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The nonlinear subharmonic resonance of an orthotropic rectangular laminated composite plate is studied. Based on the theory of high-order shear laminates, von Karman's geometric relation for the large deformation of plates, and Hamilton's principle, the nonlinear dynamic equations of a rectangular, orthotropic composite laminated plate subjected to the transverse harmonic excitation are established. According to the displacement boundary conditions, the modal functions that satisfy the boundary conditions of the rectangular plate are selected. The two-degree-of-freedom ordinary differential equations that describe the vibration of the rectangular plate are obtained by the Galerkin method. The multiscale method is used to obtain an approximate solution to the resonance problem. Both the amplitude-frequency equation and the average equations in the Cartesian coordinate form are obtained. The amplitude-frequency curves, bifurcation diagrams, phase diagrams, and time history diagrams of the rectangular plate under different parameters are obtained numerically. The influence of relevant parameters, such as excitation amplitude, tuning parameter, and damping coefficient, on the nonlinear dynamic response of the system is analyzed.

Composite laminates have many advantages, such as high specific strength, high specific stiffness, and good fatigue resistance. Because laminated materials are often made into thin-walled structures, they are prone to large deformation under various external loads, resulting in nonlinear dynamic characteristics that exhibit complex geometries. Under certain excitation conditions, harmonic resonance may occur, which also has a significant impact on the accuracy of the structure. Therefore, it is necessary to analyze the nonlinear harmonic resonance characteristics of laminated composite plates.

Although the nonlinear vibration characteristics of composite structures have been studied for many years, research on the free vibration of laminated plates and shell structure still retains the attention of many scholars. Wang et al. [

Analytical and numerical methods have been widely used in the study of the dynamic characteristics of composite structures [

Nonlinear vibration of the laminated plates exhibits different characteristics with the change in boundary conditions. Studies on thin plates having different boundary conditions provide the following results. An analysis of the nonlinear dynamics of a clamped-clamped FGM circular cylindrical shell subjected to an external excitation and uniform temperature change was presented by Zhang et al. [

Internal resonance is also unique to nonlinear systems and is different from linear systems. Internal resonance will occur when the two natural frequencies of the system satisfy a certain relationship. The unique internal resonance phenomenon of the nonlinear system will excite the original nonexcited modes due to the energy transfer between the modes. Nayfeh and Mook [

Secondary resonance is a phenomenon particular to the nonlinear system, which includes superharmonic and subharmonic resonance. Many scholars have studied the secondary resonance of nonlinear systems. The subharmonic resonance of FGM truncated conical shell under aerodynamics and in-plane force is investigated by the method of multiple scales by Yang et al. [

In this study, the subharmonic resonance characteristics of a two-degree-of-freedom laminated composite plate subjected to transverse harmonic excitations are investigated. The innovation of this paper lies in that the nonlinear vibration modeling of the thin plates with arbitrary boundary shapes and boundary conditions and the nonlinear vibration of the plates under different boundary conditions can be studied by assuming the corresponding mode function. The rectangular plate with simply supported boundary condition studied in this study is only a specific case when the boundary shape is determined and the boundary is acted on by no external force. In the absence of internal resonance, the low-order and high-order modes are uncoupled, and so they are studied separately. Based on the theory of higher-order shear deformation plate and von Karman’s geometric relationship, the nonlinear dynamic equations are established by using Hamilton’s principle. The ordinary differential equations for the vibration of the rectangular plate were derived by two-order discretization using the Galerkin method. The multiscale method is applied to obtain an approximate solution to the resonance problem. Both the amplitude-frequency response equation and the average equations in rectangular coordinates are obtained. In addition, the nonlinear dynamic responses of the two-order modes with system parameters are compared concretely.

The mechanical model of the special orthotropic symmetric rectangular laminated plate that is simply supported on four sides is shown in Figure

Mechanical model of a composite rectangular laminated plate.

The linear constitutive relation of each laminate is as follows:

Based on the higher-order shear deformation plate theory, the displacement fields are

According to the von Karman nonlinear geometric relation

For the assumed displacement field in (

The governing equations describing the vibration of rectangular plate are obtained by the Hamilton principle:

In this study, all the applied forces on the boundary are zero, That is to say,

Equations (

For the sake of convenience in writing, the transverse lines above the physical quantities are omitted. The boundary conditions of the simply supported plate can be expressed as

Due to the fact that the higher-order modes are not easily excited in structural vibration, the first two modes are taken for truncation analysis. Based on the displacement boundary conditions, the first two-order modal functions are selected as follows:

Since the out-of-plane vibration is dominant in the vibration system, the in-plane vibrations are ignored in this study. The inertia term is ignored and the modal functions (

The multiscale method is used for the approximate solution of the vibration equations. Firstly, the small parameter

The approximate solutions of (

The operators can be defined as

By substituting the expressions (

Order

The solutions of (

Subharmonic resonance occurs when the relationship between the excitation frequency and the first natural frequency of the system satisfies the relation

Substituting (

By substituting (_{2} = 0 into (

One can also write

The external excitation frequency and second natural frequency of the system satisfy the following relationship:

Similar to the first order, the amplitude-frequency response equation of subharmonic resonance for the second-mode mode is given by

The average equations in the form of rectangular coordinates of the subharmonic resonance for the second mode can be obtained as follows:

The values of parameters related to the laminated materials are, respectively,

Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes.

Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients.

Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies.

Figure

Figure

Figure

Based on the average (

Bifurcation diagram for the first-order mode via external excitation.

In Figure

Phase portrait on plane

Phase portrait on plane

Phase portrait on plane

Phase portrait on plane

Phase portrait on plane

Similarly, the Runge–Kutta algorithm is utilized for the numerical simulation of the average equations (

Bifurcation diagram for the second-order mode via external excitation.

Figure

Phase portrait on plane

Phase portrait on plane

Phase portrait on plane

In this work, the subharmonic resonance of a rectangular laminated plate under harmonic excitation is studied. The vibration equations of the system are established using von Karman’s nonlinear geometric relation and Hamilton's principle. The amplitude-frequency equations and the average equations in rectangular coordinates are obtained by using the multiscale method. The amplitude-frequency equations and average equations are numerically simulated in order to obtain the influence of system parameters on the nonlinear characteristics of vibration. The following conclusions can be obtained:

The results show that there are many similar characteristics in the nonlinear response of the uncoupled two-order modes, which are excited separately when subharmonic resonance occurs. Under the same amplitude of external excitation, the amplitudes of the two-order modes increase as the excitation frequency increases along with both the subharmonic domains. For the same excitation frequency, the steady-state solutions corresponding to the large amplitudes are almost independent of the external excitation. The difference is that the steady-state solution with smaller amplitude of the first-order mode increases with the increase in the excitation amplitude, while that of the second-order mode decreases with the increase in the excitation amplitude.

From the bifurcation diagram, it can be concluded that the system may experience many kinds of vibration states with the change in the amplitude of external excitation, such as quasiperiodic, periodic, and chaotic motion. The vibration of laminated plates can be controlled by adjusting the amplitude of the external excitation.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest.

The authors sincerely acknowledge the financial support of the National Natural Science Foundation of China (Grants nos. 11862020, 11962020, and 11402126) and the Inner Mongolia Natural Science Foundation (Grants nos. 2018LH01014 and 2019MS05065).