1. Introduction
Spinning disks are widely applied in a rotor machinery, such as aero engines, gas turbines, and so on. The thick disk is commonly adopted in the traditional rotor structures to achieve great structural stiffness and it can be considered as a rigid body in the vibration analysis. To meet the requirements of high spinning speed and light weight, however, thin disks are increasingly used in practical engineering applications. In such cases, the flexibility and the deformation of the disk can no longer be ignored. Theoretically, the thin disk can be modeled as an elastic annular thin plate, whose vibration behaviors have been extensively investigated [
1,
2,
3,
4,
5].
By employing the finite element method, Pan et al. [
6] studied the vibration of rotor bearing-disk system subjected to three forces. Yang et al. [
7] developed a thermal stress stiffening method to investigate the vibration behavior of spinning flexible disks. Maretic et al. [
8] proposed vibrations of spinning annular plate with two different materials. By adopting the experimental method, Kang et al. [
9] studied the vibration characteristics of spinning disk in an air-filled enclosure. The Ritz method is used by Kang et al. [
10] to study the free vibration of spinning annular plates with variable thickness. Rao et al. [
11] concerned with free vibration behaviors of an annular plate resting on Winkler foundation. Based on Mindlin plate theory, Chen et al. [
12] studied the high-frequency vibration performance of an annular plate. Tan et al. [
13] deal with the forced and free vibration of a thin annular plate with variable stiffness. Amin et al. [
14] investigated the nonlinear vibration behaviors of an FG annular plate. Wang et al. [
15] studied the free vibration of an annular plate with different edges.
Due to the spinning effect, the disk is always subjected to aerodynamic loading on its faces. To enhance structural stiffness and reduce weight, a sandwich annular plate structure could be designed in which the upper and lower faces are made of uniform solid metal and the core is porous foamed metal. Because the pores can weaken the structural stiffness, some reinforcements need to be added. GPLs, owing to their superior mechanical properties, are well suited to be the reinforcements. Recently, the vibration behavior of GPL reinforced structures for better mechanical performance has been a topic of extensive research efforts [
16,
17,
18,
19,
20]. Yang and Zhao et al. [
21,
22,
23,
24] carried out extensive research on the free vibration of rotor structures reinforced by GPLs. Based on the modified couple stress theory, Adab et al. [
25] studied the free vibration of a spinning sandwich micro-shell. Saidi et al. [
26] investigated vibrations of an FG porous GPL reinforced plate subjected to aerodynamical loading. Li et al. [
27] studied nonlinear vibrations of a sandwich FG porous GPL reinforced plate resting on Winker–Pasternak elastic foundation. Zhou et al. [
28] investigated vibrations of a GPL reinforced porous cylindrical panel under supersonic flow. Gao et al. [
29] conducted nonlinear free vibration analysis of a porous plate reinforced with GPLs. Baghlani et al. [
30] studied uncertainty propagation in free vibration of an FG porous shell with GPL reinforcement. Anamagh et al. [
31] developed a spectral-Chebyshev approach to study vibrations of an FG porous plate reinforced with GPLs. Based on a trigonometric shear deformation theory, Anirudh et al. [
32] discussed the vibration behavior of a GPL reinforced FG porous beam.
For a disk with high spinning speed, sandwich structure with a functionally graded graphene nanoplatelet reinforced porous core and stiff faces is an ideal option due to its light weight yet great structural stiffness. To the best of the authors’ knowledge, however, none of the existing studies, including those mentioned above, has discussed the dynamic behaviors of such a spinning disk. This paper aims to fill in this research gap by studying the free vibration of a spinning sandwich annular plate with FG-GPL reinforced porous core. Considering the whirl motion, the annular plate is modeled by the Kirchhoff plate theory. The differential equations of motion and free vibration results are obtained by employing the Lagrange’s equation and assumed mode method, respectively. A comprehensive study is proposed to examine the effects of the material and structural parameters on the natural frequencies of the spinning annular plate. The presented conclusions can effectively aid the design of spinning annular plates with GPL reinforced porous core.
3. Results and Discussions
In this part, the effects of material parameters on the free vibration behaviors of the spinning annular plate with porous core reinforced by GPLs are examined in detailed. Unless otherwise stated, the structural and material parameters [
34] are given in
Table 1. In addition, porosity distribution pattern X
P and GPL distribution pattern X
G are taken as an example in the subsequent analysis.
3.1. Convergence and Comparison Study
Before parametric analysis, the convergence and comparison analysis have to be conducted first.
Table 2 lists the variations of the first four natural frequencies with mode number by theoretical method (MATLAB), which shows that convergent frequencies can be obtained at
n = 6.
The finite element method is used to validate the present analysis by using commercial software ABAQUS. The functionally graded material core is divided into ten layers and the material properties of each layer are calculated by the equations in
Section 2.2. The annular plate is clamped at the inner edge, free at its outer edge, and is discretized by 4-node doubly curved general-purpose shell elements with 6 degrees of freedom. To examine the convergence of the finite element analysis,
Table 3 gives the first four natural frequencies at
Ω = 500 rad/s with different total numbers of elements
Ne = (1440, 4000, 5760, 7840) and nodes (1536, 4160, 5952, 8064).
Figure 4 displays the corresponding mesh graphs. It is clear that the free vibration results come to be converged at element number
Ne = 7840.
Table 4 and
Figure 5 give the comparison of first four natural frequencies and vibration modes by theoretical method (MATLAB) and finite element (ABAQUS) method at
Ω = 500 rad/s, respectively. It is obvious that the frequencies and vibration modes are in agreement, which shows that the present analysis is accurate.
In addition, the theoretical results are also compared with the experimental results [
35] in
Table 5, where the parameters are given in
Table 6. One can see that the theoretical calculation results are in good agreement with the experimental results, which tells that the present analysis is accurate.
3.2. Parametric Analysis
In this section, both the graphic form and tabular form are utilized to conduct the parametric analysis on the free vibration results of the spinning FG annular plate with porous core reinforced by GPLs.
Figure 6 depicts the variations of first four natural frequencies of the annular plate with spinning speed for different ratios of the core thickness to annular plate thickness. A considerable rise in the frequencies is observed as the spinning speed increases. In addition, the larger ratio of the core thickness to annular plate thickness leads to greater frequencies. It indicates that thinner faces could be adopted in the present sandwich structure to achieve better mechanical performance.
Figure 7 plots the variations of first four natural frequencies of the annular plate with spinning speed for different GPL distributions. It is seen that the GPL distribution pattern X
G provides highest frequencies, while pattern O
G has the worst enhancement effect. This implies that dispersing more GPLs around the surfaces of the core could give a hand to enhance the structural stiffness.
The variations of first four natural frequencies of the annular plate with spinning speed for different porosity distributions are presented in
Figure 8. Results show that porosity distribution pattern X
P affords greatest frequencies, while the pattern O
P gives the smallest one. It is noted that setting more pores around the surfaces of the core is effective to obtain great mechanical performance.
Since the variations of natural frequencies with spinning speed are similar, only two typical spinning speeds, 0 rad/s and 500 rad/s, are adopted in the following analysis.
Figure 9 shows the variations of first four natural frequencies of the annular plate with GPL weight fraction at different spinning speeds. One can see that the frequencies increase markedly with the GPL weight fraction. It is worth noting that adding more GPLs into the core plays a very important role in obtaining greater enhancement.
Figure 10 lists the variations of first four natural frequencies of the annular plate with GPL length-to-thickness ratio at different spinning speeds. We can see that the frequencies rise dramatically with the GPL length-to-thickness ratio. For the same content of GPLs, larger GPL length-to-thickness ratio means a thinner GPL. It can be seen that better enhancement effect occurs when thinner GPLs are added into the core.
Figure 11 gives the variations of first four natural frequencies of the annular plate with GPL length-to-width ratio at different spinning speeds, where GPL length remains constant. It is seen that the frequencies are reduced with a rise in GPL length-to-width ratios. Here it should be noted that a smaller GPL length-to-width ratio means each GPL with larger surface area, which can lead to better load transfer capacity.
The variations of first four natural frequencies of the annular plate with porosity coefficient at different spinning speeds is presented in
Figure 12. One can see that the frequencies decrease in general with the increase of porosity coefficient. Although the larger porosity coefficient can result in light weight, it weakens the structural stiffness.