Dynamic Modeling and Vibration Characteristic Analysis of Fiber Woven Composite Shaft–Disk Rotor with Weight-Reducing Holes

Abstract

In order to achieve the goal of a lightweight shaft–disk rotor, this paper applies the fiber woven composite material to the disk structure, and at the same time considers the design of the weight-reducing holes on the porous disk. It introduces the domain decomposition and coordinate mapping technology for this, and then establishes the dynamics model of the fiber woven composite material shaft–disk rotor. The model is based on the differential quadrature finite element method, which is suitable for fiber woven composite rotors with arbitrary complex hole patterns. The validity of the model is verified by comparing the results with the literature, finite element simulation, and experiments, and the mechanism of the influence of the material parameters and pore parameters on the vibration characteristics of the system is investigated, which provides the data support and theoretical basis for the analysis of the dynamics of the fiber woven composite rotor.

1. Introduction

In practical engineering applications, in order to reduce the weight of the shaft–disk rotor, improve the vibration noise phenomenon, eliminate internal stress, and other purposes, the rotor disk is often used to open the weight reduction holes, as well as for the replacement of rotor materials and other methods. At this time, the performance enhancement effect brought about by the material change and the hole parameters on the vibration performance of the system are important aspects of rotor dynamics research.

Among the existing emerging composites, fiber woven composites, in addition to possessing many advantages such as low density, excellent physical and mechanical properties, good heat resistance, etc., also have better designability, processability, and lower manufacturing cost compared to ordinary fiber-reinforced, functional gradient, and other composites, which have broad application prospects in the fields of aerospace and rail transportation. Therefore, this material is one of the ideal materials for shaft–disk rotor preparation.

The current research on fiber woven composites is mainly based on simulation and experimentation, and generally adopts the multi-scale modeling method [1] to achieve the modeling of fiber woven composites from the microscopic scale to the mesoscopic scale (representing the volumetric unit) and then to the macroscopic scale. The research includes the physical and mechanical properties [2] and the damage behaviors [3,4] of the fiber woven composites and their structures. In terms of dynamics research, single structures such as fiber woven composite plates/shells are mostly taken as the analysis objects. Poojary and Hegde [5] carried out an experimental study on the dynamic properties of intra-ply woven carbon fiber/epoxy hybrid composites using fiber orientation as a comparative variable and found that the structural intrinsic frequency decreases with the increase in the orientation angle. Damijan et al. [6] fabricated woven carbon fiber-reinforced composites gears by using a hot-press curing method, and found that, after high week fatigue testing, woven carbon fiber-reinforced composite gears have longer fatigue lives and better mechanical properties than ordinary composite gears. The above studies are based on experiments, which are highly referable and relevant, but this also results in the singularity of the research object. Huang et al. [7] applied the three-dimensional finite element method to model the carbon fiber plain weave material and used a two-step homogenization method to obtain the mechanical properties of the material, then investigated the free vibration characteristics of the carbon fiber plain weave cylindrical shells and came up with the conclusion that increasing the total fiber volume fraction leads to a linear increase in the intrinsic frequency of the structure It can be seen that the related dynamics research also mostly adopts finite element software, which has the disadvantages of difficult modeling and being time-consuming when facing complex research objects.

For rotating members, open holes, while affecting the internal mechanical properties of the structure [8], also change their dynamic properties, such as natural frequency and critical speed. Therefore, the problem of modeling the dynamics of open-hole structures and their dynamic properties has attracted the attention of many scholars. In addition to numerical analytical methods to characterize the hole boundaries, existing studies also treat the holes as virtual structures. Wang et al. [9,10] treated irregular holes as virtual plates with zero density and Young’s modulus, treated a perforated plate as an equivalent rectangular plate with inhomogeneous thickness, and solved the free vibration characteristics of the structure by using a spectral geometry method. Furthermore, subdomain decomposition of a porous plate/shell, thus converting the open-hole problem into a coupled problem of multiple non-hole plates/shells, is also a common approach to porous structures. Such methods are not limited to specific hole types and specific plate and shell structures, and are highly applicable and operable. Kwak [11] took laminated closed cones, cylindrical shells, and annular plates containing regular holes as the object of study and proposed a new, meshless method to solve their free vibration characteristics, in which the open-hole structure is decomposed into multiple non-hole unclosed shells, which are mainly aimed at simple collocated structures with regular hole types and plates/shells. Song [12] combined the artificially connected spring technique and the domain partition integration method to develop a free vibration analytical model for arbitrarily shaped plates with different cutouts, and the analysis found that the variation in the hole size would have an irregular effect on the natural frequency of the plate. Sun et al. [13], based on the proportional boundary finite element method, the exact integration method, and the degree of freedom transformation technique, performed free vibration analysis on single and laminated single and stacked composite plates with multiple cutouts, and the results were summarized in the following table. The free vibration analysis of composite panels with multiple cut-outs is summarized by the fact that the boundary conditions, thickness-to-length ratio, and hole size have significant effects on the distribution of the intrinsic frequency of the panels with the cut-outs. Zhong et al. [14,15] divided the open-pore plywood into multiple quadrilateral/triangular units, solved the unit energy equations by means of coordinate transformation, coupled the units with the help of virtual springs, and then deduced the uniform in-plane vibration solutions for irregular plywood with complex holes under different boundaries.

Rotor systems are widely used in aviation, shipbuilding, power generation, and other fields. The related dynamics analysis is an important part of its dynamics design. The thin-disk rotor is a typical structure in the rotor system, which often works at a high rotational speed and in a high load-bearing environment. The traditional mass-point equivalent simulation of the thin disk modeling method is no longer applicable, and there is a need to take into account the flexibility of the thin-disk structure, which puts more demand on the dynamics of the rotor-modeling methodology and the breadth of the analysis. This puts forward higher requirements for rotor dynamics modeling methods and analysis. For the shaft–disk rotor, the existing research focuses on investigating its dynamic characteristics, such as vibration [16], response [17] and nonlinearity [18,19]. Heydari and Khorram [20] treated both the shaft and disk as flexible bodies and used the assumed modal method to develop a vibration model of the rotor system. They pointed out through analysis that the position of the flexible disk had a significant effect on the system’s intrinsic frequency and critical speed. Tuzzi et al. [21], on the other hand, investigated the coupling behavior of the flexible shaft–disk assembly under asymmetric support conditions. Zhao et al. [22] proposed a unified modeling method for rotating flexible shaft–disk systems based on the Kirchhoff and Euler–Bernoulli beam theories, as well as a unified modeling approach for rotating flexible shaft–disk systems. Furthermore, the dynamic modeling and characterization of complex systems, such as the shaft–disc–vane system [23,24,25], shaft–disc–drum system [26,27], and cracked rotor [28,29,30], are also directions of rotor research that are constantly being more deeply investigated.

With the rise of new materials with excellent mechanical properties, such as composites, scholars have begun to consider changing the materials of the rotor structure to meet the requirements of the system, which include light weight and high speed. Venkatachalam et al. [31,32] applied orthotropic anisotropic materials to the shaft–disk rotor structure, and, with the help of experiments and finite element methods, found that the use of composites with a fiber-laying direction of 90° could significantly reduce the vibration of the rotor. Wang et al. [33,34] proposed and validated a dynamic model of a flexible functional gradient rotor system with a variable thickness disc and comprehensively analyzed the effects of geometrical parameters, material parameters, and the shape of the disc cross-section on its vibration characteristics. They pointed out that the material parameters would affect the overall vibration characteristics of the system to a certain extent. Zhao et al. [35] proposed a modeling method for a graphene nanosheet-reinforced blade–disk system and found that it was possible to use an orthotropic anisotropic material for the rotor structure with the help of experimental and finite element methods. They investigated the impact of graphene nanosheet parameters on the system and concluded that incorporating graphene sheets can enhance the mechanical properties of the system. Furthermore, thinner and larger graphene sheets are observed to increase the structural stiffness. Cai et al. [36] also applied graphene nanosheets and porous copper–metal foams to a rotating shaft–disk blade system and performed resonance analyses and parameter interval uncertainty analyses. The effects of porosity and material parameters were comprehensively explored.

In summary, in the application of fiber woven composites, most of the simple components, such as plates, shells, beams, etc., are the macroscopic carriers of the materials, and their kinetic properties are either studied by experimental means or only with a single component as the object of study. There is still a lack of in-depth exploration of the numerical modeling and analysis of the dynamics of the complex coupled structure of the fiber woven materials; in the case of the open-pore structure, the object of study is generally a simple plate and shell structure. There is no focus on the application of simple structures to complex systems. On the whole, the existing studies seldom consider fiber woven composites, weight-reducing holes and shaft–disk rotors in an integrated way, and the modeling method of the fiber woven composite shaft–disk rotor system with weight-reducing holes has yet to be perfected.

In view of the above research deficiencies, this paper takes the fiber woven composite material shaft–disk rotor as the research object, adopts the domain decomposition and coordinate mapping technology for the porous disk, and establishes a unified dynamic model of the fiber woven composite shaft–disk system through simple beam and plate theory and the differential quadrature finite element method (DQFEM). The model has been verified by the literature, finite element simulation, and experimental comparison, and has been proven to have good convergence, accuracy, and practicability. The corresponding kinetic study was carried out to investigate the effects of material parameters and hole parameters on the rotor’s natural frequency and critical speed, and the conclusions obtained will be helpful to guide the lightweight and vibration-damping design of the shaft–disk rotor.

2. Dynamic Modeling Process of the Fiber Woven Composite Shaft–Disk Rotor

As shown in Figure 1a, the fiber woven composite shaft–disk rotor consists of a perforated ring plate and several shafts with different radii. The rotor system is placed in the absolute coordinate system C₀, and a rotational coordinate system C₁ is used to describe the rotational motion of the rotor, which rotates uniformly about the X₁ axis with an angular velocity of Ω. A body-fixed coordinate system C₂ is used to describe the displacement deformation of the perforated ring plate. The parameters of the rotor model geometry are shown in Figure 1b.

2.1. Application of Fiber Woven Composite

In this paper, fiber woven composites are applied to gear webs. Fiber woven composites are composites with fiber braids as reinforcements and resin, metal, ceramics, etc., as matrix materials, and the two are cured and molded by a curing process. The weaving process can be divided into plain, twill, satin, and more complex three-dimensional woven composites. The following study selects orthogonal plain-weave carbon fiber-reinforced resin matrix composites as the ring plate material, as shown in Figure 2. Meanwhile, the focus of the study is not on solving the mechanical parameters of the material, but on exploring the effect of its use in the rotor model. Therefore, reference [7], the Young’s modulus, shear modulus, Poisson’s ratio, and density for different total fiber volume fractions (V_f) are known quantities and are given in Figure 3.

2.2. Domain Decomposition and Coordinate Mapping Technique for Perforated Webs

In order to reduce weight, facilitate machining and handling, and enable the system’s critical speed to avoid the operating speed, holes are often cut in the rotor web. These are commonly round holes, sector holes, etc. The web structure with holes, unlike a simple ring plate, is not easy to cut in the circumferential and radial directions for dividing regular sector cells when modeling. Therefore, a domain decomposition method for the web with arbitrary complex hole patterns is given below, and a coordinate mapping technique is introduced for subsequent dynamic modeling of the rotor.

As an example, a ring plate with N round holes with inner/outer diameters of R_i/R_o can be uniformly decomposed into N sector plates with holes in the center, as shown in Figure 4. Since the modeling of the ring plate is usually performed in a rotating coordinate system, the sector plate is also divided into sub-domains in the r-θ coordinate system to ensure that the segmentation lines are easy-to-describe straight lines in the r-θ coordinate system. As in Figure 4, the i^th sector plate is decomposed into four irregular quadrilateral domains. It should be noted that the domain decomposition should avoid the tangent point of the hole-shaped boundary and the parallel line of the r/θ coordinate axis; otherwise, it will affect the accuracy of the subsequent coordinate mapping, which will lead to the computational results of the model with the problems of non-convergence and large error.

After the subdomain decomposition of the open ring plate, each subdomain can be directly regarded as a plate cell for solving the energy matrix of the cell based on the DQFEM. However, the unit is an irregular quadrilateral, which needs to be converted to a regular rectangular unit by means of the coordinate mapping technique. As shown in Figure 5, for any irregular quadrilateral cell in the r-θ coordinate system, assuming that it has p × q mapping nodes, the nodes here are only used to reflect the mapping relationship, which is different from the nodes used in the differential product in Section 2.3.

By mapping Equation (1) [37], irregular quadrilaterals in the r-θ coordinate system can be converted into square cells with −1 ≤ ξ ≤ 1 and −1 ≤ η ≤ 1 in the ξ-η coordinate system.

$$\begin{array}{l}r=R_{qp}{\psi }_{qp},\theta ={\Theta }_{qp}{\psi }_{qp},\\ R_{qp}=\left[r_{11},r_{12},\dots ,r_{ij},\dots ,r_{qp}\right],\\ {\Theta }_{qp}=\left[{\theta }_{11},{\theta }_{12},\dots ,{\theta }_{ij},\dots ,{\theta }_{qp}\right],\\ {\psi }_{qp}={\left[{\phi }_{11},{\phi }_{12},\dots ,{\phi }_{ij},\dots ,{\phi }_{qp}\right]}^{\mathrm{T}}\end{array}$$

(1)

(r_ij, θ_ij) denotes the coordinates of the mapping points distributed on the edges or inside of the quadrilateral cell. φ_ij denotes the mapping function corresponding to the ij mapping point (r_ij, θ_ij), and the specific expression [15] is

$${\phi }_{ij}\left(\xi ,\eta \right)=\left({\displaystyle \prod _{s=1,s\ne i}^p\frac{\xi -{\xi }_s}{{\xi }_{ij}-{\xi }_s}}\right)\left({\displaystyle \prod _{l=1,l\ne j}^q\frac{\eta -{\eta }_l}{{\eta }_{ij}-{\eta }_l}}\right)$$

(2)

After the ring plate irregular unit has been coordinate-mapped, the two-dimensional differential product weight coefficient matrix for modeling using the DQFEM needs to be changed accordingly. The existing coordinates (r, θ) are functions with respect to (ξ, η), and the displacement function f of the structure is a composite function of (ξ, η). The basic two-dimensional differential product weight coefficient matrices A₀₂, B₀₂ and the integral weight coefficient matrix C₀₂ are derived in the ξ-η coordinate system. And for the irregular cells in the r-θ coordinate system, the weight coefficient matrices need to be corrected by the partial derivatives of r and θ with respect to ξ and η. The correction formula is derived as follows. The partial derivatives of the displacement function f with respect to ξ and η can be expressed as:

$$\left[\begin{array}{c}\partial f/\partial \xi \\ \partial f/\partial \eta \end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{\partial r}{\partial \xi }}& {\displaystyle \frac{\partial \theta }{\partial \xi }}\\ {\displaystyle \frac{\partial r}{\partial \eta }}& {\displaystyle \frac{\partial \theta }{\partial \eta }}\end{array}\right]\left[\begin{array}{c}\partial f/\partial r\\ \partial f/\partial \theta \end{array}\right]=J\left[\begin{array}{c}\partial f/\partial r\\ \partial f/\partial \theta \end{array}\right]$$

(3)

where J is the Jacobi matrix, and the partial derivatives of the displacement function f with respect to r and θ can be obtained from the inverse of J as follows:

$$\left[\begin{array}{c}\partial f/\partial r\\ \partial f/\partial \theta \end{array}\right]=J^{-1}\left[\begin{array}{c}\partial f/\partial \xi \\ \partial f/\partial \eta \end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{\partial \theta }{\partial \eta }}& -{\displaystyle \frac{\partial \theta }{\partial \xi }}\\ -{\displaystyle \frac{\partial r}{\partial \eta }}& {\displaystyle \frac{\partial r}{\partial \xi }}\end{array}\right]/\left[\begin{array}{c}\partial f/\partial \xi \\ \partial f/\partial \eta \end{array}\right]\left({\displaystyle \frac{\partial r}{\partial \xi }}{\displaystyle \frac{\partial \theta }{\partial \eta }}-{\displaystyle \frac{\partial \theta }{\partial \xi }}{\displaystyle \frac{\partial r}{\partial \eta }}\right)$$

(4)

$${\displaystyle \frac{\partial f}{\partial r}}={\displaystyle \frac{1}{\left|J\right|}}\left({\displaystyle \frac{\partial \theta }{\partial \eta }}{\displaystyle \frac{\partial f}{\partial \xi }}-{\displaystyle \frac{\partial \theta }{\partial \xi }}{\displaystyle \frac{\partial f}{\partial \eta }}\right),{\displaystyle \frac{\partial f}{\partial \theta }}={\displaystyle \frac{1}{\left|J\right|}}\left(-{\displaystyle \frac{\partial r}{\partial \eta }}{\displaystyle \frac{\partial f}{\partial \xi }}+{\displaystyle \frac{\partial r}{\partial \xi }}{\displaystyle \frac{\partial f}{\partial \eta }}\right)$$

(5)

According to Equation (1), the partial derivative operator in Equations (4) and (5) can be obtained as:

$$\begin{array}{l}{\displaystyle \frac{\partial r}{\partial \xi }}={\displaystyle \frac{\partial 〈{\displaystyle \sum _{ij=11}^{p\times q}{r}_{ij}{\phi }_{ij}\left(\xi ,\eta \right)}〉}{\partial \xi }},{\displaystyle \frac{\partial r}{\partial \eta }}={\displaystyle \frac{\partial 〈{\displaystyle \sum _{ij=11}^{p\times q}{r}_{ij}{\phi }_{ij}\left(\xi ,\eta \right)}〉}{\partial \eta }},\\ {\displaystyle \frac{\partial \theta }{\partial \xi }}={\displaystyle \frac{\partial 〈{\displaystyle \sum _{ij=11}^{p\times q}{\theta }_{ij}{\phi }_{ij}\left(\xi ,\eta \right)}〉}{\partial \xi }},{\displaystyle \frac{\partial \theta }{\partial \eta }}={\displaystyle \frac{\partial 〈{\displaystyle \sum _{ij=11}^{p\times q}{\theta }_{ij}{\phi }_{ij}\left(\xi ,\eta \right)}〉}{\partial \eta }}\end{array}$$

(6)

The original weighting coefficient matrix expressions are:

$${\left.{\displaystyle \frac{\partial f\left(\xi ,\eta \right)}{\partial \xi }}\right|}_{ij}={\displaystyle \sum _{m=1}^M{A}_{im}}{f}_{mj}=A_{02}^{\left(1\right)}f,{\left.{\displaystyle \frac{\partial f\left(\xi ,\eta \right)}{\partial \eta }}\right|}_{ij}={\displaystyle \sum _{n=1}^N{B}_{jn}}{f}_{in}=B_{02}^{\left(1\right)}f$$

(7)

Here and in the following equation, i and j are the number of radial and circumferential nodes in the quadrilateral cell divided based on DQFEM, respectively. i = 1, 2, …, M, and j = 1, 2, …, N. The actual weight coefficients’ matrix expression is corrected to:

$${\left.{\displaystyle \frac{\partial f}{\partial r}}\right|}_{ij}={\displaystyle \frac{1}{{\left|J\right|}_{ij}}}\left[{\left({\displaystyle \frac{\partial \theta }{\partial \eta }}\right)}_{ij}{A}_{02}-{\left({\displaystyle \frac{\partial \theta }{\partial \xi }}\right)}_{ij}{B}_{02}\right],{\left.{\displaystyle \frac{\partial f}{\partial \theta }}\right|}_{ij}={\displaystyle \frac{1}{{\left|J\right|}_{ij}}}\left[-{\left({\displaystyle \frac{\partial r}{\partial \eta }}\right)}_{ij}{A}_{02}+{\left({\displaystyle \frac{\partial r}{\partial \xi }}\right)}_{ij}{B}_{02}\right]$$

(8)

For ease of presentation, the above equation can be rewritten as:

$$\begin{array}{l}{A}_2={\displaystyle \frac{1}{Y}}.\ast \left({\theta }_{\eta }.\ast A_{02}-{\theta }_{\xi }.\ast B_{02}\right),B_2={\displaystyle \frac{1}{Y}}.\ast \left(-r_{\eta }.\ast A_{02}+r_{\xi }.\ast B_{02}\right),\\ a_{\alpha }={\left[\begin{array}{cccccc}{\left({\displaystyle \frac{\partial a}{\partial \alpha }}\right)}_{11}& \dots & {\left({\displaystyle \frac{\partial a}{\partial \alpha }}\right)}_{M1}& {\left({\displaystyle \frac{\partial a}{\partial \alpha }}\right)}_{12}& \dots & {\left({\displaystyle \frac{\partial a}{\partial \alpha }}\right)}_{MN}\end{array}\right]}^{\mathrm{T}},a=r,\theta ;\alpha =\xi ,\eta ,\\ Y={\left[\begin{array}{cccccc}{\left|J\right|}_{11}& \dots & {\left|J\right|}_{M1}& {\left|J\right|}_{12}& \dots & {\left|J\right|}_{MN}\end{array}\right]}^{\mathrm{T}}\end{array}$$

(9)

where the symbol a_α denotes the partial derivative of a with respect to α and “.*” denotes the Hadamard product. Similarly, the modified matrix of integral weight coefficients is:

$$C_2=Y.\ast C_{02}$$

(10)

In order to facilitate the parametric study, four types of holes are given: (1) no holes; (2) round holes; (3) sector holes; and (4) curved holes. The shape and geometry of the various hole types are shown in Figure 6, and the geometric parameters of the different hole types should be set to ensure that the area of a single hole is the same in the subsequent parameterization study.

2.3. Energy Derivation of System Structure Units

As shown in Figure 7, the nodes on the shaft in the rotor have six degrees of freedom (u_S, v_S, w_S, φ_S,x, φ_S,y, φ_S,z), and the nodes on the disk have five degrees of freedom (u_D, v_D, w_D, φ_D,r, φ_D,θ).

2.3.1. Potential and Kinetic Energy of the Rotating Shaft Units

Based on the Timoshenko beam theory, the shaft in the rotor can be equated to a six-degree-of-freedom beam, and the elastic deformation displacement at a point (x_S, y_S, z_S) on the beam is:

$${\overline{w}}_{\mathrm{S}}=w_{\mathrm{S}}\left(x_{\mathrm{S}}\right)-y_{\mathrm{S}}{\phi }_{\mathrm{S},z}\left(x_{\mathrm{S}}\right)+z_{\mathrm{S}}{\phi }_{\mathrm{S},y}\left(x_{\mathrm{S}}\right),{\overline{u}}_{\mathrm{S}}=u_{\mathrm{S}}\left(x_{\mathrm{S}}\right)-z_{\mathrm{S}}{\phi }_{\mathrm{S},x}\left(x_{\mathrm{S}}\right),{\overline{v}}_{\mathrm{S}}=v_{\mathrm{S}}\left(x_{\mathrm{S}}\right)+y_{\mathrm{S}}{\phi }_{\mathrm{S},x}\left(x_{\mathrm{S}}\right)$$

(11)

where y_S = z_S = 0. Based on elastic mechanics, the strain at this point on the axis is deduced to be:

$${\epsilon }_{\mathrm{S},x}={\displaystyle \frac{\partial w_{\mathrm{S}}}{\partial x_{\mathrm{S}}}}-y_{\mathrm{S}}{\displaystyle \frac{\partial {\phi }_{\mathrm{S},z}}{\partial x_{\mathrm{S}}}}+z_{\mathrm{S}}{\displaystyle \frac{\partial {\phi }_{\mathrm{S},y}}{\partial x_{\mathrm{S}}}},{\gamma }_{\mathrm{S},xy}={\displaystyle \frac{\partial u_{\mathrm{S}}}{\partial x_{\mathrm{S}}}}-z_{\mathrm{S}}{\displaystyle \frac{\partial {\phi }_{\mathrm{S},x}}{\partial x_{\mathrm{S}}}}-{\phi }_{\mathrm{S},z},{\gamma }_{\mathrm{S},zx}={\displaystyle \frac{\partial v_{\mathrm{S}}}{\partial x_{\mathrm{S}}}}+y_{\mathrm{S}}{\displaystyle \frac{\partial {\phi }_{\mathrm{S},x}}{\partial x_{\mathrm{S}}}}+{\phi }_{\mathrm{S},y}$$

(12)

In turn, the potential energy of the shaft is:

$$\begin{array}{ll}{V}_{\mathrm{S}}& ={\displaystyle \frac{1}{2}}{\displaystyle \int {\displaystyle \int {\displaystyle \int \left(E_{\mathrm{S}}{\epsilon }_{\mathrm{S},x}^2+G_{\mathrm{S}}{\gamma }_{\mathrm{S},xy}^2+G_{\mathrm{S}}{\gamma }_{\mathrm{S},zx}^2\right)}}}\mathrm{d}{y}_{\mathrm{S}}\mathrm{d}{z}_{\mathrm{S}}\mathrm{d}{x}_{\mathrm{S}}\\ & ={\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{L}{\int }}\left(\begin{array}{l}{E}_{\mathrm{S}}{A}_{\mathrm{S}}{\left(\frac{\partial w_{\mathrm{S}}}{\partial x_{\mathrm{S}}}\right)}^2+E_{\mathrm{S}}{I}_{\mathrm{S}}{\left(\frac{\partial {\phi }_{\mathrm{S},z}}{\partial x_{\mathrm{S}}}\right)}^2+E_{\mathrm{S}}{I}_{\mathrm{S}}{\left(\frac{\partial {\phi }_{\mathrm{S},y}}{\partial x_{\mathrm{S}}}\right)}^2\\ +{\kappa }_{\mathrm{S}}{G}_{\mathrm{S}}{A}_{\mathrm{S}}{\left(\frac{\partial u_{\mathrm{S}}}{\partial x_{\mathrm{S}}}-{\phi }_{\mathrm{S},z}\right)}^2+{\kappa }_{\mathrm{S}}{G}_{\mathrm{S}}{A}_{\mathrm{S}}{\left(\frac{\partial v_{\mathrm{S}}}{\partial x_{\mathrm{S}}}+{\phi }_{\mathrm{S},y}\right)}^2+2G_{\mathrm{S}}{I}_{\mathrm{S}}{\left(\frac{\partial {\phi }_{\mathrm{S},x}}{\partial x_{\mathrm{S}}}\right)}^2\end{array}\right)}\mathrm{d}{x}_{\mathrm{S}}\end{array}$$

(13)

where E_S, κ_S (= 6(1 + μ_S)²/(7 + 12μ_S + 4μ_S²)) are the modulus of elasticity and shear modification factor of the shaft, respectively, and μ_S is Poisson’s ratio. Its cross-sectional area A_S = πr², moment of inertia I_S = πr⁴/4, and shear modulus G_S = E_S/2/(1 + μ_S). L and r are the length and radius of the shaft section, respectively.

Referring to my existing research [38], the kinetic energy of the shaft is:

$$T_{\mathrm{S}}={\displaystyle \frac{1}{2}}{\rho }_{\mathrm{S}}{\displaystyle \underset{0}{\overset{L_{\mathrm{S}}}{\int }}\left(\begin{array}{l}{A}_{\mathrm{S}}\left({\dot{w}}_{\mathrm{S}}^2+{\dot{u}}_{\mathrm{S}}^2+{\dot{v}}_{\mathrm{S}}^2\right)+I_{\mathrm{S}}\left(2{\dot{\phi }}_{\mathrm{S},x}^2+{\dot{\phi }}_{\mathrm{S},y}^2+{\dot{\phi }}_{\mathrm{S},z}^2\right)\\ +2\Omega A_{\mathrm{S}}\left(u_{\mathrm{S}}{\dot{v}}_{\mathrm{S}}-{\dot{u}}_{\mathrm{S}}{v}_{\mathrm{S}}\right)-2\Omega I_{\mathrm{S}}\left({\dot{\phi }}_{\mathrm{S},y}{\phi }_{\mathrm{S},z}+{\dot{\phi }}_{\mathrm{S},z}{\phi }_{\mathrm{S},y}\right)\\ +{\Omega }^2{A}_{\mathrm{S}}\left(u_{\mathrm{S}}^2+v_{\mathrm{S}}^2\right)-{\Omega }^2{I}_{\mathrm{S}}\left({\phi }_{\mathrm{S},y}^2+{\phi }_{\mathrm{S},z}^2\right)\end{array}\right)}\mathrm{d}{x}_{\mathrm{S}}$$

(14)

Ω denotes the angular velocity of the rotor, and ρ_S denotes the density of the shaft.

2.3.2. Potential and Kinetic Energy of the Disk Units

According to the first-order shear deformation theory (FSDT), the displacement of a point (x_D, r_D, θ_D) on the disk is expressed as:

$$\begin{array}{l}{\overline{u}}_{\mathrm{D}}(r_{\mathrm{D}},{\theta }_{\mathrm{D}},x_{\mathrm{D}},t)=u_{\mathrm{D}}(r_{\mathrm{D}},{\theta }_{\mathrm{D}},t)+x_{\mathrm{D}}{\phi }_{\mathrm{D},\theta }(r_{\mathrm{D}},{\theta }_{\mathrm{D}},t),\\ {\overline{v}}_{\mathrm{D}}(r_{\mathrm{D}},{\theta }_{\mathrm{D}},x_{\mathrm{D}},t)=v_{\mathrm{D}}(r_{\mathrm{D}},{\theta }_{\mathrm{D}},t)-x_{\mathrm{D}}{\phi }_{\mathrm{D},r}(r_{\mathrm{D}},{\theta }_{\mathrm{D}},t),\\ {\overline{w}}_{\mathrm{D}}(r_{\mathrm{D}},{\theta }_{\mathrm{D}},x_{\mathrm{D}},t)=w_{\mathrm{D}}(r_{\mathrm{D}},{\theta }_{\mathrm{D}},t)\end{array}$$

(15)

The corresponding strain is:

$$\begin{array}{l}{\epsilon }_{\mathrm{D},r}={\epsilon }_{\mathrm{D},r}^0+x_{\mathrm{D}}{\chi }_{\mathrm{D},r}^0={\displaystyle \frac{\partial u_{\mathrm{D}}}{\partial r_{\mathrm{D}}}}+x_{\mathrm{D}}{\displaystyle \frac{\partial {\phi }_{\mathrm{D},\theta }}{\partial r_{\mathrm{D}}}},\\ {\epsilon }_{\mathrm{D},\theta }={\epsilon }_{\mathrm{D},\theta }^0+x_{\mathrm{D}}{\chi }_{\mathrm{D},\theta }^0={\displaystyle \frac{1}{r_{\mathrm{D}}}}{\displaystyle \frac{\partial v_{\mathrm{D}}}{\partial {\theta }_{\mathrm{D}}}}+{\displaystyle \frac{u_{\mathrm{D}}}{r_{\mathrm{D}}}}+x_{\mathrm{D}}\left(-{\displaystyle \frac{1}{r_{\mathrm{D}}}}{\displaystyle \frac{\partial {\phi }_{\mathrm{D},r}}{\partial {\theta }_{\mathrm{D}}}}+{\displaystyle \frac{{\phi }_{\mathrm{D},\theta }}{r_{\mathrm{D}}}}\right),\\ {\gamma }_{\mathrm{D},r\theta }={\gamma }_{\mathrm{D},r\theta }^0+x_{\mathrm{D}}{\chi }_{\mathrm{D},r\theta }^0={\displaystyle \frac{1}{r_{\mathrm{D}}}}{\displaystyle \frac{\partial u_{\mathrm{D}}}{\partial {\theta }_{\mathrm{D}}}}+{\displaystyle \frac{\partial v_{\mathrm{D}}}{\partial r_{\mathrm{D}}}}-{\displaystyle \frac{v_{\mathrm{D}}}{r_{\mathrm{D}}}}+x_{\mathrm{D}}\left({\displaystyle \frac{1}{r_{\mathrm{D}}}}{\displaystyle \frac{\partial {\phi }_{\mathrm{D},\theta }}{\partial {\theta }_{\mathrm{D}}}}-{\displaystyle \frac{\partial {\phi }_{\mathrm{D},r}}{\partial r_{\mathrm{D}}}}+{\displaystyle \frac{{\phi }_{\mathrm{D},r}}{r_{\mathrm{D}}}}\right),\\ {\gamma }_{\mathrm{D},rx}={\displaystyle \frac{\partial w_{\mathrm{D}}}{\partial r_{\mathrm{D}}}}+{\phi }_{\mathrm{D},\theta },{\gamma }_{\mathrm{D},\theta x}={\displaystyle \frac{1}{r_{\mathrm{D}}}}{\displaystyle \frac{\partial w_{\mathrm{D}}}{\partial {\theta }_{\mathrm{D}}}}-{\phi }_{\mathrm{D},r}\end{array}$$

(16)

According to Hooke’s law, the stresses of orthogonal plain-weave carbon fiber-reinforced resin matrix composite disks are:

$$\left[\begin{array}{c}{\sigma }_{\mathrm{D},r}\\ {\sigma }_{\mathrm{D},\theta }\\ {\tau }_{\mathrm{D},r\theta }\\ {\tau }_{\mathrm{D},rx}\\ {\tau }_{\mathrm{D},\theta x}\end{array}\right]=\left[\begin{array}{ccccc}{Q}_{11}& Q_{12}& 0& 0& 0\\ Q_{21}& Q_{22}& 0& 0& 0\\ 0& 0& Q_{66}& 0& 0\\ 0& 0& 0& Q_{44}& 0\\ 0& 0& 0& 0& Q_{55}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\mathrm{D},r}\\ {\epsilon }_{\mathrm{D},\theta }\\ {\gamma }_{\mathrm{D},r\theta }\\ {\gamma }_{\mathrm{D},rx}\\ {\gamma }_{\mathrm{D},\theta x}\end{array}\right]=\left[\begin{array}{ccccc}{\displaystyle \frac{E_{11}}{1-{\mu }_{12}{\mu }_{21}}}& {\displaystyle \frac{{\mu }_{21}{E}_{11}}{1-{\mu }_{12}{\mu }_{21}}}& 0& 0& 0\\ {\displaystyle \frac{{\mu }_{12}{E}_{22}}{1-{\mu }_{12}{\mu }_{21}}}& {\displaystyle \frac{E_{22}}{1-{\mu }_{12}{\mu }_{21}}}& 0& 0& 0\\ 0& 0& G_{12}& 0& 0\\ 0& 0& 0& G_{23}& 0\\ 0& 0& 0& 0& G_{13}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\mathrm{D},r}\\ {\epsilon }_{\mathrm{D},\theta }\\ {\gamma }_{\mathrm{D},r\theta }\\ {\gamma }_{\mathrm{D},rx}\\ {\gamma }_{\mathrm{D},\theta x}\end{array}\right]$$

(17)

Since the first-order shear deformation theory is applied to the ring plate, only E₁₁, E₂₂, μ₁₂, and μ₂₁ are introduced for orthotropic anisotropic materials, and they satisfy E₁₁μ₂₁ = E₂₂μ₁₂, i.e., Q₁₂ = Q₂₁. In summary, the potential energy of the ring plate can be found to be:

$$\begin{array}{l}{V}_{\mathrm{D}}={\displaystyle \frac{1}{2}}{\displaystyle \int {\displaystyle \int {\displaystyle \int \left(\begin{array}{l}{\sigma }_{\mathrm{D},r}{\epsilon }_{\mathrm{D},r}+{\sigma }_{\mathrm{D},\theta }{\epsilon }_{\mathrm{D},\theta }+{\tau }_{\mathrm{D},r\theta }{\gamma }_{\mathrm{D},r\theta }\\ +{\kappa }_{\mathrm{D}}{\tau }_{\mathrm{D},rx}{\gamma }_{\mathrm{D},rx}+{\kappa }_{\mathrm{D}}{\tau }_{\mathrm{D},\theta x}{\gamma }_{\mathrm{D},\theta x}\end{array}\right)}{r}_{\mathrm{D}}}}\mathrm{d}{r}_{\mathrm{D}}\mathrm{d}{\theta }_{\mathrm{D}}\mathrm{d}{x}_{\mathrm{D}}\\ ={\displaystyle \frac{1}{2}}{\displaystyle \int {\displaystyle \int {\displaystyle \int \left(\begin{array}{l}{Q}_{11}{r}_{\mathrm{D}}{\left(\frac{\partial u_{\mathrm{D}}}{\partial r_{\mathrm{D}}}+x_{\mathrm{D}}\frac{\partial {\phi }_{\mathrm{D},\theta }}{\partial r_{\mathrm{D}}}\right)}^2+\frac{Q_{22}}{r_{\mathrm{D}}}{\left(\frac{\partial v_{\mathrm{D}}}{\partial {\theta }_{\mathrm{D}}}+u_{\mathrm{D}}-x_{\mathrm{D}}\left(\frac{\partial {\phi }_{\mathrm{D},r}}{\partial {\theta }_{\mathrm{D}}}-{\phi }_{\mathrm{D},\theta }\right)\right)}^2\\ +2Q_{12}\left(\frac{\partial u_{\mathrm{D}}}{\partial r_{\mathrm{D}}}+x_{\mathrm{D}}\frac{\partial {\phi }_{\mathrm{D},\theta }}{\partial r_{\mathrm{D}}}\right)\left(\frac{\partial v_{\mathrm{D}}}{\partial {\theta }_{\mathrm{D}}}+u_{\mathrm{D}}-x_{\mathrm{D}}\left(\frac{\partial {\phi }_{\mathrm{D},r}}{\partial {\theta }_{\mathrm{D}}}-{\phi }_{\mathrm{D},\theta }\right)\right)\\ +Q_{66}{r}_{\mathrm{D}}{\left(\frac{1}{r_{\mathrm{D}}}\frac{\partial u_{\mathrm{D}}}{\partial {\theta }_{\mathrm{D}}}+\frac{\partial v_{\mathrm{D}}}{\partial r_{\mathrm{D}}}-\frac{v_{\mathrm{D}}}{r_{\mathrm{D}}}+x_{\mathrm{D}}\left(\frac{\partial {\phi }_{\mathrm{D},\theta }}{\partial {\theta }_{\mathrm{D}}}-r_{\mathrm{D}}\frac{\partial {\phi }_{\mathrm{D},r}}{\partial r_{\mathrm{D}}}+{\phi }_{\mathrm{D},r}\right)\right)}^2\\ +{\kappa }_{\mathrm{D}}{Q}_{44}{r}_{\mathrm{D}}{\left(\frac{\partial w_{\mathrm{D}}}{\partial r_{\mathrm{D}}}+{\phi }_{\mathrm{D},\theta }\right)}^2+{\kappa }_{\mathrm{D}}{Q}_{55}{r}_{\mathrm{D}}{\left(\frac{1}{r_{\mathrm{D}}}\frac{\partial w_{\mathrm{D}}}{\partial {\theta }_{\mathrm{D}}}-{\phi }_{\mathrm{D},r}\right)}^2\end{array}\right)}\mathrm{d}{r}_{\mathrm{D}}\mathrm{d}{\theta }_{\mathrm{D}}\mathrm{d}{x}_{\mathrm{D}}}}\end{array}$$

(18)

where the shear correction factor κ_D = 5/6.

For a multi-component coupled geared rotor, a multi-body dynamics approach is used to calculate the kinetic energy of the disk in the system. In the absolute coordinate system C₀, the position vector of a point (x_D, r_D, θ_D) on the disk is shown in Equation (19).

$$r_{\mathrm{D}}=R_0+\left(R_{\mathrm{D}}+U_{\mathrm{D}}{A}_1\right)A_{\Omega }$$

(19)

where R₀ (= [X, Y, Z]) is the origin’s position of the rotating coordinate system C₁ in the absolute coordinate system C₀. R_D denotes the position vector of the coupling nodes of the shaft and the disk, whose value is R_D = [w_SC + x_SC, u_SC, v_SC], and the subscript “SC” denotes the coupling nodes of the shaft and the disk. A₁ and A_Ω are the coordinate transformation relation matrices from C₂ to C₁ and C₁ to C₀, and their expressions are detailed in the literature [38]. The symbol U_D in Equation (19) denotes the position vector of a point (x_D, r_D, θ_D) on the disk in the follower coordinate system with the following expression:

$$\begin{array}{ll}{U}_{\mathrm{D}}& =R_{\mathrm{DP}}+U_{\mathrm{DP}}\\ & ={\left[\begin{array}{c}{x}_{\mathrm{D}}\\ r_{\mathrm{D}}\cos {\theta }_{\mathrm{D}}\\ r_{\mathrm{D}}\sin {\theta }_{\mathrm{D}}\end{array}\right]}^{\mathrm{T}}+{\left[\begin{array}{c}{w}_{\mathrm{D}}\\ \left(u_{\mathrm{D}}+x_{\mathrm{D}}{\phi }_{\mathrm{D},\theta }\right)\cos {\theta }_{\mathrm{D}}-\left(v_{\mathrm{D}}-x_{\mathrm{D}}{\phi }_{\mathrm{D},r}\right)\sin {\theta }_{\mathrm{D}}\\ \left(u_{\mathrm{D}}+x_{\mathrm{D}}{\phi }_{\mathrm{D},\theta }\right)\sin {\theta }_{\mathrm{D}}+\left(v_{\mathrm{D}}-x_{\mathrm{D}}{\phi }_{\mathrm{D},r}\right)\cos {\theta }_{\mathrm{D}}\end{array}\right]}^{\mathrm{T}}\end{array}$$

(20)

Combining the above equations, the velocity of the node on the disk can be obtained as:

$${\dot{r}}_{\mathrm{D}}=\left(R_{\mathrm{D}}+U_{\mathrm{D}}{A}_1\right){\dot{A}}_{\Omega }+\left({\dot{R}}_{\mathrm{D}}+{\dot{U}}_{\mathrm{D}}{A}_1+U_{\mathrm{D}}{\dot{A}}_1\right)A_{\Omega }$$

(21)

This, in turn, leads to an expression for the kinetic energy of the disk as:

$$T_{\mathrm{D}}={\displaystyle \frac{1}{2}}{\rho }_{\mathrm{D}}{\displaystyle \iiint {\dot{r}}_{\mathrm{D}}{{\dot{r}}_{\mathrm{D}}}^{\mathrm{T}}}{r}_{\mathrm{D}}\mathrm{d}{r}_{\mathrm{D}}\mathrm{d}{\theta }_{\mathrm{D}}\mathrm{d}{x}_{\mathrm{D}}$$

(22)

where ρ denotes the density. The kinetic energy formula for the disk is too lengthy to show specifically.

2.4. Overall Model of the Rotor System

For the overall structure, the energy expressions (Equations (13), (14), (18) and (22)) for each structural element are converted to matrix form based on the DQFEM:

$$\begin{array}{l}{V}_j^e={\displaystyle \frac{1}{2}}{q_j^e}^{\mathrm{T}}{K}_j^e{q}_j^e,\left(j=\mathrm{S},\mathrm{D}\right)\\ q_{\mathrm{S}}^e={\left[\begin{array}{cccccc}{w}_{\mathrm{S}}& u_{\mathrm{S}}& v_{\mathrm{S}}& {\phi }_{\mathrm{S},x}& {\phi }_{\mathrm{S},y}& {\phi }_{\mathrm{S},z}\end{array}\right]}^{\mathrm{T}};\\ q_{\mathrm{D}}^e={\left[\begin{array}{ccccc}{w}_{\mathrm{D}}& u_{\mathrm{D}}& v_{\mathrm{D}}& {\phi }_{\mathrm{D},r}& {\phi }_{\mathrm{D},\theta }\end{array}\right]}^{\mathrm{T}}\end{array}$$

(23)

$$\begin{array}{l}{T}_j^e={\displaystyle \frac{1}{2}}{{\dot{q}}_j^e}^{\mathrm{T}}{M}_j^e{\dot{q}}_j^e+{\displaystyle \frac{1}{2}}{{\dot{q}}_j^e}^{\mathrm{T}}{G}_j^e{q}_j^e+{\displaystyle \frac{1}{2}}{q_j^e}^{\mathrm{T}}{C}_j^e{q}_j^e,\left(j=\mathrm{S},\mathrm{D}\right)\\ j=\mathrm{S},q_j^e={\left[q_{\mathrm{S}}^e\right]}^{\mathrm{T}};j=\mathrm{D},q_j^e={\left[\begin{array}{cc}{q}_{\mathrm{SC}}^e& q_{\mathrm{D}}^e\end{array}\right]}^{\mathrm{T}};\\ q_{\mathrm{SC}}^e={\left[\begin{array}{cccccc}{w}_{\mathrm{SC}}& w_{\mathrm{SC}}& v_{\mathrm{SC}}& {\phi }_{\mathrm{SC},x}& {\phi }_{\mathrm{SC},y}& {\phi }_{\mathrm{SC},z}\end{array}\right]}^{\mathrm{T}}\end{array}$$

(24)

where K^e, M^e, G^e, and C^e denote the cell stiffness matrix, mass matrix, gyro matrix, and damping matrix, respectively.

The coupling relationship between the shaft segments and the disk is included in the above kinetic energy expression. For the whole geared rotor system, it is also necessary to add the coupling potentials and boundary potentials of the individual shaft segments, shafts, and disks to the total potential energy, and the connections between the substructures and the classical boundary conditions are simulated using the artificial spring technique. For the connection between n shaft segments, three sets of translational springs and three sets of rotational springs are defined to simulate the connection, with spring stiffnesses k_SS,u, k_SS,v, k_SS,w, k_SS,x, k_SS,y, and k_SS,z (all of which have values of 1 × 10¹³). The corresponding potential energy V_SS is:

$$V_{\mathrm{SS}}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum k_{\mathrm{SS},m}{\left(m_{\mathrm{S},r_{i+1}}-m_{\mathrm{S},l_i}\right)}^2}},(m=w,u,v,{\phi }_x,{\phi }_y,{\phi }_z)$$

(25)

where the subscripts r_i+1 and l_i denote the rightmost node of the (I + 1)st shaft segment and the leftmost node of the i-th shaft segment, respectively. n denotes the total number of shaft segments. For the simulation of the coupling point on the shaft connected to the nodes of the inner ring plate, three sets of translational springs and two sets of rotational springs with spring stiffnesses k_S,u, k_S,v, k_S,w, k_S,x, k_S,φr, and k_S,φθ (all of which have values of 1 × 10¹³) are defined. The corresponding potential energy V_C is:

$$V_{\mathrm{C}}={\displaystyle \frac{1}{2}}{{\displaystyle \underset{0}{\overset{2\pi }{\int }}\left[{\displaystyle \sum k_{\mathrm{SD},m}{m_{\mathrm{D},a}}^2}\right]}|}_{a=R_i}{r}_{\mathrm{D}}\mathrm{d}{\theta }_{\mathrm{D}},(m=w,u,v,{\phi }_r,{\phi }_{\theta })$$

(26)

For the simulation of the boundary conditions, three sets of translational springs and three sets of rotational springs are defined with spring stiffnesses k_S,u, k_S,v, k_S,w, k_S,x, k_S,y, and k_S,z. The boundary potential energy is:

$$V_B={\displaystyle \frac{1}{2}}{\left({\displaystyle \sum k_{\mathrm{S},m,a}{m_{\mathrm{S},a}}^2+{\displaystyle \sum k_{\mathrm{S},m,b}{m_{\mathrm{S},b}}^2}}\right)|}_{a=0,b=L},(m=w,u,v,{\phi }_x,{\phi }_y,{\phi }_z)$$

(27)

We define three boundaries: free (F), simply supported (S), and solidly supported (C) with the following spring stiffness values:

$$\begin{array}{l}\mathrm{F}:k_{S,u}=k_{S,v}=k_{S,w}=0,{\phi }_{\mathrm{S},x}={\phi }_{\mathrm{S},y}={\phi }_{\mathrm{S},z}=0\\ \mathrm{S}:k_{S,u}=k_{S,v}=k_{S,w}=1\times {10}^{13},{\phi }_{\mathrm{S},x}={\phi }_{\mathrm{S},y}={\phi }_{\mathrm{S},z}=0\\ \mathrm{C}:k_{S,u}=k_{S,v}=k_{S,w}=1\times {10}^{13},{\phi }_{\mathrm{S},x}={\phi }_{\mathrm{S},y}={\phi }_{\mathrm{S},z}=1\times {10}^{13}\end{array}$$

(28)

In summary, the total potential and kinetic energies of the system are:

$$V=V_{\mathrm{S}}+V_{\mathrm{D}}+V_{\mathrm{SS}}+V_{\mathrm{C}}+V_{\mathrm{B}}$$

(29)

$$T=T_{\mathrm{S}}+T_{\mathrm{D}}$$

(30)

Based on the DQFEM, the differential equations of motion for a single gear rotor system can be derived by substituting the matrix form of the total potential and kinetic energies of the system into the Lagrangian equation:

$$\begin{array}{l}\left(M^{\mathrm{T}}+M\right)\ddot{q}+\left(G^{\mathrm{T}}-G\right)\dot{q}+\left[\left(K^{\mathrm{T}}+K\right)-\left(C^{\mathrm{T}}+C\right)\right]q=0,\\ q=\left[\begin{array}{cc}{q}_{\mathrm{S}}& q_{\mathrm{D}}\end{array}\right]\end{array}$$

(31)

where M, G, C, and K are the overall mass matrix, gyro matrix, damping matrix, and stiffness matrix assembled from the individual unit matrices, boundary potential matrix, and coupling potential matrix between the structures. By solving the eigenvalues and eigenvectors of Equation (31), the vibration characteristics of the system, such as the natural frequency and vibration mode, can be obtained.

3. Numerical Analysis

Based on the proposed DQFEM, this study utilizes MATLAB programming to calculate the vibration characteristics of a fiber woven composite shaft–disk rotor with weight-reducing holes. Initially, a convergence analysis of the proposed DQFEM model is conducted; subsequently, the computational results are compared with those obtained from the commercial finite element software ANSYS and experimental modal testing to verify the accuracy of the established model. Finally, a parametric study on the hole parameters of the disk is carried out.

3.1. Convergence Study

From Section 2.2, it can be seen that the disk has a definite number of units, so the convergence analysis mainly focuses on the number of units of the shaft section (taking the number of unit nodes of the shaft section as 3) and the number of unit nodes of the ring plate. The convergence analysis is carried out by taking the rotor made of fiber woven composite with round holes as an example, and the structure of the rotor is as follows: the length of the shaft L = 0.2 m, the radius r₁ = r₂ = 0.02 m; the thickness of the disk h_D = 0.008 m, the position L_d = 0.1 m, the inner diameter R_i = 0.02 m, the outer diameter R_o = 0.1 m, the number of round holes N = 4, the radius of the round holes r_h = 0.025 m, and the eccentricity of the round holes R_h = 0.06 m. The number of shaft segment units not used as variables is 5, and the number of ring plate unit nodes is 10. The boundaries of the rotor are set at the two ends of the rotor shaft, both of which are simply supported boundaries and are denoted as S–S.

Unless otherwise stated, the shaft in the following calculations is made of steel with Young’s modulus E = 210 GPa, Poisson’s ratio μ = 0.3, and density ρ = 7800 kg/m³, and the disk material is an orthotropic plain-woven carbon fiber-reinforced resin matrix composite when the total fiber volume fraction is equal to 0.28. The mechanical properties of the materials with different total fiber volume fractions are all given in Figure 3.

The first six orders of the natural frequencies of the fiber woven composite rotor are given in

Table 1. Convergence of the natural frequency (Hz) for fiber woven composite rotors with respect to the number of shaft units.

Natural Frequency Order	Number of Shaft Units
	1	2	3	4	5	6	7
1	565.25	565.05	565.03	565.03	565.03	565.03	565.03
2	646.35	646.35	646.35	646.35	646.35	646.35	646.35
3	696.87	696.87	696.87	696.87	696.87	696.87	696.87
4	793.65	793.65	793.65	793.65	793.65	793.65	793.65
5	1588.52	1588.52	1588.52	1588.52	1588.52	1588.52	1588.52

and

Table 2. Convergence of the natural frequency (Hz) for fiber woven composite rotors with respect to the number of nodes in the disk unit.

Natural Frequency Order	Number of Nodes in the Disk Unit
	4	6	8	10	12	14	16
1	572.86	567.67	565.23	565.03	565.01	565.00	565.01
2	666.23	649.99	646.77	646.35	646.28	646.28	646.28
3	811.67	702.49	697.43	696.87	696.78	696.74	696.73
4	824.67	794.71	793.82	793.65	793.63	793.62	793.64
5	1601.52	1589.62	1588.52	1588.36	1588.32	1588.31	1601.52

for different numbers of shaft segment units and different numbers of ring plate nodes, respectively. From the tables, it can be found that the natural frequency of the rotor converges rapidly with the increase in the number of shaft segment units and the number of ring plate nodes, and stabilizes when the number of shaft units is greater than or equal to 3 and the number of ring plate nodes is greater than or equal to 10. Therefore, the number of shaft segment units is set to 3 and the number of ring plate nodes is set to 10 in the following examples.

3.2. Verification

3.2.1. Validation of Literature and FEM

The model of the shaft–disk system is validated, and

Table 3. Comparative validation of the natural frequency (Hz) of the shaft–disk system.

Method	Dimensionless Natural Frequency Order
	1	2	3	4	5	6
Present	163.44	231.14	1109.97	1323.37	2744.02	3692.69
Zhao et al. [22]	164.17	232.24	1125.86	1331.65	-	-
Heydari et al. [20]	164.27	231.57	1124.41	1332.04	2796.71	3729.08
FEM	163.51	230.80	1107.20	1324.50	2727.20	3700.10

demonstrates the literature comparison results of the shaft–disk system with the S–S boundary, whose structural parameters are L = 0.35 m, L_d = 0.37 L, r = R_i = 0.01 m, R_o = 0.1 m, and h_d = 0.002 m and whose material parameters are E = 200 GPa, μ = 0.266, and ρ = 7860 kg/m³. The comparative values of the intrinsic frequencies are very similar, which indicates that the constructed shaft–ring plate model is equally reliable.

Table 4. Validation of coupled natural frequency (Hz) and mode shape of the fiber woven composite shaft–disk system.

Method	Order
	1	2	3	4
Present
	431.03	605.50	2455.57	4604.54
FEM
	435.42	604.59	2449.22	4509.08
Difference	−1.01%	−0.14%	0.26%	2.12%
Present (made of steel)	418.92	526.99	2352.40	4344.89

compares the coupled natural frequencies and modes of the shaft–disk rotor of the fiber woven material, and the coupled natural frequencies of the steel rotor are also given in the table. The parameter settings of the disk are the same as those in Table 4, except that L_d = 0.09 m and h_d = 0.006 m. The shaft parameters L₁ = 0.08 m, L₂ = 0.02 m, L₃ = 0.04 m, and L₄ = 0.1 m correspond to the radii r₁ = r₄ = 0.01 m, r₂ = 0.02 m, and r₃ = 0.015 m. It can be seen that: the 1st- and 4th-order coupled modes are dominated by the displacement deformation of the disk, and the deformation of each structure is generated in the 2nd- and 3rd-order modes, among which the displacements of the disk in the 2nd order are mainly determined by the coupling points on the shaft. The frequency deviation does not exceed 2.12%, while the modes are basically consistent, which directly reflects the accuracy of the present model. In addition, comparing the steel rotor, it is found that the use of fiber woven material leads to a reduction in the rotor’s 1st coupling frequency, and at the same time, it can reduce the mass of the ring plate by 82.6%, which helps to achieve the goal of light weight. Considering that there is almost no vibration displacement of the shaft in the 1st-order coupled mode of the system, its coupling characteristics are not obvious, so the subsequent parametric study is dominated by the 2nd- and 3rd-order coupled intrinsic frequencies.

Finally, the results of the Campbell diagram comparison of the 2nd- and 3rd-order coupled natural frequencies of the rotor system with fiber woven material are given in Figure 8. The curves are in excellent agreement. Taken together, it seems that the present model is equally applicable to rotating rotor systems.

3.2.2. Experimental Verification

In this section, modal experiments are carried out for the disk as well as the shaft–disk rotor, four structures (round-hole disk, sector-hole disk, shaft–disk (round-hole) rotor, and shaft–disk (sector-hole) rotor) are selected as experimental objects, and the experimental pieces are all prepared from 45 steel. The parameters of the disk in the shaft–disk rotor are the same as those of the corresponding single plate, as follows: R_i = 0.02 m, R_o = 0.2 m, h_d = 0.008 m, and N = 4, of which the round hole R_h = 0.11 m, r_h = 0.05 m, the sector hole R₁ = 0.07 m, R₂ = 0.15 m, and d_h = 0.02 m. The rotors’ shafts are the same, and the disks are located in the middle of the 2nd section of the shaft. The geometric parameters of shaft are: L₁ = 0.3 m, L₂ = 0.1 m, L₃ = 0.2 m, r₁ = 0.015 m, r₂ = 0.02 m, and r₃ = 0.015 m.

In the experiment, we used the hammering method to test the natural frequency and modal vibration pattern of the structure, and the required equipment came from the DongHua DHDAS dynamic signal test and analysis system. The experimental schematic and field diagrams are shown in Figure 9 and Figure 10.

The overall experimental process is divided into three major parts:

(1)

Experimental pre-processing. The experimental parts are processed, and according to the size of the experimental parts in the modal analysis software, the model is drawn and the measurement point numbering is completed. Then, the corresponding measurement points are drawn. The experimental parts are set as the free boundary, and the elastic rope suspension is used to simulate the boundary conditions. The simulation of the experimental parts is completed and the nodes with large deformation in each order of the modal state are selected as the response test points in order to paste the sensor during the experiment.

(2)

Experimental measurement. The experiment of disk structure adopts the method of single-point vibration pickup; the experiment of shaft–disk structure adopts the method of single-point excitation and multi-point response, so the modal test of the disk only needs to place a single acceleration sensor on the response test point, while the shaft–disk system needs to select a measurement point on the shaft and the disk and paste acceleration sensors. Afterwards, the device connection is completed and the device parameters are debugged. Finally, the force hammer is used to tap each measurement point in turn to obtain and store the response data.

(3)

Post-experiment processing. After completing the experiment, the obtained data are processed through the modal analysis software, the intrinsic frequency is calculated, and the modal vibration pattern is plotted.

Table 5. Comparative results of natural frequency (Hz) and modal experiments of shaft–disk rotors with different hole types.

Hole Type	Method	Order
		1	2	3	4
Round hole	Present
		166.34	229.09	359.62	537.02
	Experiment
		170.46	231.5	359.33	545.44
	FEM
		166.23	228.87	359.24	536.53
	Difference 1	−2.42%	−1.04%	0.08%	−1.54%
	Difference 2	0.06%	0.10%	0.11%	0.09%
Sector hole	Present
		151.61	218.34	357.90	501.74
	Experiment
		154.09	213.77	354.16	493.34
	FEM
		151.12	217.85	357.74	500.78
	Difference 1	−1.61%	2.14%	1.06%	1.70%
	Difference 2	0.32%	0.23%	0.05%	0.19%

give the experimental results and their comparisons with the model, and Difference 1 and Difference 2 in the tables represent the deviations of this method from the experimental and finite element results, respectively. The natural frequencies of the two experimental parts are in good agreement, the comparison deviation of the shaft–disk rotor is not more than 2.5%. Considering the processing error of the experimental parts, the measurement error, and the imperfect simulation of the boundary conditions, in addition to many other factors, it can be assumed that the above deviations are in the acceptable range. Taking into account that the number of measurement points is much smaller than the number of nodes in the model, the comparison of the experimental modal results is not good, but in general, the two modal patterns are basically the same. The above results further illustrate the correctness of the modeling methodology of open-plate and rotor dynamics in this chapter.

3.3. Parametric Studies

Based on the constructed rotor model, the vibration characteristics of a rotating fiber woven composite rotor with a simply supported boundary are investigated. Unless otherwise stated, the system geometrical parameters and boundary settings in the following examples are the same as those in Table 4.

Firstly, the effect of the hole types on the rotor’s Campbell diagram is explored. In order to avoid an influence of the hole’s area on the structure, the number of holes N is set as 4 uniformly, and at the same time, the shape is set to make the area of the single holes as similar as possible, with the specific dimensions as follows: round holes: R_h = 0.06 m, r_h = 0.025 m; sector-shaped holes: R₁ = 0.045 m, R₂ = 0.075 m, d_h = 0.015 m; curved holes: R₁ = 0.043 m, R₂ = 0.077 m, φ = π/6. The Campbell diagrams of the 2nd and 3rd coupled natural frequencies of the system are shown in Figure 11. It is clear that the hole type has an effect on the 2nd forward (solid line) and backward (dashed line) coupled frequency values of the system, especially for the latter. This is due to the fact that this order mode is dominated by the shaft and there is a huge difference between the modes of the disk with and without holes at the same time. The 3rd natural frequency is affected by the vibration of the shaft and the disk at the same time, and the overall frequency shows the law of “No hole > Round hole > Curved hole > Sector hole”. The corresponding 3rd critical speed has the same trend.

Figure 12 exemplifies the effect of the total fiber volume fraction V_f on the vibration characteristics of the system, with the system pore parameters set as above. In conjunction with Figure 3, the Young’s modulus, shear modulus, and density of the fiber woven material itself continue to increase as V_f grows from 0.28 to 0.34, with a small decrease in Poisson’s ratio. The joint change of these parameters makes the overall stiffness of the rotor systems with different open hole types increase, so that the 3rd coupling frequency shows a similar linear reduction law: the growth rate of the 3rd coupling forward frequency gradually slows down, and the deceleration of the backward frequency gradually grows. The comparison between the subplots reveals that rotors with different hole types have different coupling frequency reduction rates, among which the reduction rate of the rotor with sector holes is the largest.

Taking the circular hole pattern with R_h = 0.06 m and r_h = 0.02 m as an example, Figure 13, Figure 14 and Figure 15 demonstrate the effects of the radius r_h, the number of holes N, and the hole eccentricity R_h on the Campbell diagrams of the 2nd and 3rd coupling frequencies. Figure 13 shows the effect of the round hole radius r_h on the natural frequency of the system when N = 4 and R_h = 0.06 m. The change in r_h can also be regarded as the change in the hole area, which is reflected as the change in stiffness in the overall structure. From the figure, it is found that when r_h is raised from 0 to 0.03 m, the 2nd frequency increases gradually and the 3rd decreases continuously. The backward frequency deceleration of the former mutates to 0 at different rotational speeds, and the mutation rotational speed increases with the increase in r_h. The mutation phenomenon is related to the modal change, referring to Figure 15 and Figure 16, for the system with small r_h. The 1st and 2nd modal diagrams are similar, and during the increase in the rotational speed, these two natural frequencies are shifted, which leads to the mutation of the frequency.

Figure 14 displays the Campbell diagrams for fiber woven composite rotor systems with different hole numbers N. From Figure 14a, it is found that the 2nd natural frequency of the system improves with the increase in N as a whole, but then the decreasing amplitude of the backward frequency produces abrupt changes at different rotational speeds, while the smaller N is the lower the speed at which the abrupt change occurs. Figure 14b shows that the 3rd order coupling frequency of the system grows with increasing N, and the decrement amplitude increases.

Figure 15 and Figure 17 present the Campbell plots and modes of the coupling frequency of the rotor of the fiber woven composite for different hole eccentricities R_h at N = 4 and r_h = 0.015 m. With the increase in R_h, the 2nd natural frequency of the system increases, and the 3rd natural frequency changes are not obvious, while the regularity is not strong. Taken together, the effect of the hole type and hole radius on the coupling frequency of the fiber woven material rotor system is more significant.

4. Conclusions

In this paper, the application of a fiber woven composite material to the disk of a shaft–disk rotor is proposed, and the domain decomposition method and coordinate mapping technique are introduced for the porous disk. Then, a dynamic modeling method applicable to the rotor with any complex hole types is given. Based on Timoshenko beam theory and FSDT, the energy expression of the shaft–disk rotor model is derived, and a rotor dynamics model of fiber woven material applicable to arbitrary hole shapes is established by DQFEM. The constructed model is verified to have good convergence, accuracy, and practicability after comparative tests in the literature, finite element simulation, and an experiment. A subsequent parametric study based on the proposed model was carried out, and the following conclusions were drawn:

(1)

The existing fiber woven composite rotor model has good convergence and accuracy, while the disk’s mass is reduced by 82.6% and the first-order coupled natural frequency is slightly reduced compared to the steel rotor.

(2)

The 1st and 4th coupled modes of the rotor are dominated by the displacement of the disk, while the 2nd- and 3rd-order modes contain the deformation of each structure, in which the disk displacement in the 2nd mode is mainly determined by the displacement of the coupling point on the shaft.

(3)

The hole type on the disk directly affect the 3rd critical speed of the rotor, which is ranked in descending order as “no hole > round hole > curved hole > sector hole”.

(4)

Taking the rotor with circular holes as an example, the increase in the total fiber volume fraction V_f, the hole eccentricity R_h and the hole number N, as well as the decrease in the hole radius r_h, all enhance the 3rd coupled natural frequency and the critical speed of the fiver woven composite rotor system. Among them, the effect of the hole radius is the most significant.