# Influence of Piezoelectric Performance on Nonlinear Dynamic Characteristics of MFC Shells

Authors: Xiangying Guo; Pan Jiang; Dongxing Cao
Journal: Complexity
 (2019)

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## Abstract

Based on the structures of unmanned aerial vehicle (UAV) wings, nonlinear dynamic analysis of macrofiber composite (MFC) laminated shells is presented in this paper. The effects of piezoelectric properties and aerodynamic forces on the dynamic stability of the MFC laminated shell are studied. Firstly, under the flow condition of ideal incompressible fluid, the thin airfoil theory is employed to calculate the effects of the mean camber line to obtain the circulation distribution of the wings in subsonic air flow. The steady aerodynamic lift on UAV wings is derived by using the Kutta–Joukowski lift theory. Then, considering the geometric nonlinearity and piezoelectric properties of the MFC material, the nonlinear dynamic model of the MFC laminated shell is established with Hamilton’s principles and the Galerkin method. Next, the effects of electric field, external excitation force, and nonlinear parameters on the stability of the system are studied under 1 : 1 internal resonance and the effects of material parameters on the natural frequency of the structure are also analyzed. Furthermore, the influence of the aerodynamic forces and electric field on the nonlinear dynamic responses of MFC laminated shells is discussed by numerical simulation. The results indicate that the electric field and external excitation have great influence on the structural dynamic responses.


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## Body

## 1. Introduction

MFC material, which was invented by NASA in 1996, has great application prospect in many engineering structures, especially in aviation and aerospace field. MFC materials are composed of two main parts: rectangular piezoceramic fibers and interdigitated electrodes. The sheet of aligned rectangular piezoceramic fibers is used to improve flexibility and damage tolerance in comparison with the traditional monolithic piezoceramic. Interdigitated electrode patterns are attached to the top and bottom of a polyamide film to permit in-plane poling and actuation of the piezoelectric fibers. MFCs mainly have two different types, namely,d31 and d33 modes, based on different laying directions of the piezoelectric fiber material.Because of the great potential of piezoelectric composite materials, piezoelectric materials become the most commonly used smart materials in active vibration and noise control, energy harvest, and so on. Tan et al. [1] studied the dynamic characteristics of a beam system with active piezoelectric fiber-reinforced composite layers. Then, more researches are reported on the MFC materials as sensors and actuators in different structures to adjust the deformation or vibration of the system, such as rotating composite thin-walled beams [2], thin beams [3], cylindrical shells [4], and smart composite plates [5].Recently, the dynamic behaviors of the classical MFC structure are attracting more and more scholars from all over the world. Park and Kim [6] investigated the material properties of MFCs by using classical lamination theory and uniform field model. Bilgen et al. [7] built a linear distributed parameter electromechanical model for frequency-response analysis of MFC actuated clamped-free thin beams and compared their results with experimental results. Cook and Vel [8, 9] considered different stresses of a simply supported laminated plate consisting of an MFC shear actuator sandwiched between graphite/polymer layers, which were subjected to an electric field perpendicular to the poling direction.The nonlinear dynamical simulations considering large displacements are taken into account for different MFC structures, such as circular plates [10], functionally graded plates [11], laminated composite plates [12], and thin-walled structures [13]. The effects of different parameters of the structures on the natural frequencies and vibration modes are discussed in these articles. Similarly, actuation properties of MFCs under strong voltages were investigated by Williams et al. [14] through using the theoretical piezoelectric constitutive model with higher-order electric field. Zhang and Shen [15] conducted the three-dimensional analysis for rectangular 1–3 piezoelectric fiber-reinforced composite laminates with the interdigitated electrodes under electromechanical loadings. Belouettar et al. [16] investigated active control of nonlinear vibrations of piezoelectric-elastic-piezoelectric sandwich beams using the method of harmonic balance.Rafiee et al. [17] investigated the nonlinear vibration and dynamic behavior of simply supported piezoelectric functionally graded shells under electrical, thermal, mechanical, and aerodynamic loadings. Hosseini et al. [18] analyzed the nonlinear free and forced vibrations of cantilever structures resting on a nonlinear elastic foundation with a piecewise piezoelectric actuator layer bonded on the top surface. The effects of various parameters on the free and forced nonlinear responses of the system were discussed. Mareishi et al. [19] considered the geometric nonlinearity of the piezoelectric fiber-reinforced laminated composite beams and analyzed the nonlinear frequencies of the beams with simply supported and clamped boundary conditions. Rafiee et al. [20] provided numerical simulation about the nonlinear dynamics of piezoelectric nanotubes/fibers/polymer multiscale composite plates, including the effects of different parameters of single-walled carbon nanotubes (SWCNTs) and multiwalled carbon nanotubes (MWCNTs) on the linear and nonlinear natural frequencies. Ninh and Bich [21] studied the electrothermal mechanical vibration of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) cylindrical shells by the numerical analytical method. Lu et al. [22] investigated the nonlinear dynamic characteristics of the time-varying piezoelectric laminated composite plate.In regard to the analysis methods for laminated composite plates with integrated piezoelectric actuators, several studies have been performed using the classical lamination theory [23], first-order shear deformation theory [24], higher-order theories [25], and the finite element method [26]. Moreover, Prasath and Arockiarajan [27, 28] studied the effect of bonding layer volume fraction on the effective thermo-electro-elastic constants of both d33- and d31-type MFCs by using the finite element method and experiments. Zhang et al. [29] investigated the structural deformation of composite laminated thin-walled structures bonded with orthotropic MFCs by establishing the finite element (FE) model based on linear piezoelectric constitutive equations.There are many researches on the aerodynamic force of different structures. Kouchakzadeh et al. [30] analyzed the aerodynamic modeling of structures by applying the classical plate theory along with the von Karman nonlinear strains and linear piston theory. Li et al. [31] used the piezoelectric material to increase the flutter velocities of the supersonic beams and adopted the supersonic piston theory to evaluate the aerodynamic pressure. Kuo [32] investigated the influence of variable fiber spacing on the supersonic flutter of rectangular composite plates and later also [33] discussed the effects of hybrid fiber distribution on the critical buckling temperature, natural frequencies, and flutter boundary of composite laminates by using the finite element method. Zhang et al. [34] considered the aerodynamics of a deploying wing in subsonic air flow and investigated the nonlinear dynamic behaviors of deploying wings in numerical simulations.Motivated by the above considerations, the nonlinear dynamic analysis of the MFC laminated shell subjected to aerodynamic force is presented here. The effects of piezoelectric properties and aerodynamic force on the dynamic stability of the structure are studied. Nonlinear dynamic equations of the cantilever MFC laminated shell are built based on UAV wings. The effect of different forces on the dynamic behaviors of MFC laminated shells is investigated in numerical simulation. Moderating effects of piezoelectric performance on the stability of the system are also presented here, which would provide guidance in controlling strategy of the nonlinear vibration for UAV wings.


## 2. Derivation of the Aerodynamic Force on the Deploying Wing

Considering the work situation of UAVs in subsonic air flow, the thin airfoil theory is applied here to calculate aerodynamic forces. Based on the thin airfoil theory, the potential function of the flow field can be divided into two parts: one is the potential function of the original uniform flow and the other is the disturbance potential function generated by the perturbation of the airfoil convection field, which also satisfies Laplace equations. Moreover, the perturbed potential function can be obtained according to the boundary condition of the typical airfoil and the infinite boundary condition of velocity approaching zero, as shown in Figure1.Figure 1
Normal velocity of the free stream in the middle curve.Therefore, the boundary condition can be given as follows:(1)α+VnV∞=dydx,where α is the angle of attack, Vn is the normal induced velocity, V∞ is the uniform flow velocity in direction α, and dy/dx means the slope at any point in the middle arc of the airfoil.Furthermore, the disturbance potential function equations and boundary conditions can be expressed in terms of the airfoil thickness and velocity potential caused by the camber curvature and attacked angle. When the curvature of the camber line is very small, the vorticity gradient in they direction is small with respect to a small camber or curvature, as shown in Figure 2.Figure 2
Circulation distribution along the airfoil.Therefore, the total circulation of the entire airfoil is expressed as follows:(2)Γ=∫0cγdξ,and the boundary condition is transformed into the following form:(3)α+1V∞∫0cγξdξ2πξ−x=dydx,where γξ can be expanded to a Fourier series of γθ:(4)γθ=2V∞A0cotθ2+∑1∞Ansinnθ.Moreover, the boundary conditions are obtained as follows through the transformationx=b/21−cosΘ:(5a)α−A0=1π∫0πdydxdΘ,(5b)An=2π∫0πdydxcosnΘdΘ.Therefore, the total circulation can be rewritten as(6)Γ=∫0cγxdx=V∞b∫0πA01+cosΘ+∑1∞AnsinnΘsinΘ=παbV∞,and the lift per unit wingspan is expressed as(7)L=παρbV∞2.Finally, the total lift on wingspan can be obtained according to the typical airfoil shape modeled in the following sections.


## 3. Mechanical Model

Generally, the induced strain of thed33 piezoelectric constant is larger than that of the d31 piezoelectric constant for MFC materials. Therefore, a d33 MFC laminated hyperbolic shell is considered here to establish the mechanical model of the high-aspect-ratio wings for UAVs. The cylindrical coordinate system is described here with curvatures α and β on the middle surface and perpendicular to the middle surface of the shell, as shown in Figure 3. The Cartesian coordinate system Oxyz is located in the tangent plane of the thin shell. Geometric dimensions of the shell are the lengths a and b and the thickness h, and the principal radii of the curvatures are R3 and R4. The displacements of an arbitrary point within the shell are expressed as u, v, and w, respectively. w is taken as a positive vector going outward from the center of the smallest radius of the curvature. The shell is subjected to the aerodynamic force as q=fcosΩ2t. A dynamic electric field is expressed as E=EcosΩ1t and applied in the longitudinal direction of the piezoelectric fibers, as shown in Figure 4.Figure 3
Model of the MFC thin shell.Figure 4
MFC-d33 material.All piezoelectric fibers are considered to be poled in theα and β directions, which can be assumed that out-of-plane electric fields vanish (that is, e33=0). Therefore, three sets of material coefficients are used to address the constitutive characteristics of the mechanical and electrical fields as well as the coupling between these fields, as follows:(8)σp=Cpqεq−ekpEk,Di=eiqεq+kikEk,where Ek is the electric field intensity, σp is the stress, εq is the strain, Cpq is the coefficient of elasticity, Di is the electric displacement, and ekp and kik represent the piezoelectric constants.Using the nonlinear von Karman’s geometric relationship for the thin shell, the strain can be expressed as(9)ε1=ε10+ηε11+η3ε12,ε2=ε20+ηε21+η3ε22,ε4=ε40+η2ε41,ε5=ε50+η2ε51,ε6=ε60+ηε61+η3ε62,where ε10,ε11,andε12 are described as(10)ε10=∂u0∂x+w0R3+12∂w0∂x2,ε11=∂ϕ1∂x,ε12=−c1∂ϕ1∂x+∂2w0∂x2−1R3∂u0∂x,ε20=∂v0∂y+w0R4+12∂w0∂y2,ε21=∂ϕ2∂y,ε22=−c1∂ϕ2∂y+∂2w0∂y2−1R4∂v0∂y,ε40=ϕ2+∂w0∂y−v0R4,ε41=−c2ϕ2+∂w0∂y−v0R4,ε50=ϕ1+∂w0∂x−u0R4,ε51=−c2ϕ1+∂w0∂x−u0R3,ε60=∂u0∂y+∂v0∂x+∂w0∂x∂w0∂y,ε61=∂ϕ1∂y+∂ϕ2∂x,ε62=−c1∂ϕ1∂y+∂ϕ2∂x+2∂2w0∂x∂y−1R3∂u0∂y−1R4∂v0∂x.The Lame coefficients of the shellsA1andA2 are expressed as(11)A1=a11+ηR3=Λ11,A2=a21+ηR4=Λ22,where ai1,2 is the surface tensor of the shell.The displacement of an arbitrary point in the composite shell can be expressed asR and be calculated as follows:(12)dR=Λ11dα+Λ22dβ+n^dη,where n^ is the unit vector perpendicular to the middle plane of the shell, which is expressed as follows:(13)n^=g1×g2a1a2.Here, gi1,2 is the vector tangent to the cylindrical coordinate axis.Since the following analysis is carried out in the Cartesian coordinate system, it needs the relations between the cylindrical coordinate system and the Cartesian coordinate system:(14)dxdydz=a1000a20001dαdβdη.The displacement field at an arbitrary point in the composite shell is given as follows based on Reddy’s third-order theory:(15a)u=u0x,y,t+zϕ1x,y,t−43h2z3ϕ1+∂w0∂α,(15b)v=v0x,y,t+zϕ2x,y,t−43h2z3ϕ2+∂w0∂β,(15c)w=w0x,y,t,where u0, v0, and w0 are the original displacements at the midplane of the MFC shell in the Cartesian coordinate directions and ϕ1 and ϕ2 represent the rotations of transverse normal at the midplane about the y and x axes.The aerodynamic force in the shell structure can be calculated as follows:(16)q=∫lLds=παρbV∞2a.Substituting these transformations into equations (15a)–(15c) and applying Hamilton’s principle, the nonlinear governing equations of motion in terms of generalized displacements for the MFC thin shell can be obtained as follows:(17a)−a11∂3w0∂x3+a12∂w0∂x∂2w0∂x2+a13∂w0∂y∂2w0∂x∂y+a14∂w0∂x∂2w0∂y2−a15∂3w0∂x∂y2+a16∂w0∂x+a17∂2u0∂x2+a18∂2u0∂y2+a19∂2v0∂x∂y+a20∂2ϕ1∂x2+a21∂2ϕ1∂y2+a22∂2ϕ2∂x∂y+a23u0+a24ϕ1−a25E1cosΩ1t=I0u¨0,(17b)−b11∂3w0∂y3+b12∂w0∂y∂2w0∂y2+b13∂w0∂x∂2w0∂x∂y+b14∂w0∂y∂2w0∂x2−b15∂3w0∂y∂x2+b16∂w0∂y+b17∂2v0∂x2+b18∂2v0∂y2+b19∂2u0∂x∂y+b20∂2ϕ2∂x2+b21∂2ϕ2∂y2+b22∂2ϕ1∂x∂y+b23v0+b24ϕ2−b25E2cosΩ1t=I0v¨0,(17c)−c11∂4w0∂x4−c12∂4w0∂y4−c13∂4w0∂x2∂y2+c14∂2w0∂x2+c15∂2w0∂y2+c16∂w0∂x2+c17∂w0∂y2+c18∂2w0∂x2∂v0∂x+c19∂2w0∂y2∂v0∂y+c20∂w0∂x∂2u0∂x2+c21∂w0∂y∂2v0∂y2+c22w0∂2w0∂x2+c23w0∂2w0∂y2+c24∂w0∂x2∂2w0∂x2+c25∂w0∂y2∂2w0∂y2+c26∂w0∂y2∂2w0∂x2+c27∂w0∂x2∂2w0∂y2+c28∂2w0∂x2∂v0∂y+c29∂2w0∂y2∂u0∂x+c30∂w0∂x∂2v0∂x∂y+c31∂w0∂y∂2u0∂x∂y+c32∂w0∂x∂w0∂y∂2w0∂x∂y+c33∂2w0∂x∂y∂v0∂x+c34∂2w0∂x∂y∂u0∂y+c35∂w0∂y∂2v0∂x2+c36∂w0∂x∂2u0∂y2−c37∂w0∂y∂3w0∂x2∂y−c38∂3w0∂x∂y2∂w0∂x+c39∂3u0∂x3+c40∂3v0∂y3−c41∂u0∂x−c42∂v0∂y+c43∂3u0∂x∂y2+c44∂3v0∂x2∂y+c45∂3ϕ1∂x3+c46∂3ϕ2∂y3+c47∂3ϕ1∂x∂y2+c48∂3ϕ2∂x2∂y+c49∂ϕ1∂x+c50∂ϕ2∂y−c51w0+fcosΩ2t−μw˙0=I0w¨0+c1I4−c13I6∂ϕ¨1∂x+∂ϕ¨2∂y+c13I6∂2w¨0∂x2+∂w¨0∂y2,(17d)d11∂3w0∂x3+d12∂3w0∂x∂y2+d13∂w0∂x∂2w0∂y2+d14∂w0∂y∂2w0∂x∂y+d15∂w0∂x+d16∂2u0∂x2+d17∂2u0∂y2+d18∂2v0∂x∂y+d19∂2ϕ1∂x2+d20∂2ϕ1∂y2+d21∂2ϕ2∂x∂y+d22u0+d23ϕ1+d24E1cosΩ1t=I2ϕ¨1−c1I4ϕ¨1−c1I4∂w¨0∂x+c1−I4ϕ¨1+c1I6ϕ¨1+c1I6∂w¨0∂x,(17e)e11∂3w0∂y3+e12∂3w0∂x2∂y+e13∂w0∂y∂2w0∂x2+e14∂w0∂x∂2w0∂x∂y+e15∂w0∂y+e16∂2v0∂x2+e17∂2v0∂y2+e18∂2u0∂x∂y+e19∂2ϕ2∂x2+e20∂2ϕ2∂y2+e21∂2ϕ1∂x∂y+e22v0+e23ϕ2+e24E2cosΩ1t=I2ϕ¨2−c1I4ϕ¨2−c1I4∂w¨0∂y+c1−I4ϕ¨2+c1I6ϕ¨2+c1I6∂w¨0∂y,where μ in equation (17c) is the damping coefficient.The boundary conditions of the cantilever thin shell are expressed as(18a)x=0:Nxy=Mxx=Mxy−c1Pxy=Q¯x=0,(18b)x=a:Nxy=Mxx=Mxy−c1Pxy=Q¯x=0,(18c)y=0:u0=v0=w0=ϕ1=ϕ2=0,(18d)y=b:Nyy=Nxy=Myy=Mxy−c1Pxy=Q¯y=0,(18e)x=0,a:∫−h/2h/2Nxxdz=±∫−h/2h/2fdz.


## 4. Effects of the Piezoelectric Parameters

Since the polarization of MFC-d33 is located in the plane, the vibration amplitudes of the shell in the x direction also need to be analyzed as those in the z direction. Here, the nonlinear coupled vibrations are considered between the in-plane and out-of-plane of the cantilever shell in the following studies. The displacements u0, v0, w0, ϕ1, and ϕ2, which satisfy the boundary conditions for the shell, are expressed as(19a)u0=u1sinπx2cosπy,v0=v1sinπx2sinπy,ϕ1=ϕ11sinπx2cosπy,ϕ2=ϕ211−cosπx2sinπy,w0=w1tXxYy,(19b)Xx=sinλ1x−sinhλ1x+α1coshλ1x−cosλ1x,Yy=31−2yb,(19c)cosλ1acoshλ1a+1=0.Then, taking all these derived expressions in equations (18a)–(18e) and (19a)–(19c) into equations (17a)–(17e) and applying the Galerkin procedure, two-degree-of-freedom nonlinear ordinary differential equations of the MFC laminated shell with dimensionless variables are obtained as follows:(20a)u¨1+ω12u1+α11w12+α12E1cosΩ1t=0,(20b)w¨1+μ1w˙1+ω22w1+α21u1+α22u1w1+α23w1E1cosΩ1t+α24w12+α25w13+α26E1cosΩ1t=α27fcosΩ2t.The structural properties are taken as follows: the shell has 7 laminae and the total thickness is0.002∗7m. The elastic constants of the fiber and matrix are 2.1×1011Pa and 3.5×109Pa, respectively. The destiny of the shell is 1.96kg/m3, and Poisson’s ratio is 0.36. The electric inductivity is 1296, and piezoelectric coefficients are d33=509pC/N and d31=−156pC/N.Firstly, the effects of the piezoelectric parameters on the deflectionK=x1/b of the shell are analyzed. Let voltage be 40 kV/cm along the y direction, the volume of the fiber content be 18%, and K be 15°. The radii R3andR4 are divided into two groups: (i) R3=1.5m and R4=1.5∼2.5m and (ii) R3=3m and R4=3∼5m. The relations between the radii of curvature R3 and R4 and K are described in Figure 5, which shows that the curves are nearly straight when R3andR4 increase. Therefore, the piezoelectric parameters would not affect the deflection of the thin shell directly.Figure 5
K-curve radius map.
(a)
(b)Then, the voltage is increased from 20 kV/cm to 60 kV/cm and the radii are set asR3=1.5m and R4=2m, as shown in Figure 6. It indicates that the deflection of the thin shell will be amplified with the increasing voltages. The deflection of the thin shell is also affected by the volume of macrofibers, as shown in Figure 7. The deflection K increases from 1.2 to 11.7 when the volume of macrofibers varies from 0 to 20%.Figure 6
K-voltage map.Figure 7
K-volume of macrofibers.It is summarized from the above results that the voltage and volume of macrofibers play an important role in the deflection of the MFC shell, which could adjust the natural frequency of the shell. It also indicts that voltage and volume of macrofibers are useful controlling parameters for the dynamic responses of the shell.


### 5. Nonlinear Characteristic Analysis under 1 : 1 Internal Resonance

Since the resonance of the system has great influence on the structural stability, the primary parameter resonance and 1 : 1 internal resonance are considered here, and the resonance relationships are expressed as follows:(21)ω1=Ω1−εσ1,ω2=Ω2−εσ2,Ω1=Ω2=1,where σ1 and σ2 are two detuning parameters, respectively.The solution of equations (20a) and (20b) can be written as follows according to the method of multiscale:(22a)u1=u10T0,T1+εu11T0,T1+⋯,(22b)w1=w10T0,T1+εw11T0,T1+⋯.Equations (21), (22a), and (22b) are introduced into equations (20a) and (20b) to obtain the following conditions.ε
0 order:(23a)D02u10+u10=0,(23b)D02w10+w10=0.ε
1 order:(24a)D02u11+u11=−2D0D1u10−α11w102−α12E1cosΩ1t+2σ1u10,(24b)D02w11+w11=−2D0D1w10−μ1D0w10−α21u10−α22u10w10−α23w10E1cosΩ1t+2σ2w10−α24w102−α25w103−α26E1cosΩ1t+α27FcosΩ2t.Assume that the solutions of equations (24a) and (24b) are as follows:(25a)u10=∏1T1eiT0+cc,(25b)w10=∏2T1eiT0+cc,where Π1=1/2ς1eiβ1 and Π2=1/2ς2eiβ2, in which ςii=1,2 is the amplitude of vibration and βii=1,2 is the initial phase.Substituting equations (25a) and (25b) into equations (24a) and (24b) and eliminating the long term, the average equations of the polar form are obtained as follows:(26a)ς˙1=−α122E1sinβ1,(26b)ς1β˙1=−α122E1cosβ1−ς1σ1,(26c)ς˙2=−12μς2−12α21ς1sinβ1−β2−12α27f−α26E1sinβ2,(26d)ς2β˙2=12α21ς1cosβ1−β2−ς2σ2+38α25ς23+12α27f−α26E1cosβ2.When the amplitude achieves a constant nontrivial value, a steady-state vibration exists. Therefore, let the left-hand side of equations (26a)–(26d) equals to zero and eliminate β1−β2 by using the relations between trigonometric functions, and then the frequency-response functions can be obtained:(27a)α1224E12=ς12σ12,(27b)−12μς2−12α27f−α26E1sinβ22+−ς2σ2+38α25ς23+12α27f−α26E1cosβ22=14α212ς12.Based on equations (27a) and (27b), the effects of electric field, transversal excitation, and nonlinear parameters (α25) on the nonlinear amplitudes are investigated by numerical simulation. Fixing the parameters of the shell as mentioned above, and after the dimensionless calculation, the following parameters are obtained: α12=6.7, α25=14.2, α26=9.9, α27=6.9, α28=12.5, and E1=20. Figure 8 illustrates that the resonance regions in the x direction increase with the increasing electric field. The resonance region moves to the left in the z direction and the resonance frequency decreases, as shown in Figure 9. Meanwhile, the system shows the hardening spring characteristic, as shown in Figure 9. Figure 10 expresses the relationship between the resonance region and the transversal excitation, which is similar to that shown in Figure 9. The nonlinear parameters can change the soft and hard spring properties of the system, as shown in Figure 11, and the hard spring characteristic is prominent with the nonlinear parameter increase.Figure 8
Frequency-response curves in thex direction with the electric field.Figure 9
Frequency-response curves in thez direction with the electric field.Figure 10
Frequency-response curves in thez direction with transversal excitation.Figure 11
Frequency-response curves in thez direction with the nonlinear parameter.


## 5. Nonlinear Characteristic Analysis under 1 : 1 Internal Resonance

Since the resonance of the system has great influence on the structural stability, the primary parameter resonance and 1 : 1 internal resonance are considered here, and the resonance relationships are expressed as follows:(21)ω1=Ω1−εσ1,ω2=Ω2−εσ2,Ω1=Ω2=1,where σ1 and σ2 are two detuning parameters, respectively.The solution of equations (20a) and (20b) can be written as follows according to the method of multiscale:(22a)u1=u10T0,T1+εu11T0,T1+⋯,(22b)w1=w10T0,T1+εw11T0,T1+⋯.Equations (21), (22a), and (22b) are introduced into equations (20a) and (20b) to obtain the following conditions.ε
0 order:(23a)D02u10+u10=0,(23b)D02w10+w10=0.ε
1 order:(24a)D02u11+u11=−2D0D1u10−α11w102−α12E1cosΩ1t+2σ1u10,(24b)D02w11+w11=−2D0D1w10−μ1D0w10−α21u10−α22u10w10−α23w10E1cosΩ1t+2σ2w10−α24w102−α25w103−α26E1cosΩ1t+α27FcosΩ2t.Assume that the solutions of equations (24a) and (24b) are as follows:(25a)u10=∏1T1eiT0+cc,(25b)w10=∏2T1eiT0+cc,where Π1=1/2ς1eiβ1 and Π2=1/2ς2eiβ2, in which ςii=1,2 is the amplitude of vibration and βii=1,2 is the initial phase.Substituting equations (25a) and (25b) into equations (24a) and (24b) and eliminating the long term, the average equations of the polar form are obtained as follows:(26a)ς˙1=−α122E1sinβ1,(26b)ς1β˙1=−α122E1cosβ1−ς1σ1,(26c)ς˙2=−12μς2−12α21ς1sinβ1−β2−12α27f−α26E1sinβ2,(26d)ς2β˙2=12α21ς1cosβ1−β2−ς2σ2+38α25ς23+12α27f−α26E1cosβ2.When the amplitude achieves a constant nontrivial value, a steady-state vibration exists. Therefore, let the left-hand side of equations (26a)–(26d) equals to zero and eliminate β1−β2 by using the relations between trigonometric functions, and then the frequency-response functions can be obtained:(27a)α1224E12=ς12σ12,(27b)−12μς2−12α27f−α26E1sinβ22+−ς2σ2+38α25ς23+12α27f−α26E1cosβ22=14α212ς12.Based on equations (27a) and (27b), the effects of electric field, transversal excitation, and nonlinear parameters (α25) on the nonlinear amplitudes are investigated by numerical simulation. Fixing the parameters of the shell as mentioned above, and after the dimensionless calculation, the following parameters are obtained: α12=6.7, α25=14.2, α26=9.9, α27=6.9, α28=12.5, and E1=20. Figure 8 illustrates that the resonance regions in the x direction increase with the increasing electric field. The resonance region moves to the left in the z direction and the resonance frequency decreases, as shown in Figure 9. Meanwhile, the system shows the hardening spring characteristic, as shown in Figure 9. Figure 10 expresses the relationship between the resonance region and the transversal excitation, which is similar to that shown in Figure 9. The nonlinear parameters can change the soft and hard spring properties of the system, as shown in Figure 11, and the hard spring characteristic is prominent with the nonlinear parameter increase.Figure 8
Frequency-response curves in thex direction with the electric field.Figure 9
Frequency-response curves in thez direction with the electric field.Figure 10
Frequency-response curves in thez direction with transversal excitation.Figure 11
Frequency-response curves in thez direction with the nonlinear parameter.


## 6. Nonlinear Dynamic Analysis

In this section, the nonlinear dynamic behavior of the MFC laminated thin shell subjected to the aerodynamic force is conducted. A series of numerical experiments are performed through the Runge–Kutta algorithm according to the nonlinear governing equations (20a) and (20b). After the dimensionless calculation of the structural parameters, the following parameters are obtained: α11=8.2, α12=6.7, α13=4.7, μ=0.5, α21=12.7, α22=5.2, α23=8.0, α24=12.0, α25=14.2, α26=9.9, α27=6.9, α28=12.5, E1=20, x1=0.8, x2=1.0, x3=0.2, and x4=0.9. With the disturbance force f increased from 0 to 300, the bifurcation diagrams of Poincare sections in the z direction for the displacements of the middle surface of the shell are shown in Figure 12. It is found that the nonlinear responses of the shell are very complex with the increasing disturbance force f. The instability of the structure would last a long time during the process as f increases from 0 to 180, and then after short windows of periodic n motions occurring in 180–230, the motions of the system become chaotic again through the path of periodic doubling bifurcation. Therefore, the motion form of the structure can be changed by controlling the amplitude of the external excitation in the resonance case.Figure 12
Bifurcation diagrams of Poincare sections. (a)x1−f. (b) x3−f.
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(b)To reveal the specific forms of different sections in the bifurcation diagram, phase portraits, power spectra, and waveforms of the shell are depicted as shown in the following figures. In Figures13–15, x1 and x2 represent the displacement and velocity in the x direction and x3 and x4 represent the displacement and velocity in the z direction. Furthermore, Figures 13(a), 14(a), and 15(a) and Figures 13(b), 14(b), and 15(b) show, respectively, the waveforms in the planes t,x1 and t,x3, and Figures 13(c), 14(c), and 15(c) and Figures 13(d), 14(d), and 15(d) show, respectively, the two-dimensional phase portraits in the planes x1,x2 and x3,x4. Figures 13(e), 14(e), and 15(e) show the three-dimensional phase portraits in the space x1,x2,x3, while Figures 13(f), 14(f), and 15(f) present the Poincare diagram. Figure 13 shows a group of characteristics of the chaotic motion for the system when f equals 100. It also indicates that there exists energy transform between the responses of the shell in two different directions in Figure 13(e), which is caused by the nonlinear coupled terms of equations (20a) and (20b).Figure 13
Chaotic motion of the composite shell whenf equals 100. (a) Waveform in the plane t,x1. (b) Waveform in the plane t,x3. (c) Two-dimensional phase portrait in the plane x1,x2. (d) Two-dimensional phase portrait in the plane x3,x4. (e) Three-dimensional phase portrait in the plane x1,x2,x3. (f) Poincare diagram.
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(b)
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(f)Figure 14
Periodic motion of the composite shell. (a) Waveform in the planet,x1. (b) Waveform in the plane t,x3. (c) Two-dimensional phase portrait in the plane x1,x2. (d) Two-dimensional phase portrait in the plane x3,x4. (e) Three-dimensional phase portrait in the plane x1,x2,x3. (f) Poincare diagram.
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(f)Figure 15
Chaotic motion of the composite shell whenf increases to 270. (a) Waveform in the plane t,x1. (b) Waveform in the plane t,x3. (c) Two-dimensional phase portrait in the plane x1,x2. (d) Two-dimensional phase portrait in the plane x3,x4. (e) Three-dimensional phase portrait in the plane x1,x2,x3. (f) Poincare diagram.
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(f)The periodicn motions of the system are shown in Figure 14 when f increases to 205, and the phenomenon of energy transition between two modes of the shell also exists in Figure 14(e).Then, when the aerodynamic disturbance force increases to 270, the system enters into chaos, and the specific shape of the response is shown in Figure15, which is similar to the chaotic motion in Figure 13.From the above analysis, it can be seen that the aerodynamic force plays a key role in the dynamic behavior of the structure. Then, the piezoelectric parameters are used to adjust the nonlinear responses of the system. Fixing the above parameters and letting the aerodynamic disturbance forcef=270, the bifurcation diagrams of Poincare sections are obtained in the x and z directions for the displacements of the middle surface of the shell when the electric field E increases from 0 to 40 in Figure 16. It is found that the motion of the system can be adjusted from chaos to the period when E is close to 20, and when the value of E is continuously increased, the system returns back to chaotic motion again.Figure 16
Bifurcation diagram of the composite shell with the electric field increase. (a)x1−E. (b) x3−E.
(a)
(b)From the above results of the numerical simulation, it is indicted that the piezoelectric parameters could adjust the vibration responses of the structure effectively. The electric field can change the piezoelectric performance of the shell through varying the stiffness of the structure, and the resonance of the shell would be restrained.


## 7. Conclusion

Considering the large geometrical deformation and piezoelectric material properties of the shell, nonlinear dynamic behaviors of a cantileverd33 MFC shell are investigated. The aerodynamical force and the electric field are introduced and calculated. Then, the Galerkin method is employed to transform the partial differential equations into two nonlinear ordinary differential equations. Next, the influence of the electric field, external excitation force, and nonlinear parameters on the stability of the system is analyzed under 1 : 1 internal resonance. Furthermore, the effects of the material parameters on the deflection are discussed, and the complex nonlinear vibration responses of the MFC shell are simulated, including the periodic and chaotic motions.This paper innovatively analyses the coupled vibration in two directions and points out that the energy transition exists between two coupled vibration directions. It is also revealed that the electric field of the MFC shell could adjust the dynamic stability of the structure from unstable to stable which would be an effective way to control responses for MFC structures. This research work may provide a way to use the MFC material in designing the wings of UAVs in the engineering field. The dynamic behavior of the wing in subsonic air flow conditions can be controlled by adjusting the electric field when the wing is made of the MFC material. Therefore, it could ensure the stability of the wing movement and the flight safety of the UAV.



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*Source: 1970248-2019-10-09.xml*