AB-267 Dinâmica e Controle de Aeronaves
Flexı́veis 2014
Flávio Silvestre
http://www.aer.ita.br/ flaviojs/
Departamento de Mecânica do Voo
Divisão de Engenharia Aeronáutica
Instituto Tecnológico de Aeronáutica
2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
PARTE IV
Dinâmica do Voo da Aeronave Moderadamente
Flexı́vel
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Introdução
Até agora:
◮
◮
◮
entendemos a dinâmica estrutural, que foi equacionada e escrita na
base modal
deduzimos as equações do movimento da aeronave rı́gida usando a
Mecânica Lagrangeana
caracterizamos o sistema de referência dos eixos médios, e
verificamos a sua simplificação para pequenos deslocamentos
elásticos
A seguir a Mecânica Lagrangeana será empregada para chegarmos às equações da aeronave flexı́vel.
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Conclusões mais importantes
◮
Mecânica Lagrangeana:
d
dt
∂L
∂ q̇j
−
∂L
= Qj j = 1, ..., n
∂qj
onde

fx
TT (G)  fy  · ∂ TT (G) R0 | + TT (G) p| dV
Qj =
B
B
B
I
B
I
B
I
∂qi
V
fz
Z


Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Conclusões mais importantes
◮
Análise Modal:
instantaneous
aircraft CG
ω
yB
xB
(G)
global body reference frame
(linearised mean axes)
(G)
element dm
p
pd = Φη
pd
pr
zB
(G)
R0
R
xI
Flávio Silvestre
AB-267 2014
OI
yI
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Conclusões mais importantes
◮
Eixos Médios:
Z
Z
V
δp
dm = 0 (elastic linear momentum)
δt
δp
dm = 0 (elastic angular momentum)
δt
V
... e assumindo pequenos deslocamentos elásticos:
Z
pd dm = 0
p×
V
Z
pr × pd dm = 0
V
NOTA:
◮ estas condições são respeitadas pelos modos de vibração livre
◮ os eixos médios coincidem com os eixos do corpo não deformado
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Conclusões mais importantes
A seguir:
◮
◮
trabalharemos as expressões da energia cinética e potencial, bem
como adicionar um potencial de dissipação resultado do atrito
estrutural, e encontrar as equações do movimento da aeronave rı́gida
encontraremos as forças generalizadas em cada grau de liberdade
◮
◮
novidade: força generalizada nos graus de liberdade elásticos
entenderemos como o estado (ângulos de ataque, derrapagem,
velocidade, velocidades angulares) da aeronave provoca deformações,
bem como de que forma as deformaçoes causam variação das forças
que atuam na aeronave
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
T
=
1
2
Z
Z
V
Z
Z
1
1
dR0 dR0
δp δp
(ω × p) · (ω × p) dm +
·
dm +
·
dm +
dt
2 V
2 V δt δt
V dt
T1
T2
T3
Z
Z
dR0
δp
dR0 δp
·
dm +
· (ω × p) dm +
· (ω × p) dm
dt
δt
V dt
V δt
T4
T5
T6
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
◮
Kinetic energy term T1
Being u, v and w the components of the aircraft velocity vector
written in the BRF, T1 becomes:
T1
=
=
Z
1 dR0 dR0
·
dm
2 dt
dt V
1
m u2 + v 2 + w 2
2
wherein m is the aircraft total mass.
Flávio Silvestre
AB-267 2014
dR0
dt ,
when
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
◮ Kinetic energy term T2
If p has coordinates x , y and z when written in the BRF, the cross product
ω × p can be converted to the following matrix multiplication:


0
z −y
0
x ω
ω × p =  −z
y −x
0
so that the term T2 results in:
T2
=
=

T 
0
z −y
0
z −y
1
0
x  ω dm
0
x   −z
ω T  −z
2 V
y −x
0
y −x
0

  2

2
Z
y +z
−xy
−xz


1 T

−xy x 2 + z 2
−yz  dm ω
ω

 V
2
−xz
−yz x 2 + y 2
Z

Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
The term between braces is the inertia tensor J. As the vector p contains
the elastic displacements, and these are in turn time-dependent, the inertia tensor is not constant and depends on the modal amplitudes as well.
However, assuming small elastic displacements, this effect can be neglected and the inertia tensor J is assumed to be constant. Thus the term T2
simplifies to:
T2 =
Flávio Silvestre
1 T
ω Jω
2
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
◮
Kinetic energy term T3 :
Vector p can be split in its rigid part (pr ) and its part regarding the elastic
deformation (pd ). As the former is static relative to the BRF, using the
modal representation of pd (pd ≈ Φ (pr ) η) the kinetic energy term T3
simplifies to:
T3
=
=
=
Z
δpd δpd
1
·
dm
2 V δt
δt
Z
1 T
η̇
ΦT (pr ) Φ (pr ) dm η̇
2
V
1 T
η̇ µη̇
2
wherein µ is the modal mass matrix.
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
◮
Kinetic energy term T4 :
Again writing p = pr + pd ,
dR0
·
T4 =
dt
Z
V
δpd
dm
δt
From the first constraint of the mean axes (regarding the linear momentum)
the integral is null, so that:
T4 = 0
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
◮
Kinetic energy term T5 :
T5
=
=
Z
dR0
·
(ω × p) dm
dt
V
Z
dR0
· ω×
p dm
dt
V
Again from the first constraint of the mean axes the integral is null, so
that:
T5 = 0
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
◮
Kinetic energy term T6 :
Using the vectorial identity, for general vectors a, b and c, that a·(b × c) =
(c × a) · b, the kinetic energy term T6 simplifies to:
T6
=
=
δp
· ω dm
p×
δt
V
Z
δp
dm · ω
p×
δt
V
Z The enforcement of the second mean axes constraint guarantees that the
latter integral is null, so that:
T6 = 0
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
Finally the kinetic energy expression achieves the following simplified form:
T =
1
1
1
m u 2 + v 2 + w 2 + ω T Jω + η̇ T µη̇
2
2
2
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética
Nevertheless, to apply the Lagrange equations of motion, the kinetic energy
must be displayed as a function of the chosen generalised coordinates.The
components of the airspeed vector, u, v , and w , as well as the components
of the angular velocity vector, p, q and r must be written in terms of xCM ,
yCM , zCM , ψ, θ, ϕ, and their time-derivatives.
⇔
⇔
dR0
δR0
=
+ ω × R0 ⇔

 
 
dt  δt
xCM
p
ẋCM
u
 v  =  ẏCM  +  q  ×  yCM  ⇔
zCM
r
żCM
w
u = ẋCM + qzCM − ryCM
v = ẏCM + rxCM − pzCM
w = żCM + pyCM − qxCM
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Energia Cinética

⇔







0
0
p
φ̇
 q  = Tφ Tθ Tψ  0  + Tφ Tθ  θ̇  + Tφ  0  ⇔
r
ψ̇
0
0
p = φ̇ − ψ̇ sin θ
q = ψ̇ cos θ sin φ + θ̇ cos φ
r = ψ̇ cos θ cos φ − θ̇ sin φ
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
The Potential energy
The potential energy of the system U is the sum of the gravitational
potential energy, UG and the potential energy of deformation, US . The
gravitational potential energy is given by:
Z
R · G dm
UG = −
V
UG = −
Z
V
(R0 + p) dm · G
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
The Potential energy
Due to the constraint
of the mean axes regarding the linear momentum,
R
the integral V p dm is null. Written in the inertial reference frame, the
position vector R0 is given by:
R 0 |I
=
=
TT
B(G) I R0 |B
xCM
TT
B(G) I
yCM
zCM
T
Substituting above, the expression of the gravitational potential energy UG
simplifies to:
UG = mg (xCM sin θ − yCM cos θ sin φ − zCM cos θ cos φ)
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
The Potential energy
The potential energy of deformation is given by:
Z
1
US =
σ T ǫ dV
2 V
In the case of finite masses, we wrote the above relation as:
US =
1 T
q Kq
2
or in modal coordinates:
US
=
=
Flávio Silvestre
1 T
η γη
2
1 T
η µωn2 η
2
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
The structural dissipation
The structural damping forces are assumed to have a viscous nature, i.e.
linearly related to the velocity of elastic displacements, and the Rayleigh’s
dissipation function D (for details, see Bisplinghoff) can be written in terms
of the modal velocities:
1 T
η̇ β η̇
2
wherein β is the matrix of structural damping factors. Neglecting the
structural damping coupling among the elastic modes, the expression of
the dissipation function can be simplified to
D=
D = η̇ T µξωn η̇
ξ is the diagonal matrix of structural modal damping.
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
The Lagrangian Equations of Motion
Being L = T − U the Lagrangian of the system, according to the Lagrangian Mechanics the equations of motion described by the generalised
degrees of freedom read:
∂L
∂D
d ∂L
−
+
= Qi
dt ∂ q̇i
∂qi
∂ q̇i
wherein qi ∈ {xCM , yCM , zCM , ψ, θ, ϕ, η1 , · · · , ηne }. Qi is the generalised
force at the correspondent degree of freedom.
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
The Lagrangian Equations of Motion
From this point to the final form of the equations of motion, a long derivation process and algebraic manipulation is necessary, which can be
accomplished without any difficulty. Furthermore after the elimination of
the inertial coupling between rigid body and elastic degrees of freedom, the
equations of motion are formally the same as those for the rigid aircraft,
added by the ne second order differential equations of the elastic modes.
V̇
B(G)
= − ω|B(G) × V|B(G) + TB(G) I G|I +
ω̇|B(G) = −J−1 ( ω|B(G) × (J ω|B(G) )) + J−1
η̈ = −2ξωn η̇ − ωn 2 η + µ−1 Qη
Flávio Silvestre
AB-267 2014
Fext B(G)
Mext (G)
1
m
B
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Gereralised Loads
Qi =
Z
f |I ·
V
∂ R|I
dV
∂qi
f is the vector of external distributed forces per unit volume, which can be
T
written as f |B = fx fy fz
, when expressed in components of the
BRF. Then:

fx
TT (G)  fy  · ∂ TT (G) R0 | + TT (G) pr | + Φη dV
Qi =
B
B
B
I
B
I
B
I
∂q
i
V
fz
Z


Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Gereralised Loads
Fext B(G) =
M
and
Q ηk =
=
ext Z
B(G)
=
Z
Z
f |B(G) dV
V
pr × f |B(G) dV
V
T
f |B(G) TB(G) I TT
B(G) I
ZV
T
f |B(G) Φk dV
V
Flávio Silvestre
AB-267 2014
∂ (Φη)
dV
∂ηk
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Exercise
Consider a flat plate representation of a rectangular wing, with dimensions:
◮
◮
chord: 1m
span: 18m
This wing is subjected to a flow of velocity 40m/s, and an angle of attack
of 2 DEG, at see level. Consider only the first elastic mode of this wing, a
symmetric wing bending, with the following characteristics:
◮
◮
resonance frequency: 3Hz
modal shape: consider the 1st bending line of a homogeneous beam
(see Bisplinghoff), with wing tip deflection of 15cm, and associated
modal mass of 1kg m2 , no torsion
The aerodynamic is quasi-static (cℓα = 2π, ℓ = 21 ρV 2 ccℓα α), and for
accounting the wing tip effect, adopt Prandtl’s lifting line theory (see Anderson) for correcting the local cℓ α .
Flávio Silvestre
AB-267 2014
Introdução
Conclusões mais importantes
Energia Cinética
The Potential energy
The structural dissipation
The Lagrangian Equations of Motion
Gereralised Loads
Exercise
Exercise
Determine for the flight condition above:
◮
◮
◮
◮
◮
the generalised load Qη
the amplitude η
the deformed wing shape (display graphically)
find the analytical expression for the dependency of Qη with respect
to α, and normalising Qη by 21 ρV 2 Sc, find the analytical expression
for the derivative CQη ,α
for the case above, find the numerical value of CQη ,α
Flávio Silvestre
AB-267 2014