The Euler-Bernoulli beam theory posits that cross-sections perpendicular to the neutral axis before deformation remain planar and perpendicular to the deformed neutral axis after bending, as illustrated in Figure 1 .

The second moment of area for a hollow shaft with an annular cross-section is as follows

$I_z=\frac{\pi \left[{\left(2r_o\right)}^4-{\left(2r_i\right)}^4\right]}{64}=\frac{\pi \left(r_o^4-r_i^4\right)}{4}$(2-1)

The cross-sectional area is

$A=\pi \left(r_o^2-r_i^2\right)$(2-2)

The vibration governing equations of the Euler-Bernoulli beam theory are as follows

$\begin{array}{l}E{I}_z\frac{{\partial }^{4}w\left(x,t\right)}{\partial x^4}+\rho A\frac{{\partial }^{2}w\left(x,t\right)}{\partial t^2}=0\\ \omega ={\left(\frac{n\pi }{L}\right)}^2\cdot \sqrt{\frac{E{I}_z}{\rho A}}\end{array}$(2-3)

In the equation, $w\left(x,t\right)$ is the transverse displacement of the beam at position $x$, $n$ is the mode number, $L$ is the length of the beam, $E$ is the Young’s modulus of the beam material, $\rho$ is the density of the beam material, and $\omega$ is the natural frequency of the beam.

Unlike the Euler-Bernoulli beam theory, Timoshenko beam theory accounts for the effects of shear deformation, as illustrated in Figure 2 .

The governing equations of the Timoshenko beam theory are Equation (2-4).

$\begin{array}{l}E{I}_z\frac{{\partial }^{4}w\left(x,t\right)}{\partial x^4}+\rho A\frac{{\partial }^{2}w\left(x,t\right)}{\partial t^2}-k_{s}GA\left(\frac{{\partial }^{2}w\left(x,t\right)}{\partial x^2}-\frac{\partial \psi \left(x,t\right)}{\partial x}\right)=q\left(x,t\right)\\ k_{s}GA\left(\frac{\partial w\left(x,t\right)}{\partial x}-\psi \left(x,t\right)\right)=0\end{array}$(2-4)

In the equations, $G$ is the shear modulus of the beam material, $k_s$ is the shear coefficient for the Timoshenko beam, and $q\left(x,t\right)$ is the uniformly distributed load per unit length on the beam. The first equation describes the coupling relationship between the bending and transverse vibration of the beam and shear deformation; the second equation reflects the relationship between shear deformation and the rotation angle.

The schematic diagrams of the hollow shaft model and its differential element established based on the higher order shear deformation theory in this paper are shown in Figure 3 . The outer diameter of the shaft is $r_o$, the inner diameter is $r_i$, and the length is $L$.

For a hollow circular cross-sectional beam under transverse bending, the shear stress (shear strain) at any boundary of the inner and outer circumferences is zero In the cylindrical coordinate system, the shear strain between the axial and radial directions is expressed as

${\epsilon }_{rx}=\frac{1}{2}\left(\frac{\partial u_r}{\partial x}+\frac{\partial u_x}{\partial r}\right)=\frac{1}{2r}\left(y\frac{\partial u_y}{\partial x}+z\frac{\partial u_z}{\partial x}+y\frac{\partial u_x}{\partial y}+z\frac{\partial u_x}{\partial z}\right)$(2-5)

The displacement field is

$u_x=\psi \left(y,z\right)v^{\prime }\left(x\right)+\vartheta \left(y,z\right)w^{\prime }\left(x\right)-\left[\psi \left(y,z\right)+y\right]{\theta }_z\left(x\right)+\left[\vartheta \left(y,z\right)+z\right]{\theta }_y\left(x\right)$

$u_y=v$(2-6)

$u_z=w$

where

$\begin{array}{c}\vartheta \left(y,z\right)=\frac{y^{4}z}{5r_i^2{r}_o^2}+\frac{2y^2{z}^3}{5r_i^2{r}_o^2}-\frac{y^{2}z\left(r_i^2+r_o^2\right)}{3r_i^2{r}_o^2}+\frac{z^5}{5r_i^2{r}_o^2}-\frac{z^3\left(r_i^2+r_o^2\right)}{3r_i^2{r}_o^2}\\ \psi \left(y,z\right)=\frac{y^5}{5r_i^2{r}_o^2}+\frac{2y^3{z}^2}{5r_i^2{r}_o^2}-\frac{y^3\left(r_i^2+r_o^2\right)}{3r_i^2{r}_o^2}+\frac{y{z}^4}{5r_i^2{r}_o^2}-\frac{y{z}^2\left(r_i^2+r_o^2\right)}{3r_i^2{r}_o^2}\end{array}$(2-7)

Assuming the secondary stress components ( ${\sigma }_{yy},{\sigma }_{zz},{\sigma }_{yz}$) to be negligible, the constitutive equations of the beam model can be expressed as

$\begin{array}{l}{\sigma }_{xx}=E{\epsilon }_{xx}\\ {\sigma }_{xz}=2G{\epsilon }_{xz}\\ {\sigma }_{yz}=2G{\epsilon }_{yz}\end{array}$(2-8)

In the equation, $E$ is the Young’s modulus of the material, and $G$ is the shear modulus of the material. The expressions for the strain components are

${\epsilon }_{ij}=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right),\left(i,j\right)=\left(x,y,z\right)$(2-9)

The strain energy density is

$U_e=\frac{1}{2}{\sigma }_{ij}{\epsilon }_{ij}=\frac{1}{2}\left({\sigma }_{xx}{\epsilon }_{xx}+2{\sigma }_{xy}{\epsilon }_{xy}+2{\sigma }_{xz}{\epsilon }_{xz}\right)$(2-10)

The total kinetic energy of the beam is

$T=\frac{1}{2}{\displaystyle {\int }_0^L\left[m{\stackrel{\dot{}}{v}}^2+m{\stackrel{\dot{}}{w}}^2-2J_p\Omega {\theta }_y{\stackrel{\dot{}}{\theta }}_z+J_p{\Omega }^2+J_d{\stackrel{\dot{}}{\theta }}_y^2+J_d{\stackrel{\dot{}}{\theta }}_z^2\right]\text{d}x}$(2-11)

The equilibrium equations constructed are as Equation (2-12)

$\begin{array}{l}\delta \left(v\right):m\ddot{v}+E{I}_6{v}^{\left(4\right)}+E{I}_{46}{\theta }_z^{\left(3\right)}-\frac{1}{2}G\tilde{A}{v}^{″}+\frac{1}{2}G\tilde{A}{{\theta }^{\prime }}_z=0\\ \delta \left(w\right):m\ddot{w}+E{I}_6{w}^{\left(4\right)}-E{I}_{46}{\theta }_y^{\left(3\right)}-\frac{1}{2}G\tilde{A}{w}^{″}-\frac{1}{2}G\tilde{A}{{\theta }^{\prime }}_y=0\\ \delta \left({\theta }_z\right):-J_p\Omega {\stackrel{\dot{}}{\theta }}_y+J_d{\ddot{\theta }}_z-E{I}_{46}{v}^{\left(3\right)}-E{\tilde{I}}_2{{\theta }^{″}}_z-\frac{1}{2}G\tilde{A}{v}^{\prime }+\frac{1}{2}G\tilde{A}{\theta }_z=0\\ \delta \left({\theta }_y\right):J_p\Omega {\stackrel{\dot{}}{\theta }}_z+J_d{\ddot{\theta }}_y+E{I}_{46}{w}^{\left(3\right)}-E{\tilde{I}}_2{{\theta }^{″}}_y+\frac{1}{2}G\tilde{A}{w}^{\prime }+\frac{1}{2}G\tilde{A}{\theta }_y=0\end{array}$ (2-12)

where

$\begin{array}{l}\frac{1}{2}G\tilde{A}=-\frac{14}{27}G\pi r_i^2-\frac{76G\pi r_i^6}{3375r_o^4}+\frac{22G\pi r_i^4}{135r_o^2}+\frac{14}{27}G\pi r_o^2-\frac{22G\pi r_o^4}{135r_i^2}+\frac{76G\pi r_o^6}{3375r_i^4}\\ E{\tilde{I}}_2=-\frac{11}{72}\pi r_i^{4}E-\frac{7\pi r_i^{8}E}{1800r_o^4}+\frac{7\pi r_i^{6}E}{150r_o^2}+\frac{11}{72}\pi r_o^{4}E-\frac{7\pi r_o^{6}E}{150r_i^2}+\frac{7\pi r_o^{8}E}{1800r_i^4}\\ E{I}_{46}=-\frac{1}{24}\pi r_i^{4}E+\frac{7\pi r_i^{8}E}{1800r_o^4}-\frac{29\pi r_i^{6}E}{1800r_o^2}+\frac{1}{24}\pi r_o^{4}E+\frac{29\pi r_o^{6}E}{1800r_i^2}-\frac{7\pi r_o^{8}E}{1800r_i^4}\\ E{I}_6=-\frac{1}{72}\pi r_i^{4}E-\frac{7\pi r_i^{8}E}{1800r_o^4}-\frac{13\pi r_i^{6}E}{900r_o^2}+\frac{1}{72}\pi r_o^{4}E+\frac{13\pi r_o^{6}E}{900r_i^2}+\frac{7\pi r_o^{8}E}{1800r_i^4}\end{array}$(2-13)

The general methodology for developing the 1D simplified beam model in this study is illustrated in Figure 4 .

This paper combines theoretical calculations with the finite element method, employs a three-node finite element method to establish a higher order shear beam model, where each node possesses four degrees of freedom (denoted as $v,u,{\theta }_y,{\theta }_z$). The element matrices are formulated based on the beam equilibrium equations. Let the total number of nodes be $n$, the degree-of-freedom array of the beam model can then be expressed as

$\left[v_1,u_1,{\theta }_{y1},{\theta }_{z1},v_2,u_2,{\theta }_{y2},{\theta }_{z2},\cdots ,v_n,u_n,{\theta }_{yn},{\theta }_{zn}\right]$(2-14)

The global system matrix has a dimension of $4n\times 4n$. By mapping the degrees of freedom (DOF) numbering, the positions of the element matrices within the global system matrix can be determined. Assemble the global stiffness matrix, global mass matrix, and global gyroscopic matrix to establish a finite element model of a hollow shaft rotor system using higher order shear beam elements, which can easily achieve parameter settings for hollow shaft rotor systems of arbitrary dimensions.

This paper employs the state-space method to calculate the modal characteristics of a higher order shear hollow shaft rotor system. The analysis begins by establishing the dynamic model of the rotor system.

$\left\{M\right\}\left\{\ddot{x}\right\}+\left\{C\right\}\left\{\stackrel{\dot{}}{x}\right\}+\left\{K\right\}\left\{x\right\}=\left\{0\right\}$(3-1)

In the equation, $M$ is the element mass matrix, $C$ is the element damping matrix, $K$ is the element stiffness matrix, and $x$ is the axial displacement of the node. By solving the full matrix $v$ and the diagonal matrix $d$ of $\left\{M\right\},\left\{K\right\}$, the mode shapes and natural frequencies of the rotor system can be calculated.

The hollow shaft rotor system model was established using 45 steel for numerical analysis, with the specific dimensional parameters and material parameters of the model listed in Table 1 .

First, a mesh independence verification was performed for the Solid186 element model by establishing the hollow shaft rotor system as shown. Due to the thin-walled structure of the hollow shaft, the element size is influenced by the wall thickness. Therefore, this section defines the element size based on the number of element layers across the hollow shaft wall (calculated as (outer diameter − inner diameter)/number of element layers) for mesh independence verification. Different element sizes were configured to achieve varying numbers of element layers in the hollow shaft. Static modal analysis was conducted on the hollow shaft, and the first-order natural frequencies under different element sizes were recorded. Using three-layer solid elements as the reference, errors for different element sizes were calculated, as illustrated in Figure 5 . According to the mesh independence verification results: although three-layer solid elements in the thin-walled hollow shaft demonstrate higher accuracy, the computational time is significantly longer. In contrast, two-layer solid elements provide faster computation with equally minimal errors. Consequently, the finite element calculations for the hollow shaft in this study adopt a two-layer solid element configuration for mesh sizing ( Figure 6 ).

The inner diameter of the hollow shaft rotor system is fixed at 45 mm, and the outer diameter is fixed at 50 mm. Static modal calculations are performed for hollow shaft rotor systems with different length-to-diameter ratios, and the resulting curves and numerical results are presented in Table 2 and Figure 7 . The computation time was recorded: the Solid186 element finite element method took approximately 30 minutes, while the higher-order shear beam element and Timoshenko beam element methods required similar time, only around 20 seconds, the computational efficiency is significantly higher than that of the Solid186 element model. Then, the results show that for smaller length-to-diameter ratios, the computational errors between the higher order shear beam model and the Timoshenko beam model are comparable. However, as the length-to-diameter ratio increases, the error of the Timoshenko beam model rises to 14.91%, while the higher order shear beam model maintains high accuracy. This demonstrates the computational accuracy advantage of the proposed higher order shear beam model over traditional beam models. For rotor systems with small length-to-diameter ratios, the Euler-Bernoulli beam theory neglects shear effects, leading to significant computational errors. Therefore, the results from Euler-Bernoulli beam theory are excluded in the analysis of short, thick beams in this study.

A further comparative analysis is conducted on the influence of other dimensional parameters. With the length-to-diameter ratio of the hollow shaft fixed, the outer diameter set to 50 mm, and the shaft length set to 2.5 m, the inner diameter of the hollow shaft is varied to investigate the effect of the inner-to-outer diameter ratio on the computational results. These results are summarized in Table 3 and Figure 8 . The computation time was recorded: the Solid186 element finite element method took approximately 45 minutes, while the higher-order shear beam element and Timoshenko beam element methods required similar time, only around 30 seconds. Then, the calculations reveal that when the inner-to-outer diameter ratio is small (thick-walled beams), the accuracy of the higher order shear beam model is comparable to that of the Timoshenko beam model. However, as the diameter ratio increases, the computational error of the Timoshenko beam model gradually grows, while the higher order shear beam model retains high precision. This demonstrates the clear computational advantage of the proposed higher order shear beam element in analyzing thick-walled beam structures.

In rotor systems, due to factors such as manufacturing inaccuracies, the mass distribution of the rotor cannot achieve perfect rotational symmetry. Consequently, an offset of the center of mass generates centrifugal forces, which induce vibrations in the system. Unbalance response analysis is critical for implementing effective engineering measures to mitigate the effects of vibration, thereby enhancing the stability and reliability of the rotor system.

When considering the gyroscopic effect, the differential equation of motion for the rotor system is

$M\ddot{U}+\left(C+G\right)\stackrel{\dot{}}{U}+KU=F$(3-2)

where, $M$ is the element mass matrix, $C$ is the element damping matrix, $G$ is the gyroscopic matrix, $K$ is the element stiffness matrix, is the unbalance force matrix, and $U$ is the element displacement matrix, namely the unbalance response matrix.

Substituting the assumed solution with a concise $U=U_0{\text{e}}^{j\omega t}$ form into Equation (3-2) yields

$\left(-{\omega }^{2}M+j\omega \left(C+G\right)+K\right)U_0=F$(3-3)

Thus, the unbalance response of the rotor system can be obtained

$U=U_0{\text{e}}^{j\omega t}={\left(-{\omega }^{2}M+j\omega \left(C+G\right)+K\right)}^{-1}F{\text{e}}^{j\omega t}$(3-4)

A constant cross-section hollow shaft model is augmented with three concentrated masses located at the center of the rotor system, 0.2L, and 0.8L along the shaft length (where L is the total shaft length). Each concentrated mass has a distinct unbalance magnitude, with specific parameters detailed in Table 4 . Squeeze Film Dampers (SFD) are incorporated at both end supports, and their design parameters are provided in Table 5 . Using the algorithm described in Section 2.2 of this paper, an unbalance response analysis of the rotor system is performed. The resulting unbalance response curves for the three concentrated masses are shown in Figure 9 , and numerical results are summarized in Table 6 . Compared with traditional ANSYS simulations, the proposed higher order shear beam element model demonstrates superior accuracy and reliability.

In Chapter 2 of this paper, the formula for calculating element strain is derived, and the formula for the radial stress of the element is expressed as

$\sigma =E{\epsilon }_{rx}=\frac{E}{2r}\left(y\frac{\partial u_y}{\partial x}+z\frac{\partial u_z}{\partial x}+y\frac{\partial u_x}{\partial y}+z\frac{\partial u_x}{\partial z}\right)$(3-5)

The rotor system is excited at its first critical speed, and an unbalance response analysis is performed to calculate the stress distribution in the hollow shaft rotor system when the response reaches its peak amplitude. The radial stress distribution data on the cross-sections of the three concentrated masses are extracted and visualized as contour plots, as shown in Figure 10 . The contour plots reveal a symmetrical stress distribution, where the maximum radial stresses occur at the top and bottom ends of the hollow shaft with opposite directions, while the minimum stresses are observed near the central axis.

In the finite element simulation model, identical support conditions are applied, and the rotor system is excited at its first critical speed to perform an unbalance response analysis. The radial stress distribution contour plots at the cross-sections of the three concentrated masses are obtained, as shown in Figure 11 with numerical results provided in Table 7 . The results indicate that the stress distribution pattern closely resembles that of the higher order shear beam model, with stresses exhibiting symmetrical distribution. The calculated maximum radial stresses at the cross-sections of the three concentrated masses using both beam element models show strong consistency, validating the high precision of the proposed higher order shear beam element model in stress computation.