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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.

 Lecture No. Topic Module 1: Introduction 1. Mechanical vibration: Linear nonlinear systems, types of forces and responses 2. Conservative and non conservative systems, equilibrium points, qualitative analysis, potential well, centre, focus, saddle-point, cusp point 3. Commonly observed nonlinear phenomena: multiple response, bifurcations, and jump phenomena. Module 2: Derivation of nonlinear equation of motion 4. Force and moment based approach 5. Lagrange Principle 6. Extended Hamilton’s principle 7. Multi body approach 8. Linearization techniques 9. Development of temporal equation using Galerkin’s method for continuous system 10. Ordering techniques, scaling parameters, book-keeping parameter. Commonly used nonlinear equations: Duffing equation, Van der Pol’s oscillator, Mathieu’s and Hill’s equations. Module 3: Approximate solution method 11. Straight forward expansions and sources of nonuniformity 12. Harmonic Balancing method 13. Linstedt-Poincare’ method 14. Method of Averaging Module 4: Perturbation analysis method 15. Method of Averaging 16. Method of multiple scales 17. Method of multiple scales 18. Method of normal form 19. Incremental Harmonic Balance method 20. Modified Lindstedt-Poincare’ method Module 5: Stability and Bifurcation Analysis 21. Lyapunov stability criteria 22. Stability analysis from perturbed equation 23. Stability analysis from reduced equations obtained from perturbation analysis 24. Bifurcation of fixed point response, static bifurcation: pitch fork, saddle-node and trans-critical bifurcation 25. Bifurcation of fixed point response, dynamic bifurcation: Hopf bifurcation 26. Stability and Bifurcation of periodic response, monodromy matrix, poincare’ section Module 6: Numerical techniques 27. Time response, Runga-Kutta method, Wilson- Beta method 28. Frequency response curves: solution of polynomial equations, solution of set of algebraic equations, 29. Basin of attraction: point to point mapping and cell-to-cell mapping 30. Poincare’ section of fixed-point, periodic, quasi-periodic and chaotic responses. 31. Lyapunov exponents 32. FFT analysis, Fractal Dimensions Module 7: Applications 33. SDOF Free-Vibration: Duffing Equation 34. SDOF Free-Vibration: Duffing Equation 35. SDOF Forced-Vibration: Van der pol’s Equation 36. SDOF Forced-Vibration: Van der pol’s Equation 37. Parametrically excited system- Mathieu-Hill’s equation, Floquet Theory 38. Parametrically excited system- Instability regions 39. Multi-DOF nonlinear systems 40. Continuous system: Micro-cantilever beam analysis
• Mechanical Vibration

• Engineering Mechanics

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• International Journal of Nonlinear Mechanics

• Nonlinear Dynamics

• Journal of Sound and Vibration

• Journal of Vibration and Acoustics (ASME)

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