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Trimmed Aircraft Equation Worksheet
- Solve the trimmed aircraft equation for T, the lift due to the
tail, using the following data:
W = 18,000 kg, dw= 2.4 m, dt = 18.3 m, T
Notice the sign of the tail lift. In which direction does this
tail lift act? ________________.
- Now suppose we use a B17G
aircraft (http://en.wikipedia.org/wiki/B-17_Flying_Fortress) as our problem model. Assuming 100 kg has been used
for fuel, the weight of the B17G is the takeoff weight less 100 kg
- We are now going to use the following assumptions for the
values of wing lift and distance from the wing to the center of
gravity (cg): W = 35,470 kg, dw = 1.07 m. Using our
trimmed aircraft equation and the fact that the total lift is
equal to the sum of the wing lift and the tail lift, we have two
simultaneous linear equations: L= W + T and (W x dw) +
(T x dt) = 0. Remembering that the total lift is
necessarily equal to our total weight (from Problem 2 above), we
can solve our first equation for the tail lift,
T = ____________________.
- Substituting T, W, and dW into our second equation,
solve for dt,
dt = __________.
- Now let's assume something in our B17G shifts position, such
that the cg moves 0.5 m aft. Then dw becomes 0.57 m and
dt becomes 12.8 m. So, to trim the plane for a fixed
wing lift, our value of tail lift must now be changed.
Using the second equation, T becomes ____________________.
What change is necessary in tail lift due to the 0.5 m change in
From our Trimmed Aircraft slide, we realize our altitude will
change because the total lift has changed. What is the direction
of the altitude change? _________.
- Now please read the slide called Size
Effects in Airplane Aerodynamics. Given that the total lift is
directly proportional to the total wing and tail area, we can
write an equation W + T = k(Aw + At), where
Aw is the area of the wing, At is the area
of the tail, W and T are, as before, the lifts due to the wing and
tail, and k represents a constant (which includes the lift
coefficient and other aerodynamic parameters which we assume to be
held constant). We can also say that the lift due to the wing is
proportional to the area of the wing and likewise for the tail,
resulting in the equations: W = kw Aw and T
= kt At. Because the conditions determining
these constants are the same at any given time, let us allow
kw = kt = k. From our B17G data, we know
Aw, which coupled with W given in Problem 3, enables us
to find k = ______________.
- Now that we know k, let's use the tail lift which you found in
Problem 3 to determine the area of the tail, i.e., At =
abs T/k, = __________________. We can see that the area of the
tail is signigicantly smaller than the area of the wing. Why does
this make sense?
- An interesting, but simple, matrix problem involves adding a
third equation to our trimmed aircraft system. For a relatively
short time period, we can fix the aircraft weight. The three
simultaneous linear equations are: L = 32,620 kg; L - W - T = 0;
and W x dw + T x dt = 0. Substitute in the
coefficients dw and dt from Problems 3 and
4, and rewrite the system as a matrix equation.
- Using your graphing calculator, find the inverse of the
- Multiply both sides of the matrix equation on the left by the
inverse matrix and simplify.
Therefore, L = __________, W= __________, T = __________.
Compare these answers with those assumed for L in Problem 2, and
calculated for W and T in Problem 3. Do they agree? _______.