research article stability analysis of a flutter panel...

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

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)

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(𝑛𝜋𝑥𝐿) 𝑑𝑥,

C𝑎𝑚𝑛

=2𝑏𝑞0

𝛽𝑈0

(Ma2 − 2)(Ma2 − 1)

∫

𝐿

0


) sin(𝑛𝜋𝑥𝐿)𝑑𝑥,

K𝑒𝑚𝑛

= 𝐸∗𝐼𝑚2𝑛2𝜋4

𝐿4∫

𝐿

0


) sin(𝑛𝜋𝑥𝐿) 𝑑𝑥,

K𝑎𝑚𝑛

=2𝑏𝑞0

𝛽

𝑛𝜋

𝐿∫

𝐿

0


) 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)


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)


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.


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.

References

[1] F. Liu, J. Cai, Y. Zhu, H. M. Tsai, and A. S. F. Wong, “Calculationof wing flutter by a coupled fluid-structure method,” Journal ofAircraft, vol. 38, no. 2, pp. 334–342, 2001.

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