# Nonlinear Dynamic Analysis of Macrofiber Composites Laminated Shells

Authors: Xiangying Guo; Dameng Liu; Wei Zhang; Lin Sun; Shuping Chen
Journal: Advances in Materials Science and Engineering
 (2017)

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## Abstract

This work presents the nonlinear dynamical analysis of a multilayerd31 piezoelectric macrofiber composite (MFC) laminated shell. The effects of transverse excitations and piezoelectric properties on the dynamic stability of the structure are studied. Firstly, the nonlinear dynamic models of the MFC laminated shell are established. Based on known selected geometrical and material properties of its constituents, the electric field of MFC is presented. The vibration mode-shape functions are obtained according to the boundary conditions, and then the Galerkin method is employed to transform partial differential equations into two nonlinear ordinary differential equations. Next, the effects of the transverse excitations on the nonlinear vibration of MFC laminated shells are analyzed in numerical simulation and moderating effects of piezoelectric coefficients on the stability of the system are also presented here. Bifurcation diagram, two-dimensional and three-dimensional phase portraits, waveforms phases, and Poincare diagrams are shown to find different kinds of periodic and chaotic motions of MFC shells. The results indicate that piezoelectric parameters have strong effects on the vibration control of the MFC laminated shell.


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## Body

## 1. Introduction

MFC is a piezoelectric fiber material, consisting of monotonic piezoelectric material, epoxy matrix, and electrodes with a specific arrangement, which can be considered as homogenized orthotropic materials with arbitrary piezoelectric fiber angles like composite structures.There are generally three types of piezoelectric material which researches focus on mainly. The first type of the piezoelectric fiber composite material is produced by Smart Material Corp, treated as 1–3 composite. The second type is originally developed by MIT, named active fiber composite actuators. Macrofiber composite (MFC) is referred to as the third type presented by NASA Langley Research Center. Since several applications require conformable and packaged piezoelectric structures or actuators, MFC material become widely applied in both academic and industrial field, which would be used for an easier integration in smart, intelligent, or adaptive structures [1–3]. Great advantages of the MFC material, which is able to respond to changing environment and controlling structural deformation, have led to a new generation for aerospace structures especially for morphing aircrafts [4].Theoretical analyses are the basis to obtain dynamical characteristics of MFC material subjected to different excitations. In some researches, piezoelectric characteristics of MFC are studied by nonlinear constitutive equations and numerical simulation, including higher-order terms and corresponding coefficients [5–7] or by using strain dependent or effective piezoelectric coefficients in constitutive equations [8–11].Recently, different finite element models are developed to analyze the nonlinear dynamical behavior of MFC laminated materials [12–16]. Gohari et al. [17] present an analytical solution to obtain static deformation and optimal shape control of smart laminated cantilever piezoelectric composite hybrid plates and beams under several coupled loads using piezoelectric actuators. Park and Kim [18] show that the aerothermal large deflection of the composite panel can be suppressed using MFC actuators and the excessive actuation of the MFC can cause snap-through phenomena. Kim et al. [19] study the vibration suppression of an end-capped cylindrical shell structure with surface bonded macrofiber composite actuators and analyze the dynamic characteristics of the cylindrical shell structure.Dano and Jullière [20] investigate the use of macrofiber composite (MFC) actuators to actively control thermally induced deformations in composite structures. Korayem and Homayooni [21] analyze the influence of the applied voltage on vibration frequency of a multilayer piezoelectric microplate as well as the size effects on the macroplate at different boundary conditions. Li et al. [22] present the active control of random vibration for piezoelectric fiber reinforced composites laminated plates and discuss the effect of piezoelectric fiber orientation in the PFRC layers. Suresh Kumar and Ray [23] analyze the geometrically nonlinear vibrations of doubly curved smart sandwich shells integrated with a patch of 1–3 piezoelectric composites active constrained layer damping (ACLD) treatment.Experimental studies are effective in verifying theoretical results. The effective properties of piezoelectric composites on the interface material are reported by using theoretical and numerical methods in [24, 25]. Andrianov et al. [26] focus on the evaluation of the homogeneous properties of the active layers in both d31 and d33 MFCs. Biscani et al. [27] study the effective electromechanical properties of MFC transducers and compare the material properties of the system with experimental data. Prasath and Arockiarajan [28, 29] present total packing effects on the overall properties of MFC using experimental models. Based on the developed Kirchhoff plate theory, in some articles [30, 31], the actuation responses of d33 and d31 MFCs integrated smart structures are studied subjected to transverse excitations. Kashiwao et al. [32] propose an optimization method of a vibration energy harvesting system which is made of MFC piezoelectric elements.


## 2. Mechanical Model

MFC mainly consist of piezoceramic fibers, epoxy matrix, and electrodes, which have two different types of structures, namedd31 and d33 modes. Here, a d31 MFC middling thick cross-ply laminated shell is considered, where the piezomacrofiber is polarized in the thickness direction and fully embedded in the epoxy matrix as shown in Figure 1. The shell is a cantilever doubly curved cylindrical shell with a rectangular base. The orthogonal curvilinear coordinate is shown in Figure 2, and α and β curves are along the lines of curvatures on the middle surface and γ is perpendicular to the middle surface of the shell. Geometric dimensions of the shell in the curvilinear coordinate are the lengths a and b, and the thickness h, and the principal radii of the curvatures R1 and R2. The displacements of an arbitrary point within the shell in the coordinate (α,β,γ) are denoted with u, v, and w, respectively. w is taken as positive vector going outward from the center of the smallest radius of the curvature. The shell is subjected to a transverse excitation given as q=f0+f1cos⁡(Ωt), as shown in Figure 2.Figure 1
MFC structure.Figure 2
The model of MFC shell.According to Reddy’s third-order theory, the displacement fields at an arbitrary point in the composite shell are given in the following form:(1a)u=u0+γϕ1-43h2γ3ϕ1+∂w0∂α,(1b)v=v0+γϕ2-43h2γ3ϕ2+∂w0∂β,(1c)w=w0,where u0, v0, and w0 are the original displacements at the mid-plane of the MFC shell in the α,β,γ directions; ϕ1 and ϕ2 represent the rotations of transverse normal at the mid-plane about the α and β axes.Using von Karman’s geometric relationship, the strain and displacements of the MFC shell can be written as follows:(2)ε1=1A1∂u∂α+1a2∂a1∂βv+a1R1w,ε2=1A2∂v∂β+1a1∂a2∂αv+a2R2w,ε3=∂w∂γ,ε4=1A2∂w∂β+A2∂∂γvA2,ε5=1A1∂w∂α+A1∂∂γuA1,ε6=A2A1∂∂αvA2+A1A2∂∂βuA1.Here, A1, A2 are the Lame coefficients of the shell and can be expressed as A1=a11+γ/R1, A2=a21+γ/R2.The stress-strain relationships for the MFC material are given by(3a)σp=Cpqεq-ekpEk,(3b)Di=eiqεq+kikEk,where Ek is the electric field intensity and ekp represents the piezoelectric constant.All piezoelectric fibers are considered to be poled inγ (through-thickness) direction. Therefore, it can be assumed that in-plane electric fields vanish (i.e., E1=E2=0), and the following reduced constitutive equations hold:(4)σ1σ2σ3σ4σ5σ6=C11C12C13000C12C11C13000C13C13C13000000C44000000C44000000C11-C122ε1ε2ε3ε4ε5ε6+00-e3100-e3100-e330-e150-e1500000E1E2E3,where C11=E11/1-ν12ν21, C12=ν12E22/1-ν12ν21, and C44=G23, C66=G12, Eii are the stiffness modulus of the MFC shell, νij denote Poisson’s ratio, and Gij represent the shear modulus.The stiffness elements of the symmetric cross-ply composite laminated shell are expressed in terms of the stiffness coefficients as follows:(5)Aij,Bij,Dij,Eij,Fij,Hij=∑k=1N∫γkγk+1Cijk1,γ,γ2,γ3,γ4,γ6dγ,i,j=1,2,6,Aij,Dij,Fij=∑k=1N∫γkγk+1Cijk1,γ2,γ4dγ,i,j=4,5,Ii=∫-h/2h/2ργidγ,i=0,1,2,3,4,6.The relationship between curvilinear coordinate system and rectangular coordinate system isdx=a1dα, dy=a2dβ, dz=dγ. Substitute these transformations into (1a), (1b), and (1c) and then apply Hamilton’s principle; the nonlinear governing equations of motion in terms of generalized displacements for the MFC shell can be obtained as follows: (6a)a11∂2u0∂x2+a12∂2u0∂y2+a13∂2v0∂x∂y+a14∂u0∂x+a15∂v0∂x-a16∂u0∂y-a17∂v0∂y-a18∂3w0∂x2-a19∂3w0∂x∂y2-a20∂2w0∂x2+a21∂2w0∂y2-a22∂2w0∂x∂y+a23∂w0∂x+a24∂2ϕ1∂x2+a25∂2ϕ1∂y2+a26∂2ϕ2∂x∂y+a27∂ϕ1∂x+a28∂ϕ2∂x+a29∂ϕ1∂y+a30∂ϕ2∂y+a31u0+a32ϕ1=I0u¨0,(6b)b11∂2v0∂y2+b12∂2v0∂x2+b13∂2u0∂x∂y+b14∂v0∂y+b15∂u0∂y-b16∂v0∂x-b17∂u0∂x-b18∂3w0∂y3-b19∂3w0∂x2∂y-b20∂2w0∂y2+b21∂2w0∂x2-b22∂2w0∂x∂y+b23∂w0∂y+b24∂2ϕ2∂y2+b25∂2ϕ2∂x2+b26∂2ϕ1∂x∂y+b27∂ϕ2∂y+b28∂ϕ1∂y+b29∂ϕ1∂y+b30∂ϕ1∂x+b31v0+b32ϕ2=I0v¨0,(6c)-c11∂4w0∂x4-c12∂4w0∂y4-c13∂3w0∂x3-c14∂3w0∂y3-c15∂4w0∂x2∂y2+c16∂3w0∂x2∂y+c17∂3w0∂x∂y2+c18∂w0∂x2+c19∂2w0∂x2+c20∂w0∂y2+c21∂2w0∂y2+c22∂u0∂x+c23∂v0∂y+c24∂u0∂x∂2w0∂x2+c25∂2u0∂x2∂w0∂x+c26v0∂2w0∂x2+c27∂v0∂x∂w0∂x+c28w0∂2w0∂x2+c29∂v0∂y∂2w0∂x2+c30u0∂2w0∂x2+c31∂u0∂x∂w0∂x+c32∂v0∂x∂2w0∂x∂y+c33∂2v0∂x2∂w0∂y-c34u0∂2w0∂x∂y-c35∂u0∂x∂w0∂y+c36∂u0∂y∂2w0∂x∂y+c37∂2u0∂x∂y∂w0∂y-c38v0∂2w0∂x∂y-c39∂v0∂x∂w0∂y+c40∂2v0∂x∂y∂w0∂x-c41∂u0∂y∂w0∂x+c42∂2u0∂y2∂w0∂x-c43∂v0∂y∂w0∂x+c44∂u0∂x∂2w0∂y2+c45v0∂2w0∂y2+c46∂v0∂y∂w0∂y+c47w0∂2w0∂y2+c48∂v0∂y∂2w0∂y2+c49∂2v0∂2y∂w0∂y+c50u0∂2w0∂y2+c51∂u0∂y∂w0∂y+c52∂3ϕ1∂x3+c53∂3ϕ2∂y3+c54∂3ϕ1∂x2∂y+c55∂3ϕ2∂x∂y2+c56∂2ϕ2∂x2+c57∂2ϕ1∂x2+c58∂2ϕ2∂y2+c59∂2ϕ1∂y2+c60∂2ϕ1∂x∂y+c61∂2ϕ2∂x∂y-c62u0-c63v0-c64w0+fcos⁡Ωt-kw˙0=c1I4∂ϕ¨1∂x+∂ϕ¨2∂y-c12I6∂ϕ¨1∂x+∂ϕ¨2∂y+∂2w¨0∂x2+∂2w¨0∂y2+I0w¨0,(6d)d11∂2ϕ1∂x2+d12∂2ϕ1∂y2+d13∂ϕ1∂x+d14∂ϕ1∂y+d15∂ϕ2∂x+d16∂ϕ2∂y+d17∂2ϕ2∂x∂y+d18∂3w0∂x3+d19∂3w0∂x∂y2+d20∂2w0∂x2+d21∂2w0∂y2+d22∂2w0∂x∂y+d23∂w0∂x+d24u0+d25ϕ1=I2ϕ¨1-c1I4ϕ¨1-c1I4∂w¨0∂x+c1-I4ϕ¨1-c1I6ϕ¨1+c1I6∂w¨0∂x,(6e)e11∂2ϕ2∂y2+e12∂2ϕ2∂x2+e13∂ϕ2∂y+e14∂ϕ2∂x+e15∂ϕ1∂y+e16∂ϕ1∂x+e17∂2ϕ1∂x∂y+e18∂3w0∂y3+e19∂3w0∂x2∂y+e20∂2w0∂y2+e21∂2w0∂x2+e22∂2w0∂x∂y+e23∂w0∂y+e24v0+e25ϕ2=I2ϕ¨2-c1I4ϕ¨2-c1I4∂w¨0∂y+c1-I4ϕ¨2-c1I6ϕ¨2+c1I6∂w¨0∂y,where c1=4/3h2, c2=3c1, and μ in (6c) is the damping coefficient, and all the parameters of the specific expression can be found in the Appendix.The boundary conditions of the cantilever shell are expressed as(7a)x=0:Nxy=Mxx=Mxy-c1Pxy=Q-x=0,(7b)x=a:Nxy=Mxx=Mxy-c1Pxy=Q-x=0,(7c)y=0:u0=v0=w0=ϕ1=ϕ2=0,(7d)y=b:Nyy=Nxy=Myy=Mxy-c1Pxy=Q-y=0,(7e)∫-h/2h/2Nxxdz=±∫-h/2h/2qdz,x=0,a.


## 3. Perturbation Analysis

Since vibration amplitudes of the lower frequencies are much larger than that of the higher frequencies for the shell, the mainly dynamical damage or instability of structures is caused by resonances in lower frequencies. Here, the first two modes of the MFC laminated shell are considered. Thus, displacementsu0, v0, w0, φ1, and φ2, which satisfy the boundary conditions for the shell, are represented as(8a)u0=u1sin⁡πx2cos⁡πy+u2sin⁡3πx2cos⁡2πy,(8b)v0=v1sin⁡πx2sin⁡πy+v2sin⁡3πx2sin⁡2πy,(8c)ϕ1=ϕ11sin⁡πx2cos⁡πy+ϕ12sin⁡πxcos⁡2πy,(8d)φ2=φ211-cos⁡πx2sin⁡πy+φ221-cos⁡πx2sin⁡2πy,(8e)w0=w1tX1xY1y+w2tX2xY2y,where(9a)Xix=sin⁡λix-sinh⁡λix+αicosh⁡λix-cos⁡λix,(9b)Y1y=1,(9c)Y2y=31-2yb,(9d)cos⁡λiacosh⁡λia+1=0,(9e)cos⁡μjbcosh⁡μjb-1=0,(9f)αi=sinh⁡λia+sin⁡λiacosh⁡λia+cos⁡λia.To obtain the dimensionless equations, the transformation of variables and parameters are introduced as(10)u-0=u0Rh,v-0=v0Rh,w-0=w0R,φ-1=φ1,φ-2=φ2,x-=xRh,y-=yRh,z-=zh,R-1=R1R,R-2=R2R,I-i=1Li+1ρIi,F-=Rh7/2Eh7F,P-=b2El3P,μ-=1ρEa2b2h2μ,T-=1LEρT,A-=1ELA,B-=1EL2B,D-=1EL3D,E-=1EL4E,F-=1EL5F,H-=1EL7H,L=Rh.Then, taking all these derived expressions in (8a), (8b), (8c), (8d), and (8e)–(9a), (9b), (9c), (9d), (9e), and (9f) into (6a), (6b), (6c), (6d), and (6e) and applying the Galerkin procedure, a two-degree-of-freedom nonlinear ordinary differential equation of the MFC laminated shell with dimensionless is obtained as follows:(11a)w¨1+μw˙1+ω12w1+α11w12+α12w22+α13w1w2+α14w1cos⁡Ω2t+α15w2cos⁡Ω2t=α16fcos⁡Ω1t,(11b)w¨2+μw˙2+ω22w2+α21w12+α22w22+α23w1w2+α24w1cos⁡Ω2t+α25w2cos⁡Ω2t=α26fcos⁡Ω1t.Here, α14, α15, α24, and α25 present the piezoelectric coefficients of the shell.Considering the case of primary parametric and 1 : 1 internal resonance, the following relations can be established accordingly:(12)ω12=Ω12+εσ1,ω22=Ω12+εσ2,ω12≈ω22,Ω1=Ω2=1,where σ1 and σ2 are the detuning parameters.To study the oscillations and bifurcations of the established nonlinear system, the method of multiscale is a powerful tool to determine the solutions for conservative and nonconservative systems. Hence, this method is adopted to investigate the nonlinear vibration responses of the MFC shell.The uniformly approximate solutions of (11a) and (11b) are obtained as(13a)w1=w10T0,T1+εw11T0,T1+⋯,(13b)w2=w20T0,T1+εw21T0,T1+⋯,where T0=t and T1=εt.Then, the derivatives with respect tot become(14a)ddt=D0+εD1,(14b)d2dt2=D02+2εD0D1.Substituting (13a), (13b), (14a), and (14b) into (11a) and (11b) and equating the coefficients of ε to zero yield the following differential equations:ε0(15a)D0w10+Ω12w10=0,(15b)D0w20+Ω12w20=0,ε1(16a)D0w11+Ω12w11=-2D0D1w10-μD0w10-σ1w10-α11w102-α12w202-α13w10w20-α14w10cos⁡Ω2t-α15w20cos⁡Ω2t+α16fcos⁡Ω1t,(16b)D0w21+Ω12w21=-2D0D1w20-μD0w20-σ1w20-α21w102-α22w202-α23w10w20-α24w10cos⁡Ω2t-α25w20cos⁡Ω2t+α26fcos⁡Ω1t.The averaged equations in the Polar Coordinates form are obtained for the MFC laminated shell as follows:(17a)a˙1=-12μa1+12α16fsin⁡θ1,(17b)a1θ˙1=12σ1a1+14α14a1+14α15a2-12α16fcos⁡θ1,(17c)a˙2=-12μa2+12α26fsin⁡θ2,(17d)a2θ˙2=12σ2a2+14α24a1+14α25a2-12α26fcos⁡θ2.


## 4. Numerical Simulation

In this section, a series of numerical experiments are conducted for the nonlinear dynamical behavior of the MFC laminated shell. The nonlinear governing equations (17a), (17b), (17c), and (17d) are used to perform the numerical simulation through the Runge-Kutta algorithm. Whereas the transverse excitation is one of the most important factors on the nonlinear vibration of the system, the excitation f1 is chosen as a controlling parameter in the subsequent studies.Choose the following parameters:μ=0.6, α11=29.2, α12=-6.7, α13=10.7, α15=1.0, α16=25.3, α21=27.7, α22=5.2, α23=27.0, α24=1.0, α25=1.0, α26=6.9, x1=1.6, x2=0.3, x3=1.9,and x1=0.9. The frequency-response curves of the first-order and second-order are obtained through changing the initial condition of the MFC shell. The behavior shown in Figure 3 features the characteristics of soften-type systems with the piezoelectric coefficient α14 increased from 0.5 to 1.5. It is also shown that the stiffness would be decreased with the increase in α14.Figure 3
Response-frequency curves.
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(b)The bifurcation diagrams of Poincare sections for the displacements of the middle surface of the shell are shown in Figure4 with the transverse excitation increasing from 700 to 2000. The complex nonlinear vibration responses of the system are discussed corresponding to the above parameters. The system keeps the chaotic motion until f1 increased to 800, at which point period motions of the system occur. After short windows of period-n motions occur in 950–1000, the motions of the system dominantly enter into chaotic motions thereafter. Then the chaotic motion is observed again until the transverse excitation f1 equals 1500. The motion status of the MFC laminated shell alternately appears from unstable to stable and then again to unstable, which depends on the amplitude of the transverse excitation f1.Figure 4
The bifurcation diagram.
(a)
(b)To reveal the specific form of different sections in the bifurcation diagram, the response-frequency curves, phase portraits, power spectrums, and waveforms of the shell are depicted as shown in Figures5(a)–5(f). Diagrams (a) and (c) are, respectively, the wave forms on the planes t,x1 and t,x3, and Diagrams (b) and (d) are the two-dimensional phase portraits on the planes (x1,x2) and (x3,x4). Diagrams (e) show the three-dimensional phase portraits in the space (x1,x2,x3) while Diagrams (f) represent the Poincare diagram. The typical chaotic motion of the system is given as in Figure 5 when the amplitude of the transverse excitation arrives at 760.Figure 5
The chaotic motion of the shell.
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(b)
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(f)From the frequency-response curves of the system analyzed, it is known that the structure stiffness would alter when the piezoelectric coefficients of the system changed, so it would be effective to adjust the motion status of the MFC shell from unstable to stable through modulating the piezoelectric coefficients of the system. Fix the system parameters, based on which Figure5 is depicted, and only adjust the piezoelectric parameters α25 to perform numerical simulations and the results are shown in Figure 6. The bifurcation diagram reveals that the system would enter into periodic motions when α25 is increased from 0 to 1.6. The specific periodic motion form of the system is also shown in Figure 7 when α25 is equal to 2.Figure 6
The bifurcation diagram of the shell.
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(b)Figure 7
The periodic motion of the shell.
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(b)
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(f)In order to describe the influence of the piezoelectric parameters on the nonlinear vibrations of the system profoundly, another set of parameters is selected in the following:μ=0.6, α11=-5.3, α12=8.9, α13=26.4, α14=1.0, α15=1.0, α16=19.3, α21=-10.4, α22=7.4, α23=-22.4, α24=1.0, α25=1.0, α26=14.1, x1=2.6, x2=1.3, x3=4.5, and x1=5.1. It is found from the bifurcation diagram in Figure 8 that the response of the system varies from periodic motions to chaotic motions when the transversal excitation increases from 3 to 18. The different periodic and chaotic motions of the system are also given as in Figures 9-10, respectively, when f1 equals 5 and 12.Figure 8
The bifurcation diagram of the shell.
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(b)Figure 9
The periodic motion of the shell.
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(f)Figure 10
The chaotic motion of the shell.
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(f)Based on the motion of the system in Figure8, we only increase the piezoelectric parameter α25 to describe the vibration responses of the system. The bifurcation diagram of the system also shows the evolution from chaos to period-n and then back into chaos again as shown in Figure 11. It is also found that the system enters into periodic motion from chaotic motion when the piezoelectric parameter α25 increases from 1.0 to 2.0. Figure 12 shows the specific periodic motion of the system as α25 equals 1.5.Figure 11
The bifurcation diagram of the shell.
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(b)Figure 12
The periodic motion of the shell.
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(f)From the results of the numerical simulation, it is discovered that the piezoelectric parameters could adjust the vibration responses of the structure effectively. Since the piezoelectric parameters could vary the stiffness of the shell, it is generalized: once the inherent frequency of the system changes, the resonance would be restrained for piezoelectric structures.


## 5. Conclusion

In this paper, nonlinear dynamical behaviors of a simply supported cantilever MFC shell are presented, which is subjected to transverse loads. Based on known geometrical and material properties of its constituents, their electric field dependence is presented. The vibration mode-shape functions are obtained according to the boundary conditions, and then Galerkin method is employed to transform the partial differential equations into two nonlinear ordinary differential equations. The illustrative case of 1 : 1 internal resonances is considered. The externally excited system is transformed into a set of averaged equations by using the method of multiple scales. Next, the effect of the transverse excitations and the piezoelectric coefficients on the MFC laminated shell is described in numerical simulation.The results of the numerical simulation demonstrate the complex nonlinear vibration responses of the MFC shell that occurred under the transverse excitation, including the periodic and chaotic motions. The energy transformation existed between two resonances modes. The appropriate control technique of the forcing excitations contributes significantly to the responses of autonomous nonlinear systems. It is also revealed that the piezoelectric parameters of the structure could adjust the dynamic stability of the structure from unstable to stable which would be a good way to control vibration responses for MFC structures.



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