ME624A

CALCULUS OF VARIATIONS

Credits:

  

 

3L-0T-0L-0D (9 Credits)
 

Course Content:


History of calculus of variations: Discussion of certain classical and modern problems in mechanics that led the emergence and development of calculus of variations. Contributions of Bernoulli(s), Euler, Lagrange, Jacobi, Weierstrass, Hamilton, Legendre and some others will be discussed briefly; Definition of a function, function space (with examples of function spaces that are important in engineering and sciences) and so called 'functional'; Definition of the first variation of a functional and a variational derivative. Necessary condition for an extremum. Euler equation. Discussion on Null Lagrangian and natural boundary condition. Constraints and Lagrange multipliers (with examples from classical mechanics); Noether theorem and conservation laws, Weierstrass-Erdmann condition and more general jump conditions (with examples from continuum mechanics of bodies with defects); Definition of the second variation and the strong/weak minimizer. Necessary condition for a minimum, Legendre condition, Jacobi condition, conjugate points. Sufficient condition for a minimum, Weierstrass E-function. Field of extremals; Definitions of convexity, quasiconvexity, strong ellipticity, weak derivative, minimizing sequences, lower semi-continuity. Some relevant notions specific to three dimensional theory of hyperelasticity, interpretation of convexity and strong ellipticity in the context of hyperelasticity; Singular minimizers, Lavrentiev phenomenon, Gamma convergence and their relation to some problems in mechanics; Conclusion of the course with a discussion of some open problems

Lecturewise Breakup (based on 50min per lecture) (total lecture 40)


I. Introduction : (Lecture 1) (1 Lectures)

  • Overview of Course, A brief historical survey of calculus of variations. (Handout on Brief history of the calculus of variations: two tables giving timeline of events related to classical calculus of variations.). Description of Brachistochrone vs isoperimetrical problems (more like like linear space vs nonlinear spaces (eg. manifolds), constrained problems vs unconstrained etc).

II. Introduction to Function spaces and Functional Analysis: (Lecture 2–4)(6 Lectures)

  • Sets, Mappings and some Function Spaces with useful Structure, Differentiation and Integration, Functional Derivative. Mentioning Banach space and Hilbert space as well as metric spaces and, very briefly, manifolds also. Detailed definitions of C, C n , D,L p , Wk,p , etc function spaces. Some motivating examples from mechanics: elasticity, particle mechanics etc, involving such function spaces. Short review of real analysis and differentiation/integration, correspondence between variation and differential. Definition of directional derivative: Gˆateaux derivative. Highlight Gˆateaux vs Fr´echet derivative. Strong vs weak derivative.

III. (Formulation) : (Lecture 5–6) (2 Lectures)

  • Extremum of a Function defined on a Euclidean space, Local Minimum of a Function defined on a Euclidean space, Extremum of a Functional defined on a Banach space, Local Minimum of a Functional defined on a Banach space, First Variation and Second Variation. Strong vs weak minimum.

IV. (First Variation : Euler Equation: (Lecture 7–15) (9 Lectures)

  • Weak Extremal, Necessary condition for extremum: Euler equation, Null Lagrangian and its importance in mechanics (eg. elasticity), Characterization of Null Lagrangians, Some typical boundary conditions and their form in the calculus of variations: Natural Boundary conditions, Constraints and Lagrange multipliers, Isoperimetric constraints, Holonomic constraints, Non-Holonomic constraints, Transversality conditions.

V. Conservation laws and Jump conditions : (Lecture 16–21) (6 Lectures)

  • Statement of Noether's theorem (without proof!). Derivation of conservation laws in mechanics, eg: Jintegral, configurational force (Eshelby), and other integrals important in elasticity. Conservation laws, Application of Noether Theorem in One Dimension: Particle Mechanics, Application of Noether Theorem in One+One Dimension: Elastodynamics of a Hyperelastic Bar, Application of Noether Theorem in One+Three Dimensions: Elastodynamics of a Hyperelastic body. Non-smooth extremals: Lipschitz Extremal and Weierstrass-Erdmann Jump conditions. Examples from solid-solid phase transition..

VI. Minimizers : (Lecture 22–24) (3 Lectures)

  • Discussion on Necessary Condition and Sufficient Condition for a Local Minimizer in One Dimensional Variational Problem and Higher Dimensional Variational Problem. C 0 local minimizer and C 1 local minimizer. Necessary condition for a minimum, Legendre condition, Jacobi condition, conjugate points. Sufficient condition for a minimum, Weierstrass E-function. Field of extremals. .

VII. C1 local minimizer : (Lecture 25–29) (5 Lectures)

  • Necessary Condition and Sufficient Condition for C 1 local minimizer: Legendre condition in One Dimensional case, Jacobi's Conjugacy condition in One Dimensional case, Legendre-Hadamard condition in Higher Dimensional case, Strong ellipticity and Strongly elliptic operator, Jacobi Necessary condition and Jacobi Sufficient condition based on eigenvalue of Jacobi operator.

VII. C0 local minimizer : (Lecture 30–34) (5 Lectures)

  • HNecessary Condition and Sufficient Condition for C 0local minimizer: Weierstrass Necessary Condition in Higher Dimensional Case, Field of Curves, Field of Extremals, Mayer field, (Aside on Exterior Forms, Exterior Product, Manifolds, Differential Forms, Exterior Derivative and Pullbacks) , Weierstrass Sufficient Condition in One Dimensional Case, *Weierstrass Sufficient condition in Higher Dimensional case (*if time permits).

VII. Direct Methods : (Lecture 35–40) (6 Lectures)

  • Definitions: Convexity, QuasiConvexity, Rank-one Convexity, Strong Ellipticity, Weak Derivative, Minimizing sequences, Lower Semi-Continuity, (if time permits: conjugate spaces and weak*-convergence). Some relevant notions specific to three dimensional theory of hyperelasticity, Discussion of some results involving Convexity and Strong Ellipticity in the context of Hyperelasticity. Singular minimizers, Lavrentiev phenomenon and their relation to some problems in mechanics and elasticity. Discussion on the Direct Methods in the Calculus of Variations and their influence on some recent developments in the field.

References:

  1. Gelfand, I. M.; Fomin, S. V., 1963. Calculus of variations, Prentice-Hall, (Textbook)

  2. Giaquinta, M., and Hildebrandt, 2004. S. Calculus of variations I, Springer-Verlag

  3. Dacorogna, B., 1989. Direct methods in the calculus of variations, Springer.
 

ME626A

Vibration of continuous systems

Credits:

  

 

3L-0T-0L-0D (9 Credits)
 

Objective:


This course aims to setting-up initial-boundary value problems for some important and fundamental structural members viz. bars, strings, membrane and plates. Analytical and approximate solutions to these problems for various loading and boundary conditions are discussed and analyzed.

Course Content:


Vibrations in bars, strings, thin and thick beams, membranes and thin plates. Derivation of equations of motion for these structures. Computation of frequencies and mode shapes. Modal analysis. Analysis of damped and forced vibrations. Computational and experimental demonstrations.

Lecturewise Breakup


I. Introduction : (1 Lectures)

  • Linear versus nonlinear vibrations; linear vibrations: principle of superposition in modal and frequency domain.

II. Introduction to mode shapes and frequency for undamped systems.(2 Lectures)


III. Hamilton's principle and Euler-Lagrange equations.(3 Lectures)

 

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

  • Derivation of equation of motion, boundary and initial conditions, mode shapes, frequencies, orthogonality, RayleighRitz method.

V. Incorporation of forcing and damping terms in above systems (bars and strings).(3 Lectures)

 

VI. Approximate methods : (Lecture 22–24) (4 Lectures)

  • Galerkin and finite difference.

VII. Travelling wave solution, transient problem, reflections from the boundaries and simple impact problems. (3 Lectures)

 

VIII. Euler-Bernoulli and Timoshenko beams (5 Lectures)

  • Derivation of equation of motion, boundary and initial conditions, mode shapes, frequencies, orthogonality

IX. Membranes (circular disc, rectangular and spherical) :(3 Lectures)

  • Equation of motion, boundary and initial conditions, mode shapes, frequencies, orthogonality, Helmholtz equation.

X. Thin plates :(4 Lectures)

  • Equation of motion, boundary and initial conditions, mode shapes, frequencies, orthogonality.

XI. Advanced topic :(3 Lectures)

  • Select any one from (i) Elementary theory of modal analysis/testing, (ii) Shell theory , and (iii) Non-proportional damping.

XII. Computational (using FEM) or Experimental demonstrations. (5 Lectures)

  • Select any one from (i) Elementary theory of modal analysis/testing, (ii) Shell theory , and (iii) Non-proportional damping.

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.
 

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:

  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.
 

ME632A

Geophysical Fluid Dynamics

Credits

  

 

3-0-0-0 (9 Credits)
 

Course Content

To introduce analytical approaches for solving fluid dynamic problems arising in atmosphere and oceans.

Prerequisite

Basic fluid mechanics, basic ordinary and partial differential equations

Course Content

To introduce analytical approaches for solving fluid dynamic problems arising in atmosphere and oceans.

Desirable

ME681, ME631 (or equivalents)

Instructor

Ishan Sharma

Lectures per week

3 hrs

Condensed Syllabus

Equations of motion in rotating coordinate frames, Cartesian approximations, Density stratified flows and internal gravity waves, Taylor-Proudman theorem, Ekman layer, single and multiple layered shallow-water systems, Geostrophic adjustment and Thermal-wind balance, Potential vorticity, Poincare, Kelvin and Rossby waves, Kelvin-Helmholtz instability, Baroclinic instability, Wave-mean theory, 2D turbulence, chaotic advection in Stratosphere, Laplace tidal equations, Internaltides in deep oceans, tsunami waves.

Lecturewise Breakup (based on 75 min per lecture)


I. Introduction (7 Lectures)

  • Equations of motion in rotating coordinate frames.

  • Cartesian approximations: f-plane and beta-plane.

  • Effect of density stratification, Boussinesq systems, gravity waves.

  • Taylor-Proudman problem.

  • Ekman layer.

II. Inviscid Shallow-water theory (15 Lectures)

  • Shallow-water theory - single and multiple layers

  • Geostrophic adjustment and Thermal-wind balance.

  • Potential vorticity conservation.

  • Poincare, Kelvin and Rossby waves.

  • Quasi-geostrophy.

  • Simplified equations of oceans and atmosphere.

III. Instabilities, wave-mean flow interaction and turbulence (10 Lectures)

  • Kelvin-Helmholtz instability, Baroclinic instability, Eady problem.

  • Wave-mean theory, Eliassen-Palm flux.

  • 2D turbulence, inverse cascade and zonal jet formation.

IV. Advanced topics in geophysical fluid dynamics (8 Lectures)

  • Stratospheric transport.

  • Laplace tidal equations, internal tide generation in deep oceans.

  • Tsunamis.

References

  1. Atmospheric and Oceanic Fluid Dynamics, G. K. Vallis

  2. Geophysical Fluid Dynamics, J. Pedlosky