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