Dynamic modeling method for bent rotor system caused by bearing misalignment

Abstract

The invention belongs to the technical field of dynamic modeling of rotating machinery, and discloses a dynamic modeling method for a bent rotor system caused by bearing misalignment. The method comprises the following steps: firstly, establishing a quasi-statics model considering the misalignment of the four-point contact ball bearing, wherein the quasi-statics model is used for calculating a pre-tightening load generated by the misalignment of the bearing; a pre-tightening load generated by misalignment of the bearing is substituted into a motion differential equation, and initial bending deformation of the rotor is obtained; based on the initial bending deformation of the rotor, the unbalanced load borne by the rotor in the initial bending state is calculated; meanwhile, a pre-tightening load generated by bearing misalignment and an unbalanced load borne by the rotor are considered, and a bearing-rotor system dynamic model considering initial bending deformation of the rotor is established. According to the model, the influence of bearing misalignment is fully considered, the influence of bearing misalignment on the contact load is considered, the influence of bearing misalignment on the initial state of the rotor is also considered, and the vibration response prediction capacity of the rotor system can be improved.

Description

Technical Field

[0001] The invention relates to the technical field of rotating machinery dynamics modeling, and in particular to a dynamics modeling method for a bending rotor system caused by misaligned bearings. Background Art

[0002] Modern aircraft engines pursue high performance and high thrust-to-weight ratio. Their structures are becoming increasingly complex and their working conditions are becoming increasingly harsh. As a result, the number of accidents that affect the use and testing of engines due to excessive vibration of the entire engine is gradually increasing. Controlling the vibration level of the engine during operation is of great significance to improving the safety, reliability and life of the engine. Engine vibration control requires joint guarantees from many aspects. Among them, the dynamic imbalance of rotating parts is one of the main reasons that affect the excessive vibration during engine testing. Therefore, the rotor must be strictly balanced before the engine test. However, due to the power of the balancing machine, only low-speed dynamic balancing methods can be used for medium and large engine rotors. Low-speed dynamic balancing has been widely used to reduce vibrations caused by eccentric centrifugal force or vibrations acting on bearings that are consistent with the operating speed frequency. In addition to dynamic balancing, the bending degree of the rotor is also an important part of vibration control.

[0003] In the actual dynamic analysis of aircraft engines, the causes of shaft bending can be divided into three categories: the first category is initial bending caused by production process, gravity droop and installation error, the second category is general thermal bending caused by thermal deformation, and the third category is friction thermal bending caused by frictional heat generation. For aircraft engines, installation errors may lead to bearing misalignment, which in turn generates bearing misalignment preload, and ultimately leads to bending deformation of the shaft, which is an important factor affecting the dynamic response of the rotor system. During the startup of the rotor system, due to the synchronous rotation of the bending deformation and the rotor, a lateral exciting force similar to an unbalanced force is generated, thereby exciting the positive coordinated synchronous lateral vibration of the rotor system, and resonance occurs near the critical speed, and even causes the rotor system to fail to start.

[0004] In engineering applications, there is a type of rotor system vibration related to the vibration of the rotor with initial bending. Due to machining errors, poor assembly, poor alignment, deformation during operation, or thermal bending caused by thermal imbalance after parking, the rotor system has a certain initial bending deformation before starting. During the startup of the rotor system, due to the synchronous rotation of the bending deformation and the rotor, a lateral exciting force similar to the unbalanced force is generated, thereby exciting the positive coordinated synchronous lateral vibration of the rotor system, and resonance occurs near the critical speed, and even causes the rotor system to fail to start.

[0005] In recent decades, rotor bending failures in rotating machinery have attracted extensive research attention. Early studies focused on the dynamic response of bent rotors under different conditions. "Nicholas JC, Gunter EJ, Allaire PE. Effect of residual shaft bow on unbalance response and balancing of a single mass flexible rotor—Part I: Unbalance response. 1976, 98(2): 171-181." and "Shiau TN, Lee E K. The residual shaft bow effect on dynamic response of an impliedly supported rotor with disk skew and mass unbalances. 1989, 111(2): 170-178." analyzed the vibration characteristics of rotors with initial bending, emphasizing the effects of mass unbalance, disk tilt and bending conditions on critical speed and vibration amplitude. "Ehrlich F F. Handbook of rotor dynamics. Krieger, 1999." and "Rao J S. A note on Jeffcott warped rotor. Mechanism and Machine Theory, 2001, 36 (5): 563-575." extend these analyses to fault identification and the role of phase angle measurement in assessing rotor bending and mass eccentricity effects. "Kang CH, Hsu WC, Lee EK, et al. Dynamic analysis of gear-rotor system with viscoelastic supports under residual shaft bow effect. Mechanism and Machine Theory, 2011, 46 (3): 264-275." Finite element modeling was used to explore the effects of bending and support stiffness on steady-state response and critical speed. Subsequent studies have explored other influencing factors in depth.“Song GF, Yang ZJ, JiC, et al. Theoretical–experimental study on a rotor with a residual shaftbow. Mechanism and Machine Theory, 2013, 63: 50-58.” emphasized the importance of eccentricity and damping ratio in amplifying the amplitude response, while “Parkinson AG, Darlow MS, Smalley AJ. Balancing flexible rotating shafts with an initial bend. AIAA journal, 1984, 22(5): 683-689” proposed a dynamic balancing method for bent rotors under different mass imbalance conditions. “Deepthikumar MB, Sekhar AS, Srikanthan M R. Modal balancing of flexible rotors with bow and distributed unbalance. Journal of sound and vibration, 2013, 332(24): 6216-6233.” and “Deepthikumar MB, Sekhar AS, Srikanthan M R. Modal balancing of flexible rotors with bow and distributed unbalance. Journal of sound and vibration, 2013, 332(24): 6216-6233.” Theoretical and experimental methods are proposed to characterize the bending-induced vibrations and rotor shape.“Yang L, Xu H, Yu L. Study on the calculation methods of vibration characteristics of the rotor with initial bending, 2017 IEEE International Conference on Mechatronics and Automation (ICMA). IEEE, 2017: 770-775” and “Weng L, Yang Z, Cao Y. The nonlineardynamic characteristics of a cracked rotor-bearing system with initial bend deformation, 2015 7th International Conference on Intelligent Human-Machine Systems and Cybernetics. IEEE, 2015, 2: 341-344.” further studied the nonlinear dynamics and system stability in the bent rotor system, including the bending effects induced by cracks. Despite these advances, the impact of rotor bending on bearing contact characteristics remains underexplored. Given the interdependence between rotor bending (caused by bearing misalignment) and bearing contact behavior, understanding their complex interactions is critical to improving system performance and reliability.

[0006] Research on four-point contact ball bearings has mainly focused on the load, contact and vibration characteristics of healthy bearings, while the attention to misaligned four-point contact ball bearings is relatively limited. "Hamrock BJ, Anderson W J. Arched-outer-race ball-bearing analysis considering centrifugal forces. National Aeronautics and Space Administration, 1972." first proposed a bearing model for calculating contact loads and contact angles at different gasket thicknesses and rotational speeds. Subsequently, "Leblanc A, Nelias D. Ball motion and sliding friction in a four-contact-point ball bearing. Journal of Tribology, 2007, 129: 801-808." developed a quasi-static model to study ball motion and sliding. "Rivera G, Tong V C, Hong S W. Contact load and stiffness of four-point contact ball bearings under loading. International Journal of Precision Engineering and Manufacturing, 2022, 23 (6): 677-687." further studied the contact load and stiffness considering different preloads, while "Halpin JD, Tran AN. An analytical model of four-point contact rolling element ball bearings. Journal of Tribology, 2016, 138 (3): 031404." improved the bearing model and proposed the optimal speed recommendation to eliminate gyroscopic torque. "Ma SJ, Li W, Yan K, et al. A study on the dynamic contact feature of four-contact-point ball bearing. Mechanical systems and signal processing, 2022, 174: 109111." has made significant progress in the research of FPCBBs, involving dynamic contact characteristics, rotor system response, cage stability and structural optimization.In addition, “Li Y, Li W, Zhu Y, et al. Dynamic performance analysis of cage in four-point contact ball bearing. Lubricants, 2022, 10(7): 149.” and “Su Y, Zhang J, Zhang W. Theoretical and experimental approaches for flexible cage dynamic characteristics of four-point contact ball bearing considering multi-point contact behaviors. Mechanical Systems and Signal Processing, 2025, 223: 111929.” analyzed the motion of the cage and revealed the influence of radial force and the importance of cage flexibility. However, research on misaligned FPCBBs is still limited. "Li Z, Wang C, Hu X, et al. Study on the effect of angular misalignment and axial preload on the mechanical properties of four-point angular contact ball bearings. Mechanism and Machine Theory 193 (2024) 105565." studied the contact and stiffness characteristics of FPCBBs with inner ring misalignment, and "Li X, Xu Y, Liu J, et al. Vibration analysis of the propulsion shaft system considering dynamic misalignment in the outer ring. Journal of Sound and Vibration, 2024, 589: 118612" analyzed the vibration response of a ship propulsion system with an outer ring tilt, but did not consider the initial rotor bending caused by outer ring misalignment. These studies reveal a key gap: although healthy bearings have been well studied, misaligned bearings still need further exploration. Installation errors inevitably affect the vibration response of the rotor, which in turn affects the contact characteristics of the bearing. In addition, the effects of key assembly parameters and operating conditions on vibration characteristics have not been deeply studied, so more comprehensive research is needed in this area.

[0007] Traditional rotor system dynamics modeling methods usually assume that the shaft is an ideal straight line, ignoring the bending effect of the shaft caused by bearing misalignment and preload. This simplified assumption may lead to significant errors in actual engineering, especially under high speed, heavy load or complex working conditions, the bending deformation of the shaft will significantly change the dynamic characteristics of the system, thereby causing distortion in the prediction of the dynamic response of the rotor system. Specifically, the bending of the shaft will not only affect the critical speed and modal vibration shape of the rotor, but may also induce nonlinear vibration phenomena such as subharmonic resonance or chaotic vibration, thereby reducing the prediction accuracy and engineering applicability of the model.

[0008] The disadvantage of the mechanical characterization of four-point contact ball bearings is that the main challenge currently faced by the mechanical characterization of four-point contact ball bearings is that the existing mechanical models usually do not fully consider the influence of the relative displacement of the inner and outer rings of the bearing caused by installation errors. These displacements may be caused by assembly errors, thermal deformation or external loads, which will significantly change the contact stress distribution and load transfer characteristics inside the bearing. However, there is currently a lack of accurate mechanical models that can effectively characterize these displacements, especially when considering complex working conditions (such as high-speed rotation, variable loads or temperature changes), the predictive ability of the model is limited. The accurate mechanical model of the four-point contact ball bearing is the basis for calculating the bending deformation displacement of the shaft, and the shortcomings of the existing model directly affect the accuracy of the dynamic analysis of the rotor system, especially in the fields of aerospace and high-end equipment with high precision requirements.

[0009] By introducing the consideration of the initial bending of the rotor into the dynamic model of the bearing-rotor system and verifying the effectiveness of the model through experiments, the present invention solves the problem of rotor bending caused by ignoring bearing misalignment in traditional models. Although some studies have considered the impact of bearing misalignment on system vibration, they usually assume that the rotor is healthy and ignore the significant impact of the initial bending of the rotor on the vibration response, which may lead to distortion in the prediction of the vibration response of the system. The present invention calculates the load caused by bearing misalignment by establishing a quasi-static model of a four-point contact ball bearing, and on this basis obtains the initial bending deformation of the rotor, and then analyzes the impact of the initial bending of the rotor on the vibration characteristics of the system. Summary of the invention

[0010] In order to solve the above problems, the present invention provides a dynamic modeling method for a bent rotor system caused by a misaligned bearing, which considers the multi-point contact characteristics of a four-point contact ball bearing and the rotation effect of the ball, establishes a quasi-static model considering the misalignment of a four-point contact ball bearing based on the Hertz contact theory and the outer raceway control theory, and uses the Newton-Raphson algorithm to calculate the force balance equation of the ball and the preload, i.e., the abnormal load, generated by the misalignment of the bearing; based on the obtained preload, introduces it into a finite element model of the rotor to calculate the initial bending deformation of the rotor; further, calculates the unbalanced load on the rotor in the initial bending state; finally, introduces the preload and the unbalanced load generated by the misalignment of the bearing into the finite element model of the rotor, and constructs a dynamic model of a bent rotor system caused by a misaligned bearing.

[0011] The technical solution of the present invention is as follows: a dynamic modeling method for a bending rotor system caused by misaligned bearings, firstly, a quasi-static model considering the misalignment of a four-point contact ball bearing is established to calculate the preload caused by the bearing misalignment; the preload caused by the bearing misalignment is introduced into the differential equation of motion to obtain the initial bending deformation of the rotor; based on the initial bending deformation of the rotor, the unbalanced load on the rotor in the initial bending state is calculated; at the same time, the preload caused by the bearing misalignment and the unbalanced load on the rotor are considered to establish a bearing-rotor system dynamic model considering the initial bending deformation of the rotor.

[0012] The process of establishing the quasi-static model considering the misalignment of the four-point contact ball bearing is as follows:

[0013] The four-point contact ball bearing comprises two semi-inner rings, an outer ring, rolling elements and a cage;

[0014] Consider the vibration displacement of the inner ring of a four-point contact ball bearing, including three translation displacements δ ix ,δ iy ,δ iz and two rotational displacements θ ix ,θ iy , the axial displacement Δa of the center of curvature of the inner raceway where the jth ball is located ij and the radial displacement Δr of the center of curvature of the inner raceway where the jth ball is located ij The expression is;

[0015]

[0016] In the formula, is the azimuth angle of the ball, R i is the trajectory radius of the inner raceway curvature center, and its expression is,

[0017] R i =0.5d m +(ri -0.5D w )cosα a

[0018] In the formula, r i is the inner raceway curvature radius, D w is the ball diameter, α a is the initial contact angle of the bearing;

[0019] d m is the pitch diameter of the bearing, and its expression is,

[0020] d m =0.5(d o1 +d i1 )

[0021] =0.5((d o -2(r o -l ab -D w / 2-(r o -D w / 2)cosα a ))+(d i -2(r i -l abi -D w / 2-(r i -D w / 2)cosα a )))In the formula, d o1 is the diameter of the highest point of the ball, d i1 is the lowest point diameter of the ball, r o is the outer raceway curvature radius, l ab is the distance between the outer groove tip after the gasket is removed and the outer groove bottom without the gasket removed, l abi It is the distance between the inner groove tip after the gasket is removed and the inner groove bottom before the gasket is removed;

[0022] d o1 The expression of is,

[0023] d o1 =d o -2l ef ,l ef =r o -l ab -D w / 2-(r o -D w / 2)cosα a

[0024] Where, d o is the outer groove tip diameter after removing the gasket, r ois the curvature radius of the outer raceway;

[0025] Similarly, d i1 The expression is,

[0026] d i1 =d i -2(r i -l abi -D w / 2-(r i -D w / 2)cosα a )

[0027] Where, d i is the inner groove tip diameter after removing the gasket, l abi It is the distance between the inner groove tip after the gasket is removed and the inner groove bottom before the gasket is removed;

[0028] Horizontal displacement A from the center of curvature of the inner raceway on the inside to the center of curvature of the outer raceway on the right izj and the vertical displacement A from the inner raceway curvature center to the right outer raceway curvature center ixj is written as,

[0029] A ozj =Δa oj +(r i +r o -D w )sinα a ,A oxj =(r i +r o -D w )cosα a +Δr oj

[0030] Δr oj and Δa oj are respectively the radial displacement and axial displacement of the center of curvature of the outer raceway where the jth ball is located;

[0031] Therefore, the contact deformation expression between the ball and the raceway is,

[0032]

[0033] In the formula, δ il is the contact deformation between the ball and the left inner raceway, δ ir is the contact deformation between the ball and the right inner raceway, δ ol is the contact deformation between the ball and the left outer raceway, δ or is the contact deformation between the ball and the right outer raceway; g i is the gasket thickness of the inner raceway, g ois the gasket thickness of the outer raceway, θ i is the equivalent deflection angle of the inner raceway, θ o is the equivalent deflection angle of the outer raceway; X ozj is the horizontal displacement from the center of the ball after contact deformation to the center of curvature of the right outer raceway before deformation, X xj is the vertical displacement from the center of the ball after contact deformation to the center of curvature of the right outer raceway before deformation, X izj A is the horizontal displacement from the center of the ball after contact deformation to the center of curvature of the right inner raceway before deformation, oxj is the vertical displacement from the center of curvature of the left outer raceway after contact deformation to the center of curvature of the right inner raceway before deformation;

[0034]

[0035] The contact angle between the ball and the inner raceway and the outer raceway is expressed as:

[0036]

[0037] α il is the contact angle between the ball and the inner raceway on the left, α ir is the contact angle between the ball and the inner raceway on the right, α ol is the contact angle between the ball and the left outer raceway, α or is the contact angle between the ball and the right outer raceway;

[0038] Based on Hertz contact theory, the contact force between the jth ball and raceway is expressed as:

[0039] Q ilj =K ilj (δ ilj ) 1.5 H(δ ilj ),Q irj =K irj (δ irj ) 1.5 H(δ irj )

[0040] Q olj =K olj (δ olj ) 1.5 H(δ olj ),Q orj =K orj (δ orj ) 1.5 H(δ orj )

[0041] In the formula, δ ilj is the contact deformation between the jth ball and the left inner raceway, δ irjis the contact deformation between the jth ball and the right inner raceway, δ olj is the contact deformation between the jth ball and the left outer raceway, δ orj is the contact deformation between the jth ball and the right outer raceway;

[0042] Q ilj , Q irj , Q olj , Q orj K is the contact force between the ball and the inner left half raceway, the contact force between the ball and the inner right half raceway, the contact force between the ball and the outer left half raceway, and the contact force between the ball and the outer right half raceway; ilj , K irj , K olj , K orj is the contact stiffness between the ball and the inner left half raceway, the contact stiffness between the ball and the inner right half raceway, the contact stiffness between the ball and the outer left half raceway, and the contact stiffness between the ball and the outer right half raceway; H() is the Heaviside function, and its expression is,

[0043]

[0044] The ball's orbital speed ω m and the rotation speed ω R for,

[0045]

[0046] α e,m is the contact angle between the ball and the outer raceway, including α ol and α or , α i,m is the contact angle between the ball and the inner raceway, including α il and α ir ;

[0047] Where, the ball velocity vector pitch angle β b The expression of is,

[0048]

[0049] In the formula, γ b is the ratio of ball diameter to bearing pitch diameter;

[0050] The centrifugal force F of the ball cj and gyroscopic moment M gj for,

[0051]

[0052] Where, X xj is the vertical displacement of the ball to the center of curvature of the outer raceway after deformation, m b is the mass of the ball, Jb is the moment of inertia of the ball, and its expression is:

[0053]

[0054] In the formula, ρ b is the density of the ball;

[0055] According to the local force balance condition of the rolling ball, the misalignment of the four-point contact ball bearing can be expressed as:

[0056]

[0057] Based on the analysis of the rolling ball, the force analysis of the inner ring of the bearing is carried out. According to Newton's law of motion, the preload caused by the misalignment of the bearing is expressed as:

[0058]

[0059] The initial bending deformation of the rotor is obtained as follows:

[0060] Based on the calculated preload caused by bearing misalignment, the static initial bending of the rotor is calculated as

[0061] r s =F a / K

[0062] In the formula, F a is the load vector including the bearing inner ring contact load, including F x 、F y 、F z 、M x 、M y , K is the stiffness of the shaft, r s is the initial bending at the rotating disk;

[0063] r d is the vibration displacement of the center of mass of the rotating shaft disk, r is the dynamic deflection, α r is the angle between the moving coordinate system and the static coordinate system, is the initial phase of imbalance, point B represents the initial center of mass position, θ r represents the phase angle of point B in the moving coordinate system, that is, the initial bending phase angle;

[0064] In the moving coordinate system, according to the relative motion theorem, the motion differential equation of the rotating disk is established as follows:

[0065]

[0066] In the formula, m d represents the mass of the rotating disk, c d and c irepresents the viscous external damping coefficient and the internal damping coefficient of the elastic axis, k d represents the bending stiffness of the rotor; and represents angular velocity and angular acceleration; ξ, Represents the translational displacement, velocity, and acceleration of the coordinate system; for the acceleration term: The term describes the inertial force along the ξ axis; Describes the component of the centrifugal force in the ξ direction caused by the rotation of the moving coordinate system. This item reflects the centrifugal force generated by the rotation of the moving coordinate system. is the Coriolis force term, which describes the effect of the motion in the η direction on the motion in the ξ direction due to the rotation of the moving coordinate system; Describe the inertial force caused by the angular acceleration of the rotation of the moving coordinate system; describes the effect of external damping, where represents the additional effect on damping due to the rotation of the moving coordinate system; Describes the internal damping effect; k(ξ-ξ s ) represents the restoring force provided by the bending stiffness of the rotor; e d is the imbalance distance, i.e. the distance from the centroid to the centroid;

[0067] ξ s and η s The expression of is,

[0068] ξ s =r s cosθ r ,η s =r s sinθ r

[0069] r s is the initial bending deformation displacement, θ r is the initial bending deformation phase;

[0070] Since the dynamic response is solved in the fixed coordinate system, the motion differential equation in the fixed coordinate system is constructed; the relationship between the fixed coordinate system and the moving coordinate system is,

[0071]

[0072] Therefore, the differential equation of motion in the fixed coordinate system is,

[0073]

[0074] Since x = r s cos(α r +θ r )+x d ,y=r ssin(α r +θ r )+y d , x d represents the vibration lateral displacement of the rotating shaft disk, y d Represents the vibration longitudinal displacement of the rotating shaft disk, and the vibration equation of the rotating shaft disk is established;

[0075]

[0076] Therefore, the rotating disk is regarded as a concentrated mass point, and the term on the right side of the above equation is regarded as the external load F unx and F uny , the expression is as follows:

[0077]

[0078] Considering the initial deformation caused by the tilted outer ring of the four-point contact ball bearing, the dynamic model of the bearing-rotor system is established; the initial bending of the rotor refers to the deviation between the geometric center line of each cross section of the rotor and the axis of rotation, and the mass eccentricity refers to the deviation between the centroid of each cross section and its geometric center line. The finite element method is used to establish the dynamic model of the bearing-rotor system, and the shaft is simulated by the Timoshenko beam unit and the shaft disk is simulated by the concentrated mass unit;

[0079] Nonlinear contact force is used to perform static modeling on angular contact ball bearings, deep groove ball bearings and cylindrical roller bearings. For angular contact ball bearings and deep groove ball bearings, the calculation formula for bearing contact load is:

[0080]

[0081] In the formula, k bp 、N bp ,δ jp They represent the contact stiffness of the angular contact ball bearing and the deep groove ball bearing, the number of rolling elements of the angular contact ball bearing and the deep groove ball bearing, and the contact deformation of the angular contact ball bearing and the deep groove ball bearing respectively; C b is the bearing contact stiffness, r dj The distance from the contact point between the ball and the raceway to the axis of rotation of the bearing; φ jp is the azimuth angle of the rolling element, expressed as,

[0082]

[0083] Where α0 represents the initial contact angle; d b and d m Indicates the ball diameter and bearing pitch diameter; r bd Indicates the radius of the inner raceway; R bd Indicates the radius of the outer raceway;

[0084] For cylindrical roller bearings, the bearing contact force F bxc and F byc Written as,

[0085]

[0086] In the formula, φ jc and δ jc Indicates the azimuth angle of the roller and the contact deformation between the roller and the raceway; φ jc The calculation method is the same as that of deep groove ball bearings; N bc Indicates the number of rollers; k bc represents the Hertz contact stiffness, which is expressed as follows:

[0087] k bc =8.06×10 4 × b 8 / 9

[0088] One side of the rotor is connected to the motor instead of the free end, so it is regarded as a spring constraint; the double-row self-aligning ball bearing is simplified as a linear spring; the three-dimensional linear spring element has six stiffness directions in the global coordinate system, namely the translation stiffness k along the three coordinate axes x , k y , k z and the angular stiffness k along the three coordinate axes θx , k θy , k θz ; The matrix expression of the linear spring element is as follows:

[0089]

[0090] Considering the unbalanced force, initial bending and bearing contact force of the rotor, the dynamic model of the bearing-rotor system is established. The motion differential equation of the bearing-rotor system is written as

[0091]

[0092] Where M, C, G, and K represent the total mass, damping, gyro, and stiffness matrices of the system; q is the displacement, velocity and acceleration vector; F un and F b is the vector of external load and bearing force, which is expressed as,

[0093]

[0094] F b =[…,F b4 ,…F b9 ,…F b15 ,…F b18 ,…]T

[0095]

[0096] Beneficial effects of the invention: The invention proposes a dynamic modeling method for a bent rotor system caused by misaligned bearings, and considers a dynamic model of initial rotor bending caused by bearing misalignment. The model fully considers the influence of bearing misalignment, including the influence of bearing misalignment on contact load and the influence of bearing misalignment on the initial state of the rotor, and can improve the ability to predict the vibration response of the rotor system.

[0097] (1) The present invention constructs a quasi-static model of the inner and outer rings of a four-point contact ball bearing taking into account axial, radial, and angular misalignment. In the model of the present invention, the influence of the inner and outer ring positions of the bearing on the contact characteristics of the bearing is fully considered, and the contact load between the ball and the raceway is accurately calculated.

[0098] (2) During the startup of the rotor system, due to the synchronous rotation of the bending deformation and the rotor, a lateral exciting force similar to the unbalanced force is generated, thereby exciting the positive coordinated synchronous lateral vibration of the rotor system, and resonance occurs near the critical speed, and even causes the rotor system to fail to start. The present invention considers the combined effect of rotor bending deformation and rotor imbalance, constructs a system dynamics model under the action of unbalanced load, and can effectively analyze the vibration response under different combinations of bending phase and unbalanced phase.

[0099] (3) In the dynamic calculation process of the bearing-rotor system, the influence of the bearing installation error on the vibration response is fully considered, which is helpful for the prediction and suppression of the vibration response. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] Figure 1 It is a schematic diagram of a four-point contact ball bearing;

[0101] Figure 2 It is a geometrical diagram of a four-point contact ball bearing;

[0102] Figure 3 Annotated diagram of the structure: (a) geometric relationship, (b) gap relationship;

[0103] Figure 4 is the position of the ball center and raceway curvature center with or without load;

[0104] Figure 5 For ball force analysis;

[0105] Figure 6 Schematic diagram of the initial bending of the shaft caused by bearing misalignment (taking Jeffcott rotor as an example): (a) initial misalignment of the shaft, (b) instantaneous position of the disk;

[0106] Figure 7 It is a multi-support rotor system; (a) is the dimension diagram, (b) is the finite element model;

[0107] Figure 8 The flow chart for the dynamic calculation of the bearing-rotor system;

[0108] Fig. 9 is the static displacement of the rotor;

[0109] Fig.10 Spectrum comparison; (a1) is the experimental spectrum of mild tilt misalignment, (a2) is the simulation spectrum of mild tilt misalignment; (b1) is the experimental spectrum of moderate tilt misalignment, (b2) is the simulation spectrum of moderate tilt misalignment; (c1) is the experimental spectrum of severe tilt misalignment, (c2) is the simulation spectrum of severe tilt misalignment.

[0110] Fig.11 is the amplitude change of the characteristic frequency, (a) is f r The change of amplitude; (b) is f vc Amplitude changes. DETAILED DESCRIPTION

[0111] The present invention first establishes a quasi-static model that takes into account the misalignment of a four-point contact ball bearing, which is used to calculate the preload generated by the misalignment of the bearing, and on this basis, obtains the initial bending deformation of the rotor. Subsequently, the present invention proposes a dynamic model of a bearing-rotor system that takes into account the initial bending deformation of the rotor, and verifies the accuracy of the dynamic model through vibration experiments. Through this model, the present invention reveals the harmonic frequencies and combined frequencies caused by the nonlinearity of the bearing, and analyzes the effects of the inclination angle of the bearing seat, the rotation speed, and the initial bending of the rotor on the contact stress of the bearing. The results show that the inclination angle of the bearing seat significantly changes the contact stress between the ball and the raceway, while the rotation speed and the initial bending of the rotor have little effect on the contact stress at a large inclination angle. In addition, the present invention also verifies the accuracy of the model through experiments, reveals the influence of the phase difference between the initial bending and imbalance of the rotor on the vibration of the rotor, and fills the deficiency of ignoring the initial bending of the rotor in existing research.

[0112] In order to make it easier for the angular contact ball to withstand large bidirectional axial forces and to facilitate the assembly of bearings in complex rotor systems, the inner ring of the double-half inner ring three-point contact ball bearing is a split-half structure. The processing method is to remove a gasket of a specific thickness from the overall inner ring to form a split-half bearing inner ring, such as Figure 1 As shown in the figure, the bearing consists of two half inner rings, an outer ring, rolling elements and a cage.

[0113] Due to the processing method of removing a gasket of a specific thickness from the entire inner ring, the center of curvature of the inner ring's groove will also shift to a certain extent, and form a gasket angle with the inner and outer grooves, such as Figure 2As shown. The thickness of the inner raceway gasket is 2e i , the thickness of the outer raceway gasket is 2e o The relationship between the pad width and the pad angle is,

[0114]

[0115] In the formula, α si , α so is the gasket angle of the inner groove and the gasket angle of the outer groove. i 、e o is the offset of the inner raceway curvature center and the outer raceway curvature center. i 、r o is the inner raceway curvature radius and the outer raceway curvature radius; D w is the ball diameter.

[0116] In the initial state, the ball and the outer ring will also produce a contact angle, which is recorded as the initial contact angle of the outer ring α o ,like Figure 2 (c) is shown. The change of clearance has a great influence on the initial contact angle. The increase of clearance will increase the initial contact angle. When the clearance increases to a certain extent in the static state, the external contact angle will be equal to the gasket angle. When the radial clearance continues to increase, the contact angle between the ball and the inner and outer rings will be greater than the gasket angle, and the normal single-point contact state of the inner ring is achieved. In order to avoid multi-point contact in the four-point contact ball bearing during operation, the condition that must be met is that the initial contact angle is greater than the gasket angle.

[0117] Initial contact angle of bearing α a The expression of is,

[0118]

[0119] In the formula, S d Represents the total radial clearance of the bearing. Its expression is,

[0120]

[0121] In the formula, l ab is the distance between the outer groove tip after the gasket is removed and the outer groove bottom without the gasket removed, l abi It is the distance between the inner groove tip after the gasket is removed and the inner groove bottom without the gasket removed. ed is the distance from the ball vertex to the outer groove tip, l edi is the distance from the bottom point of the ball to the tip of the inner groove. For ease of display, only the detailed geometric relationship diagram related to the outer groove is drawn, as shown in Figure Figure 3 As shown. o2 is the bottom diameter of the outer groove without removing the gasket, d i2 It is the bottom diameter of the inner groove without removing the gasket.o is the outer groove tip diameter after removing the gasket, d i It is the diameter of the inner groove tip after removing the gasket.

[0122] Consider the vibration displacement of the inner ring of the bearing, including three translation displacements δ ix ,δ iy ,δ iz and two rotational displacements θ ix ,θ iy , the axial displacement Δa of the center of curvature of the inner raceway where the jth ball is located ij and radial displacement Δr ij The expression is,

[0123]

[0124] In the formula, is the azimuth angle of the ball, R i is the trajectory radius of the inner raceway curvature center, and its expression is,

[0125] R i =0.5d m +(r i -0.5D w )cosα a (5)

[0126] Where, d m is the pitch diameter of the bearing, and its expression is,

[0127]

[0128] Where, d o1 and d i1 is the diameter of the highest and lowest points of the ball, d o1 As shown in Figure (a). o1 The expression of is,

[0129] d o1 =d o -2l ef ,l ef =r o -l ab -D w / 2-(r o -D w / 2)cosα a (7)

[0130] Similarly, d i1 The expression is,

[0131] d i1 =d i -2(r i -lcd -D w / 2-(r i -D w / 2)cosα a ) (8)

[0132] The positions of the ball center and the raceway curvature center with and without load are shown in the figure, where A ixj and A izj The expression is written as

[0133] A izj =(r i +r o -D w )sinα a +Δa ij ,A ixj =(r i +r o -D w )cosα a +Δr ij (9)

[0134] Consider the installation error of the bearing seat, including three translational displacement errors δ ox ,δ oy ,δ oz and two rotational displacement errors θ ox ,θ oy , the axial displacement Δa of the center of curvature of the outer raceway where the jth ball is located oj and radial displacement Δr oj The expression is

[0135]

[0136] In the formula, R o is the trajectory radius of the center of curvature of the outer raceway, and its expression is

[0137] R o =0.5d m -l cb1 =0.5d m -(r o -0.5D w )cosα a (11)

[0138] Figure 4 In, X izj , X ozj and X xj The expression is

[0139] X izj =e i +Δz bj ,Xozj =e o +Δz bj ,X xj =Δx bj +(0.5d m -R o ) (12)

[0140] A ozj and A oxj The expression is written as

[0141] A ozj =Δa oj +(r i +r o -D w )sinα a ,A oxj =(r i +r o -D w )cosα a +Δr oj (13)

[0142] Therefore, the contact deformation expression between the ball and the raceway is

[0143]

[0144] In the formula, δ il is the contact deformation between the ball and the left inner raceway, δ ir is the contact deformation between the ball and the right inner raceway, δ ol is the contact deformation between the ball and the left outer raceway, δ or is the contact deformation between the ball and the right outer raceway. g i and g o is the gasket thickness of the inner and outer raceways (see formula (15)), θ i and θ o is the equivalent deflection angle of the inner and outer raceways (see formula (16)).

[0145] g i =2e i ,g o =2e o (15)

[0146]

[0147] The contact angle between the ball and the inner and outer raceways is expressed as

[0148]

[0149] Based on Hertz contact theory, the contact force between the ball and the raceway is expressed as

[0150]

[0151] In the formula, Q ilj , Q irj , Q olj , Q orj K is the contact force between the ball and the inner left half raceway, inner right half raceway, outer left half raceway, and outer right half raceway. ilj , K irj , K olj , K orj is the contact stiffness between the ball and the inner left half raceway, inner right half raceway, outer left half raceway, and outer right half raceway. H() is the Heaviside function, and its expression is

[0152]

[0153] The ball's orbital speed ω m and the rotation speed ω R for

[0154]

[0155] Where, the ball velocity vector pitch angle β b The expression is

[0156]

[0157] In the formula, γ b It is the ratio of the ball diameter to the bearing pitch diameter.

[0158] Based on formula (20-22), the centrifugal force F of the ball cj and gyroscopic moment M gj for

[0159]

[0160] In the formula, m b is the mass of the ball, J b is the moment of inertia of the ball, and its expression is

[0161]

[0162] In the formula, ρ b is the density of the ball.

[0163] The force state of the ball is as follows Figure 5 As shown in Figure 2, according to the local force balance condition of the steel ball, the force balance equation of the steel ball can be expressed as

[0164]

[0165] Based on the analysis of the steel ball, the force analysis of the inner ring of the bearing is carried out. According to Newton's law of motion, the inner ring load can be expressed as

[0166]

[0167] Therefore, the stiffness matrix expression of the bearing is

[0168]

[0169] For the main shaft bearing of an aircraft engine, high speed is one of its main characteristics, so the outer raceway control theory used in the present invention to analyze the movement of the ball is more in line with the actual movement state. orj , olj is equal to 2 and λ irj , ilj is equal to 0.

[0170] Due to the abnormal preload caused by bearing misalignment, the shaft will produce initial bending, such as Figure 6 (a) is shown. Figure 6 (b), r s is the deflection of the shaft at initial bending, r d is the vibration displacement of the center of mass, r is the dynamic deflection, α r is the angle between the moving coordinate system and the static coordinate system, is the initial phase of imbalance, point B represents the initial center of mass position, θ r It represents the phase angle of point B in the moving coordinate system, that is, the initial bending phase angle.

[0171] In the moving coordinate system, according to the relative motion theorem, the motion differential equation of the rotating disk is established as

[0172]

[0173] In the formula, m d represents the mass of the rotating disk, c d and c i represents the viscous external damping coefficient and the internal damping coefficient of the elastic axis, k d represents the bending stiffness of the elastic shaft. and represents angular velocity and angular acceleration. ξ、 Represents the translational displacement, velocity, and acceleration of the coordinate system. For the acceleration term: The term describes the inertial force along the ξ axis; Describes the component of the centrifugal force in the ξ direction caused by the rotation of the moving coordinate system. This item reflects the centrifugal force generated by the rotation of the moving coordinate system. is the Coriolis force term, which describes the effect of the motion in the η direction on the motion in the ξ direction due to the rotation of the moving coordinate system; Describes the inertial force caused by the angular acceleration of the rotation of the moving coordinate system. describes the effect of external damping, where represents the additional effect on damping due to the rotation of the moving coordinate system; Describes the internal damping effect. k(ξ-ξ s ) represents the restoring force provided by the bending stiffness of the elastic axis.

[0174] ξ s and η s The expression is

[0175] ξ s =r s cosθ r ,η s =r s sinθ r (29)

[0176] Since the dynamic response is solved in a fixed coordinate system, it is necessary to construct the differential equation of motion in the fixed coordinate system. The relationship between the fixed coordinate system and the moving coordinate system is

[0177]

[0178] Therefore, the differential equation of motion in the fixed coordinate system is

[0179]

[0180] Since x = r s cos(α r +θ r )+x d ,y=r s sin(α r +θ r )+y d (x d ,y d represents the vibration displacement of the shaft disk), so far the vibration equation of the bearing disk can be established.

[0181]

[0182] Therefore, the rotating disk is regarded as a concentrated mass point, and the term on the right side of equation (32) is regarded as the external load F unx and F uny , the expression is as follows:

[0183]

[0184] Considering the initial deformation caused by the tilted outer ring of the four-point contact ball bearing, a dynamic model of the bearing-rotor system is established. The initial bending of the rotor refers to the deviation between the geometric center line of each cross section of the rotor and the axis of rotation, while the mass eccentricity refers to the deviation between the center of mass of each cross section and its geometric center line. The vibration characteristics of the two are roughly similar. Therefore, the phase difference between the initial bending and the mass eccentricity is a key parameter affecting the dynamic response. The present invention uses an experimental bearing-rotor system (see Figure 7 (a)) is the research object, revealing some vibration characteristics considering initial bending. The present invention uses the finite element method to establish the dynamic model of the bearing-rotor system (see Figure 7 (b)), where the shaft is simulated using Timoshenko beam elements and the disk is simulated using concentrated mass elements.

[0185] The present invention uses nonlinear contact force to perform static modeling on angular contact ball bearings, deep groove ball bearings and cylindrical roller bearings. For angular contact ball bearings and deep groove ball bearings, the calculation formula for the bearing contact load is:

[0186]

[0187] In the formula, k bp 、N bp , δ jp (p stands for angular contact ball bearing or deep groove ball bearing) represents the contact stiffness, number of rolling elements and contact deformation of angular contact ball bearing and deep groove ball bearing respectively. jp is the azimuth angle of the rolling element, expressed as

[0188]

[0189] Where α0 represents the initial contact angle. b and d m Indicates the ball diameter and bearing pitch diameter. bd and R bd Indicates the radius of the inner and outer raceways.

[0190] For cylindrical roller bearings, the bearing contact force F bxc and F byc It can be written as

[0191]

[0192] In the formula, φ jc and δ jc Indicates the roller azimuth and the contact deformation between the roller and the raceway. jc The calculation method is the same as that of deep groove ball bearings. bc Indicates the number of rollers. k bc represents the Hertz contact stiffness, which is expressed as

[0193] k bc =8.06×10 4 × b 8 / 9 (37)

[0194] One side of the rotor is connected to the motor instead of the free end and is therefore considered a spring constraint, e.g. Figure 7 (b) In order to reduce the complexity of the model and improve the calculation speed, the double-row self-aligning ball bearing is simplified into a linear spring. This strategy ignores the nonlinear effect of the bearing while ensuring the bearing support effect. The three-dimensional spring unit has six stiffness directions in the global coordinate system, namely, the translation stiffness k along the three coordinate axes. x , k y , k z and the angular stiffness k along the three coordinate axes θx , k θy , k θz The matrix expression of the spring element is as follows:

[0195]

[0196] Considering the unbalanced force, initial bending and bearing contact force of the shaft, the dynamic model of the bearing-rotor system is established (see Figure Figure 7 (b)), the differential equation of motion of the system is written as

[0197]

[0198] where M, C, G, and K represent the total mass, damping, gyro, and stiffness matrices of the system. q is the displacement, velocity and acceleration vector. F un and F b is the vector of external load and bearing force, which is expressed as

[0199]

[0200]

[0201] In addition, the key parameter phase difference is defined as

[0202]

[0203] The damping matrix is ​​expressed as

[0204] C=α d M+β d K, (43)

[0205] Among them, α d and β d Represents the proportionality factor.

[0206] First, the abnormal contact load of the four-point contact ball bearing with an inclined outer ring is calculated using the quasi-static model. Then the calculated load is substituted into the differential equation of motion and the static initial bending deformation is solved. s , the Newmark-β method is used to calculate the dynamic response of the system. The detailed calculation process is as follows Figure 8 shown.

[0207] In order to calculate and verify the initial deformation of the shaft, the same model was established in ROMAX software. The bearing parameters are shown in Table 1.

[0208] Table 1 Bearing parameters

[0209]

[0210] In addition, Table 2 lists the stiffness values ​​of the equivalent spring in different directions.

[0211] Table 2 Spring stiffness values

[0212] When solving / verifying the static initial displacement, the abnormal contact load of the four-point contact ball bearing is assumed to be F x =100N, F y =100N,M x =100N·m,M y =100N·m. The calculation result is as follows Fig. 9 As shown, the calculation results of the model proposed in the present invention are in good agreement with the calculation results of the Romax software, proving the accuracy of the initial displacement calculated by the present invention. This lays the foundation for the subsequent dynamic response calculation.

[0213] In order to verify the proposed dynamic model of the four-point contact ball bearing-rotor system, a dedicated bearing test bench was built to measure the vibration response of the bearing seat. The tilt angle of the bearing seat can be quantitatively adjusted by shaking the handle. By placing an angle measurement sensor at the bottom of the bearing seat, the tilt angle of the bearing seat can be displayed in real time in the control cabinet. Three acceleration sensors are arranged in the bearing seat in radial, vertical and axial directions. The vertical preload is applied by shaking the handle, and the preload force is measured by the force sensor. The adjusted tilt angle represents the misalignment of the bearing seat, not the actual tilt angle of the bearing outer ring, because of the clearance fit assembly method between the base and the outer ring. The tilt angle first needs to compensate for the fit clearance, and then causes the outer ring to misalign. Therefore, it is impossible to quantitatively verify the dynamic model of the bearing-rotor system, so the present invention adopts a qualitative analysis of the dynamic response under three different tilt degrees. In the simulation process, 0.1° corresponds to a mild misalignment state, 0.12° corresponds to a moderate misalignment state, and 0.14° corresponds to a severe misalignment state.

[0214] When the vertical preload is 500N, the speed is 2000rev / min, and the unbalance is 4000g / mm, the simulated and experimental vibration responses at different inclination degrees are as follows: Fig.10 There are some characteristic frequencies in the spectrum diagram, including the rotation frequency f r , variable flexible vibration frequency f vc and its frequency multiples (2f r 、3f r , 2f vc ) and the combined frequency (f vc ±f r 、f vc ±2f r ). These same characteristic frequencies verify the bearing-rotor system dynamics model proposed in the present invention.

[0215] Frequency f r and f vc The amplitude changes as Fig.11 As shown. Fig.11 In (b), the frequency f vc The amplitude of is always monotonically increasing. However, Fig.11 In (a), the frequency f in simulation and experiment r The amplitude variation pattern of is not completely consistent. The possible reasons for this error can be summarized as follows: (1) When adjusting the tilt angle, reinstalling the bearing seat may cause a change in the bearing preload state. (2) During the experiment, the vibration signal is collected from the bearing seat. In the simulation, the vibration signal comes from the shaft node corresponding to the inner ring of the bearing. In general, the characteristic frequency f vc The same change rule of further verifies the model proposed by the present invention.

[0216] The experimental results show that the initial bending of the rotor significantly increases the vibration displacement of the rotor, and the inclination angle of the bearing seat has a significant effect on the contact stress between the ball and the raceway, while the speed and the initial bending of the rotor have little effect on the contact stress at a large inclination angle. In addition, the present invention also reveals the harmonic frequencies and combined frequencies caused by the nonlinearity of the bearing, and verifies the accuracy of the model through experiments. It is worth noting that the present invention not only focuses on the influence of bearing misalignment on the vibration of the system, but also further reveals the complex interaction between the rotor vibration and the bearing contact characteristics by introducing the phase difference analysis of the initial bending of the rotor, providing new theoretical support for the design and fault diagnosis of the bearing-rotor system.

Claims

1. A dynamic modeling method for a bending rotor system caused by misaligned bearings, characterized in that: Firstly, a quasi-static model considering the misalignment of four-point contact ball bearings is established to calculate the preload caused by bearing misalignment; the preload caused by bearing misalignment is introduced into the differential equation of motion to obtain the initial bending deformation of the rotor; based on the initial bending deformation of the rotor, the unbalanced load on the rotor in the initial bending state is calculated; at the same time, the preload caused by bearing misalignment and the unbalanced load on the rotor are considered, and a bearing-rotor system dynamics model considering the initial bending deformation of the rotor is established.

2. The method for dynamic modeling of a bending rotor system caused by misaligned bearings according to claim 1 is characterized in that: The process of establishing the quasi-static model considering the misalignment of the four-point contact ball bearing is as follows: The four-point contact ball bearing comprises two semi-inner rings, an outer ring, rolling elements and a cage; Consider the vibration displacement of the inner ring of a four-point contact ball bearing, including three translation displacements δ ix ,δ iy ,δ iz and two rotational displacements θ ix ,θ iy , the axial displacement △a of the center of curvature of the inner raceway where the jth ball is located ij and the radial displacement △r of the center of curvature of the inner raceway where the jth ball is located ij The expression is; In the formula, is the azimuth angle of the ball, R i is the trajectory radius of the inner raceway curvature center, and its expression is, R i =0.5d m +(r i -0.5D w )cosα a In the formula, r i is the inner raceway curvature radius, D w is the ball diameter, α a is the initial contact angle of the bearing; d m is the pitch diameter of the bearing, and its expression is, d m =0.5(d o1 +d i1 ) =0.5((d o -2(r o -l ab -D w / 2-(r o -D w / 2)cosα a ))+(d i -2(r i -l abi -D w / 2-(r i -D w / 2)cosα a )))In the formula, d o1 is the diameter of the highest point of the ball, d i1 is the lowest point diameter of the ball, r o is the outer raceway curvature radius, l ab is the distance between the outer groove tip after the gasket is removed and the outer groove bottom without the gasket removed, l abi It is the distance between the inner groove tip after the gasket is removed and the inner groove bottom before the gasket is removed; d o1 The expression of is, d o1 =d o -2l ef ,l ef =r o -l ab -D w / 2-(r o -D w / 2)cosα a Where, d o is the outer groove tip diameter after removing the gasket, r o is the outer raceway curvature radius; Similarly, d i1 The expression is, d i1 =d i -2(r i -l abi -D w / 2-(r i -D w / 2)cosα a ) Where, d i is the inner groove tip diameter after removing the gasket, l abi It is the distance between the inner groove tip after the gasket is removed and the inner groove bottom before the gasket is removed; Horizontal displacement A from the center of curvature of the inner raceway on the inside to the center of curvature of the outer raceway on the right izj and the vertical displacement A from the inner raceway curvature center to the right outer raceway curvature center ixj Written as, A ozj =△a oj +(r i +r o -D w )sinα a ,A oxj =(r i +r o -D w )cosα a +△r oj △r oj and △a oj are the radial displacement and axial displacement of the center of curvature of the outer raceway where the jth ball is located; therefore, the contact deformation expression between the ball and the raceway is, In the formula, δ il is the contact deformation between the ball and the left inner raceway, δ ir is the contact deformation between the ball and the right inner raceway, δ ol is the contact deformation between the ball and the left outer raceway, δ or is the contact deformation between the ball and the right outer raceway; g i is the gasket thickness of the inner raceway, g o is the gasket thickness of the outer raceway, θ i is the equivalent deflection angle of the inner raceway, θ o is the equivalent deflection angle of the outer raceway; X ozj is the horizontal displacement from the center of the ball after contact deformation to the center of curvature of the right outer raceway before deformation, X xj is the vertical displacement from the center of the ball after contact deformation to the center of curvature of the right outer raceway before deformation, X izj A is the horizontal displacement from the center of the ball after contact deformation to the center of curvature of the right inner raceway before deformation, oxj is the vertical displacement from the center of curvature of the left outer raceway after contact deformation to the center of curvature of the right inner raceway before deformation; The contact angle between the ball and the inner raceway and the outer raceway is expressed as: α il is the contact angle between the ball and the inner raceway on the left, α ir is the contact angle between the ball and the inner raceway on the right, α ol is the contact angle between the ball and the left outer raceway, α or is the contact angle between the ball and the right outer raceway; Based on Hertz contact theory, the contact force between the jth ball and raceway is expressed as: Q ilj =K ilj (d ilj ) 1.5 H(d ilj ),Q irj =K irj (d irj ) 1.5 H(d irj ) Q olj =K olj (d olj ) 1.5 H(d olj ),Q orj =K orj (d orj ) 1.5 H(d orj ) In the formula, δ ilj is the contact deformation between the jth ball and the left inner raceway, δ irj is the contact deformation between the jth ball and the right inner raceway, δ olj is the contact deformation between the jth ball and the left outer raceway, δ orj is the contact deformation between the jth ball and the right outer raceway; Q ilj , Q irj , Q olj , Q orj K is the contact force between the ball and the inner left half raceway, the contact force between the ball and the inner right half raceway, the contact force between the ball and the outer left half raceway, and the contact force between the ball and the outer right half raceway; ilj , K irj , K olj , K orj is the contact stiffness between the ball and the inner left half raceway, the contact stiffness between the ball and the inner right half raceway, the contact stiffness between the ball and the outer left half raceway, and the contact stiffness between the ball and the outer right half raceway; H() is the Heaviside function, and its expression is, The ball's orbital speed ω m and the rotation speed ω R for, α e,m is the contact angle between the ball and the outer raceway, including α ol and α or , α i,m is the contact angle between the ball and the inner raceway, including α il and α ir ; Where, the ball velocity vector pitch angle β b The expression of is, In the formula, γ b is the ratio of ball diameter to bearing pitch diameter; The centrifugal force F of the ball cj and gyroscopic moment M gj for, M gj =J b oh m oh R sinβ b Where, X xj is the vertical displacement of the ball to the center of curvature of the outer raceway after deformation, m b is the mass of the ball, J b is the moment of inertia of the ball, and its expression is: In the formula, ρ b is the density of the ball; According to the local force balance condition of the rolling ball, the quasi-static model considering the misalignment of the four-point contact ball bearing is expressed as:

3. The method for dynamic modeling of a bending rotor system caused by misaligned bearings according to claim 2 is characterized in that: Based on the analysis of the rolling ball, the force analysis of the inner ring of the bearing is carried out. According to Newton's law of motion, the preload caused by the misalignment of the bearing is expressed as:

4. The method for dynamic modeling of a bending rotor system caused by misaligned bearings according to claim 3 is characterized in that: The initial bending deformation of the rotor is obtained as follows: Based on the calculated preload caused by bearing misalignment, the static initial bending of the rotor is calculated as r s =F a / K In the formula, F a is the load vector including the bearing inner ring contact load, including F x 、F y 、F z 、M x 、M y , K is the stiffness of the shaft, r s is the initial bending at the rotating disk; r d is the vibration displacement of the center of mass of the rotating shaft disk, r is the dynamic deflection, α r is the angle between the moving coordinate system and the static coordinate system, is the initial phase of imbalance, point B represents the initial center of mass position, θ r represents the phase angle of point B in the moving coordinate system, that is, the initial bending phase angle; In the moving coordinate system, according to the relative motion theorem, the motion differential equation of the rotating disk is established as follows: In the formula, m d represents the mass of the rotating disk, c d and c i represents the viscous external damping coefficient and the internal damping coefficient of the elastic axis, k d represents the bending stiffness of the rotor; and represents angular velocity and angular acceleration; ξ, Represents the translational displacement, velocity, and acceleration of the coordinate system; for the acceleration term: The term describes the inertial force along the ξ axis; Describes the component of the centrifugal force in the ξ direction caused by the rotation of the moving coordinate system. This item reflects the centrifugal force generated by the rotation of the moving coordinate system. is the Coriolis force term, which describes the effect of the motion in the η direction on the motion in the ξ direction due to the rotation of the moving coordinate system; Describe the inertial force caused by the angular acceleration of the rotation of the moving coordinate system; describes the effect of external damping, where represents the additional effect on damping due to the rotation of the moving coordinate system; Describes the internal damping effect; k(ξ-ξ s ) represents the restoring force provided by the bending stiffness of the rotor; e d is the imbalance distance, i.e. the distance from the centroid to the centroid; ξ s and η s The expression of is, x s =r s cosθ r ,or s =r s sinth r Since the dynamic response is solved in the fixed coordinate system, the motion differential equation in the fixed coordinate system is constructed; the relationship between the fixed coordinate system and the moving coordinate system is, Therefore, the differential equation of motion in the fixed coordinate system is, Since x = r s cos(α r +θ r )+x d ,y=r s sin(α r +θ r )+y d , x d represents the vibration lateral displacement of the rotating shaft disk, y d Represents the vibration longitudinal displacement of the rotating shaft disc, and the vibration equation of the disc is established; Therefore, the rotating disk is regarded as a concentrated mass point, and the term on the right side of the above equation is regarded as the external load F unx and F uny , the expression is as follows:

5. The method for dynamic modeling of a bending rotor system caused by misaligned bearings according to claim 4 is characterized in that: Considering the initial deformation caused by the tilted outer ring of the four-point contact ball bearing, the dynamic model of the bearing-rotor system is established; the initial bending of the rotor refers to the deviation between the geometric center line of each cross section of the rotor and the axis of rotation, and the mass eccentricity refers to the deviation between the centroid of each cross section and its geometric center line. The finite element method is used to establish the dynamic model of the bearing-rotor system, and the shaft is simulated by the Timoshenko beam unit and the disk is simulated by the concentrated mass unit; Nonlinear contact force is used to perform static modeling on angular contact ball bearings, deep groove ball bearings and cylindrical roller bearings. For angular contact ball bearings and deep groove ball bearings, the calculation formula for bearing contact load is: In the formula, k bp 、N bp ,δ jp They represent the contact stiffness of the angular contact ball bearing and the deep groove ball bearing, the number of rolling elements of the angular contact ball bearing and the deep groove ball bearing, and the contact deformation of the angular contact ball bearing and the deep groove ball bearing respectively; C b is the bearing contact stiffness, r dj The distance from the contact point between the ball and the raceway to the axis of rotation of the bearing; φ jp is the azimuth angle of the rolling element, expressed as, Where α0 represents the initial contact angle; d b and d m Indicates the ball diameter and bearing pitch diameter; r bd Indicates the radius of the inner raceway; R bd Indicates the radius of the outer raceway; For cylindrical roller bearings, the bearing contact force F bxc and F byc Written as, In the formula, φ jc and δ jc Indicates the azimuth angle of the roller and the contact deformation between the roller and the raceway; φ jc The calculation method is the same as that of deep groove ball bearings; N bc Indicates the number of rollers; k bc represents the Hertz contact stiffness, which is expressed as follows: k bc ＝8.06×10 4 ×l b 8 / 9 One side of the rotor is connected to the motor instead of the free end, so it is regarded as a spring constraint; the double-row self-aligning ball bearing is simplified as a linear spring; the three-dimensional linear spring element has six stiffness directions in the global coordinate system, namely the translation stiffness k along the three coordinate axes x , k y , k z and the angular stiffness k along the three coordinate axes θx , k θy , k θz ; The matrix expression of the linear spring element is as follows: Considering the unbalanced force, initial bending and bearing contact force of the rotor, the dynamic model of the bearing-rotor system is established. The motion differential equation of the bearing-rotor system is written as Where M, C, G, and K represent the total mass, damping, gyro, and stiffness matrices of the system; q is the displacement, velocity and acceleration vector; F un and F b is the vector of external load and bearing force, which is expressed as,

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