research article stability analysis of a flutter panel...
TRANSCRIPT
-
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 ยฉ 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 Galerkinโsmethod. 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 Karmanโs 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 Hsuโs 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, ๐0
E, ๏ฟฝ, ๐
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 [18โ24]. 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 byHsuโs 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 panelโsupper 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
๐ =1
2๐๐ดโซ
๐ฟ
0
๏ฟฝฬ๏ฟฝ (๐ฅ, ๐ก)2๐๐ฅ, (1)
where ๐ค(๐ฅ, ๐ก) is the lateral displacement of the panel mea-sured in the ground fixed coordinate frame, ๐ the panel
density, ๐ด the panel cross-sectional area, and โโ โ the differ-entiation with respect to time ๐ก.
The total potential energy of the panel due to lateraldisplacements is
๐ =1
2๐ธโ๐ผ โซ
๐ฟ
0
๐ค(๐ฅ, ๐ก)2๐๐ฅ +
1
2๐ (๐ก) โซ
๐ฟ
0
๐ค(๐ฅ, ๐ก)2๐๐ฅ, (2)
where ๐ธโ = ๐ธ/(1 โ ]2) for panel and ๐ธโ = ๐ธ for beamwith Youngโs modulus ๐ธ and Poissonโs ratio ]; the momentof inertia is given by ๐ผ = ๐โ3/12 with panel width ๐ andpanel thickness โ; ๐(๐ก) is the periodic axial excitation withthe frequency ๐ and โโ indicates differentiation with respectto axial position ๐ฅ.
For material viscous damping, a Rayleigh dissipationfunction is defined as
๐ =1
2๐๐ธโ๐ผ โซ
๐ฟ
0
๏ฟฝฬ๏ฟฝ(๐ฅ, ๐ก)2๐๐ฅ. (3)
Here, ๐ 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]:
๐ (๐ฅ, ๐ก) =2๐0
๐ฝ[๐๐ค (๐ฅ, ๐ก)
๐๐ฅ+
(Ma2 โ 2)(Ma2 โ 1)
1
๐0
๐๐ค (๐ฅ, ๐ก)
๐๐ก] , (4)
where ๐0
= ๐0๐2
0/2 is the dynamic pressure, ๐
0is the
undisturbed air flow density, ๐0is the flow speed at infinity,
Ma is Mach number, and ๐ฝ = (Ma2 โ 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
๐ฟ๐nc = โโซ๐ฟ
0
๐ (๐ฅ, ๐ก) ๐๐ฟ๐ค (๐ฅ, ๐ก) ๐๐ฅ. (5)
For a simply supported panel, the modal expansion of๐ค(๐ฅ, ๐ก) can be assumed in the form
๐ค (๐ฅ, ๐ก) =
โ
โ
๐=1
๐๐ (๐ก) sin(
๐๐๐ฅ
๐ฟ) , ๐ = 1, 2, 3, . . . , (6)
where ๐ is the positive integer and ๐๐(๐ก) the generalized
coordinate. After substituting (6) into all energy expressions,the system equations-of-motion are obtained via LagrangeโsEquations
๐
๐๐ก(๐๐
๐๏ฟฝฬ๏ฟฝ) โ
๐๐
๐๐+๐๐
๐๐+๐๐ท
๐๏ฟฝฬ๏ฟฝ= ๐nc (7)
with the generalized force ๐nc = ๐๐ฟ๐nc/๐๐ฟ๐ and generalizedcoordinates vector
๐ (๐ก) = [๐1 (๐ก) ๐2 (๐ก) ๐3 (๐ก) โ โ โ ]๐
. (8)
Finally, the system equations-of-motion are given as
M๏ฟฝฬ๏ฟฝ (๐ก) + (C๐ + C๐) ๏ฟฝฬ๏ฟฝ (๐ก) + [K๐ + K๐ + K๐ (๐ก)] ๐ (๐ก)
= 0,
(9)
-
Advances in Acoustics and Vibration 3
where the elements of coefficient matrices are
M๐๐
= ๐๐ดโซ
๐ฟ
0
sin(๐๐๐ฅ๐ฟ
) sin(๐๐๐ฅ๐ฟ)๐๐ฅ,
C๐๐๐
= ๐๐ธโ๐ผ๐2๐2๐4
๐ฟ4โซ
๐ฟ
0
sin(๐๐๐ฅ๐ฟ
) sin(๐๐๐ฅ๐ฟ) ๐๐ฅ,
C๐๐๐
=2๐๐0
๐ฝ๐0
(Ma2 โ 2)(Ma2 โ 1)
โซ
๐ฟ
0
sin(๐๐๐ฅ๐ฟ
) sin(๐๐๐ฅ๐ฟ)๐๐ฅ,
K๐๐๐
= ๐ธโ๐ผ๐2๐2๐4
๐ฟ4โซ
๐ฟ
0
sin(๐๐๐ฅ๐ฟ
) sin(๐๐๐ฅ๐ฟ) ๐๐ฅ,
K๐๐๐
=2๐๐0
๐ฝ
๐๐
๐ฟโซ
๐ฟ
0
sin(๐๐๐ฅ๐ฟ
) cos(๐๐๐ฅ๐ฟ) ๐๐ฅ,
K๐๐๐(๐ก) =
๐๐๐2
๐ฟ2๐ (๐ก) โซ
๐ฟ
0
cos(๐๐๐ฅ๐ฟ
) cos(๐๐๐ฅ๐ฟ) ๐๐ฅ.
(10)
The stiffness matrix K๐(๐ก) 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
Mโโ
๐ (๐) + (C๐ + C๐)โ
๐ (๐)
+ [K๐ + K๐ + K๐ (๐)] ๐ (๐) = 0(11)
with the dimensionless parameters and coordinates:
๐ฅ =๐ฅ
๐ฟ,
๐ =๐ (๐ก)
โ,
ฮฉ =๐2โ
๐ฟ2โ๐ธโ
12๐,
๐ = ๐กฮฉ,
๐ =๐0
๐,
๐ = ๐ฮฉ,
๐cr =๐ธโ๐ผ๐2
๐ฟ2,
๐ (๐) =๐ (๐ก)
๐cr,
๐ =๐0
๐ธโ,
๐ผ =โ
๐ฟ,
๐ =๐
ฮฉ,
โ
( ) =๐ ( )
๐๐.
(12)
The elements of the N.D. coefficient matrices in (11) are
M๐๐
={
{
{
1 if ๐ = ๐
0 if ๐ ฬธ= ๐,
C๐๐๐
={
{
{
๐๐4 if ๐ = ๐
0 if ๐ ฬธ= ๐,
C๐๐๐
=
{{{
{{{
{
2โ6๐๐ (Ma2 โ 2)
๐2๐ผ2 (Ma2 โ 1)3/2if ๐ = ๐
0 if ๐ ฬธ= ๐,
K๐๐๐
={
{
{
๐4 if ๐ = ๐
0 if ๐ ฬธ= ๐,
K๐๐๐
=
{{{
{{{
{
0 if ๐ = ๐
48๐๐๐ [1 โ (โ1)๐+๐
]
โMa2 โ 1 (๐2 โ ๐2) ๐4๐ผ3if ๐ ฬธ= ๐,
K๐๐๐(๐) =
{
{
{
๐2๐ (๐) if ๐ = ๐
0 if ๐ ฬธ= ๐.
(13)
Here,M,C๐,C๐, andK๐ are constant symmetricmatrices;K๐ is a constant skew-symmetric matrix; K๐ is a symmetrictime-varying matrix with a period of 2๐/๐.
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, 27โ32]. 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 Hsuโs perturbation method, the time-invariant part of the system stiffness matrix, K๐ + K๐, needsto be diagonalized by its left and right eigenvectors [34]. Theresulting equations-of-motion are
โโ
๐ (๐) + Cฬโ
๐ (๐) + [Kฬ + Kฬ๐ (๐)] ๐ (๐) = 0 (14)
-
4 Advances in Acoustics and Vibration
with
Cฬ = X๐๐ฟ(C๐ + C๐)X
๐
Kฬ = X๐๐ฟ(K๐ + K๐)X
๐
Kฬ๐ (๐) = X๐๐ฟK๐ (๐)X๐ ,
(15a)
(K๐ + K๐)X๐ = ๐X๐
(K๐ + K๐)๐
X๐ฟ= ๐X๐ฟ
X๐๐ฟX๐ = I,
(15b)
where ๐ is the eigenvalues of K๐ + K๐ and X๐ฟand X
๐ 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 Hsuโs method. The standardform in Hsuโs method is obtained by separating the regularand perturbed items in (14) and then expanding the periodictime-varying stiffness matrix into Fourier series,
โโ
๐ (๐) + Kฬ ๐ (๐) = โCฬโ
๐ (๐) โ Kฬ๐ (๐) ๐ (๐) , (16)
where
Kฬ๐ (๐) = Kฬ๐ cos (๐๐) + Kฬ๐ sin (๐๐) . (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:
โโ
๐ (๐) + [
๐2
10
0 ๐2
2
] ๐ (๐)
= โ [
๐11
๐12
๐21
๐22
]โ
๐ (๐) โ [
๐11
๐12
๐21
๐22
] cos (๐๐) ๐ (๐) ,
๐ (๐) = ๐น cos (๐๐) ,
(18)
where
๐1= โ
17
2โ15
2โ1 โ
๐2
๐2๐
,
๐2= โ
17
2+15
2โ1 โ
๐2
๐2๐
,
(19a)
๐11= ๐(
17
2โ15
2
๐๐
โ๐2๐โ ๐2
)
+
2โ6 (Ma2 โ 2)โ๐๐
๐2๐ผ2 (Ma2 โ 1)3/2,
๐22= ๐(
17
2+15
2
๐๐
โ๐2๐โ ๐2
)
+
2โ6 (Ma2 โ 2)โ๐๐
๐2๐ผ2 (Ma2 โ 1)3/2,
๐12=
15๐๐
2โ๐2๐โ ๐2
,
๐21= โ๐12,
(19b)
๐11= ๐น(
5
2โ
3๐๐
2โ๐2๐โ ๐2
),
๐22= ๐น(
5
2+
3๐๐
2โ๐2๐โ ๐2
),
๐12= ๐น
3๐
2โ๐2๐โ ๐2
,
๐21= โ๐12,
(19c)
๐๐=
15
128๐4๐ผ3โMa2 โ 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: ๐ = 2๐1+ ฮ๐
ฮ๐ = ยฑโ๐2
11
4๐2
1
โ ๐2
11
2nd principle resonance: ๐ = 2๐2+ ฮ๐
ฮ๐ = ยฑโ๐2
22
4๐2
2
โ ๐2
22
Combination resonance: ๐ = ๐2โ ๐1+ ฮ๐
ฮ๐ = ยฑโ๐2
12
4๐1๐2
โ ๐11๐22.
(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
๐1,๐
2
๐2
๐1
๐
๐c
ร10โ6
Figure 2: N.D. natural frequency variations with respect to dynamicpressure ratio: Ma = 2, ๐ผ = 0.005, and ๐
๐= 2.47 ร 10โ6.
1 2 3 4 5 6 7 8 9 100
1
2
3
0.5
1.5
2.5 ๐c
ร10โ6
๐
2๐1
2๐1
2๐2
2ฮ๐
2ฮ๐
2ฮ๐
2ฮ๐
๐2 โ ๐1
๐
Figure 3: Stability plot for the flutter panel with respect to axialexcitation frequency and dynamic pressure ratio: Ma = 2, ๐ผ = 0.005,๐๐= 2.47 ร 10โ6, ๐ = 4.39 ร 10โ5, and ๐น = 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, ๐. Once the dynamic pressureratio exceeds the critical value that equals ๐
๐, 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
2๐1
2๐2
ฮ๐
ฮ๐ฮ๐
๐2 โ ๐1
๐
๐
ร10โ4
Figure 4: Stability plot for the flutter panel with respect to axialexcitation frequency and air/panel density ratio: Ma = 2, ๐ผ = 0.005,๐๐= 2.47 ร 10โ6, ๐ = 1.27 ร 10โ6, and ๐น = 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 ๐ < ๐
๐, 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 ฮ๐ = 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
ร10โ6
๐
๐c
๐ ร10โ4
2๐1
๐2 โ ๐1
Unstable2๐1
Unstable2๐1
Unstable2๐1
Stable2๐1
Stable2๐1
Unstable๐2โ๐1
Unstable๐2โ๐1Unstable๐2โ๐1
Stable๐2โ๐1
Stable๐2โ๐1
Figure 5: Stability plot for the flutter panel with respect to air/paneldensity ratio and dynamic pressure ratio: Ma = 2, ๐ผ = 0.005, ๐
๐=
2.47 ร 10โ6, and ๐น = 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, โCalculationof wing flutter by a coupled fluid-structure method,โ Journal ofAircraft, vol. 38, no. 2, pp. 334โ342, 2001.
[2] J. L. Hall, โA review of ballute technology for planetary aero-capture,โ in Proceedings of the 4th IAA Conference on Low CostPlanetary Missions, pp. 1โ10, Laurel, Md, USA, May 2000.
[3] H. C. Nelson and H. J. Cunningham, โTheoretical investigationof flutter of two-dimensional flat panels with one surfaceexposed to supersonic potential flow,โ NACA Technical Note3465/Report 1280, 1955.
[4] M. D. Olson, โFinite elements applied to panel flutter,โ AIAAJournal, vol. 5, no. 12, pp. 2267โ2270, 1967.
[5] P. C. Parks, โA stability criterion for panel flutter via the secondmethod of Liapunov,โ AIAA Journal, vol. 4, no. 1, pp. 175โ177,1966.
[6] J. Dugundji, โTheoretical considerations of panel flutter at highsupersonic Mach numbers.,โ AIAA Journal, vol. 4, no. 7, pp.1257โ1266, 1966.
[7] E. H. Dowell, โNonlinear oscillations of a fluttering plate,โAIAAJournal, vol. 4, no. 7, pp. 1267โ1275, 1966.
[8] E. H. Dowell, โNonlinear oscillations of a fluttering plate II,โAIAA Journal, vol. 5, no. 10, pp. 1856โ1862, 1967.
[9] R. C. Zhou, C. Mei, and J.-K. Huang, โSuppression of nonlinearpanel flutter at supersonic speeds and elevated temperatures,โAIAA Journal, vol. 34, no. 2, pp. 347โ354, 1996.
[10] D. J. Gee, โNumerical continuation applied to panel flutter,โNonlinear Dynamics, vol. 22, no. 3, pp. 271โ280, 2000.
[11] S. Tizzi, โInfluence of non-linear forces on beam behaviour influtter conditions,โ Journal of Sound and Vibration, vol. 267, no.2, pp. 279โ299, 2003.
[12] T. Iwatsubo, Y. Sugiyama, and S. Ogino, โSimple and combina-tion resonances of columns under periodic axial loads,โ Journalof Sound and Vibration, vol. 33, no. 2, pp. 211โ221, 1974.
[13] C. S. Hsu, โOn the parametric excitation of a dynamic systemhaving multiple degrees of freedom,โ ASME Journal of AppliedMechanics, vol. 30, pp. 367โ372, 1963.
[14] S. K. Sinha, โDynamic stability of a Timoshenko beam subjectedto an oscillating axial force,โ Journal of Sound andVibration, vol.131, no. 3, pp. 509โ514, 1989.
[15] S. K. Sahu and P. K. Datta, โParametric instability of doublycurved panels subjected to non-uniform harmonic loading,โJournal of Sound and Vibration, vol. 240, no. 1, pp. 117โ129, 2001.
[16] S. K. Sahu and P. K. Datta, โDynamic stability of curved panelswith cutouts,โ Journal of Sound and Vibration, vol. 251, no. 4, pp.683โ696, 2002.
[17] M. Ganapathi, B. P. Patel, P. Boisse, and M. Touratier, โNon-linear dynamic stability characteristics of elastic plates sub-jected to periodic in-plane load,โ International Journal of Non-Linear Mechanics, vol. 35, no. 3, pp. 467โ480, 2000.
[18] K. Nagai andN. Yamaki, โDynamic stability of circular cylindri-cal shells under periodic compressive forces,โ Journal of Soundand Vibration, vol. 58, no. 3, pp. 425โ441, 1978.
[19] C. Massalas, A. Dalamangas, and G. Tzivanidis, โDynamicinstability of truncated conical shells, with variable modulus ofelasticity, under periodic compressive forces,โ Journal of Soundand Vibration, vol. 79, no. 4, pp. 519โ528, 1981.
[20] V. B. Kovtunov, โDynamic stability and nonlinear parametricvibration of cylindrical shells,โ Computers & Structures, vol. 46,no. 1, pp. 149โ156, 1993.
[21] K. Y. Lam and T. Y. Ng, โDynamic stability of cylindrical shellssubjected to conservative periodic axial loads using differentshell theories,โ Journal of Sound and Vibration, vol. 207, no. 4,pp. 497โ520, 1997.
[22] T. Y. Ng, K. Y. Lam, K. M. Liew, and J. N. Reddy, โDynamicstability analysis of functionally graded cylindrical shells underperiodic axial loading,โ International Journal of Solids andStructures, vol. 38, no. 8, pp. 1295โ1309, 2001.
[23] F. Pellicano and M. Amabili, โStability and vibration of emptyand fluid-filled circular cylindrical shells under static and peri-odic axial loads,โ International Journal of Solids and Structures,vol. 40, no. 13-14, pp. 3229โ3251, 2003.
[24] M. Ruzzene, โDynamic buckling of periodically stiffened shells:application to supercavitating vehicles,โ International Journal ofSolids and Structures, vol. 41, no. 3-4, pp. 1039โ1059, 2004.
-
Advances in Acoustics and Vibration 7
[25] P. Hagedorn and L. R. Koval, โOn the parametric stability of aTimoshenko beam subjected to a periodic axial load,โ Ingenieur-Archiv, vol. 40, no. 3, pp. 211โ220, 1971.
[26] K.-J. Yang, K.-S. Hong, and F. Matsuno, โRobust adaptiveboundary control of an axially moving string under a spa-tiotemporally varying tension,โ Journal of Sound and Vibration,vol. 273, no. 4-5, pp. 1007โ1029, 2004.
[27] Z. Viderman, F. P. J. Rimrott, and W. L. Cleghorn, โPara-metrically excited linear nonconservative gyroscopic systems,โMechanics of Structures and Machines, vol. 22, no. 1, pp. 1โ20,1994.
[28] V. V. Bolotin,The Dynamic Stability of Elastic Systems, Holden-Day, 1964.
[29] C. S. Hsu, โOn approximating a general linear periodic system,โJournal of Mathematical Analysis and Applications, vol. 45, pp.234โ251, 1974.
[30] P. P. Friedmann, โNumerical methods for determining thestability and response of periodic systems with applications tohelicopter rotor dynamics and aeroelasticity,โ Computers andMathematics with Applications, vol. 12, no. 1, pp. 131โ148, 1986.
[31] O. A. Bauchau and Y. G. Nikishkov, โAn implicit Floquet anal-ysis for rotorcraft stability evaluation,โ Journal of the AmericanHelicopter Society, vol. 46, no. 3, pp. 200โ209, 2001.
[32] H. A. DeSmidt, K.W.Wang, and E. C. Smith, โCoupled torsion-lateral stability of a shaft-disk system driven throuoh a universaljoint,โ Journal of AppliedMechanics, Transactions ASME, vol. 69,no. 3, pp. 261โ273, 2002.
[33] C. S. Hsu, โFurther results on parametric excitation of adynamic system,โ ASME Journal of Applied Mechanics, vol. 32,pp. 373โ377, 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