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 Hamilton’s principle – Multi body approach – Linearization techniques – Development of temporal equation using Galerkin’s method for continuous system – Ordering techniques, scaling parameters, book-keeping parameter. Commonly used nonlinear equations: Duffing equation, Van der Pol’s oscillator, Mathieu’s and Hill’s 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 pol’s Equation – SDOF Forced-Vibration: Van der pol’s Equation – Parametrically excited system- Mathieu-Hill’s equation, Floquet Theory – Parametrically excited system- Instability regions – Multi-DOF nonlinear systems – Continuous system: Micro-cantilever beam analysis
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