Nonlinear Vibration

Overview

Contents:
Introduction - Mechanical vibration: Linear nonlinear systems, types of forces and responses - Conservative and non conservative systems, equilibrium points, qualitative analysis, potential well, centre, focus, saddle-point, cusp point - Commonly observed nonlinear phenomena: multiple response, bifurcations, and jump phenomena.
Derivation of nonlinear equation of motion : Force and moment based approach - Lagrange Principle - Extended Hamiltons principle - Multi body approach - Linearization techniques - Development of temporal equation using Galerkins method for continuous system - Ordering techniques, scaling parameters, book-keeping parameter. Commonly used nonlinear equations: Duffing equation, Van der Pols oscillator, Mathieus and Hills equations.

Approximate solution method : Straight forward expansions and sources of nonuniformity - Harmonic Balancing method - Linstedt-Poincare method - Method of Averaging
Perturbation analysis method : Method of Averaging - Method of multiple scales - Method of multiple scales - Method of normal form - Incremental Harmonic Balance method - Modified Lindstedt-Poincare method
Stability and Bifurcation Analysis : Lyapunov stability criteria - Stability analysis from perturbed equation - Stability analysis from reduced equations obtained from perturbation analysis - Bifurcation of fixed point response, static bifurcation: pitch fork, saddle-node and trans-critical bifurcation - Bifurcation of fixed point response, dynamic bifurcation: Hopf bifurcation - Stability and Bifurcation of periodic response, monodromy matrix, poincare section

Numerical techniques : Time response, Runga-Kutta method, Wilson- Beta method - Frequency response curves: solution of polynomial equations, solution of set of algebraic equations - Basin of attraction: point to point mapping and cell-to-cell mapping - Poincare section of fixed-point, periodic, quasi-periodic and chaotic responses - Lyapunov exponents - FFT analysis, Fractal Dimensions
Applications : SDOF Free-Vibration: Duffing Equation - SDOF Free-Vibration: Duffing Equation - SDOF Forced-Vibration: Van der pols Equation - SDOF Forced-Vibration: Van der pols Equation - Parametrically excited system- Mathieu-Hills equation, Floquet Theory - Parametrically excited system- Instability regions - Multi-DOF nonlinear systems - Continuous system: Micro-cantilever beam analysis