Mathematics & Statistics http://hdl.handle.net/10211.3/122490 2020-01-26T00:19:54Z Studying the Dynamics of Detonation http://hdl.handle.net/10211.3/214772 Studying the Dynamics of Detonation Bullard, Erin In this thesis, we study the dynamics of detonation using a simplified version of the reactive Euler Equations, namely Fickett's model. We explore different reaction rates and approximate solutions for this system of PDEs by using upwinding and Godunov's method. Upon exploration, we confirmed that a more complicated reaction rate produced interesting behavior, ultimately arriving at the discovery of a Hopf bifurcation. 2020-01-08T00:00:00Z Control of Satellite Angles When Orbiting Asteroids http://hdl.handle.net/10211.3/214768 Control of Satellite Angles When Orbiting Asteroids Ayala, Alexis In the article "Attitude Dynamics and Control of Satellites Orbiting Rotating Asteroids", by K.D. Kumar, systems of ordinary differential equations were derived to describe the attitude angles of a satellite, and control inputs were formulated from the linearized system . The control was applied to the nonlinear equations and the solutions were found numerically by implementing the Petzold-Gear BDF method. In this paper we present the following work based on Kumar : (1) The equations of motion for a satellite orbiting a non-symmetric mass are derived. (2) Nonlinear, ordinary differential equations that represent the torques of the three attitude angles pitch, roll, and yaw are found. (3) These equations are linearized about the zero-state of the three angles, from which two different systems are derived. One system assumes that the orbit of the satellite is non-circular, while the other assumes that it is. (4) Stability analysis is performed and two control equations are established. One assumes that the mass of the asteroid is non-symmetrical, while the other assumes the opposite. In this paper we present the following contributions: (1) Two control schemes applied through pole placement and quadratic optimal control were found. (2) The alternate control schemes were implemented on the linearized system and solved using MATLAB. (3) The solutions were found by using MATLAB's function ode45, which implements the Dormand-Prince method (an explicit Runge-Kutta formula). (4) These models were compared with three different performance tests, two of which determined the error of the controls while the other calculated how much control force effort is required to maintain the control. After running these performance tests, it was shown that Kumar's controls lead to small errors for relatively low control force effort. In addition, it was determined that Kumar's controls and quadratic optimal control with Q being the 6 by 6 identity matrix and R = 1 required the least amount of effort, the pole placement control with poles at -5 had the least amount of error but required the most force, and some optimal poles found through a minimization problem were the best compromise between the two. In addition, Kumar's control does not stabilize the pitch of the system with a circular orbit, while the other three do. 2020-01-08T00:00:00Z Optimal Vehicle Fleet Size and Routing of a Fleet of Electric Vehicles for Ride-Sharing http://hdl.handle.net/10211.3/213897 Optimal Vehicle Fleet Size and Routing of a Fleet of Electric Vehicles for Ride-Sharing Yeh, Jeffrey This thesis will explore a model for control of a fleet of vehicles, assuming perfect information about vehicle status and future passenger demands. We will describe a base model from ``Data-Driven Model Predictive Control of Autonomous Mobility-on-Demand Systems'' by R. Iglesias et al. and later propose extensions based on the application of the model to the problem of ride-sharing using electric vehicles. The first extension is to support multiple-passenger vehicles of different sizes. Next, we introduce charging stations into the model in order to maintain the battery level of the vehicles. Lastly, we extend the model allow our customers to be picked up one time step after their originally requested time. Then, we consider examples of small cases of each model to demonstrate how each model works. Finally, we include results demonstrating the applications of the models to a variety of situations, including settings based on real-world data from New York City and Sendai City. 2019-10-16T00:00:00Z Rational Walks http://hdl.handle.net/10211.3/213680 Rational Walks Simon, Anthony In this thesis a mapping from decimals into {0,1,2} is defined and described as a walk. The thesis explores visual representations of rational and irrational numbers as walks. Several interesting classes of real numbers are considered, including normal numbers and Khinchin's Constant. 2019-10-08T00:00:00Z