IIT Guwahati Course , Prof. S.K. Dwivedy

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IIT Guwahati Course , Prof. S.K. Dwivedy

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

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

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Nonlinear Vibration by Prof. S.K. Dwivedy,Department of Mechanical Engineering,IIT Guwahati.For more details on NPTEL visit httpnptel.ac.in

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- 1.Mod-01 Lec-01 Introduction of Nonlinear systems
- 2.Mod-01 Lec-02 Review of Linear vibrating systems
- 3.Mod-01 Lec-03 Phenomena associated with Nonlinear systems
- 4.Mod-01 Lec-04 Commonly observed Phenomena in Nonlinear systems
- 5.Mod-02 Lec-01 Force and Moment based Approach
- 6.Mod-02 Lec-02 Energy based approach Extended Hamiltons principle and Lagrange Priciple
- 7.Mod-02 Lec-03 Derivation of Equation of motion of nonlinear discrete system (More examples)
- 8.Mod-02 Lec-04 Derivation of Equation of motion of nonlinear continuous system 1
- 9.Mod-02 Lec-05 Derivation of Equation of motion of nonlinear continuous system 2
- 10.Mod-02 Lec-06 Ordering of nonlinear Equation of motion
- 11.Mod-03 Lec-01 Qualitative Analysis Straight forward expansion
- 12.Mod-03 Lec-02 Numerical method Straight forward expansion
- 13.Mod-03 Lec-03 Lindstedt Poincare technique
- 14.Mod-03 Lec-04 Method of multiple scales
- 15.Mod-03 Lec-05 Method of Harmonic balance
- 16.Mod-03 Lec-06 Method of averaging
- 17.Mod-03 Lec-07 Generalized Method of averaging
- 18.Mod-03 Lec-08 Krylov-Bogoliubov-Mitropolski technique
- 19.Mod-03 Lec-09 Incremental harmonic balance method and Intrinsic multiple
- 20.Mod-03 Lec-10 Modified Lindstedt Poincare technique
- 21.Mod-04 Lec-01 Stability and Bifurcation of Fixed-point response 1
- 22.Mod-04 Lec-02 Stability and Bifurcation of Fixed-point response 2
- 23.Mod-04 Lec-03 Stability and Bifurcation of Fixed-point response 3
- 24.Mod-04 Lec-04 Stability and Bifurcation of Fixed-point response 4
- 25.Mod-04 Lec-05 Stability Analysis of Periodic response
- 26.Mod-04 Lec-06 Bifurcation of Periodic response And Introduction to quasi-periodic
- 27.Mod-04 Lec-07 Quasi-Periodic and Chaotic response
- 28.Mod-05 Lec-01 Numerical methods to obtain roots of characteristic equation and time response
- 29.Mod-05 Lec-02 Numerical methods to obtain time response
- 30.Mod-05 Lec-03 Numerical methods to obtain frequency response
- 31.Mod-06 Lec-01 Free Vibration of Single degree of freedom Nonlinear systems
- 32.Mod-06 Lec-02 Free Vibration of Single degree of freedom Nonlinear systems effect of damping
- 33.Mod-06 Lec-03 Free Vibration of multi- degree of freedom Nonlinear systems with Cubic
- 34.Mod-06 Lec-04 Forced nonlinear Vibration Single degree of freedom Nonlinear systems
- 35.Mod-06 Lec-05 Forced nonlinear Vibration Single and multi- degree of freedom
- 36.Mod-06 Lec-06 Nonlinear Forced-Vibration of Single and Multi Degree-of-Freedom System
- 37.Mod-06 Lec-07 Analysis of Multi- degree of freedom system
- 38.Mod-06 Lec-08 Nonlinear Vibration of Parametrically excited system Axially loaded sandwich beam
- 39.Mod-06 Lec-09 Nonlinear Vibration of Parametrically excited system Elastic and Magneto-elastic beam
- 40.Mod-06 Lec-10 Nonlinear Vibration of Parametrically excited system with internal resonance

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