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0
or if
u:
ZG (x)
h(z)
8(y)
It has been previously shown (Section 5.4.1, p. 135) that the kinematic equation on is decoupled from the others. Thus the states that are left are
Naturally the complementary system of this longitudinal system will constitute the complementary states of the complete system. This will be true after having taken out the state of the decoupled navigational equations (ZG,YG, $). (5.148) This last system will be called the lateral system. Now the sufficient conditions for decoupling can be looked for, by searching for the second type of decoupling (Section 5.4, p. 132), that is t o say X2 = X P =~constant. This will be obtained with an equilibrium (X2 = 0) from the second system Xz undisturbed by XI,the controls U or the wind W . Clearly speaking, the question Dynamics of Flight: Equations
is to look for the conditions under which, each term of the equations of the second lateral system
become zero, independently of the values of the states of the first longitudinal system
and of the controls of the first system
and of the wind =
W O t
(U;,
U;,
w;,(6wmv;)
(5.150)
The conditions found will be those of an equilibrium of a lateral system, undisturbed by the longitudinal system. The analysis of the conditions cancelling the different terms of the lateral equations of force, moment and kinematic is found in section ( F . l ) , p. 265. These conditions are described below. The values of lateral states are equal to zero. These conditions come mainly from the aerodynamic lateral force equal to zero.
The values of lateral controls are equal to zero, coming from the aerodynamic lateral force and moments
6, = 6,
(5.152)
= 0
In consequence, due to the kinematic equation, a constant heading is obtained $
= constant
(5.153)
For the wind, the following conditions are obtained in the body frame Fb pyb, TY,
b
vyL = O
= pzb, = 0
=0
-
-
b rxu
or
U& =
o
(5.154)
which is translated in the vehicle-carried normal Earth frame Fo by either for uy: = 0 ry:
= -rxL
uxo, +vy: py:cos$ +qx:sin?I, pzzcos$+qzO,sin$
= uxC, cos$ sin $ = uxc, = 0 =
0
(5.155)
5.4 Decoupled equations
or for
U :
=
139
o - u ~ s i n $ + u ~ c o s @= 0
(5.156)
For the first case (Equation 5.155), UX; can be considered as a constant. It means that the wind gradient is oriented towards the aircraft heading. For the second case (Equation 5.156), the condition means the horizontal wind has only a component oriented towards the aircraft heading. Mass and geometric symmetry of the aircraft: two products of inertia equal zero
D=F
= 0
(5.157)
The lateral aerodynamic coefficients C Y , C1, Cn independent of the longitudinal parameters a a ,q, Va,h, ,S S,. The moments and lateral propulsive forces equal to zero and independent of the longitudinal parameters (5.158)
In practice, this condition is verified if the propulsion is symmetrical. This is not the case with an engine failure on a multi-engine, or if there is an influence of the downwash of the propeller or the gyroscopic torque of the engine’s rotating parts.
These conditions lead to a flight with horizontal wings and n o sideslip, an the vertical plane which is called pure longitudinal flight. The heading of the aircraft is constant and the vertical plan coincides with the aircraft’s symmetrical plan.
A physical approach to this decoupling can be justified later by the following arguments. The longitudinal flight is a flight in which the trajectory is situated in a plane. So that this trajectory stays in this plane, it is necessary not to have any external forces perpendicular to this plane and to have the moments of the external forces perpendicular t o this plane. By choosing one of these characteristic planes, here the plane of the aircraft symmetry, the weight in this plane imposes a zero bank angle (4 = 0) since the weight vector stays vertical by definition. The zero lateral aerodynamic force imposes zero sideslip (p = 0). The symmetrical plane is therefore mixed with the vertical plane. The vertical plane could have been chosen in the beginning to arrive at the same result but this time by the lift instead of the weight. As the moments of external forces should not be in this plane, the lateral controls are zero (6l = 6, = 0). To avoid disturbing this situation, the roll and yaw velocities ( p , r ) must be zero. To all intents and purposes, they represent the derivatives of the bank and sideslip angles The wind, in order not to disturb the longitudinal flight, must not “evolve laterally”. The wind angular velocity flwmust be perpendicular to the symmetrical plane.
(4)8).
The decoupled longitudinal equations of flight
These equations come from the preceding hypotheses (Equation 5.151, p. 138) to (Equation 5.158, p. 139). It ensues from these hypotheses, some preliminary results
achieve below.
Dynamics of Flight: Equations
140
5
-
- Simplified equations ____
Angular relations The inclination angle of the aircraft 8 is expressed by a simple form due to 0, = 4 = 0 (Equation 2.85, p. 40)) the inclination angle being the sum of the angle of attack a, and the aerodynamic climb y,
e
= a,+?,
(5.159)
Moreover, if 0, = 0 then 4 = 0 leads to (Equation 2.86, p. 40) an aerodynamic bank angle pa equal to zero Pa
(5.160)
= 0
and with p a = pa = 0 (Equation 2.87, p. 40) the aircraft azimuth aerodynamic azimuth xa
+
It can be equally noted that with the right wing axis Yb.
+ is equal to the
(5.161)
= Xa
0,= 4 = 0, the aerodynamic axis y,
coincides with
Expression of wind The wind V, and @mV$ is expressed by its components in the vehicle-carried normal Earth frame aircraft azimuth oriented F,, deduced from the vehicle-carried normal Earth frame F,, by a rotation of the heading $ of the aircraft. The frame F,, represents the vehicle-carried normal Earth frame Fo whose axis x, is aligned with the heading of the aircraft. The plane (x,, z,) coincides with the vertical plane containing the trajectory of the aircraft. Thus the transformation matrices are obtained (Section A.3.1, p. 198)
T,,
cos+
= Tbo(+,O= 4 = 0 ) =
sin$
0
0
1
(5.162)
These two matrices are obtained with the hypotheses of the longitudinal flight, d, = 0 for the first one and p, = 0 for the second one. cos0 0 -sin8
T~,= Tbo(e,+= 4 = 0) =
L a
= T3a(Xa =
+=
O,%,Pa = 0) =
cosy,
0 siny,
-siny,
0 cosy,
(5.163)
(5.164)
The heading can now be considered constant, as the wind is defined in the vehiclecarried normal Earth frame aircraft azimuth oriented F,. In this frame F,, the components of the wind velocity V, are expressed by
v;
=
(z)
= T , , V ~=
WL
(
U:
--U;
cos+ sin
+ vz sin$
++ w;
cos 1c)
(5.165)
5.4 Decoupled equations In the body frame
~
~
_
141
Fb
(5.166) In the aerodynamic frame F, U: COS Y, U:
sin
- w& sin
+ w;
cos
(5.167)
If the wind velocity Vw can be any value whatsoever inspite of the decoupling operation of longitudinal equations, this is not the case for the wind gradient ChmV; which takes on a particular form (Equation F.29, p. 269). Projected in the vehiclecarried normal Earth frame aircraft azimuth oriented F,
(amv:),
=
(
uxc, 0 -qxc,
0 qz& 0 0 0 wzc,
)
(5.168)
The components of this matrix are the known data of the wind, the gradients in the vertical plan of the aircraft's trajectory. Projected in the aerodynamic frame Fa with (Equation 5.164, p. 140), the gradient takes on the form
(CGwmv:)"
=
T,,(CGwmV~)CT,, uxo, 0 qz;
(5.169) (5.170)
The expression of acceleration
The different terms of acceleration of the second form (Equation 5.33, p. 111)
will be expressed in the following simplified forms thanks t o the decoupling hypotheses. The first term (Equation 5.35, p. 111) with pa = 0
(5.173)
Dynamics of Flight: Equations
The second term equation (5.46)) p. 112 with equation (5.151))p. 138 (5.174) The third term thanks t o the expressions of the wind previously defined (Equation 5.170, p. 141)
( & ~ v ~ ) " =v ~VQ
(
uXC,
2 + W ~ C sin , ya + COS ya sin yQ(qx&- q z L ) 0 sin2 ya - qxC, cos2 ya + cosyo.siny,(~xC,- W ~ L )
cos2 ya
-qzL
1
(5.175)
The fourth term
DVZ = (cGwmVzV,w)" = T,C(&ADV~V,)"
=
( ) du; dug dwt
(5.176)
with (GmV;)' already calculated (Equation 5.168, p. 141)) and TQccomes from equation (5.164), p. 140 dut
=
dv:
= 0
dwz
=
+ q z k t & ) + siny,(qxC,uC, - wzC,wL)
COS~,(UXC,UC,
COS yQ(wzkw;
- qxkuk) + sin yQ(uxLuk+ qzLwL)
(5.177)
Expression of the external forces Gravitational force equation (5.50)) p. 113 and equation (5.160), p. 140 mg"
= mg(
- sin y,
0 cos YQ
)
(5.178)
Aerodynamic force equation (5.52)) p. 113 and equation (5.151)) p. 138
+psv,2c;
=
$pSV,2
( ) -CD
(5.179)
OL
Propulsion force equation (5.52)) p. 113 and equation (5.151)) p. 138
F" = TQb(aQ)PQ = 0) Fb =
+ sin aQ~ , b 0 - sin aQF,b + cos aQF,b cos a, F i
(5.180)
With equation (4.104)) p. 92 and a symmetric propulsion, the following form is obtained
F" = F
cos P m Cos(aQ- a,) 0 - COS& sin(a, - a,,)
) ) ( =
Ft"
(5.181)
5.4 Decoupked __________ equations
___.______
143
______
Moment equation The moment equation of pitching (Equation 5.56, p. 115) with the decoupling hypotheses (Equation 5.151, p. 138) and (Equation 5.157, p. 139) and
p=r=O
F=D=O
(5.182)
thus
Bq
=
3pSlV; C m + M k v
(5.183)
with the decoupling hypotheses equation (5.158))p. 139 and equation (5.164))p. 140
Mky = MgY
(5.184)
Kinematic equations In the vehicle-carried normal Earth frame aircraft azimuth oriented F,, the kinematic equations linked to the kinematic velocity Vk projected in F, (Equation 5.69, p. 117) are written
V i = T,aV,"+V&
(5.185)
thus = V,cosy, +uc, y& = 0 i& = - A = -V,siny,+wL
X&
(5.186)
The kinematic equations linked t o the kinematic angular velocity
f2k
are written
9 , = 0 o
=
q
q = o
(5.187)
Decoupled longitudinal flight equations Regrouped here are the previously obtained results. This longitudinal flight takes place in a vertical plane coinciding with the aircraft's symmetrical plane. It is controlled by two force equations (propulsion and sustentation), a moment equation of pitching and two kinematic equations (vertical velocity and pitching velocity). There exists a kinematic navigation equation but decoupled from the preceding equations. The propulsion equation comes from equation (5.172), p. 141 and equation (5.178), p. 142 t o equation (5.181))p. 142, expressed in the aerodynamic frame Fa
Dynamics of Flight: Equations
5 - Simplified equations
144
_____
The sustentation equation comes from equation (5.172), p. 141 and equation (5.178), p. 142 to equation (5.181), p. 142
Thanks to equation (5.197) the term &a - q could be replaced by -?,. Pitch moment equation (5.183), p. 143
Bq = ~ p S I V ~ C r n + M ~ v
(5.190)
It must be recalled that the expression of the aerodynamic pitch velocity qa on which depend the aerodynamic coefficients are written in the body frame Fb (Equation 5.59, p. 115) q,b
= 9 - 9% b
(5.191)
From equation (F.34), p. 270, it can be written
As the second form of equation is not available on the moment equation, the wind effect appears indirectly in external efforts. The pitch moment equation (5.190), p. 144 is an example. The wind, known in frame F,, comes from qx: and indirectly influences the aerodynamic pitch moment coefficient through q i . Kinematic equations of vertical and pitch velocity (Equation 5.186, p. 143) and (Equation 5.187, p. 143) are written (5.193) (5.194) Kinematic equations of distance decoupled (Equation 5.186, p. 143) are written X&
= V ~ C O S+ U~:,
(5.195)
The angular relationship must be denoted (Equation 5.159, p. 140) 8
= ff,+y,
(5.196)
thanks to which, it is possible to eliminate the variable 9 by replacing the kinematic equation of pitch velocity in equation (5.194), by
This practice is common as it simplifies the exploitation of the equations since the inclination angle 9 does not explicitly intervene in the expression of the external efforts, contrary t o the angle of attack a.
5.4 Decoupled equations
145
Decoupled longitudinal flight equations and uniform wind velocity field In section (5.3), p. 130, it has been shown that a uniform wind velocity field can be translated by the relationhip equation (5.126), p. 130
thus hereafter (Equation 5.168, p. 141) becoming
The previous equations (Equation 5.188, p. 143) to (Equation 5.194) take on the form
with (Equation 5.57, p. 115)
as well as (Equation 5.59, p. 115) Qa b
= (2
(5.199)
The longitudinal equations (Equation 5.199, p. 145) constitute the simplest obtainable form and thus represent the final result of the preceding developments. The last simplification will be done t o show equilibrium (Equation 7.24, p. 189).
5.4.3
Decoupled lateral equations
In section (5.4.2), p. 137 the system of aircraft equations has been divided into a longitudinal system
and a complementary lateral system
It has been shown that the longitudinal system can be decoupled from the lateral system under the conditions of the second type (Section 5.4, p. 132) by the equilibrium of the lateral system. Thus, X 2 = 0 whatever XI,is obtained. This equilibrium is special, since Xae = 0, with as a consequence, each of the lateral equation states equal to zero. Dynamics of Flight: Equations
5 - Simplified equations
146
The possibility of the decoupling of this type for the lateral system is now going to be examined. The question is whether XI = 0 whatever Xz,can be obtained. By using the same approach as in the longitudinal system, a condition sufficient for decoupling is looked for by trying to cancel each term of effort, or complementary acceleration, of the longitudinal equations with whatever values the lateral states may be. The sustentation equation ( 5 . 5 2 ) , p. 113 is taken to obtain ci, = 0. First, what must be done is to cancel the component of weight (Equation 5.51, p. 113)
+ cos 8 cos 4 cos a,)
m g (sin 8 sin a,
This term will be zero independently of 7T
8= 2
4, if
and a0 = O
or if IT
a, = -
2
and 8 = 0
Besides the fact that this represents the case of unusual flight, these conditions do not permit the cancellation of the component of weight on the propulsion equation (5.51), p. 113
+
m g( - sin 8 cos a, cos ,& cos 8 sin # sin ,k10 + cos 8 cos # cos p, sin a,) With 8 = and a, = 0, the component of weight -mgcosp, a, = :, 8 = 0, the component of weight mg(sin 4 sin P0
+ cos 4 cos PO)
is obtained. With
= mg cos(4 - PO)
is obtained. Thus, it is impossible to find the conditions on the longitudinal parameters (VO, a,, q, h, 8) which will lead to the cancellation of the weight components, on both the sustentation and the propulsion equations no matter what the lateral parameter values are, especially the sideslip angle Pa and the bank angle 4. From this statement, it appears that the type of decoupling obtained for the longitudinal flight can not be reconducted on the lateral flight. It is impossible to cancel each component of external forces of the longitudinal equations whatever the values of the lateral states are. A less demanding approach consists in looking for a situation of longitudinal equilibrium that is independent of the lateral states in order to obtain longitudinal states constant. If it is admitted that the aerodynamic forces of lift and drag are independent of the lateral states (and especially the sideslip angle P), the question is to show that the components of weight, and the complementary accelerations, are equally independent of the lateral states. For the sustentation equation, the component of weight is independent of the lateral states in the first order if the bank angle # is around zero. Thus cos4 x 1 can be admitted. By assuming that the wind is zero (AA, = 0 ) and the sideslip angle Pa is around zero, the complementary acceleration (AAk) is equal to /3, cos a,p - q
+ Pa sin a O r
5.4 Decoupled equations
147
In addition if is around zero at the first order approximation, then the complementary acceleration is equal t o
This term will be independent of the lateral states P a and p , if the sideslip angle and the roll angular velocity p are small enough so that the product is of second order with respect to the pitch velocity q. This special kind of situation will be found at the time of linearization (Section 6, p. 157). For the propulsion equation, with the same hypotheses as for the sustentation equation, at the first order approximation, the component of weight is equal to Pa
mg(- sin 8 + cos 8 sin a,) Thus this expression is independent of the lateral states and 4. The complementary acceleration (AAk) is zero without any special hypotheses. This analysis allows the demonstrated result t o appear farther along than at the time of the linearization operation (Section 6.4, p. 170). Around this case of rectilinear horizontal and symmetrical flight, without wind ( P a = 4 = 0)) the lateral equations are decoupled from the longitudinal equations if the sideslip angle ( P a ) and the bank angle (4) remain small. If not, the decoupling cannot be of the second type (Section 5.4, p. 132) because of the fact that it is always necessary to compensate a component of weight as a function of the sideslip angle (pa) and the bank angle (4)) with more or less lift and drag t o obtain an equilibrium. In the most general cases, only the third type of decoupling is possible as a solution (Section 5.4, p. 132). The question here is t o define the longitudinal states with an appropriate model for the case treated. The longitudinal state is worth remembering: Va, a a , q, h, 6. If an equilibrium is studied, the longitudinal equations to the equilibrium will furnish this model. In a general manner of speaking, the influence of the altitude on the external efforts is ignored. This is a useful simplification that is hardly restrictive. p
= constant
The kinematic equation of the inclination angle (Equation 5.73, p. 118) is balanced. Thus 9 = 0 leads t o the relationship
q =
T
tan4
(5.200)
The kinematic equation of altitude (Equation 5.69, p. 117) shows that sin?,
=
-w; +Va
h Va
(5.201)
The other states V a , a a ) 8 can be expressed as a function of the lateral states, making it necessary to solve the equilibriated longitudinal equations. If the studied problem is a case of non-steady flight, the question is t o find the best adapted state for each longitudinal state as a function of its own dynamics and perturbations provoked by the lateral movement. Thus, for the simplest solution, Dynamics of Flight: Equations
5 - Simplified equations
148
either the state is assumed constant or the state is the solution of an equilibriated longitudinal equation. It can be admitted that the longitudinal states are piloted (thus constant) if that corresponds to a certain reality. The most “comfortable’) analytical solution consists in using the invariables of all the longitudinal states as a working hypothesis. This solution gives good results in most situations. Decoupled lateral equations The lateral force equation (Equation 5.52, p. 113) is written
md,
Va
-m V, +m V, +m
V, =
+
+mV,(-psina,
+ r cosa,)
sin a , COS a, sin pa cos p, (92,b - qx,)b (- (p: cos2P, + py& sin2P,) sin a ,
+ (ry:
sin2 pa + r& cos2
cos a,)
p, sin Pa (-uxL cos2 a, + VYL- W Z ; sin2 a,) + m d v i rng(sin 8 cos a , sin Pa + cos 8 sin 4 cos p, - cos 8 sin a , cos 4 sin PO) COS
4pSV:Cy
- F,bsinp, COS a,
+ Fy”COS&
- F,bsin a, sinp,
(5.202)
The component of the force of propulsion in (Equation 5.202)
-F,b sin ,8,
+
COS a a
Fy”COSpa - F ’ sin a, sin p,
can also be expressed in the form (Equation 4.104, p. 92)
The calculation (Section D.6, p. 238) of the components of (G~rnv~)~ can obtain the expressions of pz:, pyb,, r y b , , r&, U & , vy,,b WZ:. The calculation of DVZ = (CGwmV;V,)” (Equation 5.49, p . 113) can obtain the expression of dvk dvz
=
+ +
+
cos 8 cos II, cos pa(sin 8 sin 4 cos II,- sin II,cos 4) - sin sin fla (cos II,sin 8 cos 4 sin #sin II,))duG (- sin PaCOS sin II,cos 6 cos Pa(sin 8 sin 4 sin II, cos II,cos 4) - sin a a sin @,(sin 8 cos 4 sin II,- sin 4 cos II,))dv:
(- sin Pacos
+
+
(sin /?a COS aa sin 8
+ COS
/?a COS
+
8 sin 4 - sin a a sin @a
COS 8 COS 4)dwL
(5.203)
with
(2;) dw;
=
(
U
X
~
rx0,uz -qx;u;
-U ry0,vg ~
+ qZ;w;
+ p.;v;
+ wz;w;
+ vy0,vG - p.0,~;
(5.204)
The moment equations (Equation 5.56, p. 115) assuming that the aircraft is symmetric (Hypothesis 4) D = F = 0 can be written as follows. The roll moment equation is
Ap - E?:+ rq(C - B)- Epq
=
$pSt!V:CZ
+ MF,b
(5.205)
5.4 Decoupled equations
149
The yaw moment equation is
C?:- Efi + p q ( B - A ) + Erq = $pStV:Cn
+ MF,b
(5.206)
The kinematic equation of the bank angle and heading (Equation 5.73, p. 118) is
6
= p+tane(qsin4+rcos4) =p+IIsinO
(5.207) (5.208)
It must be remembered that the aerodynamic moment coefficients depend on the aerodynamic angular velocity p:, q t , r y : and r x : , (Equation 3.54, p. 56) (Section 4.3.3, P. 89)
(5.209) The atmospheric perturbations mb,, q x b , , etc, are introduced in the moment equations in this way (Section D.6, p. 238).
Decoupled lateral equations with uniform wind velocity
In (Section 5.3, p. 130), it has been shown that a uniform wind velocity field can be translated by the relationship (Equation 5.126, p. 130) thus all the terms p&, py%,
ryb,,
rxL, U X : , vy,,b
WZ;
and dv; are equal to zero. Then
the lateral force equation (5.202), p. 148 is written
mb,Va
+
mV,(-psina,
+
$pSV:Cy
+ r coscy,) + cos 8 sin 4 cos Oa - cos 8 sin eta cos 4 sin Oa) - F,bsinOacosa, + F ~ ~ c o -s F:sina,sin& ~, (5.210)
= mg (sin 8 cos cya sin /?a
The moment equations and the kinematic equation do not change but the aerodynamic angular velocity is equal to the kinematic angular velocity, then
(5.211)
5.4.4
The consequence of lateral and longitudinal decoupling
It has been shown the decoupled longitudinal equations is a second type of decoupling (Section 5.4, p. 132). Then, the pure longitudinal flight (Section 5.4.2, p. 137) gives no errors within the framework of decoupling hypotheses '. In practice, these hypotheses ~~~~~~~
2The errors are relative to the non-decoupled equations.
Dynamics of Flight: Equations
5 - Simplified equations
150
could be well verified and in that case, the longitudinal flight is perfectly modeled with the decoupled equations. On the other hand, the lateral flight is associated with the third type of decoupling, and the more the flight is lateral, the more the errors are important. The decoupling is only rigorous around the level horizontal flight without wind and with weak sideslip and bank angle. The purpose of this paragraph is to evaluate these errors. Therefore, three methods will address this issue and the numerical evaluation will be made around a flight of a large commercial transport airplane. This flight is classical cruise flight a t an altitude of 30,000 f t and a Mach number of 0.8. The first method deals with the modal approach and gives the errors on the characteristic of the modes. That is to say the errors on the eigenvalues (frequency, damping ratio and time-constant). As with the first method, the second one is linked with the modal approach, but from a magnitude viewpoint. The variation of the magnitude contribution of each mode on the aircraft response is examined through the eigenvectors. The third method requires the gramian approach to throw light on the difference of energy of the signals between the coupled and decoupled model.
Mode
- Eigenvalue
The characteristics of the longitudinal and lateral modes are examined around different equilibriums. These equilibriums correspond to different values of sideslip angle and yaw rate r. Three yaw rates are taken into account: the null value for the straight flight, and two values of 0.57 O / s and 1.15 "/s which correspond to a turning flight with a bank angle of 13 O and 26 O. The results of relative differences between coupled and decoupled model are given in figure (5.8). Generally speaking, the rapid modes like the short period and rolling convergence are not affected by the decoupling. The dutch roll is only slightly influenced (from 1 % to 2 %) by the sideslip angle but not by the yaw rate. The slow modes are more affected. Then, the phugoid and spiral modes can change 50 % from the coupled model, depending of the sideslip angle and yaw rate. It should be noted that, whatever the yaw rate is, with zero sideslip angle, the decoupling has little influence on the modes.
Magnitude contribution of mode
- Eigenvector
The previous method gives information on the frequency, damping-ratio or timeconstant. However we can imagine, for example, that the frequency does not change although the magnitude of the response of the aircraft to a perturbation changes. This magnitude can be evaluated through the components of the eigenvector which give the contribution of one mode on the response of one state under the influence of one perturbation. To simplify the calculation, the perturbations are taken as initial conditions on the states x:. The influence of the initial condition x: of the state k on the state xi, through the mode j is evaluated thanks to the product mfj of the . symbol * right eigenvector component vij by the left eigenvector component u ; ~ The denotes the transpose and conjugate of the vector. (5.212)
5.4 Decoupled equations
151
Short Period frequency
Short Period damping ratio
I
-2'5
"0
4
8
12
-5 0
I
(b) 4
8p12
::Fi Phugoid frequency
0 - -
-25
-50 0
Altitude Convergence mode
50
4
8,312
Dutch Roll frequency
oh I -2:h I
25
251
I
-
-25
-50 0
4
8 p 1 2
Dutch Roll damping ratio
-50 0
4
8 s 1 2
Rolling convergence mode
5m I I
1
2.5
-2.5Oi-i---5
0 Yaw rate r:
4
Spiral mode
8 p 1 2
-r=O.O"/sec - - - -
r=0.57 "/sec
---
r = ] . ] 5 "/sec
Figure 5.8: Decoupling influence on the modal characteristics Then, the temporal response of the state x, t o an initial condition x f , is
j=1
The superscripts and are for the coupled system and the decoupled system. Therefore, a criterion ci about the magnitude for the state xi, can be defined as
(5.213)
The numerator is a length and the denominator is the maximum modal magnitude. Then, the range of the criterion is (0, 1). The decoupling does not have the same influence on each state. So, in order to have the consequence on the whole system, Dynamics of Flight: Equations
152
5
- Simplified equations
a global criterion C g l o b a l is built with the sum of the criterion ci weighted with the coefficient pi associated with each state xi. n
E Pi i= 1
(5.214)
With
(5.215)
The rangepf variation Si of the state xi gives the maximum variation whatever the perturbation is. The different ranges of variation Si are linked together with a time scale and an altitude scale. The time scale, based on rapid modes, links the angular velocity with the angle. The rapid modes are the short period mode and the dutch roll mode, and the given time scale is 0.5 s. The altitude scale, based on the total altitude ht, links the relative velocity to the altitude. The relative velocity belongs to the angle family. The total altitude is derived from the kinetic energy theorem, so that (5.216)
With these scales, only one range of variation Si is needed to define the others. The figure (5.9) shows the variation of the global criterion C g l o b a l applied on the whole states, that is to say longitudinal and lateral. This variation depends on the sideslip angle p and the yaw rate T . Three yaw rates are considered like for the mode analysis. The four presented curves correspond to four initial angle perturbations, one on the relative velocity, one on the angle of attack, one on the sideslip angle and the last one on the bank angle. Noted the classical result: with no sideslip and no yaw rate the decoupling is perfect. Globally, the relative change of magnitude between the decoupled system and coupled system is around 20 %, except for a bank angle perturbation, for which the magnitude is higher. Another way to analyse the decoupling is to consider the longitudinal system on one side and the lateral system on the other side. For example in the figure (5.10) the two above curves correspond to the longitudinal system alone. That is to say, only the variation on the longitudinal states are considered, submitted to longitudinal perturbations. The same the two lower curves, but for the lateral system. In this case, the relative change of magnitude between the decoupled system and coupled system is lower, around 10 %,
-
Energy contribution Gramian
The energy of the response on each state can be analysed thank t o the Gramian method. This approach is a kind of mixing of the two previous methods, because the
5.4 Decoupled equations
153
Relative longitudinal velocity 100
,
2
0
4
6
8
,
7 10,
12
10
a =0.01 rad
Angle of attack
UN=0.01
,
0
2
4
5
8
1
0
P(d&)
100
-
Sideslip angle ---IT-
p =0.01 rad
--
1
2
P(deg)
1 0 0 - ---~
0
2
Bank angle ----
4
4 =0.01 rad I
6
8
10
12
Figure 5.9: Decoupling influence on the magnitude of the states, for the whole system (longitudinal and lateral)
energy of the signal is sensitive to its magnitude and frequency. A physical understanding is associated with the grey surfaces seen in the figure (5.11). The energetic length of the state z ( t ) is denoted Ilz(t)lI. This length is defined through its square value
This value is evaluated thanks to the observability Gramian. The figure (5.12) shows the variation of the energetic length applied on the whole states, that is to say longitudinal and lateral. The figure (5.13) shows the variation of the energetic length applied on the longitudinal states and on the lateral states.
Dynamics of Flight: Equations
Relative longitudinal velocity
UN=0.01
Angle of attack a =0.01 rad 100 1 --7
(4
.-
20
0
r = 1.15"/s 0
2
r = 0.57"/s r = 0. I~
4
6
8
10
12
P(deg)
Sideslip angle
100
-___
p =0.01 rad
I ' - 100
Bank angle $I =0.01 rad ~
Figure 5.10: Decoupling influence on the magnitude of the states, for the separate system (longitudinal for the two above curves and lateral for the two lower curves)
Figure 5.11: Energy contribution on the coupled and decoupled system
et-6
0
5-
m
a"
m
P
0
0
0
N
0
0
P
Lateral
o
m
o
0
m
0 0
Global criterion
o
~
Lateral
Global criterion
0
0 0
m
o
o
o
1
o
w
1
~
o
m
o
Longitudinal o
r
e,
v
n
Global criterion
\
Longitudinal
Global criterion
n
o
o
m
E -
no
P
N
0
o
o
O
I
o
C
o
O
O
O 0
m
o
W
o
--1
E
g
I QII
l
v
L
n
5
Longitudinal and lateral
\;
P
o
Global criterion
o
N
Longitudinal and lateral
W
o
Global criterion
n
a
7
C 0
Longitudinal and lateral
P
Global criterion
o
N
Longitudinal and lateral
Global criterion
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6
Linearized equations
The more or less simplified equations previously studied were all non-linear equations. These equations are exploited without any difficulty by numerical methods especially to simulate the aircraft’s movement. However, when the question is to analyse the dynamics of a system and to synthesize a control law, the majority of tools offered by the automatic control scientist can only be put into use in linear systems. This is why it is necessary to linearize the equations of the aircraft’s system in order to have a model adapted to the study of its dynamics. The Zinearined equations (Section 6, p. 157), are part of the simplified equations but their major importance in the analysis of flying qualities justifies their development in a separate chapter. These linear equations of the aircraft system provide a simplified model representing an aircraft in a certain validity range, around the initial conditions of linearization. This validity range depends on the initial conditions and the type of linearized equations. Thus the longitudinal equations usually have a larger validity range than lateral equations. The linearixation method (Section 6.1, p. 158) is first expounded, then a numerical and analytical process is proposed to exploit this method. Numerical Zinearixation (Section 6.2, p. 160) can be implemented for every flight condition with non-decoupled equations and non-analytical external effort models. This numerical linearization has a large domain of application but it does not give an inside view of the phenomena as the analytical linearization does. The analytical linearixation carried out on the decoupled longitudinal (Section 6.3, p. 161) and lateral (Section 6.4, p. 170) equations leads to a more limited implementation, due to its heavy calculations. However it has the advantage of being an explicit parametric study of dynamics and thus favorable to a physical approach to phenomena. At this point, it can be shown that linearization around a steady state flight with wind modifies the state matrix with respect to steady state flight without wind. This signifies that the modes and thus the flying qualities can be a function of wind.
157
158
6.1
6 - Linearized equations
Linearization method
The system is put in a state form (Section 5.4.1, p. 135) (6.1)
= F(X,U,W")
X
The differentiation of these equations leads to a linear system. This differentiation is made about the initial conditions defined by the equations = F(Xi,Ui,WY)
Xi
(6.2)
This differentiation is representative of a development of the Taylor series in the first order. The differentiation gives dX
+
= GRAJDFX~ dX GRADFU~ dU
+ G;r~rnFw~ dW"
(6.3)
with
and the matrix GRmFxi is the Jacobian matrix of F with respect t o the vector X (Section 3.2.1, p. 45). The rows of GRmFxi are made up of the partial derivatives of one of the components of F with respect t o each of the X components. These partial derivatives are calculated about the initial conditions Xi, Ui, WO. Thus (GWmFxiis a matrix with constant coefficient.
\ with
F n x i i Fnx2; .. . Fnxni
Fmxki =
J
dFm
d x k (Xi,Ui,WO)
with m and Ic varying between 1 and n. The matrix (GWmFu,is the Jacobian matrix of F with respect to the control vector U, calculated about the initial conditions Xi,
ui, w:.
The matrix @mFwi is the Jacobian matrix of F with respect to the wind vector calculated about the initial conditions Xi, Ui, Wy. The initial conditions Xi, Xi, Ui, WO can be any value whatsoever. This is the case for the proposed method later on, in order to look for a state of equilibrium (Section 7.2, p. 186). In general, linearization is made around a state of equilibrium ( e ) such as Xi = X, = 0 (Section 7.1, p. 180). It has been shown [6] that since the Zinearixation is made around a state of equilibrium, the linear systems from two different state bases represent the same system. However for the general condition, the linear system is not intrinsically linked W O ,
6.1 Linearazation method
159
t o a non-linear system. The linearization around a state representing a trajectory of an aircraft could have this same property, found in the linearization around a state of equilibrium. In this way, let the system (Equation 6.1) without wind for simplification X
= F(X,U,O)
A state base transformation Y = T(X) is applied t o it. Let see how the linearized system is obtained, about any point (Xi,U;),so that Xi = F(Xi,Ui,O)
In order t o linearize the system, the transformation equation is linearized
dY
= &mTidX
In this new state base the linearized system is written Y
= &mTidX
However if the state base transformation is first applied on the non-linear system and the linearized process is made later, the result differs. Thus
Y
= GRAKDTX
Thus after linear ization Y
= ~(GRADT)X~+GADT~X
The two approaches give the same result only if Xi is equal t o zero, that is t o say when the point (Xi,Ui) is an equilibrium point. If the linearization around any initial state can be made mathematically, its properties are not clearly defined. All the Xi = F ( X i , U i )are not linked t o trajectories. These trajectories are defined as the system answers to a realizable control U(t). The equations of equilibrium are written (Section 7.1.1, p. 180)
If the linearization is made around the equilibrium state, X, = Xi and U, = Ui. Most of the time, equilibrium is defined around the conditions of zero wind, WO = W,O = 0, then 0
= F(Xe,Ue,O)
(6.8)
The aerodynamic coefficients can depend on the derivatives of the state X. For example, the lift coefficient C L and the pitch coefficient C m are a function of the temporal derivative of the aerodynamic angle of attack 6,. Thus the general form of the system is the following X
= F(X,X,U,W')
(6.9)
where the effort equations have several components of X in the terms of acceleration, as is the case for the lateral equations. A practical difficulty then comes up when Dynamics of Flight: Equations
6 - Lanearazed eauatzons
160
looking for whatever initial conditions with Xi # 0. In fact, in the preceding case, it was sufficient to give Xi, Ui and Wy in order to find Xi via the equations given by F. Now, a iterative step is necessary to find Xi such as Xi = F ( X i , X i , U i , W F )
(6.10)
The linearization of (Equation 6.9) makes a new term appear dX
= cGwmFxi dX
+ ~G~ADFX; d X + ~G~ADFu; dU + GRADFw~ dW"
(6.11)
or again dX
=
- GRADFX~)-' (GRmFxidX + @mFui dU
(11
+ @mFwi
dw")
(6.12)
This linearized system is frequently shown with the following notations dX
= AxdX+BudU+BwdW"
(6.13)
If no ambiguity is possible, the notation d X , dX, du,dw" which represents the increment relative to the initial value, is often dropped and replaced by X, X, U, W" (Hypothesis 25). The matrices Ax,Bu, Bw are
6.2
AX
=
(1, - @ADFX~)-~GRADFX~
BU
=
(11 - G R A D F X ~ ) - ~ @ A D F U ~
BW
= (II~ &~FX~)-~GII~ADFW~
(6.14)
Numerical linearization
In the framework of a numerical simulation of aircraft flight, the non-linear equations X
= F(X,X,U,W")
(6.15)
are put in the calculation program. It is therefore entirely possible to construct a linearization procedure from these equations. In the following paragraphs (Section 6.3, p. 161) and (Section 6.4, p. 170), the matrices Ax, Bu, BW will be analytically constructed from the linear system. This method shows a number of interesting points but it is rather complicated to put into practice. Its application is limited to special cases. However, the numerical linear method can be put into practice whatever the case of flight and modeling of external forces might be, including a non-analytical modeling case. An example would be when the aerodynamic coefficients are given in the form of graphs. The working out of this numerical linearization method is illustrated by the linear program found in section (G.l), p. 275. The matrices Ax and B are built by columns. For example, for the matrix Ax, thanks to a numerical variation of one of the components of X , A x j , the variation induced on all of the components of X, AX(Axj) can be calculated by some terms of the acceleration. The j t h column of Ax will therefore be equal to A X ( A z , ) . In
161
6.3 Longitudinal linearized equations
this manner of provoking a variation of the angle of attack Axj = Aaa, the induced variations on the components of acceleration A X ( Aaa) will be calculated. A practical difficulty can appear in the definition of the values of Axj. If these values are too high, the difference cannot represent the local slope around the initial conditions since the terms over the first order in the Taylor series can no longer be neglected in comparison to the first order term. This first order term is the linear term represented by the matrix Ax. If these values are too low, it is possible to reach computer precision and make calculation errors. For example, a numerical linearization made on a working station for a commercial aircraft in its flight envelope, gives this range for the increment Axj in order to stay within a good accuracy. For the increment on the components of velocity U , U , w, its value should be between 10-3 m / s and 10-1 m / s . For the increment on the components of angular velocity p , Q, r , its value should be between 10-3rad/sand lO-lrad/s for p and r , and for q between 10-4rad/s and 10-3 radls. For the increment on the altitude h, its value should be between 1m and 10 m. For the increment on the inclination angle 8, its value should be between 10-2 rad and 10-1 rad. For the increment on the bank angle 4, its value should be between 10-4rad and 10-2rad. It seems useful to determine the miminum and maximum values of each increment for the case which is examined. This numerical linearization is used, in particular, for the general research of equilibrium (Section 7.2, p. 186).
6.3 Longitudinal linearized equations The analytical linearization performed in this section presents the advantage of an explicit parametric study of the dynamics of an aircraft which clearly explains the physical interpretation of phenomena. However, it is more complicated to implement than the numerical linearization. As a consequence, it is only usable for special simplified cases. This linearization is performed on the decoupled longitudinal flight equations (Section 5.4.2, p. 137). Before attacking the linearization of longitudinal equations as such (Section 6.3.2, p. 165), a certain number of preliminary linearizations must be made (Section 6.3.1, p. 161).
6.3.1
Preliminary linearizations
Reduced velocity The reduction of the velocity to a nondimensional value simplifies the writing of the linearized equations. The reduced velocity will be denoted Va and it is equal to the ratio between the velocity Va and the initial velocity Vai. The quantities AV, represent the change from the initial values (most of the time the equilibrium values) and from a physical point of view is equivalent to the quantities dVa defined in equation (6.5), p. 158
AV,
= Va - Vai etc
(6.16) Dynamics of Flight: Equations
6 - Linearired equations
162
(6.17) wind velocity (6.18) thus (6.19) etc. Etkin [5] suggests a system of equations where all of the variables are reduced t o nondimensional variables. For example, ci, p , q, T , are nondimensionalized by the the mass by the factor inertia by the factor the air density by the factor time by the factor If the reduction of the velocity simplifies the writing factor of the coefficients of linearization, the reductions of the other parameters do not have the same advantage.
6,
k,
F.
&,
&,
Linearization with respect to the Mach number The Mach number M is not part of the state of the aircraft for classical forms of equations. It is therefore necessary to show the variation of Mach AM around the initial conditions in function of the states, here AV, and Ah, for the linearization. The Mach number is defined by
M = - -Va a
(6.20)
a2 = yRT
(6.21)
with the speed of sound a
Here, y represents the adiabatic constant of the air and is equal to 1.4 under normal conditions. This y is only a local utilization and must not be confused with the climb angle y of the trajectory inclination. The differentiation of equation (6.20), p. 162 gives
dM -
M
The differentiation of equation (6.21), p. 162 gives
da 2a
=
dT - 1 dT - -dh
T
T dh
dV, Va
1 dT --dh 2T d h
Thus
dA4 A4
-
--
(6.22)
6.3 Longitudinal linearized equations
163
The temperature gradient Th (Equation 4.1, p. 88) as a reduced form and will be denoted Th
dT Th= dh
Th Th= Ti
and
(6.23)
from which the change of M from the initial condition AM = M - Mi is
(6.24) This relationship could be written
Linearization with respect to the altitude The altitude h intervenes in the expression of the external forces through the air density p. For the linearization with respect to the altitude, the air density gradient Ph must be evaluated (6.25) By definition
Laws of the standard atmosphere (Equation 4.84, p. 87) (Equation 4.85, p. 87) give
(6.26) p
=
pRT
(6.27)
Thus, after the differentiation of equation (6.27)
(6.28) from which the air density gradient
Ph
is
(6.29) its reduced form ph
= ph
-Th
Linearization of pitch velocity in pure longitudinal flight
(6.30)
6 - Linearized equations
164
The aerodynamic coefficients depend on the aerodynamic pitch velocity q t (Equa-
tion 3.54, p. 56) (Section 4.3.3, p. 89) with the expression of q:
The state of the aircraft contains the kinematic pitch velocity qk usually denoted as q. The question here is to linearize q t with respect to q and the components of the wind. pitch velocity q&. The expression of qxb, in pure longitudinal flight has been calculated in (Equation F.34, p. 270) =
qxk
- sin 8 cos e(uxL - wzL)
+ qZ; sin2 8 + qx& cos28
(6.32)
where, after linearization
AqxL = -
+
(COS 28, (wZ&- u X l i ) sin 2Oi(qZLi - qxLi)) A8 sin 8; cos O,(AuxL- Awz&) + sin2 8,AqZC, cos2 8,Aq.L
+
(6.33)
The relationship equation (5.159), p. 140 valid for pure longitudinal flight is (6.34)
Finally, by differentiating equation (6.31), the result obtained is
Aqt
= Aq-
(6.35)
Linearization of thrust With the thrust model (Equation 4.102, p. 92)
The constant k , is a characteristic of the engine. The linearization with respect to the altitude h, the aerodynamic velocity Va and the position of the throttles 6, is accomplished in this manner
dF
= km-Va dP x Sxdh
dh
+ Xk,pV,X-lG,dV + k,pV,Xd(Sx)
(6.37)
Translated into a change from the initial conditions, with the preceding equation (6.17), p. 162 and equation (6.30), p. 163 (6.38)
This relationship could be denoted
1.
6.3 Longitudinal linearixed equations
6.3.2
165
Linearization of longitudinal equations
This linearization is performed in section (G.3), p. 295 on the non-linear, decoupled, longitudinal flight equations (Section 5.4.2, p. 137) which are represented by the following form. It must be remembered that the components of XL, UL,WLL,WGLand WRL,represent the change from the initial value (Hypothesis 25), that is to say the increment relative to this initial value, symbol i .
XL = F(XL,XL,UL,W')
(6.39)
with the state vector
XLt = [Va,& a , Q , h, %I
(6.40)
ULt = [6,,6,]
(6.41)
with the control vector
and, to simplify the writing, WO is divided into three elements (6.42)
with the wind linear velocities ( L )
WLL~
(6.43)
the gradients of the wind linear velocities ( G ) (6.44)
the angular or rotational wind velocities ( R ) (6.45)
This wind vector has a restricted size in the case of longitudinal flight with respect to general cases such as lateral. After linearixation around the initial conditions, symbol i , the system takes on the form
(11 - @mFXLi)AX~= GRADFXL~AXL + GRADFuL~AUL + GAIDFWGL~AWGL +@ADFWLL~AWLL +UhmFw R L A ~ W RL (6.46) and the result is
~ ~ - @ A D F X=L ~
[1
-a& 0 0 0 1+aiy& 0 0 -;q& 1 0 0 1 0 -ay& 0 0
0 0 0 0 1
1
(6.47)
Dynamics of Flight: Equations
6 - Linecrrized equations
166
The matrix GRADFxL~, which is a part of the state matrix of the longitudinal system, takes on the form
(6.48)
The state matrix
AX, (Equation 6.13, p.
AX
160) and (Equation 6.15, p. 160)) is equal t o
= (1, - GRADFXL,)-'GRADFXL,
(6.49)
The matrix GRADFXL, is due to the effect on aerodynamic force of the angle of attack derivative A, (Equation 6.112, p. 298),(Equation G.136, p. 302) and (Equation G.162, p. 304). If this effect is null, the state matrix AX is equal t o GRADFxL,. The first row is made up of the coefficients that came from the linearization of the propulsion equation (5.188)) p. 143, the second row from the kinematic angular equation (5.197))p. 144, the third row from the moment equation (5.190))p. 144, the fourth from the kinematic altitude translation equation (5.194)) p. 144 and the last from the sustentation equation (5.189))p. 144.
REMARK6.1 The matrix GRADFXL~ expresses the longitudinal dynamics of the aircraft
and depends on the wind gradients. Thus the aircraft, when it crosses an area of established wind gradient, can see its modes and thus its dynamics modified.
These coefficients have the following general form. The first row axu results in the linearization of the propulsion equation (5.188))p. 143 made in equation (G.107)) p. 298
6.3 Longitudinal linearized equations
167
The second row axa results in the linearization of the kinematic angular equation (5.197), p. 144. All the axa are the opposite sign of axy given in (Equation 6.54, p. 168) except for
axaz = -axy, for z = (U,a, h, g ) axaq = 1 - aXyq
(6.51)
The third row axq results in the linearization of the moment equation (5.190), p. 144 made in equation (G.157), p. 304
(6.52) The fourth row axh results in the linearization of the kinematic translation of altitude equation (5.194), p. 144 made in equation (G.167), p. 305 axh, hh, The other axh
= Vai shy,, Vai COS^^, are equal to zero
=
(6.53)
The fifth row axy results in the linearization of the sustentation equation (5.189), p. 144 made in equation (G.131), p. 301
Dynamics of Flight: Equations
6 - Linearized equations
168
[
The matrix of controls &mFuLi takes on the form
GRADFUL~ =
-buy, bup
buvx -buyx b?
buym
buyx
buvm
)
(6.55)
with from equation (G.108), p. 298
(6.56)
from equation (G.132), p. 301
(6.57)
from equation (G.158), p. 304
(6.58)
The matrices interpreting the wind perturbation take on the following forms. The matrix of perturbation associated with the wind translation velocities is
(6.59)
With from equation (G.109), p. 298
bwvu = bwvw =
- sin Tai qxLi - COS yaiEx& - cos Y~~
@Li + sin yaizLLi
(6.60)
from equation (G.133), p. 301 (6.61)
6.3 Longitudinal linearized equations
169 -
from equation (G.168), p. 305
1
(6.62)
The matrix of perturbation associated with the wind translation of velocity gradients is
(GW~FWGL~ =
[
bwvux -bw?ux bw";
bwvwz -bw?wz bw?,
b ~ u x bwrwz
(6.63)
With from equation (G.llO), p. 298
(6.64) from equation (G.134), p. 301
(6.65) from equation (G.160), p. 304 bwqux =
e2p ''i Cmq sin 28, 2SV&
(6.66) The matrix of perturbation associated with the wind rotations is
(6.67)
With from equation ( G . l l l ) , p. 298 bwvqx = - COS Toi sin ?ai -
sin Y~~ + Sqpie COS2 o i c D q mV2 (6.68) Dynamics of Flight: Equations
170
6 - Linearized equations
from equation (G.135), p. 302
(6.69) from equation (G.161), p. 304
(6.70)
6.4
Lateral linearized equations
In this section, the results of the preliminary linearizations are used (Section 6.3.1, p. 161). With regard to the linearization of the longitudinal equations, two problems appear when dealing with the linearization of the lateral equations. 0
0
The longitudinal states will appear in the lateral equations under a form that is a function of the flight situation studied. Depending on the hypotheses, several results of linearization will be obtained. Generally speaking, there is no such thing as a perfect decoupling between lateral and longitudinal equations. The wind does not have the simple form it had with the longitudinal flight and the most general case should be treated. The wind is defined in the vehiclecarried normal Earth frame F,. These components need to be expressed in the body frame Fb under a linearized form (Section G.2, p. 283). This operation makes the angle of heading $, the bank angle 4 and the inclination angle 8 appear. Two of these angles create a coupling: the heading with the navigational equations and the inclination angle with the longitudinal equations.
The linearization of lateral equations is made in section (G.4), p. 305. The lateral non-linear equations (Section 5.4.3, p. 145) are represented by the following form. It must be remembered that the components of XL, Xi, Ui, WLI,WGIand WRI,represent the change from the initial value (Hypothesis 25), that is t o say the increment relative to this initial value, symbol i. XI
= F(XL,X I ,XI,U,
WO)
(6.71)
with the lateral states vector
XIt = [ P a , P , r , $ , $ ]
(6.72)
and the longitudinal states vector
XLt = [Va,%,Q,eI
(6.73)
6.4 Lateral linearized equations
171
The altitude h does not appear because the influence of its eventual variation has been neglected (Hypothesis 28). The control vector
Ult = [6&]
(6.74)
and, to simplify the writing, the wind vector WO is divided into three elements
WO
=
(E)
(6.75)
with the wind linear velocities ( L )
(6.76) the wind linear velocities gradients (G)
(6.77)
wGit
The angular or rotational wind velocities ( R )which represent the wind velocity gradients that are perpendicular to the radial axis (Section 3.2, p. 45) are
(6.78) After linearization around the initial conditions, symbol i , the system takes on the form (11 - (r;rnP~i~FXii)AXl =
cGwmFxiiAX1+ (6WmFUiiAU1+ ~ G ~ ~ F x L ~ ~ A X L +@ADFWLI~AWLI + GRADFWGI~AWGI (6.79) +(r;rnPrnFwmiA WRI
These matrices are calculated in section (G.4), p. 305 for the most general cases. The reader can refer to this if, in particular, he wishes to analyse the turning flight. To simplify things, the results of linearization around a case of rectilinear steady state flight with horizontal wing, is presented in this section. This already has great practical interest. The state matrix Ax is equal to (11 - cGwrnFX~~)-'(r;rnPmFx~~, (Equation 6.13, p. 160) and (Equation 6.15, p. 160). The matrix cGwmFx~~is due to the effect of the state derivatives, if they exist, and of components of acceleration generally depending on the product of inertia E (Equation 6.80, p. 172). If this product of inertia E is null the state matrix AX is equal to GRADFXI~.
Linearization around a rectilinear steady state flight with horizontal wing The conditions of steady state flight with horizontal wing lead to the following hypotheses: 0 0
The linearization is performed around the steady state flight,
Pai = 0.
The initial conditions (Hypothesis 26) in the sideslip angle and azimuth are zero, = pi = 0. Dynamics of Flight: Equations
6 - Linearized eauations
172 0
0
0
With the conditions of zero sideslip angle and propulsion symmetry Pm = 0, the aircraft flies with its wings horizontal, q5i = 0. And equilibrium is a case of longitudinal flight, ei - cyai = Y ~ ~ . The angural velocities of roll and yaw of equilibrium are zero, pi = ri = 0
.
A hypothesis of symmetry is made, that is non-restrictive in practice. The aircraft is geometrically symmetric (Hypothesis 4) and as pi = pi = ri = 0 then c y , = Cl, = Cn, = 0.
On the other hand, it can be assumed that the wind is known in the vehicle-carried normal Earth frame aircraft fuselage oriented F f , which is oriented towards the initial inclination angle of linearization 8,. The terms duz, dvc, dw;, are calculated in equation (5.48), p. 113. The terms uxLi, vy;,,, w&,, qx”,, q&, are calculated in equation (D.75), p. 240 to equation (D.77), p. 240. The result is
0 II1
- GRAIUJFXI~ =
0
0 0
0
1 0 0 1
(6.80) 0 0
0 0
The hypothesis has been made that the aerodynamic lateral forces do not depend on the derivative of the aerodynamic sideslip angle ,& and this leads to obtaining a “1” on the first row associated with the lateral force equation. If this hypothesis is not must be recalculated. made, the first term of (11 - GRADFXI~) The inversion of (II1 - (GWmFxii)gives
(6.81)
or
(111 - cl;mP~~FXi~)-l =
1 0
l-m
0 0
0 1
50 0
0
0
0
z o o
1 0 o 0 1 0 0 0 1
j
(6.82)
It can be noted that for the inertial product E = 0, this matrix is equal to the identity matrix 11. Usually, the right term of equation (6.81) which follows the identity matrix II1 is almost equal to zero. The multiplication of the terms on the right of equation (6.79), p. 171 by this matrix only affects the two equations of yaw and roll moment.
6.4 Lateral linearized eauations
173
The matrix GRmFxl,,which is almost the state matrix of the lateral system, takes on the form
(6.83)
The first row is made up of the coefficients that came from the linearization of the lateral force equation (5.202),p. 148. The second row is obtained from the roll moment equation (5.205), p. 148 and the third came from the yaw moment equation (5.206), p. 149. The fourth and fifth rows came from kinematic angular inclination angle equation (5.207), p. 149 and azimuth equation (5.208), p. 149. The matrix (GWrnFx~l~ represents the influence of the longitudinal states on the lateral equations. They take on the following form
(6.84)
The coefficients of the matrix (Cr;wmFxl,and the matrix (GWAIDFXLI~ have the following general form. The first row axp (Equation G.184, p. 312) results from the linearization of the lateral force equation (5.202), p. 148 hPP
=
sin 2aai 2 (qGi - qx;,) sin a,, -~ (-dwEi vai
+
cos2 a a i
+
~z;,
sin2 a,, - 2ry0,,
c o d i - du& sine,) + -(du;, cos Oi - dw& sin ei) Va i COS
Dynamics of Flight: Equations
174 ____________
6 - Linearized equations --__
The first row a;xp (Equation G.184, p. 312) of the matrix GRADFXLI~ relative t o the longitudinal states, is
(6.86) The second row axp (Equation G.196, p. 315) results from the equation of the roll moment (Equation 5.205, p. 148)
The second row axp (Equation G.197, p. 315) relative t o the longitudinal states, is axpv
c1. = 2+piStVai2
axp,,
= 0
axP,
=
-
=1
-
*Pe '
.
A
F, + A-(9,VaiA
COS^,^ sina, - zm sin&)
ri(C - B ) - Ep, A p. se2 2
A Vai ( -ryLi Clp + PY;; Clr,
+ pz;,
CZr,)
(6.88)
The third row axr (Equation G.209, p. 318) results from the linearization of yaw moment equation (Equation 5.206, p. 149)
pise2
axr,
=
1.-
axr4 '
. =1
-
axr+
=
-
2 c
B-A VaiCnp - ~ i - C
p. se2
2 c
p. se2
Vai [Cnp(wzzi - v y Z i ) - qzO,,Cnr, - q x ~ , ~ ~ n r , ]
21 c1 Vai
[qxLiCnp
+ (ux:~ - vyLi)(Cnr,
-
Cnr,)]
(6.89)
6.4 Lateral linearized equations
175
The third row itxr (Equation G.210, p. 318) of the matrix (GWADFXLI~ relative t o the longitudinal states, is
p ( B - A ) - Eri axrq = C (6.90) The fourth row ax4 (Equation G.217, p. 319) results from the linearization of the kinematic bank angle equation (5.207)) p. 149 about initial conditions with a zero bank angle 4i = 0 ax+p
= 0
ax+p
=
ax+,
1 = tanOi
ax++ = qi t a n & ax++ = 0
(6.91)
The fourth row ax4 (Equation G.217, p. 319) of the matrix G . A J D F X relative L ~ ~ t o the longitudinal states, is
(6.92)
The fifth row ax$ (Equation G.220, p. 320) results from the linearization of the kinematic azimuth equation (5.208), p. 149 about initial conditions with a zero bank angle +i = 0
ax$+
=
0
(6.93)
The fifth row ax$ (Equation G.220, p. 320) of the matrix GRADFXL~~ relative to the longitudinal states, is
(6.94) ~
Dynamzcs of Flaght: Equatzons
6 - Linearized equations
176
_______-
[
The matrix of controls @mFuli takes on the following form
GRADFUI~ =
buPl
b r bun
bUPn
b i n ] bupn
(6.95)
With issue from equation (G.176), p. 309
bupl
=
S 4 pi Vai Cy61
bup,
=
$pi-VaiCy6n m
rn
S
(6.96)
from equation (G.192), p. 314
(6.97) from equation (G.205), p. 317
(6.98)
[i i i)
The matrices expressing the atmospheric perturbation have the following form. The wind velocities of translation bwPu
0
(GBPADFwLI~ =
bwpv 0
bwPw
(6.99)
with afterwards equation (G.181), p. 311
(6.100) The wind translation velocities gradients
(6.101)
6.4 Lateral linearized equations
177
with afterwards (Equation G.182, p. 311)
= 0
bwPwz
(6.102)
with afterwards equation (G.199), p. 316
(6.103) with afterwards equation (G.213), p. 319 bwrus = 0 bwrvy = 0 bwrwz
= 0
(6.104)
The wind angular velocities
with afterwards equation (G.183), p. 311
+
S
= - +pi- l ( C y p cos 8, Cyr, sin 8,) m 1 5 bwPPz = -w;, sin aaicos 8, - cos sin 8, - +pi-[Cyr, sin 8, Vai rn bwpq, = 0 bwPqz = 0 1 5 bw& = -- u;, - cos a,,cos Oi - sin sin 8, - i p i-lCyr, cos Oi Vai m
bwpp,
+
bw&
S = - +pi-l(Cyr, cos Oi - Cyp sin Oi) m
(6.106)
with afterwards equation (G.200), p. 316 bwpp,
=
-3-
bWPpz
=
-1-
pise2
A pise2
'
A
Vai(ClpCOS Oi + CZr, sin 8,) VaiCZr, sin 8,
Dynamics of Flight: Equations
6 - Linearized eauations
178
bwp,,
=
-1-‘
bwpry
=
-1-
pise2
A Vai Clr, cos Oi
pise2 Vai (Glr, cos Oi ‘ A
-
CZp sin Oi)
(6.107)
with afterwards equation (G.214), p. 319 bwrpy =
-$L p. se2 Vai (Cnpcos Oi
bwrpz =
--1.-
C
bwrqa: = 0 bwrqz = 0 bwr,.,
=
bwrry =
pise2
‘ C
-$-1- -
pise2
C pise2
‘ C
+ Cnr, sin Oi)
VaiCnr, sin Oi
V a iCnr,
cos Oi
Vai (Cnr, cos Oi
- Cnp sin Oi)
(6.108)
Linearization about a steady state flight without wind If the preceding case is taken, but with the supposition that the initial state corresponds t o a state without wind, for example an equilibrium without wind, then all the wind terms of the type ux&, qx& etc, are zero. As a consequence, the coefficients a X & , axp+ and axr+ cancel each other out and the kinematic azimuth equation is decoupled. The linearized lateral system goes from the fifth order t o the fourth order because the azimuth angle no longer has any influence on the external efforts. Moreover the matrix G R A D F XisL ~ cancelled ~ out, which means that the longitudinal states no longer influence the lateral equations. This last result leads to the conclusion that there is a true decoupling between the longitudinal and lateral equations. The result of section (5.4.3),p. 145 is discovered again. This result is very important because it shows that for a linearization around a steady state flight without wind and with a zero bank angle 4 = 0, the true decoupled lateral equations exist without a special hypothesis on the longitudinal parameters. Among other things, it is not necessary to “pilot” the longitudinal state. Finally, it can be remarked that the coefficients axpd and axrd equally cancel out each other and that the coefficient ax& is reduced to cos Oi. The wind perturbation matrices & m F w u i (Equation 6.99, p. 176) and G R ~ F W G I ~ (Equation 6.101, p. 176) cancel out each other and the aircraft is no longer sensitive t o the wind translation velocities and their gradients.
+
7
Equations for equilibrium The last case of the simplifications of equations begun in Simplified equations (Section 5 , p. 103)) will be developed here. The equilibrium (Section 7, p. 179) is a special case of the general equations of dynamics. These equations of equilibrium generally speaking correspond to the study of the performance of the aircraft. At first, the notions of equilibrium (Section 7.1, p. 180) or pseudo-equilibrium (Section 7.1.2, p. 182) are defined. The equilibrium definition that has been chosen, is the one linked t o the state representation. Thus, the aircraft will be in equilibrium when the derivative of the state vector of the principal system X is equal to zero. Physically, this means that there is an equilibrium when all of the states that have an influence on the external efforts, or the complementary terms of acceleration, are constant. All equilibrium flights correspond to a spiral trajectory such as the turning flight. The rectilinear flight could be considered as a particular spiral trajectory with an infinite radius. The principal pseudo-equilibrium is the climbing flight. In order t o assure that the conditions for the resolution of equilibrium of a linear system are present (Section 7.1.3, p. 182)) it is necessary t o complete the equation system with as many independent equations as there are controls. In general, these conditions can be practically extended to a non-linear system that represents the aircraft. To avoid difficult numerical resolutions, it is desirable to keep in mind the decoupling phenomenon of the longitudinal and lateral movements when choosing the supplementary equations. The equilibrium conditions having been defined, a method for the numerical research of equilibrium (Section 7.2, p. 186) is suggested, based on the linearization of the equation system around any flight situation. This numerical method implemented in Fortran (Section H, p. 321), allows for the research of any kind of equilibrium or pseudo-equilibrium without any special initialisation with a free choice of the supplementary conditions of the equilibrium definition. It equally detects a poor formulation of these equilibrium conditions, for example when the conditions are not independent or when they ignore the decoupling effects. General equilibrium (Section 7.3, p. 188) is evoked when the flat and fixed Earth hypotheses are not made. Within the decoupling frame, longitudinal equilibrium
179
180
7 - Equations for equilibrium
(Section 7.4, p. 188) and lateml equilibrium (Section 7.5, p. 190) are given. These are the simplest equations of the document but they are rich with multiple practical information for the analysis of aircraft flight. However, the exploitation of these equations is not one of the aims of this book. A choice had to be made in organizing the order of the chapters equilibrium and the chapter linearixed equations. As the linearized equations is a simplified system of equations but for the analysis of the dynamics of the aircraft which includes the equilibrium, the equilibrium equations appear as more simplified than the linearized ones. That is the reason for the choice made. The inverse choice should have been made because in general, the equations are linearized around a steady state flight given by equilibrium. This problem does not change anything in the formal writing of the linearized equations, since the differentiation is made around the initial conditions which can be those of equilibrium or others.
7.1
Equilibrium notions
The physical notion of equilibrium is rather intuitive. It corresponds to a stabilized situation where “the elements” do not evolve. The transformation of this idea to a rigorous analytical definition can sometimes lead to some difficulty. The sum of the external efforts equal to zero corresponds to the equilibrium definition usually used. Pure longitudinal flight becomes part of this definition frame when the wind is zero, but the steady state turning flight is not included in this formulation. In fact, in this last case, there exists an acceleration not equal to zero which is “equilibrated” by an aerodynamic force. This case can be treated all the same by placing it in the relative frame and by examining the “relative equilibrium” case seen in this frame. Howerver this type of equilibrium will depend on the choice of the relative frame. The definitions of equilibrium in the literature are numerous. The notion of equilibrium from the state representation given by automatic control scientists will be retained.
7.1.1
Definition of equilibrium
For the system put in the state form (Equation 5.138, p. 133), the following notion of equilibrium’ will be adopted For a system in the state form X = F(X,U) There is equilibrium if X = 0 whatever time t with U = constant This equilibrium is associated with a point of equilibrium, or a singular point, defined by the state vector X, and a control vector U, such as F(Xe,Ue) = 0. For the aircraft, the role that this definition plays will be examined from a practical point of view. With the example of the aircraft, it is clear that if the three kinematic navigational equations are integrated on the representation of the state, the notion of equilibrium will be reduced to the situation of an aircraft in a fixed position with respect to the Earth2. What that means is that the aircraft is on the ground! This ‘Some authors join a stability notion to this definition. It is not the case here. 2The derivative of the geographical position 5 and y has to be zero, as well as the azimuth derivative 21 = 0.
181
7.1 Eauilibrium notions
situation holds no practical interest and experience in flight proves the existence of equilibrated flight situations, for example when the aircraft is cruising. In fact, as has been shown before, the navigational equations have been decoupled (Section 5.4.1, p. 135). Therefore, if the representative state equations used are those of effort and kinematic equations without the navigational equations, a decoupled system is generated. If this decoupled system is equilibrated, very useful kinds of equilibrium are obtained. All equilibrium cases belong to the equilibrium class of steady state level turning flight, with a particular case, the rectilinear steady state flight, which is only a turning flight with an infinite radius! While turning, the azimuth changes # 0, and it is shown more precisely that the derivative of the azimuth is = constant. This is the confirmation of the non-equilibrated navigational azimuth equation. To obtain this equilibrium, the system of equations must be decoupled. The choice of the “dominating” system that needs to be equilibrated, does not cause a problem in the case of the navigational equations, if the physical sense is refered to, when decoupling. It can be remarked that the external efforts do not depend on the navigational states. Equilibrium therefore corresponds to a situation where all the states, that have an influence on the external eforts or the acceleration terms, are constant.
4
4
REMARK 7.1 In the framework where the hypotheses of a flat and fixed Earth are not made, the decoupling of navigational equations can no longer be completely made and the equilibrium will be obtained at constant latitude, that is to say for an East or West azimuth.
The decoupling can be continued. The decoupling between the longitudinal and lateral equation (Section 5.4,p. 132) has been examined. It appears that there could be a lateral equilibrium X i a t = 0 with whatever longitudinal movement, but the opposite is not possible (Section 5.4.2,p. 137). Therefore there are only two possibilities of equilibrium: a particular equilibrium in lateral, or a general longitudinal and lateral equilibrium. A decoupling could be imagined that could be obtained by changing the base of the state. This would allow an association of the equilibrium with each sub-system. As there is no chance that the new states obtained, by these base changes, do have any physical significance, what will become of these new equilibriums? Nevertheless, nothing indicates that a representation of a new physical state cannot be found in association with a new equilibrium. For example, if there is a thrust that is independent of the velocity, a change in the variable between the velocity module and the equivalent velocity Veqsuch as poVA = pV2, would certainly allow the steady state climb to be admitted into the class of equilibrium. In the representations of classical states, this non-zero climb angle flight is not an equilibrated flight but a pseudo-equilibrated flight which will be developed in the following section. These remarks show that there is some difficulty in defining the equilibrium rigorously. This difficulty is, in part, linked to the notion of the decoupling of the system which might be dependent on the base of the state representation which is not unique. An extension of the notion of equilibrium could be made by admitting the orbits into the class of equilibrium, that is to say the periodical trajectories of period T such as X(T + t ) = X ( t ) with a constant control U = constant. Dynamics of Flight: Equations
7 - Equations for equilibrium
182
7.1.2
Pseudo-equilibrium
The notion of “pseudo-quilibrium” or relative equilibrium, is used practically speaking as it corresponds t o a partial equilibrium. What is meant here is a partition of X equal t o zero. The most common example is equilibrium at nonzero climb angle. To obtain this pseudo-equilibrium, the kinematic altitude equation h = V sin y is substracted from the equation system. Thus, it is no longer necessary to force h = 0 and the altitude is not obliged t o stay constant. This approach will accept the case of an aircraft climbing in a nearly constant climb angle y as being in a state of equilibrium. It is not possible to admit just any kind of pseudo-equilibrium since these simplifying hypotheses must be justified either experimentally or theorically. The relinquishment of one or several equations of the system for the resolution of equilibrium produces results close to reality inspite of the reduction of the validity of the model. In the case of constant climb, this is justified by the very slow variation of the air density p in function of the altitude which concludes that p is a local constant. This justification is confirmed by the large time constant of the exponential altitude convergence mode, associated with the kinematic altitude equation. Another case of pseudo-equilibrium deals with the acceleration phase on the ground during takeoff. The moment equation and lift are supposed to be equilibrated. The propulsion equation is “dynamic” with a derivative of the velocity module not equal t o zero. All these equilibriums or pseudo-equilibriums are linked t o the notion of the aircraft’s performance, just as the study of dynamics is associated with the notion of the flying qualities.
7.1.3
The conditions of equilibrium
Here, the question is how to define an equilibrium or an pseudo-equilibrium and the practical consequences that proceed from this definition. In order to do this, the conditions of the resolution of an equilibrated system must be examined. In most cases, the system is strongly non-linear and only a numerical resolution is viable. However some useful information is furnished by the resolution of a linear system. These results could be extended in general to the cases of equilibrium of non-linear systems. Thus a linear system is X
= AxX+BUU+BWW
(74
There is the special case where the wind W and the components of controls U are known around the equilibrium We, U,. The state X in equilibrium is determined by writing the conditions of equilibrium
Xe
=
o
Thus There is only one solution to equilibrium for a value of U, and We. This signifies that when the wind conditions are given, there is only one state of equilibrium for
7.1 Equilibrium notions
183
the position of the controls. Otherwise, for a stable aircraft having a linear system behaviour, it is enough to position the control surfaces to attain the only position of equilibrium, for example, defined by the velocity, the altitude, etc. This equilibrium will be achieved through the modes of the aircraft of which certain are very long and others badly damped. This can thus constitute only a rudimentary means of piloting. However this result still remains fundamental and of great practical importance for the understanding of the behaviour of the aircraft. Then the aircraft, without the pilot, recognizes by itself the vertical position through the spiral mode and the altitude through the altitude convergence mode. It even recognizes the latitude through the navigational modes but with a dynamic so low that it has no pratical sense. In the general cases, it is necessary to find the vector
z =
(E)
(7.3)
such as
x = o The aircraft system is then written
AZAZ = 0 with
AZA = [ A x B u B w ] the dimension of the square matrix Ax the dimension of Bu the dimension of Bw the dimension of AZA is therefore
is is is
nx n nxm, n x m,
nx ( n + m,
+ m,)
There will be a non-trivial solution to the equilibrium if it can be written
AZZ,
=
Z,
with the squared matrix Az non-singular of order (n to zero. The equilibrium solution is given by
(7.6)
+ mu + m,)
and 2, not equal
Therefore, in order to define the equilibrium of the aircraft, the question is to complete the system of aircraft equations AZA,by using the independent equations specifying the values of the state, the controls or the wind in order to obtain Az. The number of these independent equations must be equal to the number of controls increased by the number of wind states. ~
~
~
~~~~~
Dynamics of Flight: Equations
7 - Equations for equilibrium
184
These independant supplementary equations are given by
Thus
In the particular case seen previously (Equation 7.2, p. 182), these m, mentary equations were
+ m,
supple-
In most cases, it is possible to define the equilibrium by the values of the aircraft state. For example, in the longitudinal flight, it is possible to fix the altitude h and the velocity V. In the case of the pseudo-equilibrium, it is taken into account by fixing the first values of Zo to the values not equal to zero. z o
=
(Ze)
t 7.9)
The n first values of Z, and Xe correspond to the derivative of the aircraft state. In the case of equilibrium Xe = 0, for a pseudo-equilibrium, certain of these values can be not equal to zero. For example, in the case of pseudo-equilibrium in a non-zero climb angle y # 0 the kinematic altitude equation is written /Le
= Vesinye
(7.10)
To process this particular case, it is sufficient to fix the value of h e in X e , that is to say to intervene on the first values of Z,. Another met hod of processing the pseudo-equilibrium consists in eliminating certain equations of the system. In the preceding example, it is necessary to eliminate the kinematic altitude equation (7.10) as has been shown in the beginning of this chapter. This elimination of the equation does not modify the number of supplementary equations to add to the system in order to solve the equilibrium. Generally there are four supplementary equations, the number of controls. Thanks to this example, the consequences on the results of equilibrium due to the definition of pseudo-equilibrium will be examined. Take the eliminated equation in the system
The pseudo-equilibrium will free the relationship of equilibrium fi(X) = 0, here Ve sin ye = 0. Thus in this particular case, the climb angle Y e does not have to stay at zero and the climb or the glide of the aircraft can be considered as equilibrated or rather pseudo-equilibrated. For the other equations, the freed constraints are of the type such as the angle of attack/pitch control for the moment equation of pitch, angle of attack/velocity for the equation of sustentation, etc. In the flight situation of pseudo-equilibrium, xi can be a varying state, as in the above example the altitude h
7.1 Eauilabriurn notions
185
is varying. However to define the pseudo-equilibrium, in general, a initial value must be given t o zi. This pseudo-equilibrium will thus be defined for a value of xi, which is no more a state but a parameter of the system of equations. Here the climbing flight will be defined by a value of the climb angle y, the result of the resolution of the equilibrium, however it will be around the initial value of the altitude hi defined previously as a parameter. The altitude h is no longer a state variable but a parameter. With the numerical research of equilibrium (Section 7.2, p. 186) it is shown than only an element of Z (Equation 7.16, p. 187) is a result of equilibrium and as ki = fi(X) is no longer an equation of the system, zi is no longer a state and is no part of Z and not a result of equilibrium resolution. It has been shown that the aircraft system can be decoupled into a longitudinal and lateral system, by means of several hypotheses (Section 5.4, p. 132). When these two systems are rigorously decoupled, the search for equilibrium must take into account an independent lateral equilibrium and longitudinal equilibrium. This remark must not be neglected even in the situation where the two systems are not rigorously decoupled. This decoupling corresponds, nevertheless, to a more or less marked physical reality. If the numerical difficulty of the resolution of equilibrium is t o be avoided, the consequences of decoupling must be taken into account. With the numerical research method of equilibrium (Section 7.2, p. 186), the case of equilibrium with three longitudinal conditions and one lateral, by playing with the coupling, have been nevertheless resolved. Practically speaking, this signifies that two supplementary independent equations will be taken with the longitudinal parameters (for example h = h e , V = ),(I and with the lateral parameters (for example p = p e , 4 = 4 e ) . It must be remembered that the two longitudinal parameters must be defined in order to define the two independent supplementary equations because there are two longitudinal controls, the pitch control and the throttle. There is the same problem for the lateral as there is a control for roll and a control for yaw. If there were a supplementary control, another parameter would have to be defined. Thus for a triplane aircraft with a canard and a horizontal back tail, it is possible to define a supplementary condition for equilibrated flight. Thus while cruising in a classical aircraft, for a given altitude, if the velocity is defined as a supplementary equation of equilibrium, the angle of attack is imposed by the equilibrium. However, for a triplane, it is possible t o define the velocity and the angle of attack independently. This is the supplementary degree of freedom. The resolution of equilibrium is performed in several steps. 0
The choice of what system to equilibrate. the general equations the general equations without the navigational equations the longitudinal or lateral equations
0
The definition of the level of equilibrium.
A true equilibrium by taking X e = 0 in Z,, or a pseudo-equilibrium by taking certain components of X e not equal to zero or by eliminating certain equations from the system. 0
The characteristics of equilibrium. Dynamics of Flight: Equations
7_- - Equations for equilibrium -
_186 ____
The wind being most of the time considered as a known quantity, it is necessary t o determine the m, values of the components of the wind vector We t o equilibrium. The simplest case is equilibrium without wind, with We = 0. The m, supplementary equations which characterize equilibrium. In general, for longitudinal equations, there is a pitch control 6, and thrust control 6, and lateral control, roll 61 and yaw 6,. Therefore it is necessary t o define four independent supplementary equations. To take into account the preceding remark, it is preferable t o take two equations associated with the longitudinal states and two equations associated with the lateral states. For the longitudinal equations, it is possible to fix the altitude h and the velocity V (or the angle of attack a).However it is ill-advised to fix the pitch velocity q because this supplementary equation in the case of pure longitudinal flight is not independent, since the kinematic pitch velocity equation (& + i, = q ) gives q = 0 t o the equilibrium. However it is possible t o define a pseudo-equilibrium as a resource, for example t o a constant angle of attack c i = 0 thus i/ = qe = qe. It is possible t o fix the velocity V and the angle of attack Q as an altitude h in order that the relationship between V and Q through the lift equation will be satisfied. However if a climb or glide pseudoequilibrium flight is looked for by eliminating the altitude kinematic equation, for a given altitude the lift equation cannot be satisfied for any pair of V, a. In this case, the supplementary equations are not independent. For the lateral equations, the sideslip angle ,O and the bank angle 4 or even the yaw velocity T can be fixed. However fixing the roll velocity p must be avoided; it is practically zero in equilibrium and constitutes a weak independent supplementary equation.
7.2
Numerical research of equilibrium
In section (7.1.3), p. 182 the conditions of equilibrium of a linear system have been examined. Here a numerical research method to resolve equilibrium of the non-linear system is proposed based on the results of section (7.1.3), p. 182. The method is based on the linearization of the non-linear system around whatever known initial state denoted “i’’. (7.11) The linearization (Section 6.1, p. 158) is written
AX = A x A X + B U A U + B W A W
(7.12)
with
AX=X-Xi AU=U-Ui
AX=X-Xi AW=W-Wi
The upplementary equations are written Zce
=
CXX+DUU+DWW
(7.13)
187
7.2 Numerical research of equilibrium Linearized, they take the form of
(7.14)
0 = CXAX+DUAU+DWAW
This linearization is used for the research for the solution of equilibrium on the nonlinear system linearization for which
For true equilibrium Xe = 0 for the pseudo-equilibrium a part of Xe can be not equal t o zero, where
AX = Xe-Xi By placing as before (Equation 7.3, p. 183)
(7.16)
Z The equation is obtained
AzAZ
=
(7.17)
AZ,
with
(7.18)
(7.19) and
AZ
(7.20)
= Az-' AZ,
The difference AZ thus foresees the conditions of equilibrium conditions (i )
(e)
from the initial
(7.21) where Ze
=
AZ+Zi
(7.22)
The initial state Zi is a known quantity of the problem. The difference between the estimated state of equilibrium Ze and the initial state Zi, AZ is calculated (Equation 7.20, p. 187) thanks t o the difference between the derivative of the state vector expected Xe and its value at this step Xi. Equilibrium can thus be estimated by equation Dynamics of Flight: Equations
7 - Equations for equilibrium
188
(7.22), p. 187. This process of calculation will be done again until the convergence of the solution, by reinitializing each step Zi by Z e . An example of computer code is available in section (H), p. 321, showing how this method can be numerically implemented. The case treated assumes We = Wi = 0, with the hypotheses of a flat and fixed Earth. In the case of a transport aircraft, this method converges very quickly, whatever the case of equilibrium might be. The convergence is even assured with three supplementary longitudinal equations and only one lateral one; for example, h, a,8 and 0.The solution of the equilibrium depends on the longitudinal and lateral coupling.
7.3
General equilibrium
Here, starting with the general equations (Section 4.4, p. 94), the question is to comment the conditions of the equilibrium of the aircraft with the hypothesis of a spherical, rotating Earth. Equilibrium is defined by the derivative of the state X equal to zero, as in the case for general equations
V ' = V' = Vz . . . p =q =r . h =. ALt. 4 =$ =8
= 0 for the forces equations = 0 for the moments equations = 0 for the kinematic equations of position = 0
for the kinematic angular equations
The kinematic navigational equation (4.139), p. 98 L g G = . . . is not mentioned, since it has been decoupled from the others; LgG does not intervene either in the expression of external forces or in the expression of acceleration terms. The practical and immediate consequences of these conditions of equilibrium define flight at a constant altitude ( h = 0) and a constant latitude (A& = 0). The aircraft will fly in a circle centered on the world axis North-South, in a plan parallel to the equatorial plan and the kinematic equations of position will find V ' = Vz = 0. The values of the other parameters need a longer analysis that is not the purpose of this document. With the flat and fixed Earth hypotheses and gravity independent of the latitude, the kinematic equation is freed from latitude and a generalized equilibrium will correspond to a level turning flight.
7.4
Longitudinal equilibrium
In the framework of decoupling hypotheses (Section 5.4.2, p. 137), and relative to the second form equations (Equation 5.188, p. 143) to (Equation 5.196, p. 144), the longitudinal flight in equilibrium is translated by
This is flight with a constant altitude ( h = 0) and a zero pitch velocity ( q = dr, +qa = 0). Thus this is a rectilinear level steady state flight. The equation (5.188), p. 143 to equation (5.196), p. 144 give the following relationships to equilibrium by integrating
7.4 Longitudinal equilibrium
189
the results of equation (7.23), p. 188. With a field of uniform wind velocity (Equation 5.199, p. 145) the result is
F C O SCOS(CY, ~, - a,) - + ~ S V , ~ C=D m gsiny, F C O S ~ , sin(a, - a,) + ~ ~ S V ~=C m L gCOSya M i v ++pStV;Cm = 0
q = o
(7.24)
If the vertical wind is zero (w: = 0) then the aerodynamic climb angle is zero (?a = 0). Nevertheless, in the frame of a pseudo-equilibrium, it is possible to keep the first three equations propulsion, sustentation and moment, with 7, different than zero. Three hypotheses are often used: 0
0
0
Moderate aerodynamic climb angle such as sin x and cos ?a x 1. This hypothesis is justified for most transport aircraft and allows the decoupling of the propulsion and sustentation equations with respect to the aerodynamic climb angle. Thrust parallel to the aerodynamic velocity, which comes back to imposing a, = a,. This hypothesis justified in cruise flight, allows the decoupling of the propulsion and sustentation equations with respect to the thrust. Thrust moment with respect to the center of mass G zero A4; = 0. It is assumed here that the thrust vector goes through the center of mass. ?his hypothesis has really been verified by most combat aircraft and it is acceptable for transport aircraft.
It can also be noted that on the majority of aircraft, p, hypotheses, the equations are written
x 0. With all these
F - ~ ~ S V Z C= D mgya ~ ~ S V : C L= mg Cm = 0
(7.25)
Clearly stated Thrust minus Drag = Climb angle . Weight Lift = Weight Coefficient of aerodynamic moment = 0 It can be shown that the moment equation C m = 0 is the strongest because it is independent of climb angle ya, mass m , altitude h and velocity Va. This equation gives a relationship that is somewhat linear between the position of the pitch control 6, and the angle of attack a a . I n equilibrium, the stick pilots the angle of attack. The sustentation equation shows that the lift is constant for moderate climb. The C L being linked to the angle of attack for a given altitude p and a given mass, the angle of attack pilots the velocity in equilibrium. Finally, the propulsion equation shows that to have a positive climb angle Y,, the aircraft needs a positive propulsion bilan, thus thrust superior to drag. Dynamics of Flight: Equations
190
?
________
7.5
- Equations for equilibrium
Lateral equilibrium
In the framework of decoupling hypotheses, equilibrated lateral flight (Section 5.4.3, p. 145) is translated by
4
The kinematic angular equation (5.208), p. 149 = . . . can be decoupled if the field of wind velocity is zero; see end of section (6.4), p. 170 and section (5.4), p. 132. In this case, equilibrium is defined by the four first zero derivatives. The equation (5.202), p. 148 to equation (5.208), p. 149 give the relationships t o equilibrium by integrating the results of equation (7.26)) p. 190. With a field of zero wind velocity, the following is obtained mV,(-psina,
rq(C - B) pq(B - A)
+ +
rcosa,) = mg [sin 8 cos a, sin p, cos B(sin 4 cos p, - sin a, cos 4 sin p,)] - +psv,2cy F [cosp, sin p, - sin@, COS P, cos(aa - a,)] -
+
+
+
Epq = +pSl?V:CZ+ Mba
+
Erq = +pSl?V:Cn Mkz p = - tan 8(q sin 6 + r cos 4 )
(7.27)
It must be remembered that equilibrated longitudinal flight (Equation 5.73, p. 118) gives 0 = 0, thus 4 = r tan 4. After integration of this result in equation (7.27), p. 190 it yields
mV,r cos a,( 1
+
tan 8 cos$ t a n 4 =
mg sin 8 cos aa sin O ,, mg cos B(sin 4 cos Pa - sin a, cos 4 sin p,) +psv;cy
F [cospa sin Pm - sin pa cos ,&,cos(a, - a,)]
C-B+E-
cos 4
-
P =
-r-
tan 8 cos 4
(7.28)
It is frequent to adopt the following simplifying hypotheses: 0
The inclination angle of the aircraft is weak 8 M 0, this leads to p x 0 (last part of equation (7.27), p. 190).
0
The angle of attack is weak a, x 0.
0
The engine angle of attack is weak amM 0.
7.5 Lateral equilibrium 0
0
0
191
The engines do not create a roll moment Mka = 0. This hypothesis is well verified. The engines do not create a yaw moment Mk, = 0. This hypothesis is well verified, except for the case of engine failure on a multi-engine aircraft. The sideslipe angles p, and sinp, = p,, sin p, = pm.
pm
being weak, this gives:
COS&
= cospm = 1,
After integration of these hypotheses, the simplified equations are written in the following form
In addition, the fuselage axis xb is usually the principal axis of inertia, then E = 0 and the equation of yaw moment is reduced t o C n = 0.
Dynamics of Flight: Equations
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Part I11
Appendices
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Appendix A
Transformation matrices between frames
A.1
Transformation matrices from frames FI to FE and from FI to F,
The inertial frame FI (Section 2.1.1, p. 14), which is a Galilean frame, is a geocentric inertial axis system. The origin of the frame A being the center of the Earth, the axis “south-north” ZI is carried by the axis of the Earth’s rotation, axis XI and yz keeping a fixed direction in space. The normal Earth-fixed frame FE (Section 2.1.2, p. 15), is linked to the Earth. The origin 0 is a fixed point relative to the Earth and the axis ZE is oriented following the descending direction of gravitational attraction g,. located (Section 4.3.1, p. 82) on 0. This frame is therefore fixed relative to the Earth (Figure 2.7, p. 22). The axis zo of the vehicle-carried normal Earth frame Fo (Section 2.1.3, p. 16) is oriented towards the descending direction of the local gravitational attraction g, in G, the center of gravity of a aircraft. The axis z, is therefore the direction of the gravitation as viewed by the aircraft (Figure 2.7, p. 22). A general transformation matrix is defined for the two cases, the tranformation from FZ to FE,and from Fz t o Fo (Section 2.2.2, p. 21) and (Section 2.2.3, p. 26). lrstrotation -wt about the axis ZI 2nd rotation -Lt about the axis y k = yi
2
X’ cos(-wt)
- sin(-wt)
cos(-wt) 0
195
O 0 1
) ( =
coswt -sinwt 0
sinwt 0 coswt 0 0 1
A - Transformation matrices between frames
196
cos(-;
TLt
=
=(
0 - sin(-; -sinLt 0 cosLt
0 sin(-; - Lt) 1 0 - Lt) 0 cos(-f - Lt) 0 -cosLt 1 0 -sinLt
- Lt)
T i E = Tut T L t =
- sin Lt cos wt
sin Lt sinwt cos Lt
sin wt coswt 0
- cos Lt cos wt
cos Lt sinwt - sin Lt
(A.3)
Transformation from FZ to FE: The latitude is Lt = Lto and the stellar time is wt = wto, then the transformation matrix TiE is TIE. Transformation from FI to Fo: The latitude is Lt = LtG and the stellar time is ,; is Tz, wt = WtG, then the transformation matrix a
A.2
Transformation matrix from frames FE to F,
As defined in equation (2.14), p. 22 and equation (2.15), p. 22 the longitude of G and the latitude of G with respect to 0 are
Thus, the transformation matrix T E from ~ the normal Earth-fixed frame FE (Section 2.1.2, p. 15), to the vehicle-carried normal Earth frame Fo (Section 2.1.3, p. 16) is obtained thanks to the two transformation matrices obtained previously (Section A.1, p. 195)
Tzo =
TEZ
all
=
- sin LtG cos UtG
sin WtG - cos LtG cos WtG sin LtG sin wtG cos WtG cos LtG sin wtG 0 - sin LtG cos LtG - sin Lto cos wto sin Lto sin wto cos Lto 0 sin wto cos wto - cos Lto cos wto cos Lto sin wto - sin Lto
= sin Lto cos wto sin LtG CoswtG
+ sin wtG sin LtG sin Lto sin wto
A.2 Transformation matrix from frames FE to F,
+
a12 a13
197
cos LtG cos Lto = - sin w t sin ~ Lto cos w t o COS w t sin ~ Lto sin w t o = COS L ~ COS G w t sin ~ Lto cos w t o sin Lto sin wto cos LtG sin w t ~ - sin LtG cos Lto
+
+
+
+
a23
= - sin wto sin LtG COS w t ~ cos w t o sin LtG sin w t ~ = sin oto sin w t ~ COS w t cos ~ wto = - sin oto cos L ~ COS G w t ~ cos wto cos L ~ sin G w t ~
a31
=
a32
=
a21
a22
+
COS Lto COS wto
+
sin L ~ cos G W t G + cos Lto sin w t o sin L ~ sin G W
~ G
cos LtG - sin w t COS ~ Lto cos oto + cos w t cos ~ Lto sin w t o = COS Lto COS w t o cos LtG COS w t ~ cos Lto sin wto cos L ~ sin G wtc + sin Lto sin LtG - sin Lto
a33
+
+
Then
+
+
a13
= sin Lto sin L ~ G ( CwOtSo COS w t ~ sin w t sin ~ w t o ) cos LtG cos Lto = sin Lto (cos w t sin ~ w t o - sin w t sin ~ wto) = COS L ~ sin G Lto(cos w t cos ~ oto sin w t o sin w t ~ -) sin LtG cos Lto
a21
= sin LtG(coswt0 sinwtG - sinwto coswto)
a22
=
C O S ( ~ ~ G
a23
=
COS L
a31
=
COS
a32
=
COS Lto
a33
=
COS
all a12
+
--do)
~ sin(wtG G - wto)
Lto sin L ~ G ( Cwto O S COS w t +~ sin w t o sin w t ~ -) sin L t c cos LtG sin(wt0 - W
~ G )
Lto COS L ~ (COS G w t o COS w t +~ sin oto sin w t ~+) sin Lto COS L t c
Finally with COS LgG
sin LgG
+
= C O S ( ~ ~-Gw t o ) = coswto C O S W ~ G sinwtG sinwto = sin(wtG - w t o ) = coswto sinwtG - sinwto cosoto
The transformation matrix TE, is obtained
First row all a12 a13
sin Lto sin LtG cos LgG + cos LtG cos Lto = -sinLtosinLg~ = cos L ~ sin G Lto cos LgG - sin L ~ cos G Lto =
Dynamics of Flight: Equations
A - Transformation matrices between frames
198
Second row
Third row a31
a32 a33
A.3 A.3.1
= cos L to sin LtG cos L ~ G sin L to cos LtG
-cosLtosinLg~ = cos L to cos LtG cos LgG =
+ sin Lto sin LtG
Transformation matrix from frames F, to Fb First angular system
The rotation which allows the transformation of the vehicle-carried normal Earth frame F, to the body frame Fb corresponds to the transformation of the frame determining the orientation of one solid to another. Three angles are necessary (Section 2.2.5, p. 27) lst rotation .1c, azimuth about axis zo 2nd rotation 8 inclination angle about axis yc 3'd rotation 4 bank angle about axis xb
These three transformations are associated with two intermediate frames F, and F f .
Figure A . l : Intermediate frames The frame Fc is deduced from the vehicle-carried normal Earth frame Fo by a rotation of the azimut @ of the aircraft. The frame F,, represents the vehicle-carried normal Earth frame whose axis xc is aligned with the heading of the aircraft. The subscript ('," stands for the course or heading oriented frame. The frame F f is deduced from the course oriented frame F, by a rotation of the inclination angle 8. The subscript ((f" stands for the fuselage oriented frame.
X" = T + X C
xc= TeXf cos$
To
= Tcf =
(
C T e
-sin8
Xf = T#Xb -sin$
0
0
1
;
si;8)
0 cos8
A.3 Transformation matrix from frames F, to Fb
TOT4 = gcb
199
sin 8 sin 4 sin 8 cos 4 cos 4 - sin 4 - sin 8 cos 8 sin 4 cos 8 cos 4 cos 8
Tob = T.$TO T4 = cos 8 cos II, sin 8 sin 4 cos II,- sin II,cos 4 sin II,cos 8 sin 6 sin 4 sin II, cos II,cos 4 - sin 8 cos 8 sin 4
+
A.3.2
cos II,sin 6 cos 4 + sin 4 sin II, sin 8 cos 4 sin 11, - sin 4 cos II,
Second angular system
There exists another system of rotation which is sometimes used lStrotation $ transversal azimuth about the axis z, Znd rotation 4' lateral inclination about the axis xc 3rd rotation 8' pitch angle about the axis Yb
0 COS^' 0 sin$' cos$ sin$
cos 8' cos II,- sin II,sin 8' sin 4' sin II,cos 8' cos II,sin 8' sin 4' - cos 4' sin 8'
+
-sin# cos#
-sin$ cos$ 0
0 1
+
- cos 4' sin II, sin 8' cos II, sin II,sin 4' cos 8' cos 4' cos II, sin 8' sin II,- cos II,sin 4' cos 8'
sin 4'
cos 6' cos +'
Dynamics of Flight: Equations
A - fiansformation matrices between frames
200
A.4 A.4.1
Transformation matrix from frames F, to Fa and from F, to Fh Transformation matrix fkom frames F, to F a
The transformation of the vehicle-carried normal Earth frame Fo to the aerodynamic frame Fa is defined by three angles (Section 2.2.6, p. 31) lst rotation Xa aerodynamic azimuth angle about the axis zo 2nd rotation ?a aerodynamic climb angle about the axis yoia 3'd rotation pa aerodynamic bank angle about the axis X a
X0
COS X a COS Ta
Toa
=
sin x a COS ?a
- sin
A.4.2
COS X a
sin Ta sin p a
- sin x a COS F a
sin X a sin
+
sin pa
+ sin
Xa
sin xa sin Ta - COS X a sin p a
COS p a
COS X a COS pa
COS
sin Ta sin pa
COS p a COS X a
sin
Transformation matrix from F, to
COS pa COS r a
Fk
I
By analogy with the previous process, Tok is obtained with a substitution, in To,, of Yk, pa for pk and Xa for X k .
ya for
A.5 A.5.1
Transformation matrix from frames Fb to Fa and from Fb to Fk Transformation matrix from Fb to Fa
The transformation of the body frame Fb to the aerodynamic frame Fa is, in reality, the transformation of one vector to another, from the fuselage axis vector xb, to the
201
A.6 l+ansformataon matrix from frames Fk to Fa
aerodynamic velocity vector xa. The axis x, is carried by the aerodynamic velocity Thus only two rotations will be necessary lStrotation -cYa about the right wing axis Yb 2nd rotation Pa about the axis z , = zi
Va.
cosp,
T a a=
A.5.2
1
sincu,
0
COSCY~
- sin @a COS a,
- sin cua
COS P a
- sin
sin
Transformation matrix from Fb to
By analogy with the previous process, for cYk and P a for @ k .
A.6
0
0 -sincy,
sin @a COS ,& sin
=
0
coscy,
COS cya COS P a
Tba
-sinp,
Tkb
COS
Fk
is obtained with a substitution, in
Tab,
of
Transformation matrix from frames F' to Fa
The transformation of the kinematic frame Fk to the aerodynamic frame Fa will allow the kinematic velocity Vk to be connected to the aerodynamic velocity V , . These two velocities axe made up with the wind velocity V,. Therefore, it is not surprising to see the angles for "wind" indication appear. The transformation from frames Fk to Fa is accomplished with three rotations (Section 2.2.9, p. 35) lst rotation -a, wind angle of attack about the axis Yk 2nd rotation P,, wind sideslip angle about the axis zkio 3'd rotation p, wind bank angle about the axis Xa
TPw
=
1 0 0 cosp, 0 sinp,
-sinp, cosp,
Dynamics of Flight: Equations
A - Transformation matrices between frames
202
cosa, 0 sina,
Y a w=
cos a , cos p,
Tka
- cos a , sin 0 , cos p, - sin a,,,sin p,
cos a , sin p, sin p, - sin a , cos p,
cos p, cos p,
- sin p, cos 0 ,
- sin a , sin pWcos p, cos a , sin p,,,
sin a , sin pw sin p, cos a , cos p w
sin pW
=
sin awcos p,
0 -sins, 1 0 cosa,
+
+
The transformation of Fk to Fa is therefore defined by a , and pw, wind angle of attack and wind sideslip angle. These angles could have been defined by the inverse transformation Fa to Fk, or by the inversion of the order of rotations ( a , and pW).It is a question of convention and as for example, for an inversion on the two rotations a , and pw, this gives
cos a , cos p,
- sin p, cos p, - sin p, sin a , cos p,
cos a , sin p, sin a ,
A.7
sin p, sin p,
- cos p, sin a , cos p,
- sin p, sin a , sin p,
cos p w cos p w
- sin p, cos 0 , - cos p, sin awsin p,
sin p,,, cos crw
cos a , sin p,
Probe angle of attack and sideslip angle
h n s f o r m a t i o n matrix from the bodp frame Fb to the probe frame Fa The probe for the measurement of angle of attack and sideslip angle is usually mounted with a rotation axis parallel to the body axis zb (Section C.5, p. 220). This leads to the following transformation matrix
Xb
=
Tp,T-,;, xa= Tba xa
COSPL, - sinpas
Tb,
=
(
0 cosa;,
O
sina;,
0
0
0 -shahs; 0
rotation about
zb
rotation about Y a
COSCY~,
cos a:, cos pLs - sin cosa;, sinp;, cospL, 0 sin a;,
@As
- sin ahs cos pis
- sina;, sin@:, cos a;,
A.7 Probe angle of attack and sideslip angle
203
Due to the particular rotation axis zb and particular transformation from Fb to Fa, the angle of attack and sideslip angle measured by the probe ahs, are not exactly conventional as defined in section (A.5), p. 200. This particular rotation axis leads to an inversion in the order of rotation between cy and p. The relationships between these two sets of angles are calculated in section (C.5), p. 220.
@LS
Dynamics of Flight: Equations
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Appendix B
Angular relationships
B.l
Relations between angles of attack and sideslip angles
The matrices of transformation between the aircraft body frame Fb and the aerodynamic and kinematic frames F, and Fk, are linked by the following relationship
The calculus of recalled
is not completely developed here but the Tba structure is
'II'bk'II'k,
cosa,cosp, sin@, cosp,sina,
=
Tba
-sina, 0 cosaa
X cosp, X
The element on the first column and second row yields
for a, = 0, sinp, = sin@
+p,) and as -f 5 p 5 f
The element on the first column and third row yields COS
Pa sin a,
= sin arc cos pk cos awcos Pw - sin p, sin a k sin P k
for a, = 0 then cosp, sina, = sinak cos(pk sina, = sinak; then
a,
=
ak
for
a, = O 205
+ p,) if
+ cos a k sin a , cos pw (B.3)
and thanks to equation (B.2) lr
lr
c the column i of the state derivative matrix Adot is fulfilled do j=l,dimstate Adot(j,i)=(Xdot(j)-XOdot(j))/dXdot(i) enddo c the initial state is restored Xdot (i)=XOdot(i) enddo c the last state is restored DLtdot=Xdot(dimstate) C==O=P=O=D=PI======'============'='='==============
c the last calculation before going out C=='=IP'='=P======================'=r='==============
call calcul-state-associated
G . l Numerical linearization
281
c======================t====l=============================
c c c c
Saving Saving Saving Saving
of of of of
Alin Blin Clin Dlin
in in in in
the the the the
file file file file
fileAlin fileBlin fileclin fileDlin
...........................................................
open(l83,file='/edika/files/fileAlin',status='old'~ open(l84,file='/edika/files/fileBlin',status='old'~ open(l85,file='/edika/files/fileClin',status='old') open(l86,file='/edika/files/fileDlin',status='old'~ do j=l,dimstate do i=l,dimstate write(l83,1203) Alin(i, j) enddo enddo do j=l,diminput do i=l,dimstate write(l84,1203) Blin(i, j) enddo enddo do j=l,dimstate do i=l,dimoutput write(l85,1203) Clin(i, j) enddo enddo do j=l,diminput do i=l,dimoutput write (186,1203) Dlin(i, j enddo enddo 1203 format(8(1x,el7.11)) close (183) close (184) close (185) close (186)
Dynamics of Flight: Equations
G - Linearized equations
282 enddo enddo do i=l,dims,r do j=l,dime-r Clin-r(i,j)=clin(i,j) enddo enddo do i=l,dims-r do j=l,diminput Dlin-r(i,j)=Dlin(i,j) enddo enddo endif return end C****************************************************
subroutine equivalence(Xdot,Y) C****************************************************
c this subroutine is for the transformation from c the explicit variables (pdot, qdot, etc c to the state and output vector Xdot and Y C****************************************************
c* Declarations
*
C****************************************************
include '/edika/libincl' double precision Xdot (dimstate),Y(dimoutput) c===================================pP=================
Xdot(l)=usolGgdot Xdot(2)=vsolGgdot Xdot(3)=wsolGgdot Xdot (4) =pdot Xdot (5)=qdot Xdot (6)=rdot Xdot(7)=altitudedot Xdot(8)=tetadot Xdot (9)=phidot c supplementary states Xdot (10)=psidot Xdot(ll)=DLtdot Y(l)=altitude Y (2) =mach Y(3)=alpha Y (4)=beta Y (5) =p Y (6)=q Y (7)=r Y (8)=psi Y (9)=teta
G.2 Wind velocitg field linearization
G.2
283
Wind velocity field linearization
Wind is defined in the vehicle-carried normal Earth frame F,; the problem is t o express the linearized form of the wind in the body frame Fb. The expression of the wind velocity field (&mVL) is given in section (D.6), p. 238.
Linearization with respect to the azimuth $ The linearization with respect t o the azimuth $ in a turning flight situation presents a validity domain that is obviously limited. Linearization of equation (D.63), p. 239
Linearization of equation (D.64), p. 239
Linearization of equation (D.65), p. 239
Wind defined at the initial azimuth Dynamics of Flight: Equations
284
G - Linearized equations
The generality of the problem is not affected if the wind is supposed to be known in the vehicle- carried normal Earth frame F,, oriented by the initial azimuth of the aircraft $, (Section A.3.1, p. 198). It comes down in the previous equations to take $, = 0, so
AUX; = Aux; + (rx;, - ry”,)All, AVY: = Avy; + (ry& - rxEi)All, A W Z ~= AwzL AP: A& Ary; Am: Aqx; ArxL
= A&,
+ qz;,All,
= Aqz; - pzziAll, = Ary; (uxO,, - vy;,)A$
+ + &,,,All,
= Am: = Aqx; - m & A $ = Arx; (vy& - ux”,)All,
+
(G.10) (G.ll) (G.12) (G. 13) (G.14) (G.15) (G. 16) (G.17) (G.18)
and also (G.19)
Linearization with respect to the inclination angle 8 Linearization of equation (D.69), p. 239 taking into account the results of equation (G.10), p. 284 to equation (G.18), p. 284
Auxf, = AUX; cos28, + ( r X & - r y & ) cos2 &A$ + AwZ; sin2 Oi + (Aqx; - Aqz; - (py& - p.Li)A$) sin 8, COS 8, + (sin 2Oi(wZEi- uX;,) - COS 28,(qz;, - qxLi))A8 Avyf, = AUY; + (ry& - rX;,)A$ AwzL = Aux; sin2 8, + (rXO,, - r y & ) sin’ 8,A$ + AwzL cos’ 8, + (Aqz; - Aqx; - (pZ& - m;,)A$) sin 8, COS 8,
+ (- sin28,(w~;, - uXzi)+ c 0 s 2 8 , ( q ~ ;-~ qx”,)) A8
(G.20) (G.21)
(G.22)
Linearization of equation (D.70), p. 240
AqzL = ( AqZ; - pZiA$) cos2 8, + (Aqx; - m;, A$) sin2 8, + (AuxG - AWZ; ( r X & - ry;,)A$) sin@,cos@, - (COS 28i(wz;i - uX;,) + sin ae,(qz;i - &,,)) A8
+
(G.24)
285
G.2 Wind velocity field linearixation Aryf
+
= Ary; cos 8, (uXii- uy;,)A11, cos 8, +(Am; q~;, A?)) sin 8, ( -rY&sin 8,
+
+
+ mzi cos &)A8
(G.25)
Linearization of equation (D.71), p. 240
Amf
= (Am; -
Aqxi
Arxf
=
=
+ q~;, A$) cos 8, - (Ary; + (ux;,
(m:, sin 8, + ry;,
- uyzi)A?))sin 8,
COS &)a8
(G.26)
+
(Aqx; - m;, A@)cos2 8, (Aqz; - pZziA$) sin2 8, (Awz; - nux; - ( r X O , , - ryO,,)A11,)sine, c o d ,
+ + (cos 28,(wZii- uXEi)+ sin 28, (qZ& - qXLi))A8
+ (uY& - U X " , ) ) ~ ?cos8, )) +(Am; + &,,A$) sine, + (-rX;, (Arx;
Linearization with respect to the bank angle Linearization of equation (D.75), p. 240
(G.27)
sine,
+ p.;, cos8,)AB
sin 4,
+ r y L i cos &)A@ (G.33)
(G.28)
4
Linearization of equation (D.76), p. 240
AqZL Ary;
= Aqzi cos 4,
+ Aryf sin 4, + (-q&
= A r y i cos $i - Aqzf sin 4, - ( r y f , sin 4,
+ 4.1,
cos 4,)A+
(G.34)
Linearization of equation (D.77), p. 240
Aqx;
= AqxL cos 4,
+ Arxf sin 4, + ( -qxfi
sin 4,
+ r x f , cos $,)A+
(G.36)
Dynamics of Flight: Equations
G - Lineam'zed equations
286
Arx;
=
Arxf, cos 4, - Aqxf, sin 4, - ( r x f , sin 4,
+ qxf,, cos +,)A4
(G.37)
In order t o simplify the writing of the equations, the following notations are introduced
13; = PY; g)= qz;
- qx;
?: = rx;
- ry;
and and and and and and
- P.;
ii; = UX; - uy; 6; = uy; - wz; 6; = wz0, - ux;
AC; = Am;
- Apz;
A@; = Aqz; A?: = Arx; Aii; = Aux; A$; = Auy; ACE = Awz;
- Aqx; - Ary; - nuy; - Awz; -
Aux;
(G.38) (G.39) (G .40) (G.41) (G.42) (G.43)
Wind velocity Aeld linearization The expression of the linearized components of the wind gradient (GwmV, expressed in the body frame Fb as a function of the components expressed in the vehiclecarried normal Earth frame F,, oriented towards the initial azimuth, is finally calculated from the equation (G.29), p. 285 t o equation (G.37), p. 286 in which the equation (G.20), p. 284 t o equation (G.28), p. 285 are taken in account.
+ +
AuXL = AuX; cos2 8, Aw.; sin2 8, - AGE sine, cos6, +aux4,A+ aux+,All, auxe,A€J
+
(G.44)
with
Auy;
aux4,
= 0
aux+, auxe,
=
-&,
sin 8, COS 8,
+,?;
cos2 8,
= 6:,sin28, - @Zi~ 0 ~ 2 8 ,
+ Auy0, cos2 4, + Awz; cos2 8, sin2 4, +A& cos 8, sin 4, cos 4, + A@),sin2 4, cos 8, sin 8, + A?: sin 8, sin 4, cos 4, (G.45) +avyd, A 4 + avy+, A$ + avye, A8
= AUX; sin2 4, sin2 8,
with avYd,
=
avy+,
=
avye,
=
+
(sin 24,(wZO,,cos2 ei - uY& uX;, sin2 ei COS 24, @ ( ,; COS 8, ?Zi sin 8,))
+
+
+
sin 8,COS e,)
?Gi (- cos2 4, + sin2 8, sin2 4,) + fi;, sin 8, cos e, sin2 4, +Ei cos 8, sin 4, cos 4, - 2iiLi sin 8, sin 4, cos 4, sin 4, (sin 4, ( -G& sin 28, + @Eicos 28,) + cos 4, (-jj;, sin 8, + ?Ei COS e,))
AWZL = Aux; sin2 8, sin2 4, + AuyL sin2 4, + AwZ; cos2 8, cos2 4,
-A& cos 8, sin 4, COS 4, +A@),sin 8, cos 8, cos2 4, - A?: sin 8, sin 4, COS 4, +awzd,A+
+ awz+,AlC, + awze,AO
(G.46)
G.2 Wind velocitv field linearization
287
with = sin 24,(vyO,, - wz;,cos2 8, - uXzisin2 8, - tj;, sin 8, cos 8,)
awZ4,
+,?; sin 8 ; ) sin 8, COS 8, cos2 4, + q;, COS 8, sin 4, COS 4,
- cos 24, (fi& cos 8,
fi;,
awZ+,
=
awze,
= cos 4, (sin 4, (fi& sin 8, - F;, cos 8,)
+?Ji(sin2 8, cos2 4, - sin2 4,)
Apz,b
+
=
COS
4,( -G;, sin 28, +, @;
+ 2ii;,
sin 8, sin 4, cos 4,
COS 28,))
sin 24, (-nux: sin2 8, AvYL - AwZO,cos2 8,) 2 cos Oi(cos2diApZL sin2 +,Am;)
+ +
+
A G sin 28, sin 24, - sin 8,( ArxL cos2 4, + AryL sin2 4,) - +apz4,A+
+ apz+,A$ + apze,A8
4
(G.47)
with
=
apz+w
%, -sin 28, sin 24, + cos 8,(qXzisin2 4, + qzLi cos2 4,) 4
+ sin2 8,) + ii;, sin 8, cos 24, - cos 8, (rX& cos 24, + ry;, sin 24,) sin 8,(pZ& cos 24, + py;, -?& sin 4, cos 4,(1
apZew =
- sin 4, cos 4, (
-*:,
sin 28,
+ Q;,
-
COS 28,)
sin 24,)
Aqz& = - AG; sin 8, cos 8, cos 4, + Am; sin 8, sin 4, cos 4,( AqzL cos2 8, Aqx: sin2 8,) + Ary; cos 8, sin 4, +aqz#,A$ aqz@,All, aqze,A8
+
+
+
+
(G.48)
with
aqz4,
=
aqzrow
=
sin 4, (qz;, cos2 8, + qX;, sin2 8, - t~;, sin 8, cos 8,) + cos 4,(r&, cos 8, + p&, sin 8,) - cos 4, (mLisin2 ei + pzzi cos2 8,) + qxzi sin ei sin 4, -
+?,;
aqze,
sin 8, cos 8, cos 4,
= sin 4,( -ry;,
sin 8,
+ ii;,
COS 8, sin
4,
+ py;, cos 0,) - cos 4i(GJ,cos 28, +
sin 28,)
@Ji
sin 28, Ary& = AGJ- 2 sin 4,
+ Am: sin 8, cos 4, - sin 4, (AqzL cos28, + Aqx; sin2 8,) + Aryg cos 8, cos 4,
+ary4,A@
+ ary+,A$ + arye,A8
(G.49) Dynamics of Flight: Equations
G - Linearized equations
288 with
+ py;, sin 8,) - COS 4, (qz& cos28, + qX;, sin2 ei - G ; ~sin 8, COS 8,) sin 4,(p.Eicos2 8, + pyci sin2 8,) + qX;, sin 8, cos 4, -?& sin 8, COS 8, sin 4, + ii;, COS 8, COS 4, cos q5i ( -ry;, sin 8, + py;, cos 8,) + sin di(+E, sin 28, + GZi cos 28,)
ary4,
= - sin 4,(ry& cos 8,
arY+,
=
arye,
=
-
Am:
sin (Aux; sin2 8, - AvY; Aw,; cos2 8,) 2 cos Oi (Am: cos2 $i Apz; sin2 4,) A 0 +sin 28, sin 24, - sin 8,(Ary; cos2 4, + Ayx: sin2 4,) 4 (G.50) + a p ~ + , A 4 apy+,All, apye,A8
+
+
+
+
+
with
+
apy4,
= - sin 24,@ ( ,; cos 8, FE, sin 8,) COS 24, (uX;, sin2 8, wZ;, cos2 8,
apyll,
=
+
+
+ ,+; sin 8, COS 8, - vY;,) @o,, -sin 24, sin 28, + cos qqX;, cos2 4, + qz;, sin2 4,) 4 +.?;2 (1 + sin2 8,) sin 24, - ii;, sin 8,cos 24, - cos O,(ry$, cos 24, + rxEi sin 24,) - sin 8,(&,, cos 24, + p.& + sin 4, cos 4,( -G;, sin 28, + &, 28,) AG; sin 8, cos 8, cos 4, + Am: sin 8, sin 4, + cos 4,(Aqx; cos2 8, + AqzL sin2 8,) + Arx: COS 8, sin 4, 2
apye,
=
sin 24,)
COS
Aq&
=
+aqx4,A$
+ aqX+,A$ + aqxe,A8
(G.51)
with
+ + G;, sin 8, cos 8, + + cos 4, (m;, cos2 8, + p:, sin2 8,) + qZLisin 8, sin 4,
aqx4,
=
- sin 4, (qx;,
aqx+,
=
-
aqxe,
-?,; sin 8, cos 8, COS $, - ii;, COS 8, sin 4, = sin 4, ( - r X O , , sin 8, &, cos 8,) COS 4, (G:, sin 28, tij;, COS 28,)
Arx;
cos2 8, qZ:, sin2 8, cos 4,(rXLicos 8, py;, sine,))
+ +
+
=
-AGE sin 8, cos 8, sin 4, + ApyL sin Oi cos 4i - sin 4,(Aqx; cos2 8, Aqz; sin2 8,) Arx; cos 8, cos 4, +arxd,A+ arx+,All, + arxe,A8
+
+
+
(G.52)
G.2 Wind velocity field linearization
289
with
+ + cos2 ei +
- COS 4,(qX:,
+ tij:,
aw,
=
arw,
= sin 4, (py& p Z z i sin2 8,) qZ:, sin e, cos 4, +?$ sin 8, COS 8, sin 4, - ii; COS 8; COS 4, = COS 4; ( -rx& sin 8, p:, COS 8,) - sin 4, (gii sin 28, tij;, COS 28,)
arxe,
cos28, qZ:, sin2 8, - sin 4; ( r X & COS 8, pz& sin 8,)
sin 8, COS 8,)
+
+ +
Wind velocity field linearization relative to an initial bank angle equal to zero 4, = 0
Back t o the equation (G.44), p. 286 t o equation (G,52), p. 288 and setting zero
4, to
AUX; = nux; cos2 8, + Aw,; sin2 8, - Ag; sin 8; COS Oi
+ auX+,A$ + auxe,A8
+aux4,A+
(G.53)
with = 0
aux4, auxllr, auxe,
+
-fiti
= sin 8, COS 8, F;, cos28, = 6;sin 28, - GL COS 28,
AvYL = AvyL
+ avy4, A 4 +- avyq, A$
+- avyewAB
(G.54)
with =
awllr,
= -F& = 0
avye, AwZ;
fit,cos Oi + ?Ei sin 8,
avy4,
= AwZLcos2 8,
+ AGE sine, cos 8, + awz#,A+ + awZ+,A$ + awze,A8 (G.55)
with awz4,
=
- COS e,fi& - sin Of&
+ ?zi
awZ+, = f i sin ~ e,~ cos 8, sin2 ei awzew = -tij& sin 28, QZ,COS 28;
+
ApzL
= ApzL cos Oi - Arx; sin 8,
+ apz4,A4 + apZQwA$ + apze,,A8 (G.56) Dynamics of Flight: Equations
290
G - Linearized equations
with apz4, = VY;, - uXO,, sin2 8, - wZ;,cos2 8, - ,j;, sin 8, cos Oi 0 qzWi c o d i iiki sine, aPZ+W apze, = -rX;, cos 8, - pZLisin 8,
+
Aq&
+ Aqz; cos2 ei + Aqx; + aqz+,A$ + aqze,A8
-AGg sin 8, cos Oi
=
+aqzd,A+
sin2 Oi (G.57)
with
Ary;
+ py;,
=
ry;,
aqz+, aqze,
=
-py;, sin2 8, - p z ; , cos2 8,
=
-G;,
= A p ~ sin t 8,
cos Oi
sin 8,
aqz4,
COS 28, -
+ Ary;
&, sin 28,
+ F ; ~ sin 8, COS 8,
+ aryd,A+ + ary+,A$ + arye,A8
cos 8;
(G.58) with -qZ;, cos2 8, - qX;, sin2
ary4, ary+,
= qXZisine,
arye,
=
=
+ ii;, cos& + p~;, cos 8,
ei + G;, sin 8, cos 8,
-rY&sin 8,
= Apy; cos 4 - ArY; sin 8,
+ apy#,A+ + apy$,A$ + apys,A8(G.59)
with apyd, spy+,
+
= uX;, sin2 8, wz;, cos2 8, = qxwi O COS^, - ii;, sinei
aPY@, = AqxL
-ry&
+ @E,sin 8, COS ei - v ~ o , ,
cos8, - m;, sine,
+
+
= AG; sin Bi cos Oi Aqx; cos2 Oi A& sin2 8, +aqx$,A+ aqx+,A$ aqxe,AB
+
+
(G.60)
with aqx4,
= rXZicos 8,
+ py;
sin 8,
aqx+w = -py;, cos2 ei - pZLisin2 8, - FE, sin ei COS ei aqxe, = &i sin28, tijEi cos28,
+
Arxk
= Apy; sin 8,
+ Arx;
cos 8,
+ arx4,A+ + arx+,A$ + arxe,A8
(G.61)
G.2 Wind velocity field linearixation
291
with -qxLi cos2 8; - qZGisin2 = qzLi sin 8, - ii; cos Bi = - r X L i sin 8, pZLicos Bi
arX4, arX+, arxe,
=
ei - zijki sin ei cos 8,
+
Wind velocity Aeld linearization relative to an initial inclination angle equal to zero Oi = 0 Back t o the equation (G.44)) p. 286 to equation (G.52)) p. 288 and setting 8, t o zero = Aux;
Aux:
+ aux4,A+ + aux+wA$ + aux6,Ae
(G .62)
with
=
Avyb,
aux4,
= 0
aux+, aux6,
= ,? : = -qwi -0
+ +
+
Avy; cos2 4, AWZ; sin2 qhi AfiZ sin 4i cos 4, +avy$,A+ avy+,All) avye,A8
+
(G.63)
with
(G.64) with
Apzb,
sin 24, = 2 (Avy; - Awz;)
+ Apz: cos2 4; + Am; +apz@, A 4 + apz+, A$ + apze, A8
with aPZ4w
=
apz+w apze,
-
=
fiz,sin 244 + ij;, q X G i sin
2
+i
sin2 4, (G.65)
cos 24,
+ qzGi cos2 4i -,?:
sin 4, cos 4,
-rxLi cos 24, - r y L i sin 24, -, @: sin 4, cos 4, cos 28, Dynamics of Flight: Equations
292
G - Linearized eauations
with
aqz4, aqz$, aqze, AryL
=
- sin 4,Aqz;
= =
sin 4, -pzLi cos 4i -qz&
+
+
cos #+ GO,,sin 4,
TYO,,
= pyzi sin 4i - t5Li cos 4i
+ AryO, cos 4, + ary+,A+ + ary+,A$ + arye,AO (G .67)
with
(G.68) with apy4,
=
apyrl,
=
+ ij;, qx& cos2 4, + qz;,
apyew
=
-ryzi
-Pzi
sin 2&
cos 24i
?Gi +2 sin24, sin 24, + tjz,i sin cos q!+
sin2 4,
cos 2 4 - T X &
$i
with aqx4,
=
-qx& sin 4i
+ rXzicos 4,
aqxqw = -py& COS^, - ii& sin#, aqxe, = pzisin +i t5zi cos 4,
+
Ar& with
=
- A & , sin +i
(G.70)
+ Arxz cos 4, + arx$,A4 + arx$,A$ + arxeWAqG.7l)
G.2 W i n d velocity field linearization
293
Wind velocity field linearization relative to an initial bank angle and inclination angle equal to zero 4i = 8, = 0 Back to the equation (G.44)) p. 286 t o equation (G.52)) p. 288 and setting the initial bank angle 4, and inclination angle 8, to zero, the expressions will be simpler. These relations can be considered as the simplest relation of the wind linearization and can be used in the situation of a steady state rectilinear flight (G.72) (G.73) (G.74) (G.75) (G.76) (G.77) (G.78) (G.79) (G.80)
Linearization of dv; The term dv; is the second component of DVZ = (C~;;WADV~V,)~ (Equation 5.49, p. 113). This term appears in the lateral force equation (5.203), p. 148 and is made of three terms associated with the three components of (C~;;WADV~V,)~ (Equation 5.48, p. 113).
Advi Adv&
=
Adv&
+ Adv& + Advzw
(G.81)
-
AduO,
+du&
+du&
(- sin Pai COS a,, COS ei COS +i + cos Pai(sin Oi sin 4i cos $, - sin $J~ COS 4i) - sin a,,sin Psi (cos qisin 8, cos 4, + sin 4, sin $ i ) )
(- cos PO,cos a a i cos ei cos ll,, - sin PO,(sin Bi sin 4, cos $J,- sin qi COS 4i) - sin a,, cos Pai(cos ll,,sin 8, cos 4, + sin 4, sin ll,,)) A@, (sin Pai cos a,, cos 8, sin $, - cos Pai(sin Oi sin 4, sin I), + COS qi COS 4i) - sin a,; sin P,, (- sin ll,i sin 8, cos 4, + sin 4, cos $+)) All,
+
+dugi
(cos Pai(sin 8, cos 4, cos qi sin t)i sin 4i) - sin a,, sin PO,(- cos ll,i sin Oi sin 4, cos 4, sin t,bi)> A 4
+du&
(sin Pai cos aaisin 8; COS qi cos Pai cos Oi sin 4, cos ll,i - sin aaisin Pai COS ll,i cos Oi COS + i ) A8
+du&
+
+
(sin Paisin a,, cos 8, COS $i - cos aaisin
Pai(cos
Qi
sin Oi cos 4,
+ sin 4, sin ll,,)) ACW,
(G.82)
Dynamics of Flight: Equations
294
G
Advz (- sin Paicos a,, sin $, cos ei + cos (sin 8, sin +, sin $i cos $i cos 4,) - sin a,,sin PO,(sin 8, cos 4, sin $, - sin +i COS$1)
=
A&:"
+
(-
+du&
COS
Paicos a a i sin $! cos 8,
- sin Fa,(sin 8, sin 4, sin $,
+ cos $, cos 4,)
- sin a,, cos Pai (sin 8, cos +i sin $! - sin
+, cos $,)) A@,
(- sin Pai cos a,, cos $, cos 8,
+dug,
f cos psi(sin 8, sin 4, cos $, - sin $, cos + i ) - sin a,, sin Pai (sin 0, cos
+, cos qi + sin 4, sin $,)) A$
cos c$i sin $, - cos $, sin 4,)
+dv&
(COS Pai (sin vOi
+du&
(sin PO,cos a,, sin $, sin 9, COS$^,, cos@,sin sin $, - sin a,i sin Pai cos 8, cos (sin P,, sin a,, sin $i cos 8,
+ sin a,, sin Pai(sin 8, sin +, sin $, t cos 4, cos $,)) A+
+,
+
+dv&
- cos a,, sin Pai(sin 8, cos 4, sin $, - sin
-
Adv&
- Lineam'zed equations
Adwz
(sin Pai cos a,, sin Oi
cai
(G.83)
sin Pai cos 8, cos 4,)
- sin a,; cos paicos 0, COS c$,)AP,
+
+dw& +dw& +dw&
+, cos $,)) Aa,
+ cos Paicos 8, sin 4, - sin
(cos Pai cos a,, sin 8, - sin Paicos 8, sin +,
+dw&
+, sin $,) A8
(cos Pai cos 8, cos 4, sin a,,sin PO,cos 8, sin +,) 4 4 sin 8, sin 4i (sin Pai cos a,, cos 8, - cos sin a,, sin sin 8,cos di)A8 (- sin Pal sin a,, sin Oi - cos a,, sin PO,cos 8, cos 4i)Aaa
+ ~
(G.84)
Linearization of d v t relative to an initial sideslip angle and azirr-9th angle equal to zero pi = $, = 0 The simplification of the expressim du; is relevant when the initial sideslip angle Pi and azimuth angle $, are equal t o zero. Eventually, these conditions are usually achievable. So Adu&
+ sin a,, sin 8, cos -A$ du& cos +, + A 4 du;, sin 8; cos 4, + A0 du& cos 8; sin +, Adut COS 4, + AD, du& sin aOisin 4i + A$ du& sin Oi sin 4,
= Aduz sin 8, sin 4i - A@, du;, (cos a,,cos 0,
A~u& =
+i)
(G.85j
-A+ du& sin 4; (G.86) Advzw = Adwz cos 8, sin 4, A@, dwzi (cos a,,sin 8, - sin aai cos 8, cos 4,) +A@dw& cos 8, cos 4, - A0 dw& sin 8, sin 4, (G.87)
+
Linearization of dut- relative to an initial sideslip angle, azimuth angle and bank angle equal to zero Pi = $, = = 0
+,
G.3 Linearization of the longitudinal equations
295
The previous equation (G.85) to equation (G.87) with the bank angle equal to zero
+, = 0 , yield
Furthermore, if the the initial inclination angle is equal to zero, Oi = 0, then COS a a i Apa - duo,,All, Adv& = Adv& = A d v i Advtw = -dw$i sin aaiApa + dw& A+
Finally the expression of dv;
(G.91) (G.92)
(G .93)
equation (G.81) appears as
(G.94) The linearization of Adu;,
Adwg, Adw;
equation (5.48), p. 113 yields
(G .95) (G.96) (G.97)
G .3
Linearization of the longitudinal equations
G.3.1
Linearization of the propulsion equation
The propulsion equation (5.188), p. 143 below, expressed in the aerodynamic frame Fa, is linearized
From equation (4.104), p. 92 the term cosa,F,b F cos(aa - a,) COS Pm.
+ s i n a a F i can be written as
Linearization of the acceleration terms
Dynamics of Flight: Equations
296
G - Linearized equations
and the reduced velocities (Equation 6.16, p. 161) to (Equation 6.19, p. 162) V a , etc, are introduced by dividing each term by the initial velocity Vai, then
Linearization of the external forces - mgcosyai Ay, - !jphiSVzCDiAh - piSvaiCDiAVa
(CDb,Mi(AVae iThiAh) + CDCYaACYa + CDqAqt-3piSV: CDduaAdr- + C ~ b m A 8 m ) ( Va -F, sin(aai - a,) cos/3,Aao -$piSV:
-
i
+F,
COS(cYa;
- a,)
3/,
COS
(PhiAh + AAVa +
(G.lO1)
with equation (6.33), p. 164
Aqi = Aq
- Aqx,b
(G.102)
The drag coefficient CD is often modelized by
CO = COO+ICCL~ Then the angle of attack
(G.103)
derivative of CO
CDCY, = ~ ~ C L C L C U ~
(G. 104)
G .3 Linearization of the longitudinal equations
297
The linearization of the CD proposed here is a classic one; if the reader has another modeling of the CO, he can adapt the linearization to his particular case. With the dynamic pressure, it can be noted that qpi = $piV2 the previous expressions are simplified and the linearized equations of external forces are written
sqp e Va i
+-cD,
(- sin Oi cos Oi (Aux; - A W Z + ~ )sin2 BiAqZk + cos2 OiAqxk) (G.105)
The previous results are gathered and divided by mVai in order to obtain the linearized propulsion equation
(G.106) with
Dynamics of Flight: Equations
298
G - Linearized equations
____.__
(G. 108)
(G.109)
(G. 110)
(G.112)
If the system is linearized relative to a steady state flight with an aerodynamic climb angle equal t o zero 7, = 0, the coefficients ax and bw will be simplified. Particularly the term axv, and the term axv7. The terms bw are equal to (G.113) (G .114)
(G. 116) (G.117) (G.118)
G.3 Linearization of the longitudinal equations
299
Furthermore, if the system is linearized relative to a steady state flight without wind, all wind terms for the initial conditions (subscript i ) are equal to zero. So, the terms axv,, axvy,axv,, are simplified. The terms bw are reduced to (G. 119) (G.120) (G.121)
bwvq,
=
- cos yai sin yai
+ mV2 cos2 eicoq
(G.122) (G. 123)
G.3.2
Linearization of the sustentation equation
The sustentation equation (5.189), p. 144 below, expressed in the aerodynamic frame Fa, is linearized
+
From equation (4.104)) p. 92 the term - sina,F,b cosa,F,b can be written as -Fsin(cu, - cu,)cos~,, otherwise from equation (5.197)) p. 144 &a - q = -?,. With a process similar to those applied t o the propulsion equation (Section G.3.1, p. 295)) the sustentation equation is linearized. Linearization of the acceleration terms
then with the rearranged terms
-mV,, ATa
+
mV,, AV, (-qz:,
c
2
sin2yui - qxwicos yui) Dynamics of Flight: Equations
G - Linearized equations
300
Linearization of the external forces
- F, sin(aai - a,)
C O S ~ ,
PhiAh
+ AAVa + (G.127)
with equation (6.31), p. 164 and equation (6.33), p. 164 Aqt
= A q - Aqx;
then
1;1, - SgpiCLSmA6m - sin(aai - a,) cos&ASx SX;
(G.128)
G.3 Linearization of the longitudinal equations
301
The previous results are gathered and divided by mVai in order to obtain the linearized sustentation equation
with
(G .132)
(G.133)
Dynamics of Flight: Equations
(G.136) If the system is linearized relative to a steady state flight with an aerodynamic climb angle equal to zero y, = 0, the coefficients a x and bw will be simplified. In particular, the term axy,, and the term axyr. The terms bw are equal to (G .137) (G.138) (G.139) (G.140) (G.141) (G.142) Furthermore, if the system is linearized relative to a steady state flight without wind, all the wind terms for initial conditions (subscript i ) are equal to zero. The terms axyv, il"y7, and axya, are concerned. The element within CLq disappears. The terms bw are reduced to (G.143) (G.144) bwyux = cosyai siny,, bWYW% =
+"'pi sin 2oicLq 2mV2
- COSY,, sin yai - e s q ~ i sin 2 o i c L q
2mV2
~
(G.145) (G.146) (G.147) (G.148)
G.3.3
Linearization of the moment equation
The moment equation (5.190), p. 144 below, expressed in the body frame Fb, is linearized
Bq
=
tpSeV:Cm+M~6,
(G.149)
G.3 Linearixation of the lon.qitudina1 equations
303
with equation (5.57), p. 115 M F ~=
~ c o s ~ , ( z ~ ~-x&sina,) o s a ~
(G.150)
denoted
M F i = FZ r b
(G.151)
with
z t bM
=
b COSP~(ZM COSCY, - x&
sincu,)
(G.152)
The linearization of the moment equation gives
+
+
e
$piSIV; (CmqAq; V ai z'kF,(phiAh
+
e + CmbAb, + CmGmAGm) Va i
1 AAVa + -A6m) 62;
(G.153)
with equation (6.31), p. 164 and equation (6.33), p. 164 (G.154) Thus, with the rearranged terms
+
? !E !
B
+ -B
(
e
C m a a - -Cmq Va i
(cos 28, (waLi - zlx:,)
+ sin 28,(qz& - qx;,))
&F, A6x Cm6mAGm + 6XiB
(G.155) Thus the linearized moment equation is obtained
Dynamics of Flight: Equations
304
G - Linearized equations
(G.156)
--2S'pi
BVa;
Cmq [cos 28,(wZ:, - u x ~ + i sin 20,(qzLi- qxLi )I
(G.157)
(G.158)
(G.159)
(G. 160)
(G.161)
(G.162)
G.4 Linearization of the lateral equations
G.3.4
305
Linearization of the kinematic equations
The kinematic equations of vertical and pitch velocities (Equation 5.194, p. 144) below, are linearized (G .163) (G.164) Thanks t o equation (5.197), p. 144, this last relation (Equation G.164, p. 305) can be written &a+?,
(G .165)
= q
The linearization of the kinematic equations gives
so axh, axhr
= Vai sin yai = VaiCOSY,,
(G .167)
The others axh, bwh, buh are equal t o zero, and (G .168)
For the kinematic equation of pitch velocity (Equation G.165, p. 305), the coefficient are
amq axax busy bwaz
1 -axyq -hyX for = -buy, for = -bqz for = =
x = ( V , a ,h , y ) y = (m, z)
z = ( U , w , uz, w z , qz, q z )
(G.169)
G.4
Linearization of the lateral equations
G.4.1
Linearization of the lateral force equation
Divided by mVa the lateral force equation (5.202), p. 148 i s linearized ~
b
+ j a i -AVav, - Ap sin a,, + Ar COS aai- (risin aai+ p; COS aai)Aaa
a
- sin a,, - sin
i
( (cos2Pai - sin2 Psi )ijLi AP,
COS aai
COS Pai COS 2aai
+ sin Pai COS p,, A&)
&,, Act, - sin a a i sin 2Pai&, A@,
Dynamics of Flight: Equations
G - Linearized equations
306
+
?ki
- sin a,, (cos2 PaiAp& sin2 PaiAm;) - cos a,, sin 2PQi A@, COS a,,(cos2 ,8, ATX; + sin2 P,, Ayy;) - ((p&, cos2 P,, py;, sin2 P,,) cos a,, (TY;, sin2 P,, + TX;, cos2 PO,) sin a,,) Aa,
+ + + +
+
cos 2PQi( - - u X w 6i cos2 a,,
-
f sin 2Pai( -AuxL cos2 a,, - Am;
-
sin PQiCOS P,, sin 2aaiI.$,, Act,
+
Advz
-
vai
9
-(sin 8, cos a,, cos P,,
+ + -
+ vyLi)AD, sin2 a,, + Avy;)
wz,,6 sin2 a,,
-
vai
cos 8, sin 4, sin PO, - cos 8, sin a,, cos 4, cos ,8, )A&
+ cos 8, sin a,, sin 4, sin P,, ) A 4 9 (cos 8, cos a,, sin Pai sin 8, sin 4, cos Pai + sin 8; sin a,, cos 4, sin P,, )A8 vai 9 (sin 8, sin a,, sin P,, + cos 8, cos a,, cos 4, sin Pai)Aao 9
-(cos 8, cos 6,cos Pai vai
-
-
vai
+ + + + + +
E (- sin Paisin Pm
mv,,
E
(sin/?,,
COS
- cos Pai cos Pm cos(a,, - a,))
Pmsin(a,,
mv,, AE (cosP,, sinpm - sin& mv,,
- a,))
AD,
Aa,
COSP,cos(a,,
-
a,))
(G.170)
For this linearization, the altitude h and the air density p are assumed constant; if not, complementary terms will appear in the expressions of the aerodynamic and propulsion lateral forces.
Particular initial conditions
Some hypothesis can be made without significant consequences on the majority of flight situations. 0 0
0
The linearization is made with respect t o a steady state flight case, so
Dai = 0.
The initial conditions (Hypothesis 26) on the azimuth and sideslip angle are equal t o zero, so $,I = Pi = 0. The aircraft as a geometrical plane of symmetry (Hypothesis 4) so Cyi = 0.
G.4 Linearixation of the lateral equations 0
in
307
The aircraft as a symmetry on the propulsion forces Pm = 0.
The linearized equation of the lateral force equation (G.170), p. 306 is simplified
+
b Advz (-ux,, cos2 a,, - WZ;, sin2 aai + vyLi)APa+ Vai
9
-(sin Oi cos a,, - cos Oi sin a,, COS +,)ABa
+ + +
vai
9
-(cos Oi cos 4,A4 - sin 8, sin +i
Vai
s'm ,,
+pi-
(cyPAPa
A8)
I + -(CypApa + CyrAr, + CybAb,) VQi
+
+ p i s(CySlAS1 CySnASn)
rn
With Advz calculated in equation (G.85), p. 294 to equation (G.87), p. 294
Advt
=
+
+ + +
+
Aduz sin 8, sin 4, Adv; cos 4i + Adwz COS 8, sin +i (du;, (- COS aai COS 8, - sin a,, sin 8, cos 4,) + dv& sin a,, sin 4, +dw& (cos aai sin 8, - sin a,, cos Oi cos $+)) AB,
+
(-du& cos 4, dv& sin 8, sin 4,) Azl, (duzisin 8, cos 4i - dv;, sin 4, + dw;, COS 8, COS 4,) A+ (du&cos 8, sin 4, - dw& sin 8, sin 4,) At9
(G 172)
The components of the linearized wind gradient are calculated in section (G.2), p. 283. The aerodynamic angular velocities (Equation 3.54, p. 56) are
and
Gathering the previous results for the particular initial conditions, and with the expressions of Ap&, Ar&, &, pz& and r x & written as functions of the components of Dynamics of Flight: Equations
G - Linearized equations
308
the wind expressed in the vehicle-carried normal Earth frame F, (Equation G . l , p. 283) to (Equation G.37, p. 286) and (Equation G.44, p. 286) to (Equation G.52, p. 288), we obtain
(G.174)
+ -
(duo,, COS@, - dwGi sine,)
+
8, cos
COS (xai
*P4
=
+
&Pe
=
Va i 9 -(sin Va i
- cos Oi sin aaicos 4i)
1
Vai (dv& sin 4, - cos +i (du& sin ei + dwzi cos e,)) + Va i cos 8, cos 4i apz4, sin a,, - arx4, cos aai 9
1 9 - dwz,sin 8, sin 4i) - -sin 8, sin +i Va i Va i +apze, sin a a i - arxe, cos aai - - (duzi cos 8, sin 4,
G.4 Linearization of the lateral equations
S m
-+pi -t(arxe,Cyr,
309
+ arye,Cyr, + apyewCyp)
(G.175)
The terms apzt,, apytw, aryt,, arxt, with t = [e, 6, $3 are respectively calculated in equation (G.47), p. 287, equation (G.50), p. 288, equation (G.49), p. 287 and equation (G.52), p. 288. The terms duO,, dug, dwg,are calculated in equation (5.48), p. 113. The terms U X ~ , v, y b , , W Z & , q&, q z & , are calculated in equation (D.63), p. 239 t o equation (D.77), p. 240 as functions of the components of the wind expressed in the vehicle-carried normal Earth frame F,. The components of the control matrix
(G.176) The components of the wind perturbation matrix: the first for the linear wind velocity
(G .177) The second for the gradient of the linear wind velocity bwPuz
=
bwPvy
=
bwPwz
sin 4, -sinOiuLi va,
cos q5i Va i sin -- 4,
--
=
vai
sin 24, +2
S
- sin aai- ;pi -tcyp) m
sin 24, +(sina,, + 2 coSeiw;,
+
sin 244 2
m
S
- sin a,, - +pi-mt ~ y p )
(G.178) Finally, the third for the wind angular velocity bwPpy
=
sin $i
--
i
S -e m
- +pi bWPpz
=
cos ($i Va i
+3pi
cos
+ sin aaicos 8, sin2 q!+
(Cyp COS 8, cos2 4,
+ ~ y r sin , 8,
COS
4i)
+ sin a,;cos 6;cos2 c$~ - cos a a i sin Bi cos 4i S
t (Cyp cos ei sin2 +i
+ Cyr,
sin ei cos + i ) Dynamics of Flight: Equations
310
G
bwPqx
=
sin 4, cos eiuLi Va i +;pi
bWPqz
=
S
-e m
+ + sin
aai
sin 28, sin 24,
(Cyr, sin2 8, sin 4,
--sin 4, sin eiw& - a sin
+ cos
cos2 8, sin 4,
+ cos a,, sin2 8, sin 4,
S + + pi .t (cyr, sin 4, cos2 8, - + cyp sin 28, sin 24, + cyr, m bwPrx
=
cos 4, U;, - COS a,, Va i
--
S
- +pi -e
m
bwPry
=
equations
+ i ~ y p s i 28, n sin 24, + Cyr, sin 4, cos2 8,) sin 28, sin 24,
Va i
Qai
- Linearized
COS 8, COS
sin 4, sin2 8,)
4, - sin a a i cos2 4, sin 8,
(Cyr, cos 8, cos 4, - Cyp sin2 4, sin 8,)
sin 4, sin 8,v& - sin a a i sin2 4, sin 8, Va i
S
-+pi-[m ( ~ y r ~ c o s e ~-c ~oysp~c ~ os~4,sine,)
(G.179)
Initial conditions with horizontal wing, so a zero bank angle 4, = 0 With the hypothesis 6, = 0, the coefficients axp and bwp are simplified
+ ux,,b sin +-(cos8,dw& Va i
*P4
*Pv
=
=
1 -(du& sin Oi Va i
dv&
vi
+ wzb,, sin2aai- vy,,b COS + sinB,duO,,)+ (cos 8,duLi - dw& sin 8,) Vai
cos2 a,,
+ dw;,
COS 8,)
9 +COS 8, Va i
G.4 Linearixation of the lateral equations - $pi-t(arxe,Cyr, S
m
311
+ arye,Cyr, + apye,Cyp) (G. 180) arxt, with = [@,$,+I are respectively calculated in
The terms apzt,, apyt,, aryt,, equation (G.56), p. 289, equation (G.59), p. 290 equation (G.58), p. 290 and equation (G.61), p. 290. The terms duc, du;, dw;, are calculated in (Equation 5.48, p. 113). The terms UX;,, v y L i , wzLi, qx",, qz",, are calculated in equation (D.69), p. 239 to equation (D.71), p. 240 as functions of the components of the wind expressed in the vehicle-carried normal Earth frame Fo . The components of the control matrix do not change
The components of the wind perturbation matrix: the first for the linear wind velocity
(G.181) The second for the gradient of the linear wind velocity
bwpvy = bdwz
1
-vaiU &
(G. 182)
= 0
Finally, the third for the wind angular velocity
S
+
bwPpy = - ;pi - t ( C y p COS 8; Cyr, sin Oi) m 1 S bwPpz = -wGi sin a a i cos Oi - cos aaisin 8, - $pi-t Cyr, sin Oi m Va i bwPqz = 0
+
bwPqz = 0
1 Va i
=
-- U&
bwPry
=
- i p i - l (Cyr, cosOi - CypsinOi) m
S
-
S
cos a,,cos Oi - sin &ai sin Oi - ;pi -l Cyr, cos Oi m
bWPrx
(G.183)
Wind known at a zero inclination angle If it is assumed that the wind is known in a normal Earth-fixed frame oriented by a zero initial inclination angle Oi, the components of the wind in coefficients axp Dynamics of Flight: Equations
G - Linearized equations
312
(Equation G.180, p. 311) are simplified. It is found thanks t o the equation (G.72), p. 293 to equation (G.80), p. 293 and equation (D.69), p. 239 to equation (D.71), p. 240; this last results with Bi = 0 axpp
a&#)
sin 2aai = ( q z & - qx&) uxLi cos2 2 sin -- Vai (-dw& cosOi - du& sine,)
+
=
1 -- (du& sin 8, Va i
+ w z ~sin2 , ~ aai- u y ~ ~ COS +Va (du& cos Oi - dwEi sin Oi) i
9 + dw& cos ei)+ cos Oi + (vyLi - wzLi)sin Va i
(G.184)
G.4.2
Linearization of the roll moment equation
The linearized equation of roll moment (Equation 5.205, p. 148) appears in the following form
The roll moment of the thrust force M F ; , is assumed to be a constant with respect to the lateral states. As for the lateral force equation (3.54), p. 56
G.4 Linearixation of the lateral equations
313
then and
SO
ClrAr, ClrAr,
+
= Clr,Ary: Clr,Arx! = (Clr, Clr,)Ar - ClryAryb, - Clr,ArxL
+
(G.187)
Finally this equation is obtained AAp
-
+ -
+
+ -
-
E A + = $piSlV:Cl/3A@a (+piS12VaiClp+E$) A p + (+piS12Va,Clr- qi(C - B)) A r ( r i ( C- B) - Epi) Aq F, ( 2 $ p i S K i C l i X-(y, Va i cos& sina, - Z, sin&)
+
b piSlV: (Cl61A61 + Cl 6nA6n) sin 24,
+ AvYO,+ AwZLcos2 8,) A G sin 28, sin 24i + cos 8,(cos2 4,Am: + sin2 4,APT,;) + 4 - sin Oi(cos24,Ary; + sin2 4,Arx;) +apydWA4+ a p ~ + ~ A+$apyo,A8] sin 28, $piS12VaiClr, A c t sin 4, + Am: sin 8, cos 4, [ 2 - sin 4,(Aqzz cos2 8, + AqxL sin2 8,) + Ary; 8, 4, +ar~4,A4 + ary+wA$ + aryo,A8] $piSe2VaiClp
( n u X Lsin2 8,
COS
-
COS
$piS12VaiClr, [-ACE sin 8, cos 8, sin 4, + Am: sin 8, COS 4, AqzL sin2 8,) Arxz COS 8, COS 4,
- sin 4,(AqxL cos2 8,
+arx4, A+
+
+ arX@, A$ + arxowA81
+
(G.188)
The previous results can be gathered into
E Ap--A+
A
=
+ + + + + + + Dynamics of Flight: Equations
314
G - Linearized equations
with
pis e 2
*Pp
=
3-
*Pr
=
5
axpq =
1-
A
pis e 2
A
E vaiczp+ --Qi A
vai(czr,+ CZr,) - qi-C A- B
- ri(C - B ) - E p
A
The terms apyEw, aryiw, arxEw with F = [e, 4, $3 are respectively calculated with equation (G.50), p. 288, equation (G.49), p. 287 and equation (G.52), p. 288. The components of control matrix are
(G.192) The components of the wind perturbation matrix: the first for the linear wind velocity
(G.193) The second for the gradient of the linear wind velocity bwpll,
=
bwPvy
=
bwPwz
=
sin 28, sin2 ei - -sin 4, (GZr, - Clr,) 2 sin 244 -3- pise2 vai czp-
A
(G.194)
2
cos2 8,
A
sin 28, +2 sin + i ( ~ ~-r CZ~,)) ,
Finally, the third for the wind angular velocity bwPpy
=
--1.-
2
pise2
A
vai(CZPCOS ei cos2 + C Z ~sin , ei COS +i
+i)
G.4 Linearization of the lateral equations bwPpz bwPqx
=
-1-
'
A
Vai (Clpcos ei sin2 4i + Clr, sinei COS 4i)
pise2
Vai ( -CZr, sin2 8, sin 4i - f ~ lsin p 28, sin 24, - Clr, sin +i cos2 e,) A piS.f2 Vai( -Clry sin 4i cos2 oi + Clp sin 28, sin 24i - Clr, sin 4i sin2 ei) = -12 A =
-1-
bwPr,
=
-1-
bwprg
=
-1-
bwP,,
piSP
315
2
Va;(Clr, cos ei cos 4i - c l p sin2 4i sin ei)
pis e 2
* 2
A pise2
Vai (Clr, cos ei cos 4, - CZPcos2 4, sin e,)
A
(G.195)
Initial conditions with horizontal wing, so a zero bank angle c$~ = 0 With the hypothesis $i = 0, the coefficients are simplified. Moreover, the wind is
assumed t o be known in the normal Earth-fixed frame oriented by the initial inclination angle 8;. In other terms it is a question of using the wind results equation (G.72), p. 293 to equation (G.80), p. 293 and equation (D.75), p. 240 t o equation (D.77), p. 240, with 4i = ei = o
' axpr =
axPd
=
1 -
p.S12
'
Vai(Clr,
A
piS P -1-
'
A
+ CZr,)
Vai [Clp (wZ&
- ~i
C-B
A
- w;,)
- q z L i CZr,
+ qx;, C Z ~ , ]
Cli F, a x p ~ = 2+piSlV,,- AA VaiA(ym cos Pm sin am - zTnsin Pm) axpa = 0 ri(C- B) - E p axpq = -
+
A
axpe
=
-1-2
pi
se2 A vai (-rY;iclP + PY;pry + pz&Clr,) I
(G.197)
The components of the control matrix do not change bupl
=
bup,
=
)Fv2clal pi se 3A Vz Clan
(G.198)
The components of the wind perturbation matrix: the first for the linear wind velocity which do not change
Dynamics of Flight: Equations
316
G - Linearized eauations
The second for the gradient of the linear wind velocity
(G.199) Finally, the third for the wind angular velocity
G.4.3
bwP,,
pi se2 - - 4 -Vai A
bWPpz
=
-1-
bWPry
=
-1-
pise2
' A
(Clp cos 8;
+ Clr, sin Oi)
Vai Clr, sin 8,
pise2 Vai (Clr, cos Oi - CZp sin ei)
' A
(G.200)
Linearization of the yaw moment equation
The linearized equation of yaw moment (Equation 5.206, p. 149) appears in the following form
By analogy with the previous linearization of the roll moment equation (G.185), p. 312, it can be written
E A+ - -Ap C
= axrpA@,
+ + + + + +
+ axrpAp + axr,Ar + axrdA4 + axr+Aazl, + axrvAVa + axr,Aq + axrgA8
axr,Aaa burlAdl + bur,ASn bwruAuG bwrvAvz bwrwAwz bwruxAuxO, bwrvyAvyz bwr,,AwzOu, bwr,,ApyO, + bwrpzApyL bwrqtAqxO, + bwrqrAqZL bwrrxArxL bwrryAryOu, (G.202)
+
+
+
+
+ +
317
G.4 Linearization of the lateral equations
with
axr,
= 0
axrq =
- pi(B - A ) - Eri
C (G.204)
The terms apzt,, am
