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Nonlinear Oscillations in Mechanical Engineering
Introduction: linear and nonlinear systems, conservative and non-conservative systems; potential well, Phase planes, types of forces and responses, fixed points, periodic, quasi-periodic and chaotic responses; Local and global stability; commonly observed nonlinear phenomena: multiple response, bifurcations, jump phenomena.
Development of nonlinear governing equation of motion of Mechanical systems, linearization techniques, ordering techniques; commonly used nonlinear equations: Duffing equation, Van der Pol’s oscillator, Mathieu’s and Hill’s equations.
Analytical solution methods: Harmonic balance, perturbation techniques (Linstedt-Poincare’, method of Multiple Scales, Averaging – Krylov-Bogoliubov-Mitropolsky), incremental harmonic balance, modified Lindstedt Poincare’ techniques.
Stability and bifurcation analysis: static and dynamic bifurcations of fixed point and periodic response, different routes to chaotic response (period doubling, torus break down, attractor merging etc.), crisis.
Numerical techniques: time response, phase portrait, FFT, Poincare’ maps, point attractors, limit cycles and their numerical computation, strange attractors and chaos; Lyapunov exponents and their determination, basin of attraction: point to point mapping and cell to cell mapping, fractal dimension.
Application: Single degree of freedom systems: Free vibration-Duffing’s oscillator; primary-, secondary-and multiple- resonances; Forced oscillations: Van der Pol’s oscillator; parametric excitation: Mathieu’s and Hill’s equations, Floquet theory; effects of damping and nonlinearity. Multi degree of freedom and continuous systems.