are able to classify a given differential equation according to the criteria of dimension (ordinary vs. partial), order, linear, and homogeneous.

Graphical Solution of First-Ordinary Differential Equations: Students are familiar with the concepts of direction field and solution curves for such differential equations and can represent these graphical elements for given differential equations. They can also assign a direction field to the corresponding differential equation.

Numerical Solution of Ordinary Differential Equations: Students know what it means to solve a differential equation numerically. They are familiar with the Euler-Cauchy and Heun methods, as well as the classical Runge-Kutta method. They are able to solve a given first-order differential equation numerically using these methods. Furthermore, they can convert a higher-order differential equation into a system of first-order differential equations and know how to then apply the above numerical methods.

Analytical Solution of Ordinary Differential Equations: Students are familiar with the principle of separation of variables and can apply this to solving suitable differential equations. For linear first-order differential equations, students are familiar with the solution concept of "variation of constants." For linear higher-order differential equations with constant coefficients, they know the correct solution approach and can thus determine the solution set. Students will be familiar with the oscillation equation (free and forced) and understand the principle of resonance.

Approximate Methods: Students will be able to calculate steady-state solutions for suitable nonlinear differential equations and determine the corresponding linearized differential equation using a Taylor expansion.

Tool: Students will also be able to apply the content covered in the module practically using a tool such as Python.