maltlab Problem 2. The longitudinal dynamics of a B747 aircraft under some condi

Question

maltlab

Problem 2. The longitudinal dynamics of a B747 aircraft under some conditions are given by the state space realization x = Ax + Bu. where the state matrix is given by -0.00687 -0.0905 1.187 × 10-4-0.001026-0.4285 0 0.01395 -0.3151 773.98 -32.2 0 1. Find the eigenvalues of A using Matlab, and determine the eigenvalues that are related to the Phugoid 2. Find the natural frequency and damping ratio of the phugoid mode. Hint: Use the Matlab command 3. Find the natural frequency and damping ratio of the short period mode. Hint: Use the Matlab 4. Find the A matrix of the state space representation of the phugoid mode approximation and compare mode and the short period mode. Hint: use the Matlab command eig(A) damp(A) command damp(A) the natural frequency of the phugiod mode approximation to the natural frequency of the phugoid mode obtained from part 2

Explanation / Answer

SAVE THE FOLLOWING CODE IN MATLAB AND GET THE RESULTS-

clc

clear

close all;

A=[-.00687 0.01395 0 -32.2;-.09055 -.3151 773.98 0;-1.187*10^-4 -.001026 -.4285 0;...

0 0 1 0];

%% Part (A)

% returns diagonal matrix D of eigenvalues and matrix V whose

% columns are the corresponding right eigenvectors, so that A*V = V*D.

[V,D]=eig(A);

%% Part (B) and (C)

% returns the natural frequencies, Wn, and damping ratios,zeta, of the poles of A.

[Wn,zeta]=damp(A);

SOLUTION-

V =

0.0210 + 0.0165i 0.0210 - 0.0165i -0.9902 + 0.0000i -0.9902 + 0.0000i

0.9996 + 0.0000i 0.9996 + 0.0000i 0.1396 - 0.0006i 0.1396 + 0.0006i

-0.0001 + 0.0012i -0.0001 - 0.0012i -0.0001 + 0.0000i -0.0001 - 0.0000i

0.0011 - 0.0004i 0.0011 + 0.0004i 0.0002 + 0.0013i 0.0002 - 0.0013i

D =

-0.3737 + 0.8897i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i

0.0000 + 0.0000i -0.3737 - 0.8897i 0.0000 + 0.0000i 0.0000 + 0.0000i

0.0000 + 0.0000i 0.0000 + 0.0000i -0.0016 + 0.0438i 0.0000 + 0.0000i

0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i -0.0016 - 0.0438i

Wn =

0.9650

0.9650

0.0438

0.0438

zeta =

0.3872

0.3872

0.0362

0.0362

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