Skip to main content

Modulbeschreibung - Modelling of Dynamic Systems
(Modellieren dynamischer Systeme)

Nummer
mds
ECTS 3.0
Level intermediate
Overview In order to dimension and control technical systems as adequately as possible, it is vital that the dynamic behavior of such systems can be predicted with good accuracy. This module shows how this is achieved by using exact and numerical methods.

    1. Modeling
  • using balance equations, relevant physical laws and appropriate assumptions in order to model a dynamic process
  • deriving the corresponding differential equation

    2. Classification of differential equations
  • dimension
  • order
  • linearity
  • homogeneity

    3. Graphical solution of first order differential equations
  • direction field
  • isoclines

    4. Numerical solution of ordinary differential equations
  • Euler method
  • Heun’s method
  • Classical Runge-Kutta method
  • transforming a higher order differential equation into a system of first order differential equations

    5. Simulink
  • block representation of an ordinary differential equation
  • determining and plotting the numerical solution
  • controlling a Simulink model with a Matlab m-file

    6. Analytical solution of ordinary differential equations
  • separation of variables
  • variation of constants
  • suitable approach to linear differential equations with application to resonance phenomena
  • solving selected ordinary differential equations using a computer algebra system

    7. Methods of approximation
  • stationary solution
  • linearization

The above methods are exemplified using selected problems such as (i) outflow problems, (ii) mechanical systems composed of springs and masses, (iii) electrical RCL systems and (iv) 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),
  • Mechanik (mech), simultaneous attendance

Exam format Continuous assessment grade
Diese Seite teilen: