Modelling of Dynamic Systems

Number

mds

ECTS

3.0

Level

intermediate

Overview

In order to dimension and control technical-physical systems as adequately as possible, it is essential to accurately predict the dynamic behavior of such systems. This module shows how this can be achieved using exact and numerical methods.

Modeling

Finding balance equations or relevant physical laws and suitable assumptions for the process to be modeled
Setting up the corresponding differential equation

Classification of differential equations

Dimension
Order
Linearity
Homogeneity

Graphical solution of first-order ordinary differential equations

Directional field
Solution curves

Numerical solution of ordinary differential equations

Euler-Cauchy method
Heun method
Classical Runge-Kutta method
Conversion of a higher-order differential equation into a system of First-order differential equations

Analytical solution of ordinary differential equations

Separation of variables
Variation of constants
Suitable approach for linear differential equations with discussion of resonance phenomena
Solution of selected ordinary differential equations using a computer algebra system

Approximate methods

Stationary solution
Linearization

The above content is illustrated using selected problems (outflow problems, mechanical spring-mass systems, electrical RCL systems, or simple thermal systems).

Learning objectives

Modeling The students are able to use selected laws of physics in order to derive the differential equation describing a given dynamic process, thereby making reasonable assumptions. Classification of differential equations The students are able to categorize a given differential equation according to the criteria dimension (ordinary vs. partial), order, linearity and homogeneity. Graphical solution of first order differential equations The students are familiar with the concepts ‘direction field’ and ‘isoclines’ for such differential equations. They are also able to plot these ‘objects’ for a given first order differential equation as well as to perform corresponding mappings. Numerical solution of ordinary differential equations The students understand what is meant by the numerical solution of a differential equation. For that purpose, they know the method of Euler and Heun as well as the classical Runge-Kutta method. They are able to use these methods in order to solve a given differential equation numerically. They are further capable of transforming a higher order differential equation into a system of first order differential equations, and they know how the above methods have to be applied in this case. Simulink / Matlab The students are also able to apply the mathematical techniques treated in this module using MATLAB. They are capable to solve application orientated tasks with MATLAB, and they can use this tool for corresponding visualizations. They are further able to transform differential equations into a Simulink model and vice versa. They are also capable of using a Simulink model in order to solve a differential equation numerically, and they know how a Simulink model can be controlled with the help of a Matlab m-file. Analytical solution of ordinary differential equations The students are familiar with the method ‘separation of variables’ for the exact solution of appropriate differential equations. They can further apply the method “variation of constants” in order to solve linear differential equations of first order. They also know how to deal with linear differential equations when it comes to finding their exact solutions. After all, the students know the equation describing an oscillation (free and forced case), and they understand the principle of resonance. Methods of approximation For suitable nonlinear differential equations, the students are able to find (i) stationary solutions and (ii) corresponding linearized differential equations using a Taylor approximation.

Previous knowledge

Analysis 2 (an2)
Algebra (alg)
Mechanics (mech), concurrent visit

Exam format

Continuous assessment grade