README.md

Homework #5

due 11/17 by 11:59pm

Problem 1 due 11/9

1. Create a new github repository called '05_curve_fitting'.

a. Add rcc02007 and zhs15101 as collaborators.

b. Clone the repository to your computer.

2. Create a least-squares function called least_squares.m that accepts a Z-matrix and dependent variable y as input and returns the vector of best-fit constants, a, the best-fit function evaluated at each point $f(x_{i})$, and the coefficient of determination, r2.

[a,fx,r2]=least_squares(Z,y);

Test your function on the sets of data in script problem_2_data.m and show that the following functions are the best fit lines. Report the coefficient of determination in your README.

a. y=0.3745+0.98644x+0.84564/x

b. y= 22.47-1.36x+0.28x^2

c. y=4.0046e^(-1.5x)+2.9213e^(-0.3x)+1.5647e^(-0.05x)

d. y=0.99sin(t)+0.5sin(3t)

3. Load the data from the class dart experiment. Show your work in a script with filename dart_statistics.m in your repository.

a. Which user (0-32) was the most accurate dart thrower e.g. mean(x)+mean(y) was closest to 0 cm?

b. Which user (0-32) was the most precise dart thrower e.g. std(x)+std(y) was closest to 0 cm?

4. 100 steel rods are going to be used to support a 1000 kg structure. The rods will buckle when the load in any rod exceeds the critical buckling load

$P_{cr}=\frac{\pi^3 Er^4}{16L^2}$

where E=200e9 Pa, r=0.01 m +/-0.001 m, and L is the length of the rod which can only be controlled to 1% tolerance. Create a Monte Carlo model buckle_monte_carlo.m that predicts the mean and standard deviation of the buckling load for 100 samples with normally distributed dimensions r and L.

[mean_buckle_load,std_buckle_load]=buckle_monte_carlo(E,r_mean,r_std,L_mean,L_std)

a. What is the mean_buckle_load and std_buckle_load for L=5 m?

b. What length, L, should the beams be so that only 2.5% will reach the critical buckling load?

5. The drag coefficient for spheres such as sporting balls is known to vary as a function of the Reynolds number Re, a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces:

$Re = \frac{\rho V D}{\mu}$

where ρ= the fluid’s density (kg/m3), V= its velocity (m/s), D= diameter (m), and μ= dynamic viscosity (N⋅s/m2). The following table provides values for a smooth spherical ball:

Re (x1e-4)	C_D
2	0.52
5.8	0.52
16.8	0.52
27.2	0.5
29.9	0.49
33.9	0.44
36.3	0.18
40	0.074
46	0.067
60	0.08
100	0.12
200	0.16
400	0.19

a. Create a function sphere_drag.m that outputs the drag coefficient based on the given table and an input Reynolds number using a spline interpolation of either linear ('linear'), piecewise cubic ('pchip'), or continuous cubic spline('spline'):

[Cd_out]=sphere_drag(Re_in,spline_type)

b. Use the following physical constants to plot the drag force vs velocity for a baseball: ρ= 1.3 kg/m3, V= 4 - 40 (m/s), D= 23.5 cm, and μ=1.78e-5 Pa-s. Plot all three interpolation methods on a single plot. Show the plot in your README.

6. Evaluate the integral of the following function:

a. analytically

b. with 1-point Gauss quadrature

c. with 2-point Gauss quadrature

d. with 3-point Gauss quadrature

e. include the results in a table in your README

method	value	error
analytical	...	0%
1 Gauss point	...	..%
...	...	...