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Stability Coefficients of Lateral Motion
Yawed flight During steady yawed flight, the incident flow condition is determined by the sideslip angle /? (Fig. 3-61 b) in addition to the angle of attack a. Because of the asymmetric incident flow, in addition to lift, drag, and pitching moment, additive forces and moments are created, namely, the side force due to sideslip Y, the rolling moment due to sideslip Mx, and the yawing moment due to sideslip Mz (see Fig. 1-6). They vary linearly with /3 for small angles of sideslip. The derivatives of the dimensionless coefficients with respect to the sideslip angle are, therefore, independent of the sideslip angle. They are termed stability coefficients of lateral motion. All three coefficients for a wing are strongly dependent on the sweepback angle and the dihedral angle.
First, the wing without dihedral will be treated, followed by a discussion of the effect of the dihedral angle.
A fundamental treatment of the yawed wing was first given by Weissinger [94]. The resulting theory can be designated as simple lifting-line theory in the sense of Sec. 3-3-3. In this theory it is assumed that free vortices are shed only from the trailing edge and that these are parallel to the incident-flow direction. The inclination of the free vortex strips against the wing axis of symmetry is of secondary effect on the results of the Weissinger theory. In this theory, Weissinger [94] introduced a correction factor taking into account the effect of the wing end flaps on the rolling moment due to sideslip. Later Gronau [25] made comprehensive computations of the rolling moment due to sideslip and the yawing moment due to sideslip, mainly for swept-back and delta wings, using the method of the extended lifting-line theory (Sec. 3-34). Here, too, the effect of the free vortex strip inclination has only approximately been taken into account.
Test results from various sources for the rolling moment due to sideslip of rectangular, swept-back, and delta wings against the aspect ratio are shown in Fig. 3-64. For comparison, theoretical curves from [25] are added. Agreement between measurements and theory is good. With decreasing aspect ratio, the rolling moment due to sideslip decreases strongly. This presentation reveals further that sweepback causes a strong increase in the rolling moment due to sideslip. This means that the rolling moments due to sideslip of swept-back and delta wings, in particular, are strongly dependent on the lift coefficient (see also Kohlman [45]). Figure 3-65 gives the corresponding plots for the yawing moment due to sideslip.
A wing in asymmetric flow (yawed wing) corresponds aero dynamically to a wing of asymmetric planform (see Fig. 3-17). Based on this concept, its circulation distribution can be computed from the extended lifting-line theory (Sec. 3-3-4), or the lifting-surfaces theory (Sec. 3-3-5). However, the required computation effort is considerably greater than for symmetric incident flow because of the asymmetry of the wing planform.* The result of such a computation is the circulation distribution over the span, measured normal to the incident flow direction. With it, the total lift and the neutral-point positions are obtained from the formulas of Sec. 3-3-2. In this way the rolling moment about the experimental x axis is also determined.
The rolling moment due to sideslip, accordingly, is proportional to the total lift. The circulation distribution for the three wings without twist examined earlier has been computed by this method at three different angles of sideslip. In Fig. 3-66, the circulation distributions for (3 — 0° and /3=10° have been presented over the span coordinate, measured normal to the direction of the incident flow. For all three wings, the circulation distribution changes very little with the angle of sideslip.
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*Only after electronic computers became available has this procedure gained practical value. t 7 г з v 5 6 A —- Figure 3-64 Rolling moment due to sideslip of rectangular wings, swept-back wings, and delta wings vs. aspect ratio л; theory of Gronau. Measurements: curve 1, rectangular wing (ip = 0°), (a) from Bussmann and Kopfermann, (?) from NACA Kept. 1091. Curve 2, swept-back wing of constant chord (<p = 45°), (•) from Gronau, (■) from NACA TN 1669, O) from Jacobs. Curve 3, delta wing ( = – g-), (®) from Gronau, (s) from Lange and Wacke. |
It should be mentioned that this behavior is typical for wings without dihedral. The locations of the neutral points for three angles of sideslip are inscribed into the wing planforms. The coefficients of the rolling moment due to sideslip and the coordinates of the neutral points are compiled in Table 3-7. The yawing moment due to sideslip is caused by the difference in drag of the two wing-halves. It consists of a contribution from the profile drag and one from the induced drag. The latter
Figure 3-66 Circulation distribution of three yawed wings without twist in sideslipping, based on the lifting-surface theory of Truckenbrodt. Angle of attack a = 1, measured in the section parallel to the incident flow direction, j = rjVb. Geometric data of the wings from Table 3-5. (a) Trapezoidal wing: f = 0a; л = 2.75; Л = 0.5. (b) Swept-back wing: <*г = 50°; л = 2.75; = 0.5. (c) Delta wing: ^ = 52.4°; л = 2.31; = 0.
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Table 3-7 Coefficients of the rolling moment due to sideslip and position of the neutral point for jS = 0°, 5°, and 10° for a trapezoidal, a swept-back, and a delta wing*
^Distances are measured in the wing-fixed coordinate system from the leading-edge station of the wing middle (root) section. Table 3-7 is based on Table 3-5 (see Fig. 3-66). |
contribution is proportional to the square of the lift, precisely like the induced drag. The side force due to sideslip of a wing without dihedral can be determined approximately by considering that the profile drag of a yawed wing acts parallel to the direction of the incident flow, but the induced drag acts in the direction of the wing axis of symmetry. Consequently, in asymmetric incident flow, only the component of the profile drag су — cDp sin /3 acts in the direction of the wing-fixed lateral axis. Hence the side force slope is
(3-153)
Wing with dihedral The dihedral of a wing is understood to be the inclination of the left and the right wing-halves relative to the xy plane (Fig. 3-6lb). The dihedral angle is designated as v; in the general case v may vary along the span. The stability coefficients of yawed flight 3cy/0j3, dcMx/dj3, and dcMzld(3 of the wing are strong functions of the dihedral. For the total airplane, the contributions of the wing to the side force due to sideslip 3cy/0j3 and to the yawing moment due to sideslip are relatively small. Conversely, the contribution of the wing to the rolling moment due to sideslip of the total airplane is of decisive significance. Selection of the wing dihedral is governed exclusively by the requirement of a flight mechanically favorable value of the rolling moment due to sideslip.[21]
The aerodynamic effect of the dihedral in yawed flight is due to the angle of attack, increased by the amount da of the leading half-wing, and the angle of attack decreased by A a of the trailing half-wing. This angle Л a can be determined
as fohows: From Fig. 3-67a and Ъ, the lateral component of the incident flow Vy = V sin 13 produces on either half-wing a component normal to the wing of amount
V„ = j-Fysinv
Together with the component Vx = V cos (3 of the incident flow, the additive angle-of-attack change becomes
(3-154д)
(3-1546)
The second relationship is valid for small angles of sideslip and small dihedral angles. The exact establishment of the dihedral angle from a given wing geometry must be based on Eq. (3-11).
The lift distribution of a wing with dihedral during yawed motion may thus be determined by adding the geometric angle-of-attack distribution of the wing without dihedral to the antimetric* twist from Eq. (3-154) (see also Fig. 3-67). As in Fig. 3-67, the lift (Z/2 L-AL/2) acts on the leading wing-half, the lift (Z/2 —JZ/2) on the trailing wing-half. Z is the lift for symmetric incident flow and A LI2 is the additive lift of one wing-half in yawed motion.
For the determination of the aerodynamic forces of the two wing-halves, it has to be realized that, as Fig. 3-67 demonstrates, the resultant incident flow direction is deflected up by the angle A a on the leading wing-half but deflected down by the same angle da on the trading wing-half. These angle-of-attack changes are relative to the angles of symmetric incident flow. The resultant aerodynamic forces on the two wing-halves undergo the same direction changes. The exact determination of the side force due to sideslip, of the rolling moment due to sideslip, and of the yawing moment due to sideslip requires computation of the lift distribution on the given wing for the antimetric angle-of-attack distribution in Eq. (3-154).
Approximate expressions for the aerodynamic quantities of the yawed wing with dihedral giving an explicit account of their dependence on the dihedral angle and the total lift coefficient can be gained, however, through the following estimations: The side force due to sideslip resulting from the dihedral is, from Fig. (3-67Й),
being the additive lift of one wing-half, where, from Eq. (3-154b), Aoc=v(3. Consequently, the coefficient of the side force due to sideslip becomes
(3-155)
The coefficient (dcLjdot)v of this equation can be determined exactly only by computation of the lift distribution on a wing with antimetric twist as in Fig. 3-67c.
Translator’s note: The word antimetric, found repeatedly in the text, has been coined by the
authors to avoid an inconvenient expression like “acting or pointing in opposite directions but being of equal magnitude.”
Figure 3-67 Aerodynamics of a wing with dihedral in sideslipping, (<z) Wing planform, xу plane. (b) Dihedral yz plane, (c) Additive antimetric angle-of – attack distribution due to the dihedral a a = ± v(3, (d) Incident flow resultant and aerodynamic-forces resultant of the two wing-halves.
As an approximation, however, it may be assumed that this coefficient is equal to that of a wing without twist of aspect ratio Л/2. Equation (3-155) reveals, then, that the coefficient of the side force is proportional to the square of the dihedral angle and independent of the total lift coefficient. Introduction into Eq. (3-155) of the lift-slope value for an aspect ratio Л/2 from the extended lifting-line theory of Eq. (3-98) yields, for the unswept wing,
where к — яЛ/с’і oo.
Measurements that confirm the above formula are reported in the summary account [72].
The rolling moment due to the dihedral is (see Fig. 3-61b)
w л A L
Mx = 2 —yL
where у і designates the distance of the center of the additive lift of the half-wing, AL/2, from the wing root. For the rolling-moment coefficient cMx =MxfqAs results, corresponding to the above discussion,
(3-157)
Here, (vl)v —Уь!$ is the dimensionless distance of the center of the additive lift of the half-wing from the wing root. This equation demonstrates that the coefficient of the rolling moment due to yaw as a result of the dihedral is proportional to the dihedral angle and independent of the total lift coefficient. To a good approximation, (t}l)v can be set equal to fir = 0.424. With this value, the following approximate relationship for the unswept wing is obtained. Here (dcifda)v from Eq. (3-98) for one-half of the aspect ratio A /2 is again introduced.*
The additive yawing moment due to sideslip resulting from the dihedral is very small in general. Its sign is such that it tends to turn the leading half-wing further upstream. This comes about because, as shown in Fig. 3-67(7, the resultant aerodynamic force at the leading half-wing is being turned toward the front and at the trailing half-wing toward the rear. Measurements are given in [72].
Rolling motion A linearly variable vertical velocity Vz = coxy is obtained when the wing executes a rotary motion about the longitudinal axis as in Fig, 3-68<z (see also Fig. 3-61c). Superposition with the incident flow velocity V results, from Fig. 3-68b and c, in an additive antimetric angle-of-attack distribution
Atx{ri) — r]Qx
where Qx = coxs/V is the dimensionless angular rolling velocity. This angle-of-attack distribution produces an antimetric lift distribution along the span and consequently a moment about the x axis that always tends to inhibit the rotary motion. This moment is designated rolling moment due to roll rate or roll damping. The
^Through evaluation of Eq. (3-100) with Eq. (3-154), Eq. (3-158) may be established as solution for the elliptic wing.
asymmetric force distribution along the span furthermore produces a yawing moment, the so-called yawing moment due to roll rate. These two moments are proportional to the dimensionless rolling angular velocity Qx, making their coefficients Ьсмх№&х and dcMz/bQx independent of Qx. In determining the aerodynamic force of the two wing-halves from Fig. 3-68c, it should be noted that relative to the symmetric incident flow direction, the resultant incident flow direction of the downward-turned wing-half is deflected upward by the angle A a and that of the upward-turned wing-half deflected by the same angle A a downward. Consequently, the local aerodynamic forces on the two wing-halves undergo the same directional changes.
For the determination of the roll damping of a given wing, the antimetric circulation distribution та(т?) over the span has to be established following a procedure for the computation of the lift distribution of Sec. 3-3. Hence the roll damping is, from Table 3-1,
(3-160 b)
Accordingly, the roll-damping coefficient bcMxjdQx is independent of the total lift coefficient of the wing. The roll-damping coefficients of the three wings (trapezoidal, swept-back, delta) examined earlier are found in Table 3-5.
A simple approximate formula for the roll damping of unswept wings is obtained by setting a = p in Eq. (3-100):
Use of this approximate formula is not recommended for wings of strong sweepback. A more accurate computation should be made. Schlottmann [75] demonstrates the theoretical determination of the roll damping of slender wings by a nonlinear theory and experimental confirmation of the computed results.
The yawing moment due to roll rate tends to turn the downward-turning wing-half forward. This behavior can be understood as follows: On the downward – moving wing-half, the resultant incident flow direction is turned upward and consequently the resultant aerodynamic force turns forward. On the upward-moving wing-half, the resultant aerodynamic force consequently turns rearward. On a section у of an unswept wing, the force dD’ = dDt — dLAa = dL(oci — da) is thus obtained in direction of the undisturbed incident flow.[22] Here dDt — dL&i from Eq. (3-17). Integration produces the induced yawing moment due to roll rate:
With dL= gVTdy from Eq. (3-14) and у = FjbV and A a from Eq. (3-159), the coefficient of the yawing moment is determined as
The total circulation у is composed of the contribution of the wing in symmetric incident flow ys and of the contribution ya created by the rotary motion for aa=7] that is, y= ys + Qxya. Correspondingly, the induced angle of attack becomes щ – ais +Qxaia. Introduction of these relationships into the above equation yields
(3-162)
For wings without twist of elliptic circulation distribution, a simpler evaluation of the integral is possible. The circulation distribution is obtained from Eq. (3-65), specifically, ys with p = 1 and ya with д= 2, З,…, M. Correspondingly, the induced angles of attack are found from Eq. (3-73), ot. is with n= 1 and aia with n = 2, 3,, M. By taking into account Eq. (3-65b) with r] = cos # and dr} = — sin в d&, the integration over G <n yields the relationship
(3-163)
where ax — cL/-nA and a2 = —(2/тгА)(дсМхІд@х) from Eqs. (3-66a) and (3-666).
By introducing Eq. (3-161), the following approximate formula for the coefficient of the yawing moment due to Toll rate is finally found:
= _ J_ y^3 +4-1
8QX 4 |/£2 – f – 4 – f 2
Thus the coefficient of the yawing moment due to roll rate is proportional to the total lift coefficient.
Yawing motion Motion of the airplane about the vertical axis produces additive longitudinal velocities, of reversed signs on the two wing-halves (Fig. 3-69; see also Fig. 3-6 le). An asymmetric lift distribution over the span results, creating a yawing moment and a rolling moment. This yawing moment counteracts the rotary motion and is termed, therefore, yaw-damping or turn-damping of the wing. It is very small compared with that of the whole airplane, and therefore its computation is omitted.
The rolling moment created by the yawing motion is termed rolling moment due to yaw rate or turning rolling moment. The turning rolling moment tends to turn the forward-moving wing-half upward.
The turning rolling moment can be computed in the following way: Through the rotary motion with angular velocity coz from Fig. 3-69b, a linear distribution of the longitudinal velocity is generated along the span:
7х(у) = У-щу (3-165)
To ensure that the wing is a streamlayer of this inhomogeneous flow field, the kinematic flow condition
(3-166a)
ay + y = 0 (3-1666)
must be satisfied at each point of the wing surface. Here Vn is the component of the longitudinal velocity Vx normal to the wing chord; thus, Vn — aVx (Fig. 3-69c).
In a homogeneous flow field, Vx = V, the kinematic flow condition becomes w/T+a=0. Comparison with Eq. (3-166b) demonstrates that the inhomogeneous flow is equivalent to a homogeneous flow with the mathematical angle of attack
«t = «-^ = *(l-S,4) (3-167)
where Qz = cozsjV. Consequently, the circulation distribution for inhomogeneous flow can be computed by using the computation procedures of Sec. 3-3, but by applying an angle-of-attack distribution as in Eq. (3-167). The resulting circulation distribution is Гъ = b V7b. A wing strip of width dy thus produces a lift dL – q VxTb dy = qV( 1 —Огг])Гъ dy, and the rolling moment becomes
S S
Mx = — J ydL = — 0 f Vxrbydy
~s – s
And further, the coefficient of the rolling moment cMx =Mx/qAs is found as
і
Смх= —A/(1 — Qzri) ybri drt
-i
The circulation distribution yb at the angle of attack ab, from Eq. (3-167), may be composed as follows:
П = ayu — cc Qzya (3-168)
Here yu is the circulation distribution for a = 1, and ya that for aa — r] [see Eq. (3-160)]. For the sake of simplicity, let a wing without twist a = const, be considered. Introduction of Eq. (3-168) into the equation for the rolling-moment coefficient yields
= i^A f yur]2 drj Л f yarj dr^j ci (3-169)
This equation demonstrates that the coefficient of the rolling moment due to yaw rate is proportional to the angular velocity Qz and the total lift coefficient cL. For a wing without twist of elliptic circulation distribution, the following approximation formula is obtained with Eq. (3-98) and after evaluation of the integrals similar to those of Eqs. (3-162) and (3-163):
dcMx_ j_ L, VFTt + i
8QZ 4 "И у в +4-f2 / L
This expression is nearly independent of the aspect ratio. For the three wings that have been examined (trapezoidal, swept-back, and delta, Table 3-5), the coefficients of the rolling moment due to sideslip are listed in Table 3-8.
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Table 3-8 Coefficients of the rolling moment due to yaw rate for a trapezoidal, a swept-back, and a delta wing based on Table 3-5
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For an accurate computation of the rolling moment due to yaw rate, it must be realized that the rotary motion of the wing causes the free vortices to be shed into a lateral flow. Hence the portions of the free vortices that lie on the wing produce an addition to the lift and thus to the rolling moment due to yaw rate. A detailed computation reveals that the coefficient of the rolling moment due to yaw rate for wings of small aspect ratio (/1< 3) depends considerably on the position of the axis of rotation.
