# Nonlinear Dynamics of Mechanical Systems/ Nichtlineare Dynamik mechanischer Systeme

Wintersemester 2023/24

Lecturer:
Assistant:
Lecture start:

16.10.2023

Language:

English (questions can be answered in German)

Course ID number:

340540100

Time frame:

4 SWS, 6 ECTS

This lecture is intended for graduate and PhD students from engineering sciences and physics, who are interested in the behavior of nonlinear dynamical systems. The course makes the student familiar with nonlinear phenomena such as limit cycles, quasiperiodicity, bifurcations and chaos. These nonlinear phenomena occur in for instance biological, economical, celestial and electrical systems, but only mechanical multibody systems will be taken as examples. With the theory explained in the course one is able to understand flutter instability of wings, stick-slip vibrations, post-buckling behavior of frames and nonlinear control techniques. Exercises and examples during the course include: hunting motion of railway vehicles, forced oscillation of a nonlinear mass-spring system, instability of the Watt steam governor and symmetric and asymmetric buckling. Engineering practice as well as the standard engineering curriculum often do not exceed a linear analysis of nonlinear systems. The course pays special attention to indicate the limitations of a linear analysis. The aim of the course is to give the student a basic knowledge and understanding of nonlinear system behavior and to provide analysis tools to analyze nonlinear dynamical systems.

1. Introduction:  Notation; Literature
2. Dynamical Systems: Continuous-time systems; Discrete-time systems; Limit sets; Lyapunov stability
3. Bifurcations of Equilibria: Center Manifold; Center manifold reduction; Definition of Bifurcation; Normal forms
4. Bifurcations of Fixed Points of Discrete-time Systems; Linearization around a fixed point; One-dimensional linear discrete-time systems; Stability of fixed points of nonlinear discrete-time systems; Bifurcations of fixed points with a single eigenvalue +1; Flip bifurcation (single eigenvalue -1); Naimark-Sacker bifurcation (complex eigenvalue through unit circle); The logistic map; Horseshoes & intermittency
5. Stability and Bifurcations of Periodic Solutions; Periodicity properties; Fundamental Solution Matrix; Stability of periodic solutions; The Poincaré map; Bifurcations of periodic solutions; Harmonic Balance Method

The course material will be provided through ILIAS.

There will be a written exam (90 min.) at the end of the course where any written or printed material can be used. Further details concerning the exam will be announced during the course.

See "Stellung im Studienplan"

### Remco I. Leine

Prof. Dr. ir. habil.

Direktor

M.Sc.