User:Egm6341.s11.team1.arm/HW7

Problem 2: Solve parameters for Equation of Motion
Referenced the work of Team 3, Spring 2010.

Improvements: -Added display of absolute error to user can monitor progress of convergence -Code restructured, variables renamed, and additional comments and to make code easier to follow and inputs easier to change -Improved parts of code to reduce time for absolute error to converge to $$10^{-6}$$ -->Old code time: Longer than 1 hour -->This code time: 107 sec

Given
Equations of motion for an aircraft:

where

and

Physical parameters were taken from the Subchan and Zbikowski paper:

Initial conditions were taken from the Subchan and Zbikowski paper:

Values for the controls are given in [[media:Egm6341.s10.mtg41.djvu|S10 Mtg 41]]: $$ \underline{u} (t)= \left \lfloor T(t), \alpha\ (t) \right \rfloor \ $$

Find: z and J
Solve to find the matrix $$z \ $$ for the bunt maneuver using S10 and S11 values, as well as values for $$ J \ $$ such that:

$$ J= \int_{0}^{t_f} y(t)\, dt \ $$ such that $$ y(t_f)=0 \ $$

Solve: Using S10 values
The following MATLAB code was developed to solve for $$\underline{z} \ $$ using Newton-Raphson and Hermite-Simpson:

The solution was confirmed using the MATLAB command ode45:



Solve: Using S11 values
The different values of $$\alpha \ $$ used for the S11 solution are shown below:

The following MATLAB code executes the model using these new values of $$\alpha \ $$:



Solve: Compare J values
The following MATLAB code was used to use the MATLAB trapz command to integrate the output of Parts 1 and 2 and determine $$ J_1 \ $$ and $$ J_2 \ $$.

$$J_1 = 16860 \ $$ $$J_2 = 10824 \ $$

Meaning that since $$ J_2 < J_1 \ $$, the S11 values provide a better solution.