A method for analyzing vibration characteristics of a misaligned bearing-rotor system under a base motion

Abstract

The present application belongs to the technical field of bearing-rotor system vibration characteristic analysis, and discloses a vibration characteristic analysis method of misaligned bearing-rotor system under foundation motion. The rotating shaft is discretized into multiple beam elements; the kinetic energy expression and the potential energy expression of the beam elements under the foundation motion are obtained; the stiffness matrix, the mass matrix and the additional effect matrix of the elements are obtained after the kinetic energy expression and the potential energy expression are substituted into the Lagrange equation; the corresponding matrix of the rotating shaft is obtained. The disc is considered as a concentrated mass point, and the same modeling method as the beam element is adopted to obtain the mass matrix and the additional effect matrix of the concentrated mass point. The influence of the misalignment of the angular contact bearing on the bearing contact is considered; the contact angle of the ball under the misalignment state and the clearance between the ball and the raceway are solved based on the function method; the contact deformation between the ball and the raceway is calculated on the basis of further considering the displacement of the inner and outer rings of the bearing, and finally the supporting force of the misaligned angular contact bearing is obtained.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of bearing-rotor system vibration characteristic analysis, and particularly relates to a vibration characteristic analysis method of misaligned bearing-rotor system under foundation motion. BACKGROUND

[0002] In rotating machinery, bearings as the main supporting components, their performance is crucial to the overall stability and life of the system. However, rotating machinery is not only affected by internal unbalance excitation, but also often disturbed by foundation motion excitation. Foundation motion may be caused by external environmental factors such as sea waves of ships, maneuvering flight of aircrafts, and rugged ground of vehicles. This motion exerts additional forces on the bearing, especially the axial load, which changes the contact characteristics of the bearing and may shorten its service life. Existing researches mainly focus on the vibration characteristics under fixed foundation, ignoring the influence of foundation motion on the bearing and rotor system. Therefore, it is of great significance to study the influence of foundation motion on the nonlinear vibration of rotor-bearing system, which not only optimizes the system design, but also provides guidance for the reliable operation of the equipment.

[0003] At the same time, the misalignment of bearings caused by bearing assembly errors is also an important factor affecting the performance of rotating machinery. Existing researches have shown that the load characteristics and vibration responses of different types of bearings such as deep groove ball bearings and cylindrical roller bearings change significantly under misalignment, especially angular contact ball bearings under the action of axial force. However, most current researches only consider the influence of a single factor on the system, and few studies have simultaneously studied the dynamic model of foundation motion and bearing misalignment. Therefore, developing a dynamic model that considers foundation motion and bearing misalignment and analyzing its influence on vibration characteristics and contact characteristics will provide valuable theoretical basis for the optimization design of bearing-rotor system. SUMMARY

[0004] The present application aims to solve the problem of lacking modeling methods for angular contact bearing misalignment in the dynamic analysis technology of misaligned bearing-rotor system under foundation motion. A vibration characteristic analysis method of misaligned bearing-rotor system under foundation motion is proposed, which is a method for establishing the dynamic simulation model of bearing-rotor system under foundation motion. This method considers the additional effect matrix of shaft and disc caused by foundation motion, and establishes a contact force model of misaligned bearing based on the deformation coordination relationship of bearing. Combined with unbalanced force, additional load caused by foundation motion and bearing force, a dynamic simulation model of misaligned bearing-rotor system under foundation motion is established.

[0005] The technical scheme of the present application is as follows: a vibration characteristic analysis method of misaligned bearing-rotor system under foundation motion, comprising the following steps:

[0006] S1. According to the lumped mass method, a disc is established by using a lumped mass unit; kinetic energy expression and potential energy expression of the lumped mass unit under the basic motion are obtained; after the kinetic energy expression and the potential energy expression are substituted into the Lagrange equation, the stiffness matrix, the mass matrix and the additional effect matrix of the lumped mass unit are obtained;

[0007] S2. According to the Timoshenko beam unit theory, a rotating shaft is established by using a Timoshenko beam unit; kinetic energy expression and potential energy expression of the beam unit under the basic motion are obtained; after the kinetic energy expression and the potential energy expression are substituted into the Lagrange equation, the stiffness matrix, the mass matrix and the additional effect matrix of the beam unit are obtained;

[0008] The lumped mass unit and the beam unit are group lumped units; the stiffness matrix, the mass matrix and the additional effect matrix corresponding to the rotating shaft are obtained according to the stiffness matrix, the mass matrix and the additional effect matrix of the group lumped units;

[0009] S3. The influence of the misalignment of the angular contact bearing on the bearing contact is considered, the ball contact angle and the clearance between the ball and the raceway under the misalignment state are solved based on the function method, the contact deformation between the ball and the raceway is calculated on the basis of further considering the displacement of the inner and outer rings of the bearing, and finally the supporting force of the misaligned angular contact bearing is obtained;

[0010] S4. On the basis of the stiffness matrix, the mass matrix and the additional effect matrix of the rotating shaft / disk obtained, the supporting force of the misaligned bearing is introduced, and finally the dynamic model of the misaligned bearing-rotor system under the basic motion is established.

[0011] In the step S1, the mass matrix and the additional effect matrix of the lumped mass unit are obtained as follows:

[0012] S1.1 The translational kinetic energy and the rotational kinetic energy of the lumped mass unit under the basic motion state are

[0013]

[0014] Wherein, T t is the translational kinetic energy of the lumped mass unit, m d is the mass of the disc, v ord is the translational velocity vector of the lumped mass unit, is the translational velocity of the basic motion coordinate system in the global coordinate system, is the translational velocity of the lumped mass unit in the basic motion coordinate system, is the rotational velocity of the basis, and r is the displacement vector of the lumped mass unit in the basic motion coordinate system;

[0015]

[0016] Wherein, is the velocity matrix; T r is the rotational kinetic energy of the lumped mass unit, I d , I p is the rotational inertia and polar moment of inertia of the disc, a, b, g represent the Euler angles of the coordinate transformation and the superscript dot represents the angular velocity corresponding to the Euler angles, a represents the Euler angles;

[0017] S1.2 According to the translational kinetic energy expression and the rotational kinetic energy expression obtained in step S1.1, the total kinetic energy expression is:

[0018] T = T t + T r , (3)

[0019] wherein T is the total kinetic energy;

[0020] S1.3 According to the total kinetic energy obtained in step S1.2, the mass matrix and the additional effect matrix of the lumped mass unit are obtained based on the Lagrange equation:

[0021]

[0022] wherein q i , represents the generalized displacement and the generalized velocity; Q i represents the generalized force; U represents the potential energy;

[0023] Since a = q x , b = q y , g = q z + Ωt, the motion differential equation of the lumped mass unit is

[0024]

[0025] wherein Ω represents the rotational speed of the disc, M d , G d represent the mass matrix and the gyro matrix of the disc, q Bx , represent the angular displacement, angular velocity, angular acceleration of the disc around the x-axis, q By , represent the angular displacement, angular velocity, angular acceleration of the disc around the y-axis, q Bz , represent the angular displacement, angular velocity, angular acceleration of the disc around the z-axis; q d , represent the displacement vector, velocity vector, acceleration vector of the disc shaft; C Bzd , C Byd , C Bxd represent the additional damping matrix caused by the rotation of the foundation motion around the x, y, z axes; C dz , Cdy , C dx , H dz , H dy , H dx , K rd , K Bzd , K Bxd , K Byd , K Bxyd , K Byzd , K Bxzd represents the additional stiffness matrix caused by the basic motion; F Bd represents the additional load vector caused by the basic motion.

[0026] In the step S2, the stiffness matrix, mass matrix and additional effect matrix of the beam element are obtained as follows:

[0027] S2.1 The kinetic energy expression and potential energy expression of the Timoshenko beam element in the basic motion state are as follows:

[0028]

[0029] wherein, is the potential energy of the beam element, is the kinetic energy of the beam element, is the diameter moment of inertia and polar moment of inertia of the unit per unit length, A represents the cross-sectional area, ρ is the density of the unit, E and G represent the elastic modulus and Poisson's ratio, J represents the torsional section inertia, I x , I y respectively represent the section moment of inertia around the x-axis and the section moment of inertia around the y-axis, κ x , κ y represent the shear correction coefficient around the x-axis and the shear correction coefficient around the y-axis; u, v, w are the displacements of an arbitrary cross section in the x, y, z axes relative to the origin, θ, φ, represent the angular displacement of an arbitrary cross section, l k represents the length of the beam element; u', v', w' are the velocity of an arbitrary cross section in the x, y, z axes relative to the origin; θ', φ', represent the angular velocity of an arbitrary cross section;

[0030] S2.2 According to the kinetic energy expression and potential energy expression obtained in step S2.1, the motion differential equation of the beam element is obtained after substituting into the Lagrange equation as follows:

[0031]

[0032] wherein, represents the acceleration vector, velocity vector and displacement vector of the beam element; represents the mass matrix, gyro matrix and stiffness matrix of the beam element; CBzdsh 、C Bydsh 、C Bxdsh represents the additional damping matrix caused by the foundation motion; C dzsh 、C dysh 、C dxsh 、H dzsh 、H dysh 、H dxsh , K rdsh , K Bzdsh , K Bxdsh , K Bydsh , K Bxydsh , K Byzdsh , K Bxzdsh Represents the additional stiffness matrix caused by foundation motion; C1, K1, K2, K3 are intermediate variables.

[0033] In step S3, the supporting force of the misaligned angular contact bearing is obtained as follows:

[0034] S3.1 The coordinate expression of the raceway curvature center of an angular contact bearing in a healthy state is:

[0035]

[0036] Among them, f i 、f o Indicates the curvature coefficient of the inner raceway and the outer raceway, r b Indicates the ball radius, r p represents the bearing pitch radius, α0 represents the initial contact angle between the ball and the raceway, and x i 、z i Indicates the coordinate of the center of curvature of the inner raceway; x o 、z o Indicates the coordinates of the center of curvature of the outer raceway;

[0037] S3.2 Based on the coordinates of the raceway curvature center obtained in step S3.1, taking into account the rotor vibration and axial, radial and angular assembly errors, the equivalent misalignment is expressed as:

[0038]

[0039] Among them, α ieq , α oeq Indicates the equivalent misalignment angle of the inner ring and the outer ring, α ix , α iy Indicates the inclination angle of the inner circle around the x and y axes; α ox , α oy The tilt angle of the outer ring around the x and y axes, θ ix ,θ iy Indicates the angular displacement of the inner ring around the x and y axes due to vibration; θ ox, θ oy denote the angular displacement of the outer ring around the x, y axes due to vibration, e iaeq , e oaeq denote the equivalent axial misalignment distance of the inner ring, the equivalent axial misalignment distance of the outer ring, e ia , e oa denote the static axial misalignment distance of the inner ring, the static axial misalignment distance of the outer ring, z ia , z oa denote the axial misalignment displacement due to vibration, e ireq , e oreq denote the equivalent radial misalignment distance of the inner ring, the equivalent radial misalignment distance of the outer ring, e irx , e iry denote the static radial misalignment distance of the inner ring; e orx , e ory denote the static radial misalignment distance of the outer ring, x ir , y ir , x or , y or denote the radial misalignment displacement due to vibration;

[0040] S3.3 According to the misalignment amounts obtained in step S3.2, the coordinates of the curvature centers of the raceways in the misalignment state are calculated as,

[0041]

[0042] wherein x im , z im denote the coordinates of the curvature center of the inner raceway; x om , z om denote the coordinates of the curvature center of the outer raceway;

[0043] S3.4 According to the coordinates of the curvature centers of the raceways in the misalignment state obtained in step S3.3, the bearing clearance and the contact angle between the ball and the raceway are calculated as:

[0044]

[0045] wherein c m denotes the clearance of the misalignment bearing, a m denotes the contact angle of the bearing;

[0046] S3.5 According to the clearance of the misalignment bearing obtained in step S3.4, the contact deformation between the ball and the raceway is calculated by the following formula:

[0047]

[0048] wherein d j denotes the contact deformation of the jth ball;

[0049] S3.6 According to the contact deformation between the ball and the raceway obtained in step S3.5 and the ball contact angle obtained in step S3.4, the contact force and the contact moment of the misaligned bearing are:

[0050]

[0051] wherein F x , F x , F z represents the contact force, M x , M y represents the contact moment, k b represents the contact stiffness, and r1 represents the distance from the contact point to the axis.

[0052] In the step S4, the dynamic model of the misaligned bearing-rotor system under the base motion is established as follows:

[0053] S4.1 The stiffness matrix, the mass matrix, the gyro matrix, and the additional effect matrix of the set unit are combined, and the motion differential equation of the misaligned bearing-rotor system under the base motion is:

[0054]

[0055] wherein M, C, G, and K represent the total mass matrix, the damping matrix, the gyro matrix, and the stiffness matrix of the misaligned bearing-rotor system under the base motion. q represents the acceleration, the speed, and the displacement vector of the misaligned bearing-rotor system under the base motion; F un , F b represent the unbalance force model and the bearing force model.

[0056] The beneficial effects of the present application are that the present application simultaneously considers the influence of the base motion and the misalignment of the bearing, proposes a nonlinear restoring force calculation method of the angular contact ball bearing with 5 degrees of freedom, and introduces a dynamic model of the misaligned bearing-rotor system under the base motion. By taking the angular, radial, and axial deviations of the misaligned bearing into the contact force calculation formula, the contact characteristics and the axial force response of the bearing under different working conditions are accurately obtained. Based on the model, the rotor vibration characteristics under the combined action of the base motion and the misalignment of the bearing are further analyzed, and the influence of the increase of the base angular velocity on the bearing variable stiffness frequency amplitude is revealed. The model is closer to the actual working condition, especially in the complex system where the misalignment of the bearing and the base motion coexist, and provides a more accurate simulation basis for the optimal design and fault diagnosis of the bearing-rotor system. BRIEF DESCRIPTION OF DRAWINGS

[0057] Figure 1 is a schematic diagram of a bearing-rotor system;

[0058] Figure 2 is a schematic diagram of a beam element;

[0059] Figure 3 is a bearing geometry diagram; (a) is a bearing with the large end face facing right; (b) is a bearing with the large end face facing left.

[0060] Figure 4 is a three-dimensional spectrum diagram; (a) is a displacement spectrum diagram at bearing 1; (b) is a displacement spectrum diagram at bearing 4; (c) is a bearing contact force spectrum diagram at bearing 1; (d) is a bearing contact force spectrum diagram at bearing 4.

[0061] Figure 5 is the effect of base motion velocity on bearings; (a) is a bearing contact angle change diagram; (b) is a ball contact force change diagram; (c) is a bearing contact stiffness change diagram; (d) is a bearing z-direction contact force; (e) is a bearing y-direction contact force; (f) is a bearing x-direction contact force.

[0062] Figure 6 is a bearing contact load area. (a) is a contact force radar chart at bearing 1 when the base is stationary; (b) is a contact force radar chart at bearing 4 when the base is stationary; (c) is a contact force radar chart at bearing 1 when the base is moving; (d) is a contact force radar chart at bearing 4 when the base is moving. DETAILED DESCRIPTION

[0063] The technical solutions of the present application will be described clearly and completely below in conjunction with the drawings. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the scope of protection of the present application.

[0064] Embodiment 1

[0065] Step 1 Calculate the disc motion differential equation under the base motion state

[0066] S1.1 The bearing-rotor system is as shown in Figure 1 First, the translational kinetic energy and rotational kinetic energy of the lumped mass element under the base motion state are

[0067]

[0068] where T t is the translational kinetic energy of the lumped mass element, m d is the mass of the disc, v ord is the translational velocity vector of the lumped mass element, is the translational velocity of the base motion coordinate system in the global coordinate system, is the translational velocity of the lumped mass element in the base motion coordinate system, is the base rotational velocity, r is the displacement vector of the lumped mass unit in the base motion coordinate system.

[0069]

[0070] where T r is the rotational kinetic energy of the lumped mass unit, I d , I p are the rotational inertia and polar moment of inertia of the disc, a, b, g represent the Euler angles of the coordinate transformation, and the superscript dot represents the angular velocity corresponding to the Euler angles. According to the translational kinetic energy and rotational kinetic energy expressions obtained in step S1.1, the total kinetic energy expression is:

[0071] T = T t + T r , (3)

[0072] where T is the total kinetic energy.

[0073] S1.3 According to the total kinetic energy obtained in step 1.2, the mass matrix and the proximity effect matrix of the lumped mass unit are obtained based on the Lagrange equation:

[0074]

[0075] where q i , represent the generalized displacement and generalized velocity; Q i represents the generalized force; and U represents the potential energy.

[0076] Since a = θ x , b = θ y , g = θ z + Ωt, the motion differential equation of the lumped mass unit is

[0077]

[0078] where Ω represents the rotational speed of the disc, M d , G d represent the mass matrix and gyro matrix of the disc, θ Bx , represent the angular displacement, angular velocity, and angular acceleration of the disc around the x, y, and z axes. q d , represent the displacement vector, velocity vector, and acceleration vector of the disc axis. C Bzd , C Byd , C Bxd represent the additional damping matrix caused by the base motion. C dz , C dy , C dx , H dz , Hdy 、H dx , K rd , K Bzd , K Bxd , K Byd , K Bxyd , K Byzd , K Bxzd Represents the additional stiffness matrix caused by the foundation motion. Bd Represents the additional load vector due to foundation motion.

[0079] Step 2 Calculate the differential equation of motion of the beam element in the basic motion state

[0080] S2.1 Timoshenko beam element Figure 2 As shown. The kinetic energy and potential energy expressions of the Timoshenko beam element in the basic motion state are:

[0081]

[0082] in, is the potential energy of the unit, is the kinetic energy of the unit, is the diameter moment of inertia and polar moment of inertia of the unit per unit length, A represents the cross-sectional area, ρ is the density of the unit, E and G represent the elastic modulus and Poisson's ratio, J represents the torsional section moment of inertia, I x , I y represents the section moment around the x and y axes, κ x , κ y The shear correction coefficients u, v, and w around x and y are the displacements of any section on the x, y, and z axes relative to the origin, θ, φ, represents the angular displacement of any section, l k Indicates the length of the unit.

[0083] S2.2 Based on the kinetic energy and potential energy expressions obtained in step S2.1, substitute them into the Lagrange equation to obtain the differential equation of motion of the unit:

[0084]

[0085] in, Represents the displacement, velocity, and acceleration vectors of the beam element. K s e h Represents the mass matrix, gyro matrix, and stiffness matrix of the unit. C Bzdsh 、C Bydsh 、C Bxdsh represents the additional damping matrix caused by the foundation motion. C dzsh 、C dysh 、Cdxsh 、H dzsh 、H dysh 、H dxsh , K rdsh , K Bzdsh , K Bxdsh , K Bydsh , K Bxydsh , K Byzdsh , K Bxzdsh Represents the additional stiffness matrix caused by the foundation motion. C1, K1, K2, and K3 are intermediate variables used to simplify the expression.

[0086] Step 3 Calculation of contact force of angular contact bearing under misalignment

[0087] S3.1 Bearing geometry is as follows Figure 3 The coordinate expression of the raceway curvature center of the angular contact bearing in a healthy state is:

[0088]

[0089] Among them, f i 、f o Indicates the curvature coefficient of the inner and outer raceways, r b Indicates the ball radius, r p represents the bearing pitch radius, α0 represents the initial contact angle between the ball and the raceway, and x i 、z i 、x o 、z o Indicates the coordinates of the center of curvature of the inner and outer raceways.

[0090] S3.2 Based on the coordinates of the raceway curvature center obtained in step S3.1, taking into account the rotor vibration and axial, radial and angular assembly errors, the equivalent misalignment is expressed as

[0091]

[0092] Among them, α ieq , α oeq Indicates the equivalent misalignment angle of the inner ring and the outer ring, α ix , α iy , α ox , α oy Indicates the equivalent misalignment angle of the inner ring and the tilt angle of the outer ring around the x and y axes, θ ix ,θ iy ,θ ox ,θ oy It represents the angular displacement of the inner ring around the x and y axes due to vibration, and the angular displacement of the outer ring around the x and y axes due to vibration, e iaeq 、e oaeqdenotes the equivalent axial misalignment distance of the inner ring, the equivalent axial misalignment distance of the outer ring, e ia oa denotes the static axial misalignment distance of the inner ring, the static axial misalignment distance of the outer ring, z ia oa denotes the axial misalignment displacement due to vibration, e ireq oreq denotes the equivalent radial misalignment distance of the inner ring, the equivalent radial misalignment distance of the outer ring, e irx iry orx ory denotes the static radial misalignment distance of the inner ring, the static radial misalignment distance of the outer ring, x ir ir or or denotes the radial misalignment displacement due to vibration.

[0093] S3.3 According to the misalignment obtained in step S3.2, the coordinates of the curvature center of the raceway in the misalignment state are calculated as

[0094]

[0095] wherein x im im om om denotes the coordinate of the curvature center of the inner raceway, the coordinate of the curvature center of the outer raceway.

[0096] S3.4 According to the coordinates of the curvature center of the raceway in the misalignment state obtained in step S3.3, the bearing clearance and the contact angle between the ball and the raceway are calculated as

[0097]

[0098] wherein c m denotes the clearance of the misaligned bearing, a m denotes the contact angle of the bearing.

[0099] S3.5 According to the clearance of the misaligned bearing obtained in step S3.4, the contact deformation between the ball and the raceway can be calculated by the following formula

[0100]

[0101] wherein d j denotes the contact deformation of the jthball.

[0102] ​​​​​​​​​​​​S3.6 The contact force and contact moment of misaligned bearing are calculated based on the contact deformation between ball and raceway obtained in step S3.5 and the ball contact angle obtained in step S3.4:

[0103]

[0104] where F x , F x , F z represent the contact force, M x , M y represent the contact moment, k b represents the contact stiffness, and r1 represents the distance from the contact point to the axis.

[0105] Step 4 Dynamic modeling of misaligned bearing-rotor system under base motion

[0106] S4.1 The stiffness matrix, mass matrix, gyro matrix, and additional effect matrix of the group set unit, and the motion differential equation of the misaligned bearing-rotor system under base motion are as follows:

[0107]

[0108] where M, C, G, K represent the total mass matrix, damping matrix, gyro matrix, and stiffness matrix of the misaligned bearing-rotor system under base motion. q represents the acceleration, velocity, and displacement vector of the system. F un , F b represent the unbalance force and bearing force model.

[0109] According to the dynamic analysis model in step 4, the dynamic response of the system is solved according to the Newmark-β time domain integration method, and the dynamic response of the rotor system is analyzed. It is found that there are combination frequencies of bearing characteristic frequency and rotational frequency in the frequency spectrum.

[0110] The frequency spectrum of the bearing node x-direction displacement under different base motion speeds is as shown in Figure 4 . With the increase of base motion speed, the amplitude of rotational frequency f r generally decreases, while the amplitude of frequency f vc1 increases. In addition, the amplitude of VC frequency f vc5 of bearing 5 first increases, then decreases, and finally increases again, as shown in Figure 4 (b). This is because the base motion first offsets the misalignment effect of the bearing. Some combination frequencies also appear in the frequency spectrum, including f vc1 -f r and f vc1 +f r . In addition, Figure 4 , the frequency fvc2 As the speed increases, the amplitude of the frequency fr increases first, then remains constant, and finally Figure 4 (c) and 4(d). In addition, the frequency f vc The amplitude of the frequency increases significantly, and its amplitude is greater than the frequency f r The amplitude of .

[0111] Figure 5 The effect of base motion on bearing contact characteristics is intuitively demonstrated. As the base motion speed increases, the following phenomena are observed: (1) The maximum contact angle and maximum contact stiffness of bearing 1 decrease monotonically, while those of bearing 4 increase monotonically. The contact angle variation ranges of bearings 1 and 4 are approximately 10° and 6°, respectively. This indicates that the base motion has a significant effect on α m1 The impact of α m4 The influence of F is more significant. (2) The maximum rolling element contact force also gradually increases. The rolling element contact force of bearing 1 is greater than that of bearing 4. b1 The change of F is greater, and the base movement has a greater impact on b1 The impact of F b4 The influence of is more obvious. (3) In addition, the contact force of the bearing in the z direction and y direction also gradually increases, such as Figure 5 (d) and 5(e). The contact force F of bearing 4 in the y direction is y1 Greater than F of bearing 1 y4 (4) Compared with the contact force of the bearing in the x-direction and y-direction, the F y4 (F y1 ) is greater than F x4 (F x1 ). This is a non-uniform load phenomenon caused by base movement. Another phenomenon caused by base movement is a significant increase in the axial force of the bearing, which worsens the load environment of the bearing. (5) The reasons for the increase in axial load at different base movement speeds are summarized as follows: the axial excitation force caused by base movement gradually increases, which intensifies the axial vibration of the rotor and ultimately leads to an increase in the axial force of the bearing.

[0112] Figure 6 Shown in and Radar diagram of the bearing contact load distribution of bearings 1 and 4, considering the misalignment of bearing 1. Figure 6 In (a), the contact range is distributed between -57° to 57° and 130° to 230°, and the bearing contact force in the right area is greater than that in the left area. This is because the left area (i.e. Figure 1 The gap between the rolling element and the raceway is reduced. Figure 6 In (b), the bearing contact force is always greater than 0N. at the same time, subjected to additional forces caused by the base motion, Figure 6 (c) has only one contact interval, ranging from -62° to 62°. Figure 6 (d) has a bearing contact force that is always greater than 0 N, again due to the bearing being compressed in the axial direction, causing the rolling elements to always be in contact with the raceway.

[0113] Embodiments of the application are presented by way of example and description only, and are not intended to be exhaustive or to limit the application to the form disclosed. Many modifications and variations will be apparent to those skilled in the art upon reading this disclosure. Embodiments are chosen and described in order to best explain principles of the application and practical application, and to enable others skilled in the art to best utilize the application, with various modifications as are suited to the particular use contemplated.

Claims

1. A method for analyzing vibration characteristics of a misaligned bearing-rotor system under basic motion, characterized in that: The steps include: S1. Based on the lumped mass method, a lumped mass unit is used to construct a disk. The kinetic energy and potential energy expressions of the lumped mass unit under the basic motion are obtained. After substituting the kinetic energy and potential energy expressions into the Lagrange equation, the stiffness matrix, mass matrix, and additional effect matrix of the lumped mass unit are obtained. S2. Based on the Timoshenko beam element theory, use the Timoshenko beam element to establish the rotation axis; obtain the kinetic energy and potential energy expressions of the beam element under the foundation motion; substitute the kinetic energy and potential energy expressions into the Lagrange equation to obtain the stiffness matrix, mass matrix, and additional effect matrix of the beam element; The lumped mass unit and the beam unit are grouped units; the stiffness matrix, mass matrix and additional effect matrix corresponding to the rotating shaft are obtained according to the stiffness matrix, mass matrix and additional effect matrix of the grouped units; S3. Considering the effect of angular contact bearing misalignment on bearing contact, the ball contact angle and the clearance between the ball and raceway under misalignment are calculated using a function method. Furthermore, the contact deformation between the ball and raceway is calculated based on the displacement of the bearing inner and outer rings, ultimately obtaining the supporting force of the misaligned angular contact bearing. S4. Based on the obtained stiffness matrix, mass matrix, and additional effect matrix of the rotating shaft / disk, the support force of the misaligned bearing is introduced, and finally the dynamic model of the misaligned bearing-rotor system under basic motion is established.

2. The vibration characteristic analysis method of a misaligned bearing-rotor system under basic motion according to claim 1 is characterized in that: In step S1, the mass matrix and the additional effect matrix of the lumped mass unit are obtained as follows: S1.1 The translational kinetic energy and rotational kinetic energy of the concentrated mass unit in the basic motion state are: Among them, T t is the translational kinetic energy of the concentrated mass unit, m d is the mass of the disk, v ord is the translational velocity vector of the lumped mass unit, is the translational velocity of the base motion coordinate system in the global coordinate system, is the translational velocity of the concentrated mass unit in the base motion coordinate system, is the rotational velocity of the foundation, r is the displacement vector of the concentrated mass unit in the foundation motion coordinate system; in, is the velocity matrix; T r is the rotational kinetic energy of the concentrated mass unit, I d , I p are the moment of inertia and polar moment of inertia of the disk, α, β, and γ represent the Euler angles of coordinate transformation, and the superscript dot represents the angular velocity corresponding to the Euler angle, and a represents the Euler angle; S1.2 Based on the translational kinetic energy expression and rotational kinetic energy expression obtained in step S1.1, the total kinetic energy expression is: T=T t +T r , (3) Where T is the total kinetic energy; S1.3 Based on the total kinetic energy obtained in step S1.2, the mass matrix and additional effect matrix of the lumped mass unit are obtained based on the Lagrange equation: Among them, q i 、 represents generalized displacement and generalized velocity; Q i represents generalized force; U represents potential energy; since a=θ x ,β=θ y ,γ=θ z +Ωt, the differential equation of motion of the concentrated mass unit is Among them, Ω represents the rotation speed of the disk, M d , G d represents the mass matrix and gyro matrix of the disk, θ Bx 、 represents the angular displacement, angular velocity, and angular acceleration of the disk around the x-axis, θ By 、 represents the angular displacement, angular velocity, and angular acceleration of the disk around the y-axis, θ Bz 、 represents the angular displacement, angular velocity, and angular acceleration of the disk around the z-axis; q d 、 Represents the displacement vector, velocity vector, and acceleration vector of the disk axis; C Bzd 、C Byd 、C Bxd Represents the additional damping matrix caused by the rotation of the base motion around the x, y, and z axes; C dz 、C dy 、C dx 、H dz 、H dy 、H dx , K rd , K Bzd , K Bxd , K Byd , K Bxyd , K Byzd , K Bxzd represents the additional stiffness matrix caused by foundation motion; F Bd Represents the additional load vector due to foundation motion.

3. The vibration characteristic analysis method of a misaligned bearing-rotor system under basic motion according to claim 1 is characterized in that: In step S2, the stiffness matrix, mass matrix and additional effect matrix of the beam element are obtained as follows: S2.1 The kinetic energy and potential energy expressions of the Timoshenko beam element in the basic motion state are: in, is the potential energy of the beam element, is the kinetic energy of the beam element, is the diameter moment of inertia and polar moment of inertia of the unit per unit length, A represents the cross-sectional area, ρ is the density of the unit, E and G represent the elastic modulus and Poisson's ratio, J represents the torsional section moment of inertia, I x , I y They represent the section moment around the x-axis and the section moment around the y-axis, κ x , κ y represents the shear correction coefficient around the x-axis and the shear correction coefficient around the y-axis; u, v, w are the displacements of any section relative to the origin on the x-, y-, and z-axes, θ, φ, represents the angular displacement of any section, l k represents the length of the beam element; u′, v′ , w′ is the velocity of any section on the x, y, z axis relative to the origin; θ′, φ′, represents the angular velocity of any cross section; S2.2 Based on the kinetic energy expression and potential energy expression obtained in step S2.1, substitute them into the Lagrange equation to obtain the motion differential equation of the beam element: in, Represents the acceleration vector, velocity vector, and displacement vector of the beam element; Represents the mass matrix, gyro matrix, and stiffness matrix of the beam element; C Bzdsh 、C Bydsh 、C Bxdsh represents the additional damping matrix caused by the foundation motion; C dzsh 、C dysh 、C dxsh 、H dzsh 、H dysh 、H dxsh , K rdsh , K Bzdsh , K Bxdsh , K Bydsh , K Bxydsh , K Byzdsh , K Bxzdsh Represents the additional stiffness matrix caused by foundation motion; C1, K1, K2, K3 are intermediate variables.

4. The vibration characteristic analysis method of a misaligned bearing-rotor system under basic motion according to claim 1 is characterized in that: In step S3, the supporting force of the misaligned angular contact bearing is obtained as follows: S3.1 The coordinate expression of the raceway curvature center of an angular contact bearing in a healthy state is: Among them, f i 、f o Indicates the curvature coefficient of the inner raceway and the outer raceway, r b Indicates the ball radius, r p represents the bearing pitch radius, α0 represents the initial contact angle between the ball and the raceway, and x i 、z i Indicates the coordinate of the center of curvature of the inner raceway; x o 、z o Indicates the coordinates of the center of curvature of the outer raceway; S3.2 Based on the coordinates of the raceway curvature center obtained in step S3.1, taking into account the rotor vibration and axial, radial and angular assembly errors, the equivalent misalignment is expressed as: Among them, α ieq , α oeq Indicates the equivalent misalignment angle of the inner ring and the outer ring, α ix , α iy Indicates the inclination angle of the inner circle around the x and y axes; α ox , α oy The tilt angle of the outer ring around the x and y axes, θ ix ,θ iy Indicates the angular displacement of the inner ring around the x and y axes due to vibration; θ ox ,θ oy Indicates the angular displacement of the outer ring around the x and y axes due to vibration, e iaeq 、e oaeq Indicates the equivalent axial misalignment distance of the inner ring and the equivalent axial misalignment distance of the outer ring, e ia 、e oa Indicates the static axial misalignment distance of the inner ring and the static axial misalignment distance of the outer ring, z ia 、z oa Indicates the axial misalignment caused by vibration, e ireq 、e oreq Indicates the equivalent radial misalignment distance of the inner ring and the equivalent radial misalignment distance of the outer ring, e irx 、e iry Indicates the static radial misalignment distance of the inner ring; e orx 、e ory Indicates the static radial misalignment distance of the outer ring, x ir 、y ir 、x or 、y or Indicates radial misalignment caused by vibration; S3.3 Based on the misalignment value obtained in step S3.2, calculate the coordinates of the center of curvature of the raceway under misalignment: Among them, x im 、z im Indicates the coordinates of the center of curvature of the inner raceway; xom, z om Indicates the coordinates of the center of curvature of the outer raceway; S3.4 Based on the coordinates of the center of curvature of the raceway under the misaligned state obtained in step S3.3, calculate the bearing clearance and ball contact angle as follows: Among them, c m Indicates the clearance of misaligned bearings, α m Indicates the bearing contact angle; S3.5 Based on the clearance of the misaligned bearing obtained in step S3.4, the contact deformation between the ball and the raceway is calculated using the following formula: Among them, δ j represents the contact deformation of the jth ball; S3.6 Based on the contact deformation between the ball and raceway obtained in step S3.5 and the ball contact angle obtained in step S3.4, the contact force and contact torque of the misaligned bearing are: Among them, F x 、F x 、F z Represents the contact force, M x 、M y represents the contact torque, k b Represents the contact stiffness, and r1 represents the distance from the contact point to the axis.

5. The vibration characteristic analysis method of a misaligned bearing-rotor system under basic motion according to claim 1, characterized in that: In step S4, the dynamic model of the misaligned bearing-rotor system under basic motion is established as follows: The stiffness matrix, mass matrix, gyro matrix, and additional effect matrix of the S4.1 group element, and the motion differential equation of the misaligned bearing-rotor system under basic motion are: Among them, M, C, G, and K represent the total mass matrix, damping matrix, gyro matrix, and stiffness matrix of the misaligned bearing-rotor system under basic motion; q represents the acceleration, velocity, and displacement vector of the misaligned bearing-rotor system under basic motion; F un 、F b Represents the unbalanced force model and bearing force model.

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