Nonlinear Vibrations

The description of real systems usually leads to nonlinear differential equations. Often, the analysis of the linearised equations already provides sufficient insight into the dynamics - nevertheless, there is a multitude of practically significant phenomena, which can only be uncovered and understood by the investigation of the nonlinear equations.

The lecture will give an introduction to common methods for the treatment of nonlinear oscillatory systems and demonstrate technically significant nonlinear phenomena. Connected to this is also an introduction to the basics of kinetic stability theory.

Beyond learning about nonlinear effects, only the study of nonlinear dynamics reveals the validity limits of linear analyses and thus allows an assessment of the validity limits of linear models.

• Introduction
• Basic concepts: Dynamical systems, state space, solutions
• Stability of solutions
• Approximation methods: Harmonic balance (Galerkin), multiple time scales, averaging (method of slowly varying amplitude and phase).
• Phenomena: nonlinear resonance, self-excitation, parameter excitation, entrainment
• Branching & solution tracking
• Deterministic chaos

• Engineering Mechanics 1-3
• Mathematics 1-3
• Engineering Vibrations
• Linear Vibrations (recommended)

• S. H. Strogatz: Nonlinear Dynamics and Chaos, Westview Press (2nd Edition 2014)
• D.W. Jordan, P. Smith: Nonlinear Ordinary Differential Equations, Oxford University Pess (4th Edition 2007)
• P. Hagedorn: Non-Linear Oscillations, Oxford University Pess (1st Edition 1981)
• K. Klotter: Technische Schwingungslehre, Band 1 - Teil B: Nichtlineare Schwingungen, Springer-Verlag (3rd Edition 1978)
• J. Argyris: Die Erforschung des Chaos, Springer-Verlag (3rd Edition 2017)
• A. H. Nayfeh, B. Balachandran: Applied Nonlinear Dynamics, Wiley-VCH (1st Edition 1995)
• A. H. Nayfeh, D. T. Mook: Nonlinear oscillations, Wiley-VCH (1st Edition 1995)