Non-linear Dynamics

Code:

PRODEM018

Acronym:

DLC

Keywords
Classification	Keyword
OFICIAL	Mechanical Engineering

Instance: 2012/2013 - 2S

Active?	Yes
Responsible unit:	Applied Mechanics Section
Course/CS Responsible:	Doctoral Program in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
PRODEM	1	Syllabus since 2009/10	1	-	6,5	60	175,5

Teaching language

English

Objectives

Teach fundamental aspects of the theory of non-linear dynamical systems, as well as numerical and experimental methodologies for their analysis. The theory has applications in a wide number of fields, but emphasis will be given to mechanical systems.

Teaching Language: depends on the studenst, it can be Portuguese or English.

Program

1. Introduction. Non-linear dynamical systems.

2. Fundamental concepts
2.1 Discrete and continuous-time systems, autonomous and non-autonomous systems, phase space.
2.2 Existence and uniqueness of solutions
2.3 Equilibrium points: centres, nodes, focus and saddle points.
2.4 Lymit cycles.
2.5 Linerization of non-linear systems (Hartman-Grobman theorem)
2.6 Stability of Lyapunov, assymptotic stability, stability of Poincaré.
2.7 Two dimensional flows:Poincaré-Bendixson theorem and Bendixson's criterion

3. Methods of resolution of the equations of motion.
3.1 Perturbation method - asymptotic expansions. 3.2 Multiple Scales Method. 3.3 Harmonic balance method. 3.4 Numerical integration in the time domain. 3.5 Shooting method.3.6 Continuation method

4. Periodic motions and methods to characterize motions.
4.1 Definition. Time history. 4.2 Phase plane. 4.3 Fourier Spectrum. 4.4 Poincaré Map. 4.5 Floquet theory. 4.6 Bifurcations of periodic solutions.

5. Quasi-periodic motions
5.1 Definition. 5.2 Time history; phase plane; Fourier spectrum and Poincaré map.

6. Chaos.
6.1 Definition. Routes to chaos. 6.2 Time history; Phase plane; Fourier Spectrum and Poincaré Maps. 6.3 Lyapunov exponents.

7. Vibrations of structures in the geometrically non-linear regime.
7.1 Equations of motion of structures vibrating with large displacements. 7.2 Non-linear modes of vibration. Change of the mode shape and natural frequency. 7.3 Internal resonances.

Mandatory literature

Nayfeh, Ali Hasan; Applied nonlinear dynamics. ISBN: 0-471-59348-6
Thomsen, Jon Juel; Vibrations and stability. ISBN: 3-540-40140-7

Complementary Bibliography

Moon, Francis C.; Chaotic vibrations. ISBN: 0-471-67908-9
Bathe, Klaus-Jurgen; Finite element procedures. ISBN: 0-13-301458-4
Wiggins, Stephen; Introduction to applied nonlinear dynamical systems and chaos. ISBN: 0387-00177-8
Verhulst, Ferdinand; Nonlinear differential equations and dynamical systems. ISBN: 3-540-50628-4

Teaching methods and learning activities

Exposition of theory, with some demonstrative examples. Computational (or experimental) applications and analytical exercises will be carried out by the students.

The lectures may be in English, if the students so wish.

Software

Maple 6
The Mathworks - Matlab - Release 11.1

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	24,00
	Total:	-	0,00

Calculation formula of final grade

0.5*Exam + 0.5*works