Nonlinear and chaotic vibration and stability analysis of an aero-elastic piezoelectric FG plate under parametric and primary excitations

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Nonlinear and chaotic vibration and stability analysis of an aero-elastic piezoelectric FG plate under parametric and primary excitations Mousa Rezaee n, Reza Jahangiri School of Mechanical Engineering, University of Tabriz, P.O. Box 51665315, Tabriz, Iran

a r t i c l e i n f o

abstract

Article history: Received 19 June 2014 Received in revised form 16 December 2014 Accepted 19 January 2015 Handling Editor: M.P. Cartmell

In this study, in the presence of supersonic aerodynamic loading, the nonlinear and chaotic vibrations and stability of a simply supported Functionally Graded Piezoelectric (FGP) rectangular plate with bonded piezoelectric layer have been investigated. It is assumed that the plate is simultaneously exposed to the effects of harmonic uniaxial inplane force and transverse piezoelectric excitations and aerodynamic loading. It is considered that the potential distribution varies linearly through the piezoelectric layer thickness, and the aerodynamic load is modeled by the first order piston theory. The vonKarman nonlinear strain–displacement relations are used to consider the geometrical nonlinearity. Based on the Classical Plate Theory (CPT) and applying the Hamilton's principle, the nonlinear coupled partial differential equations of motion are derived. The Galerkin's procedure is used to reduce the equations of motion to nonlinear ordinary differential Mathieu equations. The validity of the formulation for analyzing the Limit Cycle Oscillation (LCO), aero-elastic stability boundaries is accomplished by comparing the results with those of the literature, and the convergence study of the FGP plate is performed. By applying the Multiple Scales Method, the case of 1:2 internal resonance and primary parametric resonance are taken into account and the corresponding averaged equations are derived and analyzed numerically. The results are provided to investigate the effects of the forcing/piezoelectric detuning parameter, amplitude of forcing/piezoelectric excitation and dynamic pressure, on the nonlinear dynamics and chaotic behavior of the FGP plate. It is revealed that under the certain conditions, due to the existence of bi-stable region of non-trivial solutions, system shows the hysteretic behavior. Moreover, in absence of airflow, it is observed that variation of control parameters leads to the multi periodic and chaotic motions. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction Recently, metal alloys have opened a new horizon in manufacturing advanced materials that have optimal performance under mechanical/thermal conditions and in various applications, such as, space craft, aircraft propulsion systems, atmospheric re-entry vehicles, and turbine wing protection coverage. For instance, in aerospace shuttles composites that

n

Corresponding author. Tel.: þ98 41 3339 2459; fax: þ 98 41 3335 4153. E-mail address: [email protected] (M. Rezaee).

http://dx.doi.org/10.1016/j.jsv.2015.01.025 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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M. Rezaee, R. Jahangiri / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

contain ceramic tiles are used for thermal protection against the heat produced during the re-entry to the atmosphere. The variation in the material properties between layers results in high thermal stresses concentrations, which can results delamination, crack initiation, and the structural failure. However, with the gradual variation of properties from one material to the other, the stress concentration decreases considerably. The most well-known FG materials have gradual structure from ceramic properties to metal properties. Among the non-homogeneous composite structural elements, FG panels can be used in some applications, which are susceptible to flutter due to high speed air-flow. Panel flutter is a phenomenon of self-excited oscillations of skin panels at high flow velocity, which can lead to a fatigue in the skin panels. In some research, these structural drawbacks have been reported for launch vehicles, high-speed jet engines, supersonic, and hypersonic aircrafts. Therefore, investigating such aero-elastic phenomenon has received serious attention in the past few decades in two particular areas, namely, wing flutter and panel flutter. A comprehensive numerical study of coupled mode flutter of the plates is conducted to perform investigation and identification of the linear and nonlinear flutter characteristics and boundaries and their changes due to changes within the problem parameters. Dowell [1] studied the chaotic behavior of a fluttering buckled plate and showed that near the point in the parameter space of flow velocity and in-plane load, where the flutter and buckling instability boundaries merge, chaos occurs. Chaotic behavior of viscoelastic plate under supersonic flow has been studied by Pourtakdoust and Fazelzadeh [2]. They showed that in the range of the chaotic behavior, increasing the damping, especially the nonlinear damping, makes the chaotic behavior zone to be smaller and it can change the chaotic oscillations to periodic ones. Considering the different boundary conditions, Cheng [3] employed a FEM formulation for investigating the panel flutter at hypersonic speeds. He examined a possible type of the panel behavior, including flat, buckled, LCO, periodic and chaotic motions. By evaluating the largest Lyapunov exponent, he distinguished that at low or moderately high dimensionless dynamic pressures, the fluttering panel typically takes a period-doubling route to evolve into chaos. Recently, most investigations on the flutter of the plates have been carried out considering the structural nonlinearity of the plates theoretically and numerically. In this regards, the influence of aerodynamic nonlinearities on the plate dynamics was studied by Prakash et al. [4]. Using the third order piston theory and applying the FEM, they analyzed the aero-elastic oscillations of the FG plate under the hypersonic flow condition. They showed that increasing the Mach number causes the flexural vibration to be chaotic. The nonlinear aero-elastic behavior of FG plates in supersonic flow was developed by Haddadpour et al. [5]. They showed, unlike pure metal plates, in FG plates the bending-extensional coupling has an important role on the steady state deformation of the plates and increases the flutter margin of the plates. Using the finite element method (FEM), Sohn and Kim [6] analyzed the flutter boundaries of FG plates through a linear flutter theory and they defined the static stability boundaries. Using Mori–Tanaka relations and linear piston theory, the aero-elastic behavior of homogeneous/FGM two/three dimensional flat plates under supersonic airflow was investigated by Navazi et al. [7]. In addition to airflow, aerospace structures may be subjected to variable mechanical/piezoelectrical in-plane or out-ofplane loads which may be applied over a range of frequencies. Under special conditions, when the dynamic loading is inplane compressive force, the structural parametric resonance may be occurred even in the amplitude of in-plane loads are less than the buckling load. Moreover, in the presence of parametric excitation and under certain conditions, when the plate is exposed to the primary excitation, the plate may show a complicated dynamics such as multi-periodic or chaotic motions. Therefore, it is necessary to study the nonlinear behavior of FG/FGP plates under simultaneous action of the in-plane and out-of-plane excitations in the absence/presence of the aerodynamic loading. Recently, Ramachandra and Kumar Panda [8] studied the instability of a shear deformable composite plates and they showed that the type of boundary conditions and type of loading distribution have a significant influence on the dynamic instability regions. Hu and Zhang [9] investigated the vibrations and stability of FG plates under the parametric excitations and they examined the existence or inexistence of resonance in the range of excitation frequency and showed that increasing the linear damping can prevent the occurrence of the resonance phenomenon. Ng et al. [10] studied parametric resonance of SS FG plates under biaxial harmonic in-plane loading. They found out that the volume fraction of the constituent materials (ceramic and metal) has significant effect on the parametric resonance of the FG plates. Zhang and Zhao [11] studied the nonlinear vibrations of a composite laminated cantilever rectangular plate subjected to in-plane and transversal forcing excitations. They showed that the chaotic responses are sensitive to the changing of the forcing amplitude and the damping coefficient. In another investigation, Zhang and Zhao [12], studied the nonlinear response of a symmetric cross-ply composite laminated cantilever plate under in-plane and moment excitations. Their results illustrate that over a range of in-plane force and moment amplitude, the plate may show the bifurcation and chaotic motions. Considering the simultaneous interaction of parametric resonance and fluid flow, Tezak et al. [13] analyzed the nonlinear behavior of the plates subjected to parametric excitation at the onset of flutter instability, using the first order piston theory. Flutter suppression of an isotropic SS panel via combination of internal and parametric resonances is investigated by Chin et al. [14]. They studied the static bifurcations occurrence, such as supercritical and subcritical pitchfork and saddle-node bifurcations, for different values of the detuning parameter and characterized the conditions in which the parametric excitation can provide stabilizing effect. Young and Chen [15] investigated the stability of skew plates subjected to aerodynamic and harmonic in-plane loads. They assumed that the aerodynamic pressure to be smaller than its critical value, also, they found that with a small, slowly varying in-plane harmonic force, the plate may become unstable before the aerodynamic pressure reaches to its critical value. In another study, they studied the dynamic stability of clamped fluttered plates under the harmonic in-plane forces and showed that, when the excitation amplitude in comparison with the aerodynamic force is small, the plate may become dynamically stable [16]. Young and Chen [17] studied the nonlinear Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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vibration of a cantilever skew plate when it becomes unstable. They found out that the aerodynamic damping has a stabilizing effect, however, amplitude of the forcing excitation has a destabilizing effect. The influence of the quadratic and cubic terms on nonlinear dynamic characteristics of the angle-ply composite laminated plate with parametric and external excitations is investigated by Sayed and Mousa [18]. Kim [19] studied the multimode parametric excitation of a SS plate under time-varying and non-uniform edge loading. He showed that due to the coupled excitation terms, the time dependency beside the non-uniformity of the edge loading may be affected the resonance frequencies. Tang and Chen [20] investigated the nonlinear vibrations and stability of in-plane translating viscoelastic plates under external and internal resonances. They examined the effects of the in-plane translating speed, the viscosity coefficient, and the excitation amplitude on the steady-state responses. Zhang et al. [21] developed the nonlinear dynamic behaviors of a SS 3D-Kagome truss core sandwich plate subjected to transverse and in-plane excitations. Their results exhibit the existence of the periodic, multi-periodic and chaotic responses with the variation of the excitations amplitudes. Recently, the mechanical deformations of the flexible structures such as wings and panels can be sensed and controlled using the smart piezoelectric patches used as sensors or actuators. The most commonly piezoelectric materials, which include piezoelectric lead–zirconate–titanate (PZT) and piezoelectric polyvinylidene fluoride (PVDF), which are usually bonded to the surface of a host structures. The actuators induce strain actuations in host structures with intrinsic oscillation and shape control capabilities. This capabilities cause that the applications of the piezoelectric plates have been widely developed in the two past decades [22]. In some applications, flexible wings and panels with bounded/embedded piezoelectric patches may be undergo to the large amplitude oscillations. Hence, the nonlinear dynamics, bifurcations, and chaotic behavior of such structures will play a main role in engineering, especially in aerospace applications. During the last years, several researchers have focused their attention on investigating the nonlinear and chaotic dynamics of piezoelectric plates under the resonance conditions. The nonlinear vibration of composite laminated plates have been extensively done by Zhang et al. [23] who presented the bifurcations and chaotic dynamics of a SS symmetric cross-ply composite laminated piezoelectric plates, which are simultaneously forced by the transverse, in-plane and piezoelectric excitations. They studied the influence of the excitations on the bifurcations and chaotic behaviors of such plates. Rafiee et al. [24] investigated the nonlinear dynamic stability of initially imperfect piezoelectric FG carbon nanotube reinforced composite plates under a combined thermal and electrical loadings and simultaneous interaction of parametric and external resonances. Wang et al. [25] studied the effects of a piezoelectric layer on the stability of viscoelastic plates subjected to the follower forces. They examined that the stability of the viscoelastic plates can be effectively improved by determination of optimal location for the piezoelectric layers and the most favorable voltage assignment. Song and Li [26] analyzed the active aeroelastic flutter and vibration control of the supersonic composite laminated plates with the piezoelectric patches. They observed that, the proportional feedback or the velocity feedback control algorithm, the aeroelastic flutter characteristics of the plate can be improved and that the vibration amplitudes can be reduced. Xue et al. [27] developed the vibration of a orthotropic smart plate made of silver, under the combined action of a transverse magnetic field and a transverse harmonic mechanical load. Recently, Hosseini et al. [28] analyzed the nonlinear forced vibrations of viscoelastic piezoelectric cantilevers resting on a nonlinear elastic foundation. They discussed about the effects of various parameters including the foundation coefficients, length of the piezoelectric layer, and the piezoelectric coefficients on the nonlinear responses of the system. The flutter of orthotropic sandwich smart composite plates with an electrorheological fluid layer subjected to supersonic airflow is investigated by Rahiminasab and Rezaeepazhand [29]. They presented that the electrorheological core layer has a capability to delay the onset of flutter instability. Lai et al. [30] presented an optimal control design to suppress panel flutter limit-cycle motions using piezoelectric. They showed that within the maximum suppressible dynamic pressure, limit cycle oscillations can be completely suppressed. The nonlinear aeroelastic resonant behavior of the plates under the combined actions of parametric forcing and primary piezoelectric excitations under the supersonic air flow, has not yet been investigated. The importance of this topic is that in some practical applications, especially in aerospace applications, parametric forcing excitation techniques are employed to suppress the vibration of the fluttering panels and wings [14]. Under certain conditions and in the presence of the parametric resonance, the plate may be exposed to the piezoelectric primary resonance. Reviewing the literature shows that, there is lack of studies in the field of the FG/FGP plate dynamics which is acted upon by both the aerodynamic and the harmonic in-plane loadings simultaneously. The objective of this study is to investigate large amplitude and chaotic vibrations and stability of a SS isotropic FGP aeroelastic rectangular thin plate acted upon by the harmonic in-plane forcing, out-of-plane piezoelectric excitations and transverse aerodynamic loading. Using a linear rule of constituent's mixtures, it is assumed that effective material properties of the FG layer are continuously varied through the thickness direction according to a simple power law distribution. By neglecting the effects of rotary inertia, in-plane inertia, and shear forces and based on the von Karman-type plate assumptions, and Classical Plate Theory, the Hamilton's principle is employed to derive the governing partial differential equations of motion. Moreover, in order to model the aerodynamic loading, six aero-elastic modes in stream-wise direction and one mode in span-wise direction are used and the first-order piston theory is employed. To model the piezoelectric primary excitation, it is assumed that the electric potential distribution vary linearly through the thickness of the piezoelectric layer. Utilizing the Galerkin's method, the governing equations of motion are transformed into a set of nonlinear ordinary differential equations. Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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Fig. 1. Geometric schematic of Ceramic/Metal FG plate exposed to biaxial mechanical and aerodynamic loads.

In order to validate the formulation, the Limit Cycle Oscillations and the aero-elastic stability boundaries of the SS Homogenous/FGP plate with two kinds of boundary conditions, (a) SS all immovable edges, and (b) SS all movable edges, are studied and in the case of FGP plate, the convergence study of the results have been demonstrated. The 1:2 internal resonance and primary parametric resonance are investigated by holding two aero-elastic modes in stream-wise direction and one mode in span-wise direction, which leads to a two-degrees of freedom nonlinear system. The Multiple Scales Method is utilized to obtain the averaged equations and the numerical methods are applied to demonstrate the nonlinear dynamics and chaotic behavior of the aeroelastic/elastic system under the 1:2 internal resonance and primary parametric resonance conditions. The effects of the forcing/piezoelectric excitation frequency, forcing/piezoelectric excitation amplitude, and dynamic pressure on the bifurcation are studied and the stability of stationary solutions is discussed in detail. The results reveal that under the certain conditions, existence of multiple coexisting non-trivial stable solutions leads to wellknown jump phenomenon that makes the system to show hysteretic characteristics. Additionally, in the absence of air flow the chaotic dynamics of the system against the changes of the control parameters is demonstrated.

2. Model description and formulation Consider a simply-supported rectangular Ceramic/Metal FGP thin plate with bonded piezoelectric layer, subjected to inplane excitation, piezoelectric excitation and aerodynamic loading, as shown in Fig. 1. Overall thickness of FGP plate is h þ hp , and its dimensions along x and y directions are a and b, respectively. The thickness of the host FG layer is h and the thickness of the isotropic piezoelectric layer which perfectly bonded to the entire bottom surface of the FG plate is hp . Assume that u, v and w are displacement components of an arbitrary point in the mid-plane ðz ¼ 0Þ along x, y, z directions, respectively. In Fig. 1, assume that the movable edges FGP plate is subjected to uniaxial in-plane forces in which, at edges x ¼ 0; a affected by uniform distributed compressive harmonic in-plane loading Nnxx ðtÞ ¼ N 0x þN 1x cos Ω1 t, in the stream-wise direction, where, N 0x is the constant static term, N1x is the harmonic forcing excitation amplitude, and, Ω1 represents the frequency of harmonic excitations in x direction. Moreover, because of the air-flow over the upper surface of the FGP plate at supersonic conditions along the x axis, an aerodynamic distributed transverse pressure, Δp, will act on the plate. Beside the mentioned excitations, the FGP plate motion may be affected by the piezoelectric bending moments. This actuating layer is considered to be segmented into Nxc by Nyc elements, so that only the desired portions of the piezoelectric layer are actuated by the external electrical field.

2.1. Material properties In this study, the host FG layer has been made of metal and ceramic, and material properties are assumed to vary gradually through the FG layer thickness. The metal volume fraction is changing from 100% in lower surface to 0% at the upper surface of the FG layer, and the ceramic volume fraction is varying continuously from 100% in the upper surface to 0% at the lower surface of the FG layer. Position-dependent effective material properties through the FG layer thickness follow a linear rule as PðzÞ ¼ P m V m ðzÞ þ P c V c ðzÞ

(1)

where P m and P c represent the metal and the ceramic properties, respectively, and V m and V c indicate the volume fraction of the metal and the ceramic parts. By using a simple power-law distribution, volume fraction of the ceramic part and metal Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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part in each point across the FG layer thickness is given as follows [21]:
 z  z0 1 n þ ; 0 rn r 1 V c ðzÞ ¼ 2 h V m ðzÞ ¼ 1  V c ðzÞ

(2)

where n is a non-negative real number and identifies the ceramic volume fraction index and explains the distribution of the ceramic part through the plate thickness. Also, z0 is the distance between the geometric mid-surface and physical neutral surface of FGP plate, and is given by [4] R h=2 h=2  h zEðzÞdz (3) z0 ¼ R h=2 p h=2  hp EðzÞdz Therefore, according to Eqs. (1)–(3), the effective Young modulus EðzÞ, Poisson's ratio νðzÞ and, mass density ρðzÞ across the FGP plate thickness can be expressed as ( ðEc Em ÞV c ðzÞ þ Em ;  h=2 r z rh=2 EðzÞ ¼ (4a)  h=2  hp rz r  h=2 Ep ; (

νðzÞ ¼

νp ;

(

ρðzÞ ¼

ðνc  νm ÞV c ðzÞ þ νm ;

 h=2 r z rh=2

(4b)

 h=2  hp rz r h=2

ðρc  ρm ÞV c ðzÞ þ ρm ;

ρp ;

 h=2 rz r h=2  h=2  hp rz r  h=2

:

(4c)

2.2. Aerodynamic and piezoelectric loading of FGP plate Panel flutter is a self-excited oscillation of a plate or shell when it is subjected to airflow along its surface. Flutter is induced by the aerodynamic loads, which act only on one side of the panel. The panel flutter is a dynamic instability pffiffiffi phenomenon in the supersonic flow regime (M 1 4 2) [29]. According to Fig. 1, because of the longitudinal air-flow on the upper surface of the plate, a transverse aerodynamic force pffiffiffiis generated and acts on the plate. To study the flow-induced oscillations of the plate under supersonic air-flow regime ( 2 o M 1 o 5, where M 1 is the Mach number), one should use an aerodynamic linear model along with a structural nonlinear model [3]. According to Piston theory, local pressure that acts on each point of the plate surface is due to (a) air pounding caused by transverse local motions of the plate as a piston, (b) pressure acts on the plate due to air convection. Therefore, the velocity of the transverse motion of points on the mid-plane is as follows [31]: _ wðx; y; tÞ ¼

∂wðx; y; tÞ ∂wðx; y; tÞ þU 1 ∂t ∂x

(5)

where U 1 ∂w=∂x represents the convection term due to the airflow. According to first-order Piston theory, the instantaneous dynamic pressure on the plate surface is given as follows [3]: ! ( ) ρ1 U 21 ∂w M 21  2 1 ∂w þ Δpðx; tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (6) ∂x M 21  1 U 1 ∂t M 21 1 In the above equation, ρ1 and U 1 are the air mass density and the air stream velocity, respectively. In the following, two dimensionless parameters of the flow, the dynamic pressure, λ, and the aerodynamic damping coefficient, g a , are defined as [32] sffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ1 U 21 a3 ρ1 U 1 ðM21 2Þ D11 n λ¼ ; ga ¼ ; ω ¼ (7) ; β ¼ M 21 1 3 4 Io a βD11 β Io ωn à of the FG plate and ωn is the convenient reference frequency. Moreover, for Here D11 represents the flexural rigidity Äà simplicity, for the case of M 1 aA 1, one may obtain the following approximation [33]: ðM21  2Þ μ μ ðM21  1Þ β M 1

(8)

Here μ indicates the air-plate mass ratio, which is defined as μ ¼ ρ1 a=m. Hence, the generalization for smaller M 1 is obvious [33]. Using the above simplification, the dimensionless aerodynamic damping, g a , can be expressed as Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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ga ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λμ=M1 . Substituting (7) and (8) in (6), one obtains

Δpðx; tÞ ¼


 D11 ∂w g a 1 ∂w λ þ 3 ∂x ωn a ∂t a

(9)

Like Ref. [24], in order to model the effect of the piezoelectric excitation, it is assumed that the electric potential distribution varies linearly through the thickness direction of the piezoelectric layer. For the panel type piezoelectric structures, the component of the electric field in poling direction (thickness direction) is dominant and it is defined as Ez ðx; y; tÞ ¼  φ;z , where φ is the potential electric field. Due to a very thin thickness of the piezoelectric layer, the self-induced electric potential will be much smaller than the external excitation voltage, which is applied to the actuator in its thickness direction. According to above assumptions, the relation between the uniform applied voltage V np ðtÞ and input electric field intensity within each of the actuation portions can be described as [24,34] Ezij ðtÞ ¼

V np ðtÞ hp

(10)

Then, the total electric field which generated by all of the active/un-active piezo portions can be obtained as x

Ez ðx; y; tÞ ¼

y

Nc X Nc X i¼1j¼1

h i Ezij ðtÞ ðHðx  xi1 Þ  Hðx  xi ÞÞðHðy yj  1 Þ  Hðy yj ÞÞ ( Hðx  x0 Þ ¼

1

x Z x0

0

x o x0

(11a)

(11b)

where Hð:Þ is the Heaviside step function, and xi and xi  1 are the position of ith actuator along the x-direction, and yj and yj  1 are the position of jth actuator along the y-direction. When electric fields are applied to the piezoelectric portions in the poling direction, it produces the actuation strain that generates the piezoelectric in-plane force resultants of Npxx , N pyy and Npxy and bending moments resultants of Mpxx , M pxx and M pxx , which are obtained as 9 8 p 9 8 M pxx > N > > > = Z  h=2 > = < xx < e31 >  p p Nyy M yy ¼ e32 1; ðz  z0 Þ Ez ðx; y; tÞdz (12) > > > >  h=2  hp : ; > ; : Npxy M pxy > 0 where e31 and e32 are the piezoelectric stiffness of the actuator portions.

3. The governing equations of motion Based on the CPT and considering the nonlinear Van-Karman type assumptions, the constitutive stress-strain relations for an isotropic FGP plate can be obtained. Substituting the obtained strain field, into stress constitutive relations, and integrating through the piezoelectric plate thickness, the stress resultants are given as 9 2 9 8 Np 9 8 38 > u;x þ w2;x =2 Nxx > xx > A11 A12 0 B11 B12 0 > > > > > > > > > > > > > > > > p > 7> > > 6 > > >N > > N 2 > > > yy > B21 B22 0 7> > > > > 6 A12 A22 0 yy > v;y þw;y =2 > > > > > > > > > > > > 6 7 p > > 6 > > > > = = = < < < 7 N Nxy 0 B66 7 u;x þ v;y þ w;x w;y 0 A66 0 xy 60 (13) ¼6  7 p D11 D12 0 7> M xx > > 6 B11 B12 0 > > > Mxx > >  w;xx > > > > > > > > > 6 7> > > > > > > > > > > > 6 > > > M pyy > > Myy > >  w;yy B22 0 D12 D22 0 7 B > > > > > 5> > > 4 21 > > > > > > > > > ; ; > : Mxy > : ; : Mp > 0 D66 0 0 B66 0  w;xy xy

In the above relation, Nxx , N yy and Nxy are the membrane force resultants and M xx , M yy and Mxy are the flexural bending moment resultants. Also, Aij , Bij and Dij are the elements of the extensional stiffness, bending-extensional coupling stiffness and the bending stiffness of the FG layer, respectively, and are obtained as Z þ h=2   Q ij ðzÞ〈1; ðz z0 Þ; ðz  z0 Þ2 〉dz; ði; j ¼ 1; 2; 6Þ (14) Aij ; Bij ; Dij ¼  h=2  hp

According to the Hamilton's principle and neglecting the rotary inertia and in-plane inertia effects, the nonlinear governing equations of aero-elastic motion for the FG thin plate subjected to in-plane and transvers excitations are derived. Substituting stress resultants from (13) into the equilibrium equations, leads to the nonlinear in-plane and out-of-plane governing equations in terms of the displacement field components u, v and w. A11

∂2 u ∂2 u ∂2 v ∂3 w ∂3 w ∂w ∂2 w ∂w ∂2 w ∂w ∂2 w  B11 3  ðB12 þ 2B66 Þ ¼0 þ A66 2 þ ðA12 þ A66 Þ þ A11 þ A66 þ ðA12 þA66 Þ 2 2 2 2 ∂x∂y ∂x ∂x ∂x ∂y ∂y ∂x∂y ∂x ∂y ∂x ∂x∂y

(15a)

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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A66

∂2 v ∂2 v ∂2 u ∂3 w ∂3 w ∂w ∂2 w ∂w ∂2 w ∂w ∂2 w B22 3 ðB21 þ 2B66 Þ 2 þ A66 ¼0 þA22 2 þ ðA21 þ A66 Þ þ A22 þ ðA21 þ A66 Þ ∂x∂y ∂y ∂x2 ∂y ∂y2 ∂x ∂x∂y ∂x2 ∂y ∂y ∂x ∂y B11

7

(15b)

∂3 u ∂3 v ∂3 u ∂3 v ∂4 w ∂4 w ∂4 w þ B22 3 þ ðB21 þ2B66 Þ þðB12 þ 2B66 Þ 2  D11 4 D22 4  ðD12 þD21 þ 4D66 Þ 2 2 ∂x3 ∂y ∂x∂y2 ∂x ∂y ∂x ∂y ∂x ∂y

þ A11

∂2 u ∂w ∂2 u ∂w ∂2 v ∂w ∂2 v ∂w ∂2 v ∂w ∂2 u ∂w ∂u ∂2 w þA66 2 þ A22 2 þ A66 2 þ ðA12 þ A66 Þ þðA21 þA66 Þ þA11 2 ∂x∂y ∂x ∂x∂y ∂y ∂x ∂x2 ∂x ∂x ∂y ∂x ∂y ∂y ∂x ∂y

∂u ∂2 w ∂v ∂2 w ∂v ∂2 w ∂u ∂2 w ∂v ∂2 w ∂2 w ∂2 w ∂3 w ∂w þ2A66 þ 2B66 2 þ A22 þ A12 þ 2A66 þ ðB21  B12 Þ 2 2 2 2 ∂x ∂y ∂y ∂y ∂y ∂x ∂y ∂x∂y ∂x ∂x∂y ∂x ∂y ∂x∂y2 ∂x
2 2 2 2
 3 2 2 2
2
∂ w ∂w ∂ w∂ w 3 ∂w ∂ w 3 ∂ w ∂w 1 ∂ w ∂w 1 ∂2 w ∂w 2  ðB12 þ B21 Þ 2 þ A11 þ A22 2 þ A12 2 þ A21 2 þ ðB12 B21 Þ 2 2 2 2 ∂x 2 ∂y 2 ∂y 2 ∂x ∂x ∂y ∂y ∂x ∂y ∂x ∂y ∂x ∂y 2 2
2 2
2 2 2
2
2 2 ∂ w ∂w ∂ w ∂w ∂ w ∂ w ∂w ∂ w ∂w ∂w ∂ w ∂w þA21 þA66 2 þA66 2 þðB12 þ B21 Þ  2B66 þA12 ∂y ∂x ∂x∂y ∂x∂y ∂x ∂x∂y ∂y ∂x ∂x∂y ∂y ∂x ∂y þA21

þ4A66

∂2 M pxy ∂2 M pyy ∂w ∂2 w ∂w ∂2 w ∂2 w ∂2 w ∂2 M pxx € ¼0  N pxx 2  2N pxy N pyy 2   2 þ Δpðx; tÞ I o w 2 ∂x ∂x∂y ∂y ∂x∂y ∂x∂y ∂x ∂y ∂x ∂y2

where I o is defined as the area mass density and given by Z þ h=2 Io ¼

 h=2  hp

ρðzÞdz

(15c)

(16)

Previous research on the flow induced vibrations of rectangular plates showed that the effects of coupled modes have an important role in fluttering dynamics of the plate. Here, we have focused on the nonlinear flow induced transverse vibrations of the FGP plate by considering the effects of the dominant coupled modes. According to Fig. 1, the corresponding in-plane natural/geometrical boundary conditions of the movable edges plate under in-plane loadings are Z b Z b N xx jx ¼ 0;a dy ¼ Nnxx ðtÞjx ¼ 0;a dy (17a) 0

Z

0

a 0

Z N yy jy ¼ 0;b dy ¼

b 0

N nyy ðtÞjy ¼ 0;b dy

(17b)

v¼0

at

x ¼ 0; a

(17c)

u¼0

at

y ¼ 0; b

(17d)

Also, the geometric lateral boundary conditions of the SSSS immovable/movable edges FG plate can be expressed as w ¼ Mx ¼ 0

at

x ¼ 0; a

(18a)

w ¼ My ¼ 0

at

y ¼ 0; b

(18b)

The nature of non-homogeneity of the FG plates causes bending-extensional coupling, consequently the natural boundary conditions will not satisfied exactly. However, in order to satisfy the mentioned boundary conditions, one may add some additional analytical displacements like expressions to u and v [35,36]. It is our interest to choose appropriate mode shapes in order to satisfy the geometric and the natural boundary condition for lateral vibration of the simply supported FG plate in two cases, Case 1: SSSS FG plate with all immovable (fixed in-plane) edges and, Case 2: SSSS FG plate with all movable (free in-plane) edges. The closed-form solution to the coupled nonlinear equilibrium Eq. (15c), which satisfies the associated boundary conditions, is too difficult to obtain. In this paper, similar to the earlier studies, an approximate solution to the SSSS immovable/movable edges plate is considered as: wðx; y; tÞ ¼

M X N X

    wm;n ðtÞ sin mπ x=a sin nπ y=b

(19)

m¼1n¼1

Here m and n are the number of half-wave in x and y directions, and wmn ðtÞ denotes the time dependent lateral vibration amplitude. Eqs. (15a) and (15b) are nonlinear coupled partial differential equations, and finding the exact solution for them is too complicated. Due to the coupling of the stretch and bending effects, the in-plane displacements u and v are coupled with the out of plane displacements. However, by substituting the approximate solution (19) into Eqs. (15a) and (15b), for case 2, and applying the Galerkin's procedure one can derive the expression for the in-plane displacement field components, u and v in the terms of out of plane displacement as uðx; y; tÞ ¼

M X N X m¼1n¼1

    n U mn ðtÞ cos mπ x=a sin nπ y=b þ ðη1ð1Þ N nxx þ ηð1Þ 2 N yy Þx

(20a)

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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vðx; y; tÞ ¼

M X N X m¼1n¼1

    ð2Þ n n V mn ðtÞ sin mπ x=a cos nπ y=b þ ðηð2Þ 1 N xx þ η2 N yy Þy

(20b)

ð1Þ ð2Þ ð2Þ The last terms in Eqs. (20a) and (20b) indicate the effects of the biaxial in-plane loadings, where ηð1Þ 1 , η2 , η1 and η2 can be obtained via Eqs. (17a) and (17b). In case 1, which will be used to verify the accuracy of the results of the present work with the literature, the following approximate in-plane displacement fields, u and v, are considered [5]:

uðx; y; tÞ ¼

M X N X

    um;n ðtÞ sin mπ x=a sin nπ y=b

m ¼ 1n ¼ 1

vðx; y; tÞ ¼

M X N X

    vm;n ðtÞ sin mπ x=a sin nπ y=b

(21)

m ¼ 1n ¼ 1

In Section 7, the six modes in stream-wise direction and one mode in span-wise direction have been used to validate the accuracy of the numerical results with the pervious works, Dowell [33] and Navazi et al. [7]. In order to obtain the dimensionless governing equations of motion, we introduce the transformation of the parameters and variables as u u ¼ ; a

v v ¼ ; b

w ¼

W ; h

x a

η¼ ;

y b

ζ¼ ;

t ¼ ωn t;

Rx ¼

π 2 a2 N nxx D11

;

Ry ¼

π 2 b2 N nyy D11

;

V¼

Ep d31 a2 n Vp D11

(22)

where V is the dimensionless piezoelectric voltage, Ep is the piezoelectric modulus of elasticity and d31 is the piezoelectric strain constant. For simplicity, in following, we will drop the over bars in the formulation.

4. Geometrical and material properties Unless otherwise specified, it is assumed that the material properties of the FGP plate vary exponentially through the plate thickness from purely (100%) metal properties (aluminum) ρm ¼ 2707 kg=m3 , Em ¼ 70 GPa and νm ¼ 0:3 in the bottom surface to the purely ceramic properties (zirconium) ρc ¼ 3000 kg=m3 , Ec ¼ 151 GPa and νc ¼ 0:3 in the top surface of the plate [37]. The index of volume fraction for material mixture is assumed to be n ¼ 3. Additionally, unless otherwise noted, it is assumed that the plate is square ðb=a ¼ 1Þ and the overall thickness of the FGP plate, h þ hp , is 6 mm and the thickness of the piezoelectric layer is assumed to be hp ¼ 1 mm and the FG layer thickness ratio is a=h ¼ 200. Furthermore, it is assumed that a non-viscous ideal air stream under the supersonic regime passes over the plate and there is no thermal exchange (ΔT ¼ 0) between the air and the plate, and μ=M 1 is assumed to be 0:01. The material properties of the homogeneous isotropic piezoelectric layer are assumed to be Ep ¼ 63 Gpa, ρp ¼ 0:3 kg=m3 ,νp ¼ 0:3 and e31 ¼ e32 ¼ 17:6 C=m2 . The piezoelectric guest layer has the same shape of the main host FG layer, and is divided into Nxc ¼ 20 by N yc ¼ 20 elements. It is assumed that the coordinates of the activated segments of the piezoelectric layer lie within the region 2a=4 r x r 3a=4 and 0 r yr b. When the electric fields are applied in the polling direction of portions, active piezoelectric elements produce the actuation strain that induce the in-plane loads and bending moments. By comparing the obtained results with those of the literatures, the formulation and the accuracy of the numerical results of the present work are validated. To this end, the comparison have been conducted to show the LCO, aero-elastic stability boundaries, and modal convergence study.

Fig. 2. Dimensionless LCO amplitude versus dimensionless dynamic pressure for SSSS all immovable edges homogeneous square plate.

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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Fig. 3. (a) Aero-elastic stability regions for the SSSS immovable edges homogeneous plate under combined effects of aerodynamic pressure and in-plane load and (b) comparison of aero-elastic stability boundaries for the SSSS FG plate under uniaxial static in-plane loadings for two cases: two and six modes.

Fig. 4. Convergence study for SSSS all movable edges FGP plate under uniaxial compressive in-plane edge loadings in x direction.

5. Verification and convergence study Applying the fourth-order Runge–Kutta method, fluttering motion of an isotropic immovable and movable edges SSSS FG plate is studied. The LCO amplitude, the maximum deflection of the fluttering plate, occurs around point η ¼ x=a ¼ 0:75 and ζ ¼ y=b ¼ 0:5 in both cases 1 and 2. In Fig. 2, the LCO amplitude for the case 1, a square homogeneous plate is depicted and verified by comparing with those of Dowell [33] and Navazi et al. [7]. As it is seen, the maximum transverse dimensionless vibration amplitude, W max =h, is found to increase continuously with dynamic pressure, as expected. The study has been focused on determining the aero-elastic stability boundaries of SSSS FG plate which is simultaneously under a combined biaxial static compressive uniform in-plane loads (Ry ¼ Rx ), and aerodynamic load λ. To do this, considering six aero-elastic coupled mode shapes, the related linear and nonlinear eigenvalue problems are solved to define flutter boundaries and the divergence boundaries of a FG plate, respectively.

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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In Fig. 3(a), the effect of compressive static biaxial compressive in-plane loading on the aero-elastic stability margins is assessed and compared with those of Navazi et al. [7] and Dowell [33]. As seen, there is a good agreement between the present results for μ=M 1 ¼ 0:01 and the others. It should be noted that in references [5,33] in the plot of stability boundaries, the variation of dynamic pressure, λ, is plotted against Rx =π 2 , however, in this paper, the variations of λ is plotted against Rx . It can be concluded from Fig. 3(b) that the value of critical dynamic pressure depends on the number of modes, in which the FGP plate with all movable edges (case 2) using the two stream wise modes has less stability and the flutter occurs sooner than the FGP plate with the same boundary conditions with six modes. However, the number of modes does not affect the point in which the divergence occurs. It is of interest to compare the results obtained by using different numbers of modes. As mentioned above, the value of critical dynamic pressure depends on the number of modes. To this end, in Fig. 4 the LCO amplitude of the SSSS all movable edges fluttering piezoelectric FGP plate is plotted against the dynamic pressure for different cases of the aero-elastic mode numbers, when the plate is subjected to uniaxial compressive static in-plane load in x axis. As observed, for the case of six or more modes, the solution appears converged. 6. Perturbation analysis As it can be observed from Fig. 4, when six modes are used, the nonlinear aero-elastic solution appears converged, and with only two modes the results are inaccurate. However, it has been shown by several researchers [1,13,22,38] that considering two coupled modes may be sufficient to provide qualitative results for practical applications [22]. Therefore, for simplicity on the aero-elastic dynamic behavior of the plate near the parametric resonance condition, we will retain only the first two modes in stream-wise direction (m ¼ 1; 2) and one mode in span-wise direction (n ¼ 1). Consequently, the transverse deflection will be simplified as wðx; y; tÞ ¼ w1 ðtÞ sin ðπ xÞ sin ðπ yÞ þw2 ðtÞ sin ð2π xÞ sin ðπ yÞ

(23)

Here w1 and w2 are the amplitudes of the plate vibrations in ð1; 1Þ and ð2; 1Þ modes, respectively. Using (22), the dimensionless in-plane force Rx is given as Rx ¼  Rx0 Rx1 cos Ω1 t

(24)

Here Rx0 and Rx1 are the static in-plane force and harmonic forcing excitation amplitude, respectively. In Eq. (24) Ω1 represents the frequency of harmonic parametric excitation in x direction. Moreover, the piezoelectric excitation voltage can be considered as VðtÞ ¼ V AC cos Ω2 t

(25)

where V AC is the harmonic excitation voltage amplitude and Ω2 is the frequency of harmonic primary piezoelectric excitation. Using (23)–(25) and applying the Galerkin's procedure, one can transform the coupled nonlinear partial differential Eq. (15c) into a coupled set of ordinary nonlinear differential equations in time. In the presence of the supersonic air-flow and considering the transverse deflection as (23), the nonlinear Mathieu equations of the transverse motion, can be obtained as € 1 þg a w _ 1 þ ðω21 þ a1 Rx1 cos Ω1 tÞw1 þ a2 λw2 þ a3 w31 þ a4 w1 w22 þ a5 V AC cos Ω2 t ¼ 0 w

(26a)

€ 2 þg a w _ 2 þb1 λw1 þ ðω22 þ b2 Rx1 cos Ω1 tÞw2 þb3 w21 w2 þ b4 w32 þ b5 V AC cos Ω2 t ¼ 0 w

(26b)

where b1 Rx1 and b2 Rx1 are the parametric excitation terms, a5 V AC and b5 V AC indicate the primary electrical excitation terms. ω21 and ω22 are the natural frequencies in which the effect of the static terms of in-plane forcing is included. a2 λ and b1 λ where a2 ¼  b1 , indicate air stream coupling terms. While, a3 , a4 , b3 and b4 are the structural nonlinear stiffness terms. Also, as indicated in Eq. (26), modal coupling is caused by both the cubic nonlinear terms and the air-flow dynamic pressure. It is worth to note that considering the effect of the movement of physical surface causes the coupling matrix, B, to be zero, consequentially the quadratic terms will not appear in Eq. (26). Considering that, the in-plane force excites the first mode resonance parametrically, and piezoelectric voltage excites the second mode resonance primarily, in this study, the cases ω1 ffi Ω1 =2 and ω2 ffi Ω2 which are named a parametric and primary resonance, respectively, will be considered to study the nonlinear stability of the SSSS all movable edges FGP plate. When the in-plane excitation frequency, Ω1 , is close to twice the fundamental natural frequency of the system, and the transverse excitation frequency, Ω2 , is close to the second natural frequency, any small changes in the system parameters can lead to large amplitude responses. To investigate the nonlinear behavior of the system in the case of foundational parametric resonance beside the primary resonance, detuning parameters, σ 1 and σ 2 , are introduced to demonstrate the 1:2 internal resonance and primary parametric resonance frequency relations as follows [12]:

Ω1 ¼ 2ω1 þ σ 1 ε;

Ω2 ¼ ω2 þ σ 2 ε;

Ω2 ¼ Ω1 ¼ Ω

(27)

Also we assume that there is weak aerodynamic interaction between the air-flow and the plate motion, for which λ lies in the subcritical region in Fig. 3(b) in the vicinity of the dimensionless force axis. Moreover, we consider the effects of dynamic pressure, aerodynamic damping and amplitude of the harmonic parametric and primary excitation terms to be small, where they can be written as εg a , εRx1 and εV AC , respectively. Also, all the nonlinear terms are assumed to be small in comparison Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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with the linear terms. Due to the above assumptions, one may introduce the following scale transformations: ai ¼ εαi ;

bi ¼ εβi ;

ði ¼ 1; :::; 5Þ

(28)

where ε is a small dimensionless perturbation parameter which reflects the order of nonlinearities. By substituting (27) and (28), into (26), one obtains _ 1  εα3 w31  εα4 w1 w22  εα5 V AC cos Ωt € 1 þ ω21 w1 ¼  εα1 Rx1 w1 cos Ωt  εα2 λw2  εg a w w

(29a)

€ 2 þ ω22 w2 ¼  εβ1 λw1  εβ2 Rx1 w2 cos Ωt  εg a w2  εβ2 w21 w2  εβ4 w32  εβ5 V AC cos Ωt w

(29b)

In order to apply the method of multiple scales, the first order approximate solutions of Eq. (29) may be approximated as w1 ðt; εÞ ¼ w10 ðT 0 ; T 1 Þ þ εw11 ðT 0 ; T 1 Þ; w2 ðt; εÞ ¼ w20 ðT 0 ; T 1 Þ þ εw21 ðT 0 ; T 1 Þ; T n ¼ εn t;

n ¼ 0; 1:

(30)

εD0 D1 þ :::; where Dj ¼ ∂=∂T j ; ðj ¼ 0; 1Þ, and Employing the differential operators d=dt ¼ D0 þ εD1 þ::: and d =dt ¼ substituting solution (30) into Eq. (29) and equating the coefficients of like powers of ε, leads to 2

2

D20 þ 2

ε0 -D20 w10 þ ω21 w10 ¼ 0

(30a)

ε0 -D20 w20 þ ω22 w20 ¼ 0

(30b)

ε1 -D20 w11 þ ω21 w11 ¼  2D1 D0 w10 þ σ 1 Ωw10  ga D0 w10  α1 Rx1 w10 cos ΩT 0  α2 λw20  α3 w310  α4 w10 w220  α5 V AC cos ΩT 0

(30c)

ε1 -D20 w21 þ ω22 w21 ¼  2D1 D0 w20 þ 2σ 2 Ωw20  ga D0 w20  β1 λw10  β2 Rx1 w20 cos ΩT 0  β3 w210 w20  β4 w320  β 5 V AC cos ΩT 0

(30d)

The general solution of Eqs. (30a) and (30b) are w10 ðT 0 ; T 1 Þ ¼ A1 ðT 1 ÞeiΩT 0 =2 þ A1 ðT 1 Þe  iΩT 0 =2

(31a)

w20 ðT 0 ; T 1 Þ ¼ A2 ðT 1 ÞeiΩT 0 þ A2 ðT 1 Þe  iΩT 0

(31b)

In the above equations, A1 ðT 1 Þ and A2 ðT 1 Þ are unknown complex functions, and A1 ðT 1 Þ and A2 ðT 1 Þ are their complex conjugates and they can be expressed in Cartesian form as A1 ðT 1 Þ ¼ x1 ðT 1 Þ þix2 ðT 1 Þ and A2 ðT 1 Þ ¼ x3 ðT 1 Þ þ ix4 ðT 1 Þ, where x1 , x2 , x3 and x4 are real functions of scaled time T 1 . By substituting the homogeneous solution (31a) and (31b) into Eqs. (30c) and (30d), one can obtain D20 w11 þ ω21 w11 ¼ f 1 ðA1 ; A1 ; A2 ; A2 ÞeiΩT 0 =2 þf 2 ðA1 ; A1 ; A2 ; A2 ÞeiΩT 0 þ f 3 ðA1 ; A1 ; A2 ; A2 Þe3iΩT 0 =2 þ f 4 ðA1 ; A1 ; A2 ; A2 Þe2iΩT 0 þf 5 ðA1 ; A1 ; A2 ; A2 Þe5iΩT 0 =2 þc:c: þ f 6 ðA1 ; A1 ; A2 ; A2 Þ

(32a)

D20 w21 þ ω22 w21 ¼ g 1 ðA1 ; A1 ; A2 ; A2 ÞeiΩT 0 =2 þg 2 ðA1 ; A1 ; A2 ; A2 ÞeiΩT 0 þ g 3 ðA1 ; A1 ; A2 ; A2 Þe3iΩT 0 =2 þ g 4 ðA1 ; A1 ; A2 ; A2 Þe2iΩT 0 þ g 5 ðA1 ; A1 ; A2 ; A2 Þe5iΩ1 T 0 =2 þ g 6 ðA1 ; A1 ; A2 ; A2 Þe3iΩT 0 þ c:c: þ g 7 ðA1 ; A1 ; A2 ; A2 Þ

(32b)

To eliminate the secular terms, we should have f 1 ðA1 ; A1 ; A2 ; A2 Þ ¼ I ΩD1 A1 1=2Ig a ΩA1 þ σ 1 ΩA1 1=2α2 Rx1 A1 3α4 A21 A1  2α5 A1 A2 A2 ¼ 0

(33a)

g 2 ðA1 ; A1 ; A2 ; A2 Þ ¼ 2I ΩD1 A2 þ 2σ 2 ΩA2  g a I ΩA2 2β4 A1 A1 A2 3β5 A22 A2  1=2β6 V AC ¼ 0

(33b)

Separating the real and imaginary parts of (33) and solving for dxi =dT i ði ¼ 1; :::; 4Þ, the averaged equations in Cartesian form are obtained as  1 dx1 1  α1 Rx1 x2  6α3 x21 x2  6α3 x32  4α5 x2 x23 4α5 x2 x24  ga x1 þ σ 1 x2 ¼ 2 dT 1 2Ω  dx2 1  1 α1 Rx1 x1 þ6α3 x31 þ6α3 x1 x22 þ 4α5 x1 x23 þ 4α5 x1 x24  ga x2  σ 1 x1 ¼ 2 dT 1 2Ω  1 dx3 1  2 2 2 3 ¼ 2β2 x1 x4 þ2β2 x2 x4 þ2β4 x3 x4 þ3β4 x4 þ g a x3 þ σ 2 x4 2 dT 1 2Ω  1 dx4 1  ¼  2β 2 x3 x21  2β2 x3 x22  2β4 x23 x4  3β4 x33 þ β 5 V AC þ g a x4  σ 2 x3 2 dT 1 2Ω

(34)

In the steady-state condition, the amplitudes of the oscillation tend to constant values with respect to scaled time T 1 . Under this condition, dxi =dT i -0; ði ¼ 1; :::; 4Þ, one can obtain the frequency-response, aerodynamic-response, forcing-response and voltage-response equations as sets of nonlinear algebraic equations which are solved numerically. Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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6.1. Stability analysis of the steady-state solutions In order to analyze the stability of the solutions, we suppose xi ðT 1 Þ ¼ xsi þ xpi ðT 1 Þ; i ¼ 1; :::; 4

(35)

xsi ði ¼ 1; :::; 4Þ

are the steady-state trivial and nontrivial solutions of the frequency-response equations and where xpi ði ¼ 1; :::; 4Þ represents the perturbed solution around the steady-state solution. Substituting (35) into Eq. (34) and keeping the linear parts of the equations, leads to: x_ p1 ¼ m11 xp1 þ m12 xp2 þ m13 xp3 þm14 xp4 x_ p2 ¼ m21 xp1 þ m22 xp2 þ m23 xp3 þm24 xp4 x_ p3 ¼ m31 xp1 þ m32 xp2 þ m33 xp3 þm34 xp4 x_ p4 ¼ m41 xp1 þ m42 xp2 þ m43 xp3 þm44 xp4

(36)

where mij ði ¼ 1; :::; 4Þ and ðj ¼ 1; :::; 4Þ are the coefficients due to linearization and over dots show the derivatives respect to the scaled time T 1 . According to the Routh-Hurwitz criterion, if the real part of all the eigenvalues of the coefficient matrix ½mij  are negative, the system will be stable [39]. 7. Numerical results and stability analysis In order to investigate the simultaneous effects of the primary and parametric resonances on the system response, a parametric study is performed using two coupled modes, Eq. (23), to approximate the transverse deflections of the plate. All

Fig. 5. (a) Frequency-response curves of a FGP plate for of the first mode resonance amplitude, a1 , (b) frequency-response curves of a FGP plate for of the second mode resonance amplitude, a2 , (c) frequency-response curves for a1 corresponding to the three different values of forcing excitation amplitude, and (d) frequency-response curves for a2 corresponding to the three different values of forcing excitation amplitude.

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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Fig. 6. The characteristic curves of the resonance amplitude, a1 , versus the forcing excitation amplitude for three different values of detuning parameters, σ 1 ε, in the presence of weak aerodynamic effects.

Fig. 7. (a) Bifurcation diagrams of resonance amplitude, a1 , versus the parametric excitation amplitude and (b) bifurcation diagrams of the resonance amplitude, a2 , versus the parametric excitation corresponding to the five different values of dynamic pressure.

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

14

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subsequent numerical studies are conducted to examine the effects of the dimensionless dynamic pressure, dimensionless forcing excitation, detuning parameters and dimensionless piezoelectric voltage on the homogeneous solutions of Eq. (26), qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ x21 þ x22 and a2 ¼ x23 þ x24 , near the 1:2 internal resonance and primary parametric resonance conditions. The frequency-response characteristic curves are shown in Fig. 5(a) and (b) for constant values λ ¼ 20, μ=M 1 ¼ 0:01, σ 2 ε ¼  1, Rx1 ¼ 0:4 and V AC ¼ 0:91. According to these figures, characteristic curves include the trivial and nontrivial solutions, in which the solid curves indicate the stable solutions and dashed curves indicate the unstable solutions. Fig. 5(a) exhibits a typical hardening type behavior of a nonlinear vibratory system in simultaneous presence of primary and parametric excitations. Throughout region I, when the forcing detuning parameter, σ 1 ε, is sufficiently lower than the zero point (σ 1 ε ¼ 0, which corresponds to twice the fundamental natural frequency, 2ω1 ), there is no a nontrivial solution and only a stable trivial solution exists for the resonance amplitude of the first mode, a1 . It means that, in the region I the parametric resonance of the plate will not be excited. However, because of the primary resonance effects, in which the second natural frequency is excited, and according to Fig. 5(b), in region I there is only one stable nontrivial solution for the resonance amplitude of the second mode, a2 . By continuation of σ 1 ε to increase, the values of the resonance amplitudes a1 and a2 remains zero and constant, respectively, till the first supercritical bifurcation point, A, appears, and after this point, there is one stable nontrivial periodic solution (corresponding to stable LCO) for the resonance amplitudes of the system in the first and second modes. It means that the parametric resonance of the plate will be excited (region II). In region II as σ 1 ε tend to increase slowly, the amplitudes a1 and a2 gradually increases and decreases, respectively. When σ 1 ε continues to increase, and move away from point A and after zero crossing point, the second subcritical bifurcation point, B, appears. After passing the second bifurcation point, one unstable nontrivial periodic solution (corresponding to unstable LCO) appears and the system will have two nontrivial stable and unstable solutions for a1 and a2 (region III). In such a case, if the initial states of the system lie near the attraction domain of the stable solutions, the

Fig. 8. (a) Variation of the first mode resonance amplitude, a1 , versus the dynamic pressure and (b) variation of the second mode resonance amplitude, a2 , against the dynamic pressure for four different values of Rx1 ¼ 0:2, Rx1 ¼ 0:24, Rx1 ¼ 0:28 and Rx1 ¼ 0:32.

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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system responses will be attracted to these solutions and the parametric resonance of the system will be excited. In Fig. 5(c) and (d), the frequency-responses of the system corresponding to three values of forcing excitation amplitudes of Rx1 ¼ 0:6, Rx1 ¼ 0:8 and Rx1 ¼ 1:0 are shown. It is concluded that increasing Rx1 has an destabilizing effect and makes the first and the second bifurcation points to shift far apart from each other and the width of the parametric resonance to be extended. Moreover, it makes an overall increase and/or decrease in the system stable responses. In the following, in the absence of the primary resonance (V AC ¼ 0 and σ 2 ε ¼ 0 ), the characteristic curves of the resonance amplitude, a1 , versus the forcing excitation amplitude, is presented in Fig. 6. One can find out that an increase in a forcing detuning parameter makes an overall increase in the first mode resonance amplitude and shifts the second subcritical bifurcation point to the right, however, its variation does not affect the width of the resonance region. It is noted that in such a case, the second mode amplitude will not be excited. In Fig. 7, the bifurcation diagrams are plotted for five values of the dynamic pressure of λ ¼ 0, λ ¼ 12, λ ¼ 26, λ ¼ 42 and λ ¼ 59. As observed, an increase in dynamic pressure, decreases/increases the first/second mode amplitudes, and its variation shifts the first saddle node and the second subcritical bifurcation point to the right and causes the width of the resonance region of the fist mode to become narrow. This figure reveals that an increase in λ, increases the dynamic pressure, (i.e., aerodynamic effect will be dominant to the forcing excitation) which has a stabilizing effect on the first mod amplitude. In Fig. 8(a) and (b), the variation of the resonance amplitudes a1 and a2 are plotted against the dynamic pressure, λ, for four values of the forcing excitation amplitude, Rx1 . These figures illustrate that as Rx1 increases, the influence of excitation amplitude dominates to the dynamic pressure, which makes the width of the nontrivial solutions region becomes wider. The characteristic curve of frequency-response shown in Fig. 9(a) exhibits a typical hardening type hysteretic behavior of a nonlinear aeroelastic vibration of the FGP plate, under the simultaneous presence of the primary and the parametric resonances. According to this figure, when the piezoelectric excitation frequency, Ω2 , is sufficiently lower than the second

Fig. 9. (a) Frequency-response curve for the primary resonance case if the system, (b) frequency-response curves corresponding to the three different values of dynamic pressure, and (c) frequency-response curves corresponding to the five different values of the excitation voltage.

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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mode natural frequency, ω2 , (point A in region I), there is only a stable non-trivial solution, and under this condition, the system resonates with small amplitude. As the piezoelectric detuning parameter,σ 2 ε, is gradually increased from point A, the resonance amplitude tends to increase slowly. When the detuning parameter approaches the second natural frequency, i.e., σ 2 ε ¼ 0, the amplitude of the stable non-trivial response increases rapidly until it reaches point C. After this point, a

Fig. 10. Voltage-response curve of the FGP plate, in the case of primary piezoelectric resonance.

Fig. 11. (a) A chaotic bifurcation diagram of the maximum amplitude of the first mode amplitude, a1 , (b) a chaotic bifurcation diagram of the minimum amplitude of the first mode amplitude, a1 , (c) a chaotic bifurcation diagram of the maximum amplitude of the second mode amplitude, a2 , and (d) a chaotic bifurcation diagram of the minimum amplitude of the second mode amplitude, a2 .

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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branch representing the unstable non-trivial solution bifurcates toward the reverse direction. Such a saddle node bifurcation of the non-trivial solution leads to the well-known jump phenomenon, in which the resonance amplitude at point C rapidly decreases toward the other stable nonzero attractor (point C'). When the detuning parameter is increased further, the amplitude of the system response decreases slowly from point C0 to point D in region III. On the other hand, when the excitation frequency is gradually decreased from point D toward the back, the amplitude of the system response, a2 , increases continuously following the path of DC'B' until the amplitude reaches point B' (another saddle node bifurcation point), where it rapidly increases to a coexisting stable non-trivial solution (point B). After that, the amplitude decreases by further decreasing the detuning parameter until it reaches point A. As observed, for the primary resonance case, the aeroelastic FGP plate can have two catastrophic bifurcations (points C and B) and as it is seen in Fig. 9(a), depending on the path followed on the frequency response curve due to increasing/decreasing the detuning parameter, path ABCC'D/ABCC'D, different resonant behaviors may be exist. Such hysteretic characteristics near primary resonance condition, arise because of the existence of multiple coexisting non-trivial solutions. From Fig. 9(a), it is clearly visible that there exist two stable nontrivial solutions and one unstable non-trivial solution for resonance amplitude in the range of σ B ε o σε o σ C ε (region II). The system response in this bi-stable region can appear differently depending on the system initial conditions. Variation of the frequency response curves with respect to the variations of the dynamic pressure are depicted in Fig. 9 (b). As observed in this figure, the dynamic pressure influences the system behavior mainly in the region near the resonant frequency, in which, a decrease in the value of the dynamic pressure makes an overall increase in the amplitude of the system response and allows the bi-stable region to be extended. It means that, the bi-stable region, (width of the hysteretic resonant region), tends to be reduced as the dynamic pressure increases. Moreover, the jump up/down height decreases considerably with an increases in the dynamic pressure. Fig. 9(c) shows the variations of the frequency response curves against the variation of the dimensionless piezoelectric excitation voltage. This figure shows that, in the case of V AC ¼ 0:36, only a single stable non-trivial solution exists. In other words, when a relatively small excitation voltage is applied to the system, the response amplitude will be also small, as expected. But, when the excitation voltage is relatively large, two saddle node bifurcation points will be appear and form the bi-stable region. Also, increasing the excitation voltage makes an overall increase in the resonance amplitude, so that, the jump up/down height increases significantly with an increase in the amplitude of the excitation. In Fig. 10, the characteristic curves of the voltage-response are depicted for three values of the detuning parameter σ 2 ε ¼ 1:2, σ 2 ε ¼ 1:6 and σ 2 ε ¼ 2:0. One can find out that an increase in the detuning parameter, makes an overall increases in the system response and also it causes the first and the second bifurcation points to shift toward the right and far apart from each other, so that the bi-stable region, (region of the hysteretic behavior) tends to be increased as the detuning parameter increases, i.e., increasing the detuning parameter has destabilizing effect on system response.

Fig. 12. (a) Multi periodic time history of the first mode amplitude, (b) multi periodic time history of the second mode amplitude, (c) phase portrait trajectories of x2 against x1 , and (d) phase portrait trajectories of x4 against x3 .

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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Investigating the dynamic responses of the system in various conditions reveals that in absence the aerodynamic load which has stabilizing effect bellow the critical dynamic pressure, the system will show a complicated nonlinear dynamics including the multi-periodic and chaotic motions. In the following, the fourth-order Runge–Kutta algorithm is utilized to solve the averaged Eq. (34) numerically and the multi periodic and chaotic responses of the plate in the case of 1:2 internal resonance and primary parametric resonance are analyzed. In order to study the influence of the in-plane excitation Rx1 and piezoelectric voltage excitation V AC to the system behavior in the range of multi-periodic and chaotic responses, we choose Rx1 and V AC as the controlling parameters. Studying the bifurcation diagrams given in Fig. 11, in which the maximum and minimum amplitude of a1 and a2 are plotted against the forcing control parameter, Rx1 , illustrates that the multi-periodic and chaotic motions can be globally occurred in the spread range of the control parameters variations. So that by increasing the forcing amplitude, the nature of nonlinear response changes from multi-periodic motion to the chaotic motion and vice versa. Fig. 11 is derived by assuming

Fig. 13. (a) Chaotic time history of the first mode amplitude, (b) chaotic time history of the second mode amplitude, (c) phase portrait trajectories of x2 against x1 , and (d) phase portrait trajectories of x4 against x3 .

Fig. 14. A chaotic bifurcation diagram of the maximum and minimum amplitude of the first mode, a1 .

Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i

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the other parameters as λ ¼ 0, σ 1 ε ¼ 0:5, σ 2 ε ¼ 0:3 and V AC ¼ 2:1 under the initial conditions of x1 ð0Þ ¼ 0:12, x2 ð0Þ ¼ 0, x3 ð0Þ ¼ 0:0, and x4 ð0Þ ¼ 0. Fig. 12(a)–(d) exhibits the multi-periodic time responses and phase plane trajectories of the first and the second modes for a special point on Fig. 11, Rx1 ¼ 1:9, in which the multi-periodic behavior occurs, by assuming the abovementioned parameters and initial conditions. In Fig. 13(a)–(d), the variation of the resonance amplitudes, a1 ðT 1 Þ and a2 ðT 1 Þ are plotted versus the scaled time, T 1 , for the special case of Rx1 ¼ 1:5 and corresponding to the same initial conditions. As it is expected from Fig. 11, in such a case the plate resonance will have chaotic behavior. In Fig. 14, the bifurcation diagrams of the maximum and minimum amplitude of a1 is illustrated against the voltage control parameter by considering the other parameters as λ ¼ 0, σ 1 ε ¼ 0:5, σ 2 ε ¼ 0:3 and Rx1 ¼ 4, and choosing the initial conditions as x1 ð0Þ ¼ 0:2, x2 ð0Þ ¼ 0:2, x3 ð0Þ ¼ 0:2, and x4 ð0Þ ¼  0:2. 8. Conclusions In this research, the nonlinear transverse vibration of a simply supported FGP plate which is simultaneously is subjected to harmonic uniaxial in-plane force and piezoelectric out-of-plane excitation and aerodynamic loading is investigated. Based on the CLPT, von-Karman type relations and the Hamilton's principle, the equations of motion in the presence of the geometrical nonlinearities, structural inhomogeneity and supersonic aerodynamic loadings are derived. By employing the Galerkin's approach, the governing equation are reduced to nonlinear ordinary differential equations corresponding to a two-degree-of-freedom coupled nonlinear system under the parametric and primary excitations. Then, utilizing the Multiple-scales method, the frequency, forcing, voltage and dynamic pressure response equations are derived and solved numerically. In the steady-state condition, the nonlinear trivial/nontrivial solutions are obtained and their stability are determined. The bifurcation diagrams are provided, and influences of the forcing excitation, the dynamic pressure, the voltage excitation and the detuning parameters on the parametric/primary resonance of the system are studied. It is found that, when the system is excited parametrically and the forcing detuning parameter lies in the special range, the plate resonance will be excited. Also it is shown that in part I of the resonance region, the system has two nontrivial stable and unstable solutions and in the other part, it only has one nontrivial stable solution. Also, it is shown that when the system is excited primarily and the voltage detuning parameter lies in a certain range, the system may have multiple coexisting stable non-trivial solutions and the system will show the catastrophic hysteretic resonant behavior. Additionally, we showed that (a) In the case of forcing excitation, an increase in the parametric excitation amplitude shifts the bifurcation points of the characteristic curves, in which the resonance amplitude is plotted versus the dynamic pressure, to the right side and makes the width of resonance region to become larger. (b) In the presence of low supersonic aerodynamic loading and in absence of piezoelectric excitation, an increase in the forcing detuning parameter increases the resonance amplitude and does not influence the resonance region width. (c) In the forcing-response plots, an increase in the dynamic pressure shifts the bifurcation points of the characteristic curves to the right side and decreases the width of the resonance region and has a stabilizing effect. (d) In the frequency-response plots, in which the primary resonance amplitude is plotted versus the piezoelectric detuning parameter, an increase in the dynamic pressure makes an overall decrease in the amplitude of the system response and causes the hysteretic behavior region to become narrow. (e) In the frequency-response curves, in which the primary resonance amplitude is plotted versus the piezoelectric detuning parameter, an increase in the amplitude of the voltage excitation makes an overall increase in the amplitude of the system response and causes the hysteretic behavior region to be extended. (f) In the voltage-response plots, increasing the piezoelectric detuning parameter has a destabilizing effect and shifts the bifurcation points of the characteristic curves to the right side and causes an overall increase in the amplitude of the system response. (g) In absence the aerodynamic load which has stabilizing effect bellow the critical dynamic pressure, the system will show a complicated nonlinear dynamics including the multi-periodic and chaotic motions.

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Please cite this article as: M. Rezaee, & R. Jahangiri, Nonlinear and chaotic vibration and stability analysis of an aeroelastic piezoelectric FG plate under parametric and primary excitations, Journal of Sound and Vibration (2015), http://dx.doi. org/10.1016/j.jsv.2015.01.025i