1 Introduction

In the mid-1980s, a new class of advanced composite materials, known as functionally gradient materials (FGMs), attracted much attention due to their excellent mechanical properties. The concept of FGMs was first introduced by a group of Japanese researchers who focused on the aerospace application^[1]. These composites are usually composed of two completely different phases such as metal and ceramic phases. In FGMs, the material distribution changes continually from the top surface to the bottom one. Compared with conventional composites, the improved aspects of FGMs are that they have smoother stress distribution, less stress concentration, and higher joint strength of different materials. These good aspects can meet the high demand in the modern engineering field including aerospace, mechanical engineering, civil engineering, and electronics.

As an important structural element, FGM plates play a significant role in practical engineering applications. Thus, dynamics analysis of this type of FGM structures is an integral part in structure design. Yas and Moloudi^[2] carried out a three-dimensional free vibration analysis on FGM piezoelectric annular plates. Using a four-variable plate theory, Hadji et al.^[3] analyzed the free vibration of FGM sandwich rectangular plates. Gupta et al.^[4] calculated natural frequencies of a shear deformable FGM plate under various boundary conditions with the finite element method. Utilizing the three-dimensional theory of elasticity, Jin et al.^[5] performed free vibration analysis of annular sector FGM plates. Based on the first-order shear deformation theory, Thai et al.^[6] studied the fundamental frequency of FGM sandwich plates. The frequency characteristics of FGM plates were investigated by Bernardo et al.^[7], who used a meshless method together with the finite element method to get the solutions. The above literature is related to linear dynamic analysis of FGM plates. A few studies have been carried out on the nonlinear dynamics of FGM plates. Among them, Zhang et al.^[8] researched the chaotic dynamics of simply-supported thick FGM plates utilizing the third-order shear deformation theory. Employing the Mindlin plate theory in conjunction with the modified couple stress theory, Ke et al.^[9] implemented the axisymmetric nonlinear free vibration analysis of FGM microplates. Alijani et al.^[10] adopted the Lagrange method together with the pseudo-arc-length continuation technique to obtain nonlinear dynamic response of FGM plates. Hao et al.^[11-12] studied periodic, quasi periodic, and chaotic vibrations of FGM plates under different boundary constrains. Using Reddy third-order shear deformation plate theory, Yang et al.^[13] investigated the nonlinear frequencies and transient response of cracked FGM plates. Recently, employing the classical plate theory, Wang and Zu^[14-15] analyzed the nonlinear steady-state responses and internal resonance of moving rectangular FGM plates coupled with fluid. They also carried out the dynamic thermoelastic analysis of moving rectangular FGM plates^[16].

Besides, longitudinally moving structures have been the center of attention of many researchers in the recent twenty years. Their application can be found in deployment of appendages in aerospace, transmission belts, satellite tethers, etc. Longitudinally moving beams, plates, and shells made of metal or laminated materials have been widely investigated^[17-29]. Ding and Chen^[30] reported an in-depth study on the natural frequencies of longitudinally moving beams by using the Galerkin method. With the consideration of longitudinal-transverse coupling, Yang and Zhang^[31] analyzed the nonlinear vibration of longitudinally moving beams. They gave a detailed study on the energy transfer mechanism in the structure. Recently, Wang et al.^[32-35] and Wang and Zu^[36-37] dealt with longitudinally moving plates coupled with fluid. A variety of dynamic characteristics of these coupled structures were presented including natural frequencies, mode functions, critical speed, parametric resonance, motion stability, and internal resonance.

The conception of sigmoid functionally graded materials (S-FGMs) was proposed by Chi and Chung^[38] in 2006. The merit of S-FGMs is that they can reduce stress concentration more effectively^[38]. For example, it is found that the usage of S-FGMs can dramatically reduce the stress intensity factors in a cracked structure^[39]. Bending and stress analyses of S-FGM plates have been performed by several researchers by use of various methods^[40-42]. Free vibration analysis of S-FGM beams with variable cross-sections was carried out by Atmane et al.^[43], who considered three different boundary conditions. Because of the complexity of constituent distribution of S-FGMs, no studies have been reported on nonlinear oscillations of S-FGM plates.

In this study, geometrically nonlinear oscillations of S-FGM plates with longitudinal speed are investigated. For the S-FGM plates, material properties are assumed to vary smoothly along the thickness direction and obey the sigmoid distribution rule. The von Kármán nonlinear plate theory is adopted to describe nonlinear geometrical relations. The nonlinear equation of motion is derived based on the D'Alembert's principle for moving S-FGM plates. Utilizing the Galerkin method, the equation of motion is discretized to a set of time-varying ordinary differential equations, which are then solved by using the method of harmonic balance. The obtained approximate analytical solutions are validated through the adaptive step-size fourth-order Runge-Kutta method. Besides, the stability of the steady-state solutions is examined. Finally, nonlinear frequency responses are presented for the system, and the effects of different parameters are highlighted.

2 Problem statement and formulation

Consider a fully simply-supported rectangular S-FGM plate moving in the x-axis at a constant velocity V. The plate has a length a, width b, and a thickness h. A Cartesian coordinate system (O, x, y, z) is used, and the origin O is set at one of the plate corners, as shown in Fig. 1. The S-FGM plate is a mixture of metal and ceramic. The material composition changes smoothly from pure metal on the top surface to pure ceramic at the bottom surface of the plate. Displacement components of points of the mid-plane of the plate are denoted by u, v, and w in the x-, y-, and z-directions, respectively. Besides, there is a pretension force per unit width N₀ applied on the plate in the x-axis.

In accordance with the rule of mixture, the effective material properties of S-FGM plates can be written as

(1)

where P_m and P_c are, respectively, the material properties of metal and ceramic, and V_m and V_c are volume fractions of metal and ceramic, respectively.

The volume fraction relations of constituents are given by

(2)

For the purpose of realizing the smooth distribution of stresses, Chi and Chung^[38] proposed an S-FGM, the volume fraction of which is defined as

(3)

in which N∈ [0, ∞) denotes the power-law index, and V_ci (i=1, 2) denote the volume fractions of ceramic in different areas.

Accordingly, the general Young's modulus E_i, the Poisson ratio υ_i, and the mass density ρ_i (i=1, 2) of the S-FGM plate take the form of

(4)

(5)

(6)

The S-FGM is made of alumina (ceramic) and nickel (metal). Their material properties at normal temperature are listed in Table 1.

In order to show the mechanical characteristics of the S-FGM plate, the change rule of Young's moduli against the plate thickness is given in Fig. 2. Three constituent volume fractions are considered, namely, N=1, 3, and 6. It is seen that Young's moduli exhibit the sigmoid distribution rule. Therefore, this type of FGMs is termed as S-FGM.

According to the classical thin plate theory, the strain relations of a thin S-FGM plate are given by^[44]

(7)

(8)

(9)

where ε_x, ε_y, and γ_xy are strain components of arbitrary points of the plate, χ_x, χ_y, and χ_xy are the changes in the curvature and torsion of the middle surface, ε_x⁰, ε_y⁰, and γ_xy⁰ are the middle surface strains, and z is the distance of an arbitrary point to the middle surface of the plate.

Based on the von Kármán nonlinear plate theory, the strain-displacement relations are stated as^[44]

(10)

(11)

(12)

(13)

(14)

(15)

The stress-strain relations for an S-FGM plate, under the plane stress condition, are given by

(16)

where σ_x, σ_y, and τ_xy are the in-plane stress components, and Q_jk^(ⅰ) (i=1, 2;j, k=1, 2, 6) define the reduced stiffnesses in the following form for an isotropic material (i=1 for 0≤z≤h/2; i=2 for -h/2≤z≤0) :

(17a)

(17b)

(18a)

(18b)

(19a)

(19b)

The internal force and moment of an S-FGM plate are given by

(20)

(21)

Substitution of (7) -(9) and (13) in (20) and (21) yields the constitutive equation

(22)

where N and ε are defined, respectively, as

(23)

(24)

and S takes the form

(25)

in which A_jk, B_jk, and D_jk (j, k=1, 2, 6) denote, respectively, the extensional, coupling, and bending stiffness coeffcients. They are calculated by

(26)

Using the D'Alembert's principle, the nonlinear equation of motion describing out-of-plane vibration of a longitudinally moving S-FGM plate is derived as

(27)

where c denotes the damping coefficient, and the total derivative in the first two terms has the following form due to the longitudinal speed:

(28)

In (27), F(x, y, t) denotes the transverse load which is considered as the practical point excitation taking the form^[45-49]

(29)

where F₀ is the force amplitude, δ is the Dirac delta function, ω is the circular frequency of the force, and x₀ and y₀ are the coordinates in the x-and y-directions, respectively. In this study, the excitation is applied at the center of the plate.

Applying (16), (20), (21), (28), and (29) in (27) yields the equation of motion in term of transverse displacement,

(30)

where the normal and shear stresses resulting from the in-plane deformation are neglected due to the fact that the natural frequencies of in-plane oscillation are much higher than those of out-of-plane oscillation, as reported in Ref. [50].

3 Solution techniques

In this study, the first two modes are considered, and the displacement function satisfying exactly the simply-supported boundary condition is written as

(31)

where A_{1, 1}(t) and A_{2, 1}(t) are functions of time t and denote the generalized coordinates of the first two modes.

The Galerkin method is used to discretize the governing equation to a set of ordinary differential equations. The weight function is given by

(32)

Mathematically, the Galerkin program is stated as

(33)

where Φ₍₃₀₎ denotes the equation of motion (30).

Thus, we obtain the nonlinear ordinary differential equations with respect to the generalized coordinates,

(34)

(35)

in which the over-dot stands for derivative with respect to t, and M_i and K_j(i=1, 2, …, 10; j=1, 2, …, 8) denote the integral coefficients as the function of geometrical and material parameters of the system.

To simplify the calculation, let us employ the following dimensionless variables:

(36)

where ω_{1, 1} denotes the fundamental natural frequency of the longitudinally moving S-FGM plate.

Substituting (36) into (34) yields the following dimensionless equations:

(37)

(38)

where M_i and K_j (i=1, 2, …, 10; j=1, 2, …, 8) denote new integral coefficients deriving from the above dimensionless transformation.

Using the harmonic balance method, we write the solutions to (37) and (38) in the following truncated Fourier series forms:

(39)

(40)

in which A_n and B_n (n=0, 1, …, H) stand for the Fourier coefficients, and H is the total of harmonics contained in the truncated Fourier series.

Applying (39) and (40) in (37) and (38) and then gathering each harmonic component including cos (nΩτ) and sin(nΩτ) (n=0, 1, …, H) in the resulting equations, we can obtain 4H+2 algebraic equations related to the Fourier coefficients A_n and B_n (n=0, 1, …, H). These equations are stated for H=1 as

(41)

where F_i(i=1, 2, …, 6) stand for algebraic expressions associated to unknowns A_i, B_i (i=0, 1, 2), and Ω. From (41), A_i and B_i can be obtained for a given Ω. Therefore, we can obtain the solutions of q₁ and q₂ with the use of (39) and (40).

4 Stability of steady-state solutions

In order to distinguish the stable and unstable responses in the analytical solutions, let us introduce the perturbation terms as follows:

(42)

where ΔA_i(τ) and ΔB_i(τ)(i=0, 1, 2) denote the perturbations of steady-state solutions.

Applying (42) in (37) and (38) and then gathering the harmonic components containing trigonometric functions cos (nΩτ) and sin (nΩτ) (n=0, 1), we can get a set of perturbation equations with respect to perturbation terms,

(43)

where

(44)

(45)

(46)

Implementing the Taylor series expansion for ψ at X=0 leads to

(47)

where P denotes the Jacobian matrix of the function ψ calculated in X=0.

If all the eigenvalues of the Jacobian matrix have negative real part, the corresponding steady-state responses are stable. Otherwise, if the eigenvalues of the Jacobian matrix have at least one positive real part, the corresponding steady-state responses are unstable.

5 Results and discussion

In order to validate the present model, we first consider a simply-supported rectangular homogeneous plate with the following geometrical and material parameters: a=0. 515 m, b=0.184 m, h=0. 000 3 m, E=69× 10⁹ Pa, ρ =2 700 kg/m³, and υ =0. 33. This plate has been investigated by Amabili^[51]. The natural frequencies of the homogeneous plate can be obtained by neglecting the nonlinear, damping, and external excitation terms in (34) and (35). In Table 2, the results obtained are listed together with those reported in Ref. [51], where m and n are the half-wave numbers in the length and width directions, respectively. It is seen that the present results agree well with those in the literature.

Then, we focus on a simply-supported S-FGM plate with the longitudinal speed. The S-FGM plate is made of nickel and alumina, and the geometrical parameters of the plate are the length a=0. 4 m, the width b=0. 1 m, and the thickness h=0. 001 m. This is a thin plate because the width-to-thickness ratio is b/h=100.

Figure 3 demonstrates the frequency-response relationships of the moving S-FGM plate in the neighborhood of the fundamental frequency, where V=25 m/s, N₀=1 000 N/m, F₀=5 N, c=30N·s/m³, and N=3. The maximum amplitudes of the first two generalized coordinates during the vibration period are shown. The stable and unstable responses are denoted by solid and dashed lines, respectively. It is seen that the system exhibits relatively complex frequency-response relationships. There are two peaks on the frequency-response curves of each generalized coordinate, which are different from the normal frequency response. This phenomenon indicates that there exists nonlinear mode interaction between the two modes so that an extra peak emerges in each mode. Furthermore, there are three stable branches on the frequency-response curves of each generalized coordinate, which are denoted by A, B, and C, as shown in the figure. It is also found that the nonlinear system possesses multi-solution property. At the beginning, the solutions are on the branch A of both modes. As the excitation frequency increases, the solution exhibits jump phenomenon. The solution may jump from the branch A to the branch B or C, which depends on the initial condition of the system. Additionally, it is certified that when the solution of mode q₁ is on the branch B (see Fig. 3(a)), the solution of mode q₂ is also on the branch B (see Fig. 3(b)). Another stable branch C of each mode also has this corresponding relationship.

To validate the analytical technique developed in this paper, the adaptive step-size fourth-order Runge-Kutta method is used to seek numerical solutions to the problem. Setting the initial condition as the numerical solutions can be gained via direct integration of ordinary differential equations (37) and (38). The results obtained by the approximate analytical method and those by the Runge-Kutta method are shown in Fig. 4. As can be seen, they match well with each other, which means that the method developed in the present study is accurate.

Figure 5 gives the time histories of modes q₁ and q₂, where the excitation frequency is Ω =2. 172 8 near the first peak in Fig. 3. At the beginning, the amplitude of the first mode q₁ increases, while that of the second mode q₂ decreases with the increasing time, which shows that the system energy transfers from one mode to the other. Specially, when the amplitude of the first mode q₁ gets to the maximum, the amplitude of the second mode q₂ reduces to zero, and vice versa. Overall, the time histories exhibit the energy transfer rule between the two modes.

Figures 6 and 7 show the frequency-response relationships of the moving S-FGM plate with excitation amplitudes F₀=2 N and F₀=8 N, respectively. The other system parameters are the same as those in Fig. 3. It should be noted that these curves also contain stable and unstable solutions. However, our concentration here is on the general characteristics. Thus, the stability is not shown for brevity. From Fig. 3 (F₀ =5 N), Fig. 6, and Fig. 7, we may find that as the excitation amplitude increases, the hardening spring characteristics of the system strengthen, and the resonance domain and resonant amplitude of each mode enlarge accordingly. In addition, it is seen that the extra peak of each generalized coordinate is not obvious when F₀=2 N (see Fig. 6), showing that the mode interaction cannot appear under relatively small external excitation. There is a threshold level, below which the mode interaction cannot be motivated. In fact, this is a nonlinear phenomenon that is not observed under quite small external forces.

Figures 8 and 9 are the frequency-response relationships of the moving S-FGM plate with in-plane pretension forces N₀=6 000 N/m and N₀=50 000 N/m, respectively. The other parameters are kept the same as those in Fig. 3. Through the comparison of Figs. 3, 8, and 9, it is noticed that when the in-plane pretension force is in a low-value range, the frequency-response relationships change insignificantly. However, when the in-plane pretension force is large enough, for example, N₀=50 000 N/m, the extra peaks of the coordinates appear late and their resonance domains reduce. Additionally, an obvious shrink of the extra peak in the first generalized coordinate can be observed.

In Figs. 10 and 11, the effect of speed is studied on nonlinear frequency-response relationships of the moving S-FGM plate. V=15 m/s and V=5 m/s are considered, and the other parameters are kept the same as those in Fig. 3. It is found that when the speed is small, i.e., V=5 m/s in Fig. 11, only one peak emerges in each mode, demonstrating that the nonlinear mode interaction does not take place and the two modes are uncoupled. The increase of the longitudinal speed brings occurence of nonlinear mode interaction between the two modes so that another peak emerges in each generalized coordinate, as shown in Figs. 3 and 10. In addition, comparing Fig. 3 (V=25 m/s), Fig. 10, and Fig. 11 reveals that the resonance domain of the system moves to the low-frequency region with the increase of speed. This is because the larger longitudinal speed leads to smaller natural frequencies of the moving plates^[34].

Figure 12 shows frequency-response relationships of the moving S-FGM plate with different power-law indexes. Three power-law indexes, i.e., N=1, N=3, and N=6, are incorporated. The parameters are taken as V=25 m/s, N₀= 1 000 N/m, F₀=5 N, and c=30 N· s/m³. It is seen that frequency-response curves change slightly when the power-law index varies from N=1 to N=3. However, when the power-law index varies to N=6, frequency-response curves change significantly. This indicates that large power-law index affects the vibration characteristics of the system more greatly than small power-law index. It is further found that the resonance domain of the system moves to the low-frequency region with the increase of power-law index. Moreover, the first peaks of the generalized coordinates magnify, but the second peaks shrink as the power-law index increases, showing the modulation effect of power-law index on the nonlinear mode interaction.

6 Conclusions

Geometrically nonlinear oscillations of S-FGM plates with a longitudinal speed are investigated. The material properties of S-FGM plates are assumed to vary smoothly along the thickness direction and obey the sigmoid distribution rule. Based on the D'Alembert's principle, the nonlinear equation of motion is derived for moving S-FGM plates, where the von Kármán nonlinear plate theory is adopted to describe nonlinear geometrical relations. Utilizing the Galerkin method, the equation of motion is discretized to a set of time-varying ordinary differential equations, which are then solved by using the method of harmonic balance. The obtained approximate analytical solutions are validated through the adaptive step-size fourth-order Runge-Kutta method. Besides, the stability of the steady-state solutions is examined. For the longitudinally moving S-FGM plate, nonlinear mode interaction can take place between the first two modes and two peaks emerge in the frequency-response curves. The nonlinear vibration characteristics show hardening-spring behavior of the system. Additionally, the excitation amplitude, the moving speed, and the in-plane pretention force can affect frequency-response characteristics of moving S-FGM plates. Specially, small excitation amplitude or longitudinal speed can eliminate nonlinear mode interaction phenomenon and make the lowest two modes uncoupled. Large power-law index affects vibration characteristics of S-FGM plates more greatly than small power-law index. Furthermore, the increase of power-law index makes the resonance domain move to the low-frequency region and changes the resonant amplitude of the moving S-FGM plates.