# Teaching

Teaching responsibilities in the ENDLab reflect many of our research interests, and include core and elective subjects on mechanics, dynamics, nonlinear dynamics and experimental methods. All the students get experience of being TA's during their time in the ENDLab. The courses. The courses we are involved in are listed below.

## 2.003J Dynamics and Control I

Introduction to the dynamics and vibrations of lumped-parameter models of mechanical systems. Kinematics. Force-momentum formulation for systems of particles and rigid bodies in planar motion. Work-energy concepts. Virtual displacements and virtual work. Lagrange's equations for systems of particles and rigid bodies in planar motion. Linearization of equations of motion. Linear stability analysis of mechanical systems. Free and forced vibration of linear multi-degree of freedom models of mechanical systems; matrix eigenvalue problems. Introduction to numerical methods and MATLAB to solve dynamics and vibrations problems.

Course material for 2.003J, including Prof. Peacock's lecture notes, are available through the MIT OpenCourseWare. [link]

There are also very nice video lectures presented by Prof. Sarma, which present the material in a slightly different way. [link]

## 2.050J Nonlinear Dynamics I: Chaos

Introduction to nonlinear dynamics and chaos in dissipative systems. Forced and parametric oscillators. Phase space. Periodic, quasiperiodic, and aperiodic flows. Sensitivity to initial conditions and strange attractors. Lorenz attractor. Period doubling, intermittency, and quasiperiodicity. Scaling and universality. Analysis of experimental data: Fourier transforms, Poincare sections, fractal dimension, and Lyapunov exponents. Applications to mechanical systems, fluid dynamics, physics, geophysics, and chemistry. See 12.207J/18.354J for Nonlinear Dynamics II.

Detailed course notes and psets from the material originally developed by Professor Rothman are available. [link] There is a set of Applets put together by Prof. Michael Cross at Caltech to give you some experience playing around with chaotic dynamics. [link]

## 2.032 Dynamics

Review of momentum principles. Hamilton's principle and Lagrange's equations. Three-dimensional kinematics and dynamics of rigid bodies. Study of steady motions and small deviations therefrom, gyroscopic effects, causes of instability. Free and forced vibrations of lumped-parameter and continuous systems. Nonlinear oscillations and the phase plane. Nonholonomic systems. Introduction to wave propagation in continuous systems.

A version of the course taught by Prof. Haller is available through MIT OpenCourseWare. [link]

## 2.087 Engineering Mathematics

Introduction to ordinary differential equations (ODEs). Linear systems of equations. Initial value problems, 1st and 2nd order systems. Eigenproblems, eigenvalues and eigenvectors, including complex numbers, functions, vectors and matrices. Introduces students to MATLAB and programming concepts such as data types, operators, flow control, arrays and functions.

## Lagrangian Coherent Structures

We have developed a short course (5 lectures) on the topic of Lagrangian Coherent Structures and their application to ocean transport problems. The course covers flow maps, flow map gradients, the right Cauchy-Green strain tensor and its eigenvectors and eigenvalues, Finite-Time Lyapunov Exponents (FTLE), strainlines and shearlines, numerical methods. The course notes and Matlab software are available on request.

## 2.21J Advanced Fluid Dynamics of the Environment

Fundamentals of fluid dynamics intrinsic to natural physical phenomena and/or engineering processes. Discusses a range of topics and advanced problem-solving techniques. Sample topics include a brief review of basic laws of fluid motion, scaling and approximations, creeping flows, boundary layers in high-speed flows, steady and transient, similarity method of solution, buoyancy-driven convection in porous media, dispersion in steady or oscillatory flows, physics and mathematics of linearized instability, effects of shear and stratification.

There is an excellent set of notes provided by Prof. C.C. Mei. [link]

## 2.671 Measurement and Instrumentation

Experimental techniques for observation and measurement of physical variables such as force, strain, temperature, flowrate, and acceleration. Emphasizes principles of transduction, measurement circuitry, MEMS sensors, Fourier transforms, linear and non-linear function fitting, uncertainty analysis, probability density functions and statistics, system identification, electrical impedance analysis and transfer functions, computer-aided experimentation, and technical reporting. Typical laboratory experiments involve oscilloscopes, electronic circuits including operational amplifiers, thermocouples, strain gauges, digital recorders, lasers, etc. Basic material and lab objectives are developed in lectures. Instruction and practice in oral and written communication provided.

## 18.354 Nonlinear Dynamics II: Continuum Systems

General mathematical principles of continuum systems. (1) From microscopic to macroscopic. Examples range from random walkers, to Newtonian mechanics, to option pricing. (2) Singular Perturbations. Examples include boundary layer theory, snow flakes and geophysical flows. (3) Instability. Generalize ideas from 18.353 to continuum systems. Examples from fluid mechanics, solid mechanics, astrophysics and biology. (4) Pattern formation and turbulence.

There is an extensive set of notes developed for this course by Prof. Peacock that builds on an early set of notes by Prof. Brenner (Harvard), and which have been added to by others in more recent years. If you would like a copy of these notes, just email Prof. Peacock with your request (they can't be posted online for copyright reasons right now. sorry).