Effects of Propulsion System
Propellers
Both propellers and jets can affect longitudinal static stability. Consider first the case of a propellerdriven airplane, shown schematically in Figure
8.20. As illustrated in the figure, a propeller develops not only a thrust, T, directed along its axis of rotation, but also a force, PN, directed normal to its axis. It will be shown that PN is proportional to the propeller thrust, a, and the advance ratio, J, and is directed upward for a positive angle of attack.
Since the thrust is directed along the propeller axis and rotates with the rest of the airplane, its contribution to the moment about the center of gravity does not change with a. Hence Сщ is not affected by T. On the other hand, Сщ is affected by T since, generally, the thrust line will not pass through the center of gravity. If the thrust line lies a distance of Zp above the center of gravity, then Сщ, given by Equation 8.5, is decreased by the amount
—©(t) 
(8.73) 
The normal force, PN, affects both the trim, Сщ, Сщ is increased by the amount 
and the stability, Сщ. 
—Н’ч II < 
(8.74) 
while Сщ is increased by 

Д CM =CN kr ina Iypa £ 
(8.75) 
PN 
Figure 8.20 Effect of propeller forces and slipstream on longitudinal static stability. 
516 STATIC STABILITY AND CONTROL where
r ЭСЧ
Cn"°‘ 9 a
In addition to contributing to the forces and moments affecting longitudinal motion, a propeller at an angle of attack produces a yawing moment that couples with the lateraldirectional behavior of the airplane. Let us now examine the origin of both the normal force and yawing moment. Figure 8.21a shows a side view of a propeller at an angle of attack a. It is seen that a component of the freestream velocity, Va, lies in the plane of the propeller and is directed upward. Figure 8.21 b is a view in the plane of rotation looking in the direction of flight. Measuring the displacement of a blade from the
(c) 
vertical bottom line, it is seen that a blade section experiences a velocity normal to the blade given by the sum of tar and aV sin ф. In Figure 8.21c we view a section of the blade looking in toward the hub. The section lift and drag combine vectorially to produce a thrust per unit blade length of
Since all three of these force components vary with the blade position, ф, average values are obtained by integrating each with respect to ф from 0 to 2ir and then dividing by 2tr. For example,
where В is the number of blades and R is the propeller radius.
In order to cast the equations in a tractable form, it is assumed that the angle ф is small so that
Vr — о)Г Va sin ф
With this assumption dL/dr and dDIdr become
рЦр = ip(wr – Va sin ф)2сС( (8.77)
= p(a>r – Va sin ф)сС<1 (8.78)
Ci, as well as the local velocity, varies with ф, whereas Cd is assumed to be constant. Using a quasisteady approximation (neglecting unsteady aerodynamics), Ci is written as
С, = а(/3ф) — а(/3 — tan1— —A
cor — Va sin Ф/
/3 the blade pitch angle defined in figure 6.9, will be chosen so as to give a constant С/ along the blade when a is zero.
Thus,
The differential, instantaneous pitching, and yawing moments about the y – and zaxes, respectively, will be given by
dMp = r cos ф dr
dNp = r sin ф dr
The foregoing relationships are combined and integrated over ф from 0 to 2ir and over r from 0 to R. The chords c and Cd are assumed constant, so they can be taken outside of the integration. The following results are obtained.
It is convenient to express PN and Np in terms of T and the product TR, respectively. Since the same assumptions are involved in calculating T, PN, and Np, the ratios of PN and Np to T are probably more accurate than the predicted absolute values of these quantities. Also, (aqA) cancels out, so that one needs only to estimate the average lift coefficient of the blades in order to obtain the force and moment ratios.
In steady, level flight, the thrust is equal to the airplane drag. Thus, knowing the drag, propeller geometry, advance ratio, and q, one can calculate the Ci from Equation 8.80. The result is relatively insensitive to Cd, so that a typical value of 0.01 can be assumed for this quantity.
Figure 8.22a and 8.22b presents the derivatives of PN and Np with respect to a in ratio form as a function of advance ratio for constant values of Ct. If the propeller geometry is not known, a reasonable value of C( would probably be around 0.8.
X co Figure 8.22c Correction for upwash ahead of a wing. Model of figure 8.5 assumed using an unswept elliptic wing. c0=root chord, x = distance ahead of quarterchord line. 
The direction of PN is independent of the direction of the propeller rotation. However, the sign of the moment, Np, is reversed if the propeller rotates opposite to that shown in Figure 8.21. As derived, all rotations and moments follow the righthanded coordinate system of Figure 8.1.
This method for predicting PNa and NPa is original with this text and is untested. Sufficiently complete systematic data against which to compare the predictions could not be found. An alternate method for predicting PN and Np is offered in Reference 8.13. Graphs based on this reference can be found in other sources, but they seem to be specific to particular propellers, and their general applicability appears questionable. The conclusions of Reference 8.13 may be applied, however, without these graphs. They state that a propeller behaves like a fin having the same area as the average projected side area of the propeller with an effective aspect ratio of approximately 8. The moment produced by the propeller is approximately equal to the lift on the fin multiplied by the propeller radius. “Average” side area means an average area projected in one revolution. This is given approximately by onehalf the number of blades times the projected side area of one stationary blade.
In calculating the propeller normal force, one must account for the fact that there is an upwash ahead of the wing that effectively increases the angle
of attack. Thus Equation 8.75 becomes
ACMa = CNpf(l~ea)
This can also be written as
(8.83 b)
which assumes the thrust equal to the drag. Again, ea is the rate of change of downwash with a. Ahead of the wing, ea is negative so that (1 – ea) is greater than unity. This factor is presented in Figure 8.21c. This graph was calculated using the vortex system shown in Figure 8.5. Again, in the final analysis, one should resort to wind tunnel testing of a powered model to determine these effects accurately.
The effect of the propeller slipstream on the wing and tail will not be treated here in any detail. At higher speeds, these effects are small and can usually be neglected. At the lower speeds, however, the increased q in the propeller slipstream can increase both the wing and tail lift. In addition, particularly for V/STOL (vertical/short takeoff and landing) applications, the propellerinduced velocity can become appreciable relative to the freestream velocity, resulting in significant changes in the section angles of attack of both the wing and tail. For a detailed treatment of those effects, refer to Reference 3.3
Jets
The two principal effects of a jet propulsion system entail the forces developed internally on the jet ducts and the influence of the jet exhaust on the flow field of the tail. The following is based on material in Reference 8.2.
Figure 8.23 depicts schematically the flow through a jet propulsion system. The notation is similar to that of the reference. First, it is obvious that an increment to the trim moment results from the thrust, T, which is independent of a. This increment can be written as
(8.84)
Second, as in the case of the propeller, a normal force is developed by the jet. This force, Nj, is directed normal to the exhaust velocity V, and acts at the intersection of lines parallel to V, and V, through the centers of the inlet and exhaust. This is usually close to the inlet. Applying momentum principles to the system, the magnitude of N, is found to be
Nj = mjV6
where nij is the mass flow rate through the engine. The angle в equals the
angle of attack of the thrust line a minus the downwash angle e, at the inlet. If the inlet is ahead of the wing, e, will generally be negative.
в = a — є,
From this, it follows that the normal force, N,, contributes an increment to CMa, given by
Equation 8.85 can be evaluated given the characteristics of the engine. The quantity (1 – €jj can be estimated using Figure 8.22 or Figure 8.6.
The influence of the jet exhaust on the flow field external to the jet is illustrated in Figure 8.24. Viscous shear along the edges of the jet produces an entrainment of the external flow into the jet. To the external flow, the jet appears somewhat like a distributed line of sinks. As shown in Figure 8.24, this jetinduced flow can produce a change in the local flow direction for a lifting surface in proximity to the jet.
Again, the calculation of flow inclination and the effect on a lifting surface is a difficult task and is best determined from wind tunnel tests. The effect can be appreciable and generally should not be neglected. According to Reference 8.2, it may vary with a enough to reduce the stability significantly.
From material to be found in the appendix of Reference 8.2, an approximate estimate of the inclination angle due to jet entrainment can be calculated from
(8.86)
where
CD = airplane drag coefficient n = number of engines
x = distance aft of exhaust exit plus 2.3 diameters of the jet at its exit r = radial distance from jet centerline b = wingspan A = aspect ratio
Although a does not appear in Equation 8.86, the inclination of the jet itself varies with a, which causes the distance, r, to change. For more detailed information on this, see Reference 8.2 and the original sources on this topic, which are noted in the reference.
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