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Old Yesterday, 11:45 PM
Buwa is offline
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Australia, VIC, Keysborough
Joined Oct 2008
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System Matrix output from AVL

Hi,

I have been trying to do some simulations with AVL system output lately and have some doubts to clarify.

The case Im lookin at is Supra (given in AVL/runs) trimmed for steady state glide at CL=0.7.
To make it simple, lets limit this to the longitudinal matrix.
These are the conditions at this trim state.

1. From Treft plane plot
V= 6.824
ele = -0.9714 (deg) -0.016954128 (rad)
alfa = 6.0675 (deg) 0.105897852 (rad)
CL= 0.7
CD= 0.02441
q = 0

2. From Mode Simulation
When the pertubation is removed by "-" trimmed theta found to be 0.

3. From EigenValues
Correct longitudinal matrix should have these eigenvalues.

-9.4661560 3.2088668
-9.4661560 -3.2088668
-0.13851462 0.66802061
-0.13851462 -0.66802061


When Extracted A and B matrices from the system output file, They are as follows.

A =
[[ -0.0486, 2.397 , 0.6799, 9.81 ],
[ -1.4589, -11.0072, -4.693 , 0. ],
[ -0.1097, 2.4217, -8.1535, 0. ],
[ 0. , 0. , 1. , 0. ]]

B =
[[-0.01],
[ 0.09],
[-1.03],
[ 0. ]]

Initial state vector and control vetors are,

x =
[[ 6.81985473], # V*cos(alfa)
[ 0.23781808], # V*sin(alfa)
[ 0. ],
[ 0. ]]

y = -0.016954128353872919 # Elevator deflection in Rad

Now the problem is when evaluated this does not satisfy Ax +By = 0.

Ax+By=
[[ 0.24],
[-12.57],
[ -0.15],
[ 0. ]]

I would like to get these few pointers clarified.

1. When I wrote to Prof. Drela, he suggested that gravitational accelaration need to be added in to the matrix. According to my limited understanding only places where the gravity applies is at (1,4 - first row 4th col) and (2,4 - second row 4th col). Given the theta0 is 0, values 9.81 and 0 seems to be correct. isnt it ? And eigen values are also very close.

[-9.4661216+3.20885761j, -9.4661216-3.20885761j,
-0.1385284+0.66805654j, -0.1385284-0.66805654j]

2. As B*y is ~zero, and pitch rate is known to be 0, by considering the 3rd row, M_u/M_w (1st and 2nd elements) should have the same ratios as u/w (22.07566) . But this doesnt seem to be the case.
V *np.sin(alfa) / V*np.cos(alfa) = 9.4077

3. Again according to my limited understanding element Z_q (3rd column 2nd row) should be equal to Uo. In the matrix this is -4.693. Shouldn't this be something close to -6.81985473 [V*cos(alfa)]

4. Regarding the first 2 rows and columns, to get the x_dot to zero, (u_dot and w_dot should be zero because its trimmed flight), shouldn't one of the elements has to flip the signs ? Say u and w both +, then 1st raw will be zero but second would not and vise verse.

Sorry for the long post. I've been struggling with this for some long nights now.
What am I missing here?

Any feedback is much appreciated.
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