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This work presents the nonlinear dynamical analysis of a multilayer

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 [

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 [

Recently, different finite element models are developed to analyze the nonlinear dynamical behavior of MFC laminated materials [

Dano and Jullière [

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 [

MFC mainly consist of piezoceramic fibers, epoxy matrix, and electrodes, which have two different types of structures, named

MFC structure.

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:

Using von Karman’s geometric relationship, the strain and displacements of the MFC shell can be written as follows:

The stress-strain relationships for the MFC material are given by

All piezoelectric fibers are considered to be poled in

The stiffness elements of the symmetric cross-ply composite laminated shell are expressed in terms of the stiffness coefficients as follows:

The relationship between curvilinear coordinate system and rectangular coordinate system is

The boundary conditions of the cantilever shell are expressed as

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, displacements

To obtain the dimensionless equations, the transformation of variables and parameters are introduced as

Then, taking all these derived expressions in (

Considering the case of primary parametric and 1 : 1 internal resonance, the following relations can be established accordingly:

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 (

Then, the derivatives with respect to

Substituting (

The averaged equations in the Polar Coordinates form are obtained for the MFC laminated shell as follows:

In this section, a series of numerical experiments are conducted for the nonlinear dynamical behavior of the MFC laminated shell. The nonlinear governing equations (

Choose the following parameters:

Response-frequency curves.

The bifurcation diagrams of Poincare sections for the displacements of the middle surface of the shell are shown in Figure

The bifurcation diagram.

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 Figures

The chaotic motion of the shell.

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 Figure

The bifurcation diagram of the shell.

The periodic motion of the shell.

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:

The bifurcation diagram of the shell.

The periodic motion of the shell.

The chaotic motion of the shell.

Based on the motion of the system in Figure

The bifurcation diagram of the shell.

The periodic motion of the shell.

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.

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.

The authors declare that they have no conflicts of interest.

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant nos. 11572006, 11202009, 11072008, and 10732020 and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).