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Solution Procedures 39
relaxation is applied to the distance flown in the destination segments for range credit.
5-1.4 Maximum Gross Weight
Flight conditions are specified for the sizing task and for the flight performance analysis. Mission
takeoff conditions are specified for the sizing task and for the mission analysis. Optionally for flight
conditions and mission takeoff, the gross weight can be maximized, such that the power required equals
the power available, min(PavP G−PreqPG) =0(zero power margin, minimum over all propulsion groups);
or such that the power required equals an input power, min((d + fPavP G) − PreqPG) =0(minimum over
all propulsion groups, with d an input power and f an input factor; this convention allows the power to
be input directly, f =0, or scaled with power available).
The secant method or the method of false position is used to solve for the maximum gross weight.
A tolerance ɛ and a perturbation Δ are specified. The variable is gross weight, with initial increment
of W Δ, and tolerance of 0.01Wɛ. Note that the convergence test is applied to the magnitude of the
gross-weight increment.
5-1.5 Maximum Effort
The aircraft performance can be analyzed for the specified state or a maximum-effort performance
can be identified. The secant method or the method of false position is used to solve for the maximum
effort. The task of finding maximum endurance, range, or climb is usually solved using the goldensection
method. A tolerance ɛ and a perturbation Δ are specified.
A quantity and variable are specified for the maximum-effort calculation. Tables 5-1 and 5-2
summarize the available choices, with the tolerance and initial increment used for the variables. Note
that the convergence test is applied to the magnitude of the variable increment. Optionally two quantity/
variable pairs can be specified, solved in nested iterations. The two variables must be unique. The two
variables can maximize the same quantity (endurance, range, or climb). If the variable is velocity, first
the velocity is found for the specified maximum effort; the performance is then evaluated at that velocity
times an input factor. For endurance, range, or climb, the slope of the quantity to be maximized must be
zero; hence in all cases the target is zero. The slope of the quantity is evaluated by first-order backward
difference. For the range, first the variable is found such that V/ ˙w is maximized (slope zero), and then
the variable is found such that V/ ˙w equals 99% of that maximum; for the latter the variable perturbation
is increased by a factor of 4 to ensure that the solution is found on the correct side of the maximum.
5-1.6 Trim
The aircraft trim operation solves for the controls and motion that produce equilibrium in the
specified flight state. A Newton—Raphson method is used for trim. The derivative matrix is obtained
by numerical perturbation. A tolerance ɛ and a perturbation Δ are specified.
Different trim-solution definitions are required for various flight states. Therefore one or more
trim states are defined for the analysis, and the appropriate trim state selected for each flight state of
a performance condition or mission segment. For each trim state, the trim quantities, trim variables,
and targets are specified. Tables 5-3 and 5-4 summarize the available choices, with the tolerances and
perturbations used.