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tan q =
a b + sin g
q
a2 − sin2 g + b2
a2 − sin2 g
, q 6= ±p
2 (6.44)
which allows us to compute q, knowing the pre-specified value of g and the current
value of j. Obviously, this constraint is not applicable for ‘speed on pitch’.
6.3.5 The coordinated turn constraint
In the Earth-fixed reference frame, assuming a flat Earth, the velocity vector is tangential
to the turning circle, so the centripetal acceleration, expressed in units of [g]
equals:
G =
˙y
V
g0
(6.45)
We can derive a constraint for coordinated turns by taking the lateral v˙ equation of
the nonlinear force equations (2.28), imposing the steady-state condition v˙ = 0, and
the coordination condition Fy = Fgr, i.e. no aerodynamic and/or propulsive sideforce.
Substituting equation (3.10) yields:
g0 cos q sin j + pw − ru = 0 (6.46)
From the kinematic relations (2.59), we can derive:
p = − ˙y sin q
r = ˙y cos q cos j (6.47)
Substituting these equations in equation (6.46), and replacing u and v by the corresponding
relations from equation (2.31), we find:
g0 cos q sin j − ˙yV

sin q sin a cos b + cos q cos j cos a cos b

= 0 (6.48)
Finally, by substituting equation (6.45) and re-writing terms, the resulting coordinated
turn constraint is found:
sin j = G cos q (sin a tan q + cos a cos j) (6.49)
6.3.6 Combined constraints
Equation (6.49) must be used in conjunction with (6.44) if we want to trim the aircraft
for a coordinated turning flight with a specified rate-of-climb. If these equations are
solved simultaneously, the only remaining variables to be adjusted by the numerical
trim algorithm are the angle of attack and sideslip angle and the control inputs.
According to ref.[33] the simultaneous solution equals:
tan j = G
cos b
cos a
( ˜a −˜b2) +˜b tan a
q
˜ c (1 −˜b2) + G2 sin2 b
˜a2 −˜b 2 (1 + ˜ c tan2 a)
(6.50)
where:
˜a  1 − G tan a sin b
˜b sin g
cos b
(6.51)
˜ c  1 + G2 cos2 b
92 Chapter 6. Analytical tools
u x
Specify (initial)
flight- and aircraftcondition
for trimming
Minimisation
algorithm
Store steady-state
values of x a n d u
Flight path
constraints
Aircraft model
(evaluate state equations)
Scalar cost function
J ( x ) = J ( x , u )
.
x .
Figure 6.7: Aircraft trim algorithm
The value of j found in equation (6.50) can be substituted in (6.44) to solve for q.
When the flight-path angle is zero, equation (6.50) reduces to:
tan j = G cos b
cos a − G sin a sin b
(6.52)
For skidding turns j can be selected freely by the user, so then only the rate-of-climb
constraint remains to be solved.
6.3.7 The resulting steady-state trimmed-flight algorithm
We can now develop a general aircraft-trim algorithm that determines steady-state
flight conditions by searching for the state and control vectors for which the state
derivatives ˙V , ˙a, ˙b, p˙, q˙, and r˙ are identically zero, as specified in equation (6.40).
This is achieved by means of a multivariable numerical optimisation routine that
minimises a scalar cost function by adjusting the control inputs and appropriate state
variables. The cost function J is usually equal to the sum of the weighted squares of
the time-derivatives:
J = c1 ˙V 2 + c2˙a2 + c3 ˙b 2 + c4p˙2 + c5q˙2 + c6r˙2 (6.53)
where ci, i 2 {1, 2, . . . , 6}, are weighting constants. A suitable optimisation technique
for this particular problem is the Simplex method [33].
Figure 6.7 shows a block diagram of the resulting trim algorithm. First the flight
condition must be specified. The trim program must make an initial guess for the
independent state variables and the control variables that will be adjusted during
6.4. Miscellaneous simulation issues 93
the trim process. Next, the minimisation routine which searches for those values of x
and u that minimise the cost function J will be started. The elements of these vectors,
which are adjusted by either the minimisation routine or the constraints, are updated
for each iteration step.
The state equations are then evaluated for the new values of u and x to find the
time-derivative of the state-vector, ˙x. Substituting the results in equation (6.53) yields
the new value of J, which is returned to the minimisation routine. A stop criterion
that depends upon the change of J between two iterations is used to decide when
to finish the minimisation procedure. Also, the maximum number of iterations is
　
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