Dynamics Study of Multi-Supports Rotor Systems with Bearing Clearance Considering Angular Deflections

Abstract

Bearing clearance, prevalent in multi-supports rotor systems of aero engines, exerts a significant influence on the dynamics of rotor systems, actuators, and aero engines. The essence of it lies in the complex mechanical effects between the bearing and support. These effects become more complicated when significant relative angular deflections between the bearing and support exist, which is rarely considered in previous studies. In this paper, a model of support structure with bearing clearance considering angular deflections is proposed, and a mechanical model of the multi-supports rotor system with bearing clearance is developed. The dynamic response of the multi-supports rotor system with bearing clearance is investigated by numerical calculation and experimental verification. The results indicate that, in addition to the rotational frequency, remarkable harmonic frequency components occur in the response, which are generated by the relative movement and periodical collision between the bearing and support, and the relative angular deflections between the bearing and support have a significant impact on the amplitude of them; reducing the bearing clearance or increasing the misalignment both leads to a notable increase in the amplitudes of the harmonic frequency components.

1. Introduction

Aero engines operate under extreme conditions featuring high rotational speed, elevated temperature, and heavy load [1]. Variations in operating conditions significantly alter the loading distribution of rotor-support systems, potentially leading to deformation mismatch and localized high stress. Alternatively, such variations may cause substantial displacements of supports in rotor systems, resulting in misalignments [2] and amplifying the dynamic response. To mitigate the above issues, the bearing clearance is typically incorporated between the bearing outer ring and support in the support structure [3] (where the support structure with bearing clearance is termed as a loose support), which introduces nonlinearity to the characteristics of the support structure, thereby influencing the dynamic behavior of rotor systems and even the reliability of actuators and aero engines.

The essence of the impact of bearing clearance on the dynamic response of rotor systems lies in the mechanical effects between the bearing and support, mainly including the contact force generated by radial contact and the tangential friction force caused by relative movement. The former is generally represented as diverse forms of nonlinear elastic restoring forces [3,4,5,6,7] in the model of support with bearing clearance, functionally dependent on the relative lateral displacements between the bearing and support, and the latter is conventionally characterized by the Coulomb friction model [3,7,8]. In subsequent studies, the relative velocity between the bearing and support at the radial contact surfaces is further considered, constituting an impact process accompanied by kinetic energy dissipation. The effects of the impact process are commonly accounted for by introducing a fixed coefficient of restitution [9,10] into the model of support structure with bearing clearance or by incorporating an impact force component [11] (which is dependent on relative velocity) into the nonlinear elastic restoring force.

Nevertheless, bending deformation inevitably occurs in the rotor-support system under high rotational speed, generating pronounced relative angular deflections between the bearing and support. Consequently, the contact state between the bearing outer ring and support changes from the line contact [3,4,5,6,7] (where only the lateral displacements are considered) to the point contact [12,13,14], resulting in the decline in the contact stiffness of the bearing [15,16,17] and further the influence of the dynamic response of the rotor-support system [18,19,20,21]. Therefore, the angular deflections will be considered in this paper, and the influence of angular deflections on the contact state between the bearing and support will be incorporated in the model of support structure with bearing clearance.

Bearing clearance induces friction due to relative movement or the tendency for relative movement between the bearing and support, and the type of friction varies depending on the force and relative movement state between the bearing and support. When the relative movement occurs, the type of friction is sliding friction, which can be described using different forms of mathematical descriptions [8,22,23,24]. When the bearing and support move together, the friction cannot be explicitly expressed, but the motion constraint relationship with zero relative acceleration [25] exists. In summary, changes in the relative movement state of the bearing and support under complex loading environments, as well as the resulting changes in the type of friction, are often abruptness and uncertainty. Accurate determination of the type of friction is required in numerical calculations, which is typically achieved by adding a judgment function [26] in the numerical calculations. The type of friction is determined at each step of the iterative computation, ensuring the accuracy of calculation. However, different types of friction often correspond to different constraint relationships of motion, and the iterative formulas used for calculations also differ, which leads to the conversion of iterative parameters and the adjustment of load step during calculation, significantly increasing the time of calculation.

To improve the efficiency of calculation, many researchers have adopted the method of modifying friction models [25,27], involving the integration of different types of friction and the description of them with a single nonlinear equation, which allows that only a single iterative formula is employed during numerical calculation, significantly enhancing the efficiency of calculation. However, there are considerable errors in the vicinity of the conditions for the switching of friction types and motion states regarding the modified friction model, which greatly influence the accuracy of the dynamic response in the numerical calculation. Considering that the effects of different types of friction on the rotor system can be equivalently represented as different forms of constraint equations, forming extended equations by further integrating them and coupling them with motion differential equations, conventional numerical methods can be employed for numerical calculation. Based on this, this paper will propose a numerical method for calculating the dynamic response of rotor systems with constraint equations.

In summary, the research framework and the innovations of this paper are as follows: (1) Considering the angular deflections, a model of support structure with bearing clearance is proposed to describe the various mechanical behaviors between bearing and support, which constitutes the first innovation of this paper. (2) A mechanical model of the multi-supports rotor system with bearing clearance is developed. The governing equations of the rotor system are obtained by obtaining the constraint equations used to describe different types of friction between the bearing and support. (3) To address the challenges of numerical calculation introduced by the constraint equations due to bearing clearance, a numerical method for calculating the dynamic response of rotor systems with constraint equations is proposed, representing the second major innovation of this paper. (4) Numerical calculations and experimental studies are conducted on the dynamic response of the multi-supports rotor system with bearing clearance, and the experimental results are compared and validated against the numerical conclusions.

2. Model of Support Structure with Bearing Clearance Considering Angular Deflections

The relative movement state (which means the different relationships of displacements and velocities) between the bearing and support vary significantly with the operating conditions of the rotor system due to the bearing clearance, which leads to the distinction of the contact state and the interaction force between the bearing and support, especially considering the angular deflections. The constraint effects exerted by the support on the rotor exhibit distinct characteristics, thereby significantly influencing the dynamic response of rotor systems.

In this section, the support structure is properly simplified. The interaction forces between the bearing and support under different relative movement states are derived in detail, and the model of support structure with bearing clearance considering angular deflections is proposed.

2.1. Simplification of Support Structure

The support structure of aero engines typically comprises a rolling-element bearing, a support, a retaining nut, and a flexible support assembly. Axial preload is applied to the bearing outer ring via the retaining nut, which presses it against the support, and a clearance ${\delta }_0$ exists between the bearing outer ring and support. The support is mounted to the frame through the flexible support, as depicted in Figure 1.

For analytical purposes, the support structure can be simplified as illustrated in Figure 2, where inertial effects are concentrated at two primary nodes: the bearing and support. There is a bearing clearance ${\delta }_0$ between the two nodes, and they are pressed together at the end surfaces, creating an axial preload. The combined stiffness and damping effects of the frame and flexible support are represented by an equivalent linear stiffness $k_s$ and a linear damping $c_s$.

Given the inevitable bending deformation of the rotor during operation, which induces angular deflections in the support structure, the lateral displacements in the y and z directions and the angular deflections in the ${\theta }_y$ and ${\theta }_z$ directions are considered, and the effect of gravity is neglected.

2.2. Interaction Forces Between Bearing and Support

Under minor rotor dynamic load, the axial preload induced by the retaining nut ensures synchronized motion between the bearing and support through static friction at the end surfaces. In this regime, the interfacial friction force remains sufficient to prevent relative movement, while the radial clearance eliminates contact along the cylindrical surfaces, and no additional interaction force exists between the bearing and support.

However, as the dynamic support load escalates beyond the static friction threshold, the preload can no longer maintain cohesion, leading to the relative sliding and consequent contact, and the contact force $f_n$ at the radial cylindrical surfaces occurs.

Considering the angular deflections of the support system caused by the bending deformation of rotor system, there are three contact states between the bearing and support at the radial cylindrical surface, as shown in Figure 3: (a) there is a certain relative lateral displacement between the bearing and support, while the relative angular deflection $\mathbf{\theta }$ is minimal (which can be regarded as 0), resulting in line contact between them; (b) the angular deflection between the bearing and support cannot be ignored, leading to the one point contact state; (c) the relative angular deflection between them is relatively large, causing contact at two points in opposite sides. In the three different contact states, the contact force $f_n$ also takes different forms.

For the first line contact state, the contact force $f_n$ is a lateral force passing through the central plane of the bearing. Before and after the contact, the cross-sectional schematic of the radial cylindrical surfaces of the bearing and support is shown in Figure 4. It can be observed that the relative lateral displacement $r$ between the bearing and support at the contact position exceeds the clearance ${\delta }_0$ and has a certain penetration $\Delta r$ (where $\Delta r=\left|r\right|-{\delta }_0$). The contact angle is $\epsilon$, meaning the contact occurs within the range $\left[-\epsilon /2,\epsilon /2\right]$.

The cross-sectional centers of the bearing and support are set as $O_b\left(y_b,z_b,{\theta }_{b,y},{\theta }_{b,z}\right)$ and $O_s\left(y_s,z_s,{\theta }_{s,y},{\theta }_{s,z}\right)$, respectively. According to the Lankarani theory [13] (proposed by Lankarani and Nikravesh), the contact force $f_n$ depends on the relative lateral displacement and relative lateral velocity between the bearing and support at the contact position (which refers to the relative lateral displacement and relative lateral velocity between the center of the bearing and the center of the support in the line contact state). It can be expressed as

$$f_n=-P_n{\displaystyle \frac{r}{\left|r\right|}}\cdot H\left(\left|r\right|-{\delta }_0\right)\left(1+c_c{v}_n\right)=-P_n{\displaystyle \frac{r}{\left|r\right|}}\cdot H\left(\left|r\right|-{\delta }_0\right)-c_c{v}_n{P}_n{\displaystyle \frac{r}{\left|r\right|}}\cdot H\left(\left|r\right|-{\delta }_0\right)$$

(1)

where $H\left(\cdot \right)$ is the Heaviside function, and the expression of it can be written as

$$H\left(x\right)=\left\{\begin{array}{c}1 x>0\\ 0 x\le 0\end{array}\right.$$

(2)

$c_c$ is the collision factor depending on the material parameters; and $v_n$ is the relative velocity in the normal direction at the contact point between the bearing and support, which can be represented as

$$v_n={\displaystyle \frac{v_{y}y+v_{z}z}{\left|r\right|}}={\displaystyle \frac{y\dot{y}+z\dot{z}}{\sqrt{y^2+z^2}}}$$

(3)

in the equation, $y,z$ and $\dot{y},\dot{z}$ represent the components of the relative lateral displacement and the components of the relative lateral velocity between the bearing center and the support center along the y and z directions, respectively. They can be expressed as

$$\left\{\begin{array}{l}y=y_b-y_s\\ z=z_b-z_s\end{array}\right. \left\{\begin{array}{l}\dot{y}={\dot{y}}_b-{\dot{y}}_s\\ \dot{z}={\dot{z}}_b-{\dot{z}}_s\end{array}\right.$$

(4)

In Equation (1), the first term represents the elastic restoring force generated due to the relative lateral displacement between the bearing and support at the contact point, where $P_n$ denotes the magnitude of the elastic restoring force. The second term represents the damping force that dissipates kinetic energy, which arises from the contact and collision of the bearing and support with a certain relative lateral velocity. This term can also be referred to as the collision force. According to the Hertz contact model and Persson’s theory [28,29], $P_n$ satisfies

$$P_n=E_b{\delta }_0{l}_0\cdot {\displaystyle \frac{\pi \left(1+\alpha \right)\left(b^2+1\right)b^2}{2-\left(1-\alpha \right)\left[\ln \left(b^2+1\right)+2b^4\right]-4\beta \left(b^2+1\right)b^2}}$$

(5)

$\alpha ,\beta$ are dimensionless coefficients, which depend on the elastic moduli $E_b,E_s$ and the Poisson’s ratios ${\nu }_b,{\nu }_s$ of the bearing outer ring and support, given by

$$\alpha ={\displaystyle \frac{E_s-E_b}{E_s+E_b}}, \beta ={\displaystyle \frac{E_s\left(1-{\nu }_b\right)-E_b\left(1-{\nu }_s\right)}{2\left(E_s+E_b\right)}}$$

(6)

$l_0$ is the contact width (which means the length of the contact region along the axial direction), which corresponds to the width of the bearing outer ring in the line contact state; $b$ is the tangent of the contact half-angle $\epsilon /2$, which means $b=\tan \left(\epsilon /2\right)$; and the contact angle $\epsilon$ depends on the penetration $\Delta r$, where $\Delta r={\delta }_0\left(1/\cos \epsilon -1\right)$.

Compared to line contact, the contact area of point contact is significantly reduced, particularly the contact width $l$, which is smaller than $l_0$ in the line contact state, as illustrated in Figure 5.

Meanwhile, in the point contact state, the cross-sectional shape of the contact between the bearing and support is no longer circular but elliptical (with the major axis as $R_b/\cos \left|\theta \right|$ and the minor axis as $R_b$). However, the magnitude of relative angular deflection between the bearing and the support is small ($\left|\theta \right|\approx 0 and \left|\theta \right|\ne 0$), and the cross-sectional shape of the bearing contact in the point contact state can still be approximated as circular. The magnitude of the contact force $P_n$ has a form similar to Equation (5), as follows:

$$P_n=E_b{\delta }_{0}l\cdot {\displaystyle \frac{\pi \left(1+\alpha \right)\left(b^2+1\right)b^2}{2-\left(1-\alpha \right)\left[\ln \left(b^2+1\right)+2b^4\right]-4\beta \left(b^2+1\right)b^2}}$$

(7)

where the contact width $l$ satisfies

$$l=\left\{\begin{array}{l}{\displaystyle \frac{\Delta r}{\sin \left|\theta \right|\cos \left|\theta \right|}}\approx {\displaystyle \frac{\Delta r}{\left|\theta \right|}} \mathrm{when} \Delta r\le \left|\theta \right|l_0\\ l_0 \mathrm{when} \Delta r>\left|\theta \right|l_0\end{array}\right.$$

(8)

That is, the contact width $l$ cannot exceed the width of the bearing outer ring. In addition, the expressions of the relative lateral displacement, the penetration $\Delta r$, and the relative normal velocity $v_n$ at the contact point in the point contact state also change significantly compared to the line contact state. They not only depend on the relative lateral displacement and relative lateral velocity between the bearing center and the support center, but also depend on the relative angular deflection and relative angular velocity between them. Considering that the one point contact state is a special case of the two points contact state (one point contact occurs when only one of the two contact points in the two points contact state is engaged), here, we only derive the contact forces $f_{1,n},f_{2,n}$ for the two points contact state.

For the convenience of description and computation, the vectors can be expressed in complex exponential notation, and the vectors $r$ and $\theta$ satisfy the following relationships:

$$r=y+i\cdot z, \theta ={\theta }_y+i\cdot {\theta }_z$$

(9)

where ${\theta }_y,{\theta }_z$ represents the relative angular deflection components of the bearing center and support center, while the ${\dot{\theta }}_y,{\dot{\theta }}_z$ represents the relative angular velocity components of the bearing center and support center. They respectively satisfy

$$\left\{\begin{array}{l}{\theta }_y={\theta }_{b,y}-{\theta }_{s,y}\\ {\theta }_z={\theta }_{b,z}-{\theta }_{s,z}\end{array}\right. \left\{\begin{array}{l}{\dot{\theta }}_y={\dot{\theta }}_{b,y}-{\dot{\theta }}_{s,y}\\ {\dot{\theta }}_z={\dot{\theta }}_{b,z}-{\dot{\theta }}_{s,z}\end{array}\right.$$

(10)

In the two points contact state, the relative lateral displacements $r_1,r_2$ between the bearing and support at the contact points satisfy

$$r_1=r+{\displaystyle \frac{\theta }{2}}{l}_0\cdot e^{-i{\displaystyle \frac{\pi }{2}}}, r_2=r-{\displaystyle \frac{\theta }{2}}{l}_0\cdot e^{-i{\displaystyle \frac{\pi }{2}}}$$

(11)

where the dot product of the vector $\left(\theta /2\right)l_0$ with $e^{-i\left(\pi /2\right)}$ indicates that the phase of this vector within the $yOz$ plane is lagged by $\pi /2$. Then, the penetrations $\Delta r_1,\Delta r_2$ can be expressed as

$$\Delta r_1=\left|r+{\displaystyle \frac{\theta }{2}}{l}_0\cdot e^{-i{\displaystyle \frac{\pi }{2}}}\right|-{\delta }_0, \Delta r_2=\left|r-{\displaystyle \frac{\theta }{2}}{l}_0\cdot e^{-i{\displaystyle \frac{\pi }{2}}}\right|-{\delta }_0$$

(12)

The relative normal velocities $v_{1,n},v_{2,n}$ can then be written as

$$\begin{array}{l}{v}_{1,n}={\displaystyle \frac{1}{\left|r_1\right|}}\left[v_{1,y}\left(y+{\displaystyle \frac{{\theta }_z}{2}}{l}_0\right)+v_{1,z}\left(z-{\displaystyle \frac{{\theta }_y}{2}}{l}_0\right)\right]={\displaystyle \frac{\left(\dot{y}+\frac{{\dot{\theta }}_z}{2}{l}_0\right)\left(y+\frac{{\theta }_z}{2}{l}_0\right)+\left(\dot{z}-\frac{{\dot{\theta }}_y}{2}{l}_0\right)\left(z-\frac{{\theta }_y}{2}{l}_0\right)}{\sqrt{{\left(y+\frac{{\theta }_z}{2}{l}_0\right)}^2+{\left(z-\frac{{\theta }_y}{2}{l}_0\right)}^2}}}\\ v_{2,n}={\displaystyle \frac{1}{\left|r_2\right|}}\left[v_{2,y}\left(y-{\displaystyle \frac{{\theta }_z}{2}}{l}_0\right)+v_{2,z}\left(z+{\displaystyle \frac{{\theta }_y}{2}}{l}_0\right)\right]={\displaystyle \frac{\left(\dot{y}-\frac{{\dot{\theta }}_z}{2}{l}_0\right)\left(y-\frac{{\theta }_z}{2}{l}_0\right)+\left(\dot{z}+\frac{{\dot{\theta }}_y}{2}{l}_0\right)\left(z+\frac{{\theta }_y}{2}{l}_0\right)}{\sqrt{{\left(y-\frac{{\theta }_z}{2}{l}_0\right)}^2+{\left(z+\frac{{\theta }_y}{2}{l}_0\right)}^2}}}\end{array}$$

(13)

The magnitudes of the elastic restoring forces $P_{1,n},P_{2,n}$ can be written as

$$\left\{\begin{array}{l}{P}_{1,n}=E_b{\delta }_0{l}_1\cdot {\displaystyle \frac{\pi \left(1+\alpha \right)\left(b_1^2+1\right)b_1^2}{2-\left(1-\alpha \right)\left[\ln \left(b_1^2+1\right)+2b_1^4\right]-4\beta \left(b_1^2+1\right)b_1^2}}\\ P_{2,n}=E_b{\delta }_0{l}_2\cdot {\displaystyle \frac{\pi \left(1+\alpha \right)\left(b_2^2+1\right)b_2^2}{2-\left(1-\alpha \right)\left[\ln \left(b_2^2+1\right)+2b_2^4\right]-4\beta \left(b_2^2+1\right)b_2^2}}\end{array}\right.$$

(14)

in the equation, the contact widths $l_1,l_2$ satisfy

$$l_1=\left\{\begin{array}{l}{\displaystyle \frac{\Delta r_1}{\left|\theta \right|}} \mathrm{when} \Delta r_1\le \left|\theta \right|l_0\\ l_0 \mathrm{when} \Delta r_1>\left|\theta \right|l_0\end{array}\right. , l_2=\left\{\begin{array}{l}{\displaystyle \frac{\Delta r_2}{\left|\theta \right|}} \mathrm{when} \Delta r_2\le \left|\theta \right|l_0\\ l_0 \mathrm{when} \Delta r_2>\left|\theta \right|l_0\end{array}\right.$$

(15)

The expressions for the contact forces $f_{1,n},f_{2,n}$ are given by

$$\begin{array}{l}{f}_{1,n}=-P_{1,n}{\displaystyle \frac{r_1}{\left|r_1\right|}}\cdot H\left(\left|r_1\right|-{\delta }_0\right)\left(1+c_c{v}_{1,n}\right)\\ f_{2,n}=-P_{2,n}{\displaystyle \frac{r_2}{\left|r_2\right|}}\cdot H\left(\left|r_2\right|-{\delta }_0\right)\left(1+c_c{v}_{2,n}\right)\end{array}$$

(16)

According to Figure 5, there is a certain axial distance $d$ from the bearing center plane to the contact forces $f_{1,n},f_{2,n}$, which causes the contact forces to exert angular constraints on the bearing, specifically resulting in bending moments. For the two points contact state, there are axial distances $d_1,d_2$ ($d_1=l_0/2\cdot \cos \left|\theta \right|-l_1/2\approx \left(l_0-l_1\right)/2,d_2=\left(l_0-l_2\right)/2$) from the contact forces $f_{1,n},f_{2,n}$ to the bearing center plane, and the bending moments $M_{1,n},M_{2,n}$ can be expressed as

$$\begin{array}{l}{M}_{1,n}=P_{1,n}{\displaystyle \frac{r_1}{\left|r_1\right|}}\cdot H\left(\left|r_1\right|-{\delta }_0\right)\left(1+c_c{v}_{1,n}\right)\cdot {\displaystyle \frac{\left(l_0-l_1\right)}{2}}\cdot e^{-i\left(\pi /2\right)}\\ M_{2,n}=-P_{2,n}{\displaystyle \frac{r_2}{\left|r_2\right|}}\cdot H\left(\left|r_2\right|-{\delta }_0\right)\left(1+c_c{v}_{2,n}\right)\cdot {\displaystyle \frac{\left(l_0-l_2\right)}{2}}\cdot e^{-i\left(\pi /2\right)}\end{array}$$

(17)

Comparing Figure 3b,c, the difference between the one point contact and the two points contact lies in the magnitude of the relative angular deflection $\left|\theta \right|$ between the bearing and support. It can be understood that if $\left|\theta \right|$ is small, the system remains in a one point contact state even if the relative lateral displacement $\left|r\right|$ between the bearing and support is large, with $\left|r\right|>{\delta }_0$, and the penetrations $\Delta r_1>0,\Delta r_2>0$. In this case, the expressions of the contact force $f_n$ and the moment $M_n$ depend on the relative magnitudes of the penetrations $\Delta r_1,\Delta r_2$, namely

$$f_n=\left\{\begin{array}{l}{f}_{1,n} \mathrm{when} \Delta r_1>\Delta r_2\\ f_{2,n} \mathrm{when} \Delta r_1<\Delta r_2\end{array}\right., M_n=\left\{\begin{array}{l}{M}_{1,n} \mathrm{when} \Delta r_1>\Delta r_2\\ M_{2,n} \mathrm{when} \Delta r_1<\Delta r_2\end{array}\right.$$

(18)

For the case where $\Delta r_1=\Delta r_2$ (which means $r\times \theta =0$), the bearing and support are in the two points contact state. Based on this, the expressions of the total contact force $f_n$ and moment between the bearing and support, which are applicable to different contact states, can be derived as follows:

$$f_n=\left\{\begin{array}{l}{f}_{1,n}+f_{2,n} \mathrm{when} \left|r\right|\le {\delta }_0 \mathrm{or} \Delta r_1=\Delta r_2\\ f_{1,n} \mathrm{when} \left|r\right|>{\delta }_0 \mathrm{and} \Delta r_1>\Delta r_2\\ f_{2,n} \mathrm{when} \left|r\right|>{\delta }_0 \mathrm{and} \Delta r_1<\Delta r_2\end{array}\right., M_n=\left\{\begin{array}{l}{M}_{1,n}+M_{2,n} \mathrm{when} \left|r\right|\le {\delta }_0 \mathrm{or} \Delta r_1=\Delta r_2\\ M_{1,n} \mathrm{when} \left|r\right|>{\delta }_0 \mathrm{and} \Delta r_1>\Delta r_2\\ M_{2,n} \mathrm{when} \left|r\right|>{\delta }_0 \mathrm{and} \Delta r_1<\Delta r_2\end{array}\right.$$

(19)

where the expressions of $f_{1,n},f_{2,n}$ and $M_{1,n},M_{2,n}$ are given by Equations (16) and (17).

While the bearing and support contacts at the radial cylindrical surface, they also experience relative sliding at the preloaded end surfaces, resulting in tangential frictions. The magnitude and direction of these tangential frictions depend on the relative movement state of the bearing and support at the end surfaces, as well as the relative velocity determined by it. Generally, there are two relative movement states between the bearing and support, as illustrated in Figure 6: one state involves a certain relative lateral displacement between the bearing and support, while the relative angular deflection $\left|\theta \right|$ is extremely small (which can be considered as 0), presenting the relative movement state named “translation”; the other state involves both significant relative lateral displacement and relative angular deflection, presenting the relative movement state named “tilting”.

For the “translation” state, the relative velocities $v_1,v_2$ (where subscripts 1 and 2 denote the right and left end surfaces, respectively) at the two end surfaces of the bearing and support are the same as the velocity $v$ of the bearing center relative to the support (which means $v_1=v_2=v=\dot{r}$). Consequently, the tangential frictions $f_{1,\tau },f_{2,\tau }$ generated due to the relative movement at the two end surfaces are also identical. According to the ideal dry frictional model proposed by Hartog [8], the tangential friction can be expressed as

$$f_{1,\tau }=f_{2,\tau }=-\mu N{\displaystyle \frac{v}{\left|v\right|}}$$

(20)

where the magnitude of relative velocity is $\left|v\right|=\sqrt{v_y^2+v_z^2}=\sqrt{{\left({\dot{y}}_b-{\dot{y}}_s\right)}^2+{\left({\dot{z}}_b-{\dot{z}}_s\right)}^2}$; $\mu$ is the coefficient of friction; and $N$ is the preload of the retaining nut.

For the “tilting” state, the relative velocities $v_1,v_2$ at the two end surfaces of the bearing relative to the support differ from the velocity $v$ of the bearing center relative to the support, satisfying, respectively, the following:

$$v_1=\dot{r}+{\displaystyle \frac{\dot{\theta }}{2}}{l}_0\cdot e^{-i{\displaystyle \frac{\pi }{2}}}, v_2=\dot{r}-{\displaystyle \frac{\dot{\theta }}{2}}{l}_0\cdot e^{-i{\displaystyle \frac{\pi }{2}}}$$

(21)

The tangential frictions $f_{1,\tau },f_{2,\tau }$ at the end surfaces are also different and can be expressed as

$$f_{1,\tau }=-\mu N{\displaystyle \frac{v_1}{\left|v_1\right|}}, f_{2,\tau }=-\mu N{\displaystyle \frac{v_2}{\left|v_2\right|}}$$

(22)

$v_1,v_2$ satisfy $\left|v_1\right|=\sqrt{{\left[\dot{y}+\left({\dot{\theta }}_z/2\right)l_0\right]}^2+{\left[\dot{z}-\left({\dot{\theta }}_y/2\right)l_0\right]}^2},\left|v_2\right|=\sqrt{{\left[\dot{y}-\left({\dot{\theta }}_z/2\right)l_0\right]}^2+{\left[\dot{z}+\left({\dot{\theta }}_y/2\right)l_0\right]}^2}$. Meanwhile, there is a certain axial distance from the central plane of the bearing to the tangential frictions at the end surfaces, and the tangential frictions at the two end surfaces differ in this state. They generate bending moments on the bearing, denoted as $M_{1,\tau },M_{2,\tau }$, which can be expressed as

$$M_{1,\tau }=\mu N{\displaystyle \frac{v_1}{\left|v_1\right|}}\cdot {\displaystyle \frac{l_0}{2}}\cdot e^{-i{\displaystyle \frac{\pi }{2}}}, M_{2,\tau }=-\mu N{\displaystyle \frac{v_2}{\left|v_2\right|}}\cdot {\displaystyle \frac{l_0}{2}}\cdot e^{-i{\displaystyle \frac{\pi }{2}}}$$

(23)

Equations (22) and (23) also can be applied to the “translation” state, when $v_1=v_2=v=\dot{r}$, $f_{1,\tau }=f_{2,\tau }=-\mu N\left(v/\left|v\right|\right)$, and the total moment generated satisfies $M_{\tau }=M_{1,\tau }+M_{2,\tau }=$$\mu N\left(v/\left|v\right|\right)\cdot \left(l_0/2\right)\cdot e^{-i\left(\pi /2\right)}-\mu N\left(v/\left|v\right|\right)\cdot \left(l_0/2\right)\cdot e^{-i\left(\pi /2\right)}=0$.

It is should be noted that when the relative velocity between the bearing and support at the end surfaces is zero (which means $\left|v_1\right|=0$ or $\left|v_2\right|=0$), the friction translates to static friction, and Equations (22) and (23) are no longer applicable. Moreover, as the dynamic behavior of the rotor and support evolve, the contact states between them and the corresponding constraint equations (which govern the kinematic constraints of the structural motion) also change. The detailed impact of this mechanism is further elaborated in Section 3.3.

In this case, appropriate modifications can be made to derive expressions for the total friction and moment at the end surfaces of the bearing and support under different relative movement states, namely

$$\begin{array}{l}{f}_{\tau }=f_{1,\tau }+f_{2,\tau }=-\mu N\left[H\left(\left|v_1\right|\right)\cdot {\displaystyle \frac{v_1}{\left|v_1\right|}}+H\left(\left|v_2\right|\right)\cdot {\displaystyle \frac{v_2}{\left|v_2\right|}}\right]\\ M_{\tau }=M_{1,\tau }+M_{2,\tau }=\mu N\left[H\left(\left|v_1\right|\right)\cdot {\displaystyle \frac{v_1}{\left|v_1\right|}}-H\left(\left|v_2\right|\right)\cdot {\displaystyle \frac{v_2}{\left|v_2\right|}}\right]\cdot {\displaystyle \frac{l_0}{2}}\cdot e^{-i{\displaystyle \frac{\pi }{2}}}\end{array}$$

(24)

In summary, the interaction forces between the bearing and support (which can be called the loose support force) consist of the transverse force $f_{sum}$ and the angular moment $M_{sum}$, which include the contact force $f_n$ and moment $M_{\tau }$ generated at the radial cylindrical surface, as well as the friction $f_{\tau }$ and moment $M_{\tau }$ produced by relative movement at the end surfaces. That is

$$f_{sum}=f_n+f_{\tau }, M_{sum}=M_n+M_{\tau }$$

(25)

2.3. Verification of Model Accuracy

Compared to the model of support structure with bearing clearance proposed in the previous literature, the model presented in this paper incorporates the influence of the angular deflections of the bearing and support on the loose support force, of which the key lies in the more accurate description of the contact forces between the bearing and support under different angular deflections. Here, the accuracy of the proposed model is verified by comparing it with previous contact models and FEM results.

The Johnson model [30] and the Pereira model [31] (proposed by Pereira et al.) are widely used in describing the cylindrical contacts such as those between the bearing and support [6]. Johnson applied Hertz contact theory to describe the contact force between two cylindrical bodies, and the expression of it is as follows:

$$\Delta r={\displaystyle \frac{P_n}{\pi E^{\ast }{l}_0}}\left[\ln \left({\displaystyle \frac{4\pi E^{\ast }{l}_0\delta }{P_n}}\right)-1\right]$$

(26)

in the equation, $\Delta r$ represents the relative displacement of the two cylindrical bodies. $P_n$ denotes the magnitude of the contact force. $\delta$ is the radius difference between the two cylindrical bodies. $l_0$ is the contact width of the cylindrical bodies. $E^{\ast }$ is the modified elastic modulus of the materials of the two cylindrical bodies, which depends on the elastic moduli $E_1,E_2$ and the Poisson’s ratios ${\nu }_1,{\nu }_2$ of them, given by

$${\displaystyle \frac{1}{E^{\ast }}}={\displaystyle \frac{1-{\nu }_1^2}{E_1}}+{\displaystyle \frac{1-{\nu }_2^2}{E_2}}$$

(27)

The Pereira model is an analytical cylindrical contact model for the description of the contact forces between cylindrical bodies, particularly for the interaction between the bearing and journal. So, it is also more suitable for characterizing the contact forces between the bearing and support compared to other models, and the mathematical expression is written as

$$P_n={\displaystyle \frac{\left(a\delta +b\right)L{E}^{\ast }}{\delta }}{\left(\Delta r\right)}^n$$

(28)

where $a=0.965,b=0.0965,n=Y{\delta }^{-0.005}$ for the contact between the bearing and support, and the value of $Y$ is determined by the clearance, as shown below.

$$Y=\left\{\begin{array}{l}1.51\times {\left[\ln \left(1000\delta \right)\right]}^{-0.151} \delta \in \left[0.005,0.34954\right)\mathrm{mm}\\ 0.0151\delta +1.151 \delta \in \left[0.34954,10\right)\mathrm{mm}\end{array}\right.$$

(29)

Based on the structural characteristics of the support structure with bearing clearance, the finite element model of the loose support is established, as illustrated in Figure 7. The relevant parameters are summarized in

Table 1. Parameters of the finite element model of support structure with bearing clearance.

Description	Value	Description	Value
radius of the bearing (mm)	100	radius of the support (mm)	110
thickness of the bearing (mm)	10	thickness of the support (mm)	10
$\mathrm{elastic} \mathrm{moduli} \mathrm{of} \mathrm{the} \mathrm{bearing} E_b$ (GPa)	204	$\mathrm{elastic} \mathrm{moduli} \mathrm{of} \mathrm{the} \mathrm{support} E_s$ (GPa)	204
$\mathrm{Poisson}’\mathrm{s} \mathrm{ratio} \mathrm{of} \mathrm{the} \mathrm{bearing} {\mu }_b$	0.3	$\mathrm{Poisson}’\mathrm{s} \mathrm{ratio} \mathrm{of} \mathrm{the} \mathrm{support} {\mu }_s$	0.3
$\mathrm{width} \mathrm{of} \mathrm{the} \mathrm{bearing} l_0$ (mm)	30	$\mathrm{bearing} \mathrm{clearance} {\delta }_0$ (mm)	0.05

.

A comparison between the two contact models, the contact model proposed in this paper, and the FEM results is shown in Figure 8a and

Table 2. Percentage differences between different models and FEM results at different relative displacements.

	0.01 mm	0.02 mm	0.03 mm	0.04 mm	0.05 mm
Model proposed in this paper	12.01%	0.79%	4.37%	5.59%	5.69%
Johnson model	28.70%	1.18%	-	-	-
Pereira model	30.60%	29.74%	27.59%	25.59%	23.72%

. It can be observed that the contact model proposed in this paper exhibits a better accuracy with the FEM results compared to the other two models, and the model proposed in this paper demonstrates relatively smaller percentage differences against FEM results compared to the other two models across different relative displacements.

The relationship of the relative displacement and contact force under three angular deflections $0.5’,1’,2’$ between the contact model proposed in this paper and the FEM results is further compared, as shown in Figure 8b and

Table 3. Percentage differences between the model proposed in this paper and FEM results at different angular deflections.

	0.01 mm	0.02 mm	0.03 mm	0.04 mm	0.05 mm
0.5′	10.96%	4.11%	0.60%	1.14%	1.76%
1′	2.34%	2.87%	2.02%	1.37%	10.51%
2′	9.26%	2.94%	0.70%	2.71%	4.92%

, and it can be concluded that at a certain relative displacement, the penetration and the contact force between the bearing and support increases, as the angular deflection increases. The contact model proposed in this paper can effectively describe this trend and exhibits good consistency with the FEM results. Therefore, the contact model proposed in this paper, along with the model of support structure with bearing clearance based on it, can accurately describe the complex interaction forces between the bearing and support when the angular deflections are considered.

3. Mechanical Model of Multi-Supports Rotor System with Bearing Clearance

3.1. Simplification of Rotor-Support Structure

In aero engines, the low-pressure rotor typically employs a multi-supports rotor structure supported by three supports with four bearings, as shown in Figure 1. The 1#, 2a#, and 2b# bearings support the fan rotor, while the turbine rotor is supported by the 3# bearing at the rear. The fan rotor and turbine rotor are assembled via a transition shaft, which connects the fan shaft and the turbine shaft using splines to transmit torque and cylindrical surfaces for centering, with bolts used to transfer axial forces. During operation, the fan shaft and turbine shaft connected by the transition shaft may exhibit relative angular deflection under certain load conditions. Concurrently, there is a certain radial bearing clearance between the bearing outer ring and support at the 3# bearing. Based on these structural characteristics, a mechanical model of the multi-supports rotor system with bearing clearance is established, as shown in Figure 9.

The mechanical model consists of the rotor structure and the loose support. The rotor structure rotates about the x-axis at rotational speed $\omega$, which comprises six nodes. Specifically, node 1 corresponds to the axial positions of the 1# bearing, which is represented using a flexible support element. Nodes 2 and 4 represent the fan and the turbine, which are modeled as two rigid disks (disk 1 and disk 2), each with their respective masses $m_1,m_2$ and moments of inertia $I_{p,1},I_{d,1},I_{p,2},I_{d,2}$. The axial positions of the 2a# and 2b# bearings are represented by two nodes, 3a and 3b, which correspond to the right end of the fan shaft and the left end of the turbine shaft in the flexible rotor system. The 2a# and 2b# bearing is also modeled using a flexible support element, which is connected to node 3a. An angular spring (with angular stiffness $k_{c,\theta }$) is used to simulate the angular stiffness of the transition shaft between nodes 3a and 3b, and it is assumed that the lateral displacements of these two nodes are equal. The shafts connecting the disks and bearings are modeled using flexible shaft elements. The axial position of the 3# bearing corresponds to node 5. The loose support element is utilized at this location, which includes two nodes: the bearing and support. Considering that the bearing motion is caused by the rotor motion, the displacement of the bearing node is identical to that of node 5. Therefore, these two nodes can be merged, and only the support node and the displacement of it need to be considered in addition to the six nodes of the rotor structure.

Notably, the model of multi-supports rotor system deliberately excludes contact interactions induced by factors beyond bearing clearance effects and boundary condition variations arising from any load environments.

In summary, the mechanical model of the multi-supports rotor system with bearing clearance consists of a total of seven nodes, including the six nodes of the rotor structure and the support node at the loose support location. In the analysis, the lateral displacements in the $y,z$ directions, as well as the angular deflections in the ${\theta }_y,{\theta }_z$ directions, are considered for each node. Based on this, the generalized coordinate $q$ of the mechanical model can be defined as

$$\begin{array}{l}q=\left\{y_1,z_1,{\theta }_{1,y},{\theta }_{1,z},y_2,z_2,{\theta }_{2,y},{\theta }_{2,z},y_{3a},z_{3a},{\theta }_{3a,y},{\theta }_{3a,z},y_{3b},z_{3b},{\theta }_{3b,y},{\theta }_{3b,z},\right.\\ {\left. y_4,z_4,{\theta }_{4,y},{\theta }_{4,z},y_5,z_5,{\theta }_{5,y},{\theta }_{5,z},y_s,z_s,{\theta }_{s,y},{\theta }_{s,z}\right\}}^{\mathrm{T}}\end{array}$$

(30)

In the generalized coordinate, the following constraint relationships exist.

$$y_{3a}=y_{3b},z_{3a}=z_{3b}$$

(31)

The angular stiffness $k_{c,\theta }$ of the transition shaft can be referenced from the literature [32,33,34] and can be expressed as

$$k_{c,\theta }=a_0+a_1{\theta }^2=a_0+a_1\left[{\left({\theta }_{3a,y}-{\theta }_{3b,y}\right)}^2+{\left({\theta }_{3a,z}-{\theta }_{3b,z}\right)}^2\right]$$

(32)

In the equation, $a_0,a_1$ are constants determined by the mechanical characteristics of the transition shaft.

3.2. The Influence of Misalignment

Due to the influence of non-coordinated deformation caused by machining and assembly errors, as well as various loads such as aerodynamic, thermal, and inertial loads during operation, misalignment frequently occurs in the multi-supports rotor system of aero engines, and the impact of misalignment is considered in the mechanical model of multi-supports rotor system. The following analysis will address this issue.

Assuming that the misalignment of 3# bearing compared to 1# and 2# bearings is in the y direction, it produces an initial displacement $\Delta q_m$, which includes the lateral displacements and angular deflections of nodes 3b, 4, and 5, as well as the support node in the loose support element, as shown in Figure 10. It can be expressed as Equation (33):

$$\Delta q_m={\left\{0,0,0,0,\cdots ,0,0,0,\Delta {\theta }_{3b,z,m},\Delta y_{4,m},0,0,\Delta {\theta }_{4,z,m},\delta ,0,0,\Delta {\theta }_{5,z,m},\Delta y_{s,m},0,0,\Delta {\theta }_{s,z,m}\right\}}^{\mathrm{T}}$$

(33)

where $\Delta y_{s,m}=\delta , \Delta y_{4,m}=l_{34}/\left(l_{34}+l_{45}\right)\cdot \delta , \Delta {\theta }_{3b,z,m}=\Delta {\theta }_{4,z,m}=\Delta {\theta }_{5,z,m}=\Delta {\theta }_{s,z,m}=\delta /\left(l_{34}+l_{45}\right)$. $l_{34},l_{45}$ represent the axial distances between node 3a (3b) and node 4, and between node 4 and node 5, respectively.

Both the motion and the stress state of the transition shaft will change when misalignment occurs, and it generates a pair of additional excitations of equal magnitude but in the opposite direction on the connected fan shaft and turbine shaft as the rotor rotates, which can be called misalignment excitation. The misalignment excitation can be expressed in component form as $F_{sp,y},F_{sp,z},M_{sp,y},M_{sp,z}$ and $-F_{sp,y},-F_{sp,z},-M_{sp,y},-M_{sp,z}$. According to the derivations in references [32,35], they can be represented as

$$\left\{\begin{array}{l}{F}_{sp,y}=2m_{sp}L\tan \left(\Delta {\theta }_{3b,z,m}/2\right){\omega }^2\cos 2\omega t\\ F_{sp,z}=2m_{sp}L\tan \left(\Delta {\theta }_{3b,z,m}/2\right){\omega }^2\sin 2\omega t\end{array}\right. \left\{\begin{array}{l}{M}_{sp,y}=m_{sp}{L}^2\tan \left(\Delta {\theta }_{3b,z,m}/2\right){\omega }^2\cos 2\omega t\\ M_{sp,z}=m_{sp}{L}^2\tan \left(\Delta {\theta }_{3b,z,m}/2\right){\omega }^2\sin 2\omega t\end{array}\right.$$

(34)

where $m_{sp}$ represents the mass of the transition shaft, and $L$ denotes the maximum axial distance between the transition shaft and the centering cylindrical surfaces of the fan shaft and turbine shaft.

3.3. Governing Equations of Rotor System

The multi-supports rotor system with bearing clearance can be regarded as a combination of two rotor systems connected in series, both operating at the same rotational speed: one rotor system is a single-disk (disk 1) rotor system supported by two bearings (1# and 2# bearings), including nodes 1, 2, and 3a; the other rotor system is a single-disk (disk 2) rotor system supported by one bearing (3# bearing), which is a loose support, and includes nodes 3b, 4, and 5, and a support node. These two rotor systems are connected at nodes 3a and 3b through an angular spring.

Firstly, by neglecting the loose support force in the loose support element (while still considering the mass, damping, and stiffness of the support), and also ignoring the angular spring between the two rotors, the motion differential equations of the two rotor systems can be, respectively, obtained by the finite element method [36,37,38], as follows:

$$M_1{\ddot{q}}_1+\left(C_1-\omega G_1\right){\dot{q}}_1+K_1{q}_1=Q_1$$

(35)

$$M_2{\ddot{q}}_2+\left(C_2-\omega G_2\right){\dot{q}}_2+K_2{q}_2=Q_2$$

(36)

In the equations, $M_1,C_1,G_1,K_1$ and $M_2,C_2,G_2,K_2$ represent the mass, damping, gyroscopic, and stiffness matrices of these two rotor systems; and $q_1,q_2$ represent the generalized coordinates of these two rotor systems, which can be expressed as

$$q_1={\left\{y_1,z_1,{\theta }_{1,y},{\theta }_{1,z},y_2,z_2,{\theta }_{2,y},{\theta }_{2,z},y_{3a},z_{3a},{\theta }_{3a,y},{\theta }_{3a,z}\right\}}^{\mathrm{T}}$$

(37)

$$q_2={\left\{y_{3b},z_{3b},{\theta }_{3b,y},{\theta }_{3b,z},y_4,z_4,{\theta }_{4,y},{\theta }_{4,z},y_5,z_5,{\theta }_{5,y},{\theta }_{5,z},y_s,z_s,{\theta }_{s,y},{\theta }_{s,z}\right\}}^{\mathrm{T}}$$

(38)

Also, the equation $q={\left\{q_1^{\mathrm{T}},q_2^{\mathrm{T}}\right\}}^{\mathrm{T}}$ holds. $Q_1,Q_2$ represent the unbalance excitations of these two rotor systems caused by disks, which can be expressed as

$$Q_1={\left\{\cdots ,{\left(me\right)}_1{\omega }^2\cos \omega t,{\left(me\right)}_1{\omega }^2\sin \omega t,-\left(I_{p,1}-I_{d,1}\right){\tau }_1{\omega }^2\cos \left(\omega t+{\phi }_1\right),-\left(I_{p,1}-I_{d,1}\right){\tau }_1{\omega }^2\sin \left(\omega t+{\phi }_1\right),\cdots \right\}}^{\mathrm{T}}$$

(39)

$$Q_2={\left\{\cdots ,{\left(me\right)}_2{\omega }^2\cos \omega t,{\left(me\right)}_2{\omega }^2\sin \omega t,-\left(I_{p,2}-I_{d,2}\right){\tau }_2{\omega }^2\cos \left(\omega t+{\phi }_2\right),-\left(I_{p,2}-I_{d,2}\right){\tau }_2{\omega }^2\sin \left(\omega t+{\phi }_2\right),\cdots \right\}}^{\mathrm{T}}$$

(40)

In the equation, ${\left(me\right)}_1,{\left(me\right)}_2$ represent the unbalance of the two disks, $I_{p,1},I_{d,1}$ and $I_{p,2},I_{d,2}$ denote the spindle moments of inertia and the diameter moments of inertia of the two disks, while ${\tau }_1,{\phi }_1$ and ${\tau }_2,{\phi }_2$, respectively, represent the angles of the principal axes of inertia of the two disks and their relative phase angles with respect to the unbalances.

Secondly, considering the angular spring between node 3a and node 3b, the resulting constraint effect can be equivalently represented as a pair of moments $M_c,-M_c$ with equal magnitude but with opposite directions applied at these two nodes. The components of $M_c$ in the $y,z$ directions, denoted as $M_{c,y},M_{c,z}$, can be expressed as

$$M_{c,y}=-k_{c,\theta }\left({\theta }_{3a,y}-{\theta }_{3b,y}\right), M_{c,z}=-k_{c,\theta }\left({\theta }_{3a,z}-{\theta }_{3b,z}\right)$$

(41)

The motion differential equations of the multi-supports rotor system, neglecting the misalignment and loose support force, can be written as

$$M\ddot{q}+\left(C-\omega G\right)\dot{q}+Kq=Q+F_c$$

(42)

in equation:

$$M=\left[\begin{array}{cc}{M}_1& \\ & M_2\end{array}\right],C=\left[\begin{array}{cc}{C}_1& \\ & C_2\end{array}\right],G=\left[\begin{array}{cc}{G}_1& \\ & G_2\end{array}\right],K=\left[\begin{array}{cc}{K}_1& \\ & K_2\end{array}\right],Q=\left\{\begin{array}{c}{Q}_1\\ Q_2\end{array}\right\}$$

(43)

$$F_c={\left\{0,0,0,0,\cdots ,0,0,M_{c,y},M_{c,z},0,0,-M_{c,y},-M_{c,z},\cdots ,0,0,0,0\right\}}^{\mathrm{T}}$$

(44)

Then, the influence of misalignment is considered, including the misalignment excitation $F_{sp}$ acting on nodes 3a and 3b through the transition shaft and the initial displacement $\Delta q_m$. The misalignment excitation $F_{sp}$ satisfies

$$F_{sp}={\left\{0,0,0,0,\cdots ,F_{sp,y},F_{sp,z},M_{sp,y},M_{sp,z},-F_{sp,y},F_{sp,z},-M_{sp,y},-M_{sp,z},\cdots ,0,0,0,0\right\}}^{\mathrm{T}}$$

(45)

where $F_{sp,y},F_{sp,z},M_{sp,y},M_{sp,z}$ can be referenced from Equation (34). The influence of the initial displacement is causing the origin of the lateral and angular deflection of the multi-supports rotor system to shift from its original position. As a result, the generalized coordinate $q$ in the motion differential equations is replaced by $q-\Delta q_m$. Furthermore, according to the expression of $\Delta q_m$ (Equation (33)), it can be concluded that

$$\dot{q}-\Delta {\dot{q}}_m=\dot{q}, \ddot{q}-\Delta {\ddot{q}}_m=\ddot{q}$$

(46)

Therefore, the motion differential equations considering misalignment can be expressed as

$$M\ddot{q}+\left(C-\omega G\right)\dot{q}+K\left(q-\Delta q_m\right)=Q+F_c+F_{sp}$$

(47)

Furthermore, the loose support force is considered. Section 2.2 provides a detailed derivation of the loose support force, and its expressions are given in Equations (16), (17), (19), (24), and (25). In order to facilitate its substitution into the motion differential equations, it can be expressed as

$$F_{sum}={\left\{0,0,0,0,\cdots ,f_{sum,y},f_{sum,z},M_{sum,y},M_{sum,z},-f_{sum,y},-f_{sum,z},-M_{sum,y},-M_{sum,z}\right\}}^{\mathrm{T}}$$

(48)

in equation:

$$\begin{array}{c}{f}_{sum,y}=f_{n,y}+f_{\tau ,y}, f_{sum,z}=f_{n,z}+f_{\tau ,z}\\ M_{sum,y}=M_{n,y}+M_{\tau ,y}, M_{sum,z}=M_{n,z}+M_{\tau ,z}\end{array}$$

(49)

The detailed expressions of the Equation (49) can be calculated using the equations provided in Section 2.2, which will not be reiterated here.

It should be noted that, although the tangential friction at the end surfaces between the bearing and support can be described using Equation (24), which is applicable to different relative movement states between the bearing and support, these equations cannot fully consider the changes in the form of friction and the relative movement relationship caused by the change in relative movement states.

In the initial state, the bearing and support often move together (which means the relative velocity at the left and right end surfaces is 0), and the generalized coordinates have constraint relationships as shown in Equations (50) and (51), which correspond to the cases where there is no relative movement between the bearing and support at the left and right end surfaces, respectively). Also, there is often a certain force acting between the bearing and support at both end surfaces in order to maintain these constraint relationships, which is the static friction. When the force at one of the end surfaces exceeds the maximum static friction, the constraint relationship in Equation (50) or (51) can no longer be maintained, and the force at that end surface becomes sliding friction.

$$v_{l,\tau ,y}=\dot{y}-{\displaystyle \frac{1}{2}}{\dot{\theta }}_z{l}_0=0, v_{l,\tau ,z}=\dot{z}+{\displaystyle \frac{1}{2}}{\dot{\theta }}_y{l}_0=0$$

(50)

$$v_{r,\tau ,y}=\dot{y}+{\displaystyle \frac{1}{2}}{\dot{\theta }}_z{l}_0=0, v_{r,\tau ,z}=\dot{z}-{\displaystyle \frac{1}{2}}{\dot{\theta }}_y{l}_0=0$$

(51)

$y,z,{\theta }_y,{\theta }_z$ and $\dot{y},\dot{z},{\dot{\theta }}_y,{\dot{\theta }}_z$ are the relative displacements and velocities between the bearing and support. Their relationship with the generalized coordinates of the rotor system can be expressed as

$$\begin{array}{l}y=y_5-y_s, z=z_5-z_s, {\theta }_y={\theta }_{5,y}-{\theta }_{s,y}, {\theta }_z={\theta }_{5,z}-{\theta }_{s,z}\\ \dot{y}={\dot{y}}_5-{\dot{y}}_s, \dot{z}={\dot{z}}_5-{\dot{z}}_s, {\dot{\theta }}_y={\dot{\theta }}_{5,y}-{\dot{\theta }}_{s,y}, {\dot{\theta }}_z={\dot{\theta }}_{5,z}-{\dot{\theta }}_{s,z}\end{array}$$

(52)

In summary, the loose support force between the bearing and support can be described using Equations (48) and (49) at any motion of rotor system. However, the existence of the constraint relationships shown in Equation (50) or (51) depends on whether relative movement occurs (which means $\left|v_1\right|=0$ or $\left|v_2\right|=0$ holds) or if there is a tendency for relative movement between the bearing and support at the end surfaces, which requires calculating the forces at both end surfaces and determining whether they exceed the maximum static friction. The derivation process for these forces is as follows:

When the bearing and support move together, the total force exerted on the support node by the bearing (which means node 5 of the rotor system), including the lateral forces $f_{s,y},f_{s,z}$ and the moments $M_{s,y},M_{s,z}$, acts as the external load. It can be expressed as

$$\left\{\begin{array}{c}{f}_{s,y}\\ f_{s,z}\\ M_{s,y}\\ M_{s,z}\end{array}\right\}=\left[\begin{array}{cccc}{m}_s& & & \\ & m_s& & \\ & & J_s& \\ & & & J_s\end{array}\right]\left\{\begin{array}{c}{\ddot{y}}_s\\ {\ddot{z}}_s\\ {\ddot{\theta }}_{s,y}\\ {\ddot{\theta }}_{s,z}\end{array}\right\}+\left[\begin{array}{cccc}{c}_s& & & \\ & c_s& & \\ & & c_{s,\theta }& \\ & & & c_{s,\theta }\end{array}\right]\left\{\begin{array}{c}{\dot{y}}_s\\ {\dot{z}}_s\\ {\dot{\theta }}_{s,y}\\ {\dot{\theta }}_{s,z}\end{array}\right\}+\left[\begin{array}{cccc}{k}_s& & & \\ & k_s& & \\ & & k_{s,\theta }& \\ & & & k_{s,\theta }\end{array}\right]\left\{\begin{array}{c}{y}_s\\ z_s\\ {\theta }_{s,y}\\ {\theta }_{s,z}\end{array}\right\}$$

(53)

which means

$$\left\{\begin{array}{c}{f}_{s,y}\\ f_{s,z}\\ M_{s,y}\\ M_{s,z}\end{array}\right\}=\left\{\begin{array}{c}{m}_s{\ddot{y}}_s+c_s{\dot{y}}_s+k_s{y}_s\\ m_s{\ddot{z}}_s+c_s{\dot{z}}_s+k_s{z}_s\\ J_s{\ddot{\theta }}_{s,y}+c_{s,\theta }{\dot{\theta }}_{s,y}+k_{s,\theta }{\theta }_{s,y}\\ J_s{\ddot{\theta }}_{s,z}+c_{s,\theta }{\dot{\theta }}_{s,z}+k_{s,\theta }{\theta }_{s,z}\end{array}\right\}$$

(54)

In addition to the forces at the two end surfaces (including the forces $f_l,f_r$, where the subscripts $l,r$ denote the left and right end surfaces, respectively), the total interaction force between the bearing and support also includes the contact force at the radial cylindrical surface (including the force $f_n$ and the moment $M_n$). The component forms of the forces at the two end surfaces $f_{l,\tau ,y},f_{l,\tau ,z},f_{r,\tau ,y},f_{r,\tau ,z}$ can be expressed as

$$\left\{\begin{array}{l}{f}_{l,\tau ,y}=\left(f_{s,y}+f_{n,y}\right)/2-\left(M_{s,z}+M_{n,z}\right)/l_0\\ f_{l,\tau ,z}=\left(f_{s,z}+f_{n,z}\right)/2+\left(M_{s,y}+M_{n,y}\right)/l_0\end{array}\right.$$

(55)

$$\left\{\begin{array}{l}{f}_{r,\tau ,y}=\left(f_{s,y}+f_{n,y}\right)/2+\left(M_{s,z}+M_{n,z}\right)/l_0\\ f_{r,\tau ,z}=\left(f_{s,z}+f_{n,z}\right)/2-\left(M_{s,y}+M_{n,y}\right)/l_0\end{array}\right.$$

(56)

We can take the left end surface of the bearing as an example. When the interaction force between the bearing and support at the left end surface remains below the maximum static frictional force (set as $\mu N$), which means $\sqrt{f_{l,\tau ,y}^2+f_{l,\tau ,z}^2}<\mu N$, the constraint relationship in Equation (50) holds. The same principle applies to the right end surface. Based on the above constraint relationships and the constraint relationship caused by the transition shaft (Equation (31)), the constraint equations can be written as

$$\begin{array}{l}{g}_1=v_{l,\tau ,y}\cdot H\left(\mu N-\sqrt{f_{l,\tau ,y}^2+f_{l,\tau ,z}^2}\right)=0\\ g_2=v_{l,\tau ,z}\cdot H\left(\mu N-\sqrt{f_{l,\tau ,y}^2+f_{l,\tau ,z}^2}\right)=0\\ g_3=v_{r,\tau ,y}\cdot H\left(\mu N-\sqrt{f_{r,\tau ,y}^2+f_{r,\tau ,z}^2}\right)=0\\ g_4=v_{r,\tau ,z}\cdot H\left(\mu N-\sqrt{f_{r,\tau ,y}^2+f_{r,\tau ,z}^2}\right)=0\\ g_5=\sqrt{{\left(y_{3a}-y_{3b}\right)}^2+{\left(z_{3a}-z_{3b}\right)}^2}=0\end{array}$$

(57)

The above constraint equations can be uniformly expressed as

$$G_c\left(\ddot{q},\dot{q},q,t\right)={\left\{g_1,g_2,g_3,g_4,g_5\right\}}^{\mathrm{T}}=0$$

(58)

Thus, the governing equations (which includes the motion differential equations and the constraint equations) of the multi-supports rotor system with bearing clearance can be expressed as

$$\left\{\begin{array}{c}M\ddot{q}+\left(C-\omega G\right)\dot{q}+K\left(q-\Delta q_m\right)=Q+F_c+F_{sp}+F_{sum}\\ G_c=0 \end{array}\right.$$

(59)

4. Numerical Method for Dynamic Response of Rotor Systems with Constraint Equations

For the solution of dynamic response of nonlinear systems, common numerical methods include the Harmonic Balance Method (HBM) [39], Optimal Auxiliary Functions Method (OAFM) [40,41] and many numerical integration methods. However, in the case of the multi-supports rotor system with bearing clearance, which is characterized by high degrees of freedom, HBM and OAFM often struggle to balance computational efficiency with solution accuracy, and the dynamic response cannot be analyzed using conventional numerical methods due to the presence of the constraint equations $G_c\left(\ddot{q},\dot{q},q,t\right)=0$ as shown in Equation (59). Accordingly, a numerical method based on the Newmark-β approach which can be applied in the numerical study of rotor systems with constraint equations is proposed in this section.

Assuming the motion differential equations and constraint equations of rotor systems can be written as

$$\mathrm{motion} \mathrm{differential} \mathrm{equations}: M\ddot{x}+\left(C-\omega G\right)\dot{x}+Kx=Q\left(t\right)$$

(60)

$$\mathrm{constraint} \mathrm{equations}: G_c\left(\ddot{x},\dot{x},x,t\right)=0$$

(61)

Critically, the adoption of $x$ (rather than $q$) for generalized coordinates signifies that this method extends beyond the multi-supports rotor system with bearing clearance to any rotor system with constraint equations. Set the freedom degree of the rotor system as $p$, and the number of constraint equations as $q$, which means

$$x={\left\{x_1,x_2,\cdots ,x_p\right\}}^{\mathrm{T}}$$

(62)

$$G_c\left(\ddot{x},\dot{x},x,t\right)={\left\{g_1\left(\ddot{x},\dot{x},x,t\right),g_2\left(\ddot{x},\dot{x},x,t\right),\cdots ,g_q\left(\ddot{x},\dot{x},x,t\right)\right\}}^{\mathrm{T}}=0$$

(63)

Introduce the Lagrange multipliers $\lambda ={\left\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_q\right\}}^{\mathrm{T}}$. Coupling the motion differential equations with the constraint equations, an extended system is formed, and the extended equations are as follows:

$$\left\{\begin{array}{c}M\ddot{x}+\left(C-\omega G\right)\dot{x}+Kx+{\Phi }_x^{\mathrm{T}}\lambda =Q\left(t\right)\\ G_c\left(\ddot{x},\dot{x},x,t\right)=0\end{array}\right.$$

(64)

In the equation, ${\Phi }_x$ represents the Jacobian matrix generated by constraint equations (Equation (61)), and it satisfies

$${\Phi }_x={\displaystyle \frac{\partial G_c\left(\ddot{x},\dot{x},x,t\right)}{\partial \ddot{x}}}=\left[\begin{array}{ccc}{\displaystyle \frac{\partial g_1\left(\ddot{x},\dot{x},x,t\right)}{\partial {\ddot{x}}_1}}& \dots & {\displaystyle \frac{\partial g_1\left(\ddot{x},\dot{x},x,t\right)}{\partial {\ddot{x}}_p}}\\ ⋮& \ddots & ⋮\\ {\displaystyle \frac{\partial g_q\left(\ddot{x},\dot{x},x,t\right)}{\partial {\ddot{x}}_1}}& \dots & {\displaystyle \frac{\partial g_q\left(\ddot{x},\dot{x},x,t\right)}{\partial {\ddot{x}}_p}}\end{array}\right]$$

(65)

It can be assumed that the total number of steps in the numerical calculation is $N$, and the time step size for each step is fixed as $\Delta t$. The initial values of the numerical calculation include the values of motion $x_0,{\dot{x}}_0$ and the values of the Lagrange multipliers ${\lambda }_0$. The former can be determined based on the initial motion state of the rotor system, while the latter can be set to 0.

For the step $n+1\left(n=0,1,\dots ,N-1\right)$, the preliminary values of displacement and velocity $x_{n+1}^{pre},{\dot{x}}_{n+1}^{pre}$ are first calculated based on the parameters from the previous step (step $n$), as follows:

$$\begin{array}{l}{x}_{n+1}^{pre}=x_n+\Delta t\cdot {\dot{x}}_n+{\left(\Delta t\right)}^2\cdot \left({\displaystyle \frac{1}{2}}-\beta \right){\ddot{x}}_n\\ {\dot{x}}_{n+1}^{pre}={\dot{x}}_n+\Delta t\cdot \left(1-\gamma \right){\ddot{x}}_n\end{array}$$

(66)

Then, ${\ddot{x}}_{n+1},{\lambda }_{n+1}$ are obtained by solving the following nonlinear equations:

$$\left\{\begin{array}{c}M{\ddot{x}}_{n+1}+\left(C-\omega G\right){\dot{x}}_{n+1}^{pre}+K{x}_{n+1}^{pre}+{\Phi }_{x,n+1}^{\mathrm{T}}{\lambda }_{n+1}=Q\left(t_{n+1}\right)\\ G_c\left({\ddot{x}}_{n+1},{\dot{x}}_{n+1}^{pre},x_{n+1}^{pre},t_{n+1}\right)=0\end{array}\right.$$

(67)

in equation:

$${\Phi }_{x,n+1}={\displaystyle \frac{\partial G_c\left({\ddot{x}}_{n+1},{\dot{x}}_{n+1}^{pre},x_{n+1}^{pre},t_{n+1}\right)}{\partial \ddot{x}}}$$

(68)

The nonlinear equations can be solved using the discrete Newton method, and the specific process is as follows:

It can be set that

$$F\left(z\right)=\left\{\begin{array}{l}M{\ddot{x}}_{n+1}+\left(C-\omega G\right){\dot{x}}_{n+1}^{pre}+K{x}_{n+1}^{pre}+{\Phi }_{x,n+1}^{\mathrm{T}}{\lambda }_{n+1}-Q\left(t_{n+1}\right)\\ G_c\left({\ddot{x}}_{n+1},{\dot{x}}_{n+1}^{pre},x_{n+1}^{pre},t_{n+1}\right)\end{array}\right.=0$$

(69)

where $z={\left[{\ddot{x}}_{n+1}^{\mathrm{T}} {\lambda }_{n+1}^{\mathrm{T}}\right]}^{\mathrm{T}}$, and the initial values of it are provided by the parameters from the previous step (step $n$), which means $z^{\left(0\right)}={\left[{\ddot{x}}_n^{\mathrm{T}} {\lambda }_n^{\mathrm{T}}\right]}^{\mathrm{T}}$.

The iterative computation is carried out using the following iterative formula:

$$z^{\left(k+1\right)}=z^{\left(k\right)}-J{\left(z^{\left(k\right)},h^{\left(k\right)}\right)}^{-1}F\left(z^{\left(k\right)}\right) k=0,1,2,\cdots$$

(70)

where

$$h^{\left(k\right)}={\left\{h_1^{\left(k\right)},h_2^{\left(k\right)},\cdots ,h_{p+q}^{\left(k\right)}\right\}}^T, h_j^{\left(k\right)}\ne 0 \left(j=1,2,\cdots ,p+q\right)$$

(71)

$$J\left(z^{\left(k\right)},h^{\left(k\right)}\right)=\left[\begin{array}{ccc}{\displaystyle \frac{f_1\left(z^{\left(k\right)}+h_1^{\left(k\right)}{e}_1\right)-f_1\left(z^{\left(k\right)}\right)}{h_1^{\left(k\right)}}}& \dots & {\displaystyle \frac{f_1\left(z^{\left(k\right)}+h_{p+q}^{\left(k\right)}{e}_{p+q}\right)-f_1\left(z^{\left(k\right)}\right)}{h_{p+q}^{\left(k\right)}}}\\ ⋮& \ddots & ⋮\\ {\displaystyle \frac{f_{p+q}\left(z^{\left(k\right)}+h_1^{\left(k\right)}{e}_1\right)-f_{p+q}\left(z^{\left(k\right)}\right)}{h_1^{\left(k\right)}}}& \dots & {\displaystyle \frac{f_{p+q}\left(z^{\left(k\right)}+h_{p+q}^{\left(k\right)}{e}_n\right)-f_{p+q}\left(z^{\left(k\right)}\right)}{h_{p+q}^{\left(k\right)}}}\end{array}\right]$$

(72)

in the equation, $e_j$ is the $p+q$ dimensional unit vector of the dimension $j$.

It is worth mentioning that the Jacobian matrix of constraint equations ${\Phi }_x$ is computed using an analogy to the difference method of Equation (72), which will not be elaborated here. In summary, the iterative computation continues until

$$‖J{\left(z^{\left(k\right)},h^{\left(k\right)}\right)}^{-1}F\left(z^{\left(k\right)}\right)‖/‖z^{\left(k\right)}‖\le \epsilon$$

(73)

which means the iterative computation converges. According to $z^{\left(k\right)}={\left[{\ddot{x}}_{n+1}^{\mathrm{T}} {\lambda }_{n+1}^{\mathrm{T}}\right]}^{\mathrm{T}}$, ${\ddot{x}}_{n+1}$ can be obtained, and it can be used to correct the preliminary values $x_{n+1}^{pre},{\dot{x}}_{n+1}^{pre}$ derived from Equation (66), yielding.

$$\begin{array}{l}{x}_{n+1}=x_{n+1}^{pre}+{\left(\Delta t\right)}^2\cdot \beta \cdot {\ddot{x}}_{n+1}\\ {\dot{x}}_{n+1}={\dot{x}}_{n+1}^{pre}+\Delta t\cdot \gamma \cdot {\ddot{x}}_{n+1}\end{array}$$

(74)

In summary, the numerical method for the dynamic response of rotor systems with constraint equations is divided into the following steps, and the corresponding flowchart is shown in Figure 11.

• Couple the motion differential equations with the constraint equations, and introduce the Lagrange multiplier $\lambda$ to form an extended system, of which the extended equations are given by Equation (64).
• Based on the initial motion state of the rotor system, specify the initial values $x_0,{\dot{x}}_0,{\lambda }_0$ for numerical computation.
• Begin the numerical computation. For the step $n+1\left(n=0,1,\dots ,N-1\right)$:
  • Firstly, use Equation (66) to obtain the preliminary values of displacement and velocity $x_{n+1}^{pre},{\dot{x}}_{n+1}^{pre}$;
  • Then, set the precision level $\epsilon$ and the maximum number of iterations $N$, and use the discrete Newton method to solve the nonlinear equations (Equation (69)) in order to obtain the acceleration ${\ddot{x}}_{n+1}$;
  • Based on the results from the previous step, use Equation (74) to correct the pre-liminary values $x_{n+1}^{pre},{\dot{x}}_{n+1}^{pre}$ obtained in 3.a., yielding the displacement and velocity $x_{n+1},{\dot{x}}_{n+1}$ for this step;
  • If $n+1<N$, return to 3.a. to continue the computation; otherwise, stop the computation.
• Complete the computation.

5. Numerical Study on the Dynamic Response

For the mechanical model established in Section 3, the dynamic response of the multi-supports rotor system with bearing clearance is studied by using the numerical method proposed in Section 4. The parameters are shown in

Table 4. Parameters of the multi-supports rotor system with bearing clearance.

Description	Value	Description	Value
mass of disk $m_1,m_2$ (kg)	150,120	unbalance of disk ${\left(me\right)}_1,{\left(me\right)}_2$ (g × mm)	2000,2000
spindle moment of inertia of disk $I_{p,1},I_{p,2}$ (kg × m2)	10,5.5	diameter moment of inertia of disk $I_{d,1},I_{d,2}$ (kg × m2)	8,3
length of shaft $l_{12},l_{23},l_{34},l_{45}$ (mm)	300,300, 1400,100	density of shaft material ${\rho }_{12},{\rho }_{23},{\rho }_{34},{\rho }_{45}$ (kg/m3)	4680,4680, 8240,8240
outer diameter of shaft $D_{12},D_{23},D_{34},D_{45}$ (mm)	120,140,120,140	elastic modulus of shaft material $E_{12},E_{23},E_{34},E_{45}$ (GPa)	112,112, 204,204
inner diameter of shaft $d_{12},d_{23},d_{34},d_{45}$ (mm)	80,60,100,100	Poisson ratio of shaft material ${\mu }_{12},{\mu }_{23},{\mu }_{34},{\mu }_{45}$	0.3,0.3,0.3,0.3
stiffness of 1#, 2#, and 3# bearings $k_1,k_2,k_3$ (×107 N/m)	1.5,3,1.5	damping of 1#, 2#, and 3# bearings $c_1,c_2,c_3$ (×103 N/(m/s))	1.5,2,1.5
mass of 3# bearing $m_b$ (kg)	10	mass of 3# support $m_s$ (kg)	100
elastic moduli of the 3# bearing $E_b$ (GPa)	204	Poisson’s ratio of the 3# bearing ${\mu }_b$	0.3
elastic moduli of the 3# support $E_s$ (GPa)	204	Poisson’s ratio of the 3# support ${\mu }_s$	0.3
width of the bearing outer ring $l_0$ (mm)	30	collision factor $c_c$ (s/N)	100
bearing clearance ${\delta }_0$ (mm)	0.05	coefficient of friction $\mu$	0.3
pretension force of 3# bearing $N$ (N)	5000	misalignment of 3# bearing $\delta$ (mm)	1.0

(referring to the low-pressure rotor system of an aero engine).

5.1. Dynamic Response at Different Rotational Speeds

Firstly, the steady-state dynamic response under unbalance at different rotational speeds is calculated. Since the motion state of the rotor system can be depicted more accurately and concisely using the frequency spectrum of response, the spectrum cascades of the dynamic response at different rotational speeds are extracted, as shown in Figure 12. It should be noted that the response at 3# bearing represents the motion of the support of 3# bearing. In the figure, the spectrum cascades at different rotational speeds do not include any transient components. Considering that the misalignment causes significant static displacement of the rotor (which means that the amplitude of the 0 Hz component in the response is high, and it may far exceed the amplitudes of other frequencies), the spectrum cascades of the velocity in the y direction are used to facilitate the identification and investigation of the relative magnitudes of the frequency components.

It can be observed that the dynamic response is primarily composed of the rotational frequency $f$, with the presence of the harmonic frequency $2f$ caused by misalignment. When the rotational speed is high, significant harmonic frequencies $3f,4f,5f$ appear in the response of disk 2 and 3# bearing. Since these harmonic frequencies are particularly prominent at positions near the loose support, it is speculated that this phenomenon may be related to the bearing clearance.

According to Figure 12b,c, the dynamic response exhibits different frequency distribution characteristics at different rotational speeds, which essentially stems from the varying constraint effects of the loose support on the rotor at different speeds, leading to the difference in the motion state of the rotor system.

The relative movement between the bearing and support potentially occurs only when the dynamic response is sufficiently high and the dynamic load at the loose support become relatively significant, which results in the nonlinear characteristics of the loose support force and affects the motion state of rotor system. Therefore, two rotational speeds with high amplitudes of response are selected in the low-speed and high-speed regions, respectively: point A (4500 rpm) and point B (10,500 rpm). The dynamic responses at these two speeds are calculated, as shown in Figure 13 and Figure 14.

It should be noted that the rotor deformation ignores the static displacement caused by misalignment, and the spectrum plots also show the frequency distribution of the velocity in the y direction at these three positions.

According to Figure 13, the rotor system exhibits localized vibration on the front end of the rotor at speed A, with significant lateral displacement at disk 1. The trajectories at different axial positions (at disk 1 and disk 2) are approximately circular, indicating the synchronous whirling status. The frequency spectrums are primarily dominated by the rotational frequency $f$, accompanied by the harmonic frequency $2f$ caused by misalignment. Meanwhile, the support moves together with the rotor under the influence of the whirling of rotor, and there is no relative movement between them. Since the support is only under the vibration loads transmitted from the rotor, including unbalance excitation and misalignment excitation, and it does not generate unbalance excitation itself, the motion state of it is more significantly influenced by misalignment excitation compared to the rotor system. As a result, the amplitude of the second harmonic frequency $2f$ in the spectrum plot of 3# bearing is relatively higher.

According to Figure 14, the rotor system exhibits significant bending deformation at speed B. The trajectories of different positions are non-circular, indicating the non-synchronous whirling status. The frequency spectrums are not only primarily dominated by the rotational frequency $f$ and harmonic frequency $2f$ caused by misalignment, but are also accompanied by noticeable harmonic frequency components such as $3f,4f,5f$.

By comparing the spectrum plots at different positions in Figure 14h–j, it can be observed that the closer to the loose support, the more pronounced the harmonic frequency components in the frequency spectrums. Furthermore, there is a certain amount of relative movement between the support and rotor according to Figure 14a, and the constraint effect of loose support on the motion of rotor exhibits nonlinear characteristics. The time history of the relative displacements $\left|r_1\right|,\left|r_2\right|$ of the bearing and support at the contact points on both sides is shown in Figure 15, which fluctuates periodically within a certain range near clearance ${\delta }_0$, indicating that there is periodic contact and separation (this process can be called “collision”) between the bearing and support. Therefore, it can be concluded that the relative movement and periodic collision between the bearing and support leads to the presence of significant harmonic frequency components in the dynamic response.

5.2. The Influence of Angular Deflections

Consider a situation where the influence of angular deflections in the model of support structure with bearing clearance is neglected, including the effect on the contact width and the angular moment $M_{sum}$. The spectrum cascades of the dynamic response at the disk 1, disk 2 and 3# bearing at different rotational speeds are calculated, as shown in Figure 16.

By comparing Figure 12 and Figure 16, it can be observed that the harmonic frequency components are markedly reduced when the influence of angular deflections in the model of support structure with bearing clearance is neglected.

The spectrum cascades of relative angular deflections between bearing and support at different rotational speeds are further extracted, as shown in Figure 17.

According to Figure 12 and Figure 17, it can be observed that the increased amplitude of harmonic frequency components is accompanied by relatively significant relative angular deflections at higher rotational speeds. Therefore, the relative movement and periodic collision between the bearing and support lead to the occurrence of the harmonic frequencies in the dynamic response, while the relative angular deflection between them is the critical factor contributing to the amplitudes of the harmonic frequencies. Considering the influence of angular deflection is of paramount importance in the model of support structure with bearing clearance.

5.3. The Influence of Key Parameters

The uniqueness of the dynamic response of the multi-supports rotor system with bearing clearance lies in the fact that the frequency spectrum contains a significant harmonic frequency $2f$ caused by misalignment and harmonic frequencies $3f,4f,5f$ induced by bearing clearance in addition to the rotational frequency caused by unbalance. According to Equations (14) and (34), the misalignment excitation and the loose support force vary with the change of the bearing clearance ${\delta }_0$ and the misalignment $\delta$, thereby affecting the magnitudes of these frequency components. Therefore, the influence of these two parameters on the dynamic response will be investigated by focusing on the change of the amplitudes of the rotational frequency $f$ and the harmonic frequencies $2f,3f$.

5.3.1. Bearing Clearance ${\delta }_0$

Varying the clearance ${\delta }_0$ while keeping other parameters constant, the amplitudes of the rotational frequency $f$ and the harmonic frequencies $2f,3f$ in the lateral displacement of the disk 1, disk 2 and 3# bearing at different rotational speeds are calculated, as shown in Figure 18.

It can be known that

(1)

In the low speed range (0~9000 rpm), the change of the clearance ${\delta }_0$ has minimal impact on the amplitudes of these frequency components because there is no relative displacement between the bearing and support and no loose support force generated by it.

(2)

When the rotational speed is high (9000~18,000 rpm), the amplitudes of the harmonic frequencies $2f,3f$ decrease as the clearance ${\delta }_0$ increases, which is attributed to the reduced influence of collision between the bearing and support on the dynamic response as the clearance ${\delta }_0$ increases, and the transmitted vibrational energy between the rotor and support also decreases consequently.

This trend is also reflected in the change of the rotational frequency $f$, which increases at disk 2 but decreases at 3# bearing, indicating that the vibrational energy transmitted through collision between the bearing and support is reduced, and more energy is concentrated on the rotor.

It should be noted that an increase in bearing clearance may reduce the constraint effect of the support, potentially leading to the occurrence of quasi-rigid local modes. However, such quasi-rigid modal vibrations primarily manifest as transient components that decay with time in the dynamic response. Consequently, their impact on the dynamic response of the rotor system becomes negligible during steady-state operation, which is mainly focused on in this paper.

5.3.2. Misalignment $\delta$

Varying the misalignment $\delta$ while keeping other parameters constant, the amplitudes of the rotational frequency $f$ and the harmonic frequencies $2f,3f$ in the lateral displacement of the disk 1, disk 2 and 3# bearing at different rotational speed are calculated, as shown in Figure 19.

As the misalignment $\delta$ increases, the amplitudes of the rotational frequency $f$ at different positions remain essentially unchanged, while the amplitudes of the harmonic frequencies $2f,3f$ significantly increase. This is because the increase in misalignment leads to the corresponding increase in the misalignment excitation, which directly results in the increase in the amplitude of the harmonic frequency $2f$, making the non-synchronous whirling status of rotor system more pronounced, and the collision between the bearing and support, as well as the resulting harmonic frequency $3f$, also become more evident.

In summary, the dynamic response of the multi-supports rotor system with bearing clearance primarily consists of the rotational frequency $f$ and the harmonic frequency $2f$ induced by misalignment. As the rotational speed increases and the dynamic load at the loose support rises, relative movement (especially the relative angular deflection) and collision between the bearing and support occurs, leading to the emergence of significant harmonic frequencies $3f,4f,5f$ in the dynamic response. Both the changes of the bearing clearance ${\delta }_0$ and the misalignment $\delta$ have a notable impact on the dynamic response. Reducing the bearing clearance or increasing the misalignment leads to the increase in the amplitudes of these harmonic frequencies.

6. Experimental Verification

To validate the conclusions derived from the numerical study, a test rig of the multi-supports rotor system with bearing clearance is designed based on the mechanical model illustrated in Figure 9, as shown in Figure 20. It should be noted that the distinct colors in Figure 20a serve solely to differentiate between components and are not associated with material types or other attributes.

In the test rig, the rear shaft of disk 1 is connected to the slender shaft via a transition shaft. The torque is transmitted between the transition shaft, the rear journal, and the slender shaft using splines. Additionally, there is a bearing clearance ${\delta }_0$ at the 3# bearing, making the support at 3# bearing the loose support. The right end of the 3# bearing is connected to the motor through a coupling, and the motor provides the operation torque of the multi-supports rotor system with bearing clearance (hereinafter referred to as the test rotor). The detailed parameters of the test rotor are listed in

Table 5. Parameters of the test rotor.

Description	Value	Description	Value
$\mathrm{mass} \mathrm{of} \mathrm{disk} m_1,m_2$ (kg)	9,8	$\mathrm{unbalance} \mathrm{of} \mathrm{disk} {\left(me\right)}_1,{\left(me\right)}_2$ (g × mm)	500,500
spindle moment of inertia of disk $I_{p,1},I_{p,2}$ (kg × m2)	0.055,0.036	diameter moment of inertia of disk $I_{d,1},I_{d,2}$ (kg × m2)	0.028,0.019
distance between bearings $d_{12},d_{23}$ (mm)	200,850	support stiffness of 1#,2#,3# bearings $k_1,k_2,k_3$ (×106 N/m)	4,4,2
mass of 3# bearing $m_b$ (kg)	0.5	mass of 3# frame $m_s$ (kg)	15
width of the bearing outer ring $l_0$ (mm)	30	bearing clearance ${\delta }_0$ (mm)	0.05
pretension force of 3# bearing $N$ (N)	10,000	misalignment of 3# bearing $\delta$ (mm)	2

.

6.1. Experimental Plan

To investigate the impact of bearing clearance on the dynamic response of the test rotor, it is necessary to operate the test rotor near the critical speeds of it as much as possible, which ensures that the dynamic response becomes significant and allows the loose support to exhibit nonlinear mechanical characteristics. A finite element model of the test rotor neglecting the effects of bearing clearance and misalignment is established. The critical speeds are calculated and analyzed, as illustrated in Figure 21, which shows the corresponding modal shapes at the first two critical speeds.

It can be observed that the angular deflections are substantial at the 3# bearing in the modal shapes of the first two critical speeds, which implies that regardless of which critical speed is chosen for operation, the significant relative angular deflections between the bearing and support at the 3# bearing is likely to occur, thereby inducing nonlinear mechanical characteristics of the loose support.

The test plan requires the rotor to be progressively accelerated to several predefined operating speed steps, which are near the first two critical speeds to ensure the significant angular deflections of the test rotor, as shown in Figure 22.

Sensors are installed on disk 1, disk 2, and three bearings along both horizontal and vertical directions to measure the lateral displacements of them. Panasonic HL-G108-S-J laser displacement sensors (Osaka, Japan) are used at disk 1 and disk 2, featuring an 85 mm reference distance, ±20 mm range, and 2.5 μm resolution, while DH 1A803E piezoelectric accelerometers (Taizhou, China) are employed at three bearings with ±500 g range, 0.008 g resolution, and 5 Hz–10 kHz frequency response, as shown in Figure 23.

6.2. Analysis of Experimental Data

6.2.1. Dynamic Response at Different Rotational Speeds

Firstly, the dynamic response throughout the entire test is analyzed. Similar to the numerical study, our focus is solely on the frequency spectrums of the test rotor. The spectrum cascades of the dynamic response at the disk 1, disk 2, and 3# bearing throughout the entire test are extracted, as shown in Figure 24.

It can be observed that noticeable harmonic frequencies $2f,3f,4f,5f$ occur in the dynamic response at each sensor in addition to the rotational frequency $f$. Moreover, the harmonic frequency components are more prominent in the responses near the loose support (disk 2 and 3# bearing), which is consistent with the results of numerical study.

Considering the speed steps 2 and 4 of the speed–time curve (as shown in Figure 22) correspond to 3600 rpm and 5500 rpm, which are relatively close to the first two critical speeds (3319 rpm and 5184 rpm), respectively, the amplitudes of the dynamic response is higher at these two speed steps, along with the impact of nonlinear characteristics of the loose support on the frequency spectrums more significant. Therefore, the steady-state responses at these two speed steps are selected for detailed analysis, as shown in Figure 25 and Figure 26.

According to Figure 25 and Figure 26, the trajectories of the test rotor are non-circular at different positions, indicating that the test rotor is in non-synchronous whirling status at these two speed steps. Noticeable harmonic frequencies $2f,3f,4f,5f$ are observed in the spectrum plots, which are more pronounced in the response of positions near the loose support (disk 2 and 3# bearing).

By comparing the spectrum plots of the 3# bearing at two speed steps in Figure 25c and Figure 26c, it can be observed that the amplitudes of the harmonic frequencies are relatively higher at 3600 rpm compared to those at 5500 rpm. The reason is that these two speeds are near the first two critical speeds, and the rotor deformation resembles the modal shapes as shown in Figure 21. For these two modal shapes, the angular deflections at the 3# bearing are larger in the former than in the latter, making the relative movement and collision between the bearing and support more likely to occur, which leads to more pronounced harmonic frequency components in the dynamic response.

6.2.2. The Influence of Key Parameters

In the experiment, the influence of bearing clearance and misalignment on the dynamic response of test rotor are investigated by replacing the support of 3# bearing with different fit dimensions and adjusting the misalignment of 3# bearing.

Supports with different clearances relative to the outer ring of 3# bearing are replaced, while the bearing clearance is controlled to be around 0.01 mm and 0.05 mm, respectively, and tests are conducted using the speed–time curve shown in Figure 22. Considering that the influence of the bearing clearance on the dynamic response is particularly significant in the frequency spectrum of disk 2 and 3# bearing at the speed step 3600 rpm, the spectrum plots with different clearance ${\delta }_0$ at speed step 3600 rpm are analyzed, as shown in Figure 27. It can be observed that the amplitudes of the harmonic frequencies $2f,3f,4f,5f$ increase significantly as the bearing clearance decreases.

Similarly, the misalignment of 3# bearing is adjusted to 1 mm and 3 mm, respectively, and tests are conducted using the speed–time curve shown in Figure 22. The spectrum plots with different misalignment at the speed step 3600 rpm are presented in Figure 28. By comparison, it is evident that the amplitudes of the harmonic frequencies $2f,3f,4f,5f$ both show a noticeable increase as the misalignment increases.

In summary, the multi-supports rotor system with bearing clearance is in the non-synchronous whirling status during the test. Significant harmonic frequency components are observed in the dynamic response across various rotational speeds. Reducing the bearing clearance at 3# bearing or increasing the misalignment of 3# bearing both lead to a notable increase in the amplitudes of the harmonic frequency components, which is highly consistent with the results of numerical study.

7. Conclusions

The dynamic response of multi-supports rotor systems with bearing clearance is investigated in this paper. Bearing clearance induces nonlinear forces between the bearing and support, which are more complicated when the relative angular deflections between the bearing and support are significant, and a model of support structure with bearing clearance considering angular deflections is proposed to accurately describe them. A mechanical model of the multi-supports rotor system with bearing clearance considering the misalignment and the different types of friction between the bearing and support is developed, and a numerical method is presented for the solution of the dynamic response of the rotor system with constraint equations caused by the different types of friction. Finally, numerical study and experimental verification are employed to investigate the dynamic response of the rotor system. The main conclusions are as follows:

(1)

In the model of support structure with bearing clearance, the loose support force between the bearing and support includes the radial contact force and the tangential friction force caused by relative movement, and the moments due to the relative angular deflections between the bearing and support because of the bending deformation of the rotor, and the accuracy of the model is validated by comparing it with the other models and the FEM results.

(2)

The dynamic response of the multi-supports rotor system with bearing clearance contains not only the rotational frequency $f$ caused by the unbalance of the rotor and the harmonic frequency $2f$ due to the misalignment, but also the harmonic frequencies $3f,4f,5f$ induced by the relative movement and collision between the bearing and support. Moreover, the harmonic frequency components in the dynamic response are more pronounced near the support with bearing clearance, and relative angular motion between the bearing and support is the critical factor contributing to the amplitudes of the harmonic frequencies.

(3)

Reducing the bearing clearance or increasing the misalignment both leads to a notable increase in the amplitudes of the harmonic frequency components in the dynamic response.

While this paper focuses on steady-state dynamics response under constant-speed conditions, it is acknowledged that transient behaviors during acceleration or deceleration may induce complex nonlinear coupling effects and whirl instabilities, which represent critical challenges in rotor systems under operational transitions. Although such transient analyses fall beyond the scope of the current model framework, their investigation is essential for comprehensive dynamic characterization. In future research, we will incorporate time-varying rotational speeds and experimentally validated transient models, and further quantify these effects on the dynamic response of a rotor system based on the numerical method proposed in this paper.