# Nonlinear Dynamic Responses of a Rotor-Bearing System with Eccentric Squeeze Film Dampers

Authors: Jiaqi Han; Wei Chen; Lulu Liu; Guihuo Luo; Fei Wang
Journal: International Journal of Aerospace Engineering
 (2023)

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## Abstract

A squeeze film damper (SFD) is a commonly used component in aeroengine support structures. However, eccentric SFDs are often employed in practical applications. Existing research in this field has mainly focused on simple rotor models and frequency-domain solution techniques. In this study, a rotor system tester with and without an eccentric SFD was built, and a simulation model based on the finite element method and fixed interface modal synthesis method was proposed. Then, the nonlinear dynamic response of the rotor system during run-up and run-down was studied based on the tester and simulation model. The results show that an eccentric SFD affords zero frequency and a series of double frequencies in the rotor system response. An eccentric SFD may cause the increase of the critical speed, amplitude jump, and displacement amplitude oscillation of the rotor system during run-up/run-down. Additionally, it may cause the acceleration response at the mounting bracket to significantly increase when the rotor system is suddenly accelerated.


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## Body

## 1. Introduction

In aeroengine support systems, a squeeze film damper (SFD) is commonly used in combination with a squirrel cage. Usually, SFD is arranged in parallel with a squirrel cage support, the squirrel cage support is used to bear the gravity of the rotor system, and SFD is used to attenuate the vibrations and transmit forces of the rotor systems. Figure1 displays the typical arrangement of the SFD and squirrel cage support. However, in practical applications, factors such as machining error, assembly error, preloading error, and maneuvering flight cause a static displacement of the shaft journal. Moreover, an eccentric shaft journal will afford an eccentric SFD [1, 2], which may cause a series of nonlinear vibration phenomena. These phenomena may deteriorate the vibration of the rotor system, which may cause aeroengine failure. This study mainly focuses on the nonlinear response characteristics of a rotor system caused by static eccentricity of the squirrel cage.Figure 1
Typical structure of the SFD and squirrel cage support.In literature, SFD-rotor systems have always been the focus of researchers; numerous scholars have conducted considerable theoretical and experimental research in this field, such as nonlinear response of SFD-rotor system. Zhou et al. [3] established a SFD-rotor system model with a floating ring and carried out experimental research. The dynamic characteristics of the SFD-rotor system with a floating ring were studied and compared. Wan and Jian [4] systematically analyzed the periodic motion of the rotor system with SFD nonlinear force and rub-impact nonlinear force at different speeds. Wang [5] established a mathematical model of the rotor system with SFD, carried out a sensitivity analysis on the parameters affecting the critical speed of the rotor system, and analyzed the critical speed of the rotor system by using genetic algorithm. Bonello and Pham [6] carried out simulation and experimental research on the twin-shaft rotor system with SFD, and the research shows that in the simulation prediction; when the speed ratio is the ratio of two low integers, the initial phase has a greater influence on the rotor system response. Luo et al. [7] established a finite element model of the SFD-rotor system with a cantilever disk. Based on this model, the bifurcation phenomenon at different positions on the rotor system was studied. The results show that the response characteristics of the rotor at different positions are different. Ri et al. [8] proposed an IHB method for analyzing the dynamic response of a flexible rotor-elastic support system with SFD, compared the calculation results of this method with the calculation results of the Runge–Kutta method and literature results, and finally analyzed the stability of the rotor system with SFD based on this method. Chen et al. [9] combined the harmonic balance method and Runge–Kutta method to analyze the dynamic characteristics of the SFD-rotor system with dual-frequency excitation. The research conclusion shows that the nonlinear characteristics are mainly concentrated in the resonance region, and the damper parameters and the excitation frequency have a great influence on the nonlinear characteristics. Chen et al. [10] discussed the rigid body translation motion and rigid body precession motion in the SFD–rotor system with unsymmetrical supports. The simulation results show that when the translation amplitude is much larger than the precession amplitude, the precession can be ignored, and also analyzed the relationship between rotor system bifurcation phenomenon and rotor system parameters. Ma et al. [11] established a nonlinear simulation model and a tester of the rotor system including SFD and rolling bearings and studied the influence of structural parameters on the nonlinear characteristics of the rotor system based on the simulation model. Wang et al. [12] established a five degrees of freedom nonlinear restoring force mathematical model to study the bearing misalignment of the rotor system. And this mathematical model is used to study the influence of bearing seat misalignment, coupling misalignment, and unbalance factors on the response of the rotor system at different speeds. Ri et al. [13] studied the influence of third-order nonlinear factors on the response of the rotor system. The research shows that the third-order nonlinear factors have a greater influence on the rotor response near the second-order critical speed.Furthermore, studies have been conducted on the response characteristics of eccentric SFD-rotor systems. Tonnesen [14] designed an SFD-rotor tester in a vertical installation state, studied the damping coefficient of SFD under different structural parameters, and compared the experimental and theoretical results. In their study, the relative error between the theoretically and experimentally calculated SFD damping coefficients under the condition of small static eccentricity was small. Pan and Tonnesen [15] established an SFD oil film force model considering static eccentricity based on the “short bearing, half-film” formula and compared the steady-state, finite amplitude, and limit cycle motions of a rigid, symmetric rotor system. Sykes and Holmes [16] designed a three-bearing rigid rotor and studied the response characters of the rotor-bearing system in different support conditions. They observed the subharmonic motions and attributed them to the static misalignment of SFD. Zhao et al. [17–19] established an SFD model based on the short bearing hypothesis and the “π” film cavitation model, and they investigated the stability of eccentric SFD-mounted rotor systems using a trigonometric collocation method. Additionally, they stated that subharmonic and nonsynchronous vibrations might arise if parameters of the eccentric SFD are inappropriate. Bonello et al. [20] studied the cavitation eccentric SFD model with the receptance harmonic balance method. They focused on the effects of cavitation and found that the cavitation could be promoted by increasing the static eccentricity and that cavitation might be beneficial in preventing excessive shaft vibration. Adiletta and Pietra [21] experimentally proved that the critical speed of a rotor system might increase due to static eccentricity of SFD, and the theoretical prediction based on short bearing approximation well agreed with the experimental data.In the above research on the response of a static eccentric SFD-rotor system, the shaft is treated as a simple rigid rotor, and frequency-domain techniques, such as harmonic balance method or trigonometric collocation method, are employed as the theoretical analysis method. However, as stated in reference [20], these techniques might yield results that disagree with the results of the numerical integration method due to loss of periodicity. In comparison, time-domain techniques such as the Runge–Kutta method yield the actual steady-state response after the initial transient states have died out.In this study, a rotor system tester with/without eccentric SFD is established and a static eccentric loading method was proposed. Based on this, the response characteristics of the rotor system under different SFD static eccentricity conditions are obtained. Then, a modeling method based on the finite element method and the fixed interface modal synthesis method is proposed. Using this modeling method and the improved Newmark-β solution algorithm, the nonlinear dynamic responses of the rotor-bearing system with and without static eccentricity are obtained, analyzed, and compared. Finally, the nonlinear response of a rotor system with/without eccentric SFD during run-up and run-down is studied based on experimental and simulation methods.Compared with the existing researches, the modeling method developed herein can study the dynamic response of an aeroengine rotor system under static eccentricity conditions with complex structures, such as bifurcated structure, annular cavity structure, cantilever structure, and conical shell. Furthermore, the reasons for the increase of the displacement response amplitude at the disk, the jump of the displacement response amplitude, and the increase of the external force-caused eccentric SFD were provided based on the simulation results. A measurement method for effectively locating the SFD static eccentricity fault in practical applications was proposed by monitoring the acceleration response in different directions at the mounting bracket.


## 2. Dynamic Model and Simulation Method

### 2.1. SFD with and without Static Eccentricity

The SFD of the rotor system without static eccentricity is shown in Figure2. In the static state, the shaft journal center (Oj) and the clearance circle center (Ob) of SFD coincide, and in the working state. The shaft journal whirls around the clearance circle center (Ob), and the whirl center of the shaft journal (Oe) coincides with the clearance circle center (Ob) of SFD. The SFD of the rotor system with static eccentricity is shown in Figure 3. The shaft journal center (Oj) no longer coincides with the clearance circle center (Ob), and some distance is present between the two centers in the static state. In the working state, the whirl center of the shaft journal (Oe) no longer coincides with the center (Ob) of the clearance circle, and a misalignment force directed from Ob to Oe is introduced to describe this phenomenon.Figure 2
The SFD of the rotor system without static eccentricity. (a) In the static state. (b) In the working state.
(a)(b)Figure 3
The SFD of the rotor system with static eccentricity. (a) In the static state. (b) In the working state.
(a)(b)For an SFD in an ideal state, experimental investigations have demonstrated that the short bearing approximation of SFD exhibits some consistency with the experimental data [5, 6]. Thus, the short bearing assumption is adopted herein to incorporate SFD with the rotor system. Thus, for the SFD considered herein, in the process of obtaining the nonlinear force of SFD, the short bearing assumption and Reynolds boundary conditions are selected. The nonlinear force of the SFD is shown in the following [20–22]:
(1a)fxs=−μsRL3c2x2+y2xε̇I2+εψ̇I1−yε̇I1+εψ̇I3fys=−μsRL3c2x2+y2yε̇I2+εψ̇I1+xε̇I1+εψ̇I3,(1b)ε=x2+y2/c,(1c)ε̇=xẋ+yẏ/cx2+y2,(1d)ψ̇=yẋ−xẏ/x2+y2,(1e)tanψ=y/x.Here,fxs and fys represent the nonlinear force components introduced by SFD in the horizontal and vertical, and x and y represent the horizontal and vertical displacements of the shaft journal, respectively. R represents the radius of SFD, and L represents the length along the shaft of SFD. Additionally, μs and c denote the dynamic viscosity and thickness of the oil film, respectively. I1,I2,andI3 are Summerfield integrals that can be expressed as
(2a)I1=∫θ1θ1+πcos2θ1+εcosθ3dθ,(2b)I2=∫θ1θ1+πsinθcosθ1+εcosθ3dθ,(2c)I3=∫θ1θ1+πsin2θ1+εcosθ3dθ,(2d)θ1=tan−1−ε̇εϕ̇.The supporting structure of the rotor system mainly includes the squirrel cage and SFD.Fb represents the nonlinear force introduced by the bearing on the shaft journal, which can be expressed as
(3a)Fb=−Fk+FSFD.Here,FSFD represents the nonlinear force vector introduced by SFDs, which can be calculated using Equations (1a)–(1e), and Fk represents the force vector introduced by the squirrel cage, Fk=k1x1k1y1k2x2k2y2⋯knxnknyn, k1–kn represent the stiffness of the squirrel cage, and x1–xn and y1–yn represent the displacement of the shaft journal in the horizontal and vertical directions at each squirrel cage.When SFD is in a static eccentric state, the squirrel cage experiences some displacement in the static state and the nonlinear forceFb can be expressed using the following with static eccentricity.
(3b)Fb=−Fk+FSFD+Fe.Here,Fe represents the additional force vector of the squirrel cage afforded by the static eccentricity. Fe=k1xs1k1ys1k2xs2k2ys2⋯knxsnknysn, k1–kn represent the stiffness of the squirrel cage at each SFD, n represents the number of SFDs, and xs1–xsn and ys1–ysn represent the static eccentricity of each shaft journal in the horizontal and vertical directions, respectively. The SFD parameters of this study are shown in Table 1, and the physical meaning of each parameter in Table 1 is shown in Figure 4 and is consistent with Equations (1a)–(1e).Table 1
SFD parameters.
L [mm]R [mm]c [mm]μs [10−2 Pa s]Front SFD15300.11.0752Rear SFD15300.1Figure 4
Diagram of SFD.


### 2.2. Dynamic Model of the Rotor System

Herein, the Timoshenko beam element is used to simulate the shaft, and the shaft section element is shown in Figure5. Each element contains two nodes, μe1 represents the displacement of the node 1 along the x-axis, ve1 represents the displacement of the node 1 along the y-axis, θe1 represents the rotation angle of node 1 around the x-axis, φe1 represents the rotation angle of node 1 around the y-axis, and the physical meaning of the four degrees of freedom of node 2 is consistent with that of node 1. The z-axis direction represents the axial direction. Assuming that the shaft element is an isotropic linear elastic cylinder, the cross-section of the shaft element is equal along the z-axis direction, and any cross-section in the element is still flat after deformation, but it is not necessarily perpendicular to the neutral axis, i.e., shear deformation is considered. The generalized coordinates of each element are defined by
(4)u=ue1ve1θe1φe1ue2ve2θe2φe2.Figure 5
Diagram of the Timoshenko beam element [22].According to reference [23], the nonlinear equations of motion of a nonlinear rotor system can be represented using the following.
(5)Mu¨+C+ΩGu̇+Ku=Funb+Fb.Funb represents the force vector introduced by the shaft unbalance. Fb represents the nonlinear forces introduced by the bearings herein. M, C, G, and K denote the mass, damping, stiffness, and gyroscopic matrices of the rotor system, respectively, which can be obtained by assembling the element matrix in Ref. [22]. Additionally, u is the displacement vector of the rotor-bearing system.Figure6 displays the dynamic model of the SFD-rotor system. The shaft is divided into four parts, as shown in the figure, and all parts are solid cylinders. The diameter and start/end points of each part are shown in Table 2. Disk parameters are shown in Table 3; the location, mass, IP, ID, unbalance, and initial phase of the unbalance of the disk are given in Table 3. In the simulation, the disk was simulated as a concentrated mass unit that accounts for the gyroscopic effect. Moreover, only the radial stiffness of the squirrel cages was considered, and Table 4 displays the stiffness of the squirrel cages; kxx and kyy represent the stiffness of the squirrel cages in the x-axis and the y-axis direction, respectively, kxy and kyx represent the cross stiffness of the squirrel cages; what is more, the stiffness value of each squirrel cage in Table 4 is obtained through static tests.Figure 6
Dynamic model of the SFD-rotor system.Table 2
Shaft parameters.
Shaft numberStart point (mm)End point (mm)Diameter (mm)Part 1017318Part 217364320Part 364387220Part 487295620Table 3
Disk parameters.
Disk numberLocation (mm)Mass (kg)IP (kg m2)ID (kg m2)Unbalance (g cm)Initial phase16432.510.008540.00427100Table 4
Stiffness of squirrel cages.
Bearing numberkxx (N/m)kyy (N/m)kxy (N/m)kyx (N/m)12.5×1062.5×1060022.5×1062.5×10600Based on the above listed parameters, the dynamic model of the SFD-rotor system with and without eccentric SFD can be established, which has a total of 97 nodes. Among them, the node numbers of the disk, front support, and rear support are 66, 18, and 89, respectively.


### 2.3. Modeling Method Based on the Fixed Interface Modal Synthesis

In Subsection2.2, the dynamic model of the SFD-rotor system with/without static eccentricity was established by introducing different nonlinear force vectors exerted by bearings.Equation (5) shows that the dynamic model of the SFD-rotor system is a set of nonlinear differential equations. The nonlinear response of the SFD-rotor system can be obtained by solving these differential equations. Generally, the differential equations obtained in Subsection 2.2 have large dimensions and nonlinear forces, all of which result in low computational efficiency. Herein, after considering the overall computational efficiency and accuracy of the dynamic model, fixed interface modal synthesis is employed to reduce the dimensions of the nonlinear differential equations of the SFD-rotor system and thus the computational burden.First, the degrees of freedom of the rotor system are divided into two parts. The degree of freedom with external excitation is called the interface degree of freedom, such as the degree of freedom at the rotor bearing, where the nonlinear force introduced by SFD acts. The other is called the internal degree of freedom, where only the unbalanced forces act. Then, Equation (5) can be rewritten as
(6)MIIMIJMJIMJJu¨Iu¨J+GIIGIJGJIGJJu̇Iu̇J+KIIKIJKJIKJJuIuJ=FIunb0+0FJnonl.Here,uI represents the internal degrees of freedom of the rotor system, while uJ represents the interface degrees of freedom of the rotor system. Additionally, the transformation relation between the physical and modal coordinates of the system is
(7a)uIuJ=ϕkϕc0IqkuJ=TqkuJ.Here,ϕk∈RnI×nk represents the mass normalized normal mode matrix; ϕc∈RnI×nJ represents the mass normalized constrained mode; I represents a nJ dimensional identity matrix; qk∈RnI×1 represents the normal mode coordinates; nI, nJ, and nK are the numbers of internal degrees of freedom, interface degrees of freedom, and restrained principal modes retained in the calculation, respectively; and T is defined as the transformation matrix. The transformation matrix T can be obtained by solving ϕc and ϕk.According to the fixed interface modal synthesis principle, the definition of the constrained modeϕc is as follows. When the interface degrees of freedom of the system sequentially generate unit displacements and the remaining interface degrees of freedom are constrained, a set of displacements is generated by the internal degrees of freedom of the system. Therefore, ϕc can be expressed as
(7b)ϕc=−KII−1KIJ.Based on the motion differential Equation (6), by constraining all the interface degrees of freedom of the system and ignoring the influence of the gyroscopic effect, the differential equation of motion of the rotor system when the interface degrees of freedom are constrained can be obtained as shown in the following equation.
(7c)MIIu¨I+KIIuI=0.Solving the eigenvalues and eigenvectors of Equations (7a)–(7c), the mass normalized normal mode matrix ϕk and the modal frequency Ωk can be obtained.Equations (7a), (3a), and (3b) are substituted into Equation (6), and TT in front of the expression is multiplied on both sides of the equal sign of Equation (6). The reduced differential equation of motion of the system can be obtained as shown in Equations (8a) and (8b).
(8a)IM¯IJM¯JIM¯JJq¨ku¨J+G¯IIG¯IJG¯JIG¯JJq̇ku̇J+K¯II00K¯JJqkuJ=ϕkTFIunbϕcTFIunb+0FJnonl,where
(8b)M¯JJ=MJJ+MJIϕc+ϕcTMIIϕc+MIJM¯IJ=M¯JI=ϕkTMIIϕc+MIJG¯II=ϕkTGIIϕkG¯II=GJJ+GJIϕc+ϕcTGIIϕc+GIJ+cJJG¯IJ=G¯JI=ϕkTGIIϕc+ϕkTGIJK¯II=diagΩr21≤r≤nK¯JJ=KIJ+KJIϕc+kJJ.Ωr represents the r-th modal frequency of the system obtained from Equation (7c). Moreover, FIunb represents the unbalanced force vector acting on the internal degrees of freedom. FJnonl F2 represents the external load acting on the interface degrees of freedom. Herein, FJnonl=Fb.Here, the dimension of the differential equation of motion of the rotor system is reduced from the4n order of Equation (5) to the nI+nJ order of Equation (8a).Based on the above ideas, a dynamic model of the SFD-rotor system that comprehensively considers the model accuracy and solution efficiency is established. Additionally, the implicit time-domain method based on the Newmark-β method is used to study the dynamic responses of the nonlinear rotor system with/without static eccentricity.


### 2.4. Comparison of Critical Speed of Rotor System

Before conducting simulation and experimental research on the nonlinear response of the SFD-rotor system with and without eccentric SFD, the critical speed and vibration mode of the rotor system are first compared. In the free state, the motion equation of a rotor system can be written as(9)Mu¨+Gu̇+Ku=0.In the state space it can be expressed as(10a)Aq̇+Bq=0,where
(10b)q=uu̇,q̇=u̇u¨,A=GMM0B=K00M.Letqt=q0est and substitute it into Equation (10a) to obtain:
(10c)sAq0=−Bq0.The modal frequency of the rotor system and the corresponding modal shape can be obtained by solving Equation (10c). Furthermore, in a rotor system, at different speeds, the gyroscopic effect has different effects on the damping matrix G of the rotor system. Consequently, the matrix A in Equation (10b) will also be different. Therefore, a series of speeds need to be considered and calculations need to be performed at each speed. Then, the rotational speed as the abscissa and the modal frequency at each rotational speed as the ordinate are employed to draw the Campbell diagram. The constant rpm line is drawn on the Campbell diagram. The intersection of the constant rpm line and the model frequency lines of the forward precession of each order denotes the critical speed of the rotor system. The first two critical speeds of the rotor system obtained from the established model are 2703 and 12448 rpm, respectively, while the first two critical speeds of the rotor system obtained using the ANSYS beam model are 2662 and 12073 rpm, respectively. The difference between the critical speeds is due to the difference in the meshing densities of the two models; the maximum error between the two results is 3.1%. The Campbell diagram and the first two modes of the rotor system are shown in Figures 7–9. In Figure 7, the red lines represent forward precession, black lines represent back precession, solid lines represent the result of the established model, and dashed lines represent the result of the ANSYS BEAM model.Figure 7
Campbell diagram of the rotor system.Figure 8
First-order mode shape of the rotor system. (a) Model established in this paper. (b) ANSYS beam model.
(a)(b)Figure 9
Second-order mode shape of the rotor system. (a) Model established in this paper. (b) ANSYS beam model.
(a)(b)


## 2.1. SFD with and without Static Eccentricity

The SFD of the rotor system without static eccentricity is shown in Figure2. In the static state, the shaft journal center (Oj) and the clearance circle center (Ob) of SFD coincide, and in the working state. The shaft journal whirls around the clearance circle center (Ob), and the whirl center of the shaft journal (Oe) coincides with the clearance circle center (Ob) of SFD. The SFD of the rotor system with static eccentricity is shown in Figure 3. The shaft journal center (Oj) no longer coincides with the clearance circle center (Ob), and some distance is present between the two centers in the static state. In the working state, the whirl center of the shaft journal (Oe) no longer coincides with the center (Ob) of the clearance circle, and a misalignment force directed from Ob to Oe is introduced to describe this phenomenon.Figure 2
The SFD of the rotor system without static eccentricity. (a) In the static state. (b) In the working state.
(a)(b)Figure 3
The SFD of the rotor system with static eccentricity. (a) In the static state. (b) In the working state.
(a)(b)For an SFD in an ideal state, experimental investigations have demonstrated that the short bearing approximation of SFD exhibits some consistency with the experimental data [5, 6]. Thus, the short bearing assumption is adopted herein to incorporate SFD with the rotor system. Thus, for the SFD considered herein, in the process of obtaining the nonlinear force of SFD, the short bearing assumption and Reynolds boundary conditions are selected. The nonlinear force of the SFD is shown in the following [20–22]:
(1a)fxs=−μsRL3c2x2+y2xε̇I2+εψ̇I1−yε̇I1+εψ̇I3fys=−μsRL3c2x2+y2yε̇I2+εψ̇I1+xε̇I1+εψ̇I3,(1b)ε=x2+y2/c,(1c)ε̇=xẋ+yẏ/cx2+y2,(1d)ψ̇=yẋ−xẏ/x2+y2,(1e)tanψ=y/x.Here,fxs and fys represent the nonlinear force components introduced by SFD in the horizontal and vertical, and x and y represent the horizontal and vertical displacements of the shaft journal, respectively. R represents the radius of SFD, and L represents the length along the shaft of SFD. Additionally, μs and c denote the dynamic viscosity and thickness of the oil film, respectively. I1,I2,andI3 are Summerfield integrals that can be expressed as
(2a)I1=∫θ1θ1+πcos2θ1+εcosθ3dθ,(2b)I2=∫θ1θ1+πsinθcosθ1+εcosθ3dθ,(2c)I3=∫θ1θ1+πsin2θ1+εcosθ3dθ,(2d)θ1=tan−1−ε̇εϕ̇.The supporting structure of the rotor system mainly includes the squirrel cage and SFD.Fb represents the nonlinear force introduced by the bearing on the shaft journal, which can be expressed as
(3a)Fb=−Fk+FSFD.Here,FSFD represents the nonlinear force vector introduced by SFDs, which can be calculated using Equations (1a)–(1e), and Fk represents the force vector introduced by the squirrel cage, Fk=k1x1k1y1k2x2k2y2⋯knxnknyn, k1–kn represent the stiffness of the squirrel cage, and x1–xn and y1–yn represent the displacement of the shaft journal in the horizontal and vertical directions at each squirrel cage.When SFD is in a static eccentric state, the squirrel cage experiences some displacement in the static state and the nonlinear forceFb can be expressed using the following with static eccentricity.
(3b)Fb=−Fk+FSFD+Fe.Here,Fe represents the additional force vector of the squirrel cage afforded by the static eccentricity. Fe=k1xs1k1ys1k2xs2k2ys2⋯knxsnknysn, k1–kn represent the stiffness of the squirrel cage at each SFD, n represents the number of SFDs, and xs1–xsn and ys1–ysn represent the static eccentricity of each shaft journal in the horizontal and vertical directions, respectively. The SFD parameters of this study are shown in Table 1, and the physical meaning of each parameter in Table 1 is shown in Figure 4 and is consistent with Equations (1a)–(1e).Table 1
SFD parameters.
L [mm]R [mm]c [mm]μs [10−2 Pa s]Front SFD15300.11.0752Rear SFD15300.1Figure 4
Diagram of SFD.


## 2.2. Dynamic Model of the Rotor System

Herein, the Timoshenko beam element is used to simulate the shaft, and the shaft section element is shown in Figure5. Each element contains two nodes, μe1 represents the displacement of the node 1 along the x-axis, ve1 represents the displacement of the node 1 along the y-axis, θe1 represents the rotation angle of node 1 around the x-axis, φe1 represents the rotation angle of node 1 around the y-axis, and the physical meaning of the four degrees of freedom of node 2 is consistent with that of node 1. The z-axis direction represents the axial direction. Assuming that the shaft element is an isotropic linear elastic cylinder, the cross-section of the shaft element is equal along the z-axis direction, and any cross-section in the element is still flat after deformation, but it is not necessarily perpendicular to the neutral axis, i.e., shear deformation is considered. The generalized coordinates of each element are defined by
(4)u=ue1ve1θe1φe1ue2ve2θe2φe2.Figure 5
Diagram of the Timoshenko beam element [22].According to reference [23], the nonlinear equations of motion of a nonlinear rotor system can be represented using the following.
(5)Mu¨+C+ΩGu̇+Ku=Funb+Fb.Funb represents the force vector introduced by the shaft unbalance. Fb represents the nonlinear forces introduced by the bearings herein. M, C, G, and K denote the mass, damping, stiffness, and gyroscopic matrices of the rotor system, respectively, which can be obtained by assembling the element matrix in Ref. [22]. Additionally, u is the displacement vector of the rotor-bearing system.Figure6 displays the dynamic model of the SFD-rotor system. The shaft is divided into four parts, as shown in the figure, and all parts are solid cylinders. The diameter and start/end points of each part are shown in Table 2. Disk parameters are shown in Table 3; the location, mass, IP, ID, unbalance, and initial phase of the unbalance of the disk are given in Table 3. In the simulation, the disk was simulated as a concentrated mass unit that accounts for the gyroscopic effect. Moreover, only the radial stiffness of the squirrel cages was considered, and Table 4 displays the stiffness of the squirrel cages; kxx and kyy represent the stiffness of the squirrel cages in the x-axis and the y-axis direction, respectively, kxy and kyx represent the cross stiffness of the squirrel cages; what is more, the stiffness value of each squirrel cage in Table 4 is obtained through static tests.Figure 6
Dynamic model of the SFD-rotor system.Table 2
Shaft parameters.
Shaft numberStart point (mm)End point (mm)Diameter (mm)Part 1017318Part 217364320Part 364387220Part 487295620Table 3
Disk parameters.
Disk numberLocation (mm)Mass (kg)IP (kg m2)ID (kg m2)Unbalance (g cm)Initial phase16432.510.008540.00427100Table 4
Stiffness of squirrel cages.
Bearing numberkxx (N/m)kyy (N/m)kxy (N/m)kyx (N/m)12.5×1062.5×1060022.5×1062.5×10600Based on the above listed parameters, the dynamic model of the SFD-rotor system with and without eccentric SFD can be established, which has a total of 97 nodes. Among them, the node numbers of the disk, front support, and rear support are 66, 18, and 89, respectively.


## 2.3. Modeling Method Based on the Fixed Interface Modal Synthesis

In Subsection2.2, the dynamic model of the SFD-rotor system with/without static eccentricity was established by introducing different nonlinear force vectors exerted by bearings.Equation (5) shows that the dynamic model of the SFD-rotor system is a set of nonlinear differential equations. The nonlinear response of the SFD-rotor system can be obtained by solving these differential equations. Generally, the differential equations obtained in Subsection 2.2 have large dimensions and nonlinear forces, all of which result in low computational efficiency. Herein, after considering the overall computational efficiency and accuracy of the dynamic model, fixed interface modal synthesis is employed to reduce the dimensions of the nonlinear differential equations of the SFD-rotor system and thus the computational burden.First, the degrees of freedom of the rotor system are divided into two parts. The degree of freedom with external excitation is called the interface degree of freedom, such as the degree of freedom at the rotor bearing, where the nonlinear force introduced by SFD acts. The other is called the internal degree of freedom, where only the unbalanced forces act. Then, Equation (5) can be rewritten as
(6)MIIMIJMJIMJJu¨Iu¨J+GIIGIJGJIGJJu̇Iu̇J+KIIKIJKJIKJJuIuJ=FIunb0+0FJnonl.Here,uI represents the internal degrees of freedom of the rotor system, while uJ represents the interface degrees of freedom of the rotor system. Additionally, the transformation relation between the physical and modal coordinates of the system is
(7a)uIuJ=ϕkϕc0IqkuJ=TqkuJ.Here,ϕk∈RnI×nk represents the mass normalized normal mode matrix; ϕc∈RnI×nJ represents the mass normalized constrained mode; I represents a nJ dimensional identity matrix; qk∈RnI×1 represents the normal mode coordinates; nI, nJ, and nK are the numbers of internal degrees of freedom, interface degrees of freedom, and restrained principal modes retained in the calculation, respectively; and T is defined as the transformation matrix. The transformation matrix T can be obtained by solving ϕc and ϕk.According to the fixed interface modal synthesis principle, the definition of the constrained modeϕc is as follows. When the interface degrees of freedom of the system sequentially generate unit displacements and the remaining interface degrees of freedom are constrained, a set of displacements is generated by the internal degrees of freedom of the system. Therefore, ϕc can be expressed as
(7b)ϕc=−KII−1KIJ.Based on the motion differential Equation (6), by constraining all the interface degrees of freedom of the system and ignoring the influence of the gyroscopic effect, the differential equation of motion of the rotor system when the interface degrees of freedom are constrained can be obtained as shown in the following equation.
(7c)MIIu¨I+KIIuI=0.Solving the eigenvalues and eigenvectors of Equations (7a)–(7c), the mass normalized normal mode matrix ϕk and the modal frequency Ωk can be obtained.Equations (7a), (3a), and (3b) are substituted into Equation (6), and TT in front of the expression is multiplied on both sides of the equal sign of Equation (6). The reduced differential equation of motion of the system can be obtained as shown in Equations (8a) and (8b).
(8a)IM¯IJM¯JIM¯JJq¨ku¨J+G¯IIG¯IJG¯JIG¯JJq̇ku̇J+K¯II00K¯JJqkuJ=ϕkTFIunbϕcTFIunb+0FJnonl,where
(8b)M¯JJ=MJJ+MJIϕc+ϕcTMIIϕc+MIJM¯IJ=M¯JI=ϕkTMIIϕc+MIJG¯II=ϕkTGIIϕkG¯II=GJJ+GJIϕc+ϕcTGIIϕc+GIJ+cJJG¯IJ=G¯JI=ϕkTGIIϕc+ϕkTGIJK¯II=diagΩr21≤r≤nK¯JJ=KIJ+KJIϕc+kJJ.Ωr represents the r-th modal frequency of the system obtained from Equation (7c). Moreover, FIunb represents the unbalanced force vector acting on the internal degrees of freedom. FJnonl F2 represents the external load acting on the interface degrees of freedom. Herein, FJnonl=Fb.Here, the dimension of the differential equation of motion of the rotor system is reduced from the4n order of Equation (5) to the nI+nJ order of Equation (8a).Based on the above ideas, a dynamic model of the SFD-rotor system that comprehensively considers the model accuracy and solution efficiency is established. Additionally, the implicit time-domain method based on the Newmark-β method is used to study the dynamic responses of the nonlinear rotor system with/without static eccentricity.


## 2.4. Comparison of Critical Speed of Rotor System

Before conducting simulation and experimental research on the nonlinear response of the SFD-rotor system with and without eccentric SFD, the critical speed and vibration mode of the rotor system are first compared. In the free state, the motion equation of a rotor system can be written as(9)Mu¨+Gu̇+Ku=0.In the state space it can be expressed as(10a)Aq̇+Bq=0,where
(10b)q=uu̇,q̇=u̇u¨,A=GMM0B=K00M.Letqt=q0est and substitute it into Equation (10a) to obtain:
(10c)sAq0=−Bq0.The modal frequency of the rotor system and the corresponding modal shape can be obtained by solving Equation (10c). Furthermore, in a rotor system, at different speeds, the gyroscopic effect has different effects on the damping matrix G of the rotor system. Consequently, the matrix A in Equation (10b) will also be different. Therefore, a series of speeds need to be considered and calculations need to be performed at each speed. Then, the rotational speed as the abscissa and the modal frequency at each rotational speed as the ordinate are employed to draw the Campbell diagram. The constant rpm line is drawn on the Campbell diagram. The intersection of the constant rpm line and the model frequency lines of the forward precession of each order denotes the critical speed of the rotor system. The first two critical speeds of the rotor system obtained from the established model are 2703 and 12448 rpm, respectively, while the first two critical speeds of the rotor system obtained using the ANSYS beam model are 2662 and 12073 rpm, respectively. The difference between the critical speeds is due to the difference in the meshing densities of the two models; the maximum error between the two results is 3.1%. The Campbell diagram and the first two modes of the rotor system are shown in Figures 7–9. In Figure 7, the red lines represent forward precession, black lines represent back precession, solid lines represent the result of the established model, and dashed lines represent the result of the ANSYS BEAM model.Figure 7
Campbell diagram of the rotor system.Figure 8
First-order mode shape of the rotor system. (a) Model established in this paper. (b) ANSYS beam model.
(a)(b)Figure 9
Second-order mode shape of the rotor system. (a) Model established in this paper. (b) ANSYS beam model.
(a)(b)


## 3. Experiment Devices and Test Procedure

### 3.1. Tester of the SFD-Rotor System

The rotor system tester with/without eccentric SFD is established (Figure10). The system comprises a drive system, a rotor system, a static eccentric loading system, and a data acquisition system. The drive system includes a drive motor and a control system, which is the power source of the tester for ensuring that the test rotor system can function under the set speed. The rotor system includes a shaft, disk, front support, and rear support.Figure 10
Schematic of the SFD-rotor system with static eccentricity.The support scheme of the front/rear support is a combination of the squirrel cage spring support, SFD, and rolling bearing (Figure1). The rolling bearing at the front support is a ball bearing, and that at the rear support is a roller bearing. A static eccentric loading system includes a rope retainer, rope, spring, and counterweight loading tray.The spring is employed to keep the mass of the counterweight block from participating in the vibration to ensure that the tension of the configuration block is loaded into the rotor system. Figure9 shows the specific details of the rotor tester, and Figure 11 shows the physical drawing of the tester.Figure 11
Physical drawing of rotor system.The test system includes a computer, a data acquisition card, acceleration sensors, and displacement sensors. The acceleration sensors are mounted on the mounting bracket to detect the vibration and prevent the rotor system from entering a dangerous state in the test. The displacement sensors are horizontally and vertically mounted at the disk position to detect the radial vibration of the disk during the test.Note that after the preliminary analysis of the test results, the displacement response at the disk is not sensitive enough to the change of the SFD static eccentricity because the wheel disk is placed far away from the rear support. Therefore, two displacement sensors are mounted near the rear bearing to monitor the shaft vibration near the rear support. Since the picture of the tester was taken during the preliminary test, Figure11 does not display the two additional displacement sensors. The installation positions of the additional displacement sensors are marked in red font and arrow in Figure 11.


### 3.2. Testing Method

In this study, the rear support is the research object of SFD with static eccentricity. Different static eccentricity conditions of the rotor system are imposed by changing the weight of the mass in the tray. The mass sequentially added in the tray is shown in Table5, the relationship between the static eccentricity and force can be expressed as:
(11a)εs=esc,(11b)es=Fekxx.Table 5
Loading sequence.
CaseMass change (kg)Force (N)εs10002+2.423.520.093+2.447.040.194+2.470.560.285+2.998.980.396+2.9127.40.51In the formula,εs represents the static eccentricity of SFD, Fe represents the gravity of the weight in the tray, es represents the journal displacement caused by gravity, and the installation state of the rope retainer on the shaft is shown in Figure 12.Figure 12
Rope retainer. (a) Main view. (b) Side view.
(a)(b)For each eccentricity, the displacement responses of the disk and the shaft and the acceleration responses on the support were measured during the run-up and run-down stages. The 0–5000–0 rpm speed range is selected herein, and the rotor system only passes through the first-order critical speed in this speed range. The sampling frequency of the displacement sensor in the experiment is 5000 Hz, and the sampling frequency of the acceleration sensor is 20 kHz.


### 3.3. Validation

In this subsection, the accuracy of the established dynamic model in predicting the response of the rotor system under different SFD static eccentricities is verified.At first, we compared the critical speed of the rotor system obtained by simulation and experiment. Figure13 presents the displacement response of the disk obtained by the test during run-up and simulation results of the transient response of the rotor system without static eccentricity. As shown in the figure, the first-order critical speeds of the rotor system obtained by the test and simulation analysis are 2738 and 2725 rpm, respectively. And the maximum displacements are 1.02 mm and 1.04 mm, respectively.Figure 13
Transient response of the rotor system.Additionally, the frequency-domain responses of the rotor system with and without static eccentricity are studied and compared via experiments and simulations. Considering the representativeness of the results, the spectrograms of displacement responses at the additional measuring points for cases 1, 2, and 6 in Table5 are selected for comparison. The three cases represent no static eccentricity, small static eccentricity, and large static eccentricity conditions, respectively.As shown in Figures14–16, except for the spectrogram at the supercritical speed of case 1, the spectrograms obtained from the test results and the simulation results are basically consistent under each working condition. In addition, it can be found from Figures 15 and 16 that static eccentricity leads to not only a 0 Hz peak in the response in the x-axis direction but also a series of frequency doubling components in the response in both the x- and y-axis directions. And the amplitude corresponding to the high-frequency component is more obvious under the condition of large static eccentricity.Figure 14
Spectrogram of the shaft for case 1. (a) Subcritical speed. (b) Critical speed. (c) Supercritical speed.
(a)(b)(c)Figure 15
Spectrogram of the shaft for case 2. (a) Subcritical speed. (b) Critical speed. (c) Supercritical speed.
(a)(b)(c)Figure 16
Spectrogram of the shaft for case 6. (a) Subcritical speed. (b) Critical speed. (c) Supercritical speed.
(a)(b)(c)By comparing the above three cases, the following conclusions can be drawn. The experimental and simulation results show that the static eccentricity of SFD may cause a static offset of the shaft journal in the static eccentricity direction and may afford a series of frequency doubling responses. Furthermore, the experimental and simulation results show that the larger the static eccentricity of the SFD, the greater the influence of the high-frequency component on the response of the rotor system.


## 3.1. Tester of the SFD-Rotor System

The rotor system tester with/without eccentric SFD is established (Figure10). The system comprises a drive system, a rotor system, a static eccentric loading system, and a data acquisition system. The drive system includes a drive motor and a control system, which is the power source of the tester for ensuring that the test rotor system can function under the set speed. The rotor system includes a shaft, disk, front support, and rear support.Figure 10
Schematic of the SFD-rotor system with static eccentricity.The support scheme of the front/rear support is a combination of the squirrel cage spring support, SFD, and rolling bearing (Figure1). The rolling bearing at the front support is a ball bearing, and that at the rear support is a roller bearing. A static eccentric loading system includes a rope retainer, rope, spring, and counterweight loading tray.The spring is employed to keep the mass of the counterweight block from participating in the vibration to ensure that the tension of the configuration block is loaded into the rotor system. Figure9 shows the specific details of the rotor tester, and Figure 11 shows the physical drawing of the tester.Figure 11
Physical drawing of rotor system.The test system includes a computer, a data acquisition card, acceleration sensors, and displacement sensors. The acceleration sensors are mounted on the mounting bracket to detect the vibration and prevent the rotor system from entering a dangerous state in the test. The displacement sensors are horizontally and vertically mounted at the disk position to detect the radial vibration of the disk during the test.Note that after the preliminary analysis of the test results, the displacement response at the disk is not sensitive enough to the change of the SFD static eccentricity because the wheel disk is placed far away from the rear support. Therefore, two displacement sensors are mounted near the rear bearing to monitor the shaft vibration near the rear support. Since the picture of the tester was taken during the preliminary test, Figure11 does not display the two additional displacement sensors. The installation positions of the additional displacement sensors are marked in red font and arrow in Figure 11.


## 3.2. Testing Method

In this study, the rear support is the research object of SFD with static eccentricity. Different static eccentricity conditions of the rotor system are imposed by changing the weight of the mass in the tray. The mass sequentially added in the tray is shown in Table5, the relationship between the static eccentricity and force can be expressed as:
(11a)εs=esc,(11b)es=Fekxx.Table 5
Loading sequence.
CaseMass change (kg)Force (N)εs10002+2.423.520.093+2.447.040.194+2.470.560.285+2.998.980.396+2.9127.40.51In the formula,εs represents the static eccentricity of SFD, Fe represents the gravity of the weight in the tray, es represents the journal displacement caused by gravity, and the installation state of the rope retainer on the shaft is shown in Figure 12.Figure 12
Rope retainer. (a) Main view. (b) Side view.
(a)(b)For each eccentricity, the displacement responses of the disk and the shaft and the acceleration responses on the support were measured during the run-up and run-down stages. The 0–5000–0 rpm speed range is selected herein, and the rotor system only passes through the first-order critical speed in this speed range. The sampling frequency of the displacement sensor in the experiment is 5000 Hz, and the sampling frequency of the acceleration sensor is 20 kHz.


## 3.3. Validation

In this subsection, the accuracy of the established dynamic model in predicting the response of the rotor system under different SFD static eccentricities is verified.At first, we compared the critical speed of the rotor system obtained by simulation and experiment. Figure13 presents the displacement response of the disk obtained by the test during run-up and simulation results of the transient response of the rotor system without static eccentricity. As shown in the figure, the first-order critical speeds of the rotor system obtained by the test and simulation analysis are 2738 and 2725 rpm, respectively. And the maximum displacements are 1.02 mm and 1.04 mm, respectively.Figure 13
Transient response of the rotor system.Additionally, the frequency-domain responses of the rotor system with and without static eccentricity are studied and compared via experiments and simulations. Considering the representativeness of the results, the spectrograms of displacement responses at the additional measuring points for cases 1, 2, and 6 in Table5 are selected for comparison. The three cases represent no static eccentricity, small static eccentricity, and large static eccentricity conditions, respectively.As shown in Figures14–16, except for the spectrogram at the supercritical speed of case 1, the spectrograms obtained from the test results and the simulation results are basically consistent under each working condition. In addition, it can be found from Figures 15 and 16 that static eccentricity leads to not only a 0 Hz peak in the response in the x-axis direction but also a series of frequency doubling components in the response in both the x- and y-axis directions. And the amplitude corresponding to the high-frequency component is more obvious under the condition of large static eccentricity.Figure 14
Spectrogram of the shaft for case 1. (a) Subcritical speed. (b) Critical speed. (c) Supercritical speed.
(a)(b)(c)Figure 15
Spectrogram of the shaft for case 2. (a) Subcritical speed. (b) Critical speed. (c) Supercritical speed.
(a)(b)(c)Figure 16
Spectrogram of the shaft for case 6. (a) Subcritical speed. (b) Critical speed. (c) Supercritical speed.
(a)(b)(c)By comparing the above three cases, the following conclusions can be drawn. The experimental and simulation results show that the static eccentricity of SFD may cause a static offset of the shaft journal in the static eccentricity direction and may afford a series of frequency doubling responses. Furthermore, the experimental and simulation results show that the larger the static eccentricity of the SFD, the greater the influence of the high-frequency component on the response of the rotor system.


## 4. Results and Discussions

In this section, the displacement and acceleration responses of the rotor system during run-up and run-down under different static eccentricities are discussed. The displacement response of the disk and the rear bearing of the rotor system under different static eccentricity conditions are compared by simulations. The acceleration response of the mounting bracket under different eccentricity conditions is investigated by experiments.


### 4.1. Displacement Response of the Rotor System during Run-Up and Run-Down

In this subsection, the response of the nodes at the disk and rear bearing on the shaft in the run-up and run-down process is studied. The cases include the six cases listed in Table5 as well as static eccentricities of 0.6 and 0.72. Static eccentricities of 0.6 and 0.72 correspond to cases 7 and 8, respectively. For each case, the run-up and run-down phases of the rotor system are separately analyzed. For the run-up phase, the speed of rotor system uniformly increases from 0 to 5000 rpm within 100 s, and the speed of the rotor system uniformly decreases from 5000 to 0 rpm when the speed is run-down. Note that the amplitudes discussed in this subsection are the maximum amplitude minus the minimum amplitude in each cycle.Figures17 and 18 show the time-domain response diagrams of the nodes at the rear bearing for cases 1 and 6, respectively. The figures show that the responses in the x- and y-axis directions of the support are symmetrical without eccentricity (case 1). When the static eccentricity is 0.51 (case 6), the displacement in the x-axis direction has some static offset and is clearly no longer symmetrical after the critical speed is exceeded. Additionally, the amplitude of the displacement response fluctuates after the rotor system crosses the critical speed. For the above reasons, in this subsection, the displacement value obtained by subtracting the minimum amplitude from the maximum amplitude in a cycle is employed as the analysis object.Figure 17
Time-domain response diagram of the nodes at the rear bearing in case 1. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 18
Time-domain response diagram of the nodes at the rear bearing in case 6. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figures19, 20, 21, and 22 show the variations of the node displacements at the disk and rear bearing during run-up and run-down process for the eight cases, and partial enlarged views of the key positions are shown for each figure. The figures show that with increasing static eccentricity, the displacement amplitude at the disk continuously increases while that at the rear bearing continuously decreases. Additionally, the change in the x-axis direction is greater than that in the y-axis direction in the figures. Simultaneously, after the rotor system crosses the critical speed, the fluctuation of the rotor response amplitude increases with the static eccentricity. Moreover, when the speed is far from the critical speed and the static eccentricity is large, the displacement response of the disk and rear bearing is not continuous with the speed change, and the performance at the rear bearing is more obvious than that at the disk. The following conclusions can be obtained from the above phenomenon. First, the displacement response of the node at the rear bearing decreases with increasing the static eccentricity, denoting that the whirl of the shaft journal decreases with increasing the static eccentricity and that the energy absorbed by SFD decreases, which subsequently increases the displacement response at the disk. Second, during the run-up and run-down process, the fluctuation of the rotor system over the critical speed shows that when the speed of the rotor system with static eccentricity slowly approaches the critical speed, the energy introduced by the resonance of the rotating shaft continues to increase, the shaft journal at the rear support gradually changes from a small-scale to a large-scale whirl, and the whirl center gradually approaches the center of the oil film. When the rotor system speed exceeds the critical speed, the shaft journal rapidly changes from a large-scale to a small-scale whirl and the whirl center rapidly jumps to a position far from the oil film center. These phenomena cause amplitude jumps in the rotor system, and the fluctuations become more severe with increasing static eccentricity. Third, excessive static eccentricity may cause the displacement amplitude oscillation of the rotor system when far from the critical speed. This is mainly due to the fact that that when the rotor system works far away from the critical speed in the presence of a large static eccentricity, the journal whirl track is elliptical and the eccentricity of the ellipse is large (as shown in Figure 23), the nonlinear force acting on the rotor by SFD changes dramatically, especially in the x-axis direction.Figure 19
Variations of the node displacements at the disk during run-up process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 20
Variations of the node displacements at the rear bearing during run-up process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 21
Variations of the node displacements at the rear bearing during run-down process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 22
Variations of the node displacements at the rear bearing during run-down process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 23
Acceleration response of the rear bearing during run-down process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)As a supplement, Figure24 shows the orbit of the shaft journal at the rear bearing for different cases. The figure shows that with increasing static eccentricity, the whirling orbit of the shaft journal gradually changes from a circle to an ellipse and the eccentricity of the ellipse continue to increases and the change of journal whirl track with the increase of static eccentricity is similar to reference [15]. Simultaneously, the larger the static eccentricity, the farther the whirl center is from the oil film center. From this phenomenon, we can infer that when the static eccentricity is large, the space that the journal can squeeze the oil film in the eccentric direction is limited. On the other hand, the reaction force of the oil film acting on the journal increases, and the increase of the reaction force is equivalent to increasing the stiffness of the rotor bearings, the increase in stiffness eventually leads to an increase in the critical speed of the rotor system.Figure 24
Orbit of the shaft at the rear bearing in different cases. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4. (e) Case 1. (f) Case 2. (g) Case 3. (h) Case 4.
(a)(b)(c)(d)(e)(f)(g)(h)


### 4.2. Acceleration Response of the Rotor System during Run-up and Run-down

In this subsection, the acceleration response of the mounting bracket in the run-up and run-down process is studied. Table6 illustrates the speed change of the rotor system during the test. Due to the drive system limitations, the rotor system speed increases when the speed is increased and uniformly decreases when the speed is decreased. The rotational speed range of 1500–4500 rpm is focused on herein.Table 6
Speed change of the rotor system during the test (unit: rpm).
StageStartEndRun-up010001500200025003000350040004500Run-down4500———————0Figures25 and 23 show the variations of the acceleration response at the rear bearing during the run-up and run-down process for the six cases of the experiment.Figure 25
Acceleration response of the rear bearing during run-up process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)The figures show that during the run-up process, except for a peak value of the rear mounting bracket when passing the critical speed, an amplitude appears in each sudden acceleration process of the rotor. Additionally, the amplitude increases with the static eccentricity. Moreover, thex-axis direction is more obvious. The reason for this phenomenon is that when the motor speeds up suddenly, the kinetic energy of the rotor system and the force transmitted outward through the bearing also increase rapidly. But the increase of SFD static eccentricity causes the journal whirl track to gradually change from a circle to an ellipse, and the degree of oil film being squeezed in the positive and negative directions of the x-axis is quite different when the rotor is running. Therefore, the ability of SFD to suppress the external force of the rotor system in the x-axis direction is seriously weakened with the increase of static eccentricity, which leads to a sudden increase in the acceleration response in the x-axis direction on the mounting bracket when the rotor system speeds up suddenly.For the run-down stage, due to the uniform run-down process, the peak value of the acceleration response only occurs at the critical speed and twice the critical speed. Similarly, when the rotor speed is twice the critical speed, the amplitude in thex-axis direction is more obvious than that in the y-axis direction. This is mainly due to the misalignment of the rotor system caused by the static eccentricity of the SFD, which further leads to the occurrence of double frequency.The above facts show that the eccentric SFD can cause the following problems. First, it will increase not only the acceleration response of the mounting bracket at the critical speed but also the sudden acceleration response of the mounting bracket during the acceleration process of the rotor system. Furthermore, the increase of the acceleration response of the mounting bracket during run-up is more obvious in the eccentricity direction, and this differential acceleration response in different directions can help locate eccentric SFD faults in practical applications. Additionally, since the acceleration response at the mounting bracket is directly related to the external force transmitted by the rotor system, the static eccentricity of the SFD may cause the deterioration of the SFD vibration reduction effect and the increase of the external force transmission. Second, it may afford an acceleration response peak at twice the critical speed of the rotor system, and it is more obvious in the eccentricity direction.


## 4.1. Displacement Response of the Rotor System during Run-Up and Run-Down

In this subsection, the response of the nodes at the disk and rear bearing on the shaft in the run-up and run-down process is studied. The cases include the six cases listed in Table5 as well as static eccentricities of 0.6 and 0.72. Static eccentricities of 0.6 and 0.72 correspond to cases 7 and 8, respectively. For each case, the run-up and run-down phases of the rotor system are separately analyzed. For the run-up phase, the speed of rotor system uniformly increases from 0 to 5000 rpm within 100 s, and the speed of the rotor system uniformly decreases from 5000 to 0 rpm when the speed is run-down. Note that the amplitudes discussed in this subsection are the maximum amplitude minus the minimum amplitude in each cycle.Figures17 and 18 show the time-domain response diagrams of the nodes at the rear bearing for cases 1 and 6, respectively. The figures show that the responses in the x- and y-axis directions of the support are symmetrical without eccentricity (case 1). When the static eccentricity is 0.51 (case 6), the displacement in the x-axis direction has some static offset and is clearly no longer symmetrical after the critical speed is exceeded. Additionally, the amplitude of the displacement response fluctuates after the rotor system crosses the critical speed. For the above reasons, in this subsection, the displacement value obtained by subtracting the minimum amplitude from the maximum amplitude in a cycle is employed as the analysis object.Figure 17
Time-domain response diagram of the nodes at the rear bearing in case 1. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 18
Time-domain response diagram of the nodes at the rear bearing in case 6. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figures19, 20, 21, and 22 show the variations of the node displacements at the disk and rear bearing during run-up and run-down process for the eight cases, and partial enlarged views of the key positions are shown for each figure. The figures show that with increasing static eccentricity, the displacement amplitude at the disk continuously increases while that at the rear bearing continuously decreases. Additionally, the change in the x-axis direction is greater than that in the y-axis direction in the figures. Simultaneously, after the rotor system crosses the critical speed, the fluctuation of the rotor response amplitude increases with the static eccentricity. Moreover, when the speed is far from the critical speed and the static eccentricity is large, the displacement response of the disk and rear bearing is not continuous with the speed change, and the performance at the rear bearing is more obvious than that at the disk. The following conclusions can be obtained from the above phenomenon. First, the displacement response of the node at the rear bearing decreases with increasing the static eccentricity, denoting that the whirl of the shaft journal decreases with increasing the static eccentricity and that the energy absorbed by SFD decreases, which subsequently increases the displacement response at the disk. Second, during the run-up and run-down process, the fluctuation of the rotor system over the critical speed shows that when the speed of the rotor system with static eccentricity slowly approaches the critical speed, the energy introduced by the resonance of the rotating shaft continues to increase, the shaft journal at the rear support gradually changes from a small-scale to a large-scale whirl, and the whirl center gradually approaches the center of the oil film. When the rotor system speed exceeds the critical speed, the shaft journal rapidly changes from a large-scale to a small-scale whirl and the whirl center rapidly jumps to a position far from the oil film center. These phenomena cause amplitude jumps in the rotor system, and the fluctuations become more severe with increasing static eccentricity. Third, excessive static eccentricity may cause the displacement amplitude oscillation of the rotor system when far from the critical speed. This is mainly due to the fact that that when the rotor system works far away from the critical speed in the presence of a large static eccentricity, the journal whirl track is elliptical and the eccentricity of the ellipse is large (as shown in Figure 23), the nonlinear force acting on the rotor by SFD changes dramatically, especially in the x-axis direction.Figure 19
Variations of the node displacements at the disk during run-up process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 20
Variations of the node displacements at the rear bearing during run-up process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 21
Variations of the node displacements at the rear bearing during run-down process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 22
Variations of the node displacements at the rear bearing during run-down process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)Figure 23
Acceleration response of the rear bearing during run-down process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)As a supplement, Figure24 shows the orbit of the shaft journal at the rear bearing for different cases. The figure shows that with increasing static eccentricity, the whirling orbit of the shaft journal gradually changes from a circle to an ellipse and the eccentricity of the ellipse continue to increases and the change of journal whirl track with the increase of static eccentricity is similar to reference [15]. Simultaneously, the larger the static eccentricity, the farther the whirl center is from the oil film center. From this phenomenon, we can infer that when the static eccentricity is large, the space that the journal can squeeze the oil film in the eccentric direction is limited. On the other hand, the reaction force of the oil film acting on the journal increases, and the increase of the reaction force is equivalent to increasing the stiffness of the rotor bearings, the increase in stiffness eventually leads to an increase in the critical speed of the rotor system.Figure 24
Orbit of the shaft at the rear bearing in different cases. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4. (e) Case 1. (f) Case 2. (g) Case 3. (h) Case 4.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4.2. Acceleration Response of the Rotor System during Run-up and Run-down

In this subsection, the acceleration response of the mounting bracket in the run-up and run-down process is studied. Table6 illustrates the speed change of the rotor system during the test. Due to the drive system limitations, the rotor system speed increases when the speed is increased and uniformly decreases when the speed is decreased. The rotational speed range of 1500–4500 rpm is focused on herein.Table 6
Speed change of the rotor system during the test (unit: rpm).
StageStartEndRun-up010001500200025003000350040004500Run-down4500———————0Figures25 and 23 show the variations of the acceleration response at the rear bearing during the run-up and run-down process for the six cases of the experiment.Figure 25
Acceleration response of the rear bearing during run-up process. (a) Response in thex-axis direction. (b) Response in the y-axis direction.
(a)(b)The figures show that during the run-up process, except for a peak value of the rear mounting bracket when passing the critical speed, an amplitude appears in each sudden acceleration process of the rotor. Additionally, the amplitude increases with the static eccentricity. Moreover, thex-axis direction is more obvious. The reason for this phenomenon is that when the motor speeds up suddenly, the kinetic energy of the rotor system and the force transmitted outward through the bearing also increase rapidly. But the increase of SFD static eccentricity causes the journal whirl track to gradually change from a circle to an ellipse, and the degree of oil film being squeezed in the positive and negative directions of the x-axis is quite different when the rotor is running. Therefore, the ability of SFD to suppress the external force of the rotor system in the x-axis direction is seriously weakened with the increase of static eccentricity, which leads to a sudden increase in the acceleration response in the x-axis direction on the mounting bracket when the rotor system speeds up suddenly.For the run-down stage, due to the uniform run-down process, the peak value of the acceleration response only occurs at the critical speed and twice the critical speed. Similarly, when the rotor speed is twice the critical speed, the amplitude in thex-axis direction is more obvious than that in the y-axis direction. This is mainly due to the misalignment of the rotor system caused by the static eccentricity of the SFD, which further leads to the occurrence of double frequency.The above facts show that the eccentric SFD can cause the following problems. First, it will increase not only the acceleration response of the mounting bracket at the critical speed but also the sudden acceleration response of the mounting bracket during the acceleration process of the rotor system. Furthermore, the increase of the acceleration response of the mounting bracket during run-up is more obvious in the eccentricity direction, and this differential acceleration response in different directions can help locate eccentric SFD faults in practical applications. Additionally, since the acceleration response at the mounting bracket is directly related to the external force transmitted by the rotor system, the static eccentricity of the SFD may cause the deterioration of the SFD vibration reduction effect and the increase of the external force transmission. Second, it may afford an acceleration response peak at twice the critical speed of the rotor system, and it is more obvious in the eccentricity direction.


## 5. Conclusions

Herein, an SFD-rotor system with/without eccentric SFD was established and studied via experiments and simulations. Based on the tester and verified simulation models, the displacement response of the rotor system and the acceleration response at the mounting bracket were studied. The main conclusions are as follows:(1)
Eccentric SFD causes a static offset of the journal shaft in the static eccentricity direction and may lead to a series of frequency doubling responses in the directions ofx-axis and y-axis(2)
Eccentric SFD causes an increase in the displacement response at the disk, the displacement response fluctuation after the rotor system passes the critical speed as well as the critical speed of the rotor system. Additionally, a too large static eccentricity may cause the displacement amplitude oscillation of the rotor system when it is far from the critical speed(3)
Eccentric SFD causes the acceleration response at the mounting bracket to increase at the critical speed and during the sudden acceleration phase. When the speed is twice the critical speed, additional peaks of the rotor system may be afforded and above phenomenon more obvious in the static eccentricity direction(4)
The main contribution of this paper is to discuss the main reasons for the increase of the displacement response at the disk, the increase of the critical speed of the rotor system, the jump of the displacement response of the rotor system, and the increase of the acceleration response at the mounting bracket through simulation or experiments; based on the above analysis and discussion, it can effectively explain and solve the nonlinear response problem caused by SFD static eccentricity encountered in engineeringIn future work, a further improved simulation analysis method will be developed to predict and compare the acceleration response at the mounting bracket of the rotor system.



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*Source: 8560275-2023-01-30.xml*