Mathematical modeling and dynamic analysis of carbon fiber reinforced plastic shaft-disk-bearing system considering internal damping effect

Abstract

The internal damping has significant influence on the dynamic characteristics of carbon fiber reinforced plastic drive shaft. In this paper, a dynamic model of the single-span multi-disk carbon fiber reinforced plastic shaft with elastic support is established based on layer-wise theory, damping specific capacity model, and Euler–Bernoulli beam theory. The model takes into account physical factors such as rigid body displacement caused by elastic support deformation, gyroscopic effect of concentrated mass disk, viscous internal damping of the carbon fiber reinforced plastic shaft, and lamination parameters. The model is used to analyze the natural angular frequency and critical instability speed of carbon fiber reinforced plastic shaft-disk-bearing system. The analysis results are compared with those obtained from the equivalent modulus beam theory, equivalent single layer theory, simplified homogenized beam theory, and quadrature element method. All the results demonstrate the accuracy of the proposed model, thereby validating its effectiveness. Furthermore, the model is utilized to investigate the influence of lamination parameters and mass ratio on the natural angular frequency and critical instability speed. This study provides valuable insights for the dynamics and stability design of the carbon fiber reinforced plastic shaft-disk-bearing system.

Keywords

carbon fiber reinforced plastic shaft shaft-disk-bearing system mathematical modelling dynamic analysis viscous internal damping

1. Introduction

The CFRP is increasingly used in marine equipment, new energy automobiles, aerospace, and other fields by virtue of their excellent mechanical properties and excellent designability (Chen et al., 2021). As the key component of power transmission in mechanical equipment system, the transmission shaft needs higher requirements for its weight (Rahagi et al., 2016), dynamic performance, and reliability with its development demand of large-scale, high-speed, and precision. Rotary drive components made of carbon fiber composites can greatly reduce the mass, the adverse dynamic effects, and instability caused by inertia, overcoming some limitations of metal shafts (Bavi et al., 2022). At present, more and more attention has been paid to the study of composite rotor dynamics.

Rotor dynamics is a field that focuses on the study of the natural frequency and stability characteristics of rotating machinery. It plays a crucial role in enhancing the safety and performance of entire systems (Torabi et al., 2017). The dynamic performance of a composite rotor system is influenced by factors such as the fiber orientation angle, stacking sequence, number of layers, and geometry of the composite shaft tube (Kaneria and Patel, 2023). Numerous scholars have utilized various homogenization methods to develop dynamic models for composite rotors. Experimental methods have been employed to validate these models, demonstrating their effectiveness in accurately predicting the natural frequency and instability threshold of rotors (Hajianmaleki and Qatu, 2013; Sino et al., 2007, 2008; Tsai and Massard, 1984). Pereira and Silveira (2002) and (Jacquet-Richardet et al., 2011) employed EMBT to study and optimize the dynamic performance of CFRP shafts. Their objective was to reduce the amplitude of unbalanced response and increase the critical speed range of instability. Sino et al., 2008 investigated finite element models incorporating internal damping and used SHBT theory to determine the natural frequency and instability threshold of CFRP shafts.

Composite shaft tube has high internal damping, which is identified as a crucial factor influencing stability of the rotor system (Wettergren and Olsson, 1996). Damping is typically simulated using either viscous damping or hysteretic damping. Hysteretic damping results in energy dissipation that is unaffected by vibration velocity, whereas viscous damping is influenced by vibration velocity. Many materials, including composites, exhibit damping properties that align more closely with viscous damping models (Arab et al., 2018). However, due to the challenges of incorporating a complex viscous damping model into the rotor dynamic equation, most current researches prefer to use hysteretic damping model for approximate solution (Montagnier and Hochard, 2014). Arab et al. (2017) developed a dynamic analysis model for composite rotors using ESLT, Euler–Bernoulli beam theory, and a composite hysteretic internal damping model. They examined the impact of bearing parameters on rotor stability. Barbosa et al. (2020) utilized a hysteretic damping model and SHBT theory to predict the dynamic behavior of composite multi-disc rotors. They conducted experiments to validate the model’s effectiveness in predicting critical rotation speed and unbalanced response of rotors. Ri et al. (2020) also employed a similar approach as Barbosa to establish a multi-degree of freedom model for composite axial-disc systems and analyzed their nonlinear forced vibrations. Ren et al. (2017) developed a rotor bending vibration model that takes into account the hysteretic damping of composite materials. They determined the first-order damping rule of the rotor under different layering schemes and analyzed the stability of the composite rotor in the supercritical speed range. Arab et al. (2018) discovered that increasing the laminate angle of the composite material leads to higher internal damping of the rotor and lower rotational speed, which can result in rotor instability. Montagnier and Hochard (2007) conducted a comparative study on the threshold speed of the composite shaft, focusing on the viscous and hysteretic damping models, and found that the choice of damping model significantly affects the determination of the threshold speed. These findings suggest that the internal damping of the CFRP shaft effectively reduces rotor vibration amplitude at critical and subcritical speeds. However, it can also lead to system instability at supercritical speeds. Additionally, the stiffness of the composite shaft tube significantly affects the natural frequency and stability of the rotor. The results indicate that a smaller fiber orientation angle leads to improved bending stiffness of CFRP shaft, which results in higher natural frequency and instability speed (Singh and Gupta, 1996). However, the impact of damping and stiffness of the CFRP shaft tube on the natural frequency and instability threshold of the rotor system is not consistent. Therefore, scholars prefer to study the influence of damping and stiffness on the dynamic performance of the rotor separately. This approach makes it challenging to determine the overall impact of laminate parameters on the dynamic performance of a composite rotor and provide effective guidance for rotor design.

The influence of composite laminate parameters and disk mass distribution on the natural frequency and instability speed of a multi-disk composite rotor is crucial for the design of the composite rotor system. However, there is limited research available on these topics. Therefore, this paper establishes a dynamic model of a single-span multi-disk composite rotor with elastic support using the Lagrange equation. The model takes into account various physical factors such as stacking sequence, gyroscopic effect of concentrated mass disk, and viscous internal damping of the CFRP shaft. Additionally, the study analyzes the impact of laminate parameters and disc mass ratio on the natural angular frequency and critical instability speed of the CFRP shaft-disk-bearing system. These findings provide valuable insights for the dynamics and stability design of the CFRP shaft-disk-bearing system.

2. Homogenization method of CFRP shaft

2.1. Equivalent stiffness

CFRP shafts are often fabricated by fiber winding process, and hollow cylindrical shafts are formed by fiber stacking with different orientation angles. Because there is curvature in the layering thickness of the CFRP shaft, the thin wall structure condition is usually not satisfied, but for each single-layer composite, its thickness is thin enough to be regarded as thin wall element. Therefore, in this study, layer-wise theory was used to homogenize the composite axes. Figure 1 shows the structure diagram of composite layering, with a generalized coordinate system of (x, y, z) and a local coordinate system of (1, 2, 3) for single layer.

The orientation angle of i-th layer is θ_i, and the constitutive relation in local coordinate system is (Arab et al., 2018)

[\begin{array}{l} σ_{1} \\ σ_{2} \\ τ_{12} \end{array}] = [\begin{array}{l} C_{11} & C_{12} & 0 \\ C_{12} & C_{22} & 0 \\ 0 & 0 & C_{66} \end{array}] [\begin{array}{l} ε_{1} \\ ε_{2} \\ γ_{12} \end{array}]

(1)

where

[C] = [\begin{array}{l} C_{11} & C_{12} & 0 \\ C_{12} & C_{22} & 0 \\ 0 & 0 & C_{66} \end{array}] = [\begin{array}{l} \frac{E_{11}}{1 - ν_{12} ν_{21}} & \frac{ν_{12} E_{22}}{1 - ν_{12} ν_{21}} & 0 \\ \frac{ν_{12} E_{22}}{1 - ν_{12} ν_{21}} & \frac{E_{22}}{1 - ν_{12} ν_{21}} & 0 \\ 0 & 0 & G_{12} \end{array}]

(2)

To facilitate the mechanical analysis of composite rotor, it is necessary to perform off-axis conversion according to the normal axis constitutive relationship and the orientation angle of the fiber:

[\begin{array}{l} σ_{x} \\ σ_{y} \\ τ_{x y} \end{array}] = [\begin{array}{l} \begin{array}{l} {\bar{C}}_{11} & {\bar{C}}_{12} & {\bar{C}}_{16} \end{array} \\ \begin{array}{l} {\bar{C}}_{12} & {\bar{C}}_{22} & {\bar{C}}_{26} \end{array} \\ \begin{array}{l} {\bar{C}}_{16} & {\bar{C}}_{26} & {\bar{C}}_{66} \end{array} \end{array}] [\begin{array}{l} ε_{x} \\ ε_{y} \\ γ_{x y} \end{array}]

(3)

[\bar{C}]

is the off-axis stiffness matrix calculated from the fiber orientation angle and equation (1) (Arab et al., 2018)

{\begin{cases} {\bar{C}}_{11} = C_{11} c o s^{4} θ_{i} + 2 (C_{12} + 2 C_{66}) s i n^{2} θ_{i} c o s^{2} θ_{i} + C_{22} s i n^{4} θ_{i} \\ {\bar{C}}_{12} = C_{12} (s i n^{4} θ_{i} + c o s^{4} θ_{i}) + (C_{11} + C_{22} - 4 C_{66}) s i n^{2} θ_{i} c o s^{2} θ_{i} \\ {\bar{C}}_{22} = C_{11} s i n^{4} θ_{i} + 2 (C_{12} + 2 C_{66}) s i n^{2} θ_{i} c o s^{2} θ_{i} + C_{22} c o s^{4} θ_{i} \\ {\bar{C}}_{16} = (C_{11} - C_{12} - 2 C_{66}) \sin θ_{i} c o s^{3} θ_{i} + (C_{12} - C_{22} + 2 C_{66}) s i n^{3} θ_{i} \cos θ_{i} \\ {\bar{C}}_{26} = (C_{11} - C_{12} - 2 C_{66}) s i n^{3} θ_{i} \cos θ_{i} + (C_{12} - C_{22} + 2 C_{66}) \sin θ_{i} c o s^{3} θ_{i} \\ {\bar{C}}_{66} = (C_{11} + C_{22} - 2 C_{12} - 2 C_{66}) s i n^{2} θ_{i} c o s^{2} θ_{i} + C_{66} (s i n^{4} θ_{i} + c o s^{4} θ_{i}) \end{cases}

(4)

The axial modulus of a single-layer composite was calculated from the following equation:

E_{i} = \frac{{\bar{C}}_{11} {\bar{C}}_{22} - {\bar{C}}_{12}^{2}}{{\bar{C}}_{22}}

(5)

Combined with layer-wise theory (Yang et al., 2023), the stiffness model of the composite was obtained as follows:

〈 E I 〉 = \sum_{i = 1}^{N} E_{i} π \frac{{(R_{o}^{i})}^{4} - {(R_{i}^{i})}^{4}}{4}

(6)

2.2. Viscous damping ratio of CFRP shaft

According to the previous research on the damping factor of composite rotor, the bending damping of composite rotor is also orthotropic (Yang et al., 2020). Single-layer composites can be described by a specific damping capacity (SDC) matrix consisting of elements that represent the energy dissipation properties of the material in three fundamental directions:

[ψ] = [\begin{array}{l} ψ_{L} & 0 & 0 \\ 0 & ψ_{T} & 0 \\ 0 & 0 & ψ_{L T} \end{array}]

(7)

where Ψ_L refers to longitudinal damping loss factor, Ψ_T refers to transverse damping loss factor, and Ψ_LT refers to shear damping loss factor. When the off-axis angle of the composite at layer i is θ_i, the SDC matrix of i-th layer is

{[\bar{ψ}]}_{i} = [\begin{array}{l} {\bar{ψ}}_{11} & {\bar{ψ}}_{12} & {\bar{ψ}}_{13} \\ {\bar{ψ}}_{12} & {\bar{ψ}}_{22} & {\bar{ψ}}_{23} \\ {\bar{ψ}}_{13} & {\bar{ψ}}_{23} & {\bar{ψ}}_{33} \end{array}]

(8)

where

{\begin{cases} {\bar{ψ}}_{11} = ψ_{L} c o s^{4} θ_{i} + 4 ψ_{L T} s i n^{2} θ_{i} c o s^{2} θ_{i} + ψ_{T} s i n^{4} θ_{i} \\ {\bar{ψ}}_{12} = (ψ_{L} + ψ_{T} - 4 ψ_{L T}) s i n^{2} θ_{i} c o s^{2} θ_{i} \\ {\bar{ψ}}_{22} = ψ_{L} s i n^{4} θ_{i} + 4 ψ_{L T} s i n^{2} θ_{i} c o s^{2} θ_{i} + ψ_{T} c o s^{4} θ_{i} \\ {\bar{ψ}}_{13} = (ψ_{L} - 2 ψ_{L T}) \sin θ_{i} c o s^{3} θ_{i} + (- ψ_{T} + 2 ψ_{L T}) s i n^{3} θ_{i} \cos θ_{i} \\ {\bar{ψ}}_{23} = (ψ_{L} - 2 ψ_{L T}) s i n^{3} θ_{i} \cos θ_{i} + (- ψ_{T} + 2 ψ_{L T}) \sin θ_{i} c o s^{3} θ_{i} \\ {\bar{ψ}}_{33} = (ψ_{L} + ψ_{T} - 2 ψ_{L T}) s i n^{2} θ_{i} c o s^{2} θ_{i} + ψ_{L T} (s i n^{4} θ_{i} + c o s^{4} θ_{i}) \end{cases}

(9)

According to the concept of energy dissipation, the damping loss factor can be expressed as the ratio of strain energy loss ΔU to total strain energy U during one vibration cycle, which is expressed as follows (Rueppel et al., 2017):

ψ = \frac{Δ U}{U}

(10)

where

\begin{array}{l} Δ U = \sum_{i = 1}^{N} \int_{V} \frac{1}{2} {({[σ]}_{i})}^{T} {[\bar{ψ}]}_{i} {[ε]}_{i} d V \\ U = \sum_{i = 1}^{N} \int_{V} \frac{1}{2} {({[σ]}_{i})}^{T} {[ε]}_{i} d V \end{array}

(11)

The volume of CFRP shaft is represented by V, and N is the total number of layers. [σ]_i and [ε]_i represent the stress and strain of the i-th layer. By performing modal analysis on the composite shaft-disc rotor system, the vibration shape of the rotating shaft at different modal frequencies can be determined, and the stress and strain of each layer of the CFRP shaft can be obtained. By combining Equation (10) and (11), the SDC (viscous damping) corresponding to different vibration frequencies of the composite rotating shaft can be calculated. The calculation flow chart for the damping of the composite shaft tube is illustrated in Figure 2. U_ij is the strain energy of the j element in i layer, ΔU_ij is the strain energy dissipation of the j element in i layer.

3. Dynamic model of composite rotor system

In this section, physical factors such as rigid body displacement caused by elastic support deformation, gyro effect of disc, and internal damping of CFRP shaft were introduced in detail. A schematic diagram of the elastic support composite multi-disk rotor model is shown in Figure 3. Both ends of the composite hollow shaft are supported on bearings A and B with bearing masses of m_A and m_B, supporting stiffness of k_A and k_B, and damping of c_b. The coordinates of bearing A are $(u_{A}, w_{A})$ , the turning angle of direction $u_{A}$ is $ϕ_{A}$ , and the turning angle of direction $w_{A}$ is $θ_{A}$ . At the distance $z_{i}$ from the left supporting point, the i-th disk is located, the center of shape and coordinates of the disk are $o_{i} (u_{i}, w_{i})$ , and the corresponding corners are $ϕ_{i}$ and $θ_{i}$ . The overall length of the axis is L, the rotational angular velocity is $Ω$ , and the precession angular velocity of bending due to mass eccentricity (whirling angular velocity) is ω.

In this paper, the free whirling equation of elastic support composite rotor was established by Lagrange equation, as shown in equation (12).

\frac{d}{d t} (\frac{\partial T}{\partial {\dot{q}}_{i}}) - \frac{\partial T}{\partial q_{i}} + \frac{\partial V}{\partial q_{i}} + \frac{\partial D}{\partial {\dot{q}}_{i}} = 0

(12)

where T and V denote the kinetic and potential energy of the system, D is the damping dissipation energy, and q _i is the broad coordinate of each unit,

q_{i} = {(u_{i}, ϕ_{i}, w_{i}, θ_{i})}^{T}

. Ignoring the effect of deformation in the z direction, the broad coordinate of the system is

q = {(u_{1}, ϕ_{1}, u_{2}, ϕ_{2}, u_{A}, u_{B}, w_{1}, θ_{1}, w_{2}, θ_{2}, w_{A}, w_{B})}^{T}

for a two-disk composite rotor system with 12 degrees of freedom.

3.1. Kinetic energy of disk and bearing

The kinetic energy of rotor system includes the kinetic energy of rigid disc, CFRP shaft and bearing, and its expression is

T = T_{d} + T_{b}

(13)

where T_d is the kinetic energy of the disk and CFRP shaft and T_b is the kinetic energy of the bearing, which can be expressed by equations (14) and (15) (Barbosa et al., 2020):

T_{d} = \frac{1}{2} \sum_{i = 1}^{N} m_{i} ({\dot{u}}_{i}^{2} + {\dot{w}}_{i}^{2}) + \frac{1}{2} \sum_{i = 1}^{N} J_{d i} ({\dot{ϕ}}_{i}^{2} + {\dot{θ}}_{i}^{2}) + Ω \sum_{i = 1}^{N} J_{p i} {\dot{ϕ}}_{i} {\dot{θ}}_{i} + J_{p s} Ω^{2}

(14)

T_{b} = \frac{1}{2} m_{A} ({\dot{u}}_{A}^{2} + {\dot{w}}_{A}^{2}) + \frac{1}{2} m_{B} ({\dot{u}}_{B}^{2} + {\dot{w}}_{B}^{2})

(15)

where m_i, J_di, and J_pi are the mass, diameter moment of inertia, and polar moment of inertia of the disk, J_ps is the polar moment of inertia of the composite axis, the composite axis is the slender axis and the mass is light, and its diameter moment of inertia is neglected. m_A and m_B are the masses of the bearing at the left and right ends, respectively. Taking equations (14) and (15) into the Lagrangian equation, the kinetic energy expression of the rotor system is obtained as

\frac{d}{d t} (\frac{\partial T}{\partial {\dot{q}}_{i}}) - \frac{\partial T}{\partial q_{i}} = (M_{m} + M_{d}) \ddot{q} + Ω M_{p} \dot{q}

(16)

M _m is the mass matrix of the rotor system, M _d contains the quadratic effect of the rotor's moment of inertia, and M _p is the gyroscopic effect matrix. They are all 12 × 12 matrices containing inertial parameters for composite multi-disk rotor systems.

3.2. Deformation energy of CFRP shaft and bearing

The displacement and angle of rotation caused by bending of the CFRP shaft is the DOF (degree of freedom) of the non-rigid body, and the displacement and angle of rotation of the CFRP shaft caused by elastic deformation of the support at both ends is the DOF of the rigid body. The broad coordinate of the composite axis in the xoz plane is $q_{u} = {(u_{1}, ϕ_{1}, u_{2}, ϕ_{2}, u_{A}, u_{B})}^{T}$ , where $q_{c u} = {(u_{1}, ϕ_{1}, u_{2}, ϕ_{2})}^{T}$ is the broad coordinate of the DOF of the non-rigid body and $q_{0 u} = {(u_{A}, u_{B})}^{T}$ is the DOF of rigid body. At any instant, the relationship between non-rigid deformation ${\tilde{q}}_{c u i} = {({\tilde{u}}_{i}, {\tilde{ϕ}}_{i})}^{T}$ caused by rigid body deformation is (Dimarogonas, 1976)

{\tilde{q}}_{c u i} = [\begin{array}{l} 1 - \frac{z_{i}}{L} & \frac{z_{i}}{L} \\ - \frac{1}{L} & \frac{1}{L} \end{array}] {\begin{cases} u_{A} \\ u_{B} \end{cases}} = Φ_{i} q_{0 u}

(17)

Therefore, the elastic deformation of the CFRP shaft is caused by the total deformation minus the rigid body DOF.

{\tilde{q}}_{c u} = q_{c u} - Φ q_{0 u}

(18)

Without considering the axial stress caused by external or transverse deformation coupling, the elastic potential energy V_s of the composite axis in the xoz plane is expressed as (Dimarogonas, 1976)

\begin{array}{l} V_{s} = \frac{1}{2} {{\tilde{q}}_{c u}}^{T} [K_{s}] {{\tilde{q}}_{c u}} = \frac{1}{2} {q_{c u} - Φ q_{0 u}}^{T} [K_{s}] {q_{c u} - Φ q_{0 u}} \\ = \frac{1}{2} {q_{u}}^{T} [\begin{array}{l} K_{s} & - K_{s} Φ \\ - Φ^{T} K_{s} & Φ^{T} K_{s} Φ \end{array}] {q_{u}} \\ = \frac{1}{2} {q_{u}}^{T} [K_{e}] {q_{u}} \end{array}

(19)

where [ K _e] is the stiffness matrix of the considered support system in the xoz plane and [ K _s] is the stiffness matrix of the CFRP shaft without considering rigid body deformation, which can be obtained from the Euler–Bernoulli beam-based impact coefficient method and expressed as (Wang and Yuan, 2021)

[K_{s}] = {[\begin{array}{l} K_{11} & K_{12} & K_{13} & K_{14} \\ K_{22} & K_{23} & K_{24} \\ K_{33} & K_{34} \\ s y m & K_{44} \end{array}]}^{- 1}

(20)

where

\begin{array}{l} K_{11} = K_{33} = \frac{z_{1}^{2} z_{2}^{2}}{3 〈 E I 〉 L} \\ K_{12} = - K_{34} = \frac{z_{1} z_{2} (z_{2} - z_{1})}{3 〈 E I 〉 L} \\ K_{13} = \frac{z_{1}^{2} (z_{2}^{2} + 2 z_{1} z_{2} - z_{1}^{2})}{6 〈 E I 〉 L} \\ K_{14} = - K_{23} = \frac{z_{1} (z_{2} + 3 z_{1}) (z_{1} - z_{2})}{6 〈 E I 〉 L} \\ K_{22} = K_{44} = \frac{z_{1}^{2} - z_{1} z_{2} + z_{2}^{2}}{3 〈 E I 〉 L} \\ K_{24} = \frac{5 z_{1}^{2} - 2 z_{1} z_{2} - z_{2}^{2}}{6 〈 E I 〉 L} \end{array}

(21)

where z₁ and z₂ are the axial coordinate values of disk 1 and 2, respectively, and <EI> is the equivalent axial stiffness of the composite rotor. Because the structure is axially symmetric with the material parameters, the elastic potential energy in the yoz plane is the same as that in the xoz plane, and the elastic potential energy of the whole composite rotating axis is obtained by superimposing the elastic potential energy of the two.

V_{s a} = \frac{1}{2} q^{T} [\begin{array}{l} K_{e} & 0 \\ 0 & K_{e} \end{array}] q

(22)

The potential energy of elastic support at both ends is

V_{b} = \frac{1}{2} {\begin{cases} u_{A} \\ u_{B} \end{cases}}^{T} [\begin{array}{l} k_{A} & 0 \\ 0 & k_{B} \end{array}] {\begin{cases} u_{A} \\ u_{B} \end{cases}} + \frac{1}{2} {\begin{cases} w_{A} \\ w_{B} \end{cases}}^{T} [\begin{array}{l} k_{A} & 0 \\ 0 & k_{B} \end{array}] {\begin{cases} w_{A} \\ w_{B} \end{cases}}

(23)

Brought axial elastic potential energy and supporting potential energy into Lagrangian equation to obtain stiffness matrix of rotor system.

\frac{\partial V}{\partial q} = \frac{\partial (V_{s a} + V_{b})}{\partial q} = K q

(24)

3.3. Damping dissipation energy

The damping dissipation contains CFRP shaft and elastic support dissipation. And the damping dissipation function of the CFRP shaft is

D_{s} = \frac{1}{2} c ({\tilde{u}}_{1}^{2} + {\tilde{w}}_{1}^{2} + {\tilde{u}}_{2}^{2} + {\tilde{w}}_{2}^{2})

(25)

where c is the damping coefficient of the composite rotor, determined by the damping ratio capacity Ψ, equivalent elastic modulus <EI>, axis mass m, and axis length l of the CFRP shaft, and its expression is (Yang et al., 2020)

c = 2 π Ψ \sqrt{\frac{48 〈 E I 〉}{m l^{3}}}

(26)

{\tilde{u}}_{i}

and

{\tilde{w}}_{i}

are the elastic deformations of the composite rotor in the u and w directions, respectively, which can be obtained by subtracting the rigid body deformations caused by the supporting ends from the total deformations

\begin{array}{l} [\begin{array}{l} {\tilde{u}}_{1} \\ {\tilde{u}}_{2} \end{array}] = [\begin{array}{l} 1 & 0 & - (1 - \frac{z_{1}}{L}) & - \frac{z_{1}}{L} \\ 0 & 1 & - (1 - \frac{z_{2}}{L}) & - \frac{z_{2}}{L} \end{array}] [\begin{array}{l} {\dot{u}}_{1} \\ {\dot{u}}_{2} \\ {\dot{u}}_{A} \\ {\dot{u}}_{B} \end{array}] \\ [\begin{array}{l} {\tilde{w}}_{1} \\ {\tilde{w}}_{2} \end{array}] = [\begin{array}{l} 1 & 0 & - (1 - \frac{z_{1}}{L}) & - \frac{z_{1}}{L} \\ 0 & 1 & - (1 - \frac{z_{2}}{L}) & - \frac{z_{2}}{L} \end{array}] [\begin{array}{l} {\dot{u}}_{1} \\ {\dot{u}}_{2} \\ {\dot{u}}_{A} \\ {\dot{u}}_{B} \end{array}] \end{array}

(27)

The damping coefficient of elastic support is c_b, the dissipation function of elastic support is

D_{b} = \frac{1}{2} c_{b} ({\dot{u}}_{A}^{2} + {\dot{u}}_{B}^{2} + {\dot{w}}_{A}^{2} + {\dot{w}}_{B}^{2})

(28)

Then, the total damping dissipation energy of the rotor system can be obtained:

D = D_{s} + D_{b}

(29)

The corresponding damping matrix is

\frac{\partial D}{\partial \dot{q}} = D \dot{q}

(30)

The free whirling differential equation of CFRP shaft-dick system with elastic support is

(M_{m} + M_{d}) \ddot{q} + (Ω M_{p} + D) \dot{q} + K q = 0

(31)

The solution of equation (31) can be written as q = X e^-λt, where X is the solution vector when the system performs synchronous motion, λ is the complex eigenvalue of the system, λ=σ+ωi (i is imaginary unit) and it is brought into equation (31), which can transform the dynamic characteristic problem of the elastic support composite multi-disc rotor system into a generalized eigenvalue problem. By analyzing the real part of λ, the critical speed of rotor system can be determined. When σ>0, rotor system is stable; when σ<0, the rotor system is unstable. The imaginary part of λ corresponds to the natural frequency of the rotor system.

4. Numerical calculation and validation

In this paper, dynamic analysis of CFRP shafts with dual discs is performed. The shaft of the composite material is carbon/epoxy and consists of 8 layers with different orientation angle. The geometrical parameters of the rotor system are shown in Table 1.

Table 1.

Geometric parameters of composite rotor system.

	Geometrical parameter	Value/m
CFRP shaft	Outer radius/Ro	0.048
	Total length/L	1.2
	Layer thickness/h	0.008
Steel disk	Inner radius/Rdi	0.048
	Outer radius/Rdo	0.15
	Thickness/e	0.05

The density of metal disk is 7850 kg/m³, and the material properties of the CFRP are shown in Table 2.

Table 2.

Properties of CFRP material (Ri et al., 2020).

Material property	Value	Material property	Value
E₁ longitudinal modulus (GPa)	172.7	ρ density (kg/m³)	1446.2
E₂ transverse modulus (GPa)	7.2	Ψ_L longitudinal damping loss factor (%)	0.45
G₁₂ shear modulus (GPa)	3.76	Ψ_T transverse damping loss factor (%)	4.22
ν₁₂ Poisson’s ratio	0.3	Ψ_LT shear damping loss factor (%)	7.05

In order to verify the validity of the proposed model, the dynamic analysis of CFRP shafts supported by isotropic bearings is conducted and compared with other theories. The bearing features are k_A = k_B = 1 × 10⁷ N/m, c_b = 50 N/m/s.

Figure 4 is Argand plot of complex frequency response of composite rotor with laminate [90₂/45/0]_S, and Figure 5(a) and (b) are corresponding Campbell plot and first-order complex frequency real part change plot. In Figure 5(a), the abbreviations FW and BW stand for forward whirl and backward whirl, respectively. The positive imaginary part of the eigenvalue represents the forward whirl angular velocity, while the negative imaginary part represents the backward whirl angular velocity (Afshari et al., 2022; Torabi and Afshari, 2016). In order to facilitate visualization on the Campbell diagram, the negative imaginary part is reversed. The natural angular frequency is determined as the smaller value between the forward and backward whirl natural angular frequencies (Afshari and Irani, 2018).

The CFRP shaft analyzed in this study is different from common CFRP shafts as it has concentrated mass disks. Therefore, the gyroscopic effect in the analysis of natural angular frequency and stability cannot be ignored. Both the forward whirl and backward whirl motion are analyzed in the Campbell diagram to study the natural angular frequency of the rotor system. Figure 4 illustrates the first 6 order whirling characteristic roots of the composite rotor system at various angular velocities. The imaginary part of the characteristic root represents the whirling angular frequency, while the real part is used to determine the stability of the rotor. By analyzing the first-order characteristic root, the natural angular frequency and unstable rotational speed of the rotor system can be determined. Figure 5(a) displays the Campbell diagram of the rotor system, which depicts the relationship between the rotation speed Ω and the imaginary part of the characteristic root. The natural angular frequency of the composite rotor system can be obtained from this figure. Figure 5(b) shows the relationship between the rotation speed Ω of the rotor system and the real part of the characteristic root. The rotation speed corresponding to a real part less than 0 represents the instability critical rotation speed of the system.

The stiffness and damping of the composite rotor are greatly influenced by the laminate parameters. In order to validate the model proposed in this study, six rotors with different laminate schemes are conducted complex frequency response analysis. The calculated natural angular frequency and critical speed of instability are compared with results from other methods mentioned in the literature, including those proposed by Pereira et al., Sino et al., Arab et al., and Kwangchol et al., as well as the method presented in this paper. The analytical results, including the methods mentioned above, are summarized in Table 3. Pereira and Silveira (2002) used an EMBT method based on Euler-Bernoulli beam theory, without considering the influence of factors such as transverse shear and layering order of the composite rotor. Sino et al., 2008 used SHBT combined with theory of Timoshenko beam, which considered the effects of transverse shear, layering order, and hysteretic internal damping. Arab et al. (2017) also considered the influence of layering order and hysteretic internal damping based on the Euler–Bernoulli beam theory of the ESLT method. (Ri et al., 2020) used QEM and theory of Timoshenko beam to consider the effects of hysteretic internal damping, transverse shear factors, Poisson coupling effects, and layering order. In this paper, layer-wise theory and Euler–Bernoulli beam theory are used, and the effects of viscous internal damping, stacking sequence, gyro effect, and non-rigid support deformation are considered.

Table 3.

Natural frequency and instability threshold of CFRP shaft with different laminate.

No.	Stacking sequence	ω_f1 (rad/s)					Instability threshold Ω (rad/s)
No.	Stacking sequence	EMBT	SHBT	ESLT	QEM	Present	EMBT	SHBT	ESLT	QEM	Present
A1	[90₂/45/0]_S	268.7	250.5	265.2	250.5	249.1	728.4	614.1	678.7	570.3	10,848.0
A2	[90/0/90/45/90/45/0/90]	268.7	251.8	266.5	251.8	250.0	729.4	619.2	686.9	574.8	10,972.1
A3	[90/45/0₂]_S	329.1	318.6	325.4	318.6	315.5	1263.3	1149.9	1158.2	1081.5	12,987.0
A4	[0₂/45₂/90₂/0₂]	329.1	319.9	325.7	319.9	316.6	1263.3	1163.0	1160.7	1094.3	12,824.7
A5	[0₂/90/45]_S	329.1	319.9	326.5	319.9	316.8	1263.3	1163.5	1169.2	1095.1	13,034.8
A6	[45/0/45/0/90/0/90/0]	329.1	322.7	327.2	322.7	319.3	1263.3	1193.3	1176.5	1123.9	12,748.3

From the results in Table 3, it is evident that there are differences in calculating the natural angular frequency and unstable rotational speed using the previous method compared to the method proposed in this paper. Regarding the natural angular frequency, the EMBT calculation yields relatively large results that do not account for the influence of stacking sequence. The ESLT method produces slightly smaller results than EMBT, but it does consider the influence of stacking sequence. The method proposed in this paper is in good agreement with the SHBT and QEM calculation results, indicating that the natural angular frequency decreases when the influence of transverse shear and stacking sequence of composites are taken into account.

The proposed method calculates significantly higher results for the critical speed of instability compared to other methods, with noticeable deviations between A1 and A2. This can be attributed to the different calculation methods of damping in the CFRP shaft and the consideration of the gyro effect of the centralized mass of the rotor. Modeling the structural damping characteristics is a complex task that often requires approximate processing. The EMBT, SHBT, ESLT, and QEM methods all utilize a hysteretic damping model that remains constant with vibration velocity. However, this paper employs a viscous damping model where the damping force is velocity-dependent. The numerical method used in this study calculates the internal damping corresponding to different modal frequencies of the composite rotating shaft, which better represents the actual damping dissipation. Additionally, considering the gyro effect of the concentrated mass of the composite rotor can further improve dynamic stability. In conclusion, the dynamic analysis model presented in this paper yields results that align well with those reported in the literature.

5. Effect of composite rotor parameters on kinetic properties

5.1. Effect of laminate parameters

The variation of the laminate parameters of the CFRP shaft affects the internal damping and stiffness of the composite shaft tube. To investigate the impact of laminate parameters, CFRP shafts with different orientation angles and stacking sequences are designed and listed in Table 4. The geometric dimensions of the rotor system in group B are consistent with those in group A. The CFRP shaft's laminate parameters are adjusted to create eight different schemes, as shown in Table 4. Among these schemes, B1 has an orientation angle of ±45°, and then the proportion of 0° layer is gradually increased until B8 consisted entirely of 0° layers. B2 and B3, as well as B4 and B5, and B6 and B7, have the same orientation angle but different stacking sequences.

Table 4.

Laminate scheme and corresponding parameters of group B.

No.	Laminate scheme	Damping specific capacity: (%)	Equivalent bending stiffness: <EI> (Nm²)
B1	[±45°]₄	1.856	30,023.62
B2	[0°₂/(±45°)₃]	1.775	98,827.99
B3	[(±45°)₃/0°₂]	1.769	133,656.20
B4	[0°₄/(±45°)₂]	1.713	178,196.17
B5	[(±45°)₂/0°₄]	1.698	224,617.82
B6	[0°₆/±45°]	1.288	269,157.78
B7	[±45°/0°₆]	1.271	303,986.00
B8	[0°]₈	0.161	372,790.37

Figures 6 –8 show the comparison of Argand diagram, Campbell diagram, and real part change of complex eigenvalue of B5 and B8.

When comparing the results in Figure 6, 7, and 8, it is observed that the impact of stiffness and damping on the natural angular frequency and critical instability speed is not uniform. To determine the specific influence pattern, the natural angular frequency and critical rotation speed of group B scheme are extracted. A comprehensive comparison is then conducted, as depicted in Figures 9 and 10.

The effect of laminate parameters on the natural angular frequency of the composite rotor system is illustrated in Figure 9. It is evident that both the orientation angle and stacking sequence greatly influence the stiffness and damping of the CFRP shaft. The general trend shows that as the ratio of the 0° angle increases, the stiffness of the CFRP shaft also increases, while the DSC decreases. When comparing different stacking sequences such as B2 and B3, B4 and B5, B6 and B7, it is observed that when ±45° is in the outer layer of the shaft and 0° is in the inner layer, the DSC of the CFRP shaft is larger but the stiffness is smaller. Considering the natural angular frequency, the stiffness of the CFRP shaft has a significant impact on it, increasing notably with higher stiffness. However, the DSC has a minimal effect on the natural angular frequency.

Figure 10 demonstrates that the B1 to B5 laminate scheme results in a decrease of 8.51% in damping specific capacity, while the stiffness increases significantly by 648.14%. Additionally, the critical instability speed shows an increase with the increase in stiffness. On the other hand, the B5 to B6 laminate scheme leads to a sudden decrease of 24.15% in damping specific capacity and an increase of 19.83% in stiffness. However, the critical instability speed of B6 shows a significant decrease. This pattern is even more pronounced in the B7 and B8 schemes, with the critical instability speed of the B8 scheme being only 286 rad/s. Therefore, the critical instability speed is closely related to the stiffness and damping of the CFRP shaft. Only when the stiffness and damping are at high levels simultaneously, can the rotor system achieve a higher critical instability speed. However, the effect of laminate parameters on stiffness and damping is not consistent. It is observed that the 0° layer helps increase the stiffness of the CFRP shaft but decreases the damping, while the ±45° layer helps increase the damping but decrease the stiffness even more. Therefore, in order to improve the critical instability speed of the composite rotor system, the proportional relationship between the 0° and ±45° layers should be considered comprehensively. However, for CFRP shaft tubes with a defined ratio of laminated angles, the inclusion of a 0° layer in the outer layer of the shaft proves beneficial in enhancing the critical instability speed of the rotor system, as depicted in B2 and B3, B4 and B5, B6 and B7 in Figure 10.

5.2. Effect of mass ratio of disks

Composite multi-disk rotor systems often have different masses for each disk, and the proportional relationship between them can affect the dynamic characteristics of the rotor system. In Table 5, we examine the impact of the disk mass ratio by designing composite rotor systems with the same total mass but varying proportions of the two disks. To eliminate the influence of laminate parameters, a standardized laminate of [90°₂/45°/0°]_S is used. Since the disk's position is symmetrical about the middle position of the rotor, we design seven schemes with the mass ratio of the left disk to the right disk as the changing parameter. The mass ratio gradually increases from 1 to 4.

Table 5.

Mass ratio parameters of composite rotor system disc in group C.

No.	C1	C2	C3	C4	C5	C6	C7
Mass ratio of disks	1	1.5	2	2.5	3	3.5	4

Using the model proposed in this paper, the complex frequency response analysis of seven schemes in Table 5 are performed to obtain the natural angular frequency and critical instability speed of each scheme. Subsequently, a comprehensive comparison was conducted, as depicted in Figure 11.

Figure 11 demonstrates that as the mass ratio δ increases, the natural angular frequency of the composite rotor system decreases from 252.2 rad/s for the C1 scheme to 249.0 rad/s for the C7 scheme. The mass ratio, however, has little effect on the natural angular frequency. Although the total mass of the rotor system remains unchanged, there is a slight change in the rotor system's diameter moment of inertia due to the variation in disk mass ratio. However, the mass ratio has a significant impact on the critical instability speed. As shown in Figure 11, when the mass ratio is 1, the critical instability speed of the composite rotor system is at its highest, but it decreases significantly with an increase in the mass ratio. This suggests that changes in mass distribution result in alterations in the axial vibration pattern of the composite material, leading to a significant change in the damping force and making the rotor system more susceptible to whirling instability.

6. Conclusion

Based on layer-wise theory, damping specific capacity model, and Euler–Bernoulli beam theory, the dynamic model of single-span multi-disc composite rotor with elastic support was established by using Lagrange equation. The natural angular frequency and critical instability speed of the composite rotor system were analyzed and compared with previous method. The results showed the following:

(1) When the inner layer of the shaft has an orientation angle of ±45° and the outer layer has an orientation angle of 0°, or when the number of 0° layer increases, the natural angular frequency of the composite rotor system increases. The DSC of the CFRP shaft has little effect on the natural angular frequency.

(2) The critical instability speed of the CFRP shaft is closely related to its stiffness and damping. Only when both the stiffness and damping are at high levels, the rotor system can achieve a higher critical instability speed. For a CFRP shaft with a specific proportion of orientation angle, designing a 0° layer in the outer layer of the shaft is beneficial for improving the critical instability speed.

(3) The mass ratio of the disk has minimal impact on the natural angular frequency, but it has a significant effect on the critical instability speed. As the mass ratio increases, the critical instability speed decreases noticeably.

Finally, the proposed model is suitable for predicting the natural frequency and instability speed of a single-span multi-disc rotor with a composite shaft tube. This paper offers valuable insights into the dynamics and stability design of the composite shaft-disk-bearing system by considering the influence of laminate parameters and disk mass distribution on dynamic behaviors. In future simulations and experiments, researchers can analyze the composite rotor dynamics using different type damping models and validate those using experimental results.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 52105095), the Natural Science Foundation of Hubei Province (No. 2023AFD054), and Xiangyang Science and Technology Project (No. 2022ABH006135 and 2022ABH006646).

ORCID iD

Mo Yang

Data availability statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Appendix

The mass matrices are as follows:

\begin{array}{l} M_{m} = [\begin{array}{l} M_{m}^{e} & 0 \\ 0 & M_{m}^{e} \end{array}], M_{d} = [\begin{array}{l} M_{d}^{e} & 0 \\ 0 & M_{d}^{e} \end{array}], M_{p} = [\begin{array}{l} M_{p}^{e} & 0 \\ 0 & M_{p}^{e} \end{array}] \\ M_{m}^{e} = [\begin{array}{l} m_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & m_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & m_{A} & 0 \\ 0 & 0 & 0 & 0 & 0 & m_{B} \end{array}], M_{d}^{e} = [\begin{array}{l} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & J_{d 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & J_{d 2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}] \\ M_{p}^{e} = [\begin{array}{l} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & J_{p 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & J_{p 2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}] \end{array}

The stiffness matrices are as follows:

\begin{array}{l} K = [\begin{array}{l} K_{e} + K_{b} & 0 \\ 0 & K_{e} + K_{b} \end{array}] \\ K_{e} = [\begin{array}{l} K_{s} & - K_{s} Φ \\ - Φ^{T} K_{s} & Φ^{T} K_{s} Φ \end{array}] \\ Φ = [\begin{array}{l} \begin{array}{l} 1 - \frac{z_{1}}{L} & \frac{z_{1}}{L} \\ - \frac{1}{L} & \frac{1}{L} \end{array} & \begin{array}{l} 1 - \frac{z_{2}}{L} & \frac{z_{2}}{L} \\ - \frac{1}{L} & \frac{1}{L} \end{array} \end{array}] \\ K_{s} = [\begin{array}{l} \frac{z_{1}^{2} z_{2}^{2}}{3 〈 E I 〉 L} & \frac{z_{1} z_{2} (z_{2} - z_{1})}{3 〈 E I 〉 L} & \frac{z_{1}^{2} (z_{2}^{2} + 2 z_{1} z_{2} - z_{1}^{2})}{6 〈 E I 〉 L} & \frac{z_{1} (z_{2} + 3 z_{1}) (z_{1} - z_{2})}{6 〈 E I 〉 L} \\ \frac{z_{1}^{2} - z_{1} z_{2} + z_{2}^{2}}{3 〈 E I 〉 L} & - \frac{z_{1} (z_{2} + 3 z_{1}) (z_{1} - z_{2})}{6 〈 E I 〉 L} & \frac{5 z_{1}^{2} - 2 z_{1} z_{2} - z_{2}^{2}}{6 〈 E I 〉 L} \\ \frac{z_{1}^{2} z_{2}^{2}}{3 〈 E I 〉 L} & - \frac{z_{1} z_{2} (z_{2} - z_{1})}{3 〈 E I 〉 L} \\ s y m & \frac{z_{1}^{2} - z_{1} z_{2} + z_{2}^{2}}{3 〈 E I 〉 L} \end{array}] \\ K_{b} = [\begin{array}{l} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & k_{A} & 0 \\ 0 & 0 & 0 & 0 & 0 & k_{B} \end{array}] \end{array}

The damping matrices are as follows:

\begin{array}{l} D = [\begin{array}{l} D_{e} & 0 \\ 0 & D_{e} \end{array}] \\ D_{e} = [\begin{array}{l} c & 0 & 0 & 0 & - (1 - \frac{z_{1}}{L}) c & - \frac{z_{1}}{L} c \\ 0 & 0 & 0 & 0 & 0 \\ c & 0 & - (1 - \frac{z_{2}}{L}) c & - \frac{z_{2}}{L} c \\ 0 & 0 & 0 \\ s y m & [{(1 - \frac{z_{1}}{L})}^{2} + {(1 - \frac{z_{2}}{L})}^{2}] c + c_{b} & [(1 - \frac{z_{1}}{L}) \frac{z_{1}}{L} + (1 - \frac{z_{2}}{L}) \frac{z_{2}}{L}] c \\ \frac{{z_{1}}^{2} + {z_{2}}^{2}}{L^{2}} c + c_{b} \end{array}] \end{array}

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