[american institute of aeronautics and astronautics 26th aiaa applied aerodynamics conference -...

Hopf Bifurcation Analysis of Typical Sections

with Structural Nonlinearities in Transonic

Flow

Elizangela Camilo,∗ Flavio D. Marques†

Universidade de Sao Paulo, Sao Carlos, SP, 13566-590, BRAZIL

Joao Luiz F. Azevedo‡

Instituto de Aeronautica e Espaco, Sao Jose dos Campos, SP, 12228-903, BRAZIL

The paper is concerned with direct aeroelastic bifurcation analyses of an

airfoil system in which both aerodynamic and structural nonlinearities are

considered. Here, structural dynamics is considered in terms of polynomial

nonlinearities. Two CFD tools are employed in the present work and they

are based on the Euler formulation. For Hopf bifurcation analysis, the CFD

code is based on structured grids for computation domain discretization.

Flutter boundaries are found with the inverse power method. Previous

work has demonstrated the scheme for both symmetric airfoil and wing

configurations with a linear structural model. The current paper presents

the first investigations of the structural nonlinearity effects on the method.

Time-marching analysis is performed and compared with direct calculation

of Hopf bifurcation points. In the time-marching case, the CFD tool solves

flows using an unstructured computational domain discretization. The re-

sults shown in the present paper are particularly concerned with the inves-

tigation of flutter boundaries and typical LCO nonlinear effects for subsonic

and transonic flows over a NACA 0012 airfoil-based typical section. The

investigation considers both time histories of the aeroelastic responses as

well as phase plane analyses.

∗Post-Doctoral Researcher; e-mail: [email protected].†Associate Professor, Aeroelasticity Laboratory, Escola de Engenharia de Sao Carlos, EESC/USP; e-mail:

[email protected].‡Senior Research Engineer, Aerodynamics Division, Comando-Geral de Tecnologia Aeroespacial,

CTA/IAE/ALA; e-mail: [email protected]. Associate Fellow AIAA.

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26th AIAA Applied Aerodynamics Conference18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-6236

Copyright © 2008 by E. Camilo, F.D. Marques and J.L.F. Azevedo. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

I. Introduction

In the last decade, nonlinear dynamic analysis developed quickly, both from a theoretical

and an experimental point of view, in a vast diversity of fields in science and engineering.

However, most aeroelastic analyses of flight vehicles are performed under the assumption of

linearity. Under this assumption, the characteristics of flutter and divergence can be obtained

using well-established tools. On the other hand, the influence of nonlinearities on modern

aircraft is becoming increasingly important and the need for more accurate predictive tools

grows stronger.

The nonlinearities in aeroelastic analyses are divided into aerodynamic and structural

ones. In this paper, the aerodynamic nonlinearities arise from the presence of shock waves

in transonic flows. For such flow conditions, the unsteady forces generated by the motion

of the shock waves have been shown to destabilize the airfoil pitching motion and affect the

bending-torsional flutter condition by lowering the flutter speed in the so-called transonic

dip phenomenon. The structural nonlinearities are subdivided into distributed and concen-

trated ones. The distributed nonlinearities are spread over the entire structure, whereas the

concentrated nonlinearities act locally. The concentrated nonlinearities are known to have

significant effects on the aeroelastic responses and to yield chaotic motion and limit cycle

oscillation (LCO), even below the flutter speed.13

Computational aeroelasticity is a relatively new field in the broader scope of aeroelastic

analysis in which loads based on Computational Fluid Dynamics (CFD) calculations are used

(see, for instance, Refs. 11, 9, 17 and 8). In this context, a significant amount of effort has

been devoted towards the numerical solution of transonic aeroelastic phenomena, not only

in the prediction of transonic dip effects,4,5 but also towards the calculation of LCO. Euler

and Navier-Stokes schemes have been coupled with structural models1,12 for such work.

Time-marching analysis of nonlinear aeroelasticity can give the detailed motion charac-

teristics. However, the cost of these calculations motivates attempts to find quicker ways of

evaluating stability, while still retaining the detailed aerodynamic information given by CFD

predictions. Due to large computational requirements associated with time domain analysis

in nonlinear aeroelasticity, recent work has been investigating bifurcation theory for com-

puting flutter points.4,5, 14 Hopf bifurcation defines a singular point in which an eigenvalue

of the system Jacobian matrix crosses the imaginary axis at the flutter condition. In this

fashion, the problem of locating a one-parameter stability boundary is reduced from multi-

ple, unsteady, time-marching calculations to a single steady-state calculation of a modified

system.

The current paper presents an investigation on the effects of concentrated structural non-

linearities on the Hopf bifurcation prediction methodology for an airfoil moving in pitch and

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plunge. In order to solve the aerodynamic problem, using the Hopf bifurcation methodology,

the Euler equations are discretized using a structured grid.4 Time-marching analyses are

also performed and their results are compared with the Hopf bifurcation predictions. In the

CFD code employed for time-marching analyses, the governing equations are integrated by

a finite volume discretization on unstructured grids.2 The results are, then, presented to

illustrate the performance of the respective schemes in resolving the nonlinear dynamics in

an efficient manner.

II. Aerodynamic Simulations

In the present study, the flow is assumed to be governed by the two-dimensional, time-

dependent Euler equations, which may be written in conservative form and Cartesian coor-

dinates as∂wf

∂t+

∂Fi

∂x+

∂Gi

∂y= 0 , (1)

where wf = (ρ, ρu, ρv, ρE)T denotes the vector of conserved variables. Fi and Gi are the

inviscid flux vectors, given by

Fi =

ρU∗

ρuU∗ + p

ρvU∗

U∗(ρE + p) + xp

, (2)

Gi =

ρV ∗

ρuV ∗

ρvV ∗ + p

V ∗(ρE + p) + yp

. (3)

In the above equations, ρ, u, v, p and E denote the density, the two Cartesian components

of velocity, the pressure and the specific total energy, respectively. U∗ and V ∗ are the two

Cartesian components of the velocity relative to the moving coordinate system, which has

local velocity components x and y, i.e.,

U∗ = u − x V ∗ = v − y. (4)

The flow solution in the Hopf bifurcation methodology is obtained using a parallel multi-

block code. A summary of the applications examined using the code may be found in Ref. 3.

A fully implicit steady solution of the Euler equations is obtained by advancing the solution

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forward in time by solving the discrete nonlinear system of equations

wn+1f −wn

f

∆t= Rf(w

n+1f ) . (5)

The term on the right-hand side, called the residual, represents the discretization of the

convective terms, given here by the approximate Riemann solver of Osher and Chakravarthy,

using MUSCL interpolation for 2nd order accuracy and the van Albada limiter (see Ref. 18).

Equation (5) is a nonlinear system of algebraic equations. These equations are solved by

an implicit method,7 the main features of which are an approximate linearization, to reduce

the size and condition number of the linear system, and the use of a preconditioned Krylov

subspace method to calculate the updates. The steady-state solver is applied to unsteady

problems within a pseudo time stepping iteration.10

For the time-marching aeroelastic solver, the 2-D Euler equations are discretized by a

finite volume procedure in an unstructured mesh. The equations are discretized in space by

a centered scheme, together with added artificial dissipation terms. The artificial dissipation

operator, D, can be written as

D = d2(wf) − d4(wf) , (6)

where d2(wf) represents the contribution of the undivided Laplacian operator, and d4(wf)

is the contribution of the biharmonic operator.11 The biharmonic operator is responsible

for providing the background dissipation to damp high frequency uncoupled error modes

and the undivided Laplacian artificial dissipation operator prevents oscillations near shock

waves. The Euler solver is integrated in time by a second-order accurate, 5-stage, explicit,

Runge-Kutta time-stepping scheme, as presented in Ref. 15.

III. Equations of Motion

The physical model considered in the present work is a typical section with pitch and

plunge degrees of freedom and free of mechanical friction. The equations of motion of this

aeroelastic system, with a linear structure, can be written in the form

dws

dt= Rs , (7)

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where

Rs =

1 0 0 0

0 1 0 12xα

0 0 1 0

0 xα 0 12r2α

−1

0

(2/µsπ)Cl

0

(4/µsπ)Cm

−

0 −1 0 0

(2ωR/U)2 0 0 0

0 0 0 −1

0 0 2r2α/(U)2 0

ws

(8)

and ws = [h, h, α, α]T . In the previous equation, h is the plunge linear displacement and

α is the incidence, or pitch angular displacement. Here, rα =√

(Iα/m) is the radius of

gyration defined in terms of the pitch moment of inertia Iα and the airfoil mass per unit

span m, xα is the offset between the center of mass and the elastic axis, µs = m/πρ∞b2 is

the airfoil-to-fluid mass ratio defined in terms of the fluid freestream density ρ∞ and the

semi-chord, b. Moreover, ωR = ωh

ωαis the ratio of the natural frequencies of plunging ωh and

pitching ωα, U = U∞bωα

is the reduced velocity defined in terms of the fluid freestream velocity

U∞, and Cl and Cm are the lift and moment coefficients about the elastic axis, respectively.

Several classes of nonlinear stiffness contributions have been studied in papers treating

the open-loop dynamics of the aeroelastic system.13 In this work, torsional polynomials have

been included in the model. Then, the linear torsional moment function is replaced by the

nonlinear function

M(α) = Kαα = Kαf(α) , (9)

where Kα is considered as a global stiffness. The functional form of f(α) can be expressed

as

f(α) = fα0+ fα1

α + fα2α2 + fα3

α3 + fα4α4 + ... + fαn

αn = fα0+ fα1

α +

n∑

i=2

fαiαi . (10)

The pitch motion structural equation is modified to include the polynomial moment curve.

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Equation (8) becomes

Rs =

1 0 0 0

0 1 0 12xα

0 0 1 0

0 xα 0 12r2α

−1

0

(2/µsπ)Cl

0

(4/µsπ)Cm − (2r2α/(U)2)(fα0

+∑n

i=2 fαiαi)

−

0 −1 0 0

(2ωR/U)2 0 0 0

0 0 0 −1

0 0 (2r2α/(U)2)fα1

0

ws

(11)

IV. Hopf Bifurcation Analysis

Consider the semi-discrete form of the coupled CFD-CSD system

dw

dt= R(w, µ) , (12)

where

w = [wf ,ws]T (13)

is a vector containing the fluid unknowns wf and the structural unknowns ws and

R = [Rf ,Rs]T (14)

is a vector containing the fluid residual Rf and the structural residual Rs. The residuals

also depend on a parameter, µ. In the case of the pitch-plunge airfoil system, i.e., the typical

section considered in the present work, one has that µ = U . For an equilibrium condition of

this system, w0(µ) satisfies R(w0, µ) = 0.

Dynamical systems theory gives criteria for an equilibrium to be stable.16 In particular,

all eigenvalues of the Jacobian matrix of Eq. (12), given by A = ∂R/∂w, must have negative

real parts. A Hopf bifurcation with respect to the µ parameter occurs in the stability limit of

the system at values of µ such that A(w0, µ) has one eigenvalue which crosses the imaginary

axis.

The authors are interested in finding the parameter, µ, for the airfoil section and assume

that stability is lost through a Hopf bifurcation. The power method16 is an algorithm

for calculating the dominant eigenvalue and eigenvector pair of any given diagonalizable

matrix, A. Its extension to the shifted inverse power method is practical for finding any

eigenvalue, provided that a good initial approximation is known. Therefore, the present

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approach assumes that the n × n matrix, A, has distinct eigenvalues, labeled λ1, λ2, ..., λn,

and consider the λj eigenvalue. Then, a constant σ can be chosen such that 1/(λj − σ) is

the dominant eigenvalue of (A − σI)−1.

Hence, for the A = ∂R/∂w matrix, and the initial guess, x0P , and constant, σ, the inverse

power method iterations are given by6

Pk+1 = (A − σI)−1xkP , (15)

xkP = Pk/‖Pk‖∞ . (16)

The shifted inverse power method can be used to calculate the critical eigenvalue in the

complex plane at a fixed freestream Mach number. By computing the location for multiple

values of freestream Mach number, the value at which the eigenvalue crosses the imaginary

axis can be obtained.

V. Torsional Polynomial Nonlinearity in Jacobian Matrix

The torsional polynomial nonlinearity is included in the A = ∂R/∂w Jacobian matrix.

The calculation of A is most conveniently done by partitioning the matrix as

∂Rf

∂wf

∂Rf

∂ws

∂Rs

∂wf

∂Rs

∂ws

=

Aff Afs

Asf Ass

. (17)

The Aff block describes the influence of the fluid unknowns in the fluid residual. The

dependence of the fluid residual on the structural unknowns is represented by Afs. The

calculation of terms of the Jacobian matrix for a linear structural model is described in Ref.

4. The addition of the structural nonlinearity does not change these two blocks, Aff and

Afs, in the calculation of Jacobian matrix presented in the linear model.

The Asf term essentially involves calculating the dependence of integrated fluid forces

on the fluid unknowns. For the pitch-plunge airfoil, the fluid variables contribute to the

structural equations through the lift and moment coefficients and, hence, again remains

unchanged. Finally, the only terms, that are different, are the ones in the exact Ass Jaco-

bian matrix, which describe the dependence of the structural equations on the structural

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unknowns. Such matrix is given by

∂Rs

∂ws

=

0 1 0 0−4(ωR)2r2

α

(U )2det0 2r2

α

detµπ∂Cl

∂α− xα

det

(

−2r2α

(U )2+ 4

µπ∂Cm

∂α

)

0

0 0 0 18xα(ωR)2

(U )2det0 −4xα

detµπ∂Cl

∂α− −4r2

α

det(U)2(∑n

i=1 ifαiαi−1) + 8

detµπ∂Cm

∂α0

, (18)

where det = r2α − x2

α.

VI. Flutter Boundaries and Time-Marching Results

Flutter boundaries have been found with the inverse power method (IPM) considering

a Hopf bifurcation point with respect to the U parameter. First, flutter boundaries for

the NACA 0012 airfoil with a linear structure and at zero incidence has been considered.

Torsional nonlinearity has been considered in the aeroelastic system and results are compared

with those considering linear structural model. The parameters for the structural model are

given in Table 1. The test case considered is called the heavy case. Here, the results have

been compare with the time-marching methodology. Time integration of the coupled fluid-

structure equations of motion, Eq. (7), is obtained using the a fourth-order Runge-Kutta

time-stepping scheme.

Table 1. Structural model parameters.

Parameter Value

rα 0.539

xα -0.2

ωR 0.343

µs 100.0 (heavy case)

xea 0.4

yea 0.0

A. Linear Structural Model Results

Flutter boundary and time-marching results have been obtained with the linear structural

model. The IPM was used to calculate airfoil flutter boundaries and the results are compared

with time-marching calculations for the same flight conditions. The results are shown in

Fig. 1, where some discrepancies can be observed between the stability boundaries obtained

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through the direct time integration of the aeroelastic equations and the IPM results.

** * * *

* * * *

*

*

Mach Number

Red

uced

Vel

ocity

0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9 IPMTime marching*

Figure 1. Flutter boundaries for heavy case with linear structure.

The results for the time-marching method have been studied through the time histories

of the solution and through phase plane analyses. The flutter boundaries of the present time-

marching method are obtained when the system begins limit cycle oscillations. For instance,

Fig. 2 presents the time history and phase plane responses for M∞ = 0.6 and U = 3.5.

In this case, the response is a small-amplitude limit cycle oscillation in pitch, indicating a

linear flutter point. In these computations, most of the runs do not need many time periods,

because system oscillations are easily identified with converging amplitude responses (see

Fig. 3(a)), before the flutter point, and diverging amplitude responses, after the flutter point

(see Fig. 3(b)), simply by looking at only a few cycles of the system response. Figures 4,

5(a) and 5(b) show LCO for M∞ = 0.82 and U = 3.68, convergent response for U = 3.65

and divergent response for U = 3.7, respectively.

Grid refinement studies have been performed in order to fully understand and validate

such results for both cases, time-marching and IPM code. In the time-marching analyses,

three unstructured grids have been considered for the grid dependence studies. Data on

these grids are presented in Table 2. Flutter points have been obtained with time history

and phase plane analyses at fixed M∞ = 0.75. The aeroelastic response for the coarse

grid and U = 3.5 is presented in Fig. 6. The system exhibits oscillations with slightly

decreasing amplitudes. Such behavior is consistent with the fact that the flutter point has

been established at U = 3.55 for this case. Before the flutter point, one can observe that

the system exhibits convergent response, as at U = 3.4, as shown in Fig. 7(a). The system

exhibits oscillation with slightly increasing amplitudes at U = 3.6, as shown in Fig. 7(b),

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0 100 200 300 400 500 600−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Pitc

h [d

eg]

Nondimensional time−0.04 −0.02 0 0.02 0.04−3

−2

−1

0

1

2

3x 10

−4

α [deg]

dα/d

t

(a) Time history (b) Phase plane

Figure 2. Pitch limit cycle for M∞ = 0.6 and U = 3.5.

0 50 100 150 200 250 300 350−0.04

−0.02

0

0.02

0.04

0.06

0.08

Pitc

h [d

eg]

Nondimensional time0 50 100 150 200 250

−15

−10

−5

0

5

10

15

20

Nondimensional time

Pitc

h [d

eg]

(a) Time history for U = 3.0 (b) Time history for U = 4.0

Figure 3. Aeroelastic responses for M∞ = 0.6.

which is a consistent behavior for values of the reduced velocity above the flutter point.

Table 2. Unstructured grids used for the mesh refinement study.

Grid name Grid size Numbers of cells on airfoil surface

coarse 13744 volumes 192

medium 17844 volumes 378

fine 35304 volumes 642

Figure 8 shows the approximate flutter boundary with the medium grid at U = 3.37.

The system exhibits a convergent response at U = 3.3, as in Fig.9(a), and an increasing

amplitude oscillation after the flutter point, as in Fig. 9(b), at U = 3.45. One can observe

that the increase in the number of cells, from coarse to medium grid, changes the flutter

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0 200 400 600 800−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

Nondimensional time

Pitc

h [d

eg]

−0.02 −0.01 0 0.01 0.02−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

α [deg]

dα/d

t


Figure 4. Pitch limit cycle for M∞ = 0.82 and U = 3.68.

0 200 400 600 800−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

Pitc

h [d

eg]

Nondimensional time0 200 400 600 800

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

Nondimensional time

Pitc

h [d

eg]


Figure 5. Aeroelastic responses for M∞ = 0.82.

point velocity in approximately 3.7%.

The same analysis with the fine grid shows the flutter point at U = 3.3, as indicated in

Fig. 10. The system exhibits convergent and divergent amplitude oscillation responses for

U = 3.2 and U = 3.4, as shown in Figs. 11(a) and 11(b), respectively. Although the number

of cells from medium to fine grids increased considerably, the flutter point with the fine grid

only varied approximately 2% with regard to the medium grid result.

It is important to note, in these grid refinement studies, the computational costs required

for the calculations. The system exhibits less dissipation in the solution with the fine grid,

as can be observed in Figs. 10 and 11. However, the computational costs with the fine grid

increases considerably. For instance, the time histories calculated up to 800 dimensionless

time units require 17 hours of CPU time for the coarse grid, 22 hours for the medium grid

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0 200 400 600 800−0.05

0

0.05

0.1

0.15

Pitc

h [d

eg]

Nondimensional time−0.06 −0.04 −0.02 0 0.02 0.04 0.06−3

−2

−1

0

1

2

3

4x 10

−4

α [deg]

dα/d

t


Figure 6. Aeroelastic response for M∞ = 0.75 and U = 3.5 with coarse unstructured grid.

0 200 400 600 800−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Nondimensional time

Pitc

h [d

eg]

0 200 400 600 800−0.1

−0.05

0

0.05

0.1

0.15

Nondimensional time

Pitc

h [d

eg]


Figure 7. Aeroelastic responses for M∞ = 0.75 with coarse unstructured grid.

and 43 hours for the fine grid, in a machine with an Intel Core 2 Duo 2.2GHz processor and

2GB of memory.

Grid refinement studies have also been performed for the IPM methodology. The grids

used are presented in Table 3. As in the time-marching case, the solutions are calculated for

flight conditions at fixed M∞ = 0.75. Flutter points have been obtained with the coarse grid

at U = 3.6925, with the medium grid at U = 3.5742, and with the fine grid at U = 3.5521.

The differences observed in the solutions obtained with these grids are also analyzed. The

difference between the reduced velocities at the flutter points obtained with coarse and

medium grids is 3.2%, and this difference is about 0.61% for the solutions with medium and

fine grids.

In the IPM code, a steady state CFD calculation, in the equilibrium point before IPM

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0 200 400 600 800−0.1

−0.05

0

0.05

0.1

0.15

Pitc

h [d

eg]

Nondimensional time−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−3

−2

−1

0

1

2

3

4x 10

−4

α [deg]

dα/d

t


Figure 8. Aeroelastic response for M∞ = 0.75 and U = 3.37 with medium unstructured grid.

0 100 200 300 400−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Pitc

h [d

eg]

Nondimensional time0 200 400 600 800

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Nondimensional time

Pitc

h [d

eg]


Figure 9. Aeroelastic responses for M∞ = 0.75 with medium unstructured grid.

iterations, is required. Then, the experience shows that the costs of computing one flutter

point with the coarse and medium grids correspond basically to the cost of one steady state

CFD computation. However, the calculations performed with the fine grid increase the com-

putational costs of the IPM iterations considerably. One can observe that the computational

costs of a steady state solution in the implicit CFD code used in the IPM methodology is

about 32 seconds for the coarse grid, 2 minutes and 10 seconds for the medium grid and 8

minutes and 30 seconds for the fine grid. All calculations have been performed in the same

machine with an Intel Core 2 Duo 2.2GHz processor with 2GB of memory.

Figure 12 shows the comparison of the various flutter points at M∞ = 0.75, obtained with

IPM and time-marching methodologies and considering different grids in the calculations. In

order to perform further calculations in this paper, the coarse grid has been chosen to obtain

13 of 28

0 200 400 600 800−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Nondimensional time

Pitc

h [d

eg]

−0.04 −0.02 0 0.02 0.04 0.06−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−4

α [deg]

dα/d

t


Figure 10. Aeroelastic response for M∞ = 0.75 and U = 3.3 with fine unstructured grid.

0 200 400 600 800−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Nondimensional time

Pitc

h [d

eg]

0 200 400 600 800−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Pitc

h [d

eg]

Nondimensional time


Figure 11. Aeroelastic responses for M∞ = 0.75 with fine unstructured grid.

the flutter boundary results with the IPM methodology. In the time-marching calculations,

the medium grid has been chosen to perform aeroelastic analyses in the remaining test cases

to be presented in the paper.

An estimate of the computational costs for a complete flutter boundary, as presented

in Fig. 1, can be obtained. The time-marching methodology requires about 1 month of

sequential calculations for the number of flight conditions considered in Fig. 1. In the IPM

calculations, the computational costs for the same number of flutter points presented in Fig. 1

is about 12 hours of sequential calculations. It is important to note that the computational

costs of the flutter boundary calculations, with both methodologies, depend on a good initial

guess of the solutions.

14 of 28

Table 3. Structured grids used for the mesh refinement study.

Grid name Grid size Numbers of cells on airfoil surface

coarse 129 × 33 96

medium 257 × 65 192

fine 513 × 129 384

*

**

Mach

Red

uced

Vel

ocity

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

Time marching - unstructured gridIPM - structured grid

*

coarse grid

medium gridfine gridcoarse grid

medium grid

fine grid

Figure 12. Flutter boundaries for M∞ = 0.75, refinement study in the time-marching and IPMmethodologies.

B. Nonlinear Structural Model Results

Flutter boundaries including torsional polynomial nonlinearities in the structure have been

calculated. Torsional polynomial nonlinearities are added according to type and incidence

considered in the model. The first structural nonlinearity, shown in Fig. 13, is given by

f(α) = 0.18α + 18000α3 . (19)

The flutter boundaries, considering the structural polynomial nonlinearity, given in Eq. (19),

are shown in Fig. 14. Time-marching calculations have been performed with this nonlinear

structural model. Figure 15 shows convergent pitch response for M∞ = 0.75 and U = 1.2.

On the other hand, Fig. 16 shows LCO at U = 1.4 for the same freestream Mach number.

The results indicate that the flutter point is at about U = 1.3 for this Mach number. In-

creasing the reduced velocity to U = 1.7, the system maintains the LCO response with high

frequency, as indicated in Fig. 17. A comparison of these results with only the linear portion

15 of 28

−0.01 −0.005 0 0.005 0.01

−0.01

−0.005

0

0.005

0.01

α [rad]

f(α)

linearnonlinear

Figure 13. Torsional polynomial nonlinearity f(α) = 0.18α + 18000α3.

* **

* ********

***

*

******

Mach Number

Red

uced

Vel

ocity

0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9 IPM - linear structureIPM - nonlinear structure

*

Figure 14. Comparison of stability boundaries for the heavy case with polynomial structuralnonlinearity given by Eq. (19).

of the polynomial, i.e., f(α) = 0.18α, can be obtained by observing the linear results, which

are shown in Fig. 18(a) for M∞ = 0.75 and U = 1.2, and in Fig. 18(b) for M∞ = 0.75 and

U = 1.4. One can see that, for this Mach number, the results indicated that the calculations

are convergent for U = 1.2, and the flutter point is at about the same value of U = 1.3

for both linear and nonlinear structures. The IPM results indicate that the flutter point for

M∞ = 0.75 is at U = 1.1407, as indicated in Fig. 14.

16 of 28

0 100 200 300 400 500 600−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Pitc

h [d

eg]


−6

−4

−2

0

2

4

6

8

10x 10

−4

α [deg]

dα/d

t


Figure 15. Aeroelastic response for f(α) = 0.18α + 18000α3 nonlinearity at M∞ = 0.75, U = 1.2.

0 200 400 600 800

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Pitc

h [d

eg]


−3

−2

−1

0

1

2

3

4x 10

−3

α [deg]

dα/d

t



For all cases previously analyzed, one can observe that the LCO amplitudes, in this

“heavy” case, typically involve very small incidences. As a form of further evaluating the

analysis procedure implemented, three news types of nonlinearities are added to the model.

These are classified according to the degree of the polynomial, and the same incidence interval

is considered, as indicated in Figs. 19(a) and (b). As before, one can observe that incidences

are kept very small in the evaluation of flutter boundaries for the heavy case.

The first structural nonlinearity is a 7-degree polynomial, P7, given by

f(α) = (3.6728 × 10−1)α − (2.8389 × 10−12)α2 + (3.4407 × 106)α3 + (1.2850 × 10−5)α4

− (8.1911 × 1012)α5 − (17.695α6) + (6.4405 × 1018)α7 . (20)

17 of 28

0 200 400 600 800

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Nondimensional time

Pitc

h [d

eg]

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3−3

−2

−1

0

1

2

3x 10

−3

α (deg)

dα/d

t



0 100 200 300 400 500 600 700−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Pitc

h [d

eg]

Nondimensional time0 50 100 150 200 250 300

−20

−15

−10

−5

0

5

10

15

20

Nondimensional time

Pitc

h [d

eg]

(a) Time history: M∞ = 0.7, U = 1.2 (b) Time history: M∞ = 0.7, U = 1.4

Figure 18. Time histories for linear portion of the polynomial, f(α) = 0.18α, for comparisonpurposes.

The second structural nonlinearity is a 11-degree polynomial, P11, given by

f(α) = (1.3076 × 10−1)α − (2.3217 × 10−11)α2 + (1.0067 × 107)α3 + (3.3485 × 10−4)α4

− (6.0815 × 1013)α5 − (1.6774 × 103)α6 + (1.7859 × 1020)α7 + (3.3998 × 109)α8

− (2.4809 × 1026)α9 − (2.4809 × 1015)α10 + (1.3070 × 1032)α11 . (21)

18 of 28

−5 0 5

x 10−4

−8

−6

−4

−2

0

2

4

6

8x 10

−4

α [rad]

f(α)

7 degree polynomial11 degree polynomial13 degree polynomiallinear

−2 −1 0 1 2 3

x 10−4

−2

−1

0

1

2

x 10−4

α [rad]

f(α)

7 degree polynomial11 degree polynomial13 degree polynomiallinear

(a) Complete view (b) Close-up view

Figure 19. Structural nonlinearity: 7-degree polynomial, 11-degree polynomial and 13-degreepolynomial.

Finally, the third structural nonlinearity is a 13-degree polynomial, P13, given by

f(α) = (3.5799 × 10−2)α + (7.9409 × 10−10)α2 + (1.4512 × 107)α3 − (1.7529 × 10−2)α4

− (1.1979 × 1014)α5 + (1.3849 × 105)α6 + (5.1166 × 1020)α7 − (4.8616 × 1011)α8

− (1.1577 × 1027)α9 + (7.7577 × 1017)α10 + (1.3179 × 1033)α11 − (4.5889 × 1023)α12

− (5.9384 × 1038)α13 . (22)

Figure 20 shows the flutter boundaries with the 7 and 11-degree polynomial nonlinear-

ities compared with the linear structure results (f(α) = α). In this case, time-marching

calculations have been performed with initial incidence equal to 0.5 deg. or 0.0087 rad. In

Fig. 21, time history and phase plane results for M∞ = 0.75 and U = 1 are presented.

For this flight condition, the system exhibits a convergent response. For M∞ = 0.75 and

U = 1.2, one obtains an LCO type response, which is consistent with the results obtained,

for this test case, with the time-marching code as indicated in Fig. 22. The limit cycle

oscillations are, however, of fairly low amplitude. The system again shows LCO response,

at comparatively larger amplitudes, when the reduced speed is increased to U = 1.5 and

U = 2. The results are shown in Figs. 23 and 24, respectively. In the inverse power method

calculations, the flutter point for M∞ = 0.75 and zero incidence, and considering the P11

structural nonlinearity, occurs at U = 1.1853.

A verification on the IPM calculation was also performed, for a very coarse grid, by

computing the complete eigenvalue spectrum of A for the heavy case, using Matlab. The

results of this calculation are shown in Fig. 25(a), for a range of values of U . A close-up

19 of 28

* **

* ********

***

*

******

+ + + + ++++++++

++++++

Mach Number

Red

uced

Vel

ocity

0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

IPM - linear structureIPM - P7 structural nonlinearityIPM - P11 structural nonlinearity

*+

Figure 20. Comparison of stability boundary for the heavy case with the P7 and P11 polyno-mial structural nonlinearities.

0 100 200 300 400 500 600 700−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

Pitc

h [d

eg]

Nondimensional time−0.015 −0.01 −0.005 0 0.005 0.01 0.015−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−4

α [deg]

dα/d

t


Figure 21. Aeroelastic response for P11 at M∞ = 0.75 and U = 1.

view of the eigenspectrum for U values ranging from 0.6 to 1.5 is plotted in Fig. 25(b). The

arrows indicate increasing values of U . The critical eigenvalue crosses the imaginary axis at

U = 1.2, which is similar to the values found from the IPM and time-marching analyses.

The next case considers the flutter boundary for the 13-degree polynomial nonlinearity.

In Fig. 26, the results are compared with those obtained with a linear structure. For the

range of M∞ = 0.5 up to 0.8, the flutter boundaries have presented a strange behavior in

the sense that the linear structure is yielding a lower flutter speed. In order to verify such

results, both the system matrix eigenspectrum and time-marching analysis calculations have

20 of 28

0 100 200 300 400 500 600 700−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Nondimensional time

Pitc

h [d

eg]

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015−3

−2

−1

0

1

2

3x 10

−4

α [deg]

dα/d

t


Figure 22. Aeroelastic response for P11 at M∞ = 0.75 and U = 1.2.

0 200 400 600 800−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Nondimensional time

Pitc

h [d

eg]

−0.02 −0.01 0 0.01 0.02−4

−3

−2

−1

0

1

2

3

4x 10

−4

α [deg]

dα/d

t



been evaluated.

The first aspect that can be verified concerns the fact that the flutter point, obtained

from the IPM approach and using the coarse grid from Table 3, for M∞ = 0.75, occurs

at approximately U = 3.4. Further calculations were also performed with a computational

grid which is even coarser than the so-called coarse grid from Table 3. This structured grid

had 96 × 24 points, and the flutter point, for the same freestream Mach number, occurs at

U = 4.29. Therefore, the system matrix eigenspectrum, for this even coarser grid, is plotted

up to U = 4.35 in Fig. 27, in order to check if any eigenvalues would cross the imaginary

axis. One can observe from the figure that an eigenvalue pair crosses the imaginary axis

towards the very end of the reduced velocity range considered in the present case. Moreover,

21 of 28

0 200 400 600 800−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Pitc

h [d

eg]

Nondimensional time−0.02 −0.01 0 0.01 0.02

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−4

α [deg]

dα/d

t


Figure 24. Aeroelastic response for P11 at M∞ = 0.75 and U = 2.

−180 −160 −140 −120 −100 −80 −60 −40 −20 0 20−150

−100

−50

0

50

100

150

Re

Im

−0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02−4

−3

−2

−1

0

1

2

3

Re

Im

(a) Complete eigenvalue spectrum (b) Close-up view of the spectrum

Figure 25. Eigenvalue spectrum of A matrix with the P11 polynomial structural nonlinearityfor M∞ = 0.75 and a range of U values.

one must also verify whether no other (isolated) eigenvalue is in the positive side of the real

axis. Figure 27 shows that, for U = 1.3, the eigenvalues along the real axis cross to the

positive side. Such behavior did not occur in P11 polynomial structural nonlinearity case,

as shown in Fig. 25(b).

Time-marching analyses have also been performed for this test case. Time histories

and phase plane results for the P13 polynomial structural nonlinearity, for M∞ = 0.75 and

U = 1.1, are shown in Figs. 28(a) and (b), respectively. One can observe from these responses

that the system exhibits initial irregular oscillations and seems to stabilize in an equilibrium

point different from zero. Figures 29(a) and (b) show the time-marching and phase plane

results, respectively, with an increase in the reduced velocity to U = 1.2. It can be observed

22 of 28

* **

* ********

***

*

******

Mach Number

Red

uced

Vel

ocity

0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

IPM - linear structureIPM - P13 structural nonlinearity

*

Figure 26. Comparison of the stability boundaries for the heavy case with the P13 polynomialstructural nonlinearity.

−0.05 0 0.05 0.1 0.15 0.2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Re

Im

Figure 27. Close-up view of the eigenvalue spectrum with the P13 polynomial structuralnonlinearity (M∞ = 0.75 and U ranging from 1.0 to 4.35).

that the system, initially, exhibits irregular oscillations and, afterwards, follows somewhat

regular oscillations with constant amplitude. The system behavior for U = 1.5 is more

complex than in the U = 1.2 case, as observed in the time history and phase plane results

in Figs. 30(a) and (b), respectively.

In the P13 case, the flutter point for M∞ = 0.75 can be estimate with the time-marching

analysis at U = 1.2. IPM calculations could not estimate this value for flutter point at this

Mach number, as shown in Fig. 26. It is important to note that Hopf bifurcation calculations

23 of 28

0 100 200 300 400 500−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Nondimensional time

Pitc

h [d

eg]

−0.015 −0.01 −0.005 0 0.005 0.01 0.015−3

−2

−1

0

1

2

3x 10

−4

α [deg]

dα/d

t



0 200 400 600 800

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Nondimensional time

Pitc

h [d

eg]

−0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02−4

−3

−2

−1

0

1

2

3

4x 10

−4

α [deg]

dα/d

t



are basically concerned with finding the point at which the system starts to exhibit limit

cycle oscillations. Some basic problems can be found in the Hopf bifurcation analyses for

the P13 nonlinear case. One of such problems can be associated with a nonlinear behavior

which is different from LCO. In other words, the results here obtained might indicate a

chaotic behavior and the present tools might not be the most appropriate to characterize

such situation. On the other hand, the equilibrium point used for the Hopf bifurcation

analysis must be further verified in the P13 nonlinear case. As one can observe in Fig. 28,

the system seems to stabilize in an equilibrium point different from zero and the equilibrium

point is assumed to be zero in the IPM calculations.

For comparison purposes, time-marching calculations considering only the linear portion

24 of 28

0 200 400 600 800

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Nondimensional time

Pitc

h [d

eg]

−0.02 −0.01 0 0.01 0.02−3

−2

−1

0

1

2

3

4x 10

−4

α [deg]

dα/d

t



0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Nondimensional time

Pitc

h [d

eg]

0 100 200 300 400 5000

0.5

1

1.5

2

2.5

Pitc

h [d

eg]

Nondimensional time

(a) Time histories for U = 1.1 (b) Time histories for U = 1.2

Figure 31. Linear portion of the P13 polynomial for M∞ = 0.75, for comparison purposes.

of the P13 polynomial (3.5799 × 10−2α) have also been performed. For M∞ = 0.75, at

U = 1.1 and U = 1.2, time histories are shown in Figs. 31(a) and (b), respectively. The

response initially diverges and goes to a non-zero equilibrium point. The system behavior

for U = 1.5 is indicated in Fig. 32. The system exhibits oscillations around a non-zero

equilibrium point. Hence, the flutter point for the linear portion of the P13 polynomial can

be considered approximately at U = 1.4.

Other types of strong polynomial nonlinearities have been evaluated. The overall usual

behavior is similar to that observed with the P13 polynomial case. Such behavior seems to

be associated to the relative small coefficient of the polynomial linear term. However, for

time-marching calculations with small coefficient of the linear structural model performed,

25 of 28

0 200 400 600 8000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Nondimensional time

Pitc

h [d

eg]

Figure 32. Linear portion of the P13 polynomial for M∞ = 0.75 and U = 1.5, for comparisonpurposes.

the system exhibits oscillations in a non-zero equilibrium point. An appropriate method to

calculate the equilibrium points, before Hopf bifurcation analyses, is necessary in order to

continue the study of flutter boundaries with the IPM methodology in these cases. Other

test cases analyzed in the context of the present research, and which are not shown here, have

indicated that flutter boundaries for zero incidence, calculated considering complete nonlin-

ear polynomials, are equal to those obtained with only the linear portion of the respective

polynomials.

VII. Concluding Remarks

Concentrated nonlinearities are shown to have significant effects on the aeroelastic re-

sponses and to yield limit cycle oscillations below the linear flutter speed. For the compu-

tations performed so far, initial results for direct Hopf bifurcation predictions of the pitch-

plunge typical section airfoil with structural nonlinearities indicate that the nonlinear terms

do not influence the bifurcation point. Time-marching analyses are compared to direct cal-

culations of Hopf bifurcation points. The results agree well and the computational tools have

show to be powerful enough to analyze nonlinear effects in aeroelasticity. Post bifurcations

calculations show the influence of nonlinear structural terms on LCO. However, further

analyses are still necessary in order to characterize the complete behavior of such nonlinear

systems, even for the simple typical section model.

Grid refinement studies indicate a good approximation between time-marching and IPM

solution procedures. The rise of the computational costs, especially with finer grids, indi-

cates the advantage of Hopf bifurcation analysis for the flutter boundary calculations. In

26 of 28

this fashion, the problem of locating a one-parameter stability boundary is reduced from

multiple, unsteady, time-marching calculations to a single steady-state calculation. Hence,

the experience has shown that it is possible to reduce the computational costs from one

month of sequential calculations with the time-marching code to some hours of sequential

calculations with the IPM code for the complete flutter boundary results presented in the

paper.

Strong nonlinearity results obtained so far have demonstrated that the Hopf bifurcation

analysis could not find the complete flutter boundary for all cases, as shown in the P13

nonlinear case. The difficulties observed seem to be associated with different nonlinear

behavior of the aeroelastic system, such as chaos. On the other hand, the time-marching

calculations have shown non-zero equilibrium points when relatively small coefficients of the

polynomial linear term are used. Further work is necessary in order to provide a more detailed

comparison of time domain predictions and direct Hopf bifurcation results for complete

flutter boundaries with nonlinear structural models. Variations of the mean, steady state

equilibrium conditions, from which computations are performed, are needed.

VIII. Acknowledgments

The authors gratefully acknowledge the partial support for this research provided by

Conselho Nacional de Desenvolvimento Cientıfico e Tecnologico, CNPq, through a Doctoral

Scholarship for the first author. Further support from CNPq was provided through the

Integrated Project Research Grant No. 312064/2006-3. Additional support to this research

was provided by Fundacao de Amparo a Pesquisa do Estado de Sao Paulo, FAPESP, through

the Thematic Project Process No. 2004/16064-9. The authors also gratefully acknowledge

the very productive interaction with Prof. Ken Badcock and Dr. M. Woodgate, who provided

the Hopf bifurcation code used in the present research.

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