### Overview

MS and PhD students in Mechanical Engineering & Mechanics can choose from a wide range of graduate-level courses offered within the department. Some degree programs, including the MS - Interdisciplinary Engineering track, also allow students to build flexible degree programs with specialized concentrations in emerging fields. Refer to the individual program descriptions for complete guidelines on course selection.

Rate equations describe phenomena across the physical sciences, including mathematical models for wave propagation, predator-prey interactions, and scalar transfer of heat and matter. Here a solution surface describes the diffusion of fibronectin produced by endothelial cells in response to the presence of a tumor in scaled coordinates. In ME/BIOE/CHE 452, students develop the analytical tools to construct, analyze, and solve ordinary and partial differential equations that arise in research and pedagogical contexts in our graduate program.

### ME452 - Mathematical Methods in Engineering I

The double pendulum is a well-known dynamical system that exhibits chaotic motion with a strong sensitivity to initial conditions. This stability plot is a map showing the time required for the second link of a swinging double pendulum to flip over based on the initial positions of the two links. Students who take ME413 learn how to solve the double pendulum and other initial value problems. They also learn data visualization techniques that can creatively illustrate the behavior of these systems under different conditions.

### ME413 - Numerical Methods in Mechanical Engineering

Complex numbers express 2D spaces in terms of a single variable, which can be exploited to solve complicated boundary value problems. This image shows the equipotential lines and streamlines of a conformal mapping used to find the fluid flow around a backward-facing step or stress concentrations in an elastic medium. Students in ME 453 equip themselves with the analytical tools of complex variable theory and solve problems involving integral transforms and special classes of partial differential equations in fluid and solid mechanics.

### ME453 - Mathematical Methods in Engineering II

Startup of a jet engine while the aircraft is on the tarmac generates two concentrated vortices with high rotation rates, one from the tarmac and the other from the fuselage, that degrade engine performance. Students who take ME 430 learn how to analyze vortex flows, as well as flows along and within a variety of engineering configurations ranging from airfoils to bearings. In doing so, they learn how to implement assumptions and approximations that lead to simplified solutions of the complete governing equations.

### ME430 - Advanced Fluid Mechanics

Students in ME464 learn about Renewable Energy technologies that include hydropower, solar, wind, tidal, and wave energies. We also discuss UN sustainability goals, energy economics (cost of energy), renewable energy resources as well as challenges in implementing renewable energy sources.

### ME464 - Renewable Energy

A spinning top is an example of a spatial rigid body that exhibits complex dynamics. Students who take MECH 425 not only learn how to derive the equations of motion for such systems using analytical methods and simulate such systems through numerical integration, but also learn techniques for gaining deeper understanding of their complex motion using tools such as stability analysis, linearization, modal analysis and conservation principles.

### MECH425 - Analytical Methods in Dynamics and Vibrations

For complete descriptions of course offerings in Mechanical Engineering & Mechanics, please refer to the University catalog.