The attached Figure 22 from Hoerner (1958) and Figure 4.8 from Newman (1977) describe the drag coefficient and added mass coefficients of elongated rotationally symmetric bodies similar to the AUV shape we will use in the 13.016 term project.
- The drag coefficient is defined with respect to the wetted area
as follows:
where
density of fluid (
for salt water), V is vehicle speed in 
, and F = drag force in N . The drag coefficient is a function of the Reynolds number and slenderness
ratio 
: 
where
, where 
kinematic viscosity of fluid ( 
for salt water). - The added mass coefficients correspond to a spheroid (or ellipsoid of revolution) of maximum diameter 2b and length 2a.
The purpose of this exercise is to construct accurate uniform cubic B-spline
approximations of the following curves:
Digitize the curve assigned to you and write a C program to produce an approximation
of your digitized data. Your program should contain a function which takes 
or 
as an argument and returns the evaluation of the curve at that point. You
should submit your code and a plot of your results to trjackso@mit.edu.
The students approximating the 
should use sampling at the following values of 
:
and should use natural spline boundary conditions.
The students approximating the added mass coefficients 
should use sampling at the following values of 
Create a plot of your curve. To evaluate the curve, you will need to parameterize
each segment i of the curve in terms of a local parameter 
.
For 
, use:
For 
and 
, use:
Given one segment of a uniform cubic B-spline curve with control points:
.
- Determine the convex hull and its area.
- Using equations for the position and first derivative at the end points in terms of control points, convert this curve segment to its Bézier form.
- Determine the convex hull of the resulting Bézier curve and find the ratio of the areas of the two convex hulls (Bézier/B-spline).
Prove that the uniform cubic B-spline basis functions
are non-negative in 
. (This property together with the summation to unity are essential for
the convex hull property of uniform B-spline curves and the variation diminishing
property.)
Get together with the other 13.016 students to discuss the project description handed out in class. As a group, decide how the responsibilities will be divided.
Once you have decided what your area will be, write a short project proposal (about 1 typed page) to inform us. This should not be long but cover what you expect to accomplish, how your work will be integrated with the other students' work, and a brief outline of how you intend to progress. Please also include what personal expectations you have for the project (what you hope to learn or achieve through this project).