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 | | From: | Gordan Sikic | | Subject: | Re: aircraft trimming and equation | | Date: | Tue, 02 Nov 2004 12:02:08 -0600 |
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 | Hi,
> Hi, someboby has ever developed a trim algoritm for trimming an aircraft in
> coordinate flight (level-wings, turning & pull-up) ??? could you tell me how
> to trim with an optimization algorithm?? and what equation you use ??
> thanks very much
>
Yes, I have done it. I calculated all missing state vector members, and
all missing control vector members in order to get trimmed state.
First step was to define complete nonlinear 6DOF mathematical model of
the aircraft in the form:
x_dot = f(x,u,t)
where:
x == state vector
u == control vector
t == time (strictly speaking this is not needed directly)
x_dot = d_x / d_t
Second step was to theoretically define "trimmed position" for all
particular situatios I wanted AC to be trimmed for. For example for
level flight (among other things), derivations of all rotational speeds
should be zero, and rotational speeds themselves also should equal to
zero, i.e.
p == q == r == p_dot == q_dot == r_dot == 0.0
next, define interconnections between state vector members, and control
vector members. For level flight, you have interconnection between
speed, angle of attack, throttle, elevator input, and eleator trim
input, as well as flapses, ..... it's up to you to decide what will be
calculated.
Define parts of state vector that are known, that need to be
constrained, and parts that need to be calculated. Same applies for
input vector. For level flight, for example, if you define altitude,
speed and mass of the AC, angle of attack should be calculated, and
parts of control vector.
Define "goal function" that will be used to evaluate position, based on
equation x_dot = f(x,u,t),
After that, I used matlab's optimization toolbox to minimize "goal
function", that boiled up into nonlinear simplex method of minimizatrion.
After theoretically solving the problem, and defining correct 6DOF
flight model (not a trivial task at all) process of trimming was
suprisingly easy to achieve.
Results were checked using two approaches. First was to feed back
calculated state vector and control vector into "x_dot = " equation and
check derivations. Other was to actualy simulate flight using calculated
x ad starting vector, and keeping u constant. If it was "trimmed flight"
there should be no (significant) changes in flight contitions.
I made calculations using MiG-21 AC model, for level flight, climbing,
coordinated turn, and coordinated turn in climb, with excelent results.
Didn't do pull-up, since during that, all elements of of x and u are
changing dramatically.
You might check Brian & Lewis "Aircraf simulation and control". They
used similar approach, but without direct constrains (they added new
equations into set of equations in order to achieve same effect).
There's also FORTRAN source at the end of the book open for examining.
Hope thes helps,
cheers,
Gordan
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