Research Article Stability Analysis of a Flutter Panel with Axial … · 2019. 12. 12. · Stability Analysis of a Flutter Panel with Axial Excitations MengPengandHansA.DeSmidt

Research ArticleStability Analysis of a Flutter Panel with Axial Excitations

Meng Peng and Hans A DeSmidt

Department of Mechanical Aerospace and Biomedical EngineeringThe University of Tennessee 606 Dougherty Engineering BuildingKnoxville TN 37996-2210 USA

Correspondence should be addressed to Meng Peng mpeng1volsutkedu

Received 6 April 2016 Revised 23 June 2016 Accepted 8 July 2016

Academic Editor Marc Thomas

Copyright copy 2016 M Peng and H A DeSmidtThis 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 excitationsThe periodic axial excitation introduces time-varyingitems into the panel systemThe numerical method based on Floquet theory and the perturbation method are utilized to solve theMathieu-Hill equationsThe system stability with respect to airpanel 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 vibrationSince flutter can cause system instability andmaterial fatiguemany scholars have carried out theoretical and experimentalanalyses on this topic Nelson and Cunningham [3] inves-tigated flutter of flat panels exposed to a supersonic flowTheir model is based on small-deflection plate theory andlinearized flow theory and the stability boundary is deter-mined after decoupling the system equations by Galerkinrsquosmethod 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 criterionDugundji [6] examined characteristics of panel flutter athigh supersonic Mach numbers and clarified the effects ofdamping edge conditions traveling and standingwavesThepanel 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 Karmanrsquos large deflection plate theory Zhouet al [9] built a nonlinear model for the panel flutter via finiteelement method including linear embedded piezoelectriclayersThe optimal control approach for the linearizedmodelwas presented Gee [10] discussed the continuation methodas 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 andor 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 boundaryconditionsThey used Hsursquos 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 panelrespectively and both models are classified as Mathieu-Hill

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2016 Article ID 7194764 7 pageshttpdxdoiorg10115520167194764

2 Advances in Acoustics and Vibration

b

L

x

w(x t)P(t)

Air flowU0 1205880

E 120588

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 [18ndash24] FurthermoreHagedorn 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 byHsursquos 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 panelrsquosupper 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

119879 =1

2120588119860int

119871

0

(119909 119905)2119889119909 (1)

where 119908(119909 119905) is the lateral displacement of the panel mea-sured in the ground fixed coordinate frame 120588 the panel

density 119860 the panel cross-sectional area and ldquosdotrdquo the differ-entiation with respect to time 119905

The total potential energy of the panel due to lateraldisplacements is

119881 =1

2119864lowast119868 int

119871

0

11990810158401015840(119909 119905)2119889119909 +

1

2119875 (119905) int

119871

0

1199081015840(119909 119905)2119889119909 (2)

where 119864lowast = 119864(1 minus ]2) for panel and 119864lowast= 119864 for beam

with Youngrsquos modulus 119864 and Poissonrsquos ratio ] the momentof inertia is given by 119868 = 119887ℎ

312 with panel width 119887 and

panel thickness ℎ 119875(119905) is the periodic axial excitation withthe frequency 120596 and ldquo1015840rdquo indicates differentiation with respectto axial position 119909

For material viscous damping a Rayleigh dissipationfunction is defined as

119877 =1

2120585119864lowast119868 int

119871

0

10158401015840(119909 119905)2119889119909 (3)

Here 120585 is the material viscous loss factor of the panelThe aerodynamic load is expressed by using the classic

quasisteady first-order piston theory [4 6 7 9 11]

119901 (119909 119905) =21199020

120573[120597119908 (119909 119905)

120597119909+

(Ma2 minus 2)(Ma2 minus 1)

1

1198800

120597119908 (119909 119905)

120597119905] (4)

where 1199020

= 12058801198802

02 is the dynamic pressure 120588

0is the

undisturbed air flow density 1198800is the flow speed at infinity

Ma is Mach number and 120573 = (Ma2 minus 1)12The flow goes against the lateral vibrations of the panel

so the nonconservative virtual work from aerodynamic loadis negative and its expression is

120575119882nc = minusint119871

0

119901 (119909 119905) 119887120575119908 (119909 119905) 119889119909 (5)

For a simply supported panel the modal expansion of119908(119909 119905) can be assumed in the form

119908 (119909 119905) =

infin

sum

119899=1

120578119899 (119905) sin(

119899120587119909

119871) 119899 = 1 2 3 (6)

where 119899 is the positive integer and 120578119899(119905) the generalized

coordinate After substituting (6) into all energy expressionsthe system equations-of-motion are obtained via LagrangersquosEquations

119889

119889119905(120597119879

120597) minus

120597119879

120597119902+120597119881

120597119902+120597119863

120597= 119876nc (7)

with the generalized force 119876nc = 120597120575119882nc120597120575119902 and generalizedcoordinates vector

119902 (119905) = [1205781 (119905) 1205782 (119905) 120578

3 (119905) sdot sdot sdot]119879

(8)

Finally the system equations-of-motion are given as

M (119905) + (C119889 + C119886) (119905) + [K119890 + K119886 + K119875 (119905)] 119902 (119905)

= 0

(9)

Advances in Acoustics and Vibration 3

where the elements of coefficient matrices are

M119898119899

= 120588119860int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871)119889119909

C119889119898119899

= 120585119864lowast119868119898211989921205874

1198714int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871) 119889119909

C119886119898119899

=21198871199020

1205731198800


int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871)119889119909

K119890119898119899

= 119864lowast119868119898211989921205874

1198714int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871) 119889119909

K119886119898119899

=21198871199020

120573

119899120587

119871int

119871

0

sin(119898120587119909119871

) cos(119899120587119909119871) 119889119909

K119875119898119899(119905) =

1198981198991205872

1198712119875 (119905) int

119871

0

cos(119898120587119909119871

) cos(119899120587119909119871) 119889119909

(10)

The stiffness matrix K119875(119905) 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 (ND)form

Mlowastlowast

119902 (120591) + (C119889 + C119886)lowast

119902 (120591)

+ [K119890 + K119886 + K119875 (120591)] 119902 (120591) = 0(11)

with the dimensionless parameters and coordinates

119909 =119909

119871

119902 =119902 (119905)

ℎ

Ω =1205872ℎ

1198712radic119864lowast

12120588

120591 = 119905Ω

120583 =1205880

120588

120585 = 120585Ω

119875cr =119864lowast1198681205872

1198712

119891 (120591) =119875 (119905)

119875cr

120590 =1199020

119864lowast

120572 =ℎ

119871

120596 =120596

Ω

lowast

( ) =119889 ( )

119889120591

(12)

The elements of the ND coefficient matrices in (11) are

M119898119899

=

1 if 119898 = 119899

0 if 119898 = 119899

C119889119898119899

=

1205851198994 if 119898 = 119899

0 if 119898 = 119899

C119886119898119899

=

2radic6120583120590 (Ma2 minus 2)

12058721205722 (Ma2 minus 1)32if 119898 = 119899

0 if 119898 = 119899

K119890119898119899

=

1198994 if 119898 = 119899

0 if 119898 = 119899

K119886119898119899

=

0 if 119898 = 119899

48120590119898119899 [1 minus (minus1)119898+119899

]

radicMa2 minus 1 (1198982 minus 1198992) 12058741205723if 119898 = 119899

K119875119898119899(120591) =

1198992119891 (120591) if 119898 = 119899

0 if 119898 = 119899

(13)

HereMC119889C119886 andK119890 are constant symmetricmatricesK119886 is a constant skew-symmetric matrix K119875 is a symmetrictime-varying matrix with a period of 2120587120596

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 27ndash32] 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 wayThe results from the perturbation (analytical) method will becompared with those from the FTM (numerical) method

To implement Hsursquos perturbation method the time-invariant part of the system stiffness matrix K119890 + K119886 needsto be diagonalized by its left and right eigenvectors [34] Theresulting equations-of-motion are

lowastlowast

119902 (120591) + Clowast

119902 (120591) + [K + K119875 (120591)] 119902 (120591) = 0 (14)


with

C = X119879119871(C119889 + C119886)X

119877

K = X119879119871(K119890 + K119886)X

119877

K119875 (120591) = X119879119871K119875 (120591)X119877

(15a)

(K119890 + K119886)X119877= 120582X119877

(K119890 + K119886)119879

X119871= 120582X119871

X119879119871X119877= I

(15b)

where 120582 is the eigenvalues of K119890 + K119886 and X119871and X

119877are 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 Hsursquos method The standardform in Hsursquos method is obtained by separating the regularand perturbed items in (14) and then expanding the periodictime-varying stiffness matrix into Fourier series

lowastlowast

119902 (120591) + K 119902 (120591) = minusClowast

119902 (120591) minus K119875 (120591) 119902 (120591) (16)

where

K119875 (120591) = K119888 cos (120596120591) + K119904 sin (120596120591) (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 beobtainedThe axial excitation force considered here is a singlefrequency cosine function

lowastlowast

119902 (120591) + [

1205962

10

0 1205962

2

] 119902 (120591)

= minus [

11988811

11988812

11988821

11988822

]lowast

119902 (120591) minus [

11989611

11989612

11989621

11989622

] cos (120596120591) 119902 (120591)

119891 (120591) = 119865 cos (120596120591)

(18)

where

1205961= radic

17

2minus15

2radic1 minus

1205902

1205902119888

1205962= radic

17

2+15

2radic1 minus

1205902

1205902119888

(19a)

11988811= 120585(

17

2minus15

2

120590119888

radic1205902119888minus 1205902

)

+

2radic6 (Ma2 minus 2)radic120583120590

12058721205722 (Ma2 minus 1)32

11988822= 120585(

17

2+15

2

120590119888

radic1205902119888minus 1205902

)

+


12058721205722 (Ma2 minus 1)32

11988812=

15120585120590

2radic1205902119888minus 1205902

11988821= minus11988812

(19b)

11989611= 119865(

5

2minus

3120590119888

2radic1205902119888minus 1205902

)

11989622= 119865(

5

2+

3120590119888

2radic1205902119888minus 1205902

)

11989612= 119865

3120590

2radic1205902119888minus 1205902

11989621= minus11989612

(19c)

120590119888=

15

12812058741205723radicMa2 minus 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 120596 = 21205961+ Δ120596

Δ120596 = plusmnradic1198962

11

41205962

1

minus 1198882

11

2nd principle resonance 120596 = 21205962+ Δ120596


22

41205962

2

minus 1198882

22

Combination resonance 120596 = 1205962minus 1205961+ Δ120596


12

412059611205962

minus 1198881111988822

(20)


0 1 2 30

05

05

1

15

15

2

25

25

3

35

4

1205961120596

2

1205962

1205961

120590

120590c

times10minus6

Figure 2 ND natural frequency variations with respect to dynamicpressure ratio Ma = 2 120572 = 0005 and 120590

119888= 247 times 10minus6

1 2 3 4 5 6 7 8 9 100

1

2

3

05

15

25 120590c

times10minus6

120590

21205961

21205961

21205962

2Δ120596

2Δ120596

2Δ120596

2Δ120596

1205962 minus 1205961

120596

Figure 3 Stability plot for the flutter panel with respect to axialexcitation frequency and dynamic pressure ratio Ma = 2 120572 = 0005120590119888= 247 times 10minus6 120583 = 439 times 10minus5 and 119865 = 05

5 Numerical Results

The ND natural frequencies due to materials and aero-dynamic loads are plotted in Figure 2 with respect tothe dynamic pressure ratio 120590 Once the dynamic pressureratio exceeds the critical value that equals 120590

119888 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 ND 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

21205961

21205962

Δ120596

Δ120596Δ120596

1205962 minus 1205961

120596

120583

times10minus4

Figure 4 Stability plot for the flutter panel with respect to axialexcitation frequency and airpanel density ratio Ma = 2 120572 = 0005120590119888= 247 times 10minus6 120590 = 127 times 10minus6 and 119865 = 05

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 airpanel density ratio is shown in Figure 4Since 120590 lt 120590

119888 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 airpanel 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 airpaneldensity ratio values Their axial excitation frequency stabilityboundaries are calculated by solving (20) for Δ120596 = 0 and theresults are plotted in Figure 5 The system stability dependson both resonances in each area divided by those boundariesThe 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 excitationsThe 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

05

05

1

15

15 25 35

2

times10minus6

120590

120590c

120583 times10minus4

21205961

1205962 minus 1205961

Unstable21205961

Unstable21205961

Unstable21205961

Stable21205961

Stable21205961

Unstable1205962minus1205961

Unstable1205962minus1205961Unstable1205962minus1205961

Stable1205962minus1205961


Figure 5 Stability plot for the flutter panel with respect to airpaneldensity ratio and dynamic pressure ratio Ma = 2 120572 = 0005 120590

119888=

247 times 10minus6 and 119865 = 05

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 frequencyairpanel density ratio and airpanel dynamic pressure ratioDifferent from the static axial force this paper introduces aperiodic axial excitation that brings the system into the time-varying domainThe axial excitation force could increase thepanel stiffness locally to overcome aerodynamic loads wheninteracting with the out-of-plane vibrationsThe study resultsin this paper indicate that the system is stable under thecombinations of the proper excitation frequency and certainairpanel 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 ldquoCalculationof wing flutter by a coupled fluid-structure methodrdquo Journal ofAircraft vol 38 no 2 pp 334ndash342 2001

[2] J L Hall ldquoA review of ballute technology for planetary aero-capturerdquo in Proceedings of the 4th IAA Conference on Low CostPlanetary Missions pp 1ndash10 Laurel Md USA May 2000

[3] H C Nelson and H J Cunningham ldquoTheoretical investigationof flutter of two-dimensional flat panels with one surfaceexposed to supersonic potential flowrdquo NACA Technical Note3465Report 1280 1955

[4] M D Olson ldquoFinite elements applied to panel flutterrdquo AIAAJournal vol 5 no 12 pp 2267ndash2270 1967

[5] P C Parks ldquoA stability criterion for panel flutter via the secondmethod of Liapunovrdquo AIAA Journal vol 4 no 1 pp 175ndash1771966

[6] J Dugundji ldquoTheoretical considerations of panel flutter at highsupersonic Mach numbersrdquo AIAA Journal vol 4 no 7 pp1257ndash1266 1966

[7] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAAJournal vol 4 no 7 pp 1267ndash1275 1966

[8] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[9] R C Zhou C Mei and J-K Huang ldquoSuppression of nonlinearpanel flutter at supersonic speeds and elevated temperaturesrdquoAIAA Journal vol 34 no 2 pp 347ndash354 1996

[10] D J Gee ldquoNumerical continuation applied to panel flutterrdquoNonlinear Dynamics vol 22 no 3 pp 271ndash280 2000

[11] S Tizzi ldquoInfluence of non-linear forces on beam behaviour influtter conditionsrdquo Journal of Sound and Vibration vol 267 no2 pp 279ndash299 2003

[12] T Iwatsubo Y Sugiyama and S Ogino ldquoSimple and combina-tion resonances of columns under periodic axial loadsrdquo Journalof Sound and Vibration vol 33 no 2 pp 211ndash221 1974

[13] C S Hsu ldquoOn the parametric excitation of a dynamic systemhaving multiple degrees of freedomrdquo ASME Journal of AppliedMechanics vol 30 pp 367ndash372 1963

[14] S K Sinha ldquoDynamic stability of a Timoshenko beam subjectedto an oscillating axial forcerdquo Journal of Sound andVibration vol131 no 3 pp 509ndash514 1989

[15] S K Sahu and P K Datta ldquoParametric instability of doublycurved panels subjected to non-uniform harmonic loadingrdquoJournal of Sound and Vibration vol 240 no 1 pp 117ndash129 2001

[16] S K Sahu and P K Datta ldquoDynamic stability of curved panelswith cutoutsrdquo Journal of Sound and Vibration vol 251 no 4 pp683ndash696 2002

[17] M Ganapathi B P Patel P Boisse and M Touratier ldquoNon-linear dynamic stability characteristics of elastic plates sub-jected to periodic in-plane loadrdquo International Journal of Non-Linear Mechanics vol 35 no 3 pp 467ndash480 2000

[18] K Nagai andN Yamaki ldquoDynamic stability of circular cylindri-cal shells under periodic compressive forcesrdquo Journal of Soundand Vibration vol 58 no 3 pp 425ndash441 1978

[19] C Massalas A Dalamangas and G Tzivanidis ldquoDynamicinstability of truncated conical shells with variable modulus ofelasticity under periodic compressive forcesrdquo Journal of Soundand Vibration vol 79 no 4 pp 519ndash528 1981

[20] V B Kovtunov ldquoDynamic stability and nonlinear parametricvibration of cylindrical shellsrdquo Computers amp Structures vol 46no 1 pp 149ndash156 1993

[21] K Y Lam and T Y Ng ldquoDynamic stability of cylindrical shellssubjected to conservative periodic axial loads using differentshell theoriesrdquo Journal of Sound and Vibration vol 207 no 4pp 497ndash520 1997

[22] T Y Ng K Y Lam K M Liew and J N Reddy ldquoDynamicstability analysis of functionally graded cylindrical shells underperiodic axial loadingrdquo International Journal of Solids andStructures vol 38 no 8 pp 1295ndash1309 2001

[23] F Pellicano and M Amabili ldquoStability and vibration of emptyand fluid-filled circular cylindrical shells under static and peri-odic axial loadsrdquo International Journal of Solids and Structuresvol 40 no 13-14 pp 3229ndash3251 2003

[24] M Ruzzene ldquoDynamic buckling of periodically stiffened shellsapplication to supercavitating vehiclesrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 1039ndash1059 2004


[25] P Hagedorn and L R Koval ldquoOn the parametric stability of aTimoshenko beam subjected to a periodic axial loadrdquo Ingenieur-Archiv vol 40 no 3 pp 211ndash220 1971

[26] K-J Yang K-S Hong and F Matsuno ldquoRobust adaptiveboundary control of an axially moving string under a spa-tiotemporally varying tensionrdquo Journal of Sound and Vibrationvol 273 no 4-5 pp 1007ndash1029 2004

[27] Z Viderman F P J Rimrott and W L Cleghorn ldquoPara-metrically excited linear nonconservative gyroscopic systemsrdquoMechanics of Structures and Machines vol 22 no 1 pp 1ndash201994

[28] V V BolotinThe Dynamic Stability of Elastic Systems Holden-Day 1964

[29] C S Hsu ldquoOn approximating a general linear periodic systemrdquoJournal of Mathematical Analysis and Applications vol 45 pp234ndash251 1974

[30] P P Friedmann ldquoNumerical methods for determining thestability and response of periodic systems with applications tohelicopter rotor dynamics and aeroelasticityrdquo Computers andMathematics with Applications vol 12 no 1 pp 131ndash148 1986

[31] O A Bauchau and Y G Nikishkov ldquoAn implicit Floquet anal-ysis for rotorcraft stability evaluationrdquo Journal of the AmericanHelicopter Society vol 46 no 3 pp 200ndash209 2001

[32] H A DeSmidt KWWang and E C Smith ldquoCoupled torsion-lateral stability of a shaft-disk system driven throuoh a universaljointrdquo Journal of AppliedMechanics Transactions ASME vol 69no 3 pp 261ndash273 2002

[33] C S Hsu ldquoFurther results on parametric excitation of adynamic systemrdquo ASME Journal of Applied Mechanics vol 32pp 373ndash377 1965

[34] L Meirovitch Computational Methods in Structural DynamicsSijthoff amp Noordhoff 1980

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



b

L

x

w(x t)P(t)

Air flowU0 1205880

E 120588

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 [18ndash24] FurthermoreHagedorn 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 byHsursquos 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 panelrsquosupper 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

119879 =1

2120588119860int

119871

0

(119909 119905)2119889119909 (1)

where 119908(119909 119905) is the lateral displacement of the panel mea-sured in the ground fixed coordinate frame 120588 the panel

density 119860 the panel cross-sectional area and ldquosdotrdquo the differ-entiation with respect to time 119905

The total potential energy of the panel due to lateraldisplacements is

119881 =1

2119864lowast119868 int

119871

0

11990810158401015840(119909 119905)2119889119909 +

1

2119875 (119905) int

119871

0

1199081015840(119909 119905)2119889119909 (2)

where 119864lowast = 119864(1 minus ]2) for panel and 119864lowast= 119864 for beam

with Youngrsquos modulus 119864 and Poissonrsquos ratio ] the momentof inertia is given by 119868 = 119887ℎ

312 with panel width 119887 and

panel thickness ℎ 119875(119905) is the periodic axial excitation withthe frequency 120596 and ldquo1015840rdquo indicates differentiation with respectto axial position 119909

For material viscous damping a Rayleigh dissipationfunction is defined as

119877 =1

2120585119864lowast119868 int

119871

0

10158401015840(119909 119905)2119889119909 (3)

Here 120585 is the material viscous loss factor of the panelThe aerodynamic load is expressed by using the classic

quasisteady first-order piston theory [4 6 7 9 11]

119901 (119909 119905) =21199020

120573[120597119908 (119909 119905)

120597119909+


1

1198800

120597119908 (119909 119905)

120597119905] (4)

where 1199020

= 12058801198802

02 is the dynamic pressure 120588

0is the

undisturbed air flow density 1198800is the flow speed at infinity

Ma is Mach number and 120573 = (Ma2 minus 1)12The flow goes against the lateral vibrations of the panel

so the nonconservative virtual work from aerodynamic loadis negative and its expression is

120575119882nc = minusint119871

0

119901 (119909 119905) 119887120575119908 (119909 119905) 119889119909 (5)

For a simply supported panel the modal expansion of119908(119909 119905) can be assumed in the form

119908 (119909 119905) =

infin

sum

119899=1

120578119899 (119905) sin(

119899120587119909

119871) 119899 = 1 2 3 (6)

where 119899 is the positive integer and 120578119899(119905) the generalized

coordinate After substituting (6) into all energy expressionsthe system equations-of-motion are obtained via LagrangersquosEquations

119889

119889119905(120597119879

120597) minus

120597119879

120597119902+120597119881

120597119902+120597119863

120597= 119876nc (7)

with the generalized force 119876nc = 120597120575119882nc120597120575119902 and generalizedcoordinates vector

119902 (119905) = [1205781 (119905) 1205782 (119905) 120578

3 (119905) sdot sdot sdot]119879

(8)

Finally the system equations-of-motion are given as

M (119905) + (C119889 + C119886) (119905) + [K119890 + K119886 + K119875 (119905)] 119902 (119905)

= 0

(9)



M119898119899

= 120588119860int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871)119889119909

C119889119898119899

= 120585119864lowast119868119898211989921205874

1198714int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871) 119889119909

C119886119898119899

=21198871199020

1205731198800


int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871)119889119909

K119890119898119899

= 119864lowast119868119898211989921205874

1198714int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871) 119889119909

K119886119898119899

=21198871199020

120573

119899120587

119871int

119871

0

sin(119898120587119909119871

) cos(119899120587119909119871) 119889119909

K119875119898119899(119905) =

1198981198991205872

1198712119875 (119905) int

119871

0

cos(119898120587119909119871

) cos(119899120587119909119871) 119889119909

(10)


Mlowastlowast

119902 (120591) + (C119889 + C119886)lowast

119902 (120591)

+ [K119890 + K119886 + K119875 (120591)] 119902 (120591) = 0(11)


119909 =119909

119871

119902 =119902 (119905)

ℎ

Ω =1205872ℎ


12120588

120591 = 119905Ω

120583 =1205880

120588

120585 = 120585Ω

119875cr =119864lowast1198681205872

1198712

119891 (120591) =119875 (119905)

119875cr

120590 =1199020

119864lowast

120572 =ℎ

119871

120596 =120596

Ω

lowast

( ) =119889 ( )

119889120591

(12)


M119898119899

=

1 if 119898 = 119899

0 if 119898 = 119899

C119889119898119899

=

1205851198994 if 119898 = 119899

0 if 119898 = 119899

C119886119898119899

=

2radic6120583120590 (Ma2 minus 2)

12058721205722 (Ma2 minus 1)32if 119898 = 119899

0 if 119898 = 119899

K119890119898119899

=

1198994 if 119898 = 119899

0 if 119898 = 119899

K119886119898119899

=

0 if 119898 = 119899

48120590119898119899 [1 minus (minus1)119898+119899

]


K119875119898119899(120591) =

1198992119891 (120591) if 119898 = 119899

0 if 119898 = 119899

(13)





lowastlowast

119902 (120591) + Clowast

119902 (120591) + [K + K119875 (120591)] 119902 (120591) = 0 (14)


with

C = X119879119871(C119889 + C119886)X

119877

K = X119879119871(K119890 + K119886)X

119877

K119875 (120591) = X119879119871K119875 (120591)X119877

(15a)

(K119890 + K119886)X119877= 120582X119877

(K119890 + K119886)119879

X119871= 120582X119871

X119879119871X119877= I

(15b)


119877are the



lowastlowast

119902 (120591) + K 119902 (120591) = minusClowast

119902 (120591) minus K119875 (120591) 119902 (120591) (16)

where

K119875 (120591) = K119888 cos (120596120591) + K119904 sin (120596120591) (17)




lowastlowast

119902 (120591) + [

1205962

10

0 1205962

2

] 119902 (120591)

= minus [

11988811

11988812

11988821

11988822

]lowast

119902 (120591) minus [

11989611

11989612

11989621

11989622

] cos (120596120591) 119902 (120591)

119891 (120591) = 119865 cos (120596120591)

(18)

where

1205961= radic

17

2minus15

2radic1 minus

1205902

1205902119888

1205962= radic

17

2+15

2radic1 minus

1205902

1205902119888

(19a)

11988811= 120585(

17

2minus15

2

120590119888

radic1205902119888minus 1205902

)

+


12058721205722 (Ma2 minus 1)32

11988822= 120585(

17

2+15

2

120590119888

radic1205902119888minus 1205902

)

+


12058721205722 (Ma2 minus 1)32

11988812=

15120585120590

2radic1205902119888minus 1205902

11988821= minus11988812

(19b)

11989611= 119865(

5

2minus

3120590119888

2radic1205902119888minus 1205902

)

11989622= 119865(

5

2+

3120590119888

2radic1205902119888minus 1205902

)

11989612= 119865

3120590

2radic1205902119888minus 1205902

11989621= minus11989612

(19c)

120590119888=

15

12812058741205723radicMa2 minus 1 (19d)




11

41205962

1

minus 1198882

11



22

41205962

2

minus 1198882

22



12

412059611205962

minus 1198881111988822

(20)


0 1 2 30

05

05

1

15

15

2

25

25

3

35

4

1205961120596

2

1205962

1205961

120590

120590c

times10minus6


119888= 247 times 10minus6

1 2 3 4 5 6 7 8 9 100

1

2

3

05

15

25 120590c

times10minus6

120590

21205961

21205961

21205962

2Δ120596

2Δ120596

2Δ120596

2Δ120596

1205962 minus 1205961

120596


5 Numerical Results


119888 two natural



1 2 3 4 5 6 7 8 9 100

1

2

3

4

21205961

21205962

Δ120596

Δ120596Δ120596

1205962 minus 1205961

120596

120583

times10minus4










1 2 3 40

05

05

1

15

15 25 35

2

times10minus6

120590

120590c


21205961

1205962 minus 1205961

Unstable21205961

Unstable21205961

Unstable21205961

Stable21205961

Stable21205961






119888=

247 times 10minus6 and 119865 = 05



Competing Interests


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











M119898119899

= 120588119860int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871)119889119909

C119889119898119899

= 120585119864lowast119868119898211989921205874

1198714int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871) 119889119909

C119886119898119899

=21198871199020

1205731198800


int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871)119889119909

K119890119898119899

= 119864lowast119868119898211989921205874

1198714int

119871

0

sin(119898120587119909119871

) sin(119899120587119909119871) 119889119909

K119886119898119899

=21198871199020

120573

119899120587

119871int

119871

0

sin(119898120587119909119871

) cos(119899120587119909119871) 119889119909

K119875119898119899(119905) =

1198981198991205872

1198712119875 (119905) int

119871

0

cos(119898120587119909119871

) cos(119899120587119909119871) 119889119909

(10)


Mlowastlowast

119902 (120591) + (C119889 + C119886)lowast

119902 (120591)

+ [K119890 + K119886 + K119875 (120591)] 119902 (120591) = 0(11)


119909 =119909

119871

119902 =119902 (119905)

ℎ

Ω =1205872ℎ


12120588

120591 = 119905Ω

120583 =1205880

120588

120585 = 120585Ω

119875cr =119864lowast1198681205872

1198712

119891 (120591) =119875 (119905)

119875cr

120590 =1199020

119864lowast

120572 =ℎ

119871

120596 =120596

Ω

lowast

( ) =119889 ( )

119889120591

(12)


M119898119899

=

1 if 119898 = 119899

0 if 119898 = 119899

C119889119898119899

=

1205851198994 if 119898 = 119899

0 if 119898 = 119899

C119886119898119899

=

2radic6120583120590 (Ma2 minus 2)

12058721205722 (Ma2 minus 1)32if 119898 = 119899

0 if 119898 = 119899

K119890119898119899

=

1198994 if 119898 = 119899

0 if 119898 = 119899

K119886119898119899

=

0 if 119898 = 119899

48120590119898119899 [1 minus (minus1)119898+119899

]


K119875119898119899(120591) =

1198992119891 (120591) if 119898 = 119899

0 if 119898 = 119899

(13)





lowastlowast

119902 (120591) + Clowast

119902 (120591) + [K + K119875 (120591)] 119902 (120591) = 0 (14)


with

C = X119879119871(C119889 + C119886)X

119877

K = X119879119871(K119890 + K119886)X

119877

K119875 (120591) = X119879119871K119875 (120591)X119877

(15a)

(K119890 + K119886)X119877= 120582X119877

(K119890 + K119886)119879

X119871= 120582X119871

X119879119871X119877= I

(15b)


119877are the



lowastlowast

119902 (120591) + K 119902 (120591) = minusClowast

119902 (120591) minus K119875 (120591) 119902 (120591) (16)

where

K119875 (120591) = K119888 cos (120596120591) + K119904 sin (120596120591) (17)




lowastlowast

119902 (120591) + [

1205962

10

0 1205962

2

] 119902 (120591)

= minus [

11988811

11988812

11988821

11988822

]lowast

119902 (120591) minus [

11989611

11989612

11989621

11989622

] cos (120596120591) 119902 (120591)

119891 (120591) = 119865 cos (120596120591)

(18)

where

1205961= radic

17

2minus15

2radic1 minus

1205902

1205902119888

1205962= radic

17

2+15

2radic1 minus

1205902

1205902119888

(19a)

11988811= 120585(

17

2minus15

2

120590119888

radic1205902119888minus 1205902

)

+


12058721205722 (Ma2 minus 1)32

11988822= 120585(

17

2+15

2

120590119888

radic1205902119888minus 1205902

)

+


12058721205722 (Ma2 minus 1)32

11988812=

15120585120590

2radic1205902119888minus 1205902

11988821= minus11988812

(19b)

11989611= 119865(

5

2minus

3120590119888

2radic1205902119888minus 1205902

)

11989622= 119865(

5

2+

3120590119888

2radic1205902119888minus 1205902

)

11989612= 119865

3120590

2radic1205902119888minus 1205902

11989621= minus11989612

(19c)

120590119888=

15

12812058741205723radicMa2 minus 1 (19d)




11

41205962

1

minus 1198882

11



22

41205962

2

minus 1198882

22



12

412059611205962

minus 1198881111988822

(20)


0 1 2 30

05

05

1

15

15

2

25

25

3

35

4

1205961120596

2

1205962

1205961

120590

120590c

times10minus6


119888= 247 times 10minus6

1 2 3 4 5 6 7 8 9 100

1

2

3

05

15

25 120590c

times10minus6

120590

21205961

21205961

21205962

2Δ120596

2Δ120596

2Δ120596

2Δ120596

1205962 minus 1205961

120596


5 Numerical Results


119888 two natural



1 2 3 4 5 6 7 8 9 100

1

2

3

4

21205961

21205962

Δ120596

Δ120596Δ120596

1205962 minus 1205961

120596

120583

times10minus4










1 2 3 40

05

05

1

15

15 25 35

2

times10minus6

120590

120590c


21205961

1205962 minus 1205961

Unstable21205961

Unstable21205961

Unstable21205961

Stable21205961

Stable21205961






119888=

247 times 10minus6 and 119865 = 05



Competing Interests


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










with

C = X119879119871(C119889 + C119886)X

119877

K = X119879119871(K119890 + K119886)X

119877

K119875 (120591) = X119879119871K119875 (120591)X119877

(15a)

(K119890 + K119886)X119877= 120582X119877

(K119890 + K119886)119879

X119871= 120582X119871

X119879119871X119877= I

(15b)


119877are the



lowastlowast

119902 (120591) + K 119902 (120591) = minusClowast

119902 (120591) minus K119875 (120591) 119902 (120591) (16)

where

K119875 (120591) = K119888 cos (120596120591) + K119904 sin (120596120591) (17)




lowastlowast

119902 (120591) + [

1205962

10

0 1205962

2

] 119902 (120591)

= minus [

11988811

11988812

11988821

11988822

]lowast

119902 (120591) minus [

11989611

11989612

11989621

11989622

] cos (120596120591) 119902 (120591)

119891 (120591) = 119865 cos (120596120591)

(18)

where

1205961= radic

17

2minus15

2radic1 minus

1205902

1205902119888

1205962= radic

17

2+15

2radic1 minus

1205902

1205902119888

(19a)

11988811= 120585(

17

2minus15

2

120590119888

radic1205902119888minus 1205902

)

+


12058721205722 (Ma2 minus 1)32

11988822= 120585(

17

2+15

2

120590119888

radic1205902119888minus 1205902

)

+


12058721205722 (Ma2 minus 1)32

11988812=

15120585120590

2radic1205902119888minus 1205902

11988821= minus11988812

(19b)

11989611= 119865(

5

2minus

3120590119888

2radic1205902119888minus 1205902

)

11989622= 119865(

5

2+

3120590119888

2radic1205902119888minus 1205902

)

11989612= 119865

3120590

2radic1205902119888minus 1205902

11989621= minus11989612

(19c)

120590119888=

15

12812058741205723radicMa2 minus 1 (19d)




11

41205962

1

minus 1198882

11



22

41205962

2

minus 1198882

22



12

412059611205962

minus 1198881111988822

(20)


0 1 2 30

05

05

1

15

15

2

25

25

3

35

4

1205961120596

2

1205962

1205961

120590

120590c

times10minus6


119888= 247 times 10minus6

1 2 3 4 5 6 7 8 9 100

1

2

3

05

15

25 120590c

times10minus6

120590

21205961

21205961

21205962

2Δ120596

2Δ120596

2Δ120596

2Δ120596

1205962 minus 1205961

120596


5 Numerical Results


119888 two natural



1 2 3 4 5 6 7 8 9 100

1

2

3

4

21205961

21205962

Δ120596

Δ120596Δ120596

1205962 minus 1205961

120596

120583

times10minus4










1 2 3 40

05

05

1

15

15 25 35

2

times10minus6

120590

120590c


21205961

1205962 minus 1205961

Unstable21205961

Unstable21205961

Unstable21205961

Stable21205961

Stable21205961






119888=

247 times 10minus6 and 119865 = 05



Competing Interests


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 30

05

05

1

15

15

2

25

25

3

35

4

1205961120596

2

1205962

1205961

120590

120590c

times10minus6


119888= 247 times 10minus6

1 2 3 4 5 6 7 8 9 100

1

2

3

05

15

25 120590c

times10minus6

120590

21205961

21205961

21205962

2Δ120596

2Δ120596

2Δ120596

2Δ120596

1205962 minus 1205961

120596


5 Numerical Results


119888 two natural



1 2 3 4 5 6 7 8 9 100

1

2

3

4

21205961

21205962

Δ120596

Δ120596Δ120596

1205962 minus 1205961

120596

120583

times10minus4










1 2 3 40

05

05

1

15

15 25 35

2

times10minus6

120590

120590c


21205961

1205962 minus 1205961

Unstable21205961

Unstable21205961

Unstable21205961

Stable21205961

Stable21205961






119888=

247 times 10minus6 and 119865 = 05



Competing Interests


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 40

05

05

1

15

15 25 35

2

times10minus6

120590

120590c


21205961

1205962 minus 1205961

Unstable21205961

Unstable21205961

Unstable21205961

Stable21205961

Stable21205961






119888=

247 times 10minus6 and 119865 = 05



Competing Interests


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Stability Analysis of a Flutter Panel with Axial … · 2019. 12. 12. · Stability Analysis of a Flutter Panel with Axial Excitations MengPengandHansA.DeSmidt