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Research ArticleStability Analysis of a Flutter Panel with Axial
Excitations
Meng Peng and Hans A. DeSmidt
Department of Mechanical, Aerospace and Biomedical
Engineering,The University of Tennessee, 606 Dougherty Engineering
Building,Knoxville, TN 37996-2210, USA
Correspondence should be addressed to Meng Peng;
[email protected]
Received 6 April 2016; Revised 23 June 2016; Accepted 8 July
2016
Academic Editor: Marc Thomas
Copyright © 2016 M. Peng and H. A. DeSmidt.This is an open
access article distributed under the Creative Commons
AttributionLicense, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is
properlycited.
This paper investigates the parametric instability of a panel
(beam) under high speed air flows and axial excitations. The idea
is toaffect out-of-plane vibrations and aerodynamic loads by
in-plane excitations.The periodic axial excitation introduces
time-varyingitems into the panel system.The numerical method based
on Floquet theory and the perturbation method are utilized to solve
theMathieu-Hill equations.The system stability with respect to
air/panel density ratio, dynamic pressure ratio, and excitation
frequencyare explored. The results indicate that panel flutter can
be suppressed by the axial excitations with proper parameter
combinations.
1. Introduction
Panel (beam) flutter usually occurs when high speed objectsmove
in the atmosphere, such as flight wings [1] and ballute[2]. This
phenomenon is a self-excited oscillation due to thecoupling of
aerodynamic load and out-of-plane vibration.Since flutter can cause
system instability andmaterial fatigue,many scholars have carried
out theoretical and experimentalanalyses on this topic. Nelson and
Cunningham [3] inves-tigated flutter of flat panels exposed to a
supersonic flow.Their model is based on small-deflection plate
theory andlinearized flow theory, and the stability boundary is
deter-mined after decoupling the system equations by
Galerkin’smethod. Olson [4] applied finite element method to the
two-dimensional panel flutter. A simply supported panel
wascalculated and an extremely accuracy approximation couldbe
obtained using only a few elements. Parks [5] utilizedLyapunov
technique to solve a two-dimensional panel flutterproblem and used
piston theory to calculate aerodynamicload. The results gave a
valuable sufficient stability criterion.Dugundji [6] examined
characteristics of panel flutter athigh supersonic Mach numbers and
clarified the effects ofdamping, edge conditions, traveling, and
standingwaves.Thepanel, Dugundji considered, is a flat rectangular
one, simplysupported on all four edges, and undergoes
two-dimensionalmidplane compressive forces.
Dowell [7, 8] explored plate flutter in nonlinear area
byemploying Von Karman’s large deflection plate theory. Zhouet al.
[9] built a nonlinear model for the panel flutter via finiteelement
method, including linear embedded piezoelectriclayers.The optimal
control approach for the linearizedmodelwas presented. Gee [10]
discussed the continuation method,as an alternate numerical method
that complements directnumerical integration, for the nonlinear
panel flutter. Tizzi[11] researched the influence of nonlinear
forces on flutterbeam. In most cases, the internal force in panel
or beamresults from either external constant loads or
geometricnonlinearities. Therefore, their models are
time-invariantsystem.
Panel is usually excited by in-plane loads resulting fromthe
vibrations generated and/or transmitted through theattached
structures and dynamics components when expe-riencing aerodynamic
loads. If the in-plane load is timedependent, the system becomes
time-varying. The topicof dynamic stability of time-varying systems
attracts manyattentions. Iwatsubo et al. [12] surveyed parametric
instabilityof columns under periodic axial loads for different
boundaryconditions.They used Hsu’s results [13] to determine
stabilityconditions and discussed the damping effect on
combinationresonances. Sinha [14] and Sahu and Datta [15, 16]
studiedthe similar problem for Timoshenko beam and curved
panel,respectively, and both models are classified as
Mathieu-Hill
Hindawi Publishing CorporationAdvances in Acoustics and
VibrationVolume 2016, Article ID 7194764, 7
pageshttp://dx.doi.org/10.1155/2016/7194764
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2 Advances in Acoustics and Vibration
b
L
x
w(x, t)P(t)
Air flowU0, 𝜌0
E, �, 𝜌
h
Figure 1: Simply supported panel (beam) subjected to an air
flowand an axial excitation.
equations. Furthermore, the stability of the nonlinear
elasticplate subjected to a periodic in-plane load was analyzed
byGanapathi et al. [17]. They solved nonlinear governing equa-tions
by using theNewmark integration scheme coupled witha modified
Newton-Raphson iteration procedure. In addi-tion, many papers have
been published for dynamic stabilityanalyses of shells under
periodic loads [18–24]. Furthermore,Hagedorn andKoval [25]
considered the effect of longitudinalvibrations and the space
distributed internal force. Thecombination resonance was analyzed
for Bernoulli-Euler andTimoshenko beams under the spatiotemporal
force. Yanget al. [26] developed a vibration suppression scheme
foran axially moving string under a spatiotemporally
varyingtension. Lyapunov method was employed to design
robustboundary control laws, but the effect of parameters of
thespatiotemporally varying tension on system stability has notbeen
fully analyzed.
Nevertheless, the published investigations on the para-metric
stability of the flutter panel (beam) with the periodi-cally
time-varying system stiffness due to axial excitations arescarce.
This paper is to explore the coactions of time-varyingaxial
excitations and aerodynamic loads on panel (beam) andconduct
parameter studies. The stability analysis is executedfirst by
Floquet theory numerically and then byHsu’s methodanalytically for
approximations.
2. System Description and Model
Theconfiguration considered herein is an isotropic thin
panel(beam) with constant thickness and cross section. As shownin
Figure 1, the panel is simply supported at both ends anda periodic
axial excitation acts on the right end. The panel’supper surface is
exposed to a supersonic flow, while theair beneath the lower
surface is assumed not to affect thepanel dynamics. Another
assumption made here is the axialstrain from the lateral
displacement is very small so that itcan be ignored. The system
model is based on the coupledeffects from the out-of-plane
(lateral) displacement of thepanel, aerodynamic loads, and
time-varying in-plane (axial)excitation forces.
The total kinetic energy of the penal due to
lateraldisplacements is
𝑇 =1
2𝜌𝐴∫
𝐿
0
�̇� (𝑥, 𝑡)2𝑑𝑥, (1)
where 𝑤(𝑥, 𝑡) is the lateral displacement of the panel mea-sured
in the ground fixed coordinate frame, 𝜌 the panel
density, 𝐴 the panel cross-sectional area, and “⋅” the
differ-entiation with respect to time 𝑡.
The total potential energy of the panel due to
lateraldisplacements is
𝑉 =1
2𝐸∗𝐼 ∫
𝐿
0
𝑤(𝑥, 𝑡)2𝑑𝑥 +
1
2𝑃 (𝑡) ∫
𝐿
0
𝑤(𝑥, 𝑡)2𝑑𝑥, (2)
where 𝐸∗ = 𝐸/(1 − ]2) for panel and 𝐸∗ = 𝐸 for beamwith Young’s
modulus 𝐸 and Poisson’s ratio ]; the momentof inertia is given by 𝐼
= 𝑏ℎ3/12 with panel width 𝑏 andpanel thickness ℎ; 𝑃(𝑡) is the
periodic axial excitation withthe frequency 𝜔 and “” indicates
differentiation with respectto axial position 𝑥.
For material viscous damping, a Rayleigh dissipationfunction is
defined as
𝑅 =1
2𝜉𝐸∗𝐼 ∫
𝐿
0
�̇�(𝑥, 𝑡)2𝑑𝑥. (3)
Here, 𝜉 is the material viscous loss factor of the panel.The
aerodynamic load is expressed by using the classic
quasisteady first-order piston theory [4, 6, 7, 9, 11]:
𝑝 (𝑥, 𝑡) =2𝑞0
𝛽[𝜕𝑤 (𝑥, 𝑡)
𝜕𝑥+
(Ma2 − 2)(Ma2 − 1)
1
𝑈0
𝜕𝑤 (𝑥, 𝑡)
𝜕𝑡] , (4)
where 𝑞0
= 𝜌0𝑈2
0/2 is the dynamic pressure, 𝜌
0is the
undisturbed air flow density, 𝑈0is the flow speed at
infinity,
Ma is Mach number, and 𝛽 = (Ma2 − 1)1/2.The flow goes against
the lateral vibrations of the panel,
so the nonconservative virtual work from aerodynamic loadis
negative and its expression is
𝛿𝑊nc = −∫𝐿
0
𝑝 (𝑥, 𝑡) 𝑏𝛿𝑤 (𝑥, 𝑡) 𝑑𝑥. (5)
For a simply supported panel, the modal expansion of𝑤(𝑥, 𝑡) can
be assumed in the form
𝑤 (𝑥, 𝑡) =
∞
∑
𝑛=1
𝜂𝑛 (𝑡) sin(
𝑛𝜋𝑥
𝐿) , 𝑛 = 1, 2, 3, . . . , (6)
where 𝑛 is the positive integer and 𝜂𝑛(𝑡) the generalized
coordinate. After substituting (6) into all energy
expressions,the system equations-of-motion are obtained via
Lagrange’sEquations
𝑑
𝑑𝑡(𝜕𝑇
𝜕�̇�) −
𝜕𝑇
𝜕𝑞+𝜕𝑉
𝜕𝑞+𝜕𝐷
𝜕�̇�= 𝑄nc (7)
with the generalized force 𝑄nc = 𝜕𝛿𝑊nc/𝜕𝛿𝑞 and
generalizedcoordinates vector
𝑞 (𝑡) = [𝜂1 (𝑡) 𝜂2 (𝑡) 𝜂3 (𝑡) ⋅ ⋅ ⋅]𝑇
. (8)
Finally, the system equations-of-motion are given as
M�̈� (𝑡) + (C𝑑 + C𝑎) �̇� (𝑡) + [K𝑒 + K𝑎 + K𝑃 (𝑡)] 𝑞 (𝑡)
= 0,
(9)
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Advances in Acoustics and Vibration 3
where the elements of coefficient matrices are
M𝑚𝑛
= 𝜌𝐴∫
𝐿
0
sin(𝑚𝜋𝑥𝐿
) sin(𝑛𝜋𝑥𝐿)𝑑𝑥,
C𝑑𝑚𝑛
= 𝜉𝐸∗𝐼𝑚2𝑛2𝜋4
𝐿4∫
𝐿
0
sin(𝑚𝜋𝑥𝐿
) sin(𝑛𝜋𝑥𝐿) 𝑑𝑥,
C𝑎𝑚𝑛
=2𝑏𝑞0
𝛽𝑈0
(Ma2 − 2)(Ma2 − 1)
∫
𝐿
0
sin(𝑚𝜋𝑥𝐿
) sin(𝑛𝜋𝑥𝐿)𝑑𝑥,
K𝑒𝑚𝑛
= 𝐸∗𝐼𝑚2𝑛2𝜋4
𝐿4∫
𝐿
0
sin(𝑚𝜋𝑥𝐿
) sin(𝑛𝜋𝑥𝐿) 𝑑𝑥,
K𝑎𝑚𝑛
=2𝑏𝑞0
𝛽
𝑛𝜋
𝐿∫
𝐿
0
sin(𝑚𝜋𝑥𝐿
) cos(𝑛𝜋𝑥𝐿) 𝑑𝑥,
K𝑃𝑚𝑛(𝑡) =
𝑚𝑛𝜋2
𝐿2𝑃 (𝑡) ∫
𝐿
0
cos(𝑚𝜋𝑥𝐿
) cos(𝑛𝜋𝑥𝐿) 𝑑𝑥.
(10)
The stiffness matrix K𝑃(𝑡) resulting from the periodicaxial
excitation introduces a periodically time-varying iteminto the
system, so (9) is identified as a set of coupledMathieu-Hill
equations. Subsequently, the system equations-of-motion are
transformed into the nondimensional (N.D.)form
M∗∗
𝑞 (𝜏) + (C𝑑 + C𝑎)∗
𝑞 (𝜏)
+ [K𝑒 + K𝑎 + K𝑃 (𝜏)] 𝑞 (𝜏) = 0(11)
with the dimensionless parameters and coordinates:
𝑥 =𝑥
𝐿,
𝑞 =𝑞 (𝑡)
ℎ,
Ω =𝜋2ℎ
𝐿2√𝐸∗
12𝜌,
𝜏 = 𝑡Ω,
𝜇 =𝜌0
𝜌,
𝜉 = 𝜉Ω,
𝑃cr =𝐸∗𝐼𝜋2
𝐿2,
𝑓 (𝜏) =𝑃 (𝑡)
𝑃cr,
𝜎 =𝑞0
𝐸∗,
𝛼 =ℎ
𝐿,
𝜔 =𝜔
Ω,
∗
( ) =𝑑 ( )
𝑑𝜏.
(12)
The elements of the N.D. coefficient matrices in (11) are
M𝑚𝑛
={
{
{
1 if 𝑚 = 𝑛
0 if 𝑚 ̸= 𝑛,
C𝑑𝑚𝑛
={
{
{
𝜉𝑛4 if 𝑚 = 𝑛
0 if 𝑚 ̸= 𝑛,
C𝑎𝑚𝑛
=
{{{
{{{
{
2√6𝜇𝜎 (Ma2 − 2)
𝜋2𝛼2 (Ma2 − 1)3/2if 𝑚 = 𝑛
0 if 𝑚 ̸= 𝑛,
K𝑒𝑚𝑛
={
{
{
𝑛4 if 𝑚 = 𝑛
0 if 𝑚 ̸= 𝑛,
K𝑎𝑚𝑛
=
{{{
{{{
{
0 if 𝑚 = 𝑛
48𝜎𝑚𝑛 [1 − (−1)𝑚+𝑛
]
√Ma2 − 1 (𝑚2 − 𝑛2) 𝜋4𝛼3if 𝑚 ̸= 𝑛,
K𝑃𝑚𝑛(𝜏) =
{
{
{
𝑛2𝑓 (𝜏) if 𝑚 = 𝑛
0 if 𝑚 ̸= 𝑛.
(13)
Here,M,C𝑑,C𝑎, andK𝑒 are constant symmetricmatrices;K𝑎 is a
constant skew-symmetric matrix; K𝑃 is a symmetrictime-varying
matrix with a period of 2𝜋/𝜔.
3. Mathematical Methods for Stability Analysis
Due to the periodic axial excitations, (11) becomes a
period-ically linear time-varying system. Floquet theory is able
toassess the stability of this type of systems through
evaluatingthe eigenvalues of the Floquet transition matrix
(FTM)numerically [24, 27–32]. The FTM method can obtain allunstable
behaviors of a system but at the cost of intensivelynumerical
computations, so the perturbation method orig-inally developed by
Hsu [13, 33] is modified in this paper toapproximate the system
stability boundary in an efficient way.The results from the
perturbation (analytical) method will becompared with those from
the FTM (numerical) method.
To implement Hsu’s perturbation method, the time-invariant part
of the system stiffness matrix, K𝑒 + K𝑎, needsto be diagonalized by
its left and right eigenvectors [34]. Theresulting
equations-of-motion are
∗∗
𝑞 (𝜏) + C̃∗
𝑞 (𝜏) + [K̃ + K̃𝑃 (𝜏)] 𝑞 (𝜏) = 0 (14)
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4 Advances in Acoustics and Vibration
with
C̃ = X𝑇𝐿(C𝑑 + C𝑎)X
𝑅
K̃ = X𝑇𝐿(K𝑒 + K𝑎)X
𝑅
K̃𝑃 (𝜏) = X𝑇𝐿K𝑃 (𝜏)X𝑅,
(15a)
(K𝑒 + K𝑎)X𝑅= 𝜆X𝑅
(K𝑒 + K𝑎)𝑇
X𝐿= 𝜆X𝐿
X𝑇𝐿X𝑅= I,
(15b)
where 𝜆 is the eigenvalues of K𝑒 + K𝑎 and X𝐿and X
𝑅are the
corresponding left and right eigenvectors, respectively,
andorthonormal to each other. I is the identity matrix.
The damping matrix and the time-varying stiffnessmatrix
resulting from the axial excitation are assumed to besmall
quantities relative to the time-invariant system stiffnessfor
better predictability through Hsu’s method. The standardform in
Hsu’s method is obtained by separating the regularand perturbed
items in (14) and then expanding the periodictime-varying stiffness
matrix into Fourier series,
∗∗
𝑞 (𝜏) + K̃ 𝑞 (𝜏) = −C̃∗
𝑞 (𝜏) − K̃𝑃 (𝜏) 𝑞 (𝜏) , (16)
where
K̃𝑃 (𝜏) = K̃𝑐 cos (𝜔𝜏) + K̃𝑠 sin (𝜔𝜏) . (17)
With (16), the stability criteria given in [13, 33] can be
appliedby setting epsilon to one.
4. Stability Boundaries for the Panelunder Flow
For the simple demonstrations of the model and the
solvingprocess developed above, only the first two modes
areconsidered so that the closed-form stability solutions can
beobtained.The axial excitation force considered here is a
singlefrequency cosine function:
∗∗
𝑞 (𝜏) + [
𝜔2
10
0 𝜔2
2
] 𝑞 (𝜏)
= − [
𝑐11
𝑐12
𝑐21
𝑐22
]∗
𝑞 (𝜏) − [
𝑘11
𝑘12
𝑘21
𝑘22
] cos (𝜔𝜏) 𝑞 (𝜏) ,
𝑓 (𝜏) = 𝐹 cos (𝜔𝜏) ,
(18)
where
𝜔1= √
17
2−15
2√1 −
𝜎2
𝜎2𝑐
,
𝜔2= √
17
2+15
2√1 −
𝜎2
𝜎2𝑐
,
(19a)
𝑐11= 𝜉(
17
2−15
2
𝜎𝑐
√𝜎2𝑐− 𝜎2
)
+
2√6 (Ma2 − 2)√𝜇𝜎
𝜋2𝛼2 (Ma2 − 1)3/2,
𝑐22= 𝜉(
17
2+15
2
𝜎𝑐
√𝜎2𝑐− 𝜎2
)
+
2√6 (Ma2 − 2)√𝜇𝜎
𝜋2𝛼2 (Ma2 − 1)3/2,
𝑐12=
15𝜉𝜎
2√𝜎2𝑐− 𝜎2
,
𝑐21= −𝑐12,
(19b)
𝑘11= 𝐹(
5
2−
3𝜎𝑐
2√𝜎2𝑐− 𝜎2
),
𝑘22= 𝐹(
5
2+
3𝜎𝑐
2√𝜎2𝑐− 𝜎2
),
𝑘12= 𝐹
3𝜎
2√𝜎2𝑐− 𝜎2
,
𝑘21= −𝑘12,
(19c)
𝜎𝑐=
15
128𝜋4𝛼3√Ma2 − 1. (19d)
The material viscous loss factor is set to zero for the
eigen-analyses in this paper. The stability boundaries can
beobtained via solving the following equations:
1st principle resonance: 𝜔 = 2𝜔1+ Δ𝜔
Δ𝜔 = ±√𝑘2
11
4𝜔2
1
− 𝑐2
11
2nd principle resonance: 𝜔 = 2𝜔2+ Δ𝜔
Δ𝜔 = ±√𝑘2
22
4𝜔2
2
− 𝑐2
22
Combination resonance: 𝜔 = 𝜔2− 𝜔1+ Δ𝜔
Δ𝜔 = ±√𝑘2
12
4𝜔1𝜔2
− 𝑐11𝑐22.
(20)
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Advances in Acoustics and Vibration 5
0 1 2 30
0.5
0.5
1
1.5
1.5
2
2.5
2.5
3
3.5
4
𝜔1,𝜔
2
𝜔2
𝜔1
𝜎
𝜎c
×10−6
Figure 2: N.D. natural frequency variations with respect to
dynamicpressure ratio: Ma = 2, 𝛼 = 0.005, and 𝜎
𝑐= 2.47 × 10−6.
1 2 3 4 5 6 7 8 9 100
1
2
3
0.5
1.5
2.5 𝜎c
×10−6
𝜎
2𝜔1
2𝜔1
2𝜔2
2Δ𝜔
2Δ𝜔
2Δ𝜔
2Δ𝜔
𝜔2 − 𝜔1
𝜔
Figure 3: Stability plot for the flutter panel with respect to
axialexcitation frequency and dynamic pressure ratio: Ma = 2, 𝛼 =
0.005,𝜎𝑐= 2.47 × 10−6, 𝜇 = 4.39 × 10−5, and 𝐹 = 0.5.
5. Numerical Results
The N.D. natural frequencies due to materials and aero-dynamic
loads are plotted in Figure 2 with respect tothe dynamic pressure
ratio, 𝜎. Once the dynamic pressureratio exceeds the critical value
that equals 𝜎
𝑐, two natural
frequencies merge together, which means they are conjugatepairs
and the system instability occurs.
The system stability with respect to the axial
excitationfrequency and the dynamic pressure ratio is shown in
Fig-ure 3. In this paper, the gray regions in stability plots
indicatethe instabilities computed by the numerical FTMmethod
andthe black dash lines are the stability boundaries calculatedfrom
the analytical perturbation method. The black solidlines in Figure
3 represent the N.D. natural frequencies. Withthe same observations
as in Figure 2, the system flutters forthe entire axial excitation
frequency range calculated here
1 2 3 4 5 6 7 8 9 100
1
2
3
4
2𝜔1
2𝜔2
Δ𝜔
Δ𝜔Δ𝜔
𝜔2 − 𝜔1
𝜔
𝜇
×10−4
Figure 4: Stability plot for the flutter panel with respect to
axialexcitation frequency and air/panel density ratio: Ma = 2, 𝛼 =
0.005,𝜎𝑐= 2.47 × 10−6, 𝜎 = 1.27 × 10−6, and 𝐹 = 0.5.
when the dynamic pressure ratio passes its critical value.
Theprinciple resonances and combination resonance given by
theperturbation method successfully match those by the FTMmethod
with a lot of computation savings.
The system stability with respect to the axial
excitationfrequency and the air/panel density ratio is shown in
Figure 4.Since 𝜎 < 𝜎
𝑐, the combination resonance and principal
resonances are clearly separated. Again, the results from
boththe FTM method and the perturbation method match eachother very
well.
It can be observed in Figures 3 and 4 that the
principalresonance of the second mode causes instabilities for
alldynamic pressure ratio and air/panel density ratio
valuesexplored here. However, it could be stabilized by
differentaxial excitation frequencies. The instabilities around
thefirst principal resonance and combination resonance can
besuppressed for some dynamic pressure ratio and air/paneldensity
ratio values. Their axial excitation frequency stabilityboundaries
are calculated by solving (20) for Δ𝜔 = 0 and theresults are
plotted in Figure 5. The system stability dependson both resonances
in each area divided by those boundaries.The panel system is only
stable when both resonances arestable, shown in the white area in
Figure 5.
6. Summary and Conclusions
This paper investigates the parametric stability of the
panel(beam) under both aerodynamic loads and axial excitations.The
dimensionless equation-of-motion is derived, includingmaterial
viscous damping, axial excitation, and aerodynamicload. The
eigen-analyses based on the first two modes aretaken as examples to
explore the stability properties of theflutter panel system with an
axial single frequency cosineexcitation. Both numerical FTM method
and analytical per-turbation method solve the problem and their
results matcheach other very well.
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6 Advances in Acoustics and Vibration
1 2 3 40
0.5
0.5
1
1.5
1.5 2.5 3.5
2
×10−6
𝜎
𝜎c
𝜇 ×10−4
2𝜔1
𝜔2 − 𝜔1
Unstable2𝜔1
Unstable2𝜔1
Unstable2𝜔1
Stable2𝜔1
Stable2𝜔1
Unstable𝜔2−𝜔1
Unstable𝜔2−𝜔1Unstable𝜔2−𝜔1
Stable𝜔2−𝜔1
Stable𝜔2−𝜔1
Figure 5: Stability plot for the flutter panel with respect to
air/paneldensity ratio and dynamic pressure ratio: Ma = 2, 𝛼 =
0.005, 𝜎
𝑐=
2.47 × 10−6, and 𝐹 = 0.5.
The panel may flutter under high-speed air flows whenits
out-of-plane dynamics couples with the aerodynamicloads. The
parameter study was conducted for the systeminstability zones with
respect to axial excitation frequency,air/panel density ratio, and
air/panel dynamic pressure ratio.Different from the static axial
force, this paper introduces aperiodic axial excitation that brings
the system into the time-varying domain.The axial excitation force
could increase thepanel stiffness locally to overcome aerodynamic
loads wheninteracting with the out-of-plane vibrations.The study
resultsin this paper indicate that the system is stable under
thecombinations of the proper excitation frequency and
certainair/panel density ratio and dynamic pressure ratio.
The perturbation method developed in this paper saveslots of
computations, which can help understand the flutterphenomenon of
the panel with axial excitations more effi-ciently.
Competing Interests
The authors declare that they have no competing interests.
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