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 Β© 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 Galerkinβsmethod. 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 Karmanβs 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 Hsuβs 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, π0
E, οΏ½, π
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 [18β24]. 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 byHsuβs 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 panelβsupper 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
π =1
2ππ΄β«
πΏ
0
οΏ½ΜοΏ½ (π₯, π‘)2ππ₯, (1)
where π€(π₯, π‘) is the lateral displacement of the panel mea-sured in the ground fixed coordinate frame, π the panel
density, π΄ the panel cross-sectional area, and ββ β the differ-entiation with respect to time π‘.
The total potential energy of the panel due to lateraldisplacements is
π =1
2πΈβπΌ β«
πΏ
0
π€(π₯, π‘)2ππ₯ +
1
2π (π‘) β«
πΏ
0
π€(π₯, π‘)2ππ₯, (2)
where πΈβ = πΈ/(1 β ]2) for panel and πΈβ = πΈ for beamwith Youngβs modulus πΈ and Poissonβs ratio ]; the momentof inertia is given by πΌ = πβ3/12 with panel width π andpanel thickness β; π(π‘) is the periodic axial excitation withthe frequency π and ββ indicates differentiation with respectto axial position π₯.
For material viscous damping, a Rayleigh dissipationfunction is defined as
π =1
2ππΈβπΌ β«
πΏ
0
οΏ½ΜοΏ½(π₯, π‘)2ππ₯. (3)
Here, π 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]:
π (π₯, π‘) =2π0
π½[ππ€ (π₯, π‘)
ππ₯+
(Ma2 β 2)(Ma2 β 1)
1
π0
ππ€ (π₯, π‘)
ππ‘] , (4)
where π0
= π0π2
0/2 is the dynamic pressure, π
0is the
undisturbed air flow density, π0is the flow speed at infinity,
Ma is Mach number, and π½ = (Ma2 β 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
πΏπnc = ββ«πΏ
0
π (π₯, π‘) ππΏπ€ (π₯, π‘) ππ₯. (5)
For a simply supported panel, the modal expansion ofπ€(π₯, π‘) can be assumed in the form
π€ (π₯, π‘) =
β
β
π=1
ππ (π‘) sin(
πππ₯
πΏ) , π = 1, 2, 3, . . . , (6)
where π is the positive integer and ππ(π‘) the generalized
coordinate. After substituting (6) into all energy expressions,the system equations-of-motion are obtained via LagrangeβsEquations
π
ππ‘(ππ
ποΏ½ΜοΏ½) β
ππ
ππ+ππ
ππ+ππ·
ποΏ½ΜοΏ½= πnc (7)
with the generalized force πnc = ππΏπnc/ππΏπ and generalizedcoordinates vector
π (π‘) = [π1 (π‘) π2 (π‘) π3 (π‘) β β β ]π
. (8)
Finally, the system equations-of-motion are given as
MοΏ½ΜοΏ½ (π‘) + (Cπ + Cπ) οΏ½ΜοΏ½ (π‘) + [Kπ + Kπ + Kπ (π‘)] π (π‘)
= 0,
(9)
Advances in Acoustics and Vibration 3
where the elements of coefficient matrices are
Mππ
= ππ΄β«
πΏ
0
sin(πππ₯πΏ
) sin(πππ₯πΏ)ππ₯,
Cπππ
= ππΈβπΌπ2π2π4
πΏ4β«
πΏ
0
sin(πππ₯πΏ
) sin(πππ₯πΏ) ππ₯,
Cπππ
=2ππ0
π½π0
(Ma2 β 2)(Ma2 β 1)
β«
πΏ
0
sin(πππ₯πΏ
) sin(πππ₯πΏ)ππ₯,
Kπππ
= πΈβπΌπ2π2π4
πΏ4β«
πΏ
0
sin(πππ₯πΏ
) sin(πππ₯πΏ) ππ₯,
Kπππ
=2ππ0
π½
ππ
πΏβ«
πΏ
0
sin(πππ₯πΏ
) cos(πππ₯πΏ) ππ₯,
Kπππ(π‘) =
πππ2
πΏ2π (π‘) β«
πΏ
0
cos(πππ₯πΏ
) cos(πππ₯πΏ) ππ₯.
(10)
The stiffness matrix Kπ(π‘) 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
Mββ
π (π) + (Cπ + Cπ)β
π (π)
+ [Kπ + Kπ + Kπ (π)] π (π) = 0(11)
with the dimensionless parameters and coordinates:
π₯ =π₯
πΏ,
π =π (π‘)
β,
Ξ© =π2β
πΏ2βπΈβ
12π,
π = π‘Ξ©,
π =π0
π,
π = πΞ©,
πcr =πΈβπΌπ2
πΏ2,
π (π) =π (π‘)
πcr,
π =π0
πΈβ,
πΌ =β
πΏ,
π =π
Ξ©,
β
( ) =π ( )
ππ.
(12)
The elements of the N.D. coefficient matrices in (11) are
Mππ
={
{
{
1 if π = π
0 if π ΜΈ= π,
Cπππ
={
{
{
ππ4 if π = π
0 if π ΜΈ= π,
Cπππ
=
{{{
{{{
{
2β6ππ (Ma2 β 2)
π2πΌ2 (Ma2 β 1)3/2if π = π
0 if π ΜΈ= π,
Kπππ
={
{
{
π4 if π = π
0 if π ΜΈ= π,
Kπππ
=
{{{
{{{
{
0 if π = π
48πππ [1 β (β1)π+π
]
βMa2 β 1 (π2 β π2) π4πΌ3if π ΜΈ= π,
Kπππ(π) =
{
{
{
π2π (π) if π = π
0 if π ΜΈ= π.
(13)
Here,M,Cπ,Cπ, andKπ are constant symmetricmatrices;Kπ is a constant skew-symmetric matrix; Kπ is a symmetrictime-varying matrix with a period of 2π/π.
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, 27β32]. 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 Hsuβs perturbation method, the time-invariant part of the system stiffness matrix, Kπ + Kπ, needsto be diagonalized by its left and right eigenvectors [34]. Theresulting equations-of-motion are
ββ
π (π) + CΜβ
π (π) + [KΜ + KΜπ (π)] π (π) = 0 (14)
4 Advances in Acoustics and Vibration
with
CΜ = XππΏ(Cπ + Cπ)X
π
KΜ = XππΏ(Kπ + Kπ)X
π
KΜπ (π) = XππΏKπ (π)Xπ ,
(15a)
(Kπ + Kπ)Xπ = πXπ
(Kπ + Kπ)π
XπΏ= πXπΏ
XππΏXπ = I,
(15b)
where π is the eigenvalues of Kπ + Kπ and XπΏand X
π are 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 Hsuβs method. The standardform in Hsuβs method is obtained by separating the regularand perturbed items in (14) and then expanding the periodictime-varying stiffness matrix into Fourier series,
ββ
π (π) + KΜ π (π) = βCΜβ
π (π) β KΜπ (π) π (π) , (16)
where
KΜπ (π) = KΜπ cos (ππ) + KΜπ sin (ππ) . (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:
ββ
π (π) + [
π2
10
0 π2
2
] π (π)
= β [
π11
π12
π21
π22
]β
π (π) β [
π11
π12
π21
π22
] cos (ππ) π (π) ,
π (π) = πΉ cos (ππ) ,
(18)
where
π1= β
17
2β15
2β1 β
π2
π2π
,
π2= β
17
2+15
2β1 β
π2
π2π
,
(19a)
π11= π(
17
2β15
2
ππ
βπ2πβ π2
)
+
2β6 (Ma2 β 2)βππ
π2πΌ2 (Ma2 β 1)3/2,
π22= π(
17
2+15
2
ππ
βπ2πβ π2
)
+
2β6 (Ma2 β 2)βππ
π2πΌ2 (Ma2 β 1)3/2,
π12=
15ππ
2βπ2πβ π2
,
π21= βπ12,
(19b)
π11= πΉ(
5
2β
3ππ
2βπ2πβ π2
),
π22= πΉ(
5
2+
3ππ
2βπ2πβ π2
),
π12= πΉ
3π
2βπ2πβ π2
,
π21= βπ12,
(19c)
ππ=
15
128π4πΌ3βMa2 β 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: π = 2π1+ Ξπ
Ξπ = Β±βπ2
11
4π2
1
β π2
11
2nd principle resonance: π = 2π2+ Ξπ
Ξπ = Β±βπ2
22
4π2
2
β π2
22
Combination resonance: π = π2β π1+ Ξπ
Ξπ = Β±βπ2
12
4π1π2
β π11π22.
(20)
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
π1,π
2
π2
π1
π
πc
Γ10β6
Figure 2: N.D. natural frequency variations with respect to dynamicpressure ratio: Ma = 2, πΌ = 0.005, and π
π= 2.47 Γ 10β6.
1 2 3 4 5 6 7 8 9 100
1
2
3
0.5
1.5
2.5 πc
Γ10β6
π
2π1
2π1
2π2
2Ξπ
2Ξπ
2Ξπ
2Ξπ
π2 β π1
π
Figure 3: Stability plot for the flutter panel with respect to axialexcitation frequency and dynamic pressure ratio: Ma = 2, πΌ = 0.005,ππ= 2.47 Γ 10β6, π = 4.39 Γ 10β5, and πΉ = 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, π. Once the dynamic pressureratio exceeds the critical value that equals π
π, 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
2π1
2π2
Ξπ
ΞπΞπ
π2 β π1
π
π
Γ10β4
Figure 4: Stability plot for the flutter panel with respect to axialexcitation frequency and air/panel density ratio: Ma = 2, πΌ = 0.005,ππ= 2.47 Γ 10β6, π = 1.27 Γ 10β6, and πΉ = 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 π < π
π, 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 Ξπ = 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.
6 Advances in Acoustics and Vibration
1 2 3 40
0.5
0.5
1
1.5
1.5 2.5 3.5
2
Γ10β6
π
πc
π Γ10β4
2π1
π2 β π1
Unstable2π1
Unstable2π1
Unstable2π1
Stable2π1
Stable2π1
Unstableπ2βπ1
Unstableπ2βπ1Unstableπ2βπ1
Stableπ2βπ1
Stableπ2βπ1
Figure 5: Stability plot for the flutter panel with respect to air/paneldensity ratio and dynamic pressure ratio: Ma = 2, πΌ = 0.005, π
π=
2.47 Γ 10β6, and πΉ = 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, βCalculationof wing flutter by a coupled fluid-structure method,β Journal ofAircraft, vol. 38, no. 2, pp. 334β342, 2001.
[2] J. L. Hall, βA review of ballute technology for planetary aero-capture,β in Proceedings of the 4th IAA Conference on Low CostPlanetary Missions, pp. 1β10, Laurel, Md, USA, May 2000.
[3] H. C. Nelson and H. J. Cunningham, βTheoretical investigationof flutter of two-dimensional flat panels with one surfaceexposed to supersonic potential flow,β NACA Technical Note3465/Report 1280, 1955.
[4] M. D. Olson, βFinite elements applied to panel flutter,β AIAAJournal, vol. 5, no. 12, pp. 2267β2270, 1967.
[5] P. C. Parks, βA stability criterion for panel flutter via the secondmethod of Liapunov,β AIAA Journal, vol. 4, no. 1, pp. 175β177,1966.
[6] J. Dugundji, βTheoretical considerations of panel flutter at highsupersonic Mach numbers.,β AIAA Journal, vol. 4, no. 7, pp.1257β1266, 1966.
[7] E. H. Dowell, βNonlinear oscillations of a fluttering plate,βAIAAJournal, vol. 4, no. 7, pp. 1267β1275, 1966.
[8] E. H. Dowell, βNonlinear oscillations of a fluttering plate II,βAIAA Journal, vol. 5, no. 10, pp. 1856β1862, 1967.
[9] R. C. Zhou, C. Mei, and J.-K. Huang, βSuppression of nonlinearpanel flutter at supersonic speeds and elevated temperatures,βAIAA Journal, vol. 34, no. 2, pp. 347β354, 1996.
[10] D. J. Gee, βNumerical continuation applied to panel flutter,βNonlinear Dynamics, vol. 22, no. 3, pp. 271β280, 2000.
[11] S. Tizzi, βInfluence of non-linear forces on beam behaviour influtter conditions,β Journal of Sound and Vibration, vol. 267, no.2, pp. 279β299, 2003.
[12] T. Iwatsubo, Y. Sugiyama, and S. Ogino, βSimple and combina-tion resonances of columns under periodic axial loads,β Journalof Sound and Vibration, vol. 33, no. 2, pp. 211β221, 1974.
[13] C. S. Hsu, βOn the parametric excitation of a dynamic systemhaving multiple degrees of freedom,β ASME Journal of AppliedMechanics, vol. 30, pp. 367β372, 1963.
[14] S. K. Sinha, βDynamic stability of a Timoshenko beam subjectedto an oscillating axial force,β Journal of Sound andVibration, vol.131, no. 3, pp. 509β514, 1989.
[15] S. K. Sahu and P. K. Datta, βParametric instability of doublycurved panels subjected to non-uniform harmonic loading,βJournal of Sound and Vibration, vol. 240, no. 1, pp. 117β129, 2001.
[16] S. K. Sahu and P. K. Datta, βDynamic stability of curved panelswith cutouts,β Journal of Sound and Vibration, vol. 251, no. 4, pp.683β696, 2002.
[17] M. Ganapathi, B. P. Patel, P. Boisse, and M. Touratier, βNon-linear dynamic stability characteristics of elastic plates sub-jected to periodic in-plane load,β International Journal of Non-Linear Mechanics, vol. 35, no. 3, pp. 467β480, 2000.
[18] K. Nagai andN. Yamaki, βDynamic stability of circular cylindri-cal shells under periodic compressive forces,β Journal of Soundand Vibration, vol. 58, no. 3, pp. 425β441, 1978.
[19] C. Massalas, A. Dalamangas, and G. Tzivanidis, βDynamicinstability of truncated conical shells, with variable modulus ofelasticity, under periodic compressive forces,β Journal of Soundand Vibration, vol. 79, no. 4, pp. 519β528, 1981.
[20] V. B. Kovtunov, βDynamic stability and nonlinear parametricvibration of cylindrical shells,β Computers & Structures, vol. 46,no. 1, pp. 149β156, 1993.
[21] K. Y. Lam and T. Y. Ng, βDynamic stability of cylindrical shellssubjected to conservative periodic axial loads using differentshell theories,β Journal of Sound and Vibration, vol. 207, no. 4,pp. 497β520, 1997.
[22] T. Y. Ng, K. Y. Lam, K. M. Liew, and J. N. Reddy, βDynamicstability analysis of functionally graded cylindrical shells underperiodic axial loading,β International Journal of Solids andStructures, vol. 38, no. 8, pp. 1295β1309, 2001.
[23] F. Pellicano and M. Amabili, βStability and vibration of emptyand fluid-filled circular cylindrical shells under static and peri-odic axial loads,β International Journal of Solids and Structures,vol. 40, no. 13-14, pp. 3229β3251, 2003.
[24] M. Ruzzene, βDynamic buckling of periodically stiffened shells:application to supercavitating vehicles,β International Journal ofSolids and Structures, vol. 41, no. 3-4, pp. 1039β1059, 2004.
Advances in Acoustics and Vibration 7
[25] P. Hagedorn and L. R. Koval, βOn the parametric stability of aTimoshenko beam subjected to a periodic axial load,β Ingenieur-Archiv, vol. 40, no. 3, pp. 211β220, 1971.
[26] K.-J. Yang, K.-S. Hong, and F. Matsuno, βRobust adaptiveboundary control of an axially moving string under a spa-tiotemporally varying tension,β Journal of Sound and Vibration,vol. 273, no. 4-5, pp. 1007β1029, 2004.
[27] Z. Viderman, F. P. J. Rimrott, and W. L. Cleghorn, βPara-metrically excited linear nonconservative gyroscopic systems,βMechanics of Structures and Machines, vol. 22, no. 1, pp. 1β20,1994.
[28] V. V. Bolotin,The Dynamic Stability of Elastic Systems, Holden-Day, 1964.
[29] C. S. Hsu, βOn approximating a general linear periodic system,βJournal of Mathematical Analysis and Applications, vol. 45, pp.234β251, 1974.
[30] P. P. Friedmann, βNumerical methods for determining thestability and response of periodic systems with applications tohelicopter rotor dynamics and aeroelasticity,β Computers andMathematics with Applications, vol. 12, no. 1, pp. 131β148, 1986.
[31] O. A. Bauchau and Y. G. Nikishkov, βAn implicit Floquet anal-ysis for rotorcraft stability evaluation,β Journal of the AmericanHelicopter Society, vol. 46, no. 3, pp. 200β209, 2001.
[32] H. A. DeSmidt, K.W.Wang, and E. C. Smith, βCoupled torsion-lateral stability of a shaft-disk system driven throuoh a universaljoint,β Journal of AppliedMechanics, Transactions ASME, vol. 69,no. 3, pp. 261β273, 2002.
[33] C. S. Hsu, βFurther results on parametric excitation of adynamic system,β ASME Journal of Applied Mechanics, vol. 32,pp. 373β377, 1965.
[34] L. Meirovitch, Computational Methods in Structural Dynamics,Sijthoff & Noordhoff, 1980.
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Journal ofEngineeringVolume 2014
Submit your manuscripts athttp://www.hindawi.com
VLSI Design
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
DistributedSensor Networks
International Journal of