Title：Perturbation Methods for Nonlinear Dynamical Systems in Engineering.
MSc/PhD course: 6 lectures of 5 hours each, 30 hours of self-study.
Lecture 1 : Introduction to the theory of stability. Stability of equilibrium solutions, and stability of periodic solutions. Linearisation. Asymptotic stablity and Lyapunov stability. Stability analysis of linear equations with variable coefficients. Floquet theory, Stability by linearisation. Blow-up techniques. Introduction to perturbation theory. Naive expansions, asymptotic expansions. The Poincaré expansion theorem.
Lecture 2 : Introduction to the multiple time-scales perturbation method for ordinary differential equations. Mathematical justification of the method. Application of the method to linear and nonlinear ordinary differential equations. Raleigh and Van der Pol equation. Introduction to the multiple scales perturbation method for partial differential equations. Mathematical justification of the method. Application of the method to linear and nonlinear partial differential equations. Vibrations of a string-like structure on a nonlinear elastic foundation. Telegraph and Klein-Gordon equations.
Lecture 3 : Application of the multiple scales perturbation method to beam-like structures. Weakly nonlinear vibrations of beams. Applicability of the Galerkin truncation method to string-like and beam-like problems. Resonance conditions and the small denominator problem. When Galerkin’s truncation method is not applicable to string-like problems to obtain accurate approximations on long time-scales, it will be shown how the multiple time-scales perturbation method in combination with the method of characteristic coordinates can be used in some cases to obtain accurate results on long time-scales. The method is applied to conveyor belt problems and to elevator cable vibrations. Theory for first order partial differential equations: method of characteristics.
Lecture 4 : Outline of the mathematical justification of the applied methods to wave and beam equations. Several examples in the field of linear and nonlinear vibrations of elastic structures (such as strings, beams, and plates) will be given to show the restrictions of the applicability of Galerkin’s truncation method. It will be made clear how infinite dimensional systems of ordinary differential equations might be studied. Introduction to singular perturbation theory for ordinary differential equations. Boundary layers and interior layers. Multiple time-scales perturbation method and the WKBJ-method.
Lecture 5 : The averaging method. The Lagrange standard form. Averaging in the periodic case. Averaging in the general case. Adiabatic invariants. Resonance manifolds. Periodic solutions. Extension of the multiple time-scales perturbation methods to difference equations. Lecture 6 : The aim of this lecture is to study autoresonance phenomena in a spacetime-varying mechanical system. The maximal amplitude of the autoresonant solution and the time of autoresonant growth of the amplitude of the modes of fast oscillations are determined. A vertically translating string with a time-varying length and a space-time-varying tension are considered. The problem can be used as a simple model to describe transversal vibrations of an elevator cable for which the length changes linearly in time. It is assumed that the axial velocity of the cable is small compared to nominal wave velocity and that the cable mass is small compared to car mass. The elevator cable is excited sinusoidally at the upper end by the displacement of the building in the horizontal direction from its equilibrium position caused by wind forces. This external excitation has a constant amplitude of order ε. It is shown that order ε amplitude excitations at the upper end result in larger order solution responses. Interior layer analysis has been provided systematically to show that there exists an unexpected timescale in the problem. For this reason, a threetimescale perturbation method is used to construct asymptotic approximations of the solutions of the initial-boundary value problem.