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Bifurcation analysis of a two-degree-of-freedomaeroelastic system with freeplay structural non-
linearity by a perturbation-incremental method
K.W. CHUNG† and C.L. CHAN
Department of Mathematics,
City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Email: makchung@cityu.edu.hk
AND
B. H. K. LEE
Aerodynamics Laboratory, National Research Council, Institute for Aerospace
Research, Ottawa, Ontario, Canada K1A 0R6
Abstract
A perturbation-incremental (PI) method is presented for the computation, contin-
uation and bifurcation analysis of Limit Cycle Oscillations (LCO) of a two-degree-of-
freedom aeroelastic system containing a freeplay structural nonlinearity. Both stable
and unstable LCOs can be calculated to any desired degree of accuracy and their sta-
bilities are determined by the Floquet theory. Thus, the present method is capable
of detecting complex aeroelastic responses such as periodic motion with harmonics,
period-doubling, saddle-node bifurcation, Neimark-Sacker bifurcation and the coexis-
tence of limit cycles. Emanating branch from a period-doubling bifurcation can be
constructed. This method can also be applied to any piecewise linear systems.
Keywords: Aeroelasticity, freeplay model, limit cycle oscillation, continuation, period-
doubling bifurcation.
† Corresponding author. Tel.: (852)27888671; fax: (852) 27888561.
1
1. Introduction
The study of the dynamic behaviour of aircraft structures is crucial in flut-
ter analysis since it provides useful information in the design of aircraft wings
and control surfaces. Concentrated structural nonlinearities can have significant
effects on the aeroelastic responses of aerosurfaces even for small vibrational am-
plitudes. One particular concentrated structural nonlinearity that has received
much attention is the bilinear or freeplay spring, which is a representative of
worn or loose control surface hinges.
A freeplay nonlinearity in pitch was first studied by Woolston et al. [1]
and Shen [2]. They showed that Limit Cycle Oscillation (LCO) might occur
well below the linear flutter boundary. McIntosh et al. [3] performed experi-
mental work with a wind tunnel model having two degrees of freedom. They
found that the behaviour of the airfoil was extremely dependent on the initial
pitch deflection. Yang and Zhao [4] considered LCO of a two-dimensional air-
foil in incompressible flow using the Theodorsen function. Two stable LCOs
of different amplitudes were detected for some airspeeds. Hauenstein et al. [5]
investigated theoretically and experimentally a rigid wing flexibly mounted at
its root with freeplay nonlinearities in both pitch and plunge degrees of free-
dom. They obtained excellent agreement between theoretical and experimental
results, and concluded that chaotic motion did not occur with a single root non-
linearity. However, Price et al. [6,7] pointed out that such conclusion was not
true. Tang and Dowell [8] analyzed the flutter instability and forced response of
a helicopter blade wind-tunnel model with no rotation. For a narrow range of
2
airspeeds, the system exhibited both LCO and chaotic behaviour. Kim and Lee
[9] investigated the dynamics of a flexible airfoil with a freeplay non-linearity.
They observed that LCO and chaotic motions were highly influenced by the
pitch-plunge frequency ratio.
An aeroelastic model with freeplay nonlinearity has been investigated using
analytical methods based on describing function and harmonic balance methods.
By using the Incremental Harmonic Balance (IHB) method, Lau and Zhang [10]
studied nonlinear vibrations of piecewise-linear systems in which the freeplay
nonlinearity is a special case. The main drawback of using harmonic balance
methods to investigate freeplay nonlinearity is that the second derivative of an
approximate solution obtained by such methods is continuous while that of the
exact solution is discontinuous at the switching points where changes in lin-
ear subdomains occur. Such inconsistency between the exact and approximate
solutions may lead to serious error in the prediction and analysis. To over-
come this drawback, Liu et al. [11] introduced the Point Transformation (PT)
method which could track the system behaviour to the exact point where the
change in linear subdomains occurred. Moreover, complex non-linear aeroelastic
behaviour such as periodic motion with harmonics, periodic doubling, chaotic
motion and the coexistence of stable limit cycles can be detected. However, The
PT method is not capable of finding unstable periodic solutions and thus is not
suitable for performing parametric continuation.
Recently, Chan et al. [12] presented a Perturbation-Incremental (PI) method
for the study of limit cycles and bifurcation analysis of strongly nonlinear au-
3
tonomous oscillators with arbitrary large bifurcation values. The PI method
is a semi-analytical and numerical process which incorporates salient features
from both the perturbation method and the incremental approach. This method
was later extended to investigate coupled nonlinear oscillators [13,14] and delay
differential equations [15].
In this paper, we extend the PI method to the continuation and bifurcation
analysis of an aeroelastic model with freeplay nonlinearity. In fact, the method
can also be applied to any piecewise-linear system. Both stable and unstable
periodic solutions can be calculated and their stabilities are determined by us-
ing the Floquet theory. The paper is organized as follows. A brief description
of an aeroelastic model with freeplay non-linearity is given in Section 2. The
PI method is described in Section 3. Section 4 deals with the computation of
stability of a LCO. Bifurcation analysis is considered in Section 5, followed by
conclusions in Section 6 and two appendices.
4
Nomenclature
ah non-dimensional distance from air-
foil mid-chord to elastic axis
t time
b airfoil semi-chord U free-stream velocity
Ch, Ca damping coefficients in plunge and
in pitch
U∗ non-dimensional velocity, U∗ =
U/(bωα)
CL(τ), CM (τ) aerodynamic lift and pitching mo-
ment coefficients
U∗L linear flutter speed
f non-dimensional linear frequency X eight-dimensional vector variable
Ḡ(h), M̄(α) nonlinear plunge and pitch stiffness
terms
xa non-dimensional distance from the
airfoil elastic axis to the centre of
mass
h plunge displacement α pitch angle of airfoil
Ia airfoil mass moment of inertia
about elastic axis
ε1, ε2 constants in Wagner’s function
Kξ, Kα stiffness in plunge and in pitch,
Kξ = Kh
ζa, ζξ viscous damping ratios in pitch and
in plunge
m airfoil mass µ airfoil/air mass ratio, µ =
m/(πρb2)
P (τ) externally applied forces ξ nondimensional plunge displace-
ment, ξ = h/b
p(t) forces acting on the airfoil τ non-dimensional time, τ = Ut/b
Q(τ) externally applied moments φ(τ) Wagner’s function
R1, R3 amplitudes of the first and the third
harmonics in the pitch motion
ψ1, ψ2 constants in Wagner’s function
r(t) moments acting on the airfoil ω fundamental frequency of the mo-
tion
ra radius of gyration about the elastic
axis
ωξ, ωα natural frequencies in plunge and in
pitch
S airfoil static moment about the
elastic axis
ω̄ frequency ratio, ω̄ = ωξ/ωα
2. The mathematical model
Figure 1 shows a sketch of a two-degree-of-freedom (2-dof) airfoil motion
in plunge and pitch. The plunge deflection is denoted by h, positive in the
downward direction, and α is the pitch angle about the elastic axis, positive
nose up. The elastic axis is located at a distance ahb from the mid-chord,
while the mass centre is located at a distance xab from the elastic axis, where
5
b is the airfoil semi-chord. Both distances are positive when measured towards
the trailing edge of the airfoil. The aeroelastic equations of motion for linear
springs have been derived by Fung [16]. For nonlinear restoring forces, the
coupled bending-torsion equations for the airfoil can be written as follows:
mḧ + Sα̈ +Chḣ+ Ḡ(h) = p(t), (1)
Sḧ + Iαα̈+ Cαα̇+ M̄ (α) = r(t), (2)
where the symbolsm, S, Ch, Iα and Cα are the airfoil mass, airfoil static moment
about the elastic axis, damping coefficient in plunge, wing mass moment of
inertia about elastic axis, and torsion damping coefficient, respectively. Ḡ(h)
and M̄ (α) are the nonlinear plunge and pitch stiffness terms, and p(t) and r(t)
are the forces and moments acting on the airfoil, respectively. Defining ξ = h/b,
Kξ = Kh, xα = S/bm, ωξ = (Kξ/m)1/2, ωα = (Kα/Iα)1/2, rα = (Iα/mb2)1/2,
ζξ = C/2(mKh)1/2 and ζα = C/2(IαKα)1/2, equations (1) and (2) can be
written in nondimensional form [17] as follows:
ξ′′ + xαα′′ + 2ζξω̄
U∗ξ′ + (
ω̄
U∗)2G(ξ) = − 1
πµCL(τ ) +
P (τ )bmU2
, (3)
xαr2αξ′′ + α′′ + 2
ζαU∗
α′ +1U∗2
M (α) =2
πµr2αCM(τ ) +
Q(τ )mU2r2α
, (4)
where G(ξ) = Ḡ(h)/Kξ and M (α) = M̄ (α)/Kα.
In equations (3) and (4), U∗ is a nondimensional velocity defined as U∗ =
U/bωα and ω̄ = ωξ/ωα, where ωξ and ωα are the uncoupled plunging and pitch-
ing mode natural frequencies, respectively, U is the free-stream velocity, and
the ′ denotes differentiation with respect to the non-dimensional time τ defined
6
Fig. 1. Schematic of airfoil with 2 dof motion.
as τ = Ut/b. CL(τ ) and CM (τ ) are the lift and pitching moment coefficients,
respectively, and µ is the airfoil/air mass ratio (m/πρ2). For incompressible
flow, Fung [16] gives the following expressions for CL(τ ) and CM(τ ):
CL(τ ) =π(ξ′′ − ahα′′ + α′) + 2π{α(0) + ξ′(0) + [12− ah]α′(0)}φ(τ )
+ 2π∫ τ
0
φ(τ − σ)[α′(σ) + ξ′′(σ) + (12− ah)α′′(σ)]dσ, (5)
CM(τ ) =π(12
+ ah){α(0) + ξ′(0) + (12− ah)α′(0)}φ(τ )
+ π(12
+ ah)∫ τ
0
φ(τ − σ){α′(σ) + ξ′′(σ) + (12− ah)α′′(σ)}dσ
+π
2ah(ξ′′ − ahα′′) − (
12− ah)
π
2α′ − π
16α′′, (6)
where the Wagner function φ(τ ) is given by
φ(τ ) = 1 − ψ1e−ε1τ − ψ2e−ε2τ (7)
and the constants ψ1 = 0.165, ψ2 = 0.335, ε1 = 0.0455 and ε2 = 0.3 are obtained
from Jones [18]. P (τ ) and Q(τ ) are the externally applied forces and moments,
7
respectively, and they are set to zero in this study.
Due to the presence of the integral terms in the integro-differential equations
(3) and (4), it is cumbersome to integrate them numerically. A set of simpler
equations was derived by Lee et al. [17] , and they introduced four new variables
w1 =∫ τ
0
e−ε1(τ−σ)α(σ)dσ, w2 =∫ τ
0
e−ε2(τ−σ)α(σ)dσ,
w3 =∫ τ
0
e−ε1(τ−σ)ξ(σ)dσ, w4 =∫ τ
0
e−ε2(τ−σ)ξ(σ)dσ. (8)
The resulting set of eight first-order ordinary differential equations by a suitable
transformation is given as
dX/dτ = f (X, τ ) (9)
where X = {x1, x2, . . . , x8} = {α, α′, ξ, ξ′, w1, w2, w3, w4} ∈ R8. Explicitly, the
system can be written as
x′1 = x2,
x′2 = a21x1 + a22x2 + a23x3 + a24x4 + a25x5 + a26x6 + a27x7,
+ a28x8 + j(d0(ω̄
U∗)2G(x3) − c0(
1U∗
)2M (x1)),
x′3 = x4,
x′4 = a41x1 + a42x2 + a43x3 + a44x4 + a45x5 + a46x6 + a47x7
+ a48x8 + j(c1(1U∗
)2M (x1) − d1(ω̄
U∗)2G(x3)),
8
x′5 = x1 − ε1x5,
x′6 = x1 − ε2x6,
x′7 = x3 − ε1x7,
x′8 = x3 − ε2x8, (10)
where the coefficients j, a21, ..., a28, a41, ..., a48, c0, d0, c1 and d1 are related
to the system parameters and their expressions are given in Appendix A.
The structural nonlinearities are represented by the non-linear functions
G(x3) and M (x1) in system (10). Generally speaking, for a freeplay model,
M (x1) is given by
M (x1) =
M0 + x1 − αf , x1 < αf ,
M0 +Mf (x1 − αf ), αf ≤ x1 ≤ αf + δ,
M0 + x1 − αf + δ(Mf − 1), x1 > αf + δ,
(11)
where M0, Mf , αf and δ are constants. Here, we give the expressions for
M (x1) in the pitch degree of freedom. Similar expressions for G(x3) in the
plunge motion can be written by replacing x1 with x3. A sketch of the freeplay
model is given in Fig. 2.
Consider the eight dimensional system given in (10) for a freeplay spring in
pitch and a linear spring in plunge, where M (x1) is given by (11) and G(x3) =
x3. According to the three linear branches of the bilinear function for a freeplay
model, the phase space X ∈ R8 can be divided into three regions, Ri(i = 1, 2, 3),
9
Moment M(α)
Displacement α
R1 R2
R3
αf
αf+δ
M0+α−α
f
M0+M
f(α−α
f)
M0+α−α
f+δ(M
f−1)
M0
Fig. 2. General sketch of a freeplay stiffness.
where each corresponds to a linear system:
X ′ = AX + F1 in R1 = {X ∈ R8|x1 < αf}, (12a)
X ′ = BX + F2 in R2 = {X ∈ R8|αf < x1 < αf + δ}, (12b)
X ′ = AX + F3 in R3 = {X ∈ R8|x1 > αf + δ}. (12c)
Here A and B are 8×8 constant matrices, and F1, F2 and F3 are 8×1 constant
vectors. The elements of A, B and Fi(i = 1, 2, 3) are determined by the system
parameters of the coupled aeroelastic equations, and they are given by
A =
A1 A2
A3 A4
, B =
B1 A2
A3 A4
(13)
and F1 = (M0 − αf )F , F2 = (M0 −Mfαf )F , F3=(M0 − αf + δ0(Mf − 1))F .
The elements of the 4 × 4 block matrices, Ai(i = 1, 2, 3, 4), B1 and the vector
F are defined in Appendix B with β = 1.
10
3. The Perturbation-Incremental (PI) method
Consider the freeplay model shown in Fig. 2. Let the Z −Y plane represent
the eight-dimensional phase space, where Z = {x1} and Y = {x2, x3, x4, x5, x6,
x7, x8}. Let Z1 and Z2 denote the switching subspaces Z = αf and Z = αf + δ,
respectively, where the linear systems change. The Z − Y phase space is now
divided by Z1 and Z2 into three regions R1, R2 and R3 as shown in Fig. 3(a).
The system response can then be predicted by following a general phase path.
Assuming that a motion initially starts at a point X1 in one of the switching
subspaces (say Z1) as shown in Fig. 3(a), the trajectory passes through R2, hits
Z2 atX2 and enters intoR3. Then, it returns toR2 throughX3 in Z2 and hits Z1
at X4. It enters into R1 and hits Z1 again atX5. The pointsXi, (i = 1, 2, 3, 4, 5)
are called switching points as they are located in the switching subspaces. Let
t1 be the travelling time of the trajectory fromX1 to X2 in region R2. Similarly,
let t2, t3 and t4 be the travelling times of the trajectory in regions R3, R2 and
R1, respectively. We note that the points X1 and X5 define a Poincaré map
in Z1. The trajectory becomes a LCO if X1 coincides with X5 (see Fig. 3(b)).
Since the system of equations in each region is strictly linear, the exact solutions
in R1, R2 and R3 can be expressed analytically. Therefore, for a given point X1
in Z1, X5 can be determined analytically.
The main idea of the PI method is to convert a LCO to an equilibrium point
of a Poincaré map in a switching subspace and consider a system of variational
equations of the map for parametric continuation. Same as in [7,11], the non-
dimensional velocity U∗ is used as the bifurcation parameter.
11
Z
Y
R1
R2
R3
αf
αf+δ
X1
X2
X3
X4
X5
t1
t2
t3
t4
(a)
Z
X1
X2
X4
X3
t1
t2
t3
t4
Y
(b)
Fig. 3. General trajectory (a) and a period-one trajectory (b) of system (10) with a
freeplay stiffness in pitch.
The procedure of the PI method is divided into two steps. The first step is
to obtain an initial solution for the continuation of the bifurcation parameter in
the second step.
3.1 Perturbation step
For a smooth dynamical system, small LCO can be obtained through Hopf
bifurcation. However, Hopf bifurcation theorems cannot be applied to a piece-
wise-linear system due to its low differentiability. Nevertheless, a piecewise-
linear system can undergo bifurcations which have similarities (but also dis-
crepancies) with the Hopf bifurcation [19]. Limit cycle bifurcation from center
in symmetric piecewise-linear systems was investigated in [20]. In fact, system
(10) with structural non-linearities defined in (11) is a symmetric piecewise-
linear system. A symmetric LCO may be obtained in the following way.
Assume that a pair of eigenvalues of the system in region R2 become pure
12
imaginary (say λ = ±iω, ω > 0) at a specific value of the bifurcation parameter
U∗ and u∼1
± iu∼2
are the corresponding eigenvectors. A periodic solution in the
linear subspace spanned by u∼1
and u∼2
can be expressed as
r∼(t) = p′eiωt(u
∼1+ iu
∼2) + p̄′e−iωt(u
∼1− iu
∼2) + u
∼0
= (p1 cos ωt− p2 sinωt)u∼1− (p2 cos ωt+ p1 sinωt)u∼2
+ u∼0, (14)
where p1 = p′+p̄′
2 ∈ R, p2 =p′−p̄′
2i ∈ R and u∼0∈ R8. Since r
∼(t) lies in region
R2, it follows from (12b) that
u∼0
=
−B−1F2 if det(B) 6= 0,
u∼3
+ u∼4
if det(B) = 0,(15)
where u∼3
satisfies the equation Bu∼3
+ F2 = 0 and u∼4is an eigenvector of the
zero eigenvalue. If the linear subspace spanned by u∼1
and u∼2
intersects both
the switching subspaces Z1 and Z2, then there exists a unique periodic solution
touching both Z1 and Z2 with maximal amplitude. Assume that, at t = 0, r∼(0)
is the switching point at Z1 (see Fig. 4). Then, r∼(T2 ) is the switching point at
Z2 where T is the period. It follows from (14) that
αf = p1u11 − p2u21 + u01
αf + δ = −p1u11 + p2u21 + u01
=⇒
p1u11 − p2u21 = −δ/2 (16a)
u01 = αf + δ/2 (16b)
where ui1 (i = 0, 1, 2) is the first component of u∼i. Since the tangents at the
switching points are orthorgonal to the Z−axis, the first component of ṙ∼(0) is
13
zero. Therefore, we have, from (14),
p1u21 + p2u11 = 0. (17)
From (16a) and (17), we obtain
p1 =−δu11
2(u211 + u221)
and p2 =δu21
2(u211 + u221)
. (18)
The periodic solution with maximal amplitude can be uniquely determined from
(15), (16b) and (18). As the bifurcation parameter is varied from the critical
value, a symmetric LCO traversing all three regions Ri(i = 1, 2, 3) may suddenly
appear.
Z
R3R2R1
Yα
f αf+δ
r∼(0) r
∼(T
2)
Fig. 4. Periodic solution with maximal amplitude in the linear subspace spanned by
u∼1
and u∼2
.
This step gives the location of the switching points and travelling time of an
initial LCO, which will be used in the incremental step.
3.2 Parameter incremental method – a Newton-Raphson procedure
14
To investigate the continuation in U∗, we note that, for a general LCO
traversing all three regions, the number of switching points is not restricted to
four. For example, the LCO in Fig. 5(a) contains six switching points. The
complete loop covering the entire region also contains a smaller loop, covering
the two regions R1 and R2. A complete loop is classified as a period-m LCO if
it covers the entire region m times. Although the LCO of Fig. 5(a) is of period-
one, the presence of a smaller loop indicates that it has a harmonic component.
In the subsequent sections, p-n-h denotes a period-n LCO with harmonics. In
Fig. 5(b), the LCO contains eight switching points and is of period-two.
Z
X1
X2
X3
X4
X5
X6
(a)
Y
Z
X1
X6
X5
X2
X3
X4
X7
X8
Y
(b)
Fig. 5. The LCOs contain (a) six and (b) eight switching points.
Assume that a LCO contains n switching points Xi(1 ≤ i ≤ n). Let
r∼i
(t)(1 ≤ i ≤ n) be the segment of LCO between the switching points Xi
and Xi+1 with Xn+1 = X1 and lie in the region Rpi(pi ∈ {1, 2, 3}). Let
λhj (j = 1, 2, . . . , 8) and v∼hjbe the eigenvalues and eigenvectors, respectively, of
the 8 × 8 matrices A if h = 1, 3 and B if h = 2. Then, r∼i
(t) can be expressed
15
analytically as
r∼i
(t) = v∼pi0
+8∑
j=1
kijeλpij tv
∼pij, 1 ≤ i ≤ n (19)
where kij ∈ R and v∼pij∈ R8. To simplify the calculation, the time t in r
∼i(t) is
defined in such a way that it counts only the time travelled in region Rpi. Since
r∼i
(t) is the trajectory from Xi to Xi+1(Xn+1 = X1) with travelling time ti, we
have
Xi = r∼i(0) = r
∼i−1(ti−1), 1 ≤ i ≤ n, (20)
with subscript 80′ replaced by 8n′ (i.e. r∼0
(t0) = r∼n(tn)). This replacement of
subscript 80′ by 8n′ will also be used in subsequent formulae derived from (20).
Substituting (19) into (12), we obtain
v∼pi0
= −C−1pi Fpi , 1 ≤ i ≤ n (21)
where Cpi =
A if pi = 1, 3,
B if pi = 2.
Furthermore, substituting (19) and (21) into (20), we obtain
Xi =8∑
j=1
kijv∼pij−C−1pi Fpi =
8∑
j=1
ki−1jeλpi−1j ti−1v
∼pi−1j−C−1pi−1Fpi−1 ,
1 ≤ i ≤ n. (22)
The period of a LCO is given by T =∑n
i=1 ti. Let Xi be expressed as
Xi =
αi
s∼∗i
where αi =
αf if Xi ∈ Z1,
αf + δ if Xi ∈ Z2,and s
∼∗i∈ R7. (23)
16
If the system parameters such as aij, U∗ in (10) are given, the matrices A, B
are determined and, thus, the eigenvalues λij , eigenvectors v∼ijin (22) can be
found. For a general LCO satisfying (22), assume that the switching subspaces
in which Xis lie are all given, i.e. αis are given for 1 ≤ i ≤ n. The LCO can be
determined by solving the 8×2×n = 16n equations in (22) with the unknowns
kij (8n of them), s∼∗i(7n) and ti (n) where 1 ≤ i ≤ n and 1 ≤ j ≤ 8. If one of
the switching points (say X1) is given, the complete loop including all the other
switching points and its period can be determined from (22). To consider the
continuation in U∗, a small increment of U∗ to U∗ + ∆U∗ in (22) corresponds
to small changes of the following quantities
kij → kij + ∆kij, s∼∗i→ s
∼∗i
+ ∆s∼∗i
and ti → ti + ∆ti.
To obtain a neighbouring solution, equations (22) are expanded in Taylor’s
series about an initial solution and linearized incremental equations are derived
by ignoring all the non-linear terms of small increments as below
αi
s∼∗i
+ ∆s∼∗i
=
8∑
j=1
kijv∼pij−C−1pi Fpi +
8∑
j=1
∆kijv∼pij
=8∑
j=1
ki−1jeλpi−1j ti−1v
∼pi−1j−C−1pi−1Fpi−1
+8∑
j=1
∆ki−1jeλpi−1jti−1 v∼pi−1j
+ ∆ti−18∑
j=1
ki−1jeλpi−1j ti−1λpi−1jv∼pi−1j
, 1 ≤ i ≤ n. (24)
Initially, a stable period-1 LCO with n = 4 can be easily located from
the perturbation step. As the bifurcation parameter U∗ varies, the LCO may
17
undergo various bifurcations such as symmetry breaking and period doubling,
and n may become very large.
To solve (19) in an efficient way, the following matrix dimension reduction
technique is used for large n.
3.3 Matrix dimension reduction
Let vi0l and vijl (1 ≤ i ≤ n and 1 ≤ j, l ≤ 8) be the l-th component of
C−1i Fi and v∼ij, respectively. Define vectors Ki, ∆Ki, Lilm, L
(λ)ilm and matrices
Mim, M(λ)im as
Ki = (ki1 ki2 · · · ki8)T , ∆Ki = (∆ki1 ∆ki2 · · · ∆ki8)T ,
Lilm = (eλi1tmvi1l eλi2tmvi2l · · · eλi8tmvi8l),
L(λ)ilm = (λi1e
λi1tmvi1l λi2eλi2tmvi2l · · · λi8eλi8tmvi8l),
Mim = (eλi1tm v∼i1eλi2tm v
∼i2· · · eλi8tm v
∼i8)
and M (λ)im = (λi1eλi1tm v
∼i1λi2e
λi2tm v∼i2
· · · λi8eλi8tm v∼i8).
With the above definitions, it follows from (24) that
αi =vi−101 +8∑
j=1
ki−1jeλpi−1j ti−1vpi−1j1 +
8∑
j=1
∆ki−1jeλpi−1jti−1vpi−1j1
+ ∆ti−18∑
j=1
ki−1jeλpi−1j ti−1vpi−1j1
=⇒ ∆ti−1 =(αi − vi−101 − Lpi−11i−1Ki−1 − Lpi−11i−1∆Ki−1)/L(λ)pi−11i−1Ki−1
(25)
and
Mpi−1i−1∆Ki−1 −Mpi0∆Ki +M(λ)pi−1i−1Ki−1∆ti−1 = Ri (26)
18
where Ri =∑8
j=1 kijv∼pij−C−1pi Fpi −
∑8j=1 ki−1je
λpi−1 jti−1 v∼pi−1j
+C−1pi−1Fpi−1 .
Substituting (25) into (26), we obtain
Si−1∆Ki−1 =Mpi0∆Ki + Ti (27a)
=⇒ ∆Ki−1 =S−1i−1Mpi0S−1i Mpi+10 · · ·S
−1n−1Mpn0∆Kn
+n∑
j=1
S−1i−1Mpi0S−1i Mpi+10 · · ·S
−1j−1Tj (27b)
where Si−1 = Mpi−1i−1 − M(λ)pi−1i−1Ki−1Lpi−11i−1/L
(λ)pi−11i−1Ki−1 and Ti =
Ri +M
(λ)pi−1i−1
Ki−1
L(λ)pi−11i−1
Ki−1(vi−101 + Lpi−11i−1Ki−1 − αi). For i = 1 in (27a), ∆Kn =
S−1n (Mp10∆K1 + T1). Substituting the above equation into (27b) and let i = 2,
we obtain
∆K1 = (I8 − S−11 Mp20S−12 Mp30 · · ·S
−1n−1Mpn0S
−1n Mp10)
−1
×n∑
j=1
S−11 Mp20S−12 Mp30 · · ·S
−1j−1Tj (28)
where I8 is the 8 × 8 identity matrix and S0 = Sn.
Once ∆K1 is found from (28), the other unknowns ∆Ki, ∆si and ∆ti can be
obtained from (27b), (25) and (24), respectively. Therefore, solving the system
of 16n equations in (24) involves only the computation of 8 × 8 matrices. The
values of kij, s∼∗i
and ti are updated by adding the original values and the corre-
sponding incremental values. The iteration process continues until the residue
terms are less than a desired degree of accuracy. The entire incremental process
proceeds by adding the ∆U∗ increment to the converged value of U∗ using the
previous solution as the initial approximation until a new converged solution is
obtained.
19
4. Stability of LCO
Let Zqi (qi ∈ {1, 2}) be the switching subspace in which Xi lies. In the
present incremental method, a LCO is considered as an equilibrium point of a
Poincaré map on Zp1 . For a general LCO with n switching pointsXi (1 ≤ i ≤ n),
a Poincaré map F1 on Zp1 can be defined by
X ′ = F1(X) (29)
where X ′ ∈ Zq1 is obtained from the solution of motion (19) with initial point
X and X1 is an equilibrium point of F1. We note that α1 in X1 = (α1 s∼∗1)T
remains unchanged when the bifurcation parameter U∗ varies. Let Σ be the
seven-dimensional subspace of Zp1 defined as
Σ = {s∼1
∈ Σ |X =
α1
s∼1
∈ Zq1}.
To simplify the computation, we defined the reduced Poincaré map on Σ from
(29) as
s∼′1
= F (s∼1
) (30)
where X = (α1 s∼1)T , X ′ = (α1 s∼
′1)T and s
∼1in X1 = (α1 s∼
∗1) is an equilibrium
point of F . The stability of a LCO is determined by the eigenvalues of the first
derivative of the reduced Poincaré map F evaluated at s∼∗1. Bifurcations occur
when the linearized map is degenerate, i.e. at least one eigenvalue has unit
modulus [21,22]. The first derivative DF is given by DF = ∂F∂ s∼1
which can be
computed directly by using implicit differentiation. For a general LCO with n
20
switching points, it follows from the chain rule that
DF =∂F
∂s∼n
∂s∼n
∂s∼n−1
· · ·∂s∼2∂s∼1
. (31)
To compute∂ s∼i+1∂ s∼i
for 1 ≤ i ≤ n and noting that∂ s∼n+1∂ s∼n
= ∂F∂ s∼n
, we partially
differentiate both sides of Xi = r∼i(0) from (20) with respect to sil where sil is
the l-th entry of s∼i
and obtain
∂Xi∂sil
= e∼l+1
=8∑
j=1
∂kpij∂sil
v∼pij
= Mpi0∂Kpi∂sil
=⇒∂Kpi∂sil
= M−1pi0e∼l+1
where 1 ≤ l ≤ 7 and e∼l+1
is the 8×1 unit vector with one at the (l+1)th entry
and zero elsewhere. Partially differentiating both sides of Xi+1 = r∼i(ti) from
(20) with respect to sil, we further obtain
0 =8∑
j=1
[∂kpij∂sil
vpij1eλpijti + kpijvpij1λpije
λpijti∂ti∂sil
]
= Lpi1i∂Kpi∂sil
+ L(λ)pi1iKpi∂ti∂sil
=⇒ ∂ti∂sil
=−Lpi1iM−1pi0e∼l+1
L(λ)pi1i
Kpi
and
∂si+1m∂sil
=8∑
j=1
[∂kpij∂sil
vpijm+1eλpij ti + kpijvpijm+1λpije
λpijti∂ti∂sil
]
= Lpim+1i∂Kpi∂sil
+ L(λ)pim+1iKpi∂ti∂sil
=⇒ ∂si+1m∂sil
= (Lpim+1i −L
(λ)pim+1i
KpiLpi1i
L(λ)pi1i
Kpi)M−1pi0e∼l+1
(32)
where 1 ≤ m ≤ 7. The stability of a LCO can be determined by evaluating the
eigenvalues of DF from (31) and (32). If all the eigenvalues lie inside the unit
21
circle C, the equilibrium point s∼∗1
is stable. As the bifurcation parameter is
varied, eigenvalues may pass through C, at which point a bifurcation occurs. If
an eigenvalue crosses C at +1, a symmetry-breaking or saddle-node bifurcation
may occur. It is a period-doubling bifurcation if an eigenvalue crosses C at −1.
Other than ±1, the bifurcation is called Neimark-Sacker bifurcation.
5. Results and discussions
To compare with the previous results in [7,11], the system parameters under
consideration are chosen as
µ = 100, ah = −1/2, xα = 1/4, ζξ = ζα = 0, rα = 0.5 and ω̄ = 0.2.
The non-linear restoring force M (x1) is given by (11) with
M0 = 0, Mf = 0, δ = 0.5◦ and αf = 0.25◦,
and the plunge is linear with G(x3) = x3. The linear flutter speed U∗L = 6.2851
is determined by solving the aeroelastic system for M0 = δ = αf = 0. For
U∗ > U∗L, some of the eigenvalues in regions R1, R2 and R3 have positive real
parts. Thus, the solution is divergent. As U∗ decreases below U∗L, the real parts
of all eigenvalues of the system in R1 and R3 are negative, but some eigenvalues
in R2 may have positive real parts. Hence, for U∗ < U∗L, the aeroelastic system
admits various non-linear behaviours.
To obtain an initial guess from the perturbation step, we observe that,
for U∗ slightly less that U∗L = 6.2851, a pair of complex eigenvalues of the
system (matrix B) in R2 have positive real part. These two eigenvalues be-
22
come pure imaginary at U∗1 ' 0.1691 U∗L where λ = ±ωi = ±0.1651i and
the corresponding eigenvectors u∼1
+ iu∼2
up to a scalar are given by u∼1
=
(−0.1250 0.0001 0.1245 0.0048 − 0.1971 − 0.3206 0.0292 0.2776)T and u∼2
=
(−0.0006 − 0.0206 − 0.0291 0.0206 0.7028 0.1746 − 0.7462 − 0.2499)T . It
follows from (18) that p1 = 2.0000 and p2 = 0.0096. We further observe
that, in (12b), F2 = 0 and matrix B has a zero eigenvalue, i.e. det(B) = 0.
From (15) and (16b), u∼0
is simply the eigenvector of B with zero eigenvalue
such that u01 = αf + δ/2 = 0.5. It is, therefore, given by u∼0= (0.5 0 −
0.2825 0 10.9890 1.6667 − 6.2096 − 0.9418)T . The travelling time between the
two switching point is T2 =πω = 19.0284. With this initial solution, equation
(24) is employed to construct the continuation curves in U∗.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.8
1
1.2
1.4
1.6
1.8
2
U*/U*L
Max
(α)
3
4
5
Fig. 6. Continuation curve of symmetric LCO. • Neimark-Sacker bifurcation. �Symmetry-breaking bifurcation.
From the incremental step, we observe that an unstable symmetric LCO is
born at U∗1 . The continuation curve of the symmetric LCO is given in Fig.
6. Initially, one eigenvalue of the first derivative DF is outside the unit circle.
23
As U∗ decreases to U∗2 = 0.1310U∗L, a saddle-node bifurcation occurs where
an eigenvalue leaves the unit circle at +1. As U∗ increases again and max(α)
increases, the LCO encounters a Neimark-Sacker (Secondary Hopf) bifurcation
at U∗3 = 0.1353U∗L (label 3) where a pair of eigenvalues enter the unit circle at
points other than ±1. The LCO is stable until a subcritical symmetry-breaking
bifurcation occurs at U∗4 = 0.2194U∗L (label 4) where an eigenvalue leaves the
unit circle at +1. At this value, the stable LCO merges with two other unstable
asymmetric LCOs and becomes unstable. As U∗ increases further, another
subcritical symmetry-breaking bifurcation occurs at U∗5 = 0.6880U∗L (label 5)
and the LCO becomes stable again. The amplitude continues to grow without
a bound as U∗ tends to U∗L. The initial switching point X1, period and stability
of the stable symmetric LCO at U∗ = 0.7U∗L are given in Table 1. A phase
portrait of this LCO is shown in Fig. 7 and is compared to the result obtained
by using the fourth order Runge-Kutta method. They are in good agreement.
Table 1
The initial switching point, period and stability of symmetric LCO at U∗ = 0.7U∗L.
Type of motion : p-1 (symmetric) Period : 72.0471
Initial switching point : (0.25 0.0462 − 5.5757 0.1435 4.4438 0.2507 − 124.6945 − 19.8591)Floquet multipliers : 0.7494, 0.1231, 0.0443 ± 0.0183i, 0.0377, 0, 0
Next, we consider the emanating curve arisen from one of the asymmet-
ric LCOs born at U∗4 as depicted in Fig. 8. On the emanating curve, four
period-doubling (PD) bifurcations are found at U∗6 = 0.2132U∗L (label 6), U∗7 =
0.21216U∗L (label 7), U∗9 = 0.2511U∗L (label 9) and U
∗10 = 0.5280U∗L (label 10);
24
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
α
α’
Fig. 7. Symmetric LCO at U∗ = 0.7U∗L. —–, Runge-Kutta method. ×, Perturbation-Incremental method.
one Neimark-Sacker bifurcation at U∗8 = 0.1968U∗L (label 8); two saddle-node
bifurcations at U∗11 = 0.1950U∗L and U
∗12 = 0.7575U
∗L. We note that the period-1
LCO is stable in the ranges (U∗8 , U∗9 ), (U∗10, U∗12) and all of these LCOs contain
harmonics. A phase portrait of the stable asymmetric LCO at U∗ = 0.7U∗L,
which coexists with that of Fig. 7, is shown in Fig. 9. The initial switching
point, period and stability of this LCO are given in Table 2.
Table 2
The initial switching point, period and stability of one of the asymmetric LCO at
U∗ = 0.7U∗L.
Type of motion : p-1-h (asymmetric) Period : 81.9851
Initial switching point : (0.25 0.0356 -2.8777 0.1719 3.9468 0.5756 -103.5019 -11.5272)
Floquet multipliers : -0.1944±0.1286i, 0.0495, 0.0240, 0.0353, 0, 0
The bifurcation results in Figs. 6 and 8 give insight into the numerical results
obtained in [7, 11]. The bifurcation diagrams in Fig. 11 of [7] and Fig. 6 of [11]
25
0.2 0.3 0.4 0.5 0.6 0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
U*/U*L
Ma
x(α
)
4
5
10
8
967
(a)
0.18 0.2 0.22 0.24 0.26 0.28 0.30.8
0.85
0.9
U*/U*L
Ma
x(α
)
4
8
9
67
(b)
Fig. 8. (a) Continuation curve of one of the asymmetric LCO. (b) Enlarged diagram
near U∗ = 0.2U∗L. • Neimark-Sacker bifurcation. N Period-doubling bifurcation. �Symmetry-breaking bifurcation.
are obtained from trajectories with a fixed initial point when U∗ varies. It was
pointed out in [11] that, in Fig. 6, there is a small reduction in amplitude for
pitch when the bifurcation parameter is increased to cross U∗ = 0.732U∗L and
the solution becomes a simple period-1 LCO. From Figs. 6 and 8 above, the
reason for the reduction is that the trajectory jumps from one of the coexisting
asymmetric LCO to the symmetric LCO. Furthermore, the discontinuity of the
bifurcation curve in Fig. 6 of [11] for the range 0.53U∗L ≤ U∗ ≤ 0.732U∗L is due
to the jump of the trajectories between the two coexisting asymmetric LCOs.
The emanating branches from the PD bifurcations of Fig. 8 at U∗6 (label
6) and U∗9 (label 9) are shown in Figs. 10(a) and 10(b), respectively. On
the emanating branch (B1) from U∗6 , two more PD bifurcations are found
at U∗11 = 0.2132U∗L (label 11) and U∗12 = 0.21217U∗L (label 12). Emanating
branches of further PD bifurcations are shown in Fig. 11 where branch Bn
(n = 1, 2, 3) contains period-2n LCOs. The emanating branches suggest a se-
26
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
−0.1
−0.05
0
0.05
0.1
α
α’
Fig. 9. One of the asymmetric LCOs at U∗ = 0.7U∗L. —–, Runge-Kutta method. ×,Perturbation-Incremental method.
quence of PD bifurcations leading to chaos. However, this phenomenon is not
observed in Fig. 6 of [11] as the LCOs on these branches are all unstable. On
the emanating branch (B1) from U∗9 (see Fig. 10(b)), three saddle-node bi-
furcations (U∗13 = 0.2484U∗L, U∗16 = 0.5548U∗L, U
∗17 = 0.4879U∗L) and two PD
bifurcations (U∗14 = 0.2489U∗L (label 14), U∗15 = 0.5546U∗L (label 15)) are found.
The period-2 LCOs on the curve segments (U∗13, U∗14), (U
∗17, U
∗10) and (U
∗15,
U∗16) are stable. Emanating branches of further PD bifurcations are shown in
Figs. 12(a)-(c) where branch Bn (n = 1, 2, 3) contains period-2n LCOs. In
Fig. 12(a), branches B2 and B3 are visually indistinguishable. In Figs. 12(b)
and (c), two sequences of stable PD bifurcations leading to chaos are observed.
Since the bifurcation values between consecutive PD bifurcations are very small,
it is expected that full chaos will be developed soon after the occurrence of PD
bifurcations at around U∗ = 0.25U∗L and 0.55U∗L. The former case was reported
in [11]. A phase portrait of the stable period-4-h LCO at U∗ = 0.25U∗L (label
27
0.212 0.2122 0.2124 0.2126 0.2128 0.213 0.2132 0.21340.882
0.883
0.884
0.885
0.886
0.887
0.888
0.889
0.89
0.891
U*/U*L
Ma
x(α
)
B1
12
11
6
7
(a)0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
U*/U*L
Ma
x(α
)
10
9
15
14
B1
(b)
Fig. 10. Emanating branches from the period-doubling bifurcations of Fig. 8. NPeriod-doubling bifurcation.
16) on branch B2 of Fig. 12(b) is shown in Fig. 13. The initial switching point,
period and stability of this LCO are given in Table 3.
Table 3
The initial switching point, period and stability of the period-4-h LCO at U∗ = 0.25U∗L
on branch B2 of Fig. 12(b).
Type of motion : p-4-h Period : 171.3142
Initial switching point : (0.25 0.0616 -0.9719 -0.0437 9.1392 0.8084 -10.6221 -2.6389)
Floquet multipliers : -0.2016±0.6601i, 0.0004, 0.0004, 0.0002, 0, 0
In Fig.7 of [11], period-2-h LCOs are found between 0.325U∗L ≤ U∗ ≤
0.466U∗L by the point transformation method. However, these LCOs are not
contained in any of the continuation curves we constructed previously. To con-
struct the continuation curve relating to these LCOs, we first obtain the LCO
at, say U∗ = 0.4U∗L by using the Runge-Kutta method. The phase portrait and
information of this LCO are given in Fig. 14 and Table 4, respectively. With
this p-2-h LCO as the initial solution, we apply the incremental step and obtain
28
0.2122 0.2124 0.2126 0.2128 0.213 0.21320.8865
0.887
0.8875
0.888
0.8885
0.889
0.8895
0.89
U*/U*L
Max
(α)
B3
B1
B211
12
7
Fig. 11. Emanating branches of PD bifurcation arisen from U∗6 . N Period-doublingbifurcation.
the continuation curve C1 as depicted in Fig. 15. It is interesting to note that
this curve forms a closed loop. From Fig. 7 of [11], it seems that, as U∗ in-
creases, the p-2-h LCO passes through the chaotic region and becomes period-1
at U∗ = 0.53U∗L where a large drop with a factor of two in the period occurs.
However, from Fig. 15, the p-2-h LCO in fact disappears at U∗ = 0.4652U∗L
due to a saddle-node bifurcation. Therefore, the results obtained from the PI
method give us a clearer and more accurate picture about the global bifurcations
of the freeplay model as this method is able to capture the unstable solutions
while the point transformation method is not capable of doing so.
Table 4
The initial switching point, period and stability of the period-2-h LCO at U∗ = 0.4U∗L.
Type of motion : p-2-h Period : 120.3980
Initial switching point : (0.25 0.0451 -2.2714 0.0319 8.2265 0.5330 -36.8109 -7.6077)
Floquet multipliers : -0.3429, 0.0889, -0.0175, 0.0042, 0.0047, 0, 0
29
0.25 0.3 0.35 0.4 0.45 0.5 0.550.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
U*/U*L
Max
(α)
9
14
10
15
B1
B2
B3
0.248 0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.2520.912
0.914
0.916
0.918
0.92
0.922
0.924
U*/U*L
Ma
x(α
)
B1
B2
B3
(b)
14
0.5525 0.553 0.5535 0.554 0.5545 0.5551.246
1.247
1.248
1.249
1.25
1.251
U*/U*L
Ma
x(α
)
B1
B3
B2
(c)
Fig. 12. Emanating branches of PD bifurcations arisen from U∗9 . N Period-doublingbifurcation.
6. Conclusion
A perturbation-incremental (PI) method has been developed to investigate
30
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
−0.1
−0.05
0
0.05
0.1
α
α’
Fig. 13. Period-4-h LCO at U∗ = 0.25U∗L on branch B2 of Fig. 12(b). —–, Runge-
Kutta method. ×, Perturbation-Incremental method.
0 0.2 0.4 0.6 0.8 1−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
α
α’
Fig. 14. Period-2-h LCO at U∗ = 0.4U∗L. —–, Runge-Kutta method. ×, Perturbation-Incremental method.
31
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
U*/U*L
Max
(α)
10
15
14
9
C1
Fig. 15. A separated continuation curve C1 which forms a closed loop. N Period-doubling bifurcation.
the dynamic response of a self-excited two-degree of freedom aeroelastic system
with structural non-linearity represented by a freeplay stiffness. The PI method
overcomes the main disadvantage of the harmonic balance (HB) method in that
an approximate LCO obtained by the present method is piecewise continuous
which agrees qualitatively with the exact solution while that obtained by the HB
method is smooth, thus providing an accurate prediction of the switching points
where the changes in linear subdomains occur. The present method is also able
to compute unstable LCOs and gives a full picture of the global bifurcation.
The continuation curves in Figs. 6, 8, 10, 12 and 15 obtained by the PI
method provide insight into the previous bifurcation results found in [7,11].
For instance, in Fig. 6 of [11], a sudden reduction of pitch amplitude at
U∗ = 0.732U∗L is due to a change of the trajectory from one of the coexisting
asymmetric LCOs to the symmetric LCO and the discontinuity of the bifurca-
32
tion curve in the range 0.53U∗L ≤ U∗ ≤ 0.732U∗L is a result of the switching of
trajectories between the two coexisting asymmetric LCOs.
In the incremental step, a matrix dimension reduction technique is presented
to reduce a system of 16n equations with the same number of unknowns to the
computation of iterative matrices with dimension 8×8. Although this technique
reduces greatly the computational time, it works as long as U∗ is used as the
bifurcation parameter since the eigenvalues and eigenvectors of matrices A and
B in (13) can be determined before the iterations. However, in the vicinity of
a saddle-node bifurcation where U∗ can no longer be used as the bifurcation
parameter, U∗ and thus the eigenvalues, eigenvectors of matrices A, B become
unknowns whose values have to be constantly updated in each iterative process.
In that case, the dimension of the iterative matrices will be much larger than
8 × 8 although the dimension technique can still be applied.
Neimark-Sacker (Secondary Hopf) bifurcation is detected in Figs. 6 and
8. The continuation of such bifurcation may also be made possible by the PI
method. We recall that a LCO can be considered as an equilibrium point of a
Poincaré map in a switching subspace. Then, a quasi-periodic solution can be
regarded as an invariant curve in the switching subspace. Previous techniques
developed in [12-15] for the computation of limit cycles can be employed to
compute invariant curves as they have similar features. This will be pursued in
future research.
From the illustrative examples presented in this paper, it is clearly demon-
strated that analytic predictions from the PI method are in excellent agreement
33
with those resulting from the point transformation (PT) method and a numer-
ical time-integration scheme. Although the investigation is concentrated on an
aeroelastic system with a structural freeplay non-linearity in the pitch degree-
of-freedom, the analysis can readily be extended to include non-linearities in
both degree-of-freedom and to hysteresis models.
Acknowledgments
This work was supported by the strategic research grant 7001557 of the City
University of Hong Kong.
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37
Appendix A: Definitions of coefficients
a21 = j(−d5c0 + c5d0), a41 = j(d5c1 − c5d1),
a22 = j(−d3c0 + c3d0), a42 = j(d3c1 − c3d1),
a23 = j(−d4c0 + c4d0), a43 = j(d4c1 − c4d1),
a24 = j(−d2c0 + c2d0), a44 = j(d2c1 − c2d1),
a25 = j(−d6c0 + c6d0), a45 = j(d6c1 − c6d1),
a26 = j(−d7c0 + c7d0), a46 = j(d7c1 − c7d1),
a27 = j(−d8c0 + c8d0), a47 = j(d8c1 − c8d1),
a28 = j(−d9c0 + c9d0), a48 = j(d9c1 − c9d1),
where j, ci (i = 0, 1, 2, . . . , 9) and di (i = 0, 1, 2, . . . , 9) are defined as
j =1
c0d1 − c1d0,
c0 = 1 +1µ, c1 = xα −
ahµ,
c2 =2µ
(1 − ψ1 − ψ2), c3 =1µ
(1 + (1 − 2ah)(1 − ψ1 − ψ2)),
c4 =2µ
(ε1ψ1 + ε2ψ2), c5 =2µ
(1 − ψ1 − ψ2 + (12− ah)(ε1ψ1 + ε2ψ2)),
c6 =2µε1ψ1(1 − ε1(
12− ah)), c7 =
2µε2ψ2(1 − ε2(
12− ah)),
c8 = −2µε21ψ1, c9 = −
2µε22ψ2,
d0 =xαr2α
− ahµr2α
, d1 = 1 +1 + 8a2h8µr2α
,
d2 = −1 + 2ahµr2α
(1 − ψ1 − ψ2),
38
d3 =1 − 2ah2µr2α
− (1 + 2ah)(1 − 2ah)(1 − ψ1 − ψ2)2µr2α
,
d4 = −1 + 2ahµr2α
(ε1ψ1 + ε2ψ2),
d5 = −1 + 2ahµr2α
(1 − ψ1 − ψ2) −(1 + 2ah)(1 − 2ah)(ψ1ε1 − ψ2ε2)
2µr2α,
d6 = −(1 + 2ah)ψ1ε1
µr2α(1 − ε1(
12− ah)),
d7 = −(1 + 2ah)ψ2ε2
µr2α(1 − ε2(
12− ah)),
d8 =(1 + 2ah)ψ1ε21
µr2α, d9 =
(1 + 2ah)ψ2ε22µr2α
.
Appendix B: Definitions of matrices and vectors
A1 =
0 1 0 0
a21 − jc0( 1U∗ )2 a22 a23 + jd0β( ω̄U∗ )
2 a24
0 0 0 1
a41 + jc1( 1U∗ )2 a42 a43 − jd1β( ω̄U∗ )
2 a44
A2 =
0 0 0 0
a25 a26 a27 a28
0 0 0 0
a45 a46 a47 a48
A3 =
1 0 0 0
1 0 0 0
0 0 1 0
0 0 1 0
A4 =
−ε1 0 0 0
0 −ε2 0 0
0 0 −ε1 0
0 0 0 −ε2
39
B1 =
0 1 0 0
a21 − jc0Mf ( 1U∗ )2 a22 a23 + jd0β( ω̄U∗ )
2 a24
0 0 0 1
a41 + jc1Mf ( 1U∗ )2 a42 a43 − jd1β( ω̄U∗ )
2 a44
F =
0
−jc0( 1U∗ )2
0
jc1( 1U∗ )2
0
0
0
0
40
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