The dynamics of a rock drill
Date
2001
Authors
Davidova, Adriana
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Abstract
This thesis focuses on a model for the movement of a rotary drill, such as that used in oil drilling rigs. Such a rotary drill is typically composed of three intermeshed gears. During operation the drill hits and bounces off the rock, chipping the surface at contact. In order to design more efficient and durable drills, one may ask, for different parts of the wheel, what the long term distribution of hits is, should a stable one exist. To answer this question, A. Lasota and K. Rusek analyzed the movement of one of the wheels, which simplifies the model to two dimensions. Using the Birkhoff ergodic theorem they were able to find an estimate for the long term distribution of hits by computing long orbit averages. However, due to the nature of this method, computer round off error is compounded throughout the numerical process, contributing greatly to the uncertainty of the estimate. This motivates the need for an alternative way of computing the density function for the distribution of hits. The main results of this thesis are a confirmation of the Lasota and Rusek's work, and an alternative and more efficient way of obtaining refined density distribution estimates.
We will begin by discussing the Birkhoff ergodic method, and Lasota and Rusek's model and results. Then, a Markov approximation scheme, known as Ulam's method of computing the density function will be presented. The similar estimate for the density function produced using Ulam's method will support Lasota and Rusek's results. More refined estimates will be computed using the latter method, providing numerical evidence of its stability. Furthermore, we will derive a formula for the drilling efficiency in terms of the rotational speed of the drill. Numerically we will determine the rotational speed that yields maximum drilling efficiency, and observe the asymptotic behaviour of the efficiency for large speeds.