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Module Author: Rondal F. Taylor
Author Contact: firstname.lastname@example.org
Funded by: National Science Foundation
Vehicles (cars and trucks) which contain dangerous cargos may need to be analyzed and critical values of vertical displacement, pitch, velocities, and accelerations computed. Mathematical models of moving vehicles can be developed from elementary concepts based on spring-mass-damper arrangements with imposed displacements at tire-roadway contact points. Optimally designed speed bumps added to the simulation may make it is possible to identify vehicles carrying unusually massive cargo for example. Discrete linear vehicle models can be constructed with multiple degrees of freedom representing the primary motions including vertical translation and pitch. Time history solutions of the differential equations using MATLAB can be performed and parameters in the system varied. Students in engineering and applied mathematics would find many aspects of this problem useful in order to gain real-world experience with an important class of simulation models. Level of student would be third or fourth year.
Module Author: Ronald Taylor
Funded By: National Science Foundation
The mathematical models discussed herein have application to aerospace and mechanical engineering. An elastic panel with one side exposed to a supersonic flow is representative of an environment experienced by wing and fuselage structures of vehicles such as the space shuttle during re-entry. High speed flow and structural imperfections or damage may lead to catastrophic failures. A simplified dynamic beam model formulated as a partial differential equation in (x,t) will be given which includes the effect of edge compression and air loading. The method of Galerkin and similar techniques reduce such a model to a system of equations which can be studied on various levels of computational and mathematical complexity. This includes (1) Kx = f (system of linear algebraic equations), (2) Mx’’+ Kx = Q (system of second order ODEs), and (3) λ2M = Kx (generalized eigenvalue problem). Structural damage or imperfections can be studied as changes to the K matrix in Kx=f or as perturbations to both M and K in the ODE and eigenvalue problems. By working through the suggested problems and project ideas outlined, undergraduate students gain experience with standard methods for reduction of the governing PDE to a finite system of ODEs and the efficient solution of matrix computation problems where a significant number changes are being made to the system matrices. The primary software used is MATLAB with the Symbolic Math Toolbox for derivation of system matrices and then solution of the resulting equations. Investigations include the study of the nonsymmetrix matrix eigenvalue problems which are deficient in eigenvectors. Level of complexity of this module can be varied so that selected problem sets projects are appropriate for intermediate level undergraduates or advanced students in applied mathematics, physics or engineering.
Module Author: David G. Robertson
Author Contact: email@example.com
Funded By: The National Science Foundation
This module is an introduction to the physics of white dwarfs and neutron stars. The differential equations that describe the equilibrium states of these objects are developed. An overview of techniques for the numerical solution of ordinary differential equations is presented.
Module Author: John Philips
An exploration of oscillatory motion and vibrations of extended objects. Starting from simple harmonic motion, as introduced in an introductory Physics class, the module builds a model of extended objects vibrating and develops some methods of solution for such a model. Skills used and developed include: Describing forces in physical systems (Review from Physics), Energy in physical systems (Review from Physics), Solutions of dierential equations (Part of Differential Equations), Using a Computer algebra system to solve complicated equations (Extension from Computational Science), Useful visualizations of large data sets (Part of Scientifc Visualization), Identifying and using independent functions (Part of vector spaces and function spaces in many math classes).
Funded By: W. M. Keck Foundation
This module provides an introduction to spontaneous magnetization in ferromagnetic materials. It studies this phenomenon by using the Ising model and Monte Carlo modeling techniques. Although the topic matter is probably not familiar to most students, the techniques used are approachable by anyone who has completed a typical first year introductory physics course. Students should be comfortable with the interpretation of integrals and derivatives, though there is no need to evaluate them analytically during the module. Some comfort with the concepts of random numbers and random selections is also useful. It is possible to approach this module in a hands-on manner, where provided sample code is used to perform the calculations and the student’s focus is on interpretation of the results. In this case, minimal programming skills are required. In situations where it is warranted, it is also possible to have the students generate all of the required computational codes. For classes making this choice, the students will need strong programming skills. Any number of intermediate choices is also possible. The difference is obviously one of what the instructor feels is the appropriate focus for the students in the class.
Module Authors: Marc Goulet, Alex Smith, Paul J. Thomas, Brandon Barrette
Author Contact: firstname.lastname@example.org
Funded By: Battelle
The first comprehensive theory of gravitational orbits was developed by Newton. The orbits, or gravitational trajectories, are conic sections and arise as solutions to a second order linear differential equation. Newton assumed that time was absolute and the universe was described by Euclidean geometry. After Newton, Riemann developed the mathematics that describes the geometry of curved spaces [Spivak, 1979]. Einstein adapted this mathematics to describe gravity as the curvature of four-dimensional spacetime [Pais, 1982]. One early success was the application of this theory to explain the advance of the perihelion of Mercury. Soon afterwards, Schwarzschild found the first exact solution to Einstein's field equations. Subsequently this solution and others were interpreted as modeling the gravitational field surrounding a black hole.
Module Authors: Patrick Shields
Author Contact: email@example.com
Funded By: National Science Foundation (9952806)
In your study of general physics you were probably introduced to projectile motion as an application of the equations of constant acceleration. But did you also gain an appreciation for just how pervasive projectile motion is in our world? Whether you are performing a leap in ballet, shooting baskets in your driveway, diving off the three-meter platform, hitting a baseball, or catching a pass in football, the motion of the object in flight can be modeled as projectile motion. For some of these examples, it is sufficient to model the motion exactly as you did in general physics - constant acceleration with no effects due to the atmosphere. We will begin with just such a model, but for games like baseball, golf, or tennis that model won't be good enough. So we will quickly extend our model to include the effects of moving through a fluid medium - the air.
Module Author: Richard Gass
Author Contact: firstname.lastname@example.org
Funded By: National Science Foundation (0618252)
Nano-science is one of the fastest growing fields in both physics and engineering. It is now possible to design and fabricate devices whose physical dimensions are on the nanometer scale and whose quantum properties can be tuned as desired. This notebook will study the quantum mechanics of what are known as reduced dimensional structures.
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