Three-Dimensional Solution for the Vibration Analysis of Functionally Graded Rectangular Plate with/without Cutouts Subject to General Boundary Conditions

Functionally graded materials (FGMs) structures are increasingly used in engineering due to their superior mechanical and material properties, and the FGMs plate with cutouts is a common structural form, but research on the vibration characteristics of FGMs plate with cutouts is relatively limited. In this paper, the three-dimensional exact solution for the vibration analysis of FGMs rectangular plate with circular cutouts subjected to general boundary conditions is presented based on the three-dimensional elasticity theory. The displacement field functions are expressed as standard cosine Fourier series plus auxiliary cosine series terms satisfying the boundary conditions in the global coordinate system. The plate with circular cutout is discretized into four curve quadrilateral sub-domains using the p-version method, and then the blending function method is applied to map the closed quadrilateral region to the computational space. The characteristic equation is obtained based on the Lagrangian energy principle and Rayleigh–Ritz method. The efficiency and reliability of proposed method are verified by comparing the present results with those available in the literature and FEM methods. Finally, a parametric study is investigated including the cutout sizes, the cutout positions, and the cutout numbers from the free vibration characteristic analysis and the harmonic analysis. The results can serve as benchmark data for other research on the vibration of FGMs plates with cutouts.


Introduction
The laminated composites are widely used in various engineering applicationssuch as aerospace, mechanical, civil, and automotive engineering-due to high specific strength and stiffness, light weight, and good thermal stability. However, stress-induced failures may occur through large in-plane stress, interlayer slip or transverse normal stress [1,2]. In order to overcome the adverse effects of laminated composites in mechanical properties, the engineering application of functionally graded materials (FGMs) was first proposed in 1984 by a group of aerospace scientists, due to a need for a type of material that can withstand high temperature difference in a space plane project [3]. The FGMs are a new type of advanced composite materials which are generally formed by two materials with smooth and continuous variation in specific direction from one surface to another, thus eliminating inter-laminar problems. The FGMs have received major attention since FGMs can effectively overcome these problems of traditional laminated composites. The corresponding specific material properties are obtained by the gradual variation in material properties and structure over volume fraction. The FGMs are designed to meet varying functionalities, and the FGMs plates and shells are the major structures owing to the wide variety of applications involved. Therefore, the study of the vibration characteristic existing research, the main type of cutout is circular cutout. In addition, there is no report which investigated the vibration solution of FGMs rectangular plate with cutouts based on the three-dimensional elasticity theory. The novelty of the present paper lies in the attempt to establish a unified theoretical analysis model of the vibration characteristics of the FGMs rectangular plate with/without cutouts, and provide the three-dimensional exact solution with general boundary conditions. The material properties of FGMs plates are supposed to vary continuously along the thickness direction in power-law distributions in terms of volume fraction. The highlight of the proposed approach is that the p-version method and the blending function method are employed to discretize the domain and map the closed region to the computational space. All displacements of the FGMs plates are expanded in the form of standard cosine Fourier series plus auxiliary cosine series terms which can improve the convergence speed and reduce the computational complexity. The threedimensional elasticity theory is combined with Rayleigh-Ritz method to solve the vibration problem of FGMs plate with cutouts. Numerical examples have been studied to verify the convergence, efficiency, and accuracy of the method and the predicted results have been compared with the theoretical solutions. The effects of cutout ratios, cutout positions, and cutout numbers on the natural frequencies are further explored by a parametric study in detail.

Description of the Problem
In this paper, a FGMs rectangular plate with circular cutout composed of two isotropic materials is considered. Figure 1a presents the three-dimensional geometric model of the structure. In order to describe geometric model clearly, the two coordinate systems are established independently on the mid-plane of the structure, the Cartesian coordinate system (O-xyz) for the rectangular plate region which is located in the corner of the plate, and the cylindrical coordinate system (Oc-x c y c z c ) for the cutout region which is located in the center of the circular cutout. The x-coordinate is taken along the length of the plate and yand z-coordinates are taken along the width and thickness directions. The length, width, and thickness of the plate are denoted by the symbols a, b, and h, respectively, and the radius of the cutout is denoted by r. The symbols u, v, and w denote the displacement components in the x, y, and z directions, respectively. report which investigated the vibration solution of FGMs rectangular plate with cutouts based on the three-dimensional elasticity theory. The novelty of the present paper lies in the attempt to establish a unified theoretical analysis model of the vibration characteristics of the FGMs rectangular plate with/without cutouts, and provide the three-dimensional exact solution with general boundary conditions. The material properties of FGMs plates are supposed to vary continuously along the thickness direction in power-law distributions in terms of volume fraction. The highlight of the proposed approach is that the pversion method and the blending function method are employed to discretize the domain and map the closed region to the computational space. All displacements of the FGMs plates are expanded in the form of standard cosine Fourier series plus auxiliary cosine series terms which can improve the convergence speed and reduce the computational complexity. The three-dimensional elasticity theory is combined with Rayleigh-Ritz method to solve the vibration problem of FGMs plate with cutouts. Numerical examples have been studied to verify the convergence, efficiency, and accuracy of the method and the predicted results have been compared with the theoretical solutions. The effects of cutout ratios, cutout positions, and cutout numbers on the natural frequencies are further explored by a parametric study in detail.

Description of the Problem
In this paper, a FGMs rectangular plate with circular cutout composed of two isotropic materials is considered. Figure 1a presents the three-dimensional geometric model of the structure. In order to describe geometric model clearly, the two coordinate systems are established independently on the mid-plane of the structure, the Cartesian coordinate system (O-xyz) for the rectangular plate region which is located in the corner of the plate, and the cylindrical coordinate system (Oc-xcyczc) for the cutout region which is located in the center of the circular cutout. The x-coordinate is taken along the length of the plate and y-and z-coordinates are taken along the width and thickness directions. The length, width, and thickness of the plate are denoted by the symbols a, b, and h, respectively, and the radius of the cutout is denoted by r. The symbols u, v, and w denote the displacement components in the x, y, and z directions, respectively.  For the FGMs composed of two types of isotropic components, such as ceramics and  metals, according to the Voigt mixing rule, the equivalent materials properties can be  expressed as where E is Young's modulus, µ is Poisson's ratio, and ρ is density, the subscripts c and m represents the ceramic and metal material, V f is the volume fraction. The volume fraction, thus the variation of the material properties of each component can be obtained by assuming to be different function distributions, such as power-law functions (P-FGM), exponential functions (E-FGM), or sigmoid functions (S-FGM) [35]. In this paper, the FGMs with power-law scheme is selected as the research object.
The volume fraction V f can be defined as where z is the thickness coordinate, and p is gradient index and takes only positive values. When p = 0, the FGMs degenerates into ceramic material and when p = ∞ indicates a fully metal material. Figure 2 shows the curve of volume fraction variation along the thickness direction corresponding to the different gradient index. The volume fraction can be effectively controlled by changing the value of p, and different kinds of FGMs can be designed by changing the above parameters according to different functional requirements. For the FGMs composed of two types of isotropic components, such as ceramics and metals, according to the Voigt mixing rule, the equivalent materials properties can be expressed as where E is Young's modulus, μ is Poisson's ratio, and ρ is density, the subscripts c and m represents the ceramic and metal material, Vf is the volume fraction. The volume fraction, thus the variation of the material properties of each component can be obtained by assuming to be different function distributions, such as power-law functions (P-FGM), exponential functions (E-FGM), or sigmoid functions (S-FGM) [35]. In this paper, the FGMs with power-law scheme is selected as the research object.
The volume fraction Vf can be defined as where z is the thickness coordinate, and p is gradient index and takes only positive values. When p = 0, the FGMs degenerates into ceramic material and when p = ∞ indicates a fully metal material. Figure 2 shows the curve of volume fraction variation along the thickness direction corresponding to the different gradient index. The volume fraction can be effectively controlled by changing the value of p, and different kinds of FGMs can be designed by changing the above parameters according to different functional requirements. In this paper, the FGMs plate is considered to be made of aluminum (Al) and alumina (Al2O3), the material properties for ceramic and metallic constituents of FGMs plate are listed in Table 1. In this paper, the FGMs plate is considered to be made of aluminum (Al) and alumina (Al 2 O 3 ), the material properties for ceramic and metallic constituents of FGMs plate are listed in Table 1.

Kinematic Equations
The strain-displacement relations of the structure can be obtained as follows based on the linear, small-strain three-dimensional elasticity theory γ yz = ∂w ∂y where ε x , ε y , ε z , γ yz , γ xz , and γ xy are the normal and shear strain.
According to the theory of three-dimensional constraint of a linear elasticity, the corresponding stress-strain relations of the three-dimensional structure are written as where C ij (z) (i, j = 1, 2, . . . , 6) are material elastic constants, for isotropic materials, they can be defined as

Boundary Conditions
As illustrated in Figure 1b, three groups of boundary springs are factitiously distributed along the edges to simulate different boundary conditions. The symbols ku, kv, and kw are used to indicate the stiffness of the springs, and through adopting appropriate values of the boundary spring stiffness, the classical boundary conditions and elastic boundary conditions can be achieved. The general boundary conditions mainly include free (F), simply supported (S), clamped (C). The expressions of the different boundary conditions along the edge x = 0 are given as follows.
Free boundary condition: Simply supported boundary condition: Clamped boundary condition: Elastic restraint boundary condition:

Energy Equations
In the light of Hamilton's principle, the governing equation of the structure and the boundary conditions are derived in the following work. In this paper, the plate domain and the cutout domain are separated, and the energy functions of the FGMs plate and the cutouts are established independently. The kinetic energy function of the FGMs plate and cutouts can be expressed as The total linear elastic strain energy function is depicted as The potential energy stored in the boundary springs is expressed as

Region Mapping
As described in the above model, the coordinate system of the plate is Cartesian coordinate system, and the coordinate system of the cutout is cylindrical coordinate system. The expression of the displacement components is different for the plate and cutout domain, which leads to the complexity of the energy integration. In order to simplify the solution process, a unified coordinate system is needed. The main purpose of this paper to deal with the vibration of rectangular plate with circular cutout is region mapping. The p-version of the finite element method is applied to discretize the plate with cutout into four curve quadrilateral sub-domains as presented in Figure 3a. The length and width of the rectangular plate are a and b, and the radius of the circular cutout is r, respectively. The sub-domain 1 is regarded as a closed region composed of four curves in the x-y coordinate system. The purpose of region mapping is to map the closed quadrilateral region into a unit square region, as described in Figure 3b,c.
The closed quadrilateral region consists of four curves, and the curve sides are in the parametric form: where −1 ≤ ξ ≤ 1, −1 ≤ η ≤ 1; the four vertex-nodes are x i , y i (i = 1, 2, 3, 4), respectively. The blending function method proposed by Gordon and Hall [36] is well suited for the purpose to map the closed quadrilateral region to the computational space.
p-version of the finite element method is applied to discretize the plate with cutout into four curve quadrilateral sub-domains as presented in Figure 3a. The length and width of the rectangular plate are a and b, and the radius of the circular cutout is r, respectively. The sub-domain 1 is regarded as a closed region composed of four curves in the x-y coordinate system. The purpose of region mapping is to map the closed quadrilateral region into a unit square region, as described in Figure 3b,c. ( ) The closed quadrilateral region consists of four curves, and the curve sides are in the parametric form: 1 1 , respectively. The blending function method proposed by Gordon and Hall [36] is well suited for the purpose to map the closed quadrilateral region to the computational space.
The mapping functions are given as Substituting the circular equations and the three linear equations into Equations (23) and (24), the following results can be obtained by  The mapping functions are given as Substituting the circular equations and the three linear equations into Equations (23) and (24), the following results can be obtained by In the light of above region mapping and coordinate transformation relationship, the displacement components of transformation matrix from the global coordinate system to the local coordinate system are related by The inverse of the Jacobian transformation matrix is

Solution Procedure
In this paper, the Rayleigh-Ritz method is used due to it is applicable to arbitrary boundary conditions without requiring any special procedures. Thus, it is very important to construct an admissible displacement function field because the accuracy and the convergence of the solution depend on the accuracy of the expression of the admissible displacement function. In this paper, the improved Fourier series method is further extended to the three-dimensional vibration analysis of FGMs rectangular plate with cutouts. According to the author's previous research, the admissible displacement functions are consistent with [37], and expressed in the form of complete trigonometric Fourier series, thus the auxiliary terms are also in the form of trigonometric Fourier series. The threedimensional admissible displacement functions of the FGMs plate are expressed as three variables separated along the x, y, and z directions as the unknown Fourier coefficients, M, N, and Q are the truncation numbers with respect to variables x, y, and z directions, respectively. In order to unify the form of the admissible displacement functions and simplify the mathematical processing, the supplementary functions are defined as The total energy of the FGMs rectangular plate with circular cutout is defined by subtracting the energy of cutout domain part from the entire plate domain. Thus, the Lagrangian energy function of the structure can be expressed as The subscripts "p" and "i" denote the energy of the plate and cutout domains, respectively. For the plate without cutout, the energy equation is Substituting Equations (20)-(22), (27), and (32)-(34) and into Equation (41), and by minimizing the Lagrangian energy functional ∏ with respect to each unknown coefficients to be zero, we can get the equation The standard eigenvalue equation of motion for rectangular plate with circular cutout can be expressed in the form of matrix where K, M, and X are the stiffness matrices, mass matrices and the unknown Fourier coefficients matrices, respectively. All of the natural frequencies and mode shapes of the three-dimensional FGMs rectangular plates with circular cutouts can be obtained by solving Equation (44).

Results and Discussion
In this section, according to the unified theoretical analysis model established above, several examples for the three-dimensional vibration analysis of FGMs plate with/without cutouts are presented to illustrate the accuracy and reliability of the proposed method. Firstly, a suitable spring stiffness value is investigated, and then the convergence, efficiency and validation are checked. Secondly, the vibration modal experiment of an aluminum square plate with a center circular cutout is conducted to verify the correctness of the proposed method. Finally, a parametric study of the FGMs plate with cutouts is carried out from free vibration characteristics and harmonic response analysis, including the cutout sizes, cutout positions, and cutout numbers.
For simplicity, the boundary conditions of the structure are described in the form of character combination, unless other stated, the non-dimensional frequency parameter is expressed as:

Determination of the Spring Stiffness
In this paper, three groups of linear springs are introduced to simulate different kinds of boundary conditions by changing the values of spring stiffness. The accuracy of the solutions is strongly affected by the selection of appropriate spring stiffness values. Therefore, in this section, the FGMs square plate without cutout is taken as an example to study the determination of the spring stiffness. The variations of the first three nondimensional frequency parameters of FGMs square plate versus different spring stiffness are given in Figure 4. The geometric dimensions and the material parameters are as follows: a = b = 1 m, h = 0.05 m, and p = 1. The boundary conditions of the plate are defined as: the edge y = 0 and y = b are completely free and the x = 0 and x = a are elastic supported by one group of spring constrain varying from 10 −3 D c to 10 12 D c . From Figure 4, it can be seen that the non-dimension frequency parameters are unchanged and approaches 0 when the spring stiffness is smaller than 10 −1 D c , and when the spring stiffness is varied from 10 −1 D c to 10 7 D c , the frequency parameters increase rapidly. Finally, when the spring stiffness exceeds 10 7 D c , the frequency parameters will approach their utmost and tend to be stable. Therefore, we can arrive at the conclusion that the free boundary conditions and clamped boundary conditions can be simulated by assigning the spring stiffness value to be 0 or 10 7 D c.  Figure 4, it can be seen that the non-dimension frequency parameters are unchanged and approaches 0 when the spring stiffness is smaller than 10 −1 Dc, and when the spring stiffness is varied from 10 −1 Dc to 10 7 Dc, the frequency parameters increase rapidly. Finally, when the spring stiffness exceeds 10 7 Dc, the frequency parameters will approach their utmost and tend to be stable. Therefore, we can arrive at the conclusion that the free boundary conditions and clamped boundary conditions can be simulated by assigning the spring stiffness value to be 0 or 10 7 Dc.

Convergence of the Method
Convergence property for the free vibration analysis of FGMs plate with circular cutout is examined in terms of the limited number of terms in the displacement expressions in actual calculation to verify the accuracy and efficiency of the proposed method. Table  2 shows the convergence studies of the first six non-dimensional frequency parameters of a FGMs square plate with a central circular cutout with different truncated numbers, and the data of [38] are also given out in Table 2. The truncated number of admissible displacement function components in Fourier series expansion is expressed as M × N × Q, and the

Convergence of the Method
Convergence property for the free vibration analysis of FGMs plate with circular cutout is examined in terms of the limited number of terms in the displacement expressions in actual calculation to verify the accuracy and efficiency of the proposed method. Table 2 shows the convergence studies of the first six non-dimensional frequency parameters of a FGMs square plate with a central circular cutout with different truncated numbers, and the data of [38] are also given out in Table 2. The truncated number of admissible displacement function components in Fourier series expansion is expressed as M × N × Q, and the truncated number of this paper is from 3 to 10. It can be observed that the maximum error with [38] is 1.7898%, and the main reason of the error is that the first order shear theory is adopted by [38]. It can be concluded that the proposed method has fast convergence and good stability, and the truncation numbers will be set as M = N = Q = 10 in the following studies. In this section, in order to verify the proposed method is also suitable for solving the vibration characteristics of FGMs plate without cutouts, the comparison study of a FGMs square plate under SSSS boundary condition will be carried out by the present method and other method presented in [25]. Table 3 shows the first eight non-dimensional frequency parameters of FGMs square plate which was studied by Huang et al. The values of the gradient indexes are taken to be 0, 1, 2, 5, 10. The symbol '-' indicates that the frequencies were not considered in the reference work. In the table, the thickness-length ratio is taken as 0.1 and 0.2, it is moderate-thick plate structure. From the comparison, we can see a consistent agreement of the results taken from the current method and the referential data. From the tables, it is obvious that these data show a similar behavior, that is the frequency parameters decrease with the increase of gradient index. When the thickness-length ratio is 0.1, the gradient index increases from 0 to 10, the first-order non-dimensional frequency parameter decreases by 36.95%, and when the thickness-length ratio is 0.1, the first-order non-dimensional frequency parameter decline rate is 38.44%. The main reason is that the increase of gradient index leads to the decrease of the volume fraction of the corresponding ceramic components, which reduces the stiffness of the structure, and finally leads to the decrease of frequency. In addition, the increase of thickness-length ratios leads to decrease of the frequency parameters for all of the cases considered. By way of illustration, as the thickness-length ratio increases, the first-order non-dimensional frequency parameter drops by 8.19% when the gradient index is 0. According to the relationship between the non-dimensional frequency parameters calculation formula and thickness, it can be known that the natural frequencies increase with the increase of thickness. The increase of thickness will increase the stiffness and mass of rectangular plate, but the increase of stiffness plays a decisive role in the influence of natural frequency. For the next comparison study, a FGMs rectangular plate with a central circular cutout under CCCC-F boundary condition is examined. In Table 4, the first fix non-dimensional frequency parameters are obtained. It is observed that the frequencies are in excellent agreement with those given in [38], which verifies the accuracy and efficiency of the proposed model. The effect of gradient index on the frequency parameters of structure with cutout is consistent with that structure without cutout also can be concluded from the table below. Therefore, the effect of material parameters on structure with cutout is independent of the geometric size of the structure. In addition, it can be seen that the frequency parameters show an increasing trend with the increase of the structural aspect ratio, but it is not a linear trend. Take the gradient index is 0 as an example, when the aspect ratio is 1.5, the first-order non-dimensional frequency parameter is 1.67 times that when the aspect ratio is 1, while it is 2.7 times when the aspect ratio is 2. This is mainly because when the aspect ratio increases, the overall mass and stiffness of the structure will also increase, and the stiffness is the main factor affecting frequency parameters.
Numerous results of the first six non-dimensional frequency parameters are demonstrated in Table 5 for the FGMs square plate with central circular cutout under different boundary conditions. The square plate with boundary restrains, including SSSS-F, SCSC-F, SFSF-F, FCFC-F, FFCF-F, FCCC-F, FSCS-F, and FCCF-F are considered. From the table below, the influence of boundary conditions on the frequency parameters is obvious, the stronger the boundary restrains, the higher the corresponding frequency parameters, this can be clearly confirmed from SFSF-F, SSSS-F, and SCSC-F boundary conditions. Comparing three groups of boundary conditions-FCFC-F, FCCF-F, and FCCC-F-it can be concluded that the clamped boundary constrain plays an important role in the frequency parameters, while frequency parameter of the opposite side constraint is larger than that of the adjacent side constraint.  The first six mode shapes of the FGMs square plate with central circular cutout under SSSS-F boundary condition when the thickness-length ratio is 0.01 and 0.2 are presented in Figures 5 and 6, respectively. From the graphs below, for thin plate structure, the lower order mode shapes are mainly transverse vibration, while for moderate thick plate structure, the shear deformation along the thickness direction gradually appears. The three dimensions of the elastic plate structure have complex deformation forms for different modes, therefore the analysis based on the three-dimensional elastic theory can fully consider the influence of the shear deformation in the thickness direction on the vibration characteristics of the structure.

Experimental Study
In this part, an experimental study of plate with central cutout is conducted to further verify the validity of the proposed method. The experimental setup-including the hammer, accelerometer sensor, charge adapter, dynamic data acquisition instrument, and computer-is shown in Figure 7. In view of the available situation, the gradient index is considered to be infinity, thus the functionally graded material degenerates to be completely aluminum. A square plate with central circular cutout under FFFF-F and CCCC-F boundary condition are examined. It is impossible to realize the complete free boundary condition in the actual experimental environment, two small holes are opened on the edge of the structure, and elastic rubber ropes are used to hang the structure on the frame, as shown in Figure 8a. For the CCCC-F boundary condition, the experimental model adopts two thicker L-shaped plates and arranges bolts uniformly around the rectangular plate structure to simulate the fixed boundary condition, as shown in Figure 8b. The parameters of dimension and the material of the structure are given in Table 6. Table 7 shows the first six natural frequencies of the structure obtained by the present method and the experiment. Through the comparative analysis of experiments and the calculation of the proposed method, the difference is 4.754% for the worst case, which is acceptable. The main reason for the error lies in two aspects. Firstly, the difference of boundary restrains between the experimental simulation and the theoretical calculation will cause a certain error. Secondly, when knocking with a hammer, it requires that the knocking direction is completely perpendicular to the panel surface, the knocking force should be constant, and the hammer shall be evacuated quickly when the knocking is finished to avoid secondary knocking, which is difficult to ensure in the process of experiment. The experimental frequency values are obtained from the vibration analysis software by searching the peak within a certain range of the frequency, and three are some interference items near desired the frequency value, the peak value is automatically identified and selected by the computer, this is also the reason for the error. The experimental values are smaller than the theoretical value, the main reason is due to the additional mass of the accelerometer which is attached to the panel.

Experimental Study
In this part, an experimental study of plate with central cutout is conducted to further verify the validity of the proposed method. The experimental setup-including the hammer, accelerometer sensor, charge adapter, dynamic data acquisition instrument, and computer-is shown in Figure 7. In view of the available situation, the gradient index is considered to be infinity, thus the functionally graded material degenerates to be completely aluminum. A square plate with central circular cutout under FFFF-F and CCCC-F boundary condition are examined. It is impossible to realize the complete free boundary condition in the actual experimental environment, two small holes are opened on the edge of the structure, and elastic rubber ropes are used to hang the structure on the frame, as shown in Figure 8a. For the CCCC-F boundary condition, the experimental model adopts two thicker L-shaped plates and arranges bolts uniformly around the rectangular plate structure to simulate the fixed boundary condition, as shown in Figure 8b. The parameters of dimension and the material of the structure are given in Table 6. Table 7 shows the first six natural frequencies of the structure obtained by the present method and the experiment. Through the comparative analysis of experiments and the calculation of the proposed method, the difference is 4.754% for the worst case, which is acceptable. The main reason for the error lies in two aspects. Firstly, the difference of boundary restrains between the experimental simulation and the theoretical calculation will cause a certain error. Secondly, when knocking with a hammer, it requires that the knocking direction is completely perpendicular to the panel surface, the knocking force should be constant, and the hammer shall be evacuated quickly when the knocking is finished to avoid secondary knocking, which is difficult to ensure in the process of experiment. The experimental frequency values are obtained from the vibration analysis software by searching the peak within a certain range of the frequency, and three are some interference items near desired the frequency value, the peak value is automatically identified and selected by the computer, this is also the reason for the error. The experimental values are smaller than the theoretical value, the main reason is due to the additional mass of the accelerometer which is attached to the panel.

Parametric Study
In this section, the parametric study of three-dimensional vibration characteristics of the FGMs plate with cutouts is carried out. Based on the existing literature, the structural vibration characteristics of different functionally graded material parameters are different, and the predecessors have done a lot of research on this. This section emphasizes the study of the influence of the parameters of the cutout on the free vibration characteristics and harmonic response analysis of the structure, including the cutout sizes, cutout positions, and number of the cutout. The FGMs square plate with circular cutouts under CCCC-F boundary conditions is taken as the analysis object, and the geometric parameters and materials parameters are set as follows: a = b = 1 m, h = 0.1 m, p = 1.
First, the variation of the non-dimensional frequency parameters with respect to diverse cutout sizes is investigated. Table 8 presents the first six frequency parameters of the FGMs square plate with a central circular cutout, and the cutout size ratios (r/a) vary from 0 to 0.25. For the small values of cutout size ratio, the frequency parameters of the structure with and without cutout are almost the same. It is found that the change trend of the low-order modal frequency parameters of the structure is relatively simple, a minimum value for the first five modes exists and the frequency parameters first decrease and then increase when the cutout ratio rise, while the change of the high-order frequency parameters is more complicated. The reason may be due to the weight of mass loss and stiffness loss on the frequency parameters is different with the increase of cutout size ratio. Then, the harmonic analysis is used to analyze the steady-state response of the FGMs plate with cutouts under simple harmonic excitation. In order to overcome the problem of numerical instability caused by structural resonance at the modal frequency of the external excitation, the damping factor will be introduced in the form of complex Young's modulus, thus E = E(1 + jη), η = 0.01. Assuming that a simple harmonic force is applied to point A along the z-axis, and the magnitude of the force is 1 N. The coordinates of the excitation force application point A and the selected response observation point B are (0.5 m, 0.8 m, 0.05 m) and (0.8 m, 0.8 m, 0.05 m), respectively. Figure 9a,b provide the results obtained from the preliminary analysis of the displacement response curve of the excitation force application point and the response observation point with frequency in the range of 0-5000 Hz. The range of the cutout size ratio is from 0 to 0.25 with a step of 0.025, and the displacement response is H = 20 * log(w). From the graph below we can see that there has been a slight rise in the displacement response with the gradual increase of the cutout size rate. This is mainly because the stiffness of the excitation force application point and the observation point is weakened by the introduction of the cutout. The resonance peak of the displacement response will shift left and right with the increase of cutout size ratio. To further explain, in Figure 9a, the first-order resonance peak frequency is 1225 Hz when the structure is without cutout, the first-order resonance peak frequency is 1221 Hz when the cutout ratio is 0.05, and the first-order resonance peak frequency is 1246 Hz when the cutout ratio is 0.1, in the case of a larger cutout ratio, it can be clearly seen that the first-order resonance peak frequency is increasing, which is consistent with the change trend of the data in Table 8. nance peak of the displacement response will shift left and right with the increase of cutout size ratio. To further explain, in Figure 9a, the first-order resonance peak frequency is 1225 Hz when the structure is without cutout, the first-order resonance peak frequency is 1221 Hz when the cutout ratio is 0.05, and the first-order resonance peak frequency is 1246 Hz when the cutout ratio is 0.1, in the case of a larger cutout ratio, it can be clearly seen that the first-order resonance peak frequency is increasing, which is consistent with the change trend of the data in Table 8.
(a) Point of A (b) Point of B Figure 9. The displacement response for FGMs square plate with diverse cutout sizes.
The following part of the study is concerned with the position of the cutout. The radius of the cutout is 0.1 m, and the position of cutout varies along the x-axis. The table below illustrates the first six non-dimensional frequency parameters of the FGMs square plate with different cutout positions. In Table 9 Figure 9. The displacement response for FGMs square plate with diverse cutout sizes.
The following part of the study is concerned with the position of the cutout. The radius of the cutout is 0.1 m, and the position of cutout varies along the x-axis. The table below illustrates the first six non-dimensional frequency parameters of the FGMs square plate with different cutout positions. In Table 9, when the cutout position x c = 0.5, it means that the cutout is located in the center of rectangular plate. The table reveals that as the cutout position gradually approaches the edge of the structure, the fundamental frequency parameter of the structure gradually declines, the second order frequency parameters of the structure gradually increases, while the higher order frequency parameters changes are more complicated. The results of the correlational analysis of displacement response for FGMs square plate with different cutout positions are shown in Figure 10. From the graph below we can see that in the frequency range of 0-3000 Hz, the vibration displacement response at the resonance peak changes slightly, for the excitation force application point, the amplitudes of the first-order resonance peaks are −378.658 dB, −378.957 dB, −379.589 dB, and −379.962 dB, respectively. While the excitation frequency is greater than 3000 Hz, the vibration displacement response at the resonance peak changes obviously. In actual engineering structures, it is often necessary to evenly arrange multiple cutouts on the structure, so in the final part of the study, the influence of the number of cutout on the vibration characteristics of the structure is investigated. Table 9 provides the first six non-dimensional frequency parameters of the FGMs square plate with different cutout numbers, the cutouts are evenly distributed along the x-axis direction. The radius of the cutouts is 0.05 m, and the other parameters remain the same with the previous data. What can be clearly seen in Table 10 is the decrease with the increasing in the number of cutouts for the frequency parameters of all orders. The results of the harmonic response correlational analysis of are presented in Figure 11. The graph shows that there has been a small change for the amplitude of the displacement response with the increasing of the cutout numbers, while all resonance peaks gradually shift to the left.   In actual engineering structures, it is often necessary to evenly arrange multiple cutouts on the structure, so in the final part of the study, the influence of the number of cutout on the vibration characteristics of the structure is investigated. Table 9 provides the first six non-dimensional frequency parameters of the FGMs square plate with different cutout numbers, the cutouts are evenly distributed along the x-axis direction. The radius of the cutouts is 0.05 m, and the other parameters remain the same with the previous data. What can be clearly seen in Table 10 is the decrease with the increasing in the number of cutouts for the frequency parameters of all orders. The results of the harmonic response correlational analysis of are presented in Figure 11. The graph shows that there has been a small change for the amplitude of the displacement response with the increasing of the cutout numbers, while all resonance peaks gradually shift to the left. tional analysis of are presented in Figure 11. The graph shows that there has been a small change for the amplitude of the displacement response with the increasing of the cutout numbers, while all resonance peaks gradually shift to the left.   Figure 11. The displacement response for FGMs square plate with different cutout numbers.

Conclusions
The aim of the present research is to establish a unified three-dimensional solution to deal with the vibration characteristics of FGMs plate with/without circular cutouts. The material properties vary continuously along the thickness direction according to the power-law distribution. The artificial spring technology is used to simulate the general boundary conditions by setting three groups of linear springs and assigning them with appropriate spring stiffness values. Due to relatively complicated governing differential equations and domain of the problem, the p-version of the finite element method is applied to discretize the plate with cutout into four curve quadrilateral sub-domains, and then map the closed quadrilateral region to the computational space by the blending function method. The independent coordinate coupling relationship is used to derive the Jacobian relationship matrix of the rectangular plate domain and the circular cutout domain, finally the Lagrangian energy equation is used to solve the differential equation. In the analysis of numerical examples, it is found that the calculation results of this method are in good agreement with other results through comparison with the existing literature and finite element simulation analysis results, which verifies that the method proposed in this paper is reliable. Then the effects of cutout sizes, cutout positions, and cutout numbers on the frequency parameters of FGMs plate with cutout are studied and discussed, and all of these factors will have an impact on the frequency parameters. The proposed method can be applicable to solve the vibration of complex shape plate with cutouts, and the numerical results can be useful for future research.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.