# Dynamic Behavior Analysis of Intermediate Bearing-Squeeze Film Dampers-Rotor System under Constant Maneuvering Overload

Authors: Nan Zheng; Mo-li Chen; Gui-Huo Luo; Zhi-Feng Ye
Journal: Shock and Vibration
 (2021)

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## Abstract

Under the flight maneuvering of an aircraft, the maneuvering load on the rotor is generated, which may induce the change of dynamic behavior of aeroengine rotor system. To study the influence on the rotor dynamic behavior of constant maneuvering overload, a nonlinear dynamic model of bearing-rotor system under arbitrary maneuver flight conditions is presented by finite element method. The numerical integral method is used to investigate the dynamic characteristics of the rotor model under constant maneuvering overload, and the simulation results are verified by experimental works. Based on this, the dynamic characteristics of a complex intermediate bearing-squeeze film dampers- (SFD-) rotor system during maneuvering flight are analyzed. The simulation results indicate that the subharmonic components are amplified under constant maneuvering overload. The amplitude of the combined frequency components induced by the coupling of the inner and outer rotors is weakened. The static displacements of the rotor caused by the additional excitation force are observed. Besides, the period stability of the movement of the rotor deteriorates during maneuver flight. The design of counterrotation of the inner and outer rotors can effectively reduce the amplitude of subharmonic under constant maneuvering overload.


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## Body

## 1. Introduction

The maneuverability is an essential indicator in evaluating the performance of military aircraft. For military drones, it is unnecessary to consider the pilot’s ability to withstand maneuvering loads, and the maneuvering overload will be significantly increased to improve the maneuverability in the future. Additional excitation load will be generated acting on the aeroengine rotor system under the condition of maneuver flight, which may affect the dynamic behavior of the rotor [1, 2]. As the maneuvering overload increases, such effect may be magnified.In recent years, significant attention has been paid to the dynamic behavior of the rotor system during maneuvering flight. Xu and Liao [3] studied the dynamic behavior of Jeffcott rotor with squeeze film dampers (SFD) under the condition of maneuver flight. The numerical simulation results manifested that the additional maneuvering load induced by maneuvering flight changes the vibration characteristics of the rotor system. The SFD can effectively suppress the transient vibration during the maneuvering process. Ananthan and Leishman [4] investigated the dynamic behavior of a turbo-shaft engine rotor system of a helicopter during maneuver flight. The numerical simulation results showed that the five types of maneuvering flight, that is, dive, jump, pulling up during descent, roll, and reverse roll, can affect the vibration amplitude of the rotor and generate unstable additional loads. Sakata et al. [5, 6] analyzed the steady-state vibration of a simply supported flexible rotor with rotary foundation through the methods of mathematical modeling, simulation calculation, and experimental methods.Furthermore, Brentner et al. [7, 8] studied the noise problem of rotorcraft in maneuvering flight and developed a new system for maneuvering noise prediction. Yang et al. [9] studied the nonlinear dynamic response of a cracked rotor with rigid support during maneuvering flight, and three ways for the vibration responses leading to chaos were observed. Lin and Meng [10, 11] studied the vibration behavior of Jeffcott rotor under the conditions of initial bending, asymmetric stiffness, and rub-impact fault in the environment of maneuvering flight. Zhu and Chen [12, 13] established a dynamic model of an aircraft rotor system in arbitrary maneuvering flight action using Lagrange equations, in which the added damping, stiffness, and maneuvering load were obtained, and the influence of maneuver flight on the movement of rotor was investigated. Hou and Chen [14–16] established a dynamic model of a bearing-rotor system under maneuver flight conditions with faults such as rub-impact and cracks. The nonlinear dynamics behavior of the system was analyzed through numerical integral and analytic methods. The subharmonic resonance and irregular motions caused by maneuver were observed. Gao et al. [17] established a flexible asymmetric rotor system with ball bearings and SFD during maneuver flight. The vibration characteristics in conditions of accelerative motion, angular motion, and pitching flight were studied.Relative simple models, such as Jeffcott and lumped parameter rotor, are widely adopted in the above mentioned literature, which can efficiently disclose the nonlinear characteristics of rotor system. However, the analysis cannot precisely reflect the vibration behavior of the actual aeroengine complex rotor system. Using the finite element theory, the transient response of a rotor system with SFD during maneuver flight was investigated by Han and Ding [18], but a relative simple single rotor model was aimed in their research. Besides, there are few researches on the dynamic behavior of the rotor system during maneuver flight by experimental method.This paper aims to establish a nonlinear dynamic model of bearing-rotor system under arbitrary maneuver flight conditions using FEM. The added damping, stiffness, and maneuvering load generated by maneuver flight are obtained. The numerical integral method is used to investigate the vibration behavior of the rotor under constant maneuvering overload, and the simulation results are verified by experiments. Based on this, the vibration characteristics of a complex intermediate bearing-SFD-rotor system under constant maneuvering overload are analyzed through the above methods.


## 2. Mathematical Model

### 2.1. Finite Element Formulation

#### 2.1.1. Disk

To obtain the dynamic differential equations of a disk during maneuvering flight, a coordinate system is presented in Figure1. OXYZ is the ground coordinate system, which represents the center of gravity, flight velocity, and acceleration of the plane. OXYZ is the coordinate system used to describe the whirl of the rotor around its own axis. Assuming that the engine rotor is installed on the aircraft with one end, which coincides with the center of gravity of the aircraft, then OXYZ is also the relative coordinate system of the aircraft. The degree-of-freedom of the disk can be expressed as(1)qd=xyθxθy.Figure 1
Coordinate system of disk.And the kinetic energy of a disk of rotor system under arbitrary maneuvering flight actions can be written as(2)T=Tt+Tr,where Tt and Tr are the translational and rotational kinetic energy of the disk, respectively. And the translational kinetic energy during the movement of the rotor can be expressed as [13](3)Tt=12mr˙Tr˙+vBTvB−rTωB2r+2r˙TωBr+2vBTr˙+2vBTωBr,where m denotes the quality of the disk, r represents the relative position of the disk in OXYZ coordinate system, and vB and ωB denote the velocity and angular velocity of the aircraft in OXYZ, respectively. r, vB, and ωB can be expressed as follows:(4)r=x+ecosΩt+φy+esinΩt+φz,vB=X˙BY˙BZ˙B,ωB=0−ωB,zωB,yωB,z0−ωB,x−ωB,yωB,x0,where Ω denotes the rotating speed of the rotor, φ is the initial phase angle, and X˙B, Y˙B, Z˙B and ωB,x, ωB,y, ωB,z are the velocities of the aircraft along each coordinate axis and the angular velocities around each coordinate axis in OXYZ coordinate system, respectively.The rotational kinetic energy of the disk during the movement of the rotor is [13](5)Tr=12Idθ˙x+ωB,x2+12Idθ˙y+ωB,y2+12IpΩ2+2Ωθy+θB,yθ˙x+ωB,x,where θB,y denotes the angular displacement of the aircraft around Y axis in OXYZ coordinate system, and θy+θB,y≈θy, θy+θB,y′≈θ˙y+ωB,y, Id and Ip are the diameter and polar moment of inertia of the disk, respectively. Considering the damping of the disk as a linear function of velocity, and assuming that the generalized damping matrix of the disk is Cd, the generalized stiffness matrix of the shaft at the position of the disk is Kd. Then, the dissipation energy of the disk can be expressed as equation (6) by the Rayleigh energy dissipation function. And the elastic potential energy of the flexible shaft at the position of the disk can be expressed as(6)D=12q˙dTCdq˙d,V=12qdTKdqd.Substitute equations (1)–(6) into the Lagrange equations for nonconservative systems, and the dynamic differential equations of the disk during arbitrary maneuver flight actions can be deduced as(7)Mdq¨d+Cd+ΩGd+Cdbq˙d+Kd+Kdbqd=Fu+Fdb+Qd,Cdb=0−2mωB,z002mωB,z00000000000,Kdb=−mωB,y2+ωB,z2mωB,xωB,y−ω˙B,z00mωB,xωB,y+ω˙B,z−mωB,x2+ωB,z20000000000,Fdb=−mX¨B−ωB,zY˙B+ωB,yZ˙B−mzωB,xωB,z+ω˙B,y−mY¨B+ωB,zX˙B−ωB,xZ˙B−mzωB,yωB,z−ω˙B,x−IpΩωB,y−Idω˙B,xIpΩωB,x−Idω˙B,y,where Md and Gd are the inertial matrix and gyro matrix of the disk with maneuver flight ignored, respectively. Their formulas are omitted in this paper, which can be found in Ref. [19]. Cdb, Kdb, and Fdb denote the added damping matrix, stiffness matrix, and maneuvering load caused by maneuver flight. Fu represents the unbalanced force of the disk, and Qd denotes the generalized external forces acting on the disk.


#### 2.1.2. Shaft

The schematic diagram of a Timoshenko beam element is shown in Figure2. Similar to the disk, the kinetic energy of an arbitrary microelement segment of length dξ can be expressed as follows:(8)dT=dTt+dTr=12ρeAer˙eTr˙e+vBTvB−reTωB2re+2r˙eTωBre+2vBTr˙e+2vBTωBredξ+12dIdθ˙e+ωB,x2+12dIdφ˙e+ωB,y2+12dIpΩ2+2Ωθ˙e+ωB,xφe+θB,y,re=xeyeze,xeyeθeφe=N100N2N300N40N1−N200N3−N400−N⌣1N⌣200−N⌣3N⌣40N⌣100N⌣2N⌣300N⌣4qe,qe=x1y1θx1θy1x2y2θx2θy2T,where qe is the nodal displacement vector, and Nii=1−4 and N⌣ii=1−4 represent the shape functions and the expressions, which are referred to in Ref. [19]. re denotes the relative position of the centroid of the microelement segment in oxyz coordinate system. Then, the kinetic energy expression of the beam element can be obtained by integrating equation (8) along the axis oz.Figure 2
Schematic diagram of beam element.
(a)(b)Similar to the derivation process of a disk, the dynamic differential equations of the beam element under arbitrary maneuver flight conditions can be expressed as(9)Meq¨e+Ce+ΩGe+Cebq˙e+Ke+Kebqe=Feb+Qe,where Me, Ge, and Ke are the inertial matrix, gyro matrix, and stiffness matrix of the beam element, and the expressions are referred to in Ref. [19]. In this work, the internal damping of the material is ignored, and Ce is the external damping of the beam element. Qe denotes the generalized external forces applied to the beam element. Ceb, Keb, and Feb denote the added damping, stiffness, and maneuvering load caused by maneuver flight, which can be expressed as follows:(10)Ceb=2ρeAeωB,zH1,Keb=ρeAeωB,y2+ωB,z2H2+ρeAeωB,x2+ωB,z2H3+ρeAeωB,xωB,y−ω˙B,zH4+ρeAeωB,xωB,y+ω˙B,zH5,Feb=ρeAeY˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA1−X˙BωB,z+Z˙BωB,x+ω˙B,xze−Y¨B−ωB,yωB,zzeA1X˙BωB,z−Z˙BωB,x−ω˙B,xze+Y¨B+ωB,yωB,zzeA2Y˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA2Y˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA3−X˙BωB,z+Z˙BωB,x+ω˙B,xze−Y¨B−ωB,yωB,zzeA3X˙BωB,z−Z˙BωB,x−ω˙B,xze+Y¨B+ωB,yωB,zzeA4Y˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA4+ρeIe2ΩωB,x−ω˙B,yB12ΩωB,y+ω˙B,xB1−2ΩωB,y+ω˙B,xB22ΩωB,x−ω˙B,yB22ΩωB,x−ω˙B,yB32ΩωB,y+ω˙B,xB3−2ΩωB,y+ω˙B,xB42ΩωB,x−ω˙B,yB4,H1=0A11A1200−A13A140A1100A12A1300A14−A1200−A22−A2300−A240−A12A2200−A23A2400−A13A2300−A33A340A1300A23A3300A34−A1400−A24−A3400−A440−A14A2400−A34A440,H2=−A1100−A12−A1300−A140000000000000000−A1200−A22−A2300−A24−A1300−A23−A3300−A340000000000000000−A1400−A24−A3400−A44,H3=000000000−A11A1200−A13A1400A12−A2200A23−A24000000000000000000−A13A2300−A33A3400A14−A2400A34−A44000000000,H4=0A11−A1200A13−A14000000000000000000A12−A2200A23−A2400A13−A2300A33−A34000000000000000000A14−A2400A34−A440,H5=00000000A1100A12A1300A14−A1200−A22−A2300−A240000000000000000A1300A23A3300A34−A1400−A24−A3400−A4400000000,where(11)Ai=∫0leNidξ,Bi=∫0leN⌣idξ,Aij=∫0leNiNjdξ,Bij=∫0leN⌣iN⌣jdξ,i=1∼4,j=1∼4,in which ρe, Ae, and Ie denote the material density, the cross-sectional area, and the area moment of the shaft inertia, respectively.Rearranging the differential equations of motion of the disks and beam elements, the dynamic differential equations of the rotor system under arbitrary maneuver flight actions can be expressed as(12)Mq¨+C+ΩG+Cbq˙+K+Kbq=Fu+Fb+Q,where M, C, G, and K represent the inertial matrix, damping matrix, gyro moment matrix, and stiffness matrix of the rotor system without maneuver flight conditions and bending-torsion coupling effects considered, respectively. Cb, Kb, and Fb are the added damping matrix, stiffness matrix, and maneuvering load vectors caused by maneuver flight. Q denotes the generalized external force, which is applied to the rotor system containing the bearing force at the support stage.


### 2.2. Nonlinear Forces of the Bearing

In the process of the whirling of the rotor, the nonlinear forces of rolling ball bearing can be written as follows [20]:(13a)fxb=−kn∑j=1NbuθjξHuθjcosθj,fyb=−kn∑j=1NbuθjξHuθjsinθj,(13b)θj=2πj−1Nb+ωbmt,(13c)uθj=xcosθj+ysinθj−γ2,(13d)Huθj0,uθj≤0,uθj,uθj>0,(13e)ωbm=ωinr1+ωoutr2r1+r2,where kn denotes the contact stiffness between the roller and the ring. Nb is the number of rollers. θj denotes the circumferential position of the j-th roller at time t. And γ is the radial clearance. uθj is the elastic contact deformation in the radial direction of the j-th roller. Furthermore, ωbm denotes the rotational speed of the revolution of rollers. r1 and r2 are the internal and external radius of the bearing. ωin and ωout represent the rotational speeds of the internal and external rings. Generally, ξ=3/2 is set for rolling ball bearing.Substituting the nonlinear forces of bearing into the generalized external forceQ in equation (12), the nonlinear dynamic model of the bearing-rotor system can be obtained.


## 2.1. Finite Element Formulation

### 2.1.1. Disk

To obtain the dynamic differential equations of a disk during maneuvering flight, a coordinate system is presented in Figure1. OXYZ is the ground coordinate system, which represents the center of gravity, flight velocity, and acceleration of the plane. OXYZ is the coordinate system used to describe the whirl of the rotor around its own axis. Assuming that the engine rotor is installed on the aircraft with one end, which coincides with the center of gravity of the aircraft, then OXYZ is also the relative coordinate system of the aircraft. The degree-of-freedom of the disk can be expressed as(1)qd=xyθxθy.Figure 1
Coordinate system of disk.And the kinetic energy of a disk of rotor system under arbitrary maneuvering flight actions can be written as(2)T=Tt+Tr,where Tt and Tr are the translational and rotational kinetic energy of the disk, respectively. And the translational kinetic energy during the movement of the rotor can be expressed as [13](3)Tt=12mr˙Tr˙+vBTvB−rTωB2r+2r˙TωBr+2vBTr˙+2vBTωBr,where m denotes the quality of the disk, r represents the relative position of the disk in OXYZ coordinate system, and vB and ωB denote the velocity and angular velocity of the aircraft in OXYZ, respectively. r, vB, and ωB can be expressed as follows:(4)r=x+ecosΩt+φy+esinΩt+φz,vB=X˙BY˙BZ˙B,ωB=0−ωB,zωB,yωB,z0−ωB,x−ωB,yωB,x0,where Ω denotes the rotating speed of the rotor, φ is the initial phase angle, and X˙B, Y˙B, Z˙B and ωB,x, ωB,y, ωB,z are the velocities of the aircraft along each coordinate axis and the angular velocities around each coordinate axis in OXYZ coordinate system, respectively.The rotational kinetic energy of the disk during the movement of the rotor is [13](5)Tr=12Idθ˙x+ωB,x2+12Idθ˙y+ωB,y2+12IpΩ2+2Ωθy+θB,yθ˙x+ωB,x,where θB,y denotes the angular displacement of the aircraft around Y axis in OXYZ coordinate system, and θy+θB,y≈θy, θy+θB,y′≈θ˙y+ωB,y, Id and Ip are the diameter and polar moment of inertia of the disk, respectively. Considering the damping of the disk as a linear function of velocity, and assuming that the generalized damping matrix of the disk is Cd, the generalized stiffness matrix of the shaft at the position of the disk is Kd. Then, the dissipation energy of the disk can be expressed as equation (6) by the Rayleigh energy dissipation function. And the elastic potential energy of the flexible shaft at the position of the disk can be expressed as(6)D=12q˙dTCdq˙d,V=12qdTKdqd.Substitute equations (1)–(6) into the Lagrange equations for nonconservative systems, and the dynamic differential equations of the disk during arbitrary maneuver flight actions can be deduced as(7)Mdq¨d+Cd+ΩGd+Cdbq˙d+Kd+Kdbqd=Fu+Fdb+Qd,Cdb=0−2mωB,z002mωB,z00000000000,Kdb=−mωB,y2+ωB,z2mωB,xωB,y−ω˙B,z00mωB,xωB,y+ω˙B,z−mωB,x2+ωB,z20000000000,Fdb=−mX¨B−ωB,zY˙B+ωB,yZ˙B−mzωB,xωB,z+ω˙B,y−mY¨B+ωB,zX˙B−ωB,xZ˙B−mzωB,yωB,z−ω˙B,x−IpΩωB,y−Idω˙B,xIpΩωB,x−Idω˙B,y,where Md and Gd are the inertial matrix and gyro matrix of the disk with maneuver flight ignored, respectively. Their formulas are omitted in this paper, which can be found in Ref. [19]. Cdb, Kdb, and Fdb denote the added damping matrix, stiffness matrix, and maneuvering load caused by maneuver flight. Fu represents the unbalanced force of the disk, and Qd denotes the generalized external forces acting on the disk.


### 2.1.2. Shaft

The schematic diagram of a Timoshenko beam element is shown in Figure2. Similar to the disk, the kinetic energy of an arbitrary microelement segment of length dξ can be expressed as follows:(8)dT=dTt+dTr=12ρeAer˙eTr˙e+vBTvB−reTωB2re+2r˙eTωBre+2vBTr˙e+2vBTωBredξ+12dIdθ˙e+ωB,x2+12dIdφ˙e+ωB,y2+12dIpΩ2+2Ωθ˙e+ωB,xφe+θB,y,re=xeyeze,xeyeθeφe=N100N2N300N40N1−N200N3−N400−N⌣1N⌣200−N⌣3N⌣40N⌣100N⌣2N⌣300N⌣4qe,qe=x1y1θx1θy1x2y2θx2θy2T,where qe is the nodal displacement vector, and Nii=1−4 and N⌣ii=1−4 represent the shape functions and the expressions, which are referred to in Ref. [19]. re denotes the relative position of the centroid of the microelement segment in oxyz coordinate system. Then, the kinetic energy expression of the beam element can be obtained by integrating equation (8) along the axis oz.Figure 2
Schematic diagram of beam element.
(a)(b)Similar to the derivation process of a disk, the dynamic differential equations of the beam element under arbitrary maneuver flight conditions can be expressed as(9)Meq¨e+Ce+ΩGe+Cebq˙e+Ke+Kebqe=Feb+Qe,where Me, Ge, and Ke are the inertial matrix, gyro matrix, and stiffness matrix of the beam element, and the expressions are referred to in Ref. [19]. In this work, the internal damping of the material is ignored, and Ce is the external damping of the beam element. Qe denotes the generalized external forces applied to the beam element. Ceb, Keb, and Feb denote the added damping, stiffness, and maneuvering load caused by maneuver flight, which can be expressed as follows:(10)Ceb=2ρeAeωB,zH1,Keb=ρeAeωB,y2+ωB,z2H2+ρeAeωB,x2+ωB,z2H3+ρeAeωB,xωB,y−ω˙B,zH4+ρeAeωB,xωB,y+ω˙B,zH5,Feb=ρeAeY˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA1−X˙BωB,z+Z˙BωB,x+ω˙B,xze−Y¨B−ωB,yωB,zzeA1X˙BωB,z−Z˙BωB,x−ω˙B,xze+Y¨B+ωB,yωB,zzeA2Y˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA2Y˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA3−X˙BωB,z+Z˙BωB,x+ω˙B,xze−Y¨B−ωB,yωB,zzeA3X˙BωB,z−Z˙BωB,x−ω˙B,xze+Y¨B+ωB,yωB,zzeA4Y˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA4+ρeIe2ΩωB,x−ω˙B,yB12ΩωB,y+ω˙B,xB1−2ΩωB,y+ω˙B,xB22ΩωB,x−ω˙B,yB22ΩωB,x−ω˙B,yB32ΩωB,y+ω˙B,xB3−2ΩωB,y+ω˙B,xB42ΩωB,x−ω˙B,yB4,H1=0A11A1200−A13A140A1100A12A1300A14−A1200−A22−A2300−A240−A12A2200−A23A2400−A13A2300−A33A340A1300A23A3300A34−A1400−A24−A3400−A440−A14A2400−A34A440,H2=−A1100−A12−A1300−A140000000000000000−A1200−A22−A2300−A24−A1300−A23−A3300−A340000000000000000−A1400−A24−A3400−A44,H3=000000000−A11A1200−A13A1400A12−A2200A23−A24000000000000000000−A13A2300−A33A3400A14−A2400A34−A44000000000,H4=0A11−A1200A13−A14000000000000000000A12−A2200A23−A2400A13−A2300A33−A34000000000000000000A14−A2400A34−A440,H5=00000000A1100A12A1300A14−A1200−A22−A2300−A240000000000000000A1300A23A3300A34−A1400−A24−A3400−A4400000000,where(11)Ai=∫0leNidξ,Bi=∫0leN⌣idξ,Aij=∫0leNiNjdξ,Bij=∫0leN⌣iN⌣jdξ,i=1∼4,j=1∼4,in which ρe, Ae, and Ie denote the material density, the cross-sectional area, and the area moment of the shaft inertia, respectively.Rearranging the differential equations of motion of the disks and beam elements, the dynamic differential equations of the rotor system under arbitrary maneuver flight actions can be expressed as(12)Mq¨+C+ΩG+Cbq˙+K+Kbq=Fu+Fb+Q,where M, C, G, and K represent the inertial matrix, damping matrix, gyro moment matrix, and stiffness matrix of the rotor system without maneuver flight conditions and bending-torsion coupling effects considered, respectively. Cb, Kb, and Fb are the added damping matrix, stiffness matrix, and maneuvering load vectors caused by maneuver flight. Q denotes the generalized external force, which is applied to the rotor system containing the bearing force at the support stage.


## 2.1.1. Disk

To obtain the dynamic differential equations of a disk during maneuvering flight, a coordinate system is presented in Figure1. OXYZ is the ground coordinate system, which represents the center of gravity, flight velocity, and acceleration of the plane. OXYZ is the coordinate system used to describe the whirl of the rotor around its own axis. Assuming that the engine rotor is installed on the aircraft with one end, which coincides with the center of gravity of the aircraft, then OXYZ is also the relative coordinate system of the aircraft. The degree-of-freedom of the disk can be expressed as(1)qd=xyθxθy.Figure 1
Coordinate system of disk.And the kinetic energy of a disk of rotor system under arbitrary maneuvering flight actions can be written as(2)T=Tt+Tr,where Tt and Tr are the translational and rotational kinetic energy of the disk, respectively. And the translational kinetic energy during the movement of the rotor can be expressed as [13](3)Tt=12mr˙Tr˙+vBTvB−rTωB2r+2r˙TωBr+2vBTr˙+2vBTωBr,where m denotes the quality of the disk, r represents the relative position of the disk in OXYZ coordinate system, and vB and ωB denote the velocity and angular velocity of the aircraft in OXYZ, respectively. r, vB, and ωB can be expressed as follows:(4)r=x+ecosΩt+φy+esinΩt+φz,vB=X˙BY˙BZ˙B,ωB=0−ωB,zωB,yωB,z0−ωB,x−ωB,yωB,x0,where Ω denotes the rotating speed of the rotor, φ is the initial phase angle, and X˙B, Y˙B, Z˙B and ωB,x, ωB,y, ωB,z are the velocities of the aircraft along each coordinate axis and the angular velocities around each coordinate axis in OXYZ coordinate system, respectively.The rotational kinetic energy of the disk during the movement of the rotor is [13](5)Tr=12Idθ˙x+ωB,x2+12Idθ˙y+ωB,y2+12IpΩ2+2Ωθy+θB,yθ˙x+ωB,x,where θB,y denotes the angular displacement of the aircraft around Y axis in OXYZ coordinate system, and θy+θB,y≈θy, θy+θB,y′≈θ˙y+ωB,y, Id and Ip are the diameter and polar moment of inertia of the disk, respectively. Considering the damping of the disk as a linear function of velocity, and assuming that the generalized damping matrix of the disk is Cd, the generalized stiffness matrix of the shaft at the position of the disk is Kd. Then, the dissipation energy of the disk can be expressed as equation (6) by the Rayleigh energy dissipation function. And the elastic potential energy of the flexible shaft at the position of the disk can be expressed as(6)D=12q˙dTCdq˙d,V=12qdTKdqd.Substitute equations (1)–(6) into the Lagrange equations for nonconservative systems, and the dynamic differential equations of the disk during arbitrary maneuver flight actions can be deduced as(7)Mdq¨d+Cd+ΩGd+Cdbq˙d+Kd+Kdbqd=Fu+Fdb+Qd,Cdb=0−2mωB,z002mωB,z00000000000,Kdb=−mωB,y2+ωB,z2mωB,xωB,y−ω˙B,z00mωB,xωB,y+ω˙B,z−mωB,x2+ωB,z20000000000,Fdb=−mX¨B−ωB,zY˙B+ωB,yZ˙B−mzωB,xωB,z+ω˙B,y−mY¨B+ωB,zX˙B−ωB,xZ˙B−mzωB,yωB,z−ω˙B,x−IpΩωB,y−Idω˙B,xIpΩωB,x−Idω˙B,y,where Md and Gd are the inertial matrix and gyro matrix of the disk with maneuver flight ignored, respectively. Their formulas are omitted in this paper, which can be found in Ref. [19]. Cdb, Kdb, and Fdb denote the added damping matrix, stiffness matrix, and maneuvering load caused by maneuver flight. Fu represents the unbalanced force of the disk, and Qd denotes the generalized external forces acting on the disk.


## 2.1.2. Shaft

The schematic diagram of a Timoshenko beam element is shown in Figure2. Similar to the disk, the kinetic energy of an arbitrary microelement segment of length dξ can be expressed as follows:(8)dT=dTt+dTr=12ρeAer˙eTr˙e+vBTvB−reTωB2re+2r˙eTωBre+2vBTr˙e+2vBTωBredξ+12dIdθ˙e+ωB,x2+12dIdφ˙e+ωB,y2+12dIpΩ2+2Ωθ˙e+ωB,xφe+θB,y,re=xeyeze,xeyeθeφe=N100N2N300N40N1−N200N3−N400−N⌣1N⌣200−N⌣3N⌣40N⌣100N⌣2N⌣300N⌣4qe,qe=x1y1θx1θy1x2y2θx2θy2T,where qe is the nodal displacement vector, and Nii=1−4 and N⌣ii=1−4 represent the shape functions and the expressions, which are referred to in Ref. [19]. re denotes the relative position of the centroid of the microelement segment in oxyz coordinate system. Then, the kinetic energy expression of the beam element can be obtained by integrating equation (8) along the axis oz.Figure 2
Schematic diagram of beam element.
(a)(b)Similar to the derivation process of a disk, the dynamic differential equations of the beam element under arbitrary maneuver flight conditions can be expressed as(9)Meq¨e+Ce+ΩGe+Cebq˙e+Ke+Kebqe=Feb+Qe,where Me, Ge, and Ke are the inertial matrix, gyro matrix, and stiffness matrix of the beam element, and the expressions are referred to in Ref. [19]. In this work, the internal damping of the material is ignored, and Ce is the external damping of the beam element. Qe denotes the generalized external forces applied to the beam element. Ceb, Keb, and Feb denote the added damping, stiffness, and maneuvering load caused by maneuver flight, which can be expressed as follows:(10)Ceb=2ρeAeωB,zH1,Keb=ρeAeωB,y2+ωB,z2H2+ρeAeωB,x2+ωB,z2H3+ρeAeωB,xωB,y−ω˙B,zH4+ρeAeωB,xωB,y+ω˙B,zH5,Feb=ρeAeY˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA1−X˙BωB,z+Z˙BωB,x+ω˙B,xze−Y¨B−ωB,yωB,zzeA1X˙BωB,z−Z˙BωB,x−ω˙B,xze+Y¨B+ωB,yωB,zzeA2Y˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA2Y˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA3−X˙BωB,z+Z˙BωB,x+ω˙B,xze−Y¨B−ωB,yωB,zzeA3X˙BωB,z−Z˙BωB,x−ω˙B,xze+Y¨B+ωB,yωB,zzeA4Y˙BωB,z−Z˙BωB,y−ω˙B,yze−X¨B−ωB,xωB,zzeA4+ρeIe2ΩωB,x−ω˙B,yB12ΩωB,y+ω˙B,xB1−2ΩωB,y+ω˙B,xB22ΩωB,x−ω˙B,yB22ΩωB,x−ω˙B,yB32ΩωB,y+ω˙B,xB3−2ΩωB,y+ω˙B,xB42ΩωB,x−ω˙B,yB4,H1=0A11A1200−A13A140A1100A12A1300A14−A1200−A22−A2300−A240−A12A2200−A23A2400−A13A2300−A33A340A1300A23A3300A34−A1400−A24−A3400−A440−A14A2400−A34A440,H2=−A1100−A12−A1300−A140000000000000000−A1200−A22−A2300−A24−A1300−A23−A3300−A340000000000000000−A1400−A24−A3400−A44,H3=000000000−A11A1200−A13A1400A12−A2200A23−A24000000000000000000−A13A2300−A33A3400A14−A2400A34−A44000000000,H4=0A11−A1200A13−A14000000000000000000A12−A2200A23−A2400A13−A2300A33−A34000000000000000000A14−A2400A34−A440,H5=00000000A1100A12A1300A14−A1200−A22−A2300−A240000000000000000A1300A23A3300A34−A1400−A24−A3400−A4400000000,where(11)Ai=∫0leNidξ,Bi=∫0leN⌣idξ,Aij=∫0leNiNjdξ,Bij=∫0leN⌣iN⌣jdξ,i=1∼4,j=1∼4,in which ρe, Ae, and Ie denote the material density, the cross-sectional area, and the area moment of the shaft inertia, respectively.Rearranging the differential equations of motion of the disks and beam elements, the dynamic differential equations of the rotor system under arbitrary maneuver flight actions can be expressed as(12)Mq¨+C+ΩG+Cbq˙+K+Kbq=Fu+Fb+Q,where M, C, G, and K represent the inertial matrix, damping matrix, gyro moment matrix, and stiffness matrix of the rotor system without maneuver flight conditions and bending-torsion coupling effects considered, respectively. Cb, Kb, and Fb are the added damping matrix, stiffness matrix, and maneuvering load vectors caused by maneuver flight. Q denotes the generalized external force, which is applied to the rotor system containing the bearing force at the support stage.


## 2.2. Nonlinear Forces of the Bearing

In the process of the whirling of the rotor, the nonlinear forces of rolling ball bearing can be written as follows [20]:(13a)fxb=−kn∑j=1NbuθjξHuθjcosθj,fyb=−kn∑j=1NbuθjξHuθjsinθj,(13b)θj=2πj−1Nb+ωbmt,(13c)uθj=xcosθj+ysinθj−γ2,(13d)Huθj0,uθj≤0,uθj,uθj>0,(13e)ωbm=ωinr1+ωoutr2r1+r2,where kn denotes the contact stiffness between the roller and the ring. Nb is the number of rollers. θj denotes the circumferential position of the j-th roller at time t. And γ is the radial clearance. uθj is the elastic contact deformation in the radial direction of the j-th roller. Furthermore, ωbm denotes the rotational speed of the revolution of rollers. r1 and r2 are the internal and external radius of the bearing. ωin and ωout represent the rotational speeds of the internal and external rings. Generally, ξ=3/2 is set for rolling ball bearing.Substituting the nonlinear forces of bearing into the generalized external forceQ in equation (12), the nonlinear dynamic model of the bearing-rotor system can be obtained.


## 3. Experimental Verification for the Bearing-Rotor System

### 3.1. Test Rig Description

To validate the correctness of the dynamic model presented in this paper, a test rig for simulating the maneuvering flight of a bearing-rotor system is proposed and designed, as shown in Figure3. The rotor system is fixed at one end of the rotating arm structure using an installing seat. The other end of the rotating arm is mounted with balancing weights. During the experiments, the rotating arm structure is driven by the driving motor through the reduction gear box. It leads to the bearing-rotor system to rotate around the central spindle of the test rig. Through the different mounting holes reserved on the installing seat, the rotor can provide different pitch angles, as shown in Figure 4. When the pitch angle is set to 0°, the maneuver flight action of hovering around the central spindle is simulated. When the pitch angle is 90°, the maneuver flight action of rolling is performed. And if the pitch angle is between 0° and 90°, the synthetic maneuver of hovering and rolling can be studied. The test rig can apply the additional centrifugal force and gyro moment to the rotor system caused by maneuver flight. The maneuvering overload of the centrifugal force can be adjusted through controlling the rotating speed of the rotating arm structure.Figure 3
Test rig for simulating maneuver flight of a bearing-rotor system.Figure 4
Diagrams of pitch angle. (a) 0°. (b) 30°. (c) 90°.
(a)(b)(c)Figure5 shows the bearing-rotor system, which is composed of a motor, a flexible coupling, three disks, and two supports. Rolling bearings are adopted for the supports, and the three disks contain two simply supported disks and one cantilever disk. Inertia properties and unbalance configuration of the disks are summarized in Table 1. Related parameters of bearings are provided in Table 2.Figure 5
Bearing-rotor system.Table 1
Inertial properties and unbalance configuration.
NameDisk 1Disk 2Disk 3Mass (kg)0.40.40.5Polar moment of inertia (g·m2)0.000380.000380.00041Unbalance (10−5 kg·m)445Table 2
Parameters of the bearings.
NameBearing 1Bearing 2Radius of outer ring (mm)3030Radius of inner ring (mm)1010No. of rollers88Radial clearance (μm)66To validate the correctness of the dynamic model, nine kinds of maneuver flight conditions are investigated by numerical simulation and experimental methods, respectively, as shown in Table3, where g denotes the gravitational acceleration. And the pitch angle is set to 30° when the synthetic maneuver of hovering and rolling is studied.Table 3
Maneuver flight parameters.
Flight maneuversOverload (g)Z˙B (m/s)ωB,y (rad/s)Y˙B (m/s)ωB,z (rad/s)1Hovering1012.128.08002Hovering2017.1511.43003Hovering302114004Rolling100012.128.085Rolling200017.1511.436Rolling300021147Synthetic1010.576.064.038Synthetic2014.859.908.575.729Synthetic3018.1912.1210.57


### 3.2. Numerical and Experimental Results

The finite element model is established, which consists of 40 elements, including 35 beam elements, 3 disk elements, and 2 supporting elements, as shown in Figure6. The geometric dimensions and information of each element are listed in Table 4.Figure 6
The mesh diagram of the bearing-rotor system.Table 4
Dimension and elements information of the bearing-rotor system.
Node numberAxial location (m)Bearing/diskElement numberOuter diameter (m)Inner diameter (m)10.001Disk 310.010020.01520.010030.03030.010040.05140.010050.06650.010060.08160.010070.096Bearing 270.010080.10180.012090.11690.0120100.133100.0120110.148110.0120120.163120.0120130.178130.0120140.193140.0120150.208Disk 2150.0120160.225160.0120170.240170.0120180.255180.0120190.270190.0120200.285200.0120210.300210.0120220.315220.0120230.330Disk 1230.0120240.341240.0120250.356250.0120260.371260.0120270.386270.0120280.401280.0100290.406Bearing 1290.0100300.419300.0100310.436310.0100320.451320.0100330.466330.0100340.480340.0100350.490350.0100360.500The Newmark-β method and Newton-Raphson method are used to solve the differential equations of motion. For the Newmark-β method, two parameters are set as α=0.3 and β=0.6. Without maneuver flight condition, the waterfall spectra plots for the vibration of disk 3 with the rotor operating in 20–600 rad/s using numerical simulation and experimental methods are shown in Figure 7. The comparison results manifest that the characteristics of the response obtained by numerical simulation and experimental approach agree well with each other. Compared with the numerical simulation results, the amplitudes of the double frequency and triple frequency components in the experimental test are more extensive due to the misalignment of shafting and coupling or asymmetry of the stiffness of bearings, which is not considered in the simulation calculation.Figure 7
Waterfall spectra plots for the vibration of disk 3 without maneuver flight. (a) Numerical simulation. (b) Experimental results.
(a)(b)The waterfall spectra plots for the vibration of disk 3 with the rotor operating in 20–600 rad/s under the maneuver flight conditions of hovering, rolling, and the synthetic maneuver of hovering and rolling with the overload of 20 g are shown in Figures8–10, respectively. The frequency component ωg in experimental results is induced by the internal gear transmission in the reduction gearbox, which is not discussed in this work. By comparing the vibration response under conditions of straight flight and maneuver flight, a zero-frequency vibration component is observed during maneuver flight; that is, the static displacements of the rotor are generated under constant maneuvering overload. The static displacements in the horizontal direction caused by the additional centrifugal force do not vary with the rotating speed of the rotor. And the static displacements in the vertical direction induced by the additional gyro moment rise as the rotating speed increases. Especially, when the aircraft makes the flight maneuver of rolling, the rotation direction of the rotor is the same as its revolution. No maneuvering gyro torque is generated; therefore, there is no static displacement in the vertical direction, as shown in Figures 9(c) and 9(d).Figure 8
Waterfall spectra plots for the vibration of disk 3 during hovering. (a) Horizontal response, numerical simulation. (b) Horizontal response, experimental results. (c) Vertical response, numerical simulation. (d) Vertical response, experimental results.
(a)(b)(c)(d)Figure 9
Waterfall spectra plots for the response of disk 3 during rolling. (a) Horizontal response, numerical simulation. (b) Horizontal response, experimental results. (c) Vertical response, numerical simulation. (d) Vertical response, experimental results.
(a)(b)(c)(d)Figure 10
Waterfall spectra plots for the vibration of disk 3 during the synthetic maneuver of hovering and rolling. (a) Horizontal response, numerical simulation. (b) Horizontal response, experimental results. (c) Vertical response, numerical simulation. (d) Vertical response, experimental results.
(a)(b)(c)(d)With comparison of the influence of maneuver flight actions on the magnitude of the static displacements in numerical simulation and experimental works, Figure11 shows the variation of the static displacements of disk 3 with the amount of overload under the maneuver flight conditions of hovering, rolling, and the synthetic maneuver when the rotor operates at 600 rad/s. It can be observed that the magnitude of the static displacements through numerical simulation and experimental methods is basically consistent.Figure 11
Variation result of the static displacements of disk 3 with overload under maneuver flight conditions. (a) Horizontal magnitude, numerical simulation. (b) Horizontal magnitude, experimental results. (c) Vertical magnitude, numerical simulation. (d) Vertical magnitude, experimental results.
(a)(b)(c)(d)According to the above result analysis, the dynamic model of the bearing-rotor system under arbitrary maneuver flight conditions presented in this paper has the superior ability to predict the dynamic behavior of the rotor system under constant maneuvering overload.


## 3.1. Test Rig Description

To validate the correctness of the dynamic model presented in this paper, a test rig for simulating the maneuvering flight of a bearing-rotor system is proposed and designed, as shown in Figure3. The rotor system is fixed at one end of the rotating arm structure using an installing seat. The other end of the rotating arm is mounted with balancing weights. During the experiments, the rotating arm structure is driven by the driving motor through the reduction gear box. It leads to the bearing-rotor system to rotate around the central spindle of the test rig. Through the different mounting holes reserved on the installing seat, the rotor can provide different pitch angles, as shown in Figure 4. When the pitch angle is set to 0°, the maneuver flight action of hovering around the central spindle is simulated. When the pitch angle is 90°, the maneuver flight action of rolling is performed. And if the pitch angle is between 0° and 90°, the synthetic maneuver of hovering and rolling can be studied. The test rig can apply the additional centrifugal force and gyro moment to the rotor system caused by maneuver flight. The maneuvering overload of the centrifugal force can be adjusted through controlling the rotating speed of the rotating arm structure.Figure 3
Test rig for simulating maneuver flight of a bearing-rotor system.Figure 4
Diagrams of pitch angle. (a) 0°. (b) 30°. (c) 90°.
(a)(b)(c)Figure5 shows the bearing-rotor system, which is composed of a motor, a flexible coupling, three disks, and two supports. Rolling bearings are adopted for the supports, and the three disks contain two simply supported disks and one cantilever disk. Inertia properties and unbalance configuration of the disks are summarized in Table 1. Related parameters of bearings are provided in Table 2.Figure 5
Bearing-rotor system.Table 1
Inertial properties and unbalance configuration.
NameDisk 1Disk 2Disk 3Mass (kg)0.40.40.5Polar moment of inertia (g·m2)0.000380.000380.00041Unbalance (10−5 kg·m)445Table 2
Parameters of the bearings.
NameBearing 1Bearing 2Radius of outer ring (mm)3030Radius of inner ring (mm)1010No. of rollers88Radial clearance (μm)66To validate the correctness of the dynamic model, nine kinds of maneuver flight conditions are investigated by numerical simulation and experimental methods, respectively, as shown in Table3, where g denotes the gravitational acceleration. And the pitch angle is set to 30° when the synthetic maneuver of hovering and rolling is studied.Table 3
Maneuver flight parameters.
Flight maneuversOverload (g)Z˙B (m/s)ωB,y (rad/s)Y˙B (m/s)ωB,z (rad/s)1Hovering1012.128.08002Hovering2017.1511.43003Hovering302114004Rolling100012.128.085Rolling200017.1511.436Rolling300021147Synthetic1010.576.064.038Synthetic2014.859.908.575.729Synthetic3018.1912.1210.57


## 3.2. Numerical and Experimental Results

The finite element model is established, which consists of 40 elements, including 35 beam elements, 3 disk elements, and 2 supporting elements, as shown in Figure6. The geometric dimensions and information of each element are listed in Table 4.Figure 6
The mesh diagram of the bearing-rotor system.Table 4
Dimension and elements information of the bearing-rotor system.
Node numberAxial location (m)Bearing/diskElement numberOuter diameter (m)Inner diameter (m)10.001Disk 310.010020.01520.010030.03030.010040.05140.010050.06650.010060.08160.010070.096Bearing 270.010080.10180.012090.11690.0120100.133100.0120110.148110.0120120.163120.0120130.178130.0120140.193140.0120150.208Disk 2150.0120160.225160.0120170.240170.0120180.255180.0120190.270190.0120200.285200.0120210.300210.0120220.315220.0120230.330Disk 1230.0120240.341240.0120250.356250.0120260.371260.0120270.386270.0120280.401280.0100290.406Bearing 1290.0100300.419300.0100310.436310.0100320.451320.0100330.466330.0100340.480340.0100350.490350.0100360.500The Newmark-β method and Newton-Raphson method are used to solve the differential equations of motion. For the Newmark-β method, two parameters are set as α=0.3 and β=0.6. Without maneuver flight condition, the waterfall spectra plots for the vibration of disk 3 with the rotor operating in 20–600 rad/s using numerical simulation and experimental methods are shown in Figure 7. The comparison results manifest that the characteristics of the response obtained by numerical simulation and experimental approach agree well with each other. Compared with the numerical simulation results, the amplitudes of the double frequency and triple frequency components in the experimental test are more extensive due to the misalignment of shafting and coupling or asymmetry of the stiffness of bearings, which is not considered in the simulation calculation.Figure 7
Waterfall spectra plots for the vibration of disk 3 without maneuver flight. (a) Numerical simulation. (b) Experimental results.
(a)(b)The waterfall spectra plots for the vibration of disk 3 with the rotor operating in 20–600 rad/s under the maneuver flight conditions of hovering, rolling, and the synthetic maneuver of hovering and rolling with the overload of 20 g are shown in Figures8–10, respectively. The frequency component ωg in experimental results is induced by the internal gear transmission in the reduction gearbox, which is not discussed in this work. By comparing the vibration response under conditions of straight flight and maneuver flight, a zero-frequency vibration component is observed during maneuver flight; that is, the static displacements of the rotor are generated under constant maneuvering overload. The static displacements in the horizontal direction caused by the additional centrifugal force do not vary with the rotating speed of the rotor. And the static displacements in the vertical direction induced by the additional gyro moment rise as the rotating speed increases. Especially, when the aircraft makes the flight maneuver of rolling, the rotation direction of the rotor is the same as its revolution. No maneuvering gyro torque is generated; therefore, there is no static displacement in the vertical direction, as shown in Figures 9(c) and 9(d).Figure 8
Waterfall spectra plots for the vibration of disk 3 during hovering. (a) Horizontal response, numerical simulation. (b) Horizontal response, experimental results. (c) Vertical response, numerical simulation. (d) Vertical response, experimental results.
(a)(b)(c)(d)Figure 9
Waterfall spectra plots for the response of disk 3 during rolling. (a) Horizontal response, numerical simulation. (b) Horizontal response, experimental results. (c) Vertical response, numerical simulation. (d) Vertical response, experimental results.
(a)(b)(c)(d)Figure 10
Waterfall spectra plots for the vibration of disk 3 during the synthetic maneuver of hovering and rolling. (a) Horizontal response, numerical simulation. (b) Horizontal response, experimental results. (c) Vertical response, numerical simulation. (d) Vertical response, experimental results.
(a)(b)(c)(d)With comparison of the influence of maneuver flight actions on the magnitude of the static displacements in numerical simulation and experimental works, Figure11 shows the variation of the static displacements of disk 3 with the amount of overload under the maneuver flight conditions of hovering, rolling, and the synthetic maneuver when the rotor operates at 600 rad/s. It can be observed that the magnitude of the static displacements through numerical simulation and experimental methods is basically consistent.Figure 11
Variation result of the static displacements of disk 3 with overload under maneuver flight conditions. (a) Horizontal magnitude, numerical simulation. (b) Horizontal magnitude, experimental results. (c) Vertical magnitude, numerical simulation. (d) Vertical magnitude, experimental results.
(a)(b)(c)(d)According to the above result analysis, the dynamic model of the bearing-rotor system under arbitrary maneuver flight conditions presented in this paper has the superior ability to predict the dynamic behavior of the rotor system under constant maneuvering overload.


## 4. Complex Dual Rotor System

The dynamic model of the bearing-rotor system under arbitrary maneuver flight conditions has been established and verified by experimental works. Based on this, the dynamic behavior of a complex intermediate bearing-SFD-rotor system under constant maneuvering overload is investigated using the finite element model proposed in this paper.


### 4.1. Nonlinear Forces of the Supports

The nonlinear oil film forces of SFD in the horizontal and vertical directions can be obtained by Reynolds equation [21, 22], which are presented as follows:(14a)fxsfd=−μRL3c2x2+y21/2xε˙I2+εψ˙I1+yε˙I1+εψ˙I3,fysfd=−μRL3c2x2+y21/2yε˙I2+εψ˙I1−xε˙I1+εψ˙I3,(14b)ε=x2+y2c,(14c)ε˙=xx˙+yy˙cx2+y2,(14d)ψ˙=yx˙−xy˙x2+y2,(14e)tanψ=xy,where x and y denote the eccentric position of the journal in the Cartesian coordinate system. R, L, and c are the radius, length, and radial clearance of SFD, respectively, and μ denotes the dynamic viscosity. Ijj=1−3 are Sommerfeld integrals.Similar to the nonlinear forces of rolling ball bearing as equations (13a)–(13e), the nonlinear forces of intermediate bearing can be represented as equations (15a) and (15b).(15a)fxb=−kn∑j=1NbuθjξHuθjcosθj,fyb=−kn∑j=1NbuθjξHuθjsinθj,(15b)uθj=xir−xorcosθj+yir−yorsinθj−γ2,where the superscripts “ir” and “or” represent the internal and external rings of the bearing, respectively.


### 4.2. Calculation Model

The intermediate bearing-SFD-rotor system is presented in Figure12. The rotor system is composed of two shafts, rotating around the same axis oz, which are linked by an intermediate bearing. There are two disks in each rotor, disks 1 and 4 in the inner rotor, and disks 2 and 3 in the outer rotor. The rotors are supported by four supports, and the supporting form of supports I, II, and III is elastic support + rolling ball bearing + SFD. Support IV is the intermediate bearing with internal ring connected with the inner rotor and external ring linked with the outer rotor. Inertia properties and unbalance configuration of the disks are shown in Table 5. Table 6 shows the radial stiffness of elastic supports. And the parameters of SFDs and intermediate bearing are shown in Tables 7 and 8. The material density of the shafts is 7810 kg/m3, Young’s modulus is 196 GPa, and the shear modulus is 75.5 GPa. The speed ratio of the rotor system is defined as λ=ω2/ω1, where ω1 and ω2 are the rotation speeds of the inner and outer rotors, respectively, and λ=1.6 in this work.Figure 12
Structural diagram of the dual rotor system.Table 5
Inertial properties and unbalance configuration of disks.
NameInner rotorOuter rotorDisk 1Disk 4Disk 2Disk 3Mass (kg)2.412.383.171.57Polar moment of inertia (kg·m2)0.00820.00820.01460.0059Unbalance (10−5 kg·m)221.53Table 6
Radial stiffness of elastic supports.
NameIIIIIIStiffness (N/m)2.9 × 1074.4 × 1072 × 107Table 7
Parameters of SFDs.
NameInner rotorOuter rotorSFD ISFD IISFD IIIRadius (mm)251835Length (mm)131314Radial clearance (mm)0.250.180.35Dynamic viscosity (10−2 Pa·s)1.0752——Table 8
Parameters of the intermediate bearing.
NameRadius of inner ring (mm)Radius of outer ring (mm)No. of rollersRadial clearance (μm)Value9.3514.1596


### 4.3. Response Analysis

The finite element model of the dual rotor system is established, and the geometric dimensions and information of each element are listed in Table9.Table 9
Dimension and elements information of the dual rotor system.
Node numberAxial location (m)Bearing/diskElement numberOuter diameter (m)Inner diameter (m)10.100Disk 110.018020.13020.018030.16030.018040.19040.018050.22050.018060.25560.018070.28970.018080.32380.018090.35990.0180100.384100.0180110.409110.0180120.433120.0180130.445Bearing 1130.0180140.490140.0220150.560150.0220160.650160.0220170.753170.0220180.820180.0220190.958190.0220201.030200.0220211.065Bearing 4210.0220221.087220.0220231.100230.0220241.140Disk 4240.0220251.170250.0220261.190260.0220271.211270.0170281.215Bearing 2280.0140291.220290.0140301.230End of the inner rotor310.642300.0350.03320.655310.0350.03330.679Bearing 3320.0350.03340.687330.0380.03350.711340.0380.03360.737350.0380.03370.762Disk 2360.0380.03380.808370.0380.03390.854380.0380.03400.899390.0380.03410.945400.0380.03420.960Disk 3410.0380.03430.980420.0380.03441.000430.070.03451.020440.060.03461.040450.060.03471.064Bearing 4460.060.03481.074Taking hovering as the research object, the dynamic behavior of the complex intermediate bearing-SFD-rotor system under constant maneuvering overload is analyzed. The maneuvering parameters of the aircraft are shown in Table10. Except for the maneuvering parameters mentioned here, the values of other parameters are set to 0.Table 10
Parameters of maneuver flight.
ManeuversvB,z (m/s)ωB,y (rad/s)Overload (g)Horizontal turn98110Solving the dynamic differential equations under the conditions of straight flight and maneuver flight, respectively, Figures13 and 14 show the waterfall spectra plots for the vibration of disks 3 and 4. And the spectrograms for the vertical vibration of disk 4 are shown in Figure 15. For the condition of straight flight, the frequency components mainly include the fundamental frequencies ω1 and ω2 and some combined frequency components such as 2ω2−ω1, 3ω2−2ω1, caused by the coupling effect of the inner and outer rotors through the intermediate bearing, and the subharmonic vibration induced by the nonlinear characteristics of SFDs, which is defined as ωs. Compared with the straight flight condition, the combined frequency components are reduced, and the subharmonic vibration component is amplified under the maneuvering flight of hovering; see Figures 13(d), 14(d), and 15. It can be observed that the amplitude of the subharmonic component is relatively small when the rotating speed of the inner rotor is below 400 rad/s, the 3rd critical speed of the rotor system. However, after the rotating speed exceeds 400 rad/s, the subharmonic vibration is rapidly amplified with the continuous increase of the rotational speed, and it will far exceed the amplitude of fundamental frequency vibration when the rotating speed of the inner rotor reaches 550 rad/s, as shown in Figure 15(h). Besides, similar to the bearing-rotor system, static displacements of the rotor can be observed under constant maneuvering overload. For 10 g overload in this work, the magnitude of the static displacement of the rotor in the horizontal direction caused by the additional centrifugal force has exceeded the resonance amplitude at the 3rd critical speed of the rotor system. Therefore, for the future aircraft with high maneuverability and overload, the clearance of rotors and stators needs to be specially designed to prevent rub-impact fault.Figure 13
Waterfall spectra plots for the vibration of disk 3. (a) Horizontal response, straight flight. (b) Horizontal response, hovering. (c) Vertical response, straight flight. (d) Vertical response, hovering.
(a)(b)(c)(d)Figure 14
Waterfall spectra plots for the vibration of disk 4. (a) Horizontal response, straight flight. (b) Horizontal response, hovering. (c) Vertical response, straight flight. (d) Vertical response, hovering.
(a)(b)(c)(d)Figure 15
Spectrograms for the vertical vibration of disk 4. (a) 300 rad/s, straight flight. (b) 300 rad/s, hovering. (c) 400 rad/s, straight flight. (d) 400 rad/s, hovering. (e) 500 rad/s, straight flight. (f) 500 rad/s, hovering. (g) 550 rad/s, straight flight. (h) 550 rad/s, hovering.
(a)(b)(c)(d)(e)(f)(g)(h)The bifurcation diagrams of the vibration response varying with the rotational speed of disk 4 are shown in Figure16. The orbits and Poincare maps for the vibration response of disk 4 are illustrated in Figures 17 and 18, respectively. By comparison, there is little difference in periodicity between the two conditions of straight flight and hovering at low speeds. However, as the rotating speed of the rotor increases, the vibration response periodicity will deteriorate under the condition of maneuver flight. Significantly, after the rotating speed of the inner rotor exceeds the 3rd critical speed of the rotor system, the movement of the rotor system gradually develops from pseudo-period to chaos under constant maneuvering overload.Figure 16
Bifurcation diagrams for the vibration of disk 4. (a) Horizontal response, straight flight. (b) Horizontal response, hovering. (c) Vertical response, straight flight. (d) Vertical response, hovering.
(a)(b)(c)(d)Figure 17
Orbit of disk 4. (a) 300 rad/s, straight flight. (b) 300 rad/s, hovering. (c) 400 rad/s, straight flight. (d) 400 rad/s, hovering. (e) 500 rad/s, straight flight. (f) 500 rad/s, hovering. (g) 550 rad/s, straight flight. (h) 550 rad/s, hovering.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 18
Poincare maps of disk 4. (a) 300 rad/s, straight flight. (b) 300 rad/s, hovering. (c) 400 rad/s, straight flight. (d) 400 rad/s, hovering. (e) 500 rad/s, straight flight. (f) 500 rad/s, hovering. (g) 550 rad/s, straight flight. (h) 550 rad/s, hovering.
(a)(b)(c)(d)(e)(f)(g)(h)The coaxial co- or counterrotating rotors exhibit different vibration characteristics [21, 23]. Under the condition of maneuver flight of hovering, the vibration characteristics of the dual rotor system in the case of counterrotation of the inner and outer rotors λ=−1.6 are analyzed in this work. In comparison with the two points, the waterfall spectra plots for the vibration of disks 3 and 4 are shown in Figures 19 and 20, respectively. The comparison results manifest that there are some differences in the static displacements between the two points under constant maneuvering overload. It is because the additional gyro torque acting on the rotor caused by maneuver flight is different in the two situations. And the magnitude of the static displacements relates to the specific structure of the rotor system, such as the axial locations of disks and bearings, the inertia properties of each disk, and the stiffness of each bearing.Figure 19
Vertical response of disk 3. (a)λ=1.6. (b)λ=−1.6.
(a)(b)Figure 20
Vertical response of disk 4. (a)λ=1.6. (b) λ=−1.6.
(a)(b)The spectrograms for the vertical vibration of disks 3 and 4 are shown in Figures21 and 22, respectively. Compared with the corotating dual rotor systems, the amplitude of subharmonic can be reduced in varying degrees under constant maneuvering overload in the case of counterrotation, as highlighted in Figures 21(e)–21(h) and 22(e)–22(h). Therefore, the design of counterrotation of the inner and outer rotors can effectively reduce the amplitude of subharmonic induced by large maneuvering overload.Figure 21
Spectrograms for the vertical vibration of disk 3. (a) 300 rad/s,λ=1.6. (b) 300 rad/s, λ=−1.6. (c) 400 rad/s, λ=1.6. (d) 400 rad/s, λ=−1.6. (e) 500 rad/s, λ=1.6. (f) 500 rad/s, λ=−1.6. (g) 550 rad/s, λ=1.6. (h) 550 rad/s, λ=−1.6.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 22
Spectrograms for the vertical vibration of disk 4. (a) 300 rad/s,λ=1.6. (b) 300 rad/s, λ=−1.6. (c) 400 rad/s, λ=1.6. (d) 400 rad/s, λ=−1.6. (e) 500 rad/s, λ=1.6. (f) 500 rad/s, λ=−1.6. (g) 550 rad/s, λ=1.6. (h) 550 rad/s, λ=−1.6.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4.1. Nonlinear Forces of the Supports

The nonlinear oil film forces of SFD in the horizontal and vertical directions can be obtained by Reynolds equation [21, 22], which are presented as follows:(14a)fxsfd=−μRL3c2x2+y21/2xε˙I2+εψ˙I1+yε˙I1+εψ˙I3,fysfd=−μRL3c2x2+y21/2yε˙I2+εψ˙I1−xε˙I1+εψ˙I3,(14b)ε=x2+y2c,(14c)ε˙=xx˙+yy˙cx2+y2,(14d)ψ˙=yx˙−xy˙x2+y2,(14e)tanψ=xy,where x and y denote the eccentric position of the journal in the Cartesian coordinate system. R, L, and c are the radius, length, and radial clearance of SFD, respectively, and μ denotes the dynamic viscosity. Ijj=1−3 are Sommerfeld integrals.Similar to the nonlinear forces of rolling ball bearing as equations (13a)–(13e), the nonlinear forces of intermediate bearing can be represented as equations (15a) and (15b).(15a)fxb=−kn∑j=1NbuθjξHuθjcosθj,fyb=−kn∑j=1NbuθjξHuθjsinθj,(15b)uθj=xir−xorcosθj+yir−yorsinθj−γ2,where the superscripts “ir” and “or” represent the internal and external rings of the bearing, respectively.


## 4.2. Calculation Model

The intermediate bearing-SFD-rotor system is presented in Figure12. The rotor system is composed of two shafts, rotating around the same axis oz, which are linked by an intermediate bearing. There are two disks in each rotor, disks 1 and 4 in the inner rotor, and disks 2 and 3 in the outer rotor. The rotors are supported by four supports, and the supporting form of supports I, II, and III is elastic support + rolling ball bearing + SFD. Support IV is the intermediate bearing with internal ring connected with the inner rotor and external ring linked with the outer rotor. Inertia properties and unbalance configuration of the disks are shown in Table 5. Table 6 shows the radial stiffness of elastic supports. And the parameters of SFDs and intermediate bearing are shown in Tables 7 and 8. The material density of the shafts is 7810 kg/m3, Young’s modulus is 196 GPa, and the shear modulus is 75.5 GPa. The speed ratio of the rotor system is defined as λ=ω2/ω1, where ω1 and ω2 are the rotation speeds of the inner and outer rotors, respectively, and λ=1.6 in this work.Figure 12
Structural diagram of the dual rotor system.Table 5
Inertial properties and unbalance configuration of disks.
NameInner rotorOuter rotorDisk 1Disk 4Disk 2Disk 3Mass (kg)2.412.383.171.57Polar moment of inertia (kg·m2)0.00820.00820.01460.0059Unbalance (10−5 kg·m)221.53Table 6
Radial stiffness of elastic supports.
NameIIIIIIStiffness (N/m)2.9 × 1074.4 × 1072 × 107Table 7
Parameters of SFDs.
NameInner rotorOuter rotorSFD ISFD IISFD IIIRadius (mm)251835Length (mm)131314Radial clearance (mm)0.250.180.35Dynamic viscosity (10−2 Pa·s)1.0752——Table 8
Parameters of the intermediate bearing.
NameRadius of inner ring (mm)Radius of outer ring (mm)No. of rollersRadial clearance (μm)Value9.3514.1596


## 4.3. Response Analysis

The finite element model of the dual rotor system is established, and the geometric dimensions and information of each element are listed in Table9.Table 9
Dimension and elements information of the dual rotor system.
Node numberAxial location (m)Bearing/diskElement numberOuter diameter (m)Inner diameter (m)10.100Disk 110.018020.13020.018030.16030.018040.19040.018050.22050.018060.25560.018070.28970.018080.32380.018090.35990.0180100.384100.0180110.409110.0180120.433120.0180130.445Bearing 1130.0180140.490140.0220150.560150.0220160.650160.0220170.753170.0220180.820180.0220190.958190.0220201.030200.0220211.065Bearing 4210.0220221.087220.0220231.100230.0220241.140Disk 4240.0220251.170250.0220261.190260.0220271.211270.0170281.215Bearing 2280.0140291.220290.0140301.230End of the inner rotor310.642300.0350.03320.655310.0350.03330.679Bearing 3320.0350.03340.687330.0380.03350.711340.0380.03360.737350.0380.03370.762Disk 2360.0380.03380.808370.0380.03390.854380.0380.03400.899390.0380.03410.945400.0380.03420.960Disk 3410.0380.03430.980420.0380.03441.000430.070.03451.020440.060.03461.040450.060.03471.064Bearing 4460.060.03481.074Taking hovering as the research object, the dynamic behavior of the complex intermediate bearing-SFD-rotor system under constant maneuvering overload is analyzed. The maneuvering parameters of the aircraft are shown in Table10. Except for the maneuvering parameters mentioned here, the values of other parameters are set to 0.Table 10
Parameters of maneuver flight.
ManeuversvB,z (m/s)ωB,y (rad/s)Overload (g)Horizontal turn98110Solving the dynamic differential equations under the conditions of straight flight and maneuver flight, respectively, Figures13 and 14 show the waterfall spectra plots for the vibration of disks 3 and 4. And the spectrograms for the vertical vibration of disk 4 are shown in Figure 15. For the condition of straight flight, the frequency components mainly include the fundamental frequencies ω1 and ω2 and some combined frequency components such as 2ω2−ω1, 3ω2−2ω1, caused by the coupling effect of the inner and outer rotors through the intermediate bearing, and the subharmonic vibration induced by the nonlinear characteristics of SFDs, which is defined as ωs. Compared with the straight flight condition, the combined frequency components are reduced, and the subharmonic vibration component is amplified under the maneuvering flight of hovering; see Figures 13(d), 14(d), and 15. It can be observed that the amplitude of the subharmonic component is relatively small when the rotating speed of the inner rotor is below 400 rad/s, the 3rd critical speed of the rotor system. However, after the rotating speed exceeds 400 rad/s, the subharmonic vibration is rapidly amplified with the continuous increase of the rotational speed, and it will far exceed the amplitude of fundamental frequency vibration when the rotating speed of the inner rotor reaches 550 rad/s, as shown in Figure 15(h). Besides, similar to the bearing-rotor system, static displacements of the rotor can be observed under constant maneuvering overload. For 10 g overload in this work, the magnitude of the static displacement of the rotor in the horizontal direction caused by the additional centrifugal force has exceeded the resonance amplitude at the 3rd critical speed of the rotor system. Therefore, for the future aircraft with high maneuverability and overload, the clearance of rotors and stators needs to be specially designed to prevent rub-impact fault.Figure 13
Waterfall spectra plots for the vibration of disk 3. (a) Horizontal response, straight flight. (b) Horizontal response, hovering. (c) Vertical response, straight flight. (d) Vertical response, hovering.
(a)(b)(c)(d)Figure 14
Waterfall spectra plots for the vibration of disk 4. (a) Horizontal response, straight flight. (b) Horizontal response, hovering. (c) Vertical response, straight flight. (d) Vertical response, hovering.
(a)(b)(c)(d)Figure 15
Spectrograms for the vertical vibration of disk 4. (a) 300 rad/s, straight flight. (b) 300 rad/s, hovering. (c) 400 rad/s, straight flight. (d) 400 rad/s, hovering. (e) 500 rad/s, straight flight. (f) 500 rad/s, hovering. (g) 550 rad/s, straight flight. (h) 550 rad/s, hovering.
(a)(b)(c)(d)(e)(f)(g)(h)The bifurcation diagrams of the vibration response varying with the rotational speed of disk 4 are shown in Figure16. The orbits and Poincare maps for the vibration response of disk 4 are illustrated in Figures 17 and 18, respectively. By comparison, there is little difference in periodicity between the two conditions of straight flight and hovering at low speeds. However, as the rotating speed of the rotor increases, the vibration response periodicity will deteriorate under the condition of maneuver flight. Significantly, after the rotating speed of the inner rotor exceeds the 3rd critical speed of the rotor system, the movement of the rotor system gradually develops from pseudo-period to chaos under constant maneuvering overload.Figure 16
Bifurcation diagrams for the vibration of disk 4. (a) Horizontal response, straight flight. (b) Horizontal response, hovering. (c) Vertical response, straight flight. (d) Vertical response, hovering.
(a)(b)(c)(d)Figure 17
Orbit of disk 4. (a) 300 rad/s, straight flight. (b) 300 rad/s, hovering. (c) 400 rad/s, straight flight. (d) 400 rad/s, hovering. (e) 500 rad/s, straight flight. (f) 500 rad/s, hovering. (g) 550 rad/s, straight flight. (h) 550 rad/s, hovering.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 18
Poincare maps of disk 4. (a) 300 rad/s, straight flight. (b) 300 rad/s, hovering. (c) 400 rad/s, straight flight. (d) 400 rad/s, hovering. (e) 500 rad/s, straight flight. (f) 500 rad/s, hovering. (g) 550 rad/s, straight flight. (h) 550 rad/s, hovering.
(a)(b)(c)(d)(e)(f)(g)(h)The coaxial co- or counterrotating rotors exhibit different vibration characteristics [21, 23]. Under the condition of maneuver flight of hovering, the vibration characteristics of the dual rotor system in the case of counterrotation of the inner and outer rotors λ=−1.6 are analyzed in this work. In comparison with the two points, the waterfall spectra plots for the vibration of disks 3 and 4 are shown in Figures 19 and 20, respectively. The comparison results manifest that there are some differences in the static displacements between the two points under constant maneuvering overload. It is because the additional gyro torque acting on the rotor caused by maneuver flight is different in the two situations. And the magnitude of the static displacements relates to the specific structure of the rotor system, such as the axial locations of disks and bearings, the inertia properties of each disk, and the stiffness of each bearing.Figure 19
Vertical response of disk 3. (a)λ=1.6. (b)λ=−1.6.
(a)(b)Figure 20
Vertical response of disk 4. (a)λ=1.6. (b) λ=−1.6.
(a)(b)The spectrograms for the vertical vibration of disks 3 and 4 are shown in Figures21 and 22, respectively. Compared with the corotating dual rotor systems, the amplitude of subharmonic can be reduced in varying degrees under constant maneuvering overload in the case of counterrotation, as highlighted in Figures 21(e)–21(h) and 22(e)–22(h). Therefore, the design of counterrotation of the inner and outer rotors can effectively reduce the amplitude of subharmonic induced by large maneuvering overload.Figure 21
Spectrograms for the vertical vibration of disk 3. (a) 300 rad/s,λ=1.6. (b) 300 rad/s, λ=−1.6. (c) 400 rad/s, λ=1.6. (d) 400 rad/s, λ=−1.6. (e) 500 rad/s, λ=1.6. (f) 500 rad/s, λ=−1.6. (g) 550 rad/s, λ=1.6. (h) 550 rad/s, λ=−1.6.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 22
Spectrograms for the vertical vibration of disk 4. (a) 300 rad/s,λ=1.6. (b) 300 rad/s, λ=−1.6. (c) 400 rad/s, λ=1.6. (d) 400 rad/s, λ=−1.6. (e) 500 rad/s, λ=1.6. (f) 500 rad/s, λ=−1.6. (g) 550 rad/s, λ=1.6. (h) 550 rad/s, λ=−1.6.
(a)(b)(c)(d)(e)(f)(g)(h)


## 5. Conclusions

A nonlinear dynamic model of the bearing-rotor system under arbitrary maneuver flight conditions is modeled in this paper, and the added damping, stiffness, and maneuvering load generated by maneuver flight are obtained. The numerical integral method is used to investigate the vibration behavior of the rotor under constant maneuvering overload. Moreover, the simulation results are compared and verified by the experimental results. Based on this, the vibration characteristics of a complex intermediate bearing-SFD-rotor system under constant maneuvering overload are analyzed. Conclusions are as follows:(1)
The maneuver flight action can lead to some static displacements of the rotor. It is found that the static displacements in the horizontal direction caused by the additional centrifugal force do not change with the rotating speed of the rotor. And the static displacements in the vertical direction induced by the additional gyro moment increase with the rise in the rotating speed. For the future aircraft with high maneuverability and overload, the clearance of rotors and stators needs to be specially designed to prevent rub-impact fault induced by the static displacements.(2)
Compared with the condition of straight flight, the combined frequency components induced by the coupling of the inner and outer rotors through the intermediate bearing are weakened. In contrast, the subharmonic vibration component caused by the nonlinear characteristics of SFDs is amplified under constant maneuvering overload. The amplitude of subharmonic vibration may far exceed the fundamental frequency vibration under certain maneuver flight conditions, which may do great harm to the bearings of rotor system, and it needs to be paid great attention in the design of aeroengine.(3)
The periodicity of motion of the rotor deteriorates during maneuver flight. For the dual rotor system under the maneuver action of hovering studied in this paper, when the rotating speed exceeds the 3rd critical speed of the rotor system, the movement of the rotor system develops from pseudo-period to chaos gradually as the rotational speed continues to increase.(4)
The co- and counterrotating dual rotor systems have different vibration characteristics under constant maneuvering overload. The design of counterrotation of the inner and outer rotors can effectively reduce the amplitude of subharmonic induced by large maneuvering overload.



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*Source: 5512409-2021-05-03.xml*