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Chapter 4
Dynamical Equations for Flight
Vehicles
These notes provide a systematic background of the derivation of the equations of motion for a flight vehicle, and their linearization. The relationship between dimensional stability derivatives and dimensionless aerodynamic coefficients is presented, and the principal contributions to all important stability derivatives for flight vehicles having left/right symmetry are explained.
4.1 Basic Equations of Motion
The equations of motion for a flight vehicle usually are written in a body-fixed coordinate system. It is convenient to choose the vehicle center of mass as the origin for this system, and the orientation of the (right-handed) system of coordinate axes is chosen by convention so that, as illustrated in Fig. 4.1:
- the x-axis lies in the symmetry plane of the vehicle^1 and points forward;
- the z-axis lies in the symmetry plane of the vehicle, is perpendicular to the x-axis, and points down;
- the y-axis is perpendicular to the symmetry plane of the vehicle and points out the right wing.
The precise orientation of the x-axis depends on the application; the two most common choices are:
- to choose the orientation of the x-axis so that the product of inertia
Ixz =
m
xz dm = 0
(^1) Almost all flight vehicles have bi-lateral (or, left/right) symmetry, and most flight dynamics analyses take advan- tage of this symmetry.
38 CHAPTER 4. DYNAMICAL EQUATIONS FOR FLIGHT VEHICLES
The other products of inertia, Ixy and Iyz , are automatically zero by vehicle symmetry. When all products of inertia are equal to zero, the axes are said to be principal axes.
- to choose the orientation of the x-axis so that it is parallel to the velocity vector for an initial equilibrium state. Such axes are called stability axes.
The choice of principal axes simplifies the moment equations, and requires determination of only one set of moments of inertia for the vehicle – at the cost of complicating the X- and Z-force equations because the axes will not, in general, be aligned with the lift and drag forces in the equilibrium state. The choice of stability axes ensures that the lift and drag forces in the equilibrium state are aligned with the Z and X axes, at the cost of additional complexity in the moment equations and the need to re-evaluate the inertial properties of the vehicle (Ix, Iz , and Ixz ) for each new equilibrium state.
4.1.1 Force Equations
The equations of motion for the vehicle can be developed by writing Newton’s second law for each differential element of mass in the vehicle,
d F~ = ~a dm (4.1)
then integrating over the entire vehicle. When working out the acceleration of each mass element, we must take into account the contributions to its velocity from both linear velocities (u, v, w) in each of the coordinate directions as well as the ~Ω × ~r contributions due to the rotation rates (p, q, r) about the axes. Thus, the time rates of change of the coordinates in an inertial frame instantaneously coincident with the body axes are
x˙ = u + qz − ry y ˙ = v + rx − pz z ˙ = w + py − qx
x
z
y
Figure 4.1: Body axis system with origin at center of gravity of a flight vehicle. The x-z plane lies in vehicle symmetry plane, and y-axis points out right wing.
40 CHAPTER 4. DYNAMICAL EQUATIONS FOR FLIGHT VEHICLES
x
x
y 1
f
z , zf 1
f ψ ψ y^1 x^1
y 1 θ
θ
x
z z
2
1 2
, y (^2)
x
z
2
φ
φ
, x
2
z
y
y 2
(a) (b) (c)
Figure 4.2: The Euler angles Ψ, Θ, and Φ determine the orientation of the body axes of a flight vehicle. (a) Yaw rotation about z-axis, nose right; (b) Pitch rotation about y-axis, nose up; (c) Roll rotation about x-axis, right wing down.
The orientation of the body axis system is specified by starting with the inertial system, then, in the following order performing:
- a positive rotation about the zf axis through the heading angle Ψ to produce the (x 1 , y 1 , z 1 ) system; then
- a positive rotation about the y 1 axis through the pitch angle Θ to produce the (x 2 , y 2 , z 2 ) system; and, finally
- a positive rotation about the x 2 axis through the bank angle Φ to produce the (x, y, z) system.
Thus, if we imagine the vehicle oriented initially with its z-axis pointing down and heading North, its final orientation is achieved by rotating through the heading angle Ψ, then pitching up through angle Θ, then rolling through angle Φ. This sequence of rotations in sketched in Fig. 4.2.
Since we are interested only in the orientation of the gravity vector in the body axis system, we can ignore the first rotation.^2 Thus, we need consider only the second rotation, in which the components of any vector transform as
x 2 y 2 z 2
cos Θ 0 − sin Θ 0 1 0 sin Θ 0 cos Θ
xf yf zf
and the third rotation, in which the components transform as
x y z
0 cos Φ sin Φ 0 − sin Φ cos Φ
x 2 y 2 z 2
(^2) If we are interested in determining where the vehicle is going – say, we are planning a flight path to get us from New York to London, we certainly are interested in the heading, but this is not really an issue as far as analysis of the stability and controllability of the vehicle are concerned.
4.1. BASIC EQUATIONS OF MOTION 41
Thus, the rotation matrix from the inertial frame to the body fixed system is seen to be
x y z
0 cos Φ sin Φ 0 − sin Φ cos Φ
cos Θ 0 − sin Θ 0 1 0 sin Θ 0 cos Θ
xf yf zf
cos Θ 0 − sin Θ sin Θ sin Φ cos Φ cos Θ sin Φ sin Θ cos Φ − sin Φ cos Θ cos Φ
xf yf zf
The components of the gravitational acceleration in the body-fixed system are, therefore,
gx gy gz
cos Θ 0 − sin Θ sin Θ sin Φ cos Φ cos Θ sin Φ sin Θ cos Φ − sin Φ cos Θ cos Φ
g 0
(^) = g 0
− sin Θ cos Θ sin Φ cos Θ cos Φ
The force equations can thus be written as
X
Y
Z
(^) + mg 0
− sin Θ cos Θ sin Φ cos Θ cos Φ
(^) = m
u˙ + qw − rv v ˙ + ru − pw w ˙ + pv − qu
where (X, Y, Z) are the components of the net aerodynamic and propulsive forces acting on the vehicle, which will be characterized in subsequent sections.
4.1.2 Moment Equations
The vector form of the equation relating the net torque to the rate of change of angular momentum is
G^ ~ =
L
M
N
m
(~r × ~a) dm (4.13)
where (L, M, N ) are the components about the (x, y, z) body axes, respectively, of the net aerody- namic and propulsive moments acting on the vehicle. Note that there is no net moment due to the gravitational forces, since the origin of the body-axis system has been chosen at the center of mass of the vehicle. The components of Eq.(4.13) can be written as
L =
m
(yz¨ − z ¨y) dm
M =
m
(z x¨ − xz¨) dm
N =
m
(xy¨ − yx¨) dm
where ¨x, ¨y, and ¨z are the net accelerations in an inertial system instantaneously coincident with the body axis system, as given in Eqs. (4.4).
When Eqs. (4.4) are substituted into Eqs. (4.14), the terms in the resulting integrals are either linear or quadratic in the coordinates. Since the origin of the body-axis system is at the vehicle c.g.,
4.2. LINEARIZED EQUATIONS OF MOTION 43
- Over a fairly broad range of flight conditions of practical importance, the aerodynamic forces and moments are well-approximated as linear functions of the state variables; and
- Normal flight situations correspond to relatively small variations in the state variables; in fact, relatively small disturbances in the state variables can lead to significant accelerations, i.e., to flight of considerable violence, which we normally want to avoid.
Finally, we should emphasize the caveat that these linear analyses are not good approximations in some cases – particularly for spinning or post-stall flight situations.
Thus, we will consider
- Perturbations from a longitudinal trim condition;
- Using stability axes;
so we can describe the state variables as
u = u 0 + u(t), p = p(t) v = v(t), q = q(t) w = w(t), r = r(t) θ = Θ 0 + θ(t), Φ= φ(t) (4.19)
Variables with the subscript 0 correspond to the original equilibrium (trim) state. Note that only the axial velocity u and pitch angle θ have non-zero equilibrium values. The trim values of all lateral/directional variables (v, p, r, and Φ) are zero because the initial trim condition corresponds to longitudinal equilibrium; the equilibrium value of w is zero because we are using stability axes; and the equilibrium pitch rate q is assumed zero as we are restricting the equilibrium state to have no normal acceleration.
The equations for the unperturbed initial equilibrium state then reduce to
X 0 − mg 0 sin Θ 0 = 0 Z 0 + mg 0 cos Θ 0 = 0 M 0 = L 0 = Y 0 = N 0 = 0
and we want to solve linear approximations to the equations
X 0 + ∆X − mg 0 sin (Θ 0 + θ) = m ( ˙u + qw − rv) Y 0 + ∆Y + mg 0 cos (Θ 0 + θ) sin φ = m ( ˙v + r(u 0 + u) − pw) Z 0 + ∆Z + mg 0 cos (Θ 0 + θ) cos φ = m ( ˙w + pv − q(u 0 + u))
and
∆L = Ix p˙ + (Iz − Iy ) qr − Ixz (pq + ˙r) ∆M = Iy q˙ + (Ix − Iz ) rp + Ixz
p^2 − r^2
∆N = Iz r˙ + (Iy − Ix) pq + Ixz (qr − p˙)
Since we assume that all perturbation quantities are small, we can approximate
sin (Θ 0 + θ) ≈ sin Θ 0 + θ cos Θ 0 cos (Θ 0 + θ) ≈ cos Θ 0 − θ sin Θ 0
44 CHAPTER 4. DYNAMICAL EQUATIONS FOR FLIGHT VEHICLES
and
sin Φ = sin φ ≈ φ cos Φ = cos φ ≈ 1
Thus, after making these approximations, subtracting the equilibrium equations, and neglecting terms that are quadratic in the small perturbations, the force equations can be written
∆X − mg 0 cos Θ 0 θ = m u˙ ∆Y + mg 0 cos Θ 0 φ = m ( ˙v + u 0 r) ∆Z − mg 0 sin Θ 0 θ = m ( ˙w − u 0 q)
and the moment equations can be written
∆L = Ix p˙ − Ixz r˙ ∆M = Iy q˙ ∆N = Iz r˙ − Ixz p˙
4.3 Representation of Aerodynamic Forces and Moments
The perturbations in aerodynamic forces and moments are functions of both, the perturbations in state variables and control inputs. The most important dependencies can be represented as follows. The dependencies in the equations describing the longitudinal state variables can be written
∆X =
∂X
∂u
u +
∂X
∂w
w +
∂X
∂δe
δe +
∂X
∂δT
δT
∆Z =
∂Z
∂u
u +
∂Z
∂w
w +
∂Z
∂ w˙
w˙ +
∂Z
∂q
q +
∂Z
∂δe
δe +
∂Z
∂δT
δT
∆M =
∂M
∂u
u +
∂M
∂w
w +
∂M
∂ w˙
w˙ +
∂M
∂q
q +
∂M
∂δe
δe +
∂M
∂δT
δT
In these equations, the control variables δe and δT correspond to perturbations from trim in the elevator and thrust (throttle) settings. Note that the Z force and pitching moment M are assumed to depend on both the rate of change of angle of attack w˙ and the pitch rate q, but the dependence of the X force on these variables is neglected.
Also, the dependencies in the equations describing the lateral/directional state variables can be written
∆Y =
∂Y
∂v
v +
∂Y
∂p
p +
∂Y
∂r
r +
∂Y
∂δr
δr
∆L =
∂L
∂v
v +
∂L
∂p
p +
∂L
∂r
r +
∂L
∂δr
δr +
∂L
∂δa
δa
∆N =
∂N
∂v v +
∂N
∂p p +
∂N
∂r r +
∂N
∂δr δr +
∂N
∂δa δa
In these equations, the variables δr and δa represent the perturbations from trim in the rudder and aileron control settings.
46 CHAPTER 4. DYNAMICAL EQUATIONS FOR FLIGHT VEHICLES
and the small-disturbance equations for lateral/directional motions: [ d dt
− Yv
]
v − Ypp + [u 0 − Yr ] r − g 0 cos Θ 0 φ = Yδr δr
−Lvv +
[
d dt
− Lp
]
p −
[
Ixz Ix
d dt
]
r = Lδr δr + Lδa δa
−Nvv −
[
Ixz Iz
d dt
]
p +
[
d dt
− Nr
]
r = Nδr δr + Nδa δa
4.3.1 Longitudinal Stability Derivatives
In order to solve the equations describing longitudinal vehicle motions, we need to be able to evaluate all the coefficients appearing in Eqs. (4.31). This means we need to be able to provide estimates for the derivatives of X, Z, and M with respect to the relevant independent variables u, w, w˙, and q. These stability derivatives usually are expressed in terms of dimensionless aerodynamic coefficient derivatives. For example, we can express the stability derivative Xu as
Xu ≡
m
∂X
∂u
m
∂u
[QSCX ] =
QS
mu 0
[2CX 0 + CX u] (4.33)
where
CX u ≡
∂CX
∂(u/u 0 )
is the derivative of the dimensionless X-force coefficient with respect to the dimensionless velocity u/u 0. Note that the first term in the final expression of Eq. (4.33) arises because the dynamic pressure Q is, itself, a function of the flight velocity u 0 + u. Similar expressions can be developed for all the required derivatives.
Derivatives with respect to vertical velocity perturbations w are related to aerodynamic derivatives with respect to angle of attack α, since
α = tan−^1
( (^) w u
w u 0
Then, for example
Zw ≡
m
∂Z
∂w
m
∂(u 0 α)
[QSCZ ] =
QS
mu 0
CZ α (4.36)
Derivatives with respect to pitch rate q are related to aerodynamic derivatives with respect to dimensionless pitch rate ˆq ≡ 2 ¯cqu 0. Thus, for example
Mq ≡
Iy
∂M
∂q
Iy
2 u 0 ˆq ¯c
) (^) [QS¯cCm] = QSc¯^2 2 Iy u 0
Cmq (4.37)
where
Cmq ≡ ∂Cm ∂ ˆq
is the derivative of the dimensionless pitching moment coefficient with respect to the dimensionless pitch rate ˆq. In a similar way, dimensionless derivatives with respect to rate of change of angle of attack ˙α are expressed in terms of the dimensionless rate of change ˆα˙ = 2 c¯u^ α˙ 0.
4.3. REPRESENTATION OF AERODYNAMIC FORCES AND MOMENTS 47
Variable X Z M
u Xu = (^) muQS 0 [2CX 0 + CX u] Zu = (^) muQS 0 [2CZ 0 + CZ u] Mu = (^) IQSy uc¯ 0 Cmu
w Xw = (^) muQS 0 CX α Zw = (^) muQS 0 CZ α Mw = QS Iy u¯c 0 Cmα
w ˙ X (^) w˙ = 0 Z (^) w˙ = 2 QSmu¯c 2 0 CZ α˙ M (^) w˙ = QS¯c
2 2 Iy u^20 Cm^ α˙ q Xq = 0 Zq = 2 QSmuc¯ 0 CZ q Mq = QSc¯
2 2 Iy u 0 Cmq
Table 4.1: Relation of dimensional stability derivatives for longitudinal motions to dimensionless derivatives of aerodynamic coefficients.
Expressions for all the dimensional stability derivatives appearing in Eqs. (4.31) in terms of the dimensionless aerodynamic coefficient derivatives are summarized in Table 4.1.
Aerodynamic Derivatives
In this section we relate the dimensionless derivatives of the preceding section to the usual aerody- namic derivatives, and provide simple formulas for estimating them. It is natural to express the axial and normal force coefficients in terms of the lift and drag coefficients, but we must take into account the fact that perturbations in angle of attack will rotate the lift and drag vectors with respect to the body axes. Here, consistent with Eq. (4.35), we define the angle of attack as the angle between the instantaneous vehicle velocity vector and the x-axis, and also assume that the propulsive thrust is aligned with the x-axis. Thus, as seen in Fig. 4.3, we have to within terms linear in angle of attack
CX = CT − CD cos α + CL sin α ≈ CT − CD + CLα CZ = −CD sin α − CL cos α ≈ −CDα − CL
Here the thrust coefficient
CT ≡
T
QS
where T is the net propulsive thrust, assumed to be aligned with the x-axis of the body-fixed system. Since all the dimensionless coefficients in Eqs. (4.39) are normalized by the same quantity QS, the representations of forces and force coefficients are equivalent.
Speed Derivatives
We first consider the derivatives with respect to vehicle speed u. The derivative
CX u = CT u − CDu (4.41)
represents the speed damping, and
CDu = M
∂CD
∂M
4.3. REPRESENTATION OF AERODYNAMIC FORCES AND MOMENTS 49
since the drag coefficient contribution vanishes when evaluated at the initial trim condition, where α = 0. The dependence of lift coefficient on speed arises due to compressibility and aeroelastic effects. We will neglect aeroelastic effects, but the effect of compressibility can be characterized as
CLu = M
∂CL
∂M
where M is the flight Mach number. The Prandtl-Glauert similarity law for subsonic flow gives
CL =
CL|M=
1 − M^2
which can be used to show that ∂CL ∂M
M
1 − M^2
CL 0 (4.51)
whence
CZ u = −
M^2
1 − M^2
CL 0 (4.52)
Use of the corresponding form of the Prandtl-Glauert rule for supersonic flow results in exactly the same formula. We then have for the dimensional stability derivative
Zu = −
QS
mu 0
[
2 CL 0 +
M^2
1 − M^2
CL 0
]
Finally, the change in pitching moment coefficient Cm with speed u is generally due to effects of compressibility and aeroelastic deformation. The latter will again be neglected, so we have only the compressibility effect, which can be represented as
Cmu = M
∂Cm ∂M
so we have
Mu =
QS¯c Iy u 0
MCmM (4.55)
Angle-of-Attack Derivatives
As mentioned earlier, the derivatives with respect to vertical velocity w are expressed in terms of derivatives with respect to angle of attack α. Since from Eq. (4.39) we have
CX = CT − CD + CLα (4.56)
we have CX α = CT α − CDα + CLαα + CL = −CDα + CL 0 (4.57)
since we assume the propulsive thrust is independent of the angle of attack, i.e., CT α = 0. Using the parabolic approximation for the drag polar
CD = CDp +
CL^2
πeAR
we have
CDα =
2 CL
πeAR
CLα (4.59)
50 CHAPTER 4. DYNAMICAL EQUATIONS FOR FLIGHT VEHICLES
and
Xw =
QS
mu 0
CL 0 −
2 CL 0
πeAR
CLα
Similarly, for the z-force coefficient Eq. (4.39) gives
CZ = −CDα − CL (4.61)
whence CZ α = −CD 0 − CLα (4.62)
so
Zw = −
QS
mu 0
(CD 0 + CLα) (4.63)
Finally, the dimensional derivative of pitching moment with respect to vertical velocity w is given by
Mw =
QSc¯ Iy u 0
Cmα (4.64)
Pitch-rate Derivatives
The pitch rate derivatives have already been discussed in our review of static longitudinal stability. As seen there, the principal contribution is from the horizontal tail and is given by
Cmq = − 2 η
ℓt ¯c VH at (4.65)
and CLq = 2ηVH at (4.66)
so CZ q = −CLq = − 2 ηVH at (4.67)
The derivative CX q is usually assumed to be negligibly small.
Angle-of-attack Rate Derivatives
The derivatives with respect to rate of change of angle of attack α˙ arise primarily from the time lag associated with wing downwash affecting the horizontal tail. This affects the lift force on the horizontal tail and the corresponding pitching moment; the effect on vehicle drag usually is neglected.
The wing downwash is associated with the vorticity trailing behind the wing and, since vorticity is convected with the local fluid velocity, the time lag for vorticity to convect from the wing to the tail is approximately
∆t =
ℓt u 0
The instantaneous angle of attack seen by the horizontal tail is therefore
αt = α + it − ε = α + it −
[
ε 0 +
dε dα (α − α˙∆t)
]
52 CHAPTER 4. DYNAMICAL EQUATIONS FOR FLIGHT VEHICLES
Variable Y L N
v Yv = (^) muQS 0 Cy β Lv = (^) IQSbx u 0 Clβ Nv = (^) IQSbz u 0 Cnβ
p Yp = 2 QSbmu 0 Cy p Lp = QSb
2 2 Ix u 0 Clp^ Np^ =^
QSb^2 2 Iz u 0 Cnp r Yr = 2 QSbmu 0 Cy r Lr = QSb
2 2 Ix u 0 Clr^ Nr^ =^
QSb^2 2 Iz u 0 Cnr
Table 4.2: Relation of dimensional stability derivatives for lateral/directional motions to dimension- less derivatives of aerodynamic coefficients.
where
Clp ≡
∂Cl ∂ pˆ
is the derivative of the dimensionless rolling moment coefficient with respect to the dimensionless roll rate ˆp.^4
Expressions for all the dimensional stability derivatives appearing in Eqs. (4.32) in terms of the dimensionless aerodynamic coefficient derivatives are summarized in Table 4.2.
Sideslip Derivatives
The side force due to sideslip is due primarily to the side force (or “lift”) produced by the vertical tail, which can be expressed as
Yv = −QvSv ∂CLv ∂αv
αv (4.79)
where the minus sign is required because we define the angle of attack as
αv = β + σ (4.80)
where positive β = sin−^1 (v/V ) corresponds to positive v. The angle σ is the sidewash angle de- scribing the distortion in angle of attack at the vertical tail due to interference effects from the wing and fuselage. The sidewash angle σ is for the vertical tail what the downwash angle ε is for the horizontal tail.^5
The side force coefficient can then be expressed as
Cy ≡
Y
QS
Qv Q
Sv S
∂CLv ∂αv
(β + σ) (4.81)
whence
Cy β ≡
∂Cy ∂β
= −ηv
Sv S
av
dσ dβ
(^4) Note that the lateral and directional rates are nondimensionalized using the time scale b/(2u 0 ) – i.e., the span di- mension is used instead of the mean aerodynamic chord which appears in the corresponding quantities for longitudinal motions. (^5) Note, however, that the sidewash angle is defined as having the opposite sign from the downwash angle. This is because the sidewash angle can easily augment the sideslip angle at the vertical tail, while the induced downwash at the horizontal tail always reduces the effective angle of attack.
4.3. REPRESENTATION OF AERODYNAMIC FORCES AND MOMENTS 53
where
ηv =
Qv Q
is the vertical tail efficiency factor.
The yawing moment due to side slip is called the weathercock stability derivative, and is caused by both, the vertical tail side force acting through the moment arm ℓv and the destabilizing yawing moment produced by the fuselage. This latter effect is analogous to the destabilizing contribution of the fuselage to the pitch stiffness Cmα, and can be estimated from slender-body theory to be
Cnβ
f use =^ −^2
V
Sb
where V is the volume of the equivalent fuselage – based on fuselage height (rather than width, as for the pitch stiffness). The yawing moment contribution due to the side force acting on the vertical tail is Nv = −ℓvYv
so the corresponding contribution of the vertical tail to the weathercock stability is
Cnβ
V =^ ηvVv^ av
dσ dβ
where
Vv =
ℓvSv bS
is the tail volume ratio for the vertical tail.
The sum of vertical tail and fuselage contributions to weathercock stability is then
Cnβ = ηvVv av
dσ dβ
V
Sb
Note that a positive value of Cnβ corresponds to stability, i.e., to the tendency for the vehicle to turn into the relative wind. The first term on the right hand side of Eq. (4.87), that due to the vertical tail, is stabilizing, while the second term, due to the fuselage, is destabilizing. In fact, providing adequate weathercock stability is the principal role of the vertical tail.
The final sideslip derivative describes the effect of sideslip on the rolling moment. The derivative Clβ is called the dihedral effect, and is one of the most important parameters for lateral/directional stability and handling qualities. A stable dihedral effect causes the vehicle to roll away from the sideslip, preventing the vehicle from “falling off its lift vector.” This requires a negative value of Clβ.
The dihedral effect has contributions from: (1) geometric dihedral; (2) wing sweep; (3) the vertical tail; and (4) wing-fuselage interaction. The contribution from geometric dihedral can be seen from the sketch in Fig. 4.4. There it is seen that the effect of sideslip is to increase the velocity normal to the plane of the right wing, and to decrease the velocity normal to the plane of the left wing, by the amount u 0 β sin Γ, where Γ is the geometric angle of dihedral. Thus, the effective angles of attack of the right and left wings are increased and decreased, respectively, by
∆α =
u 0 β sin Γ u 0 = β sin Γ (4.88)
4.3. REPRESENTATION OF AERODYNAMIC FORCES AND MOMENTS 55
Λ
Λ β
β
Λ
Figure 4.5: Effect of wing sweep dihedral effect. Sideslip increases the effective dynamic pressure on the right wing panel, and decreases it by the same amount on the left wing panel.
Note that the contribution of sweep to dihedral effect is proportional to wing lift coefficient (so it will be more significant at low speeds), and is stabilizing when the wing is swept back.
The contribution of the vertical tail to dihedral effect arises from the rolling moment generated by the side force on the tail. Thus, we have
Clβ =
z v′ b
Cy β (4.96)
where z v′ is the distance of the vertical tail aerodynamic center above the vehicle center of mass. Using Eq. (4.82), this can be written
Clβ = −ηv
z′ vSv bS
av
dσ dβ
At low angles of attack the contribution of the vertical tail to dihedral effect usually is stabilizing. But, at high angles of attack, z v′ can become negative, in which case the contribution is de-stabilizing.
The contribution to dihedral effect from wing-fuselage interference will be described only qualita- tively. The effect arises from the local changes in wing angle of attack due to the flow past the fuselage as sketched in Fig. 4.6. As indicated in the figure, for a low-wing configuration the presence of the fuselage has the effect of locally decreasing the angle of attack of the right wing in the vicinity of the fuselage, and increasing the corresponding angles of attack of the left wing, resulting in an unstable (positive) contribution to Clβ. For a high-wing configuration, the perturbations in angle of attack are reversed, so the interference effect results in a stable (negative) contribution to Clβ.
As a result of this wing-fuselage interaction, all other things being equal, a high-wing configuration needs less geometric dihedral than a low-wing one. This effect can be seen by comparing the geo- metric dihedral angle of a high-wing aircraft with a similar vehicle having a low-wing configuration. For example, the high-wing Lockheed C-5A actually has negative dihedral (or anhedral), while the low-wing Boeing 747 has about 5 degrees of dihedral; see Fig. 4.7.
Finally, it is interesting to consider the dihedral stability of the first powered airplane, the Wright Flyer; a three-view drawing is shown in Fig. 4.8. The Wright Flyer has virtually no fuselage (and, in any event, the biplane configuration of the wings is nearly symmetric with respect to all the bracing, etc.), so there is no wing-fuselage interference contribution to Clβ. Also, the wing is unswept, so there is no sweep contribution. In fact, the wings have a slight negative dihedral, so the craft has a net unstable dihedral effect. The Wright brothers did not consider stability a necessary property for
56 CHAPTER 4. DYNAMICAL EQUATIONS FOR FLIGHT VEHICLES
low wing
high wing
Figure 4.6: Effect of wing-fuselage interference on dihedral effect; figure corresponds to positive sideslip with vehicle viewed from behind. The presence of the fuselage alters the flow due to sideslip locally in the vicinity of the wing. Note that the resulting perturbations in angle of attack for a high- wing configuration are opposite in sign to those for a low-wing configuration, with this phenomenon contributing to stabilizing dihedral effect for the high-wing configuration.
(a) Boeing 747 (b) Lockheed C-5A
Figure 4.7: Illustration of effect of wing-fuselage interference on dihedral effect. The Boeing 747 and Lockheed C-5A have wings with nearly the same sweep angle, but the low-wing 747 requires significantly more geometric dihedral than the high-wing C-5A. Note: the (smaller) high-wing C- in the foreground of the photograph on the right requires less negative dihedral (anhedral) than the C-5A because it has an un-swept wing.
