# I Derivatives

Equation 10.17b governs primarily the angular acceleration about the roll axis. The stability derivatives Clft, C(., Clf, the control derivatives Ch, C)s, and the parameters ix and ixz are needed to evaluate this equation.

From the definition of ix given by Equation 10.16a, for the Cherokee 180,

ix — 24.6

The product of inertia Ixz is unavailable for the example airplane. If the jc-axis is a principle axis, Ixz is equal to zero, so it is reasonable to assume that Ixz is small. Therefore, it will be assumed that

ixz = 0

я

This derivative, known as “dihedral effect,” has been previously covered in some depth. Ccan be calculated on the basis of Equations 8.108 to 8.112. The Cherokee 180 is a low-wing airplane having a dihedral angle of 7.5°. The

wing is unswept, untapered, and has an aspect ratio of 5.625, From Figure 8.39, the contribution to Clfi from the wing equals -0.0895/rad. To this we add 0.0092 to account for the presence of the fuselage. Thus, for the wing-fuselage combination,

Ch = —0.0803/rad

The vertical tail placed above the center of gravity also contributes to Ctf). At a sideslip angle of /3, the vertical tail develops a side force in the negative у direction equal to

А У = ~VtqSvavp

Acting at a height above the center of gravity of Zv, this increment in the у force gives rise to a rolling moment equal to

Д/ = – r),qSvavAvf}

Thus, in coefficient form,

Qfv = – V,^fav (10.23)

For the Cherokee 180,

С, э = -0.0195/rad

Adding the contribution of the vertical tail to that of the wing-fuselage combination gives a total dihedral effect of

Clf> = -0.0998/rad

An effect that is not included here, but that can be important, particularly at low speeds with flaps down, is illustrated in Figure 10.5. As a propeller – driven airplane slips to the right, the slipstream trails to the left, causing an increased lift on the left wing. This can result in a significant increase in Q. This interaction is difficult to predict in a general way. It is best obtained by means of wind tunnel testing with a powered model.

C’p

The roll damping coefficient can be calculated on the basis of Figure 8.36 for both the wing and horizontal tail. For the Cherokee’s wing, Ct. = -0.420. For the horizontal tail, A, = 4.10 and A, = 1.0. Thus, from Figure 8.36, Ctf = -0.335. However, this value is based on the tail area and span. To base С/р on the wing’s dimension, we note that

/, = V, qS, b, || C, p:

Figure 10.5 Effect of slipstream on dihedral effect. |

For the Cherokee,

Thus the total Ct/ due to the wing and horizontal tail is equal to -0.426.

The vertical tail also contributes to C(_. As recommended in Chapter Eight, Figure 8.36 can also be applied to the vertical tail as if it extended below the fuselage to the same extent as above. The value obtained is then halved and corrected according to Equation 10.24. For the Cherokee the increment to Ci – from the vertical tail is approximately half of that from the horizontal tail. Thus, for the Cherokee 180,

Ci. = -0.429

*P

Ci,

The rate of change of rolling moment with yawing velocity was considered in Chapter Eight. Ctf is composed of two contributions, one from the wing and the other from the vertical tail. According to Equation 8.103 for the

Cherokee 180, the wing’s contribution to Qr will equal

= 0.181

The contribution to С/, from the vertical tail is given by Equation 8.105.

Hence, the total Q. for the Cherokee 180 is estimated to equal

Clf = 0.198

Aileron roll control was covered in Chapter Eight. C, e can be estimated on the basis of Figure 8.30b. For the Cherokee 180,

A = 1.0 A = 5.625 Xi = 0.603 jc2= 1.0

— = 0.193

c

Using these equations and a linear interpolation for aspect ratio for values of A between 4 and 6 gives a value for C,. from Figure 8.30b of

°a

Ch = -0.0531/rad

°a

Since the maximum aileron deflection for the Cherokee is 30° up and 15° down, the total Sa of 45° gives a predicted

C,, = 0.0417

dmax

C/8 can be found from Equations 8.101 and 8.102. For the Cherokee 180, CYs was estimated to equal 0.117/rad. Therefore,

= 0.0105/rad

N Derivatives

cNf>

The change of yawing moment coefficient was covered in Chapter Eight. Equation 8.90 expresses the contribution from the vertical tail as

CNvp = і?(Кас(1 – єр)

= 0.094/rad

for ле Cherokee.

The contributions to CNfl from both the propeller and fuselage are obtained from the values estimated in Chapter Nine for CM„- There, the total Cua from the fuselage and propeller was estimated to equal 0.153/rad. To obtain CNfl, the sign must be reversed and multiplied by c/b. Thus, due to the fuselage and propeller,

CN0 = -0.0268/rad

Thus, for the Cherokee 180,

Сщ = 0.0672/rad

CN is given by Equation 8.107. For the Cherokee 180 at a trim lift coefficient of 0.543, this equation reduces to

CN – = -0.0905

this contribution comes from the wing. There will also be a small contribution from the vertical tail, given by

к

Y* b or, for the Cherokee,

CN. =-0.017

Pv

Thus, the total becomes

CN – = -0.0735

iyP

Сц}

CNf can be obtained directly from the vertical tail and propeller contributions to CYf. It is left to you to show that

CNf = – CYf+CYf± (10.25)

or, for the Cherokee,

CNf = -0.188(0.435) – 0.0232(0.236)

= -0.0873

The adverse aileron yaw is difficult to estimate. In view of the differential aileron deflections used on the Cherokee, it will be assumed that CNs – 0.

cNgr

Cn, is given by

<Ч = – Сч£ (10.26)

Thus, for the Cherokee,

C4 = -0.0509/rad

In addition to the stability derivatives, the parameter iz is needed to evaluate Equation 10.17c. From Equation 10.16b, for the Cherokee 180,

iz — 1.18

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