introduction to theoretical aeroelasticity for the aircraft · draft introduction to theoretical...
TRANSCRIPT
DRAFT
Introduction to Theoretical Aeroelasticity for theAircraft
Luigi Morino c⃝ Franco Mastroddi c⃝
March 2018
Contents
An Introductory Example – Panel flutter 1
Structural operator: solution of the structural problem (no aerodynamics) . . . . . . . 1
Structural and aerodynamic operators: solution of the aeroelastic problem . . . . . . . 3
The role of Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Response in critical conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1 Basic Issues on Structural Dynamics for Aeroelasticity 13
1.1 Discrete Structural Dynamics and Lagrange Equations . . . . . . . . . . . . . . . 13
1.2 Co-ordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.1 Finite Elements vs Modes of Vibration . . . . . . . . . . . . . . . . . . . 19
1.2.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3 ODE for Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.1 Second order systems of linear ODE for structural problems: free un-
damped response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4 Second order systems of linear ODE for structural problems: driven undamped
response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Thermodynamics and constitutive equations 31
2.1 Second Principle of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Thermodynamics of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
i
2.2.1 Entropy Evolution Equation . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Thermodynamics of Solids∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Entropy Evolution Equation∗ . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Elastic Solids∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.3 Alternative Formulation – Free Energy∗ . . . . . . . . . . . . . . . . . . . 39
2.4 Constitutive equations for isotropic fluids . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Stokesian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Constitutive equations for isotropic solids ∗ . . . . . . . . . . . . . . . . . . . . . 42
2.5.1 Linearly-elastic and Hookean solids . . . . . . . . . . . . . . . . . . . . . . 42
2.5.2 Navier equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Incompressible Unsteady Potential Flows 49
3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Incompressible Potential Flows (Non-Lifting Case) . . . . . . . . . . . . . . . . . 50
3.2.1 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Irrotational and Potential Flows . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.3 Differential Formulation for the Velocity Potential . . . . . . . . . . . . . 52
3.2.4 Boundary Integral Representation for Poisson Equation∗ . . . . . . . . . . 53
3.2.5 Boundary Integral Formulation for Velocity Potential ∗ . . . . . . . . . . 54
3.2.6 The Value of E on S; Integral Equation . . . . . . . . . . . . . . . . . . . 55
3.2.7 Integral Equation; Discretization ∗ . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Incompressible potential flows (Lifting Case) . . . . . . . . . . . . . . . . . . . . 58
3.3.1 Wakes in Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.2 Wake Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.3 Formulation for Potential Flows . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.4 Integral Equation; Wake Transport . . . . . . . . . . . . . . . . . . . . . 62
3.3.5 Wake Generation; Trailing-Edge Condition . . . . . . . . . . . . . . . . . 64
3.3.6 Discretization of Incompressible-Flow Formulation . . . . . . . . . . . . . 65
ii
4 Compressible Unsteady Potential Flows 73
4.1 Compressible Potential Flows Model . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Airplane Potential Aerodynamics Integral formulation . . . . . . . . . . . . . . . 81
4.3.1 Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Unsteady Aerodynamic Modeling for Fixed-Wing Aeroelasticity 93
5.1 Generalized-Aerodynamic Force matrix . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1.1 Boundary-condition discretization: matrix E1 . . . . . . . . . . . . . . . . 95
5.1.2 Uncompressible and compressible BEM: matrix E2 . . . . . . . . . . . . . 97
5.1.3 Bernoulli-theorem discretization: matrix E3 . . . . . . . . . . . . . . . . . 97
5.1.4 Generalized-force Projection: matrix E4 . . . . . . . . . . . . . . . . . . . 100
5.1.5 p-dependency of the GAF matrix . . . . . . . . . . . . . . . . . . . . . . . 100
5.2 Steady and Unsteady Lifting-Surface Methods . . . . . . . . . . . . . . . . . . . . 101
5.2.1 Incompressible lifting-surface method . . . . . . . . . . . . . . . . . . . . 102
5.2.2 Subsonic lifting-surface method . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.3 2-D Analytical solution: Theodorsen theory and Wagner problem . . . . . 105
5.3 Finite-state approximations for the GAF matrix . . . . . . . . . . . . . . . . . . 111
5.3.1 Fixed-pole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3.2 Varing-poles approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3.3 The linearized Lagrange equations for the aeroelasticity using the finite-
state approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Stability and Response Problems for Fixed-Wing Aeroelasticity 121
6.1 Aeroelastic Stability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.1.1 Some issues on stability of linear representations of aeroelastic systems . . 122
iii
6.1.2 Stability via numerical iterations: k and p− k methods . . . . . . . . . . 124
6.1.3 Stability via Finite-State approximation . . . . . . . . . . . . . . . . . . . 131
6.1.4 The role of the critical eigenvector in the stability analysis . . . . . . . . . 132
6.2 Aeroelastic response problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2.1 Static Aeroelastic response of a wing in horizontal flight . . . . . . . . . . 134
6.2.2 Static Aeroelastic response of a wing to an aileron step angle: aileron
effectiveness and reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.3 Dynamic Aeroelastic response to an aileron step angle: finite-state and
not finite-state approximation . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.2.4 Dynamic Aeroelastic response to a deterministic or stochastic gust . . . . 144
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7 Some Issues on Aeroservoelasticity of Fixed Wings 157
7.1 Aeroelastic model for the control theory . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 Syntesis of an optimal controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8 Nonlinear Aeroelasticity and Bifurcations via Singular-Perturbation Meth-
ods 169
8.1 Duffing equation model - A simple example of MTS application . . . . . . . . . 170
8.1.1 Multiple-time scales method applied to Duffing equation . . . . . . . . . . 171
8.2 Limit cycle and Hopf bifurcation via Multiple-Time Scales method . . . . . . . . 177
8.2.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.2.2 Discussion of nonlinear stability . . . . . . . . . . . . . . . . . . . . . . . . 182
8.2.3 Some issues on nonlinear aeroelastic control . . . . . . . . . . . . . . . . . 184
8.3 An aeroelastic example: nonlinear panel flutter . . . . . . . . . . . . . . . . . . . 188
8.3.1 Metodo di Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
iv
A International requirements for gust responce criteria 199
A.1 FAR (Pratt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
A.2 JAR 25.341 - Gust and turbulence loads . . . . . . . . . . . . . . . . . . . . . . . 200
A.3 ACJ 25.341(B) - Strength and deformation (Interpretative material) . . . . . . . 202
v
An Introductory Example
i – Panel flutter
In the present introductory chapter the problem of the aeroelastic behavior of a wing skin panel
in a supersonic flow has been presented. This model, althougth associated to a specific research
branch born and developed in the past 50’s (Refs. [1, 2]), can be considered as particoularly
valid in a teaching point of view: indeed, most of the issues concerning modelling, discretization
and methods of analysis in linear (and nonlinear, Refs. [3, 11]) aeroelasticity can be introduced
with it.
Consider a simply supported panel or, more simply, a beam (see Fig. 1) described by time-
space partial differential equation
EIw′′′′
+Nw′′+ kw + ρw = P (w) (1)
with boundary coditions w(0, t) = 0, w(l, t) = 0, w′′(0, t) = 0, and w
′′(0, t) = 0, and initial
condition w(x, t) = w0(x) and w(x, t) = w0(x), with w0(x) and = w0(x) assigned x functions.
Note that, in this model, the function P (w) represents the aerodynamic distributed load which
is typically dependent on the displacement solution w(t) in aeroelastic applications.
Structural operator: solution of the structural problem (no aero-dynamics)
Before studying panel flutter, consider free vibrations (no aerodynamic, P ≡ 0) which include
only the structural operator here defined as LS• := EI(•)′′′′ + N(•)′′ + k(•). Indeed, the
eigenvalues λn and the eigenfunctions ϕn(x) assiciated to the structural operator LS are defined
by the differential problem in the x variable only
LSϕn(x) = λnϕn(x) n = 1, 2, ....∞ (2)
1
2
Figure 1: Panel-flutter beam-like model
with the same boundary conditions assumed for the panel (i.e., simply supported; the depen-
dency of ϕn(x) by x will be not indicated for simplicity in the following). Note that functions
ϕn(x) = sinnπx
l(3)
satisfy the assigned boundary conditions (ϕn(0) = 0, ϕn(l) = 0, ϕn(0)′′= 0, and ϕn(l)
′′= 0).
Moreover, in order to satisfy the Eqs. 2 and 3, one has for the eigenvalues
λn = EI(nπ
l)4 −N(
nπ
l)2 + k (4)
Indeed, any structural operator, because of its intrinsic properties, will always give real and
positive eigenvalues (see Eq. 4) and orthogonal eigenfunctions ϕn, which means, by introducing
the inner-product operation ⟨f(x), g(x)⟩ :=∫ l0 f(x)g(x)dx, that
⟨ϕn, ϕm⟩ = δnm ⟨ϕm, ϕm⟩ (5)
Thus, the free-vibrations beam problem (with no aerodynamics, Eq. 1 with P = 0), becomes
LSw + ρw = 0 (6)
Because of the orthogonality (implying undependency) of the eigenfunctions, one can exactly
express the solution in terms of eigenfunctions:
w(x, t) =∞∑n=1
wn(t)ϕn(x) (7)
so, using Eqs. 2 and 7, Eq. 6 becomes:
0 = LSw + ρw = LS
( ∞∑n=1
ϕnwn
)+
∞∑n=1
ϕnwn
=∞∑n=1
ϕn (λnwn + ρwn) (8)
AEROELASTIC MODELING – L. Morino F. Mastroddi c⃝ 3
Thus, because of the orthogonality (Eq. 5) of the functions ϕn, the proevious equation, once
projected to any function ϕm, gives ρwm + λnwm = 0, or
wm + ω2mwm = 0 m = 1, 2, ...∞ (9)
These represent ordinary differential second order equations of infinite and de-coupled linear
harmonic oscillators having natural frequency ωm, with ω2m := λm/ρ > 0. By this result, one
can conclude that the mathatematica concepts of eigenfunctions and eigenvalues correspond to
the physical concepts of natural modes and natural frequencies of vibration for a linearly elastic
structure.
Structural and aerodynamic operators: solution of the aeroelasticproblem
Now assume a supersonic flow (P = 0). Use 2D steady state Supersonic Potential Flow Theory
(Glauert-Ackeret, Ref. [1])
Pressure: P = −λdwdx
(note that this correspond to the 2-D partial-differential equation model for the potential flow
(M2∞ − 1)φxx − φyy = 0). Namely, a further operator, namelt, an aerodynamic operator1
LA(•) := λd(•)dx
= 0 (10)
can been introduced in the equation. Thus, the final differential problem becomes
EIw′′′′
+Nw′′+ λw′ + kw + ρw = 0 (11)
or, using the introduced operators symbols,
LSw + LAw + ρw = 0
Although in presence of unsteady aerodynamic load, let us use again Normal Modes of Vi-
bration, namely, the eigenfunctions ϕn, for solving the problem. This item is widely applied in
Aeroelasticity.
By substituting Eqs. 7 and 3 into Eq. 11, one has
∞∑n=1
(EI
n4π4
l4−N
n2π2
l2+ k
)wn sin
nπx
l
+ λ∞∑n=1
wn cosnπx
l+ ρ
∞∑n=1
w sinnπx
l= 0 (12)
1It should be noticed that the introduced aerodynamic operator is not self-adjoint as the structural operator.
4
By projecting onto ϕk = sin kπxl – Galerkin Method – and considering the orthogonality Eq.
5 ⟨sin
nπx
l, sin
kπx
l
⟩= δnk
l
2(13)
one has
ρω2kwk +
∞∑n=1
⟨LAϕn, ϕk⟩wn + ρwk = 0 k = 1, 2, ...∞ (14)
or
ρω2kwk + λ
∞∑n=1
aknwn + ρwk = 0 k = 1, 2, ...∞ (15)
where akn are entries of an aerodynamic matrix (see later for definition).
Note that Galerkin approach is typically considered an approximation: in the limit n, k −→∞would be exact.
CONTINUUM −→︸︷︷︸exact
DISCRETE −→︸︷︷︸approx.
FINITE DIMENSION
Next, for the sake of simplicity, all the assumed summation on the eigenfunctions (modes) are
truncated up to N = 2. This corresponds to discretize the problem in space and this is a typical
assumption in aeroelasticity whenever ona can acept to study the given problem in a limited
bandwidth: namely, the bandwith defied by the eigenvalues/natural-frequecies correspondng to
the assumed modes, the two in the present case.
From the Eq. 15 one has
akn :=2
l
∫ l
0
nπ
lsin
kπx
lcos
nπx
ldx (16)
or, in the general case:
akn =
∫ l
0
ϕkϕ′n
(ϕk, ϕk)dx = [ϕk, ϕn]
l0 −
∫ l
0ϕ′kϕndx (17)
= −ank ←− Anti− symmetric matrix
Therefore, by defining Λ := λa12/ρ, one has the final system of two second order ordinary
differential equations{w1
w2
}+
([ω21 00 ω2
2
]+ Λ
[0 1−1 0
]){w1
w2
}= 0 (18)
Solve: system of differential equations:
wk = wkest[
s2 + ω21 +Λ
−Λ s2 + ω22
]{w1
w2
}= 0
AEROELASTIC MODELING – L. Morino F. Mastroddi c⃝ 5
Det = 0, so
(s2 + ω21)(s
2 + ω22) + Λ2 = 0
Therefore,
s4 + (ω21 + ω2
2)s2 + ω2
1ω22 + Λ2 = 0
Therefore,
s2 = −ω21 + ω2
2
2±
√√√√(ω21 + ω2
2
2
)2
− ω21ω
22 − Λ2
s2 = −ω21 + ω2
2
2±
√√√√(ω21 − ω2
2
2
)2
− Λ2
For Λ = 0
s2 = −ω21 + ω2
2
2± ω2
1 − ω22
2
s2 = −ω21,−ω2
2
As Λ is increased : s2 = (average value) + ...
CRITICAL VALUE is when the discriminant is zero - the flutter boundary is:
ΛFL =∣∣∣ω2
1−ω22
2
∣∣∣note: Λ is proportional to 1
2 ρU2∞
→ the closer the two roots are, the earlier flutter occurs.
Then, one may use (to response solution)
wi =4∑
k=1
ckw(k)i eskt i = 1, 2 (19)
where w(k) are the eigenvectors corresponding to sk.
The role of Damping
Starting from the panel flutter undamped model
EIw′′′′ +Nw′′ + kw + λw′ = −ρw
a damping contribution could be taken into account as in the following:
• Structural damping is the least clear (viscous is clear but difficult to calculate)
6
• From piston theory, it can be shown that damping adds a term of type µw to left hand
side.
• Also E = complex in the frequency domain - viscoelastic material.
global viscous effects =⇒ ηEIw′′′′ + µw
Then, one obtains for the discretized problem
(EIk4π4
l4− Nk2π2
l2+ k)wk + λ
∑n
aknwn + (ηEIk4π4
l4+ µ)wk + ρwk = 0
where
ζn :=
(ηEI
n4π4
l4+ µ
)/ρ
So, for N = 2 :
w1 + ζ1w1 + ω21w1 +
λa12ρ
w2 = 0
w2 + ζ2w2 + ω22w2 −
λa12ρ
w1 = 0
• Let us study the stability (Det = 0)
{s2 + ζ1s+ ω2
1 Λ−Λ s2 + ζ1s+ ω2
2
}= 0
(s2 + ζ1s+ ω21)(s
2 + ζ1s+ ω22) + Λ2 = 0
(s4 + (ζ1 + ζ2)s3 + (ω2
1 + ω22 + ζ1ζ2)s
2 + (ζ1ω22 + ζ2ω
21)s+ ω2
1ω22 + Λ2 = 0
• We want to find the stability boundary- this is where flutter occurs (not concerned with
how stable); at the stability boundary → 4 roots with damping. Assume: s = ±iω
ω4 − i(ζ1 + ζ2)ω3 − (ω2
1 + ω22 + ζ1ζ2)ω
2 + i(ζ1ω22 + ζ2ω
21)ω + ω2
1ω22 + Λ2 = 0
Thus (equating to zero imaginary and real part respectively)
ω2F =
ζ1ζ1 + ζ2
ω22 +
ζ2ζ1 + ζ2
ω21
Therefore, ΛFL is given by
Λ2FL = −ω4
F + (ω21 + ω2
2 + ζ1ζ2)ω2F − ω2
1ω22
Undamped case:
ω2FL =
ω21 + ω2
2
2
Λ2FL =
(ω22 − ω2
1
2
)2
AEROELASTIC MODELING – L. Morino F. Mastroddi c⃝ 7
• If we have ζ1 = ζ2 (occurs if structural damping is equal to zero), ζ2ζ1ϵ(1, 16) min. and
max. values. Then,
ω2FL =
ω21 + ω2
2
2
And,
Λ2FL = [−(ω
21 + ω2
2
2)2 + (ω2
1 + ω22)(
ω22 + ω2
1
2)− ω2
1ω22 + ζ1ζ2ω
2FL]
Λ2FL = [(
ω22 − ω2
1
2)2 + ζ1ζ2ω
2FL]
Response in critical conditions
Note that using
w(x, t) =2∑
n=1
sin
(nπx
l
)wn(t) (20)
and considering the free response given by Eq. in critical condition (i.e., Λ ≡ ΛF ) when s1,2 =
±iωF one has
w(x, t) ≃ 2[(c1Rw
(F )1R − c1Iw
(F )1I
)cos(ωF t)−
(c1Iw
(F )1R + c1Rw
(F )1I
)sin(ωF t)
]sin
(1πx
l
)+ 2
[(c2Rw
(F )2R − c2Iw
(F )2I
)cos(ωF t)−
(c2Iw
(F )2R + c2Rw
(F )2I
)sin(ωF t)
]sin
(2πx
l
)where the contribution to the time response of the stable moodes are neglected. Note that
the quantitative measure of the contribution of the eigenfunction or structural modes ϕn(x) =
sin(nπx/l) to the free respones in critical conditions is represented, for any given initial conditions
(i.e., coefficients c1,2), by the components of the critical eigenvector w(F ).
Exercise
Let us consider the equation of a simple supported beam undergone to a supersonic
flow:
EI
(1 + ζ
∂
∂t
)wIV +NwII + λwI + kw = −ρw − µw + p(x, t) (21)
where EI is the bending stiffness, ζ is the modal structural damping, N is the
buckilng load, λ is an aerodynamic coefficient (given by the Ackeret theory), µ is an
aerodynamic damping, k is the constant associated with the elastic floor, ρ is the
beam linear mass density, and p(x, t) is an external applied load.
8
The continuous model has been discretized in the space throught the orthogonal
eigenfunctions of the structural operator L(•) := EI∂4(•)/∂x4 +N∂2(•)/∂x2 + k(•)defined as the functions ϕn(x), satisfying the given boundary conditions, and such
as Lϕn(x) = λnϕn where λn are the corresponding eigenvalues and ω2n := λn/ρ are
the squares of the natural (angular) frequencies of vibrations. The eigenfunctions
are given by ϕn(x) = sin(nπxl
)where l is the panel length.
Thus, the analytic solution of the problem is given by
w(x, t) =∞∑n=1
wn(t)ϕn(x) (22)
as any unknown functions wn(t) can be analytically determined for any given function
w(x, t). Next, applying the Galerkin method, i.e., projecting the Eq. 21 on the
generic function ϕm(x), one has: 2
∞∑n=1
{{[EI
(nπ
l
)4
−N
(nπ
l
)2
+ k
]wn(t) +
[ζEI
(nπ
l
)4
+ µ
]wn(t)
+ ρwn(t)
}⟨ϕn(x), ϕm(x)⟩+ λwn(t)
⟨ϕIn(x), ϕm(x)
⟩}= ⟨p(x, t), ϕm(x)⟩
Furthermore, if the problem is truncaed in a finite dimension N, i.e., the summation
is exthended up to N and the followinf matrix
anm :=
⟨ϕIn(x), ϕm(x)
⟩l/2
=2
l
∫ l
0
nπ
lsin
mπx
lcos
nπx
ldx ≡ −amn
is introduced, one obtains for the case N = 2 (two modes description{w1
w2
}+
[ζ1 00 ζ2
]{w1
w2
}+
([ω21 00 ω2
2
]+ Λ
[0 1−1 0
]){w1
w2
}=
{p1p2
}
where
ζn :=[ζEI(nπ/l)4 + µ
]/ρ
ω2n :=
(EI(nπ/l)4 −N(nπ/l)2 + k
)/ρ
Λ := λa12/ρ
pm =
⟨p(x, t)
ρl/2, ϕm(x)
⟩Next, considering the following data:
2As the internal product between two function is defined as ⟨a(x), b(x)⟩ :=∫ l
0a(x) b(x) dx, one has, because
of orthogonality, ⟨ϕn(x), ϕm(x)⟩ = l/2 δnm.
AEROELASTIC MODELING – L. Morino F. Mastroddi c⃝ 9
ζ1 = 0.01 s−1 ζ2 = 0.01 s−1
ω21 = 100000(rad/s)2 ω2
1 = 150000(rad/s)2
give a response to the following questions:
• Study the linear stability (p(x, t) = 0) of the system with respect to the param-
eter Λ displaying the obtained root loci considereing and not considering the
presence of the damping (for the undamped case, compare the obtained results
with the analytic prediction).
• The free response (note that the obtained eigenvectors are two complex conju-
gated pairs)
{w1(t)w2(t)
}=
2∑n
cn
{w
(n)1
w(n)2
}esnt + C.C.
where 2 is the number of the obtained eigenvalue pairs sn,
{w
(n)1
w(n)2
}the corre-
sponding eigenvectors, and cn complex constants depending on the prescribed
initial conditions Evaluate the free response when (stable case) Λ = 0.8ΛF (ΛF
is the flutter margin) with initial conditions
w(x, 0) = sin
(πx
l
)w(x, 0) = 0
and compare the obtained results with those that one would have obtained with
air-free and damping-free vibrations (Λ = ζ1 = ζ2 = 0).
• Study the forced response with initial condition equal to zero and with load
given by (impulsive gust load)
p(x, t)
ρl/2= P0δ(t)
with P0 = 1.
• Is it possible to find, in unstable conditions (Λ > ΛF , e.g., Λ = 1.2ΛF ), suitable
initial conditions such that the consequent time response does not diverge?
10
Bibliography
[1] Ashley, H., and Zartarian, G., “Piston Theory- A New Aerodynamic Tool for the Aeroe-
lastician”, Journal of the Aeronautical Sciences, Vol. 23, No. 12, Dec. 1956, pp. 1109-1118.
[2] Bolotin, V.V., Nonconservative Problems of the Theory of Elastic Stability, The Macmillan
Company, New York, 1963, pp. 274-312.
[3] Morino, L., “A Perturbation Method for Treating Nonlinear Panel Flutter
[4] Morino, L., and Kuo, C.C., “Detailed Extension of Perturbation Method for Panel Flut-
ter”, ASRL TR-164-2, March 1971, Aeroelastic and Structures Research Lab., Dept. of
Aeronautics and Astronautics, Massachussetts Institute of Technology, Cambridge, Mass.
Problems,” AIAA Journal, Vol. 7, No. 3, pp. 405-411, 1969.
[5] Smith, L. L., and Morino, L., “Stability Analysis of Nonlinear Differential Autonomous
System with Applications to Flutter,” AIAA Journal, Vol. 14, No. 3, 1976, pp. 333-341.
[6] Davis, W. H. Jr., “Nonlinear Flutter of Panels and Wings by Lie Transformation Pertur-
bation Method,” Master Thesis, College of Engineering, Boston University, Boston, Mass.,
1973.
[7] Dowell, E. H., “Nonlinear Oscillation of a Fluttering Plate,” AIAA Journal, Vol. 4, No. 7,
July 1966, pp. 1267-1265.
[8] Dowell, E. H., “Nonlinear Oscillation of a Fluttering Plate II”, AIAA Journal, Vol. 5, 1967,
pp. 1856-1862.
[9] Virgin, L.N., and Dowell, E.H., “Nonlinear Aeroelasticity and Chaos,” Chapter 15, in Ed.:
S. N. Atluri, Computational Nonlinear Mechanics in Aerospace Engineering, Progress in
Aeronautics and Astronautics, America Institute of Aeronautics and Astronautics, Wash-
ington D.C., 1992, pp. 531-546.
[10] Sipcic, S. R., Morino, L., “Dynamic Behavior of Fluttering Two-Dimensional Panels on an
Airplane in Pull-Up Maneuver,” AIAA Journal, Vol. 29, No. 8, August 1991, pp. 1304-1312.
11
12
[11] Sipcic, S.R., “The Chaotic Response of a Fluttering Panel: The Influence of Maneuvering,”
Nonlinear Dynamics, Vol, 1, pp. 243-264, 1990.