research article nonlinear vibrations of fgm cylindrical panel … · 2019. 7. 31. · research...

Research ArticleNonlinear Vibrations of FGM Cylindrical Panel with SimplySupported Edges in Air Flow

Y. X. Hao,1 W. Zhang,2 S. B. Li,3 and J. H. Zhang1

1College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing 100192, China2College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China3College of Science, Civil Aviation University of China, Tianjin 300300, China

Correspondence should be addressed to W. Zhang; [email protected]

Received 6 June 2014; Accepted 9 December 2014

Academic Editor: Mahmut Reyhanoglu

Copyright © 2015 Y. X. Hao et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Chaotic and periodic motions of an FGM cylindrical panel in hypersonic flow are investigated. The cylindrical panel is alsosubjected to in-plane external loads and a linear temperature variation in the thickness direction. The temperature dependentmaterial properties of panel which are assumed to be changed through the thickness direction only can be determined by a simplepower distribution in terms of the volume fractions. With Hamilton’s principle for an elastic body, a nonlinear dynamical modelbased on Reddy’s first-order shear deformation shell theory and von Karman type geometric nonlinear relationship is derived inthe form of partial equations. A third-order piston theory is adopted to evaluate the hypersonic aerodynamic load. Here, Galerkin’smethod is employed to discretize this continuous nonlinear dynamic system to ordinary differential governing equations involvingtwo degrees of freedom.The chaotic and periodic response are studied by the direct numerical simulation method for influences ofdifferentMach number and the value of in-plane load.The bifurcations, Poincare section, waveform, and phase plots are presented.

1. Introduction

With the continuous variation of the material propertiesalong the thickness, functionally gradedmaterials (FGM) canbe used in high temperature gradient environments especiallywhen they are made of metal and ceramic. The metal cankeep a certain extent of toughness and ceramics have superiorheat resistant ability. So they usually act as thermal protectionstructures in spacecraft and other structural components inhigh temperature environments [1, 2]. It is well known that,due to the combined load of airflow and heating, the flexiblepanels might exhibit large aerothermal deflections [3]. Inan extensive search of panel flutter literature, a number ofinvestigations were dedicated to FGM plates with supersonicor hypersonic flow regimes. Considered a curved skin panelwith geometrical imperfection, Abbas et al. [4] gave itsflutter in unsteady flowbynumerical simulation andGalerkinmethod. With the help of structural nonlinear and the third-order piston theory, the governing equations were derived.The effects of the system parameters on the flutter werediscussed in detail. Using the linear approach, the flutter

of rectangular flat plates in supersonic flow and thermalenvironment was studied by Prakash and Ganapathi [5].The plate was subjected to the two-dimensional aerodynamicforce. They showed that under real flight conditions heatingcaused by aerodynamics is enormous. Sohn and Kim [6]took a static and dynamic stability study on the panel underaerodynamic force as well as thermal loads. Ibrahim et al.[7, 8] investigated the thermal buckling and nonlinear flutterof thin FGM panels under the action of aerothermoelasticityby the finite element method.Then they presented the resultsfor different factors. Accounting for both the geometric andaerodynamic nonlinearities, Prakash et al. [9] studied thenonlinear flutter of FGMplates under high supersonic airflowin frequency domain and time domain, respectively. Theinfluence of various parameters including the geometrics andphysics on the flutter of FGM plates was discussed. Hos-seini and Fazelzadeh [10] used the numerical and analyticalmethodologies to study postcritical and vibration behaviorsof the FGM panels in a supersonic air flow. Ibrahim et al. [8]analyzed nonlinear flutter and thermal buckling of an FGM

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015, Article ID 246352, 14 pageshttp://dx.doi.org/10.1155/2015/246352

2 International Journal of Aerospace Engineering

panel under the combined effect of elevated temperatureconditions and aerodynamic loading. Lee and Kim [11] dealtwith the aerothermopostbuckling behaviors of the FGMpanel in supersonic air flow. Limit-cycle vibration was foundin this study by Newmark time integration method andGuyan reduction technique. Mey et al. [12] found that, inmany studies which are about the flutter characteristics ofFGM plates, the linear or nonlinear theory in conjunctionwith the first-order piston theory used to approximate theaerodynamic pressure was considered. The first-order pistontheory is valid for sufficiently high supersonicMach numbers(√2 < 𝑀

∞< 5) on surfaces with small geometric character-

istics.Since the first report of flutter instability for circular

cylindrical shells, the studies of the aeroelastic stability ofcylindrical shells in axial flow received extensive attention[13]. Numerous studies on the cylindrical shells focused ontheir flutter. Marzocca et al. [14] reviewed the correspondingresearches that analyse the dynamic behavior of curved andflat panels exposed to supersonic flow fields. Hosseini etal. [15] analyzed the nonlinear response of FGM curvedpanels in high temperature supersonic air flows. The effectsof curved panel height-rise and volume fraction index on thenonlinear dynamical behavior of the panel which is subjectedto aerothermoelastic loads are investigated.

Librescu et al. [16] presented a theoretical investigationof the flutter and postflutter of the long thin-walled circularcylindrical panels in supersonic/hypersonic flow field. Forthe problems of the flutter boundaries about the simplysupported functionally graded truncated conical shell sub-jected to supersonic air flow, Mahmoudkhani et al. [17] madeaerothermoelastic analysis to predict the flutter boundaries.The flutter boundaries were obtained for the FGM conicalshells with different semivertex cone angles, different temper-ature distributions, and different volume fraction indices.

When it comes to FGM cylindrical panel identification,damage detection, and the control of the dynamics, it isnecessary to investigate their complex nonlinear flutter inhypersonic air flow in great detail [18]. However, to the bestof the authors’ knowledge, works on dynamic instability ofFGM cylindrical panel subjected to supersonic/hypersonicflow, including the effects of thermal load and in-planeloads, appear to be scarce in the open literature. Manyinteresting researches used the linear shell theory. Thereare a few literatures on the nonlinear dynamic behavior ofFGM cylindrical panel taking into account the aerodynamicnonlinearities.

In the present research, the bifurcations and chaoticdynamics of the hypersonic FGM cylindrical panel subjectedto thermal and mechanical loads are investigated by applyinggeometrical nonlinear and the third-order piston theory.Materials properties of the constituents are graded in thethickness direction according to a power law distribution.Only transverse nonlinear oscillations of the FGM cylindri-cal panel are considered; the equations of motion can bereduced into a two-degree-of-freedom nonlinear system. Bythe numerical method, the nonlinear dynamical equationsare analyzed to find the nonlinear responses of the system.

x

R

y

z

L

𝜃

Figure 1: The model of an FGM cylindrical panel with simplysupported edges and the coordinate system.

2. Theoretical Formulation

2.1. Model of the FGM Cylindrical Panel. Consider a simplysupported hypersonic FGM circular cylindrical panel of alength 𝐿, thickness ℎ, midsurface radius𝑅, and angular width𝜃. This panel is also subjected to the in-plane harmonicexcitations. Cartesian coordinate𝑂𝑥𝑦𝑧 is adopted to describethe deformations of the FGM circular cylindrical panel. Thecoordinates 𝑥 and 𝑦 are in longitudinal and tangential direc-tions, respectively. The coordinate 𝑧 is taken to be positiveoutward radially, as shown in Figure 1. The displacementsof an arbitrary point are denoted by 𝑢(𝑥, 𝑦, 𝑧, 𝑡), V(𝑥, 𝑦, 𝑧, 𝑡),and 𝑤(𝑥, 𝑦, 𝑧, 𝑡). Assume that 𝑢

0(𝑥, 𝑦, 𝑧, 𝑡), V

0(𝑥, 𝑦, 𝑧, 𝑡), and

𝑤0(𝑥, 𝑦, 𝑧, 𝑡) are the displacements of a point in the middle

plane in the axial, circumferential, and radial directions,respectively. The rotations of the transverse normal to themidplane about 𝑦 and 𝑥 axes are assumed to be 𝜙

𝑥and

𝜙𝑦, respectively. The in-plane excitation of the FGM panel

distributed uniformly along the 𝑥 direction at the end ofpanels 𝑥 = 0 and 𝑥 = 𝐿 is of the form 𝑝 = −(𝑝

0− 𝑝1(𝑡)),

where𝑝0is the static in-plane preload and𝑝

1(𝑡) is part of time

dependent that has the form 𝑝1(𝑡) = −𝑝

1cos(𝜔

1𝑡). Here, 𝜔

1

is the frequency of excitation along axial direction.Assume that the panel material is made of a composite

of the ceramics and metals. The material properties ofconstituents of the panel including density 𝜌, elastic moduli𝐸, and thermal coefficient of expansion 𝛼 are temperaturedependent and can be expressed as [19]

�̃� = 𝑃0(𝑃−1𝑇−1

+ 1 + 𝑃1𝑇 + 𝑃2𝑇2

+ 𝑃3𝑇3

) , (1)

where 𝑃0, 𝑃−1, 𝑃1, 𝑃2, and 𝑃

3are temperature dependent coef-

ficients and 𝑇 is the environment temperature.The effective material properties 𝑃 vary continuously in

the 𝑧 direction by following a simple power law in terms ofthe volume fractions and can be expressed as

𝑃 = 𝑃𝑐𝑉𝑐+ 𝑃𝑚𝑉𝑚, (2)

where 𝑃𝑐and 𝑃

𝑚indicate, respectively, the properties of the

ceramic and metal and 𝑉𝑐and 𝑉

𝑚are their volume fractions

and have the following relationship:

𝑉𝑐+ 𝑉𝑚= 1; (3)

International Journal of Aerospace Engineering 3

the metal volume fraction 𝑉𝑚can be written as

𝑉𝑚(𝑧) = (

𝑧

ℎ+1

2)𝜂

, 𝜂 ≥ 0, (4)

where 𝜂 is the volume fraction exponent which can depict thematerial variation profile through the thickness. Accordingto Zhang et al. [20], linear temperature change is consideredwhich has the form

𝑇 (𝑧) =𝑇𝑡+ 𝑇𝑏

2+𝑇𝑡− 𝑇𝑏

ℎ𝑧, (5)

where 𝑇𝑡and 𝑇

𝑏indicate the temperature of the top and

bottom surfaces of the panel, respectively. The FGM circularcylindrical panel is subjected to a uniform temperaturevariation Δ𝑇 = 𝑇−𝑇

0, where 𝑇

𝑏is the reference temperature.

2.2. Aerodynamics Loading. The linear piston theory is validfor Mach numbers changing from √2 to 5 and for higherMach numbers the nonlinear piston theory must be usedin Mei et al. [21]. To study the nonlinear oscillation offunctionally gradedmaterial cylindrical panel under a hyper-sonic air flow, one should use third-order piston theory.Piston theory aerodynamics, which is used in problems ofoscillating airfoils advanced by Lighthill [22] and later usedas an aeroelastic tool by Ashley and Zartarian [23], is apopular modeling technique for supersonic and hypersonicaeroelastic analyses.

For a cylindrical panel, which is exposed to an externalhypersonic flow field parallel to the centerline of the panelon the surface, the aerodynamic pressure Δ𝑃 using the third-order piston theory is expressed by Amabili [13] and Cao andZhao [24] as

Δ𝑃 = −2𝑝∞

𝑀𝑎

[(𝜕𝑤

𝜕𝑥+

1

𝑈∞

𝜕𝑤

𝜕𝑡)

+(1 + 𝛾)𝑀

𝑎

4(𝜕𝑤

𝜕𝑥+

1

𝑈∞

𝜕𝑤

𝜕𝑡)

2

+(1 + 𝛾)𝑀2

𝑎

12(𝜕𝑤

𝜕𝑥+

1

𝑈∞

𝜕𝑤

𝜕𝑡)

3

] ,

(6)

where 𝛾 is the adiabatic exponent and 𝑀𝑎and 𝑡 are the

Mach number and time, respectively. The free-stream staticpressure is given as

𝑃∞=1

2𝜌∞𝑈2

∞, 𝑈∞= 𝑀𝑎𝑎∞, (7)

where 𝜌∞, 𝑈∞, and 𝑎

∞are the free-stream air density,

velocity, and the free-stream speed of sound, respectively.

2.3. Geometry and Constitutive Relations. The displacementcomponents for the FGM cylindrical panel based on Reddy’sfirst-order shear deformation theory [25] can be representedas

𝑢 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑢0(𝑥, 𝑦, 𝑡) + 𝑧𝜙

𝑥(𝑥, 𝑦, 𝑡) ,

V (𝑥, 𝑦, 𝑧, 𝑡) = V0(𝑥, 𝑦, 𝑡) + 𝑧𝜙

𝑦(𝑥, 𝑦, 𝑡) ,

𝑤 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑤0(𝑥, 𝑦, 𝑡) .

(8)

It is assumed that the transverse normal stress is negligibleand normals are not vertical to the midplane after defor-mation. Substituting displacement components into VonKarman nonlinear strains-displacement relations, the strainsin terms of middle-surface displacements are given as

𝜀𝑥𝑥=𝜕𝑢0

𝜕𝑥+1

2(𝜕𝑤

𝜕𝑥)

2

+ 𝑧𝜕𝜙𝑥

𝜕𝑥,

𝜀𝑦𝑦=𝜕V0

𝜕𝑦+1

2(𝜕𝑤

𝜕𝑦)

2

+ 𝑧𝜕𝜙𝑦

𝜕𝑦+𝑤

𝑅,

𝛾𝑥𝑦=𝜕𝑢0

𝜕𝑦+𝜕V0

𝜕𝑥+ 𝑧(

𝜕𝜙𝑥

𝜕𝑦+𝜕𝜙𝑦

𝜕𝑥) +

𝜕𝑤

𝜕𝑥

𝜕𝑤

𝜕𝑦,

𝛾𝑦𝑧= 𝜙𝑦+𝜕𝑤

𝜕𝑦−V0

𝑅,

𝛾𝑧𝑥= 𝜙𝑥+𝜕𝑤

𝜕𝑥.

(9)

The constitutive relations of the panel in which thethermal effects due to temperature difference are consideredcan be written as

{{{{{

{{{{{

{

𝜎𝑥𝑥

𝜎𝑦𝑦

𝜎𝑦𝑧

𝜎𝑧𝑥

𝜎𝑥𝑦

}}}}}

}}}}}

}

=

{{{{{

{{{{{

{

𝑄11

𝑄12

0 0 0

𝑄21

𝑄22

0 0 0

0 0 𝑄44

0 0

0 0 0 𝑄55

0

0 0 0 0 𝑄66

}}}}}

}}}}}

}

×

{{{{{

{{{{{

{

{{{{{

{{{{{

{

𝜀𝑥𝑥

𝜀𝑦𝑦

𝛾𝑦𝑧

𝛾𝑧𝑥

𝛾𝑥𝑦

}}}}}

}}}}}

}

−

{{{{{

{{{{{

{

𝛼𝑥𝑥

𝛼𝑦𝑦

0

0

2𝛼𝑥𝑦

}}}}}

}}}}}

}

Δ𝑇

}}}}}

}}}}}

}

,

(10)

where 𝑄𝑖𝑗(𝑖 = 1, 2 and 𝑗 = 1, 2; 𝑖 = 4, 5, 6 and 𝑗 = 4, 5, 6) are

the elastic constants which can be expressed as

𝑄11= 𝑄22=

𝐸 (𝑧)

1 − ] (𝑧)2,

𝑄12= 𝑄21=] (𝑧) 𝐸 (𝑧)1 − ] (𝑧)2

,

𝑄44= 𝑄55= 𝑄66=

𝐸 (𝑧)

2 (1 + ] (𝑧)).

(11)

The thermal expansion coefficient 𝛼 can be given as

𝛼𝑥𝑥= 𝛼𝑦𝑦= 𝛼, 𝛼

𝑥𝑦= 0. (12)


2.4. Equations of Motion. By using Hamilton’s principle, themotion equations in terms of midplane displacements areobtained as follows [20]:

(𝐴11𝐼2− 𝐵11𝐼1)𝜕2𝑢0

𝜕𝑥2+ (𝐴66𝐼2− 𝐵66𝐼1)𝜕2𝑢0

𝜕𝑦2

+ (𝐴12𝐼2+ 𝐴66𝐼2− 𝐵12𝐼1− 𝐵66𝐼1)𝜕2V0

𝜕𝑥𝜕𝑦

+ (𝐵11𝐼2− 𝐷11𝐼1)𝜕2𝜙𝑥

𝜕𝑥2+ (𝐵66𝐼2− 𝐷66𝐼1)𝜕2𝜙𝑥

𝜕𝑦2

+ (𝐵12𝐼2+ 𝐵66𝐼2− 𝐷12𝐼1− 𝐷66𝐼1)𝜕2𝜙𝑦

𝜕𝑥𝜕𝑦

+ [(𝐴12𝐼2− 𝐵12𝐼1)1

𝑅+ 𝐴55𝐾𝐼1]𝜕𝑤0

𝜕𝑥

+ (𝐴11𝐼2− 𝐵11𝐼1)𝜕𝑤0

𝜕𝑥

𝜕2𝑤0

𝜕𝑥2

+ (𝐴66𝐼2− 𝐵66𝐼1)𝜕𝑤0

𝜕𝑥

𝜕2

𝑤0

𝜕𝑦2

+ (𝐴12𝐼2+ 𝐴66𝐼2− 𝐵12𝐼1− 𝐵66𝐼1)𝜕𝑤0

𝜕𝑦

𝜕2𝑤0

𝜕𝑥𝜕𝑦

+ 𝐴55𝐾𝐼1𝜙𝑥= (𝐼0𝐼2− 𝐼2

1) �̈�0,

(13a)

(𝐴66𝐼2− 𝐵66𝐼1)𝜕2V0

𝜕𝑥2+ (𝐴22𝐼2− 𝐵22𝐼1)𝜕2V0

𝜕𝑦2

+ [(𝐴21+ 𝐴66) 𝐼2− (𝐵21+ 𝐵66) 𝐼1]𝜕2𝑢0

𝜕𝑥𝜕𝑦

+ (𝐵66𝐼2− 𝐷66𝐼1)𝜕2𝜙𝑦

𝜕𝑥2+ (𝐵22𝐼2− 𝐷22𝐼1)𝜕2𝜙𝑦

𝜕𝑦2

+ [(𝐵21 + 𝐵66) 𝐼2− (𝐷21 + 𝐷66) 𝐼1]𝜕2𝜙𝑥

𝜕𝑥𝜕𝑦

+ [(𝐴22𝐼2− 𝐵22𝐼1)1

𝑅+ 𝐴44𝐾𝐼1]𝜕𝑤0

𝜕𝑦

+ (𝐴66𝐼2− 𝐵66𝐼1)𝜕𝑤0

𝜕𝑦

𝜕2𝑤0

𝜕𝑥2

+ (𝐴22𝐼2− 𝐵2𝐼1)𝜕𝑤0

𝜕𝑦

𝜕2

𝑤0

𝜕𝑦2

+ [(𝐴21+ 𝐴66) 𝐼2− (𝐵21+ 𝐵66) 𝐼1]𝜕𝑤0

𝜕𝑥

𝜕2𝑤0

𝜕𝑥𝜕𝑦

+ 𝐾𝐴44𝐼1𝜙𝑦+ 𝐼2𝐾𝐴44

𝑅(𝜙𝑦+𝜕𝑤

𝜕𝑦−V0

𝑅)

− 𝐴44𝐾𝐼1

V0

𝑅= (𝐼2𝐼0− 𝐼2

1) V̈0,

(13b)

𝐴21

𝜕𝑢0

𝜕𝑥

𝜕2𝑤0

𝜕𝑦2+ 𝐴11

𝜕𝑢0

𝜕𝑥

𝜕2𝑤0

𝜕𝑥2+ 2𝐴66

𝜕𝑢0

𝜕𝑦

𝜕2𝑤0

𝜕𝑥𝜕𝑦

+ (𝐴21+ 𝐴66)𝜕2𝑢0

𝜕𝑥𝜕𝑦

𝜕𝑤0

𝜕𝑦+ 𝐴11

𝜕2𝑢0

𝜕𝑥2𝜕𝑤0

𝜕𝑥

+ 𝐴66

𝜕2𝑢0

𝜕𝑦2𝜕𝑤0

𝜕𝑥+ 2𝐴66

𝜕V0

𝜕𝑥

𝜕2𝑤0

𝜕𝑥𝜕𝑦+ 𝐴22

𝜕V0

𝜕𝑦

𝜕2𝑤0

𝜕𝑦2

+1

𝑅𝐴21

𝜕𝑢0

𝜕𝑥+1

𝑅𝐴22

𝜕V0

𝜕𝑦+ 𝐴12

𝜕V0

𝜕𝑦

𝜕2𝑤0

𝜕𝑥2

+ 𝐴22

𝜕2V0

𝜕𝑦2𝜕𝑤0

𝜕𝑦+ 𝐴66

𝜕2V0

𝜕𝑥2𝜕𝑤0

𝜕𝑦

+ (𝐴12+ 𝐴66)𝜕2V0

𝜕𝑥𝜕𝑦

𝜕𝑤0

𝜕𝑥+ 𝐴21

𝑤

𝑅

𝜕2𝑤

𝜕𝑥2

+ 𝐴55𝐾𝜕2𝑤0

𝜕𝑥2+ 𝐴44𝐾𝜕2𝑤0

𝜕𝑦2+

3

2𝑅𝐴12(𝜕𝑤

𝜕𝑥)

2

+3

2𝑅𝐴22(𝜕𝑤

𝜕𝑦)

2

+ 𝐴22

𝑤

𝑅

𝜕2𝑤

𝜕𝑦2

+ 𝐴22

𝑤

𝑅2+ 2 (𝐴

21+ 2𝐴66)𝜕𝑤0

𝜕𝑥

𝜕2𝑤0

𝜕𝑥𝜕𝑦

𝜕𝑤0

𝜕𝑦

+3

2𝐴22(𝜕𝑤0

𝜕𝑦)

2

𝜕2𝑤0

𝜕𝑦2

+ (1

2𝐴21+ 𝐴66)(

𝜕𝑤0

𝜕𝑥)

2

𝜕2𝑤0

𝜕𝑦2

+ (𝐴66+1

2𝐴21)(

𝜕𝑤0

𝜕𝑦)

2

𝜕2𝑤0

𝜕𝑥2

+3

2𝐴11(𝜕𝑤0

𝜕𝑥)

2

𝜕2𝑤0

𝜕𝑥2

+ (1

𝑅𝐵21+ 𝐴55𝐾)

𝜕𝜙𝑥

𝜕𝑥+ 𝐵21

𝜕𝜙𝑥

𝜕𝑥

𝜕2𝑤0

𝜕𝑦2

+ 2𝐵66

𝜕𝜙𝑥

𝜕𝑦

𝜕2𝑤0

𝜕𝑥𝜕𝑦+ (𝐵21+ 𝐵66)𝜕2𝜙𝑥

𝜕𝑥𝜕𝑦

𝜕𝑤0

𝜕𝑦

+ 𝐵11

𝜕2𝜙𝑥

𝜕𝑥2𝜕𝑤0

𝜕𝑥+ 𝐵11

𝜕𝜙𝑥

𝜕𝑥

𝜕2𝑤0

𝜕𝑥2

+ 𝐵66

𝜕2𝜙𝑥

𝜕𝑦2𝜕𝑤0

𝜕𝑥+ (

1

𝑅𝐵22+ 𝐴44𝐾)

𝜕𝜙𝑦

𝜕𝑦

+ 𝐵22

𝜕𝜙𝑦

𝜕𝑦

𝜕2𝑤0

𝜕𝑦2+ 2𝐵66

𝜕𝜙𝑦

𝜕𝑥

𝜕2𝑤0

𝜕𝑥𝜕𝑦+ 𝐵66

𝜕2𝜙𝑦

𝜕𝑥2𝜕𝑤0

𝜕𝑦

+ 𝐵12

𝜕𝜙𝑦

𝜕𝑦

𝜕2𝑤0

𝜕𝑥2+ 𝐵22

𝜕2𝜙𝑦

𝜕𝑦2𝜕𝑤0

𝜕𝑦

+ (𝐵12+ 𝐵66)𝜕2𝜙𝑦

𝜕𝑥𝜕𝑦

𝜕𝑤0

𝜕𝑥


+ 𝑁𝑇

𝑥𝑥

𝜕2𝑤0

𝜕𝑥2+ 𝑁𝑇

𝑦𝑦

𝜕2𝑤0

𝜕𝑦2− 𝐾

𝐴44

𝑅

𝜕V0

𝜕𝑦+𝑁𝑇𝑦𝑦

𝑅

− 𝑃𝜕2𝑤0

𝜕𝑥2+ Δ𝑃 − 𝜇�̇�

0= 𝐼0�̈�0,

(13c)

(𝐴11𝐼1− 𝐵11𝐼0)𝜕2𝑢0

𝜕𝑥2+ (𝐴66𝐼1− 𝐵66𝐼0)𝜕2𝑢0

𝜕𝑦2

+ (𝐴12𝐼1+ 𝐴66𝐼1− 𝐵12𝐼0− 𝐵66𝐼0)𝜕2V0

𝜕𝑥𝜕𝑦

+ (𝐵11𝐼1− 𝐷11𝐼0)𝜕2𝜙𝑥

𝜕𝑥2+ (𝐵66𝐼1− 𝐷66𝐼0)𝜕2𝜙𝑥

𝜕𝑦2

+ (𝐵12𝐼1+ 𝐵66𝐼1− 𝐷12𝐼0− 𝐷66𝐼0)𝜕2𝜙𝑦

𝜕𝑥𝜕𝑦

+ [(𝐴12𝐼1− 𝐵12𝐼0)1

𝑅+ 𝐴55𝐾𝐼0]𝜕𝑤0

𝜕𝑥

+ (𝐴11𝐼1− 𝐵11𝐼0)𝜕𝑤0

𝜕𝑥

𝜕2𝑤0

𝜕𝑥2

+ (𝐴66𝐼1− 𝐵66𝐼0)𝜕𝑤0

𝜕𝑥

𝜕2𝑤0

𝜕𝑦2

+ (𝐴12𝐼1+ 𝐴66𝐼1− 𝐵12𝐼0− 𝐵66𝐼0)𝜕𝑤0

𝜕𝑦

𝜕2𝑤0

𝜕𝑥𝜕𝑦

+ 𝐴55𝐾𝐼0𝜙𝑥= (−𝐼

0𝐼2+ 𝐼2

1) ̈𝜙𝑥,

(13d)

(𝐴66𝐼1− 𝐵66𝐼0)𝜕2V0

𝜕𝑥2+ (𝐴22𝐼1− 𝐵22𝐼0)𝜕2V0

𝜕𝑦2

+ [(𝐴21+ 𝐴66) 𝐼1− (𝐵21+ 𝐵66) 𝐼0]𝜕2𝑢0

𝜕𝑥𝜕𝑦

+ (𝐵66𝐼1− 𝐷66𝐼0)𝜕2𝜙𝑦

𝜕𝑥2+ (𝐵22𝐼1− 𝐷22𝐼0)𝜕2𝜙𝑦

𝜕𝑦2

+ [(𝐵21 + 𝐵66) 𝐼1− (𝐷21 + 𝐷66) 𝐼0]𝜕2𝜙𝑥

𝜕𝑥𝜕𝑦

+ [(𝐴22𝐼1− 𝐵22𝐼0)1

𝑅+ 𝐴44𝐾𝐼0]𝜕𝑤0

𝜕𝑦

+ (𝐴66𝐼1− 𝐵66𝐼0)𝜕𝑤0

𝜕𝑦

𝜕2𝑤0

𝜕𝑥2

+ (𝐴22𝐼1− 𝐵2𝐼0)𝜕𝑤0

𝜕𝑦

𝜕2

𝑤0

𝜕𝑦2

+ [(𝐴21+ 𝐴66) 𝐼1− (𝐵21+ 𝐵66) 𝐼0]𝜕𝑤0

𝜕𝑥

𝜕2𝑤0

𝜕𝑥𝜕𝑦

+ 𝐴44𝐾𝐼0𝜙𝑦− 𝐼0𝐴44𝐾V0

𝑅

+ 𝐼1

𝐴44

𝑅𝐾(𝜙𝑦+𝜕𝑤

𝜕𝑦−V0

𝑅) = (−𝐼

2𝐼0+ 𝐼2

1) ̈𝜙𝑦,

(13e)

where 𝜇 is the damping coefficient, all kinds of the stiffnesselements 𝐴

𝑖𝑗, 𝐵𝑖𝑗, and 𝐷

𝑖𝑗of the FGM cylindrical panel are

denoted by

(𝐴𝑖𝑗, 𝐵𝑖𝑗, 𝐷𝑖𝑗) = ∫

ℎ/2

−ℎ/2

𝑄𝑖𝑗(1, 𝑧, 𝑧

2

) 𝑑𝑧, 𝑖, 𝑗 = 1, 2, 6, (14a)

𝐴𝑖𝑗= ∫ℎ/2

−ℎ/2

𝑄𝑖𝑗𝑑𝑧, 𝑖, 𝑗 = 4, 5, (14b)

and various mass inertia terms 𝐼𝑖in (12) can be defined as

𝐼𝑖= ∫ℎ/2

−ℎ/2

𝑧𝑖

𝑝 (𝑧) 𝑑𝑧, 𝑖 = 0, 1, 2. (15)

Thermal force resultants due to temperature rise arefunctions of the temperature and coefficient of thermalexpansion equation. They can be calculated by

{

{

{

𝑁𝑇𝑥𝑥

𝑁𝑇𝑦𝑦

}

}

}

= −∫ℎ/2

−ℎ/2

[𝑄11

𝑄12

0

𝑄21

𝑄22

0]{

{

{

𝛼𝑥𝑥

𝛼𝑦𝑦

0

}

}

}

Δ𝑇𝑑𝑧. (16)

Here, the shear correction factor 𝐾 that is introduced byReddy [25] and Kadoli and Ganesan [26] is equal to 5/6.

For simply supported hypersonic FGM circular cylindri-cal panel with rectangular base, the first twomode shapes thatsatisfy the boundary conditions are assumed to be

𝑢0= 𝑢1(𝑡) cos(𝜋𝑥

𝐿) sin(

3𝜋𝑦

𝑏)

+ 𝑢2(𝑡) cos(3𝜋𝑥

𝐿) sin(

𝜋𝑦

𝑏) ,

(17a)

V0= V1(𝑡) sin(𝜋𝑥

𝐿) cos(

3𝜋𝑦

𝑏)

+ V2(𝑡) sin(3𝜋𝑥

𝐿) cos(

𝜋𝑦

𝑏) ,

(17b)

𝑤0= 𝑤1(𝑡) sin(𝜋𝑥

𝐿) sin(

3𝜋𝑦

𝑏)

+ 𝑤2(𝑡) sin(3𝜋𝑥

𝐿) sin(

𝜋𝑦

𝑏) ,

(17c)

𝜙0= 𝜙1(𝑡) cos(𝜋𝑥

𝐿) sin(

3𝜋𝑦

𝑏)

+ 𝜙2(𝑡) cos(3𝜋𝑥

𝐿) sin(

𝜋𝑦

𝑏) ,

(17d)

𝜓0= 𝜓1(𝑡) sin(𝜋𝑥

𝐿) cos(

3𝜋𝑦

𝑏)

+ 𝜓2(𝑡) sin(3𝜋𝑥

𝐿) cos(

𝜋𝑦

𝑏) ,

(17e)


where 𝑢1, V1, 𝑤1, 𝜙1, 𝜓1and 𝑢

2, V2, 𝑤2, 𝜙2, 𝜓2are the time

dependence amplitudes of the first two modes. The constant𝑏 can be represented by 𝑏 = 𝑅𝜃.

Compared to the transverse inertia term, the influences ofthe in-plane and rotary inertia terms on the vibration of thepanel are small and can be neglected; see [27, 28]. Followingthe work given in [20], the displacement components of𝑢0, V0, 𝜙𝑥, and 𝜙

𝑦can be expressed in terms of 𝑤

0. Applying

the Galerkin method on (12), one can obtain a coupled setof nonlinear ordinary differential equations in time that cantake the form as follows:

�̈�1+ 𝑐1�̇�1+ 𝜂101𝑤1+ 𝑃

1cos (𝜔

1𝑡) 𝑤1+ 𝑔102𝑤2

1

+ 𝑔103𝑤1𝑤2+ 𝑔104𝑤2

2+ 𝑔105𝑤3

1+ 𝑔106𝑤2

1𝑤2

+ 𝑔107𝑤2

2𝑤1+ 𝑔108𝑤3

2− Δ𝑝1= 0,

(18a)

�̈�2+ 𝑐2�̇�2+ 𝜂201𝑤2+ 𝑃

2cos (𝜔

1𝑡) 𝑤2+ 𝑔202𝑤2

1

+ 𝑔203𝑤1𝑤2+ 𝑔204𝑤2

2+ 𝑔205𝑤3

1+ 𝑔206𝑤2

1𝑤2

+ 𝑔207𝑤3

2+ 𝑔208𝑤2

2𝑤1− Δ𝑝2= 0,

(18b)

where 𝜂101

= 𝑔101

+𝑁1and 𝜂201

= 𝑔201

+𝑁2. The coefficients

𝑁1and 𝑁

2are the results of thermal stress resultants and

the static components of in-plane preloads.The variablesΔ𝑝1

and Δ𝑝2are related to the aerodynamic pressure; they can be

expressed as

Δ𝑝1

= 𝑀𝑎(𝐶𝑚111

𝑤1+ 𝐶𝑚112

𝑤2)

+ 𝑀2

𝑎(𝐶𝑚121

𝑤2

1+ 𝐶𝑚122

𝑤1𝑤2+ 𝐶𝑚123

𝑤2

2)

+𝑀3

𝑎(𝐶𝑚131

𝑤3

1+ 𝐶𝑚132

𝑤2

1𝑤2+ 𝐶𝑚133

𝑤1𝑤2

2+ 𝐶𝑚134

𝑤3

2)

+ 𝐷𝑚101

�̇�3

1+ 𝐷𝑚102

�̇�3

2+ 𝐷𝑚103

�̇�2

1�̇�2

+ 𝐷𝑚104

�̇�1�̇�2

2+ 𝐷𝑚105

�̇�2

1

+ 𝐷𝑚106

�̇�2

2+ 𝐷𝑚107

�̇�1�̇�2+ 𝐷𝑚108

�̇�1

+𝑀𝑎[(𝐷𝑚111

𝑤1+ 𝐷𝑚112

𝑤2) �̇�1

+ (𝐷𝑚113

𝑤1+ 𝐷𝑚114

𝑤2) �̇�2

+ (𝐷𝑚115

𝑤1+ 𝐷𝑚116

𝑤2) �̇�2

1

+ (𝐷𝑚117

𝑤1+ 𝐷𝑚118

𝑤2) �̇�2

2

+ (𝐷𝑚119

𝑤1+ 𝐷𝑚1110

𝑤2) �̇�1�̇�2]

+ 𝑀2

𝑎[(𝐷𝑚121

𝑤2

1+ 𝐷𝑚122

𝑤1𝑤2+ 𝐷𝑚123

𝑤2

2) �̇�1

+ (𝐷𝑚124

𝑤2

1+ 𝐷𝑚125

𝑤1𝑤2+ 𝐷𝑚126

𝑤2

2) �̇�2] ,

(19a)

Δ𝑝2

= 𝑀𝑎(𝐶𝑚211

𝑤1+ 𝐶𝑚212

𝑤2)

+ 𝑀2

𝑎(𝐶𝑚221

𝑤2

1+ 𝐶𝑚222

𝑤1𝑤2+ 𝐶𝑚223

𝑤2

2)

+𝑀3

𝑎(𝐶𝑚231

𝑤3

1+ 𝐶𝑚232

𝑤2

1𝑤2+ 𝐶𝑚233

𝑤1𝑤2

2+ 𝐶𝑚234

𝑤3

2)

+ 𝐷𝑚201

�̇�3

1+ 𝐷𝑚202

�̇�3

2+ 𝐷𝑚203

�̇�2

1�̇�2+ 𝐷𝑚204

�̇�1�̇�2

2

+ 𝐷𝑚205

�̇�2

1+ 𝐷𝑚206

�̇�2

2+ 𝐷𝑚207

�̇�1�̇�2+ 𝐷𝑚208

�̇�2

+𝑀𝑎[(𝐷𝑚211

𝑤1+ 𝐷𝑚212

𝑤2) �̇�1

+ (𝐷𝑚213

𝑤1+ 𝐷𝑚214

𝑤2) �̇�2

+ (𝐷𝑚215

𝑤1+ 𝐷𝑚216

𝑤2) �̇�2

1

+ (𝐷𝑚217

𝑤1+ 𝐷𝑚218

𝑤2) �̇�2

2

+ (𝐷𝑚219

𝑤1+ 𝐷𝑚2110

𝑤2) �̇�1�̇�2]

+ 𝑀2

𝑎[(𝐷𝑚221

𝑤2

1+ 𝐷𝑚222

𝑤1𝑤2+ 𝐷𝑚223

𝑤2

2) �̇�1

+ (𝐷𝑚224

𝑤2

1+ 𝐷𝑚225

𝑤1𝑤2+ 𝐷𝑚226

𝑤2

2) �̇�2] .

(19b)All the coefficients in (18a) and (18b) can be affected

by geometric and physical parameters and temperature fieldof the panel. And the coefficients in (19a) and (19b) aredependent on geometric and free-stream air density and free-stream speed of sound. They are too long to be listed outin the paper for abbreviation. This is a nonlinear dynamicsystem which includes cubic and quadratic terms. To studythe nonlinear aeroelastic behavior of the FGM cylindricalpanel in hypersonic flow, the obtained nonlinear ordinarydifferential equations can be solved by the fourth-orderRunge-Kutta method.

3. Numerical Results and Discussions

In order to validate the numerical results presented in thisstudy, a comparison is shown in Figure 2 for a simplysupported intact aluminum-zirconia FGM square plate withlength 0.2m and thickness 0.01m. A suddenly uniformtransverse load of intensity of 𝑞

0= 1MPa is applied on it.

The material properties vary in the thickness by followinga simple power law and the power law exponent is takenas 𝑛 = 0.2. It means that the plate is zirconia-rich on thetop surface and aluminum-rich at the bottom surface. InFigure 2, temporal evolution curve of center deflection of thesquare plate is given by Reddy’s finite element results (see[29]) and the results of this study. The dimensionless timeand center deflection are defined as 𝑡 = 𝑡√𝐸

𝑏/(𝜌𝑏𝑎2) and

𝑤 = 𝑤𝑐𝐸𝑏ℎ/(𝑞0𝑎2), respectively. According to this figure,

close agreements between present methodology and Reddy’sfinite element results can be observed. Although the plateis subjected to the aerodynamic force in this study and notuniform transverse load, as external pressures, both of themplay the same role on the plate.


w

t

0

−8

−16

−24

−32

−40

0 2 4 6 8 10

Figure 2: Comparison of nondimensional center deflection of the square plate with simply supported edges under a suddenly applied uniformload by Reddy’s FEM results [29] and present (—: present; ◼: Reddy’s).

w1

P̂

1

0.6

0.4

0.2

0

−0.2

−0.4

0.75 0.8 0.85 0.9 0.95

(a)

w2

P̂

1

0.75 0.8 0.85 0.9 0.95

15

10

5

0

−5

−10

(b)

w1

P̂

1

0.1

0

−0.1

−0.2

0.85 0.86 0.87 0.88 0.89

(c)

w2

P̂

1

0.85 0.86 0.87 0.88 0.89

6

2

−2

−6

(d)

Figure 3: (a) and (b) depict the bifurcation diagrams of 𝑃1versus the first two modes 𝑤

1and 𝑤

2which are given when the volume fraction

index is taken to be 0.5. The Mach number is taken to be 5.0. The magnitude and frequency of the in-plane excitation are taken to be 𝑃1∈

[5.5 × 107

, 7.79 × 107

] and 161Hz, respectively. (c) and (d) are partially enlarged details for (a) and (b), respectively. The value of horizontalcoordinate is defined as �̂�

1= 𝑃1/7.79 × 107.


w1

×10−4

2

1

0

−1

−2

−0.03 −0.028 −0.026 −0.024

ẇ1

(a)

−0.03

−0.028

−0.026

−0.024

1850 1860 1870 1880

t

w1

(b)

w2

0.02

0.01

0

−0.01

−0.02

−0.25 −0.2 −0.15 −0.1 −0.05 0

ẇ2

(c)

w2

−0.25

−0.2

−0.15

−0.1

−0.05

0

1850 1860 1870 1880

t

(d)

w2

w 1

×10−4

0.2

0

−0.2

2

0

−2 −0.03

−0.028

−0.026

−0.024

̇

w1

(e)

w1

×10−4

×10−5

5

4

−7 −6.5 −6 −5.5

ẇ1

(f)

Figure 4:The periodic response of the FGM cylindrical panel occurs when the magnitude of the in-plane excitation is taken to be 5.85×107.The volume fraction index is taken to be 0.5. The Mach number is taken to be 5.0. (a) The phase portrait on plane (𝑤

1, �̇�1); (b) the time

response on the planes (𝑡, 𝑤1); (c) the phase portrait on plane (𝑤

2, �̇�2); (d) the responses on the planes (𝑡, 𝑤

2); (e) three-dimensional phase

portrait in space (𝑤1, �̇�1, 𝑤2); (f) the Poincare section on plane (𝑤

1, �̇�1).


0.2

0.1

0

−0.1

−0.2

−0.06 −0.04 −0.02 0 0.02

(a)

−0.06

−0.04

−0.02

0

0.02

1850 1860 1870 1880

t

w1

(b)

w2

20

10

0

−10

−20

−10 −5 0 5

ẇ2

(c)

1850 1860 1870 1880

t

w2

10

0

−10

(d)

w2

w 1−0.06

−0.04

−0.020

0.02

5

0

−5

−10

0.2

0

−0.2

̇

w1

(e)

w1 ×10−5

0.7

0.6

0.5

0.4

0.3

−2.5 −2 −1.5 −1 −0.5 0

ẇ1

(f)

Figure 5: The multiple-periodic motion of the FGM cylindrical panel occurs when the magnitude of the in-plane excitation is taken to be6.70 × 107. The volume fraction index is taken to be 0.5. The Mach number is taken to be 5.0.


w1

4

2

0

−2

−4

−1 −0.5 0 0.5 1 1.5

ẇ1

(a)

1850 1860 1870 1880

t

w1

1

0.5

0

−0.5

−1

(b)

w2

60

20

−20

−60

−60 −20 20 60

ẇ2

(c)

w2

60

20

−20

−60

1850 1860 1870 1880

t

(d)

w2

w 1

50

0

−50

4

2

0

−2

−4 −1

0

1

̇

w1

(e)

w1

15

5

−5

−15

−1.5 −1 −0.5 0 0.5 1

×10−3

ẇ1

(f)

Figure 6: The chaotic motion of the FGM cylindrical panel occurs when the magnitude of the in-plane excitation is taken to be 7.79 × 107.The volume fraction index is taken to be 0.5. The Mach number is taken to be 5.0.


w1

P̂

1

0.4

0.2

0

−0.2

−0.4

0.75 0.8 0.85 0.9 0.95

(a)

w2

P̂

1

0.75 0.8 0.85 0.9 0.95

10

0

−10

(b)

Figure 7: (a) and (b) depict the bifurcation diagrams of 𝑃1versus 𝑤

1and 𝑤

2which are given when the Mach number is taken to be 10.0. The

value of horizontal coordinate has the same definition as that in Figure 3.

In this study, Al2O3and Ti-6Al-4V are chosen as the

two constituent materials of the FGM cylindrical panel. Theproperties for these two constituentmaterials can be found inShen [30]. For simplicity, it is assumed that Poisson’s ratio is aconstant as the value of ] = 0.3 and the reference temperatureremained a constant as 300K. The simply supported FGMcylindrical panel has the following geometries: length 𝐿 =2.4m, radius of curvature 𝑅 = 1.5m, thickness ℎ = 0.002m,and angular width 𝜃 = (𝜋/2)𝑅. In all of the numericalanalyses, the temperature difference through the thickness ofthe panel was assumed to have attained a steady state 100K.In all of the calculations, the initial conditions are fixed at thevalue of 𝑤

1= 0.0004, �̇�

1= −0.00025, 𝑤

2= 0.0005, and

�̇�2= 0.00006. In addition, the flow field characteristics are

as follows: the free-stream air density 𝜌∞= 1.225 kg/m3, the

free-stream speed of sound 𝑎∞= 340.3m/s, and the adiabatic

exponent 𝛾 = 1.4; see Librescu et al. [31]. In given conditions,the influences of the in-plane excitation and Mach numberson the nonlinear dynamic responses of the FGM cylindricalpanel in hypersonic flow are investigated. In order to performanalysis of the chaotic and periodic responses for the FGMcylindrical panel, the bifurcation diagrams are depicted. Thedimensionless transverse amplitudes are defined as 𝑤 = 𝑤/ℎhere. To study convenience, the overbar is dropped in thefollowing research.

Firstly, the Mach number and frequency of the in-planeexcitation are taken to be 5.0 and 161Hz, respectively. Andthe volume fraction index is as 𝑁 = 0.5. Figure 3 plots thebifurcation diagrams of transverse amplitude of the first twomodes by changing the in-plane loads with other parametersfixed. In Figures 3(a) and 3(b), the bifurcation diagrams ofthe 𝑃1(𝑃1∈ [5.5 × 107, 7.79 × 107]) versus 𝑤

1and 𝑤

2are

given. Chaotic and periodic motions have been detected, asindicated in Figure 3, which shows the very complex and richnonlinear dynamics.

As the amplitude of excitation increases within regionof 𝑃1

∈ [5.5 × 107, 6.63 × 107], periodic motion occurs

for the FGM cylindrical panel. Furthermore, a region ofmultiperiodicmotion exists when the amplitude of excitationvaried from 6.63 × 107 to 6.80 × 107. The system appearsto undergo complicated bifurcations where the system is inthe instability region. Within the apparently chaotic regionwhere the amplitude of excitation changes from 6.80 × 107to 7.79 × 107, there are some regions where the response isperiodic. Figures 3(c) and 3(d) are partially enlarged detailsfor Figures 3(a) and 3(b), respectively.

To better understand nonlinear dynamical behaviors,phase portraits, time responses, the Poincare section, andthree-dimensional phase portrait are illustrated. Figure 4illustrates nonlinear dynamic response for the FGM cylindri-cal panel when the forcing excitation is 5.85 × 107. The two-dimensional phase plane portraits are given in Figures 4(a)and 4(c) by (𝑤

𝑗, �̇�𝑗). Figures 4(b) and 4(d) denote the time

responses on the planes (𝑡, 𝑤𝑗), respectively. Here subscript

“𝑗” indicates the first two modes and takes the value 𝑗 = 1, 2.Figures 4(e) and 4(f) represent the three-dimensional phaseportrait in space (𝑤

1, �̇�1, 𝑤2) and the Poincare section on

plane (𝑤1, �̇�1), respectively (see [20, 32, 33]). It is known that

if the Poincare section displays a finite number of points, themotion can be periodic. Since there is single point in Poincaresections given in Figure 4(f), it can be concluded that thereexists periodic oscillations motion for the FGM cylindricalpanel in this case. Figure 5 suggests that themultiple-periodicmotion of the FGM cylindrical panel occurs when the in-plane excitationchanges to 6.70 × 107. Figure 6 illustrates theexistence of the chaotic motion for this system when theforcing excitation is 7.79 × 107.

Figure 7, where in-plane excitation is also taken as thebifurcation parameter, shows the effect of in-plane excitationon the dynamics of the FGM cylindrical panel when theMach is 10. Compared to the bifurcation diagrams given inFigure 3 whose Mach is 5, we can see that the appearance ofthe bifurcation point is almost in the same locations. In theinstability region, the chaotic and multiperiodic motion may


w1

4

2

0

−2

−4

−1 −0.5 0 0.5 1 1.5

ẇ1

(a)w1

−1

−0.5

0

0.5

1

1.5

1850 1860 1870 1880

t

(b)

w2

60

20

−20

−60

−60 −20 20 60

ẇ2

(c)

w2

60

20

−20

−60

1850 1860 1870 1880

t

(d)

w2

w 1

40

0

−40

5

0

−5 −1

0

1

2

̇

w1

(e)

w1

15

5

−5

−15

−1 −0.5 0 0.5 1

×10−3

ẇ1

(f)

Figure 8: The chaotic motion of the FGM cylindrical panel occurs when the magnitude of the in-plane excitation is taken to be 7.79 × 107.The volume fraction index is taken to be 0.5. The Mach number is taken to be 10.0.


be detected. Figure 8 indicates that the chaotic motion of theFGM cylindrical panel occurs.

4. Conclusions

The nonlinear dynamics of an FGM cylindrical panel undera hypersonic flow are presented. The material properties aregraded continuously throughout the thickness of the panelaccording to the power law function and are temperaturedependent. A third-order piston theory is applied for thehypersonic aerodynamic load.TheVonKarman larger deflec-tion theory in conjunction with energy approach is used toobtain the equations of motion. The bifurcation diagrams,phase portraits, time responses, and the Poincare section areemployed to understand the periodic and chaotic motions ofthe cylindrical panel. It is obtained that, with the change ofin-plane load parameter, the different nonlinear dynamics ofthe FGMpanel occur. Different nonlinear dynamic behaviorsalternate from stable periodic motion to instable chaoticmotions in hypersonic flow. In addition, from the phaseportraits and time responses, it can be seen that when thechaotic motion occurs, their phase portraits resemble eachother in appearance for different Mach.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors acknowledge the financial support of theNational Natural Science Foundation of China throughGrants nos. 11272063, 11102226, and 11472298, the ScienceFoundation of Beijing Municipal Education Commissionthrough Grant no. cit&tcd201304112, and Foundation ofTianjin City through Grant no. 13JCQNJC04400.

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