Objective:

This course will introduce the students to the basics of nonlinear dynamics with a specific emphasis on second order systems representing vibration problems. Computer based assignments and tests will be used to complement the in-class evaluations. Use of symbolic algebra packages and computations using MATLAB will be encouraged..

Course Content: (Precise syllabus for publication in course bulletin)

Introduction to concept of trajectories, phase space, singular points and limit cycle; Linear stability analysis and introduction to bifurcations; Analytical methods including perturbation techniques, and heuristic approaches like harmonic balance and equivalent linearization; Stability of periodic solutions: Floquet's theory, Hill's and Mathieu's equations; Nonlinear free and forced responses of the Duffing's and van der Pol equation; Introduction to chaos and Lyapunov exponents.

Lecturewise Breakup

I. Overview of linear vibrations and contrasting with nonlinear vibrations : (1-2 Lectures)

II. Various sources and type of nonlinearities in mechanical systems(2 Lectures)

III. Introduction to phase space and trajectories using pendulum as an example; phase space for conservative systems.(2-3 Lectures)

IV. Axial and torsional vibrations in bars, transverse vibrations in strings (5 Lectures)

V. Linear stability analysis and local phase space(3-4 Lectures)

VI. Basic bifurcations in 2-dimensional systems with some discussion about extensions to higher dimensions (1-2 Lectures )

VII. Perturbation methods for almost periodic solutions (Regular Perturbation, Poincare-Linstedt method, Method of Averaging, Method of Multiple Scales) with free vibration of Duffing and van der Pol Equation as an example(4-6 Lectures)

VIII. Heuristic methods (Harmonic Balance, Equivalent linearization, Galerkin and Collocation Techniques) (2-3 Lectures)

IX. Numerical approaches to get branches of solutions (continuation)(1 Lectures)

X. Floquet theory for parametric systems: discussion of some examples of parametric excitation, relevance to stability of periodic solutions, Meissner Equation, Mathieu-Hill equation, numerical computation of Floquet multipliers (4-6 Lectures)

XI. Forced vibration study of the Duffing oscillator with possible study of the sub-harmonic (1:3) resonance(2-3 Lectures)

XII. Stroboscopic and Poincare maps as an alternate means to study nonlinear vibrations (1 Lectures)

XIII. Detailed study of the logistic map illustrating chaos and the concept of Lyapunov exponent.s (2-3 Lectures)

XIV. Numerical computation of Lyapunov exponents for maps and flows (1 Lectures)

References:

1. Elements of Vibration Analysis, L. Meirovitch, 2nd edition, McGraw Hill Education (India), 1986

2. Methods of Analytical Dynamics, L. Meirovitch,, Dover publications, 2010.

3. Vibration of Continuous Systems, S. S. Rao, John Wiley & Sons, 2007.

4. Vibration and Waves in Continuous Mechanical Systems, P. Hegedorn and A. DasGupta, Wiley, 2007.

5. Wave Motion in Elastic Solids, K. F. Graff, Dover Publications, 1991.

6. Mechanics of Continua and Wave Dynamics, L. Brekhovskikh and V. Goncharov, SpringerVerlag, 1985.