X Derivatives and Parameters

Equation 9.33a contains the terms p, CXu, CXa, and Сц. Given the wing area, mean aerodynamic chord, air mass density, and airplane mass, the dimensionless “mass,” p, can be calculated from Equation 9.31. Сц is simply the trim lift coefficient and can be determined from

n – mg Сц QoS

where q0 = ipU02.

Expressions for the derivative CXu are presented in Equation 9.25. These expressions involve the trim drag coefficient Cq, and the trim pitch angle 0O. Since we are using stability axes, ©0 is simply the climb angle for the trim conditions. Сц, is a function of Сц and is given by

с Л+£к

Cd° S irAe

We will again use the Cherokee 180 pictured in Figure 3.62 as an example. For the trim condition, straight and level flight at a true airspeed of 50m/s (111.8mph) at a standard altitude of 1500 m (4921ft) is selected. For this airplane,

A = 5.625

Iy = 1693 kg – m2 (1249 slug-ft2)

W = 10,680.0 N (2400 lb)

S= 14.86 m2 (160 ft2) б = 1.60 m (5.25 ft)

An / of 0.5 m2 (5.38 ft2) (see discussion of Table 4.4) and an e of 0.6 (see discussion following Equation 4.33) will be assumed.

From Figure 2.3, the standard mass density at 1500 m equals 1.058 kg/m2. Thus,

q0 = j(1.058)(50)2 = 1323.0 N/m2

Thus the trim lift and drag coefficients become

Сц = 0.543 Сц, = 0.0615

Since @o is zero, CXu for this propeller-driven airplane becomes

Cxu = -0.185

From Equation 9.26,

Cx„ – Cl,, – Cn

= c^-c,„


CLa was calculated previously for the Cherokee wing-tail combination as equal to 4.50/rad. Using Equation 8.71, an increment to CLa due to the fuselage is estimated to equal 0.13/rad. The propeller contribution to CLa can be estimated using Figure 8.22a together with the estimated CDo. Assuming a C, of 0.6 and a / of 1.0,


-jr — 0.8/rad

ACla = 0.8 CDo
= 0.05

The total CLa is equal to the sum of these separate contributions.

CLa = 4.68/rad CXa = 0.0637/rad

The gross weight corresponds to a mass of 1089 kg. Thus, for the given gross weight and moment of inertia,

fi = 86.6 /v = 210.0

Using the foregoing numerical constants, Equation 9.33a becomes

173.0Й + 0.185м -0.0637a +0.5430 =0 (9.36)

Z Derivatives and Parameters

In addition to some of the terms just considered, Equation 9.33b contains the terms Cz„, Cz„-, and CZs – To these we will add the derivative C/a.

The derivative Cz„, because of the use of stability axes, is simply given by

Cza — — CLa (9.37)

Cz. can be obtained by using Equation 8.49. At the trimmed condition,

AZ — r),q0S, a,

Based on the experimental data of Reference 8.9, an tj, of 1.0 appears reasonable. At the cruise thrust coefficient, tj, is slightly greater than unity at low a values and less than unity at the higher a values. .The horizontal tail volume for the Cherokee is approximately 0.392. Thus,

Cz,, — — 2.88

The increment in the Z force resulting from d can be determined from Equation 9.34. At the trim condition,

AZ = r),q0STaT(~ e„^-)d


In Equation 9.39, d is with respect to the dimensionless time t, whereas in the preceding equation for AZ, the derivative is with respect to real time.

ea is estimated in Chapter Eight for the Cherokee to equal 0.447. Thus, Cz for the Cherokee is estimated to be

CZa = – 1.29/rad-air sec

The increment in Z resulting from an elevator deflection will be

AZ = – r),q0S, a,T Se

For a stabilator configuration, the effective lift curve slope is greater than a, (with the tab fixed) because of the linked tab. This increased slope was given earlier as

Thus, Equation 9.40, for the stabilator configuration, becomes

CZs = – Vl^at(l-Tk’) (9.41)

For the Cherokee, kt = —1.5. The effective т was estimated previously to equal 0.44. Thus,

CZt = – 0.934/rad

With the foregoing constants, Equation 9.33b for the Cherokee 180 becomes (including a term)

1.09й + 175.0d + 4.68a – 170.00 = – 0.9345 (9.42)

M Derivatives and Parameters

The equation of motion governing pitching, Equation 9.33 c, requires the values of Сщ, См^ and Сщ. In addition, we will include the term Сщ.

Сщ can be estimated on the basis of Equation 8.8 for a wing-tail combination. An increment to CK for the fuselage is obtained from Equation 8.72 and for a propeller from Figure 8.22a and 8.22c.

For the Cherokee these components of Сщ are estimated to equal:

ДСМ„ = -0.963/rad (wing-tail)

Д Сщ = +0.072/rad (propeller)

Д CM„ = +0.150/rad (fuselage)

Thus, for the total,

Cm. = -0.741/rad

The damping moment derivative, Сщ, is given by Equation 8.54. For the Cherokee 180,

Сщ = -7.42

CM. is equal to CZa multiplied by the tail length referenced to c.

Cm& = — 2rj, Унєа 4 a, (9.43)

For the Cherokee 180,

См„ = – 3.32/rad-air sec

Сщ is equal to Czs multiplied by the dimensionless tail length. Thus, for

the stabilator configuration,

Сщ = – 2.40/rad

Using the preceding stability derivatives for the Cherokee 180 results in the following for Equation 9.33c.

0.741a + 3.32a + 210.00 + 7.420 = – 2.40S

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