Integral Transform Methods

Participating in this lecture enables the student to understand as the mathematical basics of integral transforms as the differential equations describing the mechanical systems.
With the help of the computer workshops the student can apply the integral transform method for solving the differential equations and systems of differential equations, mentioned above, manually and also using computer algebra systems in order to analyse the results.
Doing the exercises and homework the students learn to evaluate the applicability of these methods for solving mechanical problems (e.g. the dynamic response of SDOF and MDOF systems under different types of loads considering initial conditions) and also to apply them for practical problems.

In this module Integral Transform Methods (ITM) are discussed focussing on the Laplace- the Fourier, the z- and the Wavelet- Transform. Mathematical Relations are derived and applications for solutions of ODEs, systems of ODEs, PDEs and systems of PDEs are discussed. Mechanical problems in the field of civil and mechanical engineering are assigned to the differential equations and solved using ITM. The analytical implementation in Computeralgebra Systems as well as numerical codes are discussed and practised in Computer-Seminars.
The effect related with discrete transforms are illustrated. Different methods designed to cope with them are discussed.

Content:
Repetition of several differential equations
- ordinary differential equation: single degree of freedom system (SDOF)
- system of ordinary differential equations: multiple degree of freedom systems (MDOF)
- partial differential equation: mass distributed beam
- system of partial differential equations: homogeneous halfspace
- initial conditions and loads
- repetition: complex numbers
- eulerian identity
- important terms (orthogonality, integrability, ...)
- functions and distributions (Dirac δ-function, sampling functions, Heaviside-function ...)

Laplace Transform:
- important terms (kernel, original domain, transformed domain ...)
- transformation rules
- differentiation in the transformed domain, shifting theorem, convolution theorem, ...
- transformation of differential equations and solution in the transformed domain
- examples
- inverse transform using a partial fraction decomposition
- systems of ODEs of order n
- transfer functions, unit impulse response functions
- inverse transform (complex inversion formula, Cauchy´s principal value, residua theorem)
- partial differential equations

Fourier Transform:
- transformation rules
- important functions and their transformation
- differentiation in the transformed domain, shifting theorem, convolution theorem, ...
- transformation of differential equations and solution in the transformed domain
- examples
- ODEs of order n with constant coefficients
- transfer functions
- multiplication and convolution
- systems of PDEs
- Helmholtz decomposition and homogeneous halfspace

Discrete Fourier Transform
- discrete FT and discrete IFT
- FFT/IFFT-Algorithms
- outlook: Wavelet-Transform
- windowing and filtering
- Aliasing