Y Derivatives

Equation 10.17a is the equation governing primarily the acceleration of the airplane’s mass in the у direction. It contains the stability derivatives, CYf), CYf, and CYp the control derivative CYs, and the parameters, Сц and fi. is the trim lift coefficient and needs no further explanation. The dimensionless mass /м is defined by Equation 10.16d. For the Cherokee 180 example used previously at an altitude of 1500 m,

M pSb

= 15.2 Cl, = 0.543

The side force derivative, CYfl, consists primarily of contributions from the fuselage, the vertical tail, and the normal force resulting from the propulsion system.

Referring to Figure 8.25, a positive sideslip angle of /3 results in a negative side force on the fuselage and vertical tail. The force on the fuselage can be estimated on the basis of Equation 8.71. The у force on the vertical tail will be given by

Yv = – ri, qSvavp~ ee) or

Cy„ = -7j(^a„/3(l-^) (10.18)

For the Cherokee 180, the effective aspect ratio of the vertical tail (see discussion following Equation 8.90) is estimated to equal 2.84. This value of A substituted into Equation 3.70 gives an estimated lift curve slope of 3.04/rad. The ratio SJS equals 0.0713. ep is taken to be zero, and tj, to be unity so

CY„ = —0.216/rad

The fuselage and propeller contributions to CYfi are assumed to equal their contributions to CZa that were previously estimated to total -0.18/rad. Thus the total side force derivative for the Cherokee 180 is estimated to equal

Cy„ = -0.396/rad

The side force resulting from the rolling velocity is normally small. It results primarily from the vertical tail lying above the longitudinal axis through the center of gravity. If the aerodynamic center of the vertical tail lies a distance of Z„ above the center of gravity, then it will experience, as shown in Figure 10.3, an angle of attack resulting from a roll rate, P, equal to



The direction of Да is such as to produce a negative Y force, given by

v – PZv

Yv—- Vtqbvav


In coefficient form,

n – _•? Sy Zy,, – Су, s ^vP



For the Cherokee 180, ZJb = 0.09. Thus,

CY – = -0.039

1 P


The side force due to yaw rate results from the damping force on the vertical tail and on the propulsor.

If the vertical tail is aft of the center of gravity a distance of /„, a yaw rate of R will produce on increment in the angle of attack, as shown in Figure

10.4, equal to


A side force in the у direction results, given by

У„ = VlqSvav yj*


In coefficient form, this becomes

Cy„ = 2q, avVvr (10.20)

Vv is the vertical tail volume defined previously, and r is the dimension­less yaw rate.

For the Cherokee 180, Vv = 0.031. Thus,

CY =0.188


In a similar manner, a propeller experiences an angle-of-attack change



Figure 10.4 Angle-of-attack increment at vertical tail due to yaw rate. Top view, due to R, given by

where lp represents the distance of the propeller ahead of the center of gravity. The direction of Да is opposite to that shown in Figure 10.4, so the Y force on the propeller is



In dimensionless form this becomes

For the Cherokee 180, the following quantities were estimated previously for the trim condition of 50 m/s at an altitude of 1500 m and a gross weight of 10,680 N.

—jb – = 0.8/rad Сщ = 0.0615

lplb equals 0.236. Hence

СУ – = -0.0232/rad/air sec


The total CYf for the Cherokee is therefore estimated to be

Cy? = 0.165

The control derivative giving the rate of change of side force with rudder deflection is found from

Ду = t),qSvavTr 8r

or, in coefficient form,

CySr = V, avrr ^ (10.22)

For the Cherokee 180, i> is estimated, using Figures 3.32 and 3.33 to equal 0.54. Thus,

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