This paper involves modeling and control of the Generic Transport Model (GTM), which is anotional twin-engine transport aircraft introduced by the NASA Langley Research Center. Theaircraft components, namely fuselage, wing, and horizontal and vertical stabilizers are modeled ashollow beams with constant thicknesses. The mass and stiness distributions of the componentsare approximated using cross-sectional properties, such as geometric center, cross-sectional area,area moments of inertia etc., at some nite number of sections on the respective components. Eachbeam is assumed to have one bending and one torsional displacements. The aerodynamic forcesand moments are generated using a quasi-steady theory. The equations of motion are derived usingLagrange’s Equations in quasi-coordinates. The partial dierential equations are replaced by a setof ordinary dierential equations by means of the Galerkin discretization method in conjunctionwith eigenfunctions of uniform cantilever beam and uniform shaft.
I. Introduction
Control of aircraft involves many challenges mainly due to the uncertainties encountered in aircraft dynamics.Variations in payload and its distribution, aging, errors in the predictions of the aerodynamic coecients andvariations in these coecients, external eects such as wind and gust can contribute in these uncertainties.As modern aircraft become more exible, maneuverable and autonomous, demand for control design that canaccommodate these aircraft increases. Requirements for control design go well beyond the requirements expectedfrom classical control designs, such as gain and phase margins. Adaptive control is becoming a powerful technologythat can very well respond to these demands, as it can accommodate parametric uncertainties.1 However, mostof adaptive control applications in the open literature seem to be limited to rigid aircraft, and ignore completelyinteractions of such control with exible motion of aircraft. Our objective in this paper is to design adaptivecontrol that can accommodate rigid body and exible dynamics of aircraft, as well as coupling between thesetwo dynamics. Control design will be demonstrated for a notional twin-engine transport aircraft introduced byNASA Langley Research Center, namely Generic Transport Model (GTM).
Equations of motion of exible aircraft can conveniently be derived using the Lagrangian equations of motionin quasi-coordinates described in Refs. 2 and 3. The Lagrangian equations of motion require the knowledge ofthree scalar quantities, namely, kinetic energy, potential energy, and the virtual work due to the applied forces.In our modeling approach, we regard the aircraft as a exible multibody system where the bodies are the fuselage(f), wing (w), horizontal stabilizer (h), and vertical stabilizer (v). To describe the motion of the aircraft, we rstattach a set of body axes xyz to the undeformed aircraft at a convenient point on the fuselage (not necessarilythe center of mass of the aircraft), as well as similar axes (xiyizi, i = f; w; h; v) to the exible components. Forconvenience, we assume that xfyfzf coincides with xyz. We model the fuselage, right- and left-half wing, right-and left-half horizontal stabilizer, and vertical stabilizer as hollow beams, each xed at its respective root, andsubject to one apwise bending displacement wi and one spanwise torsional displacement ’i (i = f; w; h; v). Thenthe motion of a point on the aircraft can be expressed by rigid-body translations and rotations of the body axesxyz and by the elastic deformations of the exible bodies relative to their respective body axes. From Refs. 2
1Assistant Professor, Mechanical Engineering Department, AIAA Senior Member2Research Scientist, Intelligent Systems Division, Mail Stop 269-1, AIAA Associate Fellow
1
51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<BR> 18th12 - 15 April 2010, Orlando, Florida
and 3, generic Lagrangian equations of motion in quasi-coordinates are
d
dt
(∂L
∂v
)+ ω
∂L
∂v− C
∂L
∂R= F
d
dt
(∂L
∂ω
)+ v
∂L
∂v+ ω
∂L
∂ω− E−T ∂L
∂θ= M (1)
∂
∂t
(∂L
∂vi
)− ∂L
∂ui+ Liui = Ui
∂
∂t
(∂L
∂αi
)+ Hiψi = Ψi, i = f, w, h, v
where L = T −V is the Lagrangian for the whole aircraft, in which T is the kinetic energy and V is the potentialenergy, v(t) = [U (t) V (t) W (t)]T and ω(t) = [P (t) Q(t) R(t)]T are the vectors of translational and angularvelocities of xyz, R(t) = [X(t) Y (t) Z(t)]T position vector of the origin o of xyz relative to the origin O of XY Z,θ(t) = [φ(t) θ(t) ψ(t)]T symbolic vector of Eulerian angles between xyz and XY Z, C = C(φ, θ, ψ) matrix ofdirection cosines from inertial axes XY Z to xyz, E = E(φ, θ) matrix relating the vector of Eulerian velocities θto angular velocity vector ω, ui(xi, t) = [0 0 wi(xi, t)]T and vi(xi, t) = ui(xi, t) elastic bending displacement andvelocity vectors, ψi(xi, t) = [ϕi(xi, t) 0 0]T and αi(xi, t) = ψi(xi, t) elastic torsional displacement and velocityvectors, L Lagrangian density exclusive of strain energy, Li and Hi matrices of stiffness differential operators, Fand M resultant of gravity, aerodynamic, propulsion and control force and moment vectors acting on the wholeaircraft in terms of body axes components, Ui and Ψi resultant of gravity, aerodynamic, propulsion and controlforce and moment density vectors. The terms like ω∂L/∂v indicate vector multiplication, ω× ∂L/∂v where ω isa skew symmetric vector derived from ω:
ω =
0 −R QR 0 −P−Q P 0
(2)
The body axes xyz are obtained from the inertial axes XY Z through the following sequence of rotations: 1)ψ about Z to the intermediate axes x1y1z1, 2) θ about y1 to the intermediate axes x2y2z2, and finally 3) φ aboutx2 to the body axes xyz. Hence, the matrices C and E are
C =
cosψ cos θ sinψ cos θ − sin θcosψ sin θ sinφ− sinψ cos φ sinψ sin θ sinφ+ cosψ cos φ cos θ sinφcosψ sin θ cosφ+ sinψ sinφ sinψ sin θ cosφ− cosψ sinφ cos θ cosφ
E =
1 0 − sin θ0 cosφ cos θ sinφ0 − sinφ cos θ cosφ
(3)
The force, moment, and the generalized equations of motion given in Eqs. (1) must be considered in conjunctionwith the kinematical relations
R = CTv, θ = E−1ω, ui = vi, ψi = αi (4)
The resulting equations of motion including Eqs. (1) and (4) are 12 first-order ordinary differential equationsfor the rigid-body translations and rotations, and a partial differential equation for each elastic displacementcomponent. The system is hybrid since it includes both ordinary and partial differential equations. It does notin general admit a closed-form solution, and an approximation of the solution amounts to discretization of thepartial differential equations. To this end, we assume that each elastic displacement can be expressed as a matrixof n shape functions multiplied by a vector of n generalized coordinates. Hence, we have
The shape functions can be from finite element method, Galerkin method, or any other discritization methods.However, for good accuracy at low n, the shape functions used in this paper are the eigenfunctions of a uni-form cantilever beam for the bending displacements, and the eigenfunctions of uniform shaft for the torsional
2
displacements. Rather than deriving first the hybrid equations and then descretizing them in space, it is moreadvantageous to carry out the descretization directly in the kinetic energy, potential energy, and virtual work.By the introduction of the discretization, the equations are reduced to a set of first-order nonlinear ordinarydifferential equations. The discrete generic equations of motion will have the form
d
dt
(∂L
∂v
)+ ω
∂L
∂v= F
d
dt
(∂L
∂ω
)+ v
∂L
∂v+ ω
∂L
∂ω= M (6)
d
dt
(∂L
∂si
)− ∂L
∂qi+ LiUiqi = Ui
d
dt
(∂L
∂ηi
)+ HiΨiξi = Ψi, i = f, w, h, v
where ∂L/∂R and ∂L/∂θ are omitted because for an unrestrained aircraft L does not depend on R or θ.The kinetic energy for the whole aircraft requires the knowledge of the velocity and the mass distribution at
every point of the aircraft. Velocity of a typical point on the fuselage can be approximated as
vf (rf , t) = v + (rf + uf )Tω + rTf αf + vf (7)
where rf = [xf yf zf ]T is the radius vector from of to the point in question. Similarly, velocity of a point on thewing, and stabilizers can be expressed as
where ri = [xi yi zi]T , Ci is matrix of direction cosines from xyz to xiyizi, rfi = [xfi yfi zfi]T is the radius vectorfrom the origin of xyz to the origin of xiyizi, ufi(t) = uf (xfi, t) is the elastic displacement of the fuselage atxf = xfi, Ωfi(t) = [0 (−∂uf /∂xf )(xfi, t) 0]T is the angular velocity of xiyizi due to the bending of the fuselageat xf = xfi, vfi(t) = vf (xfi, t) is the velocity of the origin of xiyizi due to the elastic velocity of the fuselage,and αfi(t) = αf (xfi, t) is the angular velocity of xiyizi due to the elastic torsional velocity of the fuselage atxf = xfi. The total kinetic energy is equal to the summation of the individual kinetic energies:
T =∑
i
Ti, Ti = 12
∫vT
i vidmi, i = f, w, h, v (9)
where dmi is the mass differential element for the component i. Introducing Eqs. (5) into Eqs. (7) and (8), andthe resulting expressions into Eqs. (9), the total kinetic energy can be written in the form
T = 12VTMV (10)
whereV = [vT ωT sT
f sTw sT
h sTv η
Tf η
Tw η
Th η
Tv ]T (11)
is the global velocity vector and M is the global mass matrix. Once T is known the ij entry of the mass matrixcan be determined from
Mij =∂2T∂Vi∂Vj
(12)
where Vi is the ith entry of the velocity vector V. It is important to mention that the mass matrix M is anonlinearly function of the generalized displacements qi, and it will change as the aircraft deforms.
Using the kinetic energy expression, we can also construct the momenta associated with the velocities:
pv =∂T∂v
, ρω =∂T∂ω
; pi =∂T∂si
, ρi =∂T∂ηi
, i = f, w, h, v (13)
We can also writep =
∂T∂V
= MV (14)
3
wherep = [pT
v ρTω pT
f pTw pT
h pTv ρ
Tf ρ
Tw ρ
Th ρ
Tv ]T (15)
is the global momentum vector.Potential energy is due to the strain energy. Similarly to the kinetic energy, the total potential energy is
the summation of strain energies of the individual bodies, which can easily expressed in terms of bending andtorsional elastic displacements:
V =∑
i
Vi, Vi = 12
∫ Li
0
[EIi
(∂2ui
∂x2i
)2
+GJi
(∂ψi
∂xi
)2]dxi, i = f, w, h, v (16)
where Li is the length, EIi fluxtural rigidity, and GJi torsional rigidity of the component i. Introducing Eqs. thefirst two of (5) into Eqs. (16), the total potential energy can be written as
V = 12
∑
i
(qTi Kiqi + ξT
i Kiξi) (17)
where
Ki =∫ Li
0
EIi∂2UT
i
∂x2i
∂2Ui
∂x2i
dxi, Ki =∫ Li
0
GJi∂ΨT
i
∂xi
∂Ψi
∂xidxi, i = f, w, h, v (18)
are the bending and torsional stiffness matrices, respectively.The virtual work can be expressed in terms of the applied forces and virtual displacements:
δW =∑
i
∫fTi δr
∗idDi + FT
e δr∗we (19)
where fi is the distributed force due to aerodynamics and gravity, Fe is the engine thrust, δr∗i is the virtualdisplacement of a typical point on the component i, δr∗we is the value of the virtual displacement δr∗w at theengine location. We assume in Eq. (19) that the engines are mounted on the wing. Virtual displacements δr∗ihave similar expressions as vi:
where δr∗ is the virtual displacement due to the quasi-displacement r∗ whose time derivative is equal to the quasi-velocity v, i.e. r∗ = v. Similarly, δθ∗ is the virtual angular displacement due to the angular quasi-displacement θ∗
whose time derivative is equal to the quasi velocity ω, i.e. θ∗
= ω. Moreover, Ufi = Uf (xfi), Ψfi = Ψf (xfi), and∆Ufi are constant matrices such that Ωfi = ∆Ufiqf . Now, substituting Eqs. (20) into Eq. (19) and separatingthe terms, we get
δW = FT δr∗ + MT δθ∗ +∑
i(UTi δqi + ΨT
i δξi) (21)
whereF =
∑
i
CTi
∫fidDi + Fe (22)
is the resultant force on the aircraft in which summation is carried out for i = f, w, h, v;
M =∫
(rf + Ufqf )ffdDf +∑
i
∫[(rfi + Ufiqf )CT
i +CTi (ri + Uiqi)]fidDi (23)
+[(rfw + ˜Ufwqf )CT
w + CTw (rwe + ˜Uweqw)
]Fe (24)
is the resultant moment on the aircraft in which rw = rwe is the radius vector from the origin of xwywzw to theengine location, Uwe = U(xwe), and summation is carried out for i = w, h, v;
Uf =∑
i
∫(rT
i Ci∆Ufi + CiUfi)T fidDi + (rTweCw∆Ufw +CwUfw)T Fe
Ψf =∑
i
∫ΨT
fi(rTi Ci +Cir
Tfi)
T fidDi + ΨTfw(rT
weCw + CwrTfw)TFe
(25)
4
is the generalized force and moment for the fuselage in which the summations are for i = w, h, v; and finally
Uw =∫
UTwfwdDw + UT
w(xwe)Fe
Ψw =∫
ΨTw rwfwdDw + ΨT
w(xwe)rweFe
Ui =∫
UTi fidDi, Ψi =
∫ΨT
i rifidDi, i = h, v(26)
are the generalized force and moment for the wing, horizontal stabilizer and vertical stabilizer.
II. Aerodynamic and Gravity Forces
To approximate the aerodynamic forces, we use a simple quasi-steady theory. We assume that only the wing, andthe stabilizers are subject to aerodynamics, and ignore contributions from the fuselage and the engine nacelle,which mostly contribute in the drag force. For convenience, instead of using the body axes, we use a different setof axes, namely aerodynamic axes to express the aerodynamic forces. However, the origins of xyz and xwaywazwa
coincide. Aerodynamic axes for the wing, xwaywazwa is obtained from the body axes xyz through a rotationabout the x by the dihedral angle Γ so that the matrix of direction cosines from xyz to xwaywazwa is
Cwa =
1 0 00 cos Γ sin Γ0 − sin Γ cos Γ
(27)
Hence, the velocity of a typical point in the quarter-cordline of the wing can be written as
vwa(rwa, t) = CwaCTw vw(rwa, t) (28)
where rwa is radius vector from the origin of xwaywazwa to the point question. Then, the angle of attack can beapproximated as
αw 'vwaz
U+ ϕw (29)
where vwaz is the zwa component of vwa. Using this angle of attack, the lift per unit length can be expressed as
where cw is the chord, CLw the lift coefficient, CLw0 the lift coefficient at αw = α0, CLαw the slope of the liftcurve, δa is the aileron angle, CLδa is the aileron effectiveness, and
q ' 12ρ(U
2 + V 2 +W 2) (31)
is an approximation for the dynamic pressure in which ρ is the air density. Similarly, the drag per unit length is
dw = qcwCDw = qcw[CDw0 + CDαw(αw − α0)] (32)
where CDw is the drag coefficient, CDw0 drag coefficient at αw = α0, and CDαw is the slope of the drag curve.Finally, distributed aerodynamic force on the wing can be written as
fwa =
`w sinαw0 − dw cosαw0
0−`w cosαw0 − dw sinαw0
' `w
αw0
0−1
− dw
10αw0
(33)
where αw0 = vwaz/U . Distributed aerodynamic forces on the horizontal and vertical stabilizers can be expressedby following the same steps.
The gravity forces per unit volume are simply
fig = CiC
00ρig
, i = f, w, h, v (34)
where ρi is the mass unit length for the component i and g is the gravitational acceleration.
5
III. Discrete State Equations
Recognizing that the potential energy V does not depend on the velocities, we can cast the discrete equations ofmotion in the first-order form
The first half of Eqs. (35) are kinematical relations and the second half are the force and moment equations. Notethat both velocities and momenta appear in Eqs. (35) so that the equations must be considered in conjunctionwith the momenta-velocities relation given in Eq. (14). ∂T/∂qi terms appearing in the equations are responsiblefor the centripetal and Coriolis effects on the bending of the flexible beams. Control inputs enter into the equationsthrough the engine thrust and the aerodynamic force densities. We assume that the engine thrust has a componentonly in x direction so that Fe = [Fe 0 0]T , and that aileron angles on the right- and left-half wing have the samemagnitudes, but opposite directions. Hence, the control input vector can be written as u = [Fe δa δe δr ]T whereδe and δr are the elevator and rudder angles, respectively.
Note that the aircraft model includes six cantilever beams and each beam has one bending displacement andone torsional displacement. Since we use n shape functions to represent each displacement, the number of totalelastic degrees of freedom for the whole aircraft is 12n. With the addition of six rigid body degrees of freedom,the number of total degrees of freedom is 6 + 12n, and hence, the order of Eqs. (35) is 2(6 + 12n).
IV. Undamped Vibration in Vacuum
To have an idea about the structural characteristics of the aircraft, we first address the undamped vibration ofthe unrestrained aircraft in vacuum. The governing equations will be obtained from Eq. (35) by setting F = 0,M = 0, Ui = 0, and Ψi = 0 for all i = f, w, h, v, and by linearizing the resulting equations about q = 0 andV = 0 where q = [RT θT qf qT
w qTh qT
v ξTf ξ
Tw ξ
Th ξ
Tv ]T is the global displacement vector. The linearized equations
can be written asM0q +Kq = 0 (36)
where M0 is equal to M evaluated at q = 0 and
K = diag[0 0 Kf Kw Kh Kv Kf Kw Kh Kv] (37)
is the global stiffness matrix. Differential Equation (36) admits the solution q(t) = qeiωt where ω is a constantscalar and q is a constant vector. Considering this solution, the system reduces to the algebraic eigenvalueproblem
M0λq = Kq (38)
where λ = ω2 is an eigenvalue and q is an eigenvector. Both M0 and K are (6+12n)× (6+12n) matrices so thatthe system has 6 + 12n eigenvalues and eigenvectors. The first six eigenvalues are zero and the correspondingeigenvectors involve only the rigid body displacements. Remaining eigenvectors involve both rigid body andelastic displacements.
6
V. Model Reduction
The order of the discrete equations of motion is relatively high. Control design for such a system usually expe-riences difficulties due to high order. The number of degrees of freedom for the whole aircraft can be reducedby reducing the total elastic degrees of freedom. Hence, a small order model retaining a high degree of accuracywould considerably easy the control design. To this end, we use the approach presented in Ref. (4) in which thefirst m eigenvectors of undamped vibration in vacuum beyond the first six rigid body eigenvectors are used toapproximate the elastic displacements.
Let us now consider the following matrix whose columns are the first m eigenvectors beyond the six eigenvectorsfor the rigid body degrees of freedom:
D = [q7 q8 . . . q7+m] =[Dr
De
](39)
where Dr is a 6 ×m matrix. Then the elastic displacements can be approximated as
qe(t) = Deζ(t), Ve(t) = Deζ(t) = Deν(t) (40)
whereqe = [qf qT
w qTh qT
v ξTf ξ
Tw ξ
Th ξ
Tv ]T (41)
is the elastic displacement vector and
Ve = [sTf sT
w sTh sT
v ηTf η
Tw η
Th η
Tv ]T (42)
is the elastic velocity vector, ζ = [ζ1 . . . ζm]T the reduced elastic displacement vector, and ν = ζ the reducedelastic velocity vector. The reduced kinetic energy expression T can be obtained by substituting Eqs. (40) intoEq. (10). Similarly, the reduced potential energy expression V can be obtained by substituting Eqs. (40) intoEq. (17):
V = 12 (Deζ)TKeDeζ = 1
2ζT Kζ, K = DT
e KeDe (43)
where K is the reduced stiffness matrix and Ke = diag[Kf Kw Kh Kv Kf Kw Kh Kv] is a stiffness matrix.Reduced virtual work expression can be obtained by substituting Eqs. (40) into Eqs. (19):
δWR = FT δr∗ + MT δθ∗ + UTe Deδζ (44)
where Ue = [UTf UT
w UTh UT
v ΨTf ΨT
w ΨTh ΨT
v ]T is a vector of generalized forces and moments.
R = CTvθ = E−1ω
ζ = ν
pv = −ωpv + F (45)ρω = −V pv − ωρω + Mpν = ∂T /∂ζ − Kζ + U
where
pν =∂T∂ν
(46)
is the reduced elastic momentum vector, and U = DTe Ue is the reduced generalized force vector. Equations (45)
can be cast in the following compact form:
˙q = CV˙p = −Kq + DV + F
(47)
where
q =
Rθζ
, V =
vων
, p =
pv
ρω
pν
= MV (48)
7
are the displacement, velocity and momenta vectors,
C =
CT 0 00 E−1 00 0 I
, K =
0 0 00 0 00 0 K
, D =
0 pv 0pv pω 00 0 0
(49)
in which I is an m ×m identity matrix and
F =
FM
∂T /∂ζ + U
(50)
is the force vector. The ij entry of the reduced mass matrix M can be obtained from
Mij =∂2T∂Vi∂Vj
(51)
where Vi is the ith entry of the reduced velocity vector V.
VI. Perturbation Solution
Note that the reduced state equations, Eqs. (47), are highly nonlinear. When the interest lies in the trim solutionsin steady flights such as steady level flight, steady turn maneuver, etc., Eqs. (47) can directly be used. However,if the interest lies in the dynamical analysis and the control design, then the nonlinearity in the equations makethe problem quite difficult. Another difficulty is due to the fact that the equations are both in velocities andmomenta, which makes it necessary to use the momenta-velocities relation, the third of Eqs. (48), to eliminateeither velocities or momenta in the equations. However, this elimination requires the inversion of the mass matrixM, which is not constant matrix. To obviate these difficulties, we use a perturbation solution in the form
where the variables with bars are nominal parts, and the variables with tildes are perturbation about the nominalvalues. Expanding the momenta-velocities relation, the third of Eqs. (48), into Taylor series and keeping onlythe first two terms, we get
p = p + p = MV + MV + MV (53)
Separating the orders of magnitude, we get
p = MV, p = MV + MV (54)
The perturbation in momenta, the second of Eqs. (54) can be cast in the form
p = H[
qV
](55)
where H is a constant matrix. Substituting the perturbation solutions into Eqs. (52) and separating the ordersof magnitudes we get the equations of motion for nominal and perturbation dynamics. The nominal equations ofmotion are
˙q = CV˙p = −Kq + DV + F
(56)
Choosing a state vector x = [qT VT ]T , the perturbation equations can be written in the state-space form
˙x = Ax +Bu + f (x) (57)
where x = [qT VT ]T is the perturbation part of the state vector, u is the perturbation part of the control inputvector, A and B are the coefficient matrices corresponding to the first-order terms of the Taylor series expansion,which can be obtained from
A =∂g∂x
∣∣∣∣x=x,u=u
, B =∂g∂u
∣∣∣∣x=x,u=u
(58)
8
in which x = [qT VT ]T is the nominal part of the state vector, u is the control input vector, and g nonlinearfunction of x and u in the form
g =[
CVH−1(−Kq + DV + F)
](59)
Furthermore, f (x) appearing in Eq. (57) corresponds to the quadratic and cubic terms from the Taylor seriesexpansion. We can write the ith entry of f (x) as
fi(x) = 12
∞∑
j,k=1
∂2gi
∂xj∂xk
∣∣∣∣x=x,u=u
xjxk + 16
∞∑
j,k,m=1
∂3gi
∂xj∂xk∂xm
∣∣∣∣x=x,u=u
xjxkxm (60)
where gi is the ith entry of g, and xj and xj are the jth entry of x and x, respectively.
VII. Trim Solutions
In this section, we will show how trim solutions can be obtained using the equations for the nominal dynamicsgiven in Eqs. (56). These equations are highly nonlinear, but this not introduce any difficulty in obtaining trimsolutions for steady maneuvers. A nominal state x corresponding to a desired maneuver is first postulated andEqs. (56) are solved using inverse dynamics. We will demonstrate this for a steady level flight.
Steady Level Flight: In this maneuver, aircraft nominally maintains 1) a constant forward velocity VX , 2) aconstant altitude h, 3) zero roll and yaw angles, and constant pitch angle, and 4) constant elastic displacementsso that
R =
VX t0h
, θ =
0θ0
, qi = constant, ξi = constant, i = f, w, h, v (61)
From the first three of Eqs. (35), we have
v = C ˙R =
cos θ 0 − sin θ0 1 0
sin θ 0 cos θ
VX
00
= VX
cos θ0
sin θ
, ω = E ˙θ = 0, ˙qi = 0, ˙ξi = 0 (62)
To keep these nominal velocities constant, we must require ˙v = 0, ˙ω = 0, ˙si = 0, ˙ηi = 0 for i = f, w, h, v, fromwhich we also have ˙pv = 0, ˙ρω = 0, ˙pi = 0, ˙ρi = 0 for i = f, w, h, v. Furthermore, the centripetal and Coriolisforces depend on ω so that ∂T/∂qi = 0 for all i = f, w, h, v. The remaining equations that must be satisfied are
0 = F
0 = M0 = −Kiqi + Ui
0 = −Kiξi + Ψi, i = f, w, h, v (63)
Once a forward velocity is chosen for the steady level flight, the corresponding trim solution can be obtained bysolving these equations for the nominal pitch angle θ, static deformations qi and ξi for all i = f, w, h, v as wellas nominal control input vector, u.
VIII. Numerical Model
In this paper, numerical results will be given for the NASA’s Generic Transport Model (GTM). The structures offuselage, wing and horizontal and vertical stabilizers are modeled as hollow beams with constant thicknesses. Toapproximate the mass and stiffness distributions of the bodies, we use the cross-sectional properties of the model,such as geometric center, cross-sectional area, area moments of inertia etc., at some finite number of sections onthe respective components. A distribution of the geometric centers on the individual components are shown in
9
Fig. 1. For each component i, we fit the respective geometric centers to a straight line and choose this straightline to be xi axis (Fig. 1). We assume that xi coincides with the elastic axis of the component i.
The orientation of the component body axes xiyizi can be obtained from xyz by a sequence of rotations. Thesame rotation sequence used in construction of C will be used for the construction of individual Ci. The Eulerangles for each of the flexible components are listed in Table 1:
In this numerical example, we consider a steady level flight at Mach 0.8 and 30, 000 ft. Speed of sound at thisaltitude is a = 994.66 so that VX = 0.8 × 994.66 = 795.73 ft/s. We use n = 5 shape functions for each of thedisplacements so that the total degrees of freedom of the system is 6 + 12n = 66, and hence, the order of thediscrete system, Eqs. (45), is 2(6 + 12n) = 132. From Eqs. (63), the trim solution for this flight is
θ =
[00.05560
]rad, qf =
−0.506497−0.063472−0.013547−0.003964−0.002508
ft, qw =
−0.0161290.0004920.0000690.0000130.000001
ft, qh =
−0.0003210.0000090.000001
−0.000000−0.000000
ft,
qv = 0, ξf = 0, ξw =
0.003503−0.000537−0.000075−0.000030−0.000010
rad, ξh =
0.000430−0.000048−0.000000
0.0000040.000005
rad, ξv = 0, u =
5326.200.02450
lb
rad(64)
Stability of the aircraft for this steady level flight can be addressed by computing the eigenvalues of thecoefficient matrix A. The first 4 of 132 eigenvalues are zero and the remaining are nonzero with negative realparts. First 40 eigenvalues are listed in Table 1. The first 12 eigenvalues are dominated by the rigid body motionof the aircraft, are similar to the eigenvalues encountered in flight dynamics. The remaining eigenvalues aredominated by the elastic motion and are similar to the eigenvalues encountered in aeroelasticity.
For a reduced order system, we choose m = 2 so that the total number of degrees of freedom of the reducedorder system is 6 +m = 8, and the order of the system is 2(6 + m) = 16. Trim solution for the reduced ordersystem is
θ =
00.05430
rad, ζ =
[0.124781
−0.225485
]ft, u =
5179.300.03020
lb
rad (65)
IX. Conclusions
This paper presents a model development for the NASA’s Generic Transport Model (GTM) Aircraft. The aircraftcomponents are treated as hollow beams with constant thicknesses. The equations of motions are presented inthe form of a set of nonlinear ordinary differential equations. These equations are used to obtain trim solutionfor steady flight and to address the stability of aircraft about the desired flight. They will be used to design anadaptive control to steer the aircraft to follow desired unsteady flights.
References
[1] Tao, G. Adaptive control design and analysis, John Wiley & Sons, Hoboken, NJ, 2003.
[2] Meirovitch, L. and Tuzcu, I., June 2003, “Integrated Approach to the Dynamics and Control of ManeuveringFlexible Aircraft,” NASA CR-2003-211748.
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[3] Meirovitch, L. and Tuzcu, I., “Unified Theory for the Dynamics and Control of Maneuvering Flexible Air-craft,” AIAA Journal, Vol. 42, No. 4, 2004, pp. 714-727.
[4] Meirovitch, L. and Tuzcu, I., “Time simulations of the Response of Maneuvering Flexible Aircraft,“ AIAAJournal of Guidance, Control, and Dynamics, Vol. 27, No. 5, 2004, 814-828.
