ME627A |
Non-Linear Vibration |
Credits: |
3L-0T-0L-0D (9 Credits) |
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:
- Elements of Vibration Analysis, L. Meirovitch, 2nd edition, McGraw Hill Education (India), 1986
- Methods of Analytical Dynamics, L. Meirovitch,, Dover publications, 2010.
- Vibration of Continuous Systems, S. S. Rao, John Wiley & Sons, 2007.
- Vibration and Waves in Continuous Mechanical Systems, P. Hegedorn and A. DasGupta, Wiley, 2007.
- Wave Motion in Elastic Solids, K. F. Graff, Dover Publications, 1991.
- Mechanics of Continua and Wave Dynamics, L. Brekhovskikh and V. Goncharov, SpringerVerlag, 1985.