Nonlinear Vibration

Most of the vibrating structure are nonlinear in nature. But for simplification of the analysis they have been considered to be linear. Hence, to actually know the response of the system one should study the nonlinear behavior of the system. Here one may encounter multiple equilibrium points or solutions which may be stable or unstable. The response may be periodic, quasiperiodic or chaotic. The present course is a simulation based course where one can visualize the response of different mechanical systems for different resonance conditions. Out of 9 modules, first 8 modules are on developing the equations of motion, solution procedure of these equations and application of them to general single and multi-degree of freedom systems. The last modules which will be covered in 3 weeks taking 3 different applications of current interest are project based and it will give a very good practical exposure. The course will be very useful for undergraduate, post graduate and PhD students in Academic institutions and also practicing engineers in Industry. INTENDED AUDIENCE :Senior under graduate or post graduate and PhD students in Mechanical Engineering can take this course under Advanced Dynamics domainPREREQUISITES : NoneINDUSTRIES SUPPORT :All the industry dealing with manufacturing, automobile, aerospace etc. will require nonlinear vibration analysis to improve their productivity

Week 1:Module 1: Introduction to Nonlinear Mechanical Systems
Introduction to mechanical systems, Superposition rule, familiar nonlinear equations: Duffing equation, van der Pol’s equation, Mathieu-Hill’s equation, Lorentz system, Equilibrium points: potential functionWeek 2:Module 2: Development of Nonlinear Equation of Motion using Symbolic Software
Force and moment based Approach, Lagrange Principle, Extended Hamilton’s principle, use of scaling and book-keeping parameter for orderingWeek 3:Module 3: Solution of Nonlinear Equation of Motion
Numerical solution, Analytical solutions: Harmonic Balance method, Straight forward expansion and Lindstd-Poincare’ methodWeek 4:Method of Averaging, Method of multiple scales, Method of 3 generalized Harmonic Balance methodWeek 5:Module 4: Analysis of Nonlinear SDOF system with weak excitation
Free vibration of undamped and damped SDOF systems with quadratic and cubic nonlinearity, and forced vibration with simple resonanceWeek 6:Analysis of Nonlinear SDOF system with hard excitation
Nonlinear system with hard excitations, super and sub harmonic resonance conditions, Bifurcation analysis of fixed-point response Week 7:Vibration Analysis of Parametrically Excited system
Principal and combination parametric resonance conditions, Floquet theory, frequency and forced response of nonlinear parametrically excited system.Week 8:Analysis of Periodic, quasiperiodic and Chaotic System
Stability and bifurcation analysis of periodic response, analysis of quasi-periodic system, analysis of chaotic System
Week 9:Numerical Methods for Nonlinear system Analysis
Solutions of a set of nonlinear equations, Numerical Solution of ODE and DDE equations, Time response, phase portraits, frequency response, Poincare section, FFT, Lyapunov exponentWeek 10:Practical Application 1: Nonlinear Vibration Absorber
Equation of motion, Solution of EOM: Use of Harmonic Balance method, Program to obtain time and frequency responseWeek 11:Practical Application 2: Nonlinear Energy Harvester
Development of Equation of motion: symbolic software, Solution of EOM: Use of method of Multiple Scales, Program to obtain time and frequency responseWeek 12:Practical Application 3: Analysis of electro-mechanical system
Development of Equation of motion and its solution, Use of Floquet theory, Parametric instability regions, Study of periodic, quasiperiodic and chaotic response