A method for calculating the stiffness of bearings with misalignment in a spline-connected gear rotor system

Abstract

The present invention provides a method for calculating the stiffness of misaligned bearings in a spline-connected gear rotor system. Relating to the field of mechanical design technology, the calculation method comprises the following steps: establishing a quasi-static model of a deep groove ball bearing, considering the quasi-static model of the deep groove ball bearing under the coupling effect of misalignment and support load, and establishing a bearing unit stiffness matrix expression; establishing a static model of a spline-connected gear rotor-bearing system; solving the quasi-static model of the deep groove ball bearing according to the bearing unit stiffness matrix, the spur gear pair unit stiffness matrix and the spline unit stiffness matrix, and calculating the misaligned bearing stiffness. Compared with the traditional method for calculating the stiffness of a single bearing, the proposed method considers more comprehensive and more practical factors, and the calculated bearing stiffness is more reliable. The proposed method is designed to provide theoretical support for the structural design and healthy operation and maintenance of the gear-bearing transmission system.

Description

Technical Field

[0001] The present invention relates to the technical field of mechanical design, and in particular to a method for calculating the stiffness of a misaligned bearing in a spline-connected gear rotor system. Background Art

[0002] Ball bearings are often used as rotating support elements in spline-connected gear rotor systems. Under abnormal assembly conditions, the abnormal preload state caused by bearing ring misalignment greatly affects the stiffness characteristics of the bearing. In addition to the abnormal preload caused by misalignment, the bearing is also subjected to support loads caused by gear meshing under operating conditions. In order to study the effect of ring misalignment on the support bearing stiffness in spline-connected gear rotor systems, the influence of the raceway rib is first considered, and a quasi-static model of deep groove ball bearings under the coupling of abnormal preload caused by raceway misalignment and external support loads is established. Then, considering the meshing of gear pairs, spline connection and shaft flexibility in the actual transmission system, a calculation method for the support stiffness of ball bearings is proposed.

[0003] However, existing methods for calculating bearing support stiffness all calculate bearing stiffness based on known preload and external load. Existing methods only take a single bearing as the research object and do not consider the actual transmission structure. However, due to the complexity of the gear transmission system and the nonlinearity of the ball-raceway contact, the actual support load of the deep groove ball bearing in the system is difficult to directly determine. Therefore, the calculation of bearing stiffness cannot be separated from the transmission system. In addition, when the misalignment angle of the raceway is too large, the contact ellipse between the ball and the raceway may exceed the raceway rib and be truncated by the rib, resulting in a weakening of the ball-raceway contact force. However, existing methods for calculating bearing support stiffness do not take this extreme case into account. In summary, it is necessary to invent a method for calculating bearing support stiffness that takes into account the influence of the gear transmission system and the excessive misalignment angle of the raceway. Summary of the invention

[0004] The present invention provides a method for calculating the bearing stiffness with misalignment in a spline-connected gear rotor system. The present invention mainly considers the abnormal preload caused by the misalignment of the bearing raceway, as well as the influencing factors such as gear meshing, spline connection and spindle flexibility in the actual transmission structure, thereby solving the problem that the prior art does not consider the influence of the gear transmission system and the excessive misalignment angle of the raceway. The calculated bearing stiffness is more accurate.

[0005] The technical means adopted by the present invention are as follows:

[0006] A method for calculating the stiffness of a misaligned bearing in a spline-connected gear rotor system, the method comprising the following steps:

[0007] Establish a quasi-static model of deep groove ball bearings, consider the quasi-static model of deep groove ball bearings under the coupling effect of misalignment and support load, and establish the bearing unit stiffness matrix;

[0008] Considering the influence of the rotation direction of the active gear, the stiffness matrix of the spur gear pair unit is established according to the meshing stiffness and the angle between the meshing line and the y-axis;

[0009] Considering the additional deformation of the meshing point caused by the elasticity of the tooth root, the spline unit stiffness matrix is ​​established;

[0010] The rotating shaft structure is modeled by using Timoshenko beam elements to obtain a finite element model of a spline-connected gear rotor-bearing system, and the statics equation of the system is obtained according to the finite element model of the spline-connected gear rotor-bearing system;

[0011] The quasi-static model of the deep groove ball bearing is solved according to the statics equation, the bearing unit stiffness matrix, the spur gear pair unit stiffness matrix and the spline unit stiffness matrix, and the misaligned bearing stiffness is calculated.

[0012] Furthermore, the specific steps of establishing the quasi-static model of the deep groove ball bearing are as follows:

[0013] In the case of angular misalignment, the relative axial displacement equation and the relative radial displacement equation of the inner raceway curvature center are established to obtain the relative axial displacement and relative radial displacement of the inner raceway curvature center;

[0014] The relative axial displacement of the center of curvature of the inner raceway is:

[0015]

[0016] The relative radial displacement of the center of curvature of the inner raceway is:

[0017]

[0018] Among them, r po θ is the radial distance between the bearing center and the outer raceway curvature center; to is the misalignment angle of the outer raceway; The azimuth angle of the outer ring raceway away from the center axis; is the azimuth of the jth ball; S a is the axial clearance of the bearing; S r is the radial clearance of the bearing;

[0019] The ball position is divided into a load-bearing area and a non-load-bearing area. According to the relative axial displacement and relative radial displacement of the center of curvature of the inner raceway, a load-bearing area determination formula is established to determine the ball position. The ball that satisfies the load-bearing area determination formula is located in the load-bearing area, and the ball that does not satisfy the load-bearing area determination formula is located in the non-load-bearing area.

[0020] The formula for determining the bearing area is:

[0021] (BDsinα°+B 1j ) 2 +(BDcosα°+B 2j -X 2j ) 2 >[(f i -0.5)d b ] 2 ,

[0022] Where BD is the distance between the inner and outer ring raceway curvature centers in the initial contact state; α° is the initial contact angle; X 2j F is the radial distance from the center of the ball to the center of curvature of the outer raceway; i is the curvature coefficient of the inner raceway; D b is the diameter of the ball;

[0023] When the ball is in the load-bearing area, considering that the contact ellipse between the ball and the raceway is truncated by the raceway rib due to excessive misalignment, resulting in a reduction in load-bearing capacity, the contact stiffness is rewritten using the contact ellipse truncation rate, and a calculation expression for the contact stiffness considering the rib is established;

[0024] The contact ellipse cutoff rate is:

[0025]

[0026] The contact stiffness calculation expression is:

[0027]

[0028] Among them, κ ij / oj is the ellipse eccentricity parameter; E e is the equivalent elastic modulus; is the complete elliptic integral of the first kind; is the complete elliptic integral of the second kind; ∑ρ ij / oj is the curvature and;

[0029] The contact load is obtained according to the contact stiffness calculation expression; the contact load is:

[0030]

[0031] Among them, δ ij / oj It is the contact deformation between the ball and the inner and outer raceways;

[0032] According to the contact stiffness calculation expression and the contact load, the equilibrium equation of the ball in the load-bearing area and the equilibrium equation of the ball in the non-load-bearing area are established;

[0033] The balance equation of the ball in the load-bearing area is:

[0034]

[0035] Among them, X 1j is the axial distance from the center of the ball to the center of curvature of the outer raceway; fo is the curvature coefficient of the outer raceway; F cj is the centrifugal force; M gj is the gyro torque; μ j is the average friction coefficient between the ball and the inner and outer raceways;

[0036] The balance equation of the ball in the load-bearing area is:

[0037] F cj -Q oj =0.

[0038] The abnormal preload force caused by angular misalignment is obtained, and the abnormal preload force is:

[0039]

[0040] Among them, F mistz M is the abnormal preload in the axial direction caused by misalignment; mistx and M misty is the abnormal preload torque around the x-axis and y-axis caused by misalignment; R pj is the radial distance between the bearing center and the inner raceway contact position;

[0041] According to the abnormal preload and support load, the relative axial displacement equation and relative radial displacement equation of the inner raceway curvature center are established;

[0042] The relative axial displacement equation of the inner raceway curvature center is:

[0043]

[0044] The relative radial displacement equation of the inner raceway curvature center is:

[0045]

[0046] Among them, δ z is the axial displacement of the inner ring along the z axis; θ x and θ y is the angular displacement of the inner ring around the x-axis and y-axis; δ x is the radial displacement of the inner ring along the x-axis; δ y is the radial displacement of the inner ring along the y-axis;

[0047] According to the bearing force analysis, the force balance equation of the bearing inner ring under the condition of angular misalignment is established; the force balance equation of the bearing inner ring is:

[0048]

[0049] Among them, F x and F y F is the radial support load acting on the inner ring along the x-axis and y-axis; z is the axial support load along the z axis; M x and M y are the overturning moments acting on the inner ring about the x-axis and the y-axis.

[0050] Furthermore, the expression of the bearing unit stiffness matrix is:

[0051]

[0052] Furthermore, the specific steps of establishing the stiffness matrix of the spur gear pair unit are as follows:

[0053] Based on the gear tooth load contact analysis method and the force and deformation coordination conditions of gear meshing, the gear meshing equilibrium equation is established; the gear meshing equilibrium equation is:

[0054]

[0055] Among them, λ c is the contact flexibility matrix; b is the overall flexibility matrix; I is the unit vector; F n is the normal contact force vector of the contact point; ste is the transmission error; ε is the initial separation gap vector; T p is the input torque applied to the pinion; r bp is the base circle radius of the pinion;

[0056] The meshing stiffness is obtained by solving the equilibrium equation of gear meshing; the meshing stiffness is:

[0057]

[0058] Among them, nlste is the no-load transfer error;

[0059] Considering the influence of the rotation direction of the driving gear, a symbol variable is introduced; the symbol variable is:

[0060]

[0061] Among them, ω p is the angular velocity of the pinion;

[0062] The spur gear pair is regarded as two concentrated masses, and the meshing between the two concentrated masses is simulated by springs and dampers to obtain the angle between the meshing line and the y-axis. The counterclockwise direction is positive and the clockwise direction is negative. The angle between the meshing line and the y-axis is:

[0063]

[0064] Among them, α pg is the pressure angle. pg is the relative azimuth angle of the two gears;

[0065] Considering the influence of the rotation direction of the active gear, the stiffness matrix of the spur gear pair unit is established according to the meshing stiffness and the angle between the meshing line and the y-axis; the stiffness matrix of the spur gear pair unit is:

[0066]

[0067] Among them, the vector α pg for:

[0068] α pg =[-sinψ pg cosψ pg 0 0 0 sgn×r bp sinψ pg -cosψ pg 0 0 0 sgn×r bg ].

[0069] Furthermore, the specific steps of establishing the spline unit stiffness matrix are as follows:

[0070] Based on the method of material mechanics, the involute gear teeth of the spline are simplified into a cantilever beam with variable cross-section on an elastic basis.

[0071] Based on the slicing method, the jth gear tooth is divided into n small slices, and the flexibility equation of the gear tooth cantilever beam is established; the flexibility equation of the gear tooth cantilever beam is:

[0072]

[0073] Among them, β j is the load F on the jth gear tooth j The angle with the positive direction of the y-axis; L k is the kth piece and the load F j The distance between the action points along the x direction; B is the tooth width; ν is the Poisson's ratio of the tooth material; E ν is the equivalent elastic modulus of the gear tooth material; when the ratio of tooth width to pitch circle tooth thickness is B / s p ≥5, E ν =E / (1-ν 2 ). When B / s p<5, E ν =E, E is the elastic modulus of the gear tooth material;

[0074] Considering the additional deformation of the meshing point caused by the elasticity of the tooth root; when B / s p When ≥5, the additional flexibility of the tooth root is:

[0075]

[0076] Among them, L f is the tooth root and load F j The distance between the action points along the x direction; s k is the tooth thickness corresponding to the tooth root;

[0077] When B / s p When <5, the additional flexibility of the tooth root is:

[0078]

[0079] The nonlinear contact compliance of the tooth surface meshing point is obtained, and the nonlinear contact compliance of the tooth surface meshing point is:

[0080]

[0081] Among them, B e is the effective tooth width;

[0082] According to the flexibility equation of the gear cantilever beam, the nonlinear contact flexibility and the additional flexibility equation of the tooth root, the meshing stiffness equation is established; the meshing stiffness equation is:

[0083]

[0084] The spline unit stiffness matrix includes the equivalent radial stiffness, equivalent bending stiffness and equivalent torsional stiffness of the key connection, and the equivalent radial stiffness, equivalent bending stiffness and equivalent torsional stiffness are calculated respectively according to the meshing stiffness;

[0085] The equivalent radial stiffness is:

[0086]

[0087] in, is the azimuth angle of the jth pair of gear teeth;

[0088] The equivalent bending stiffness is:

[0089]

[0090] The equivalent torsional stiffness is:

[0091]

[0092] Among them, r sb is the base circle radius of the spline.

[0093] Furthermore, the finite element model of the spline-connected gear rotor-bearing system includes: shaft I, shaft II and shaft III, shaft I and shaft II are connected by cylindrical spur gears, shaft II and shaft III are spline-connected, shaft I is the driving shaft, shaft I is divided into thirteen sections by fourteen nodes, shaft II is divided into seven sections by six nodes, shaft III is divided into twenty-four sections by twenty-five nodes, a concentrated mass unit is respectively arranged at the fourth node, the sixth node, the thirteenth node of shaft I and the seventh node of shaft II, an elastic support element is respectively arranged at the second node and the eleventh node of shaft I and the second node and the twenty-fourth node of shaft III, a small gear is arranged at the ninth node of shaft I, a large gear is arranged at the second node of shaft II, and a deep groove ball bearing is respectively arranged at the third node and the fifth node of shaft II.

[0094] Furthermore, the specific steps of solving the quasi-static model of the deep groove ball bearing are as follows:

[0095] The bearing stiffness of the bearing is estimated in advance based on the initial values ​​of the abnormal preload and the bearing load;

[0096] The bearing unit stiffness matrix, the spur gear pair unit stiffness matrix and the spline unit stiffness matrix are coupled with the stiffness matrix of the system to obtain a total stiffness matrix of the system;

[0097] The deformation and support load of the bearing node are calculated based on the static load vector and the total stiffness matrix of the system;

[0098] The initial value of the support load is replaced by the calculated support load and the above calculation process is repeated until the support load meets the preset tolerance condition, and the support stiffness of the bearing in the system is obtained.

[0099] Furthermore, the total stiffness matrix is:

[0100] K=K S +K SE +K B +K Sp +K G

[0101] The static load vector is:

[0102] F S =T G +F kn +G S +G LM +G G

[0103] Among them, K S is the shaft stiffness matrix; K SE is the stiffness matrix of the support element; K B is the bearing stiffness matrix; K Sp is the spline stiffness matrix; K G is the stiffness matrix of the spur gear pair; T G is the torque vector; F kn is the restoring force vector caused by no-load static transmission error; G S is the gravity vector of the axis; G LM is the gravity vector of the concentrated mass; G G is the gravity vector of the gear pair.

[0104] Furthermore, the pseudo-static model of the deep groove ball bearing is established based on assumptions, which include: regarding the deep groove ball bearing, spline and bearing as stiffness units, ignoring the mass of the stiffness units, ignoring the influence of temperature, lubrication and sealing elements, regarding the contact between the ball and the raceway as Hertzian elastic contact between rigid bodies, and regarding the angular misalignment between the outer ring and the inner ring as an abnormal preload state.

[0105] Compared with the prior art, the present invention has the following advantages:

[0106] The present invention takes into account the misalignment of the bearing raceway, as well as the gear meshing, spline connection and main shaft flexibility existing in the actual transmission structure, proposes a calculation method for the support stiffness matrix of the ball bearing in the spline-connected gear rotor system, and gives a detailed calculation process to make the result closer to reality.

[0107] The present invention combines the unit stiffness matrices of bearings, spur gear pairs, involute splines, rotating shafts and other components to obtain the total stiffness matrix of the spline-connected gear rotor system, and establishes a static model of the system. The stiffness calculation of traditional bearings only considers the force of a single bearing, while the present invention considers the stiffness of different components in the overall transmission system and iterates to calculate the stiffness more accurately.

[0108] The invention establishes a pseudo-static model of a ball bearing under the coupling effect of misalignment and external support load, and the model better describes the contact state between the ball and the raceway.

[0109] The present invention takes into account that excessive misalignment may cause the contact ellipse between the ball and the raceway to be truncated by the rib, resulting in reduced load capacity. The contact stiffness is rewritten using the contact ellipse truncation rate, and a contact stiffness calculation expression considering the rib is given.

[0110] The present invention simplifies the spline gear teeth into a cantilever beam with a variable cross-section based on elasticity based on the method of material mechanics. The elastic deformation of the gear teeth is considered to be the deformation caused by the bending, shearing and compression of the cantilever beam, the additional elastic deformation of the tooth root, and the contact deformation at the tooth surface meshing point. Based on the slicing method, the gear teeth are divided into several slices, each of which is deformed under the action of a unit force, and then the equivalent stiffness of the involute spline is calculated. The spline is a very important element for transmitting torque in the transmission system. The spline is sliced ​​and divided to better describe the deformation state of the spline during the transmission process, and the system stiffness is calculated accurately.

[0111] The present invention takes into account the influence of bearing raceway misalignment causing abnormal preload, as well as the influencing factors such as gear meshing, spline connection and spindle flexibility existing in the actual transmission structure, and proposes a calculation method for the support stiffness matrix of the ball bearing in the spline-connected gear rotor system. Based on this method, the bearing is coupled with the system, and the support stiffness of the bearing can be obtained by iterative calculation based on the input torque of the gear transmission system. Compared with the traditional stiffness calculation method of a single bearing, the proposed method considers more comprehensive and practical factors, and the calculated bearing stiffness is more reliable. The proposed method can provide theoretical support for the structural design and healthy operation and maintenance of the gear-bearing transmission system. BRIEF DESCRIPTION OF THE DRAWINGS

[0112] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0113] Figure 1 This is a diagram of the calculation process of the bearing stiffness of the present invention.

[0114] Figure 2 It is a schematic diagram of the misalignment of the bearing ring of the present invention.

[0115] Figure 3 This is a diagram showing the deformation coordination relationship of the ball when the bearing ring of the present invention is misaligned.

[0116] Figure 4 It is the contact ellipse and schematic diagram between the ball and the outer raceway of the present invention.

[0117] Figure 5 It is a schematic diagram showing that the contact ellipse between the ball and the inner raceway is truncated by the raceway rib according to the present invention.

[0118] Figure 6 It is the quality model diagram of the cylindrical spur gear pair of the present invention.

[0119] Figure 7 This is the spline gear tooth diagram of the present invention.

[0120] Figure 8 This is a model diagram of the spline gear teeth slice of the present invention.

[0121] Fig. 9 It is a static model diagram of the spline-connected gear rotor-bearing system of the present invention.

[0122] Fig.10 is the radial stiffness K of the misaligned angle bearing of the present invention xx The impact diagram of .

[0123] Fig.11 is the radial stiffness K of the misaligned angle bearing of the present invention yy The impact diagram of .

[0124] Fig.12 is the radial stiffness K of the misaligned angle bearing of the present invention zz The impact diagram of .

[0125] Fig.13 When the misalignment angle is 0, the input torque affects the bearing radial stiffness K xx The impact diagram of .

[0126] Fig.14 When the misalignment angle is 0, the input torque affects the bearing radial stiffness K yy The impact diagram of .

[0127] Fig.15 When the misalignment angle is 0, the input torque affects the bearing radial stiffness K zz The impact diagram of .

[0128] Fig.16 When the misalignment angle is 4 mrad, the input torque affects the bearing radial stiffness K xx The impact diagram of .

[0129] Fig.17 When the misalignment angle is 4 mrad, the input torque affects the bearing radial stiffness K yy The impact diagram of .

[0130] Fig.18 When the misalignment angle is 4 mrad, the input torque affects the bearing radial stiffness K zz The impact diagram of .

[0131] Fig.19 When the misalignment angle is 0, the speed affects the radial stiffness K of the bearing. xx The impact diagram of .

[0132] Fig. 20 When the misalignment angle is 0, the speed affects the radial stiffness K of the bearing. yy The impact diagram of .

[0133] Fig.21 When the misalignment angle is 0, the speed affects the radial stiffness K of the bearing. zz The impact diagram of .

[0134] Fig. 22 When the misalignment angle is 4 mrad, the speed affects the radial stiffness K of the bearing. xx The impact diagram of .

[0135] Fig.23 When the misalignment angle is 4 mrad, the speed affects the radial stiffness K of the bearing. yy The impact diagram of .

[0136] Fig.24 When the misalignment angle is 4 mrad, the speed affects the radial stiffness K of the bearing. zz The impact diagram of .

[0137] Description of Figure Numbers:

[0138] 1. Cylindrical spur gear; 2. Spline; 3. Centralized mass unit; 4. Elastic support element; 5. Pinion; 6. Gear; 7. Deep groove ball bearing. DETAILED DESCRIPTION

[0139] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0140] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. The following description of at least one exemplary embodiment is actually only illustrative and is by no means intended to limit the present invention and its application or use. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0141] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit exemplary embodiments according to the present invention. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprising" and / or "including" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0142] Unless otherwise specifically stated, the relative arrangement of the parts and steps described in these embodiments, the numerical expressions and numerical values ​​do not limit the scope of the present invention. At the same time, it should be clear that, for ease of description, the sizes of the various parts shown in the drawings are not drawn according to the actual proportional relationship. The technology, methods and equipment known to ordinary technicians in the relevant field may not be discussed in detail, but in appropriate cases, the technology, methods and equipment should be regarded as part of the authorization specification. In all examples shown and discussed here, any specific value should be interpreted as merely exemplary, rather than as a limitation. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar numbers and letters represent similar items in the following drawings, so once an item is defined in one drawing, it does not need to be further discussed in subsequent drawings.

[0143] like Figure 1 As shown, the present invention provides a method for calculating the stiffness of a misaligned bearing in a spline-connected gear rotor system, comprising the following steps:

[0144] Step 1. Establish a pseudo-static model of deep groove ball bearings.

[0145] Abnormal preload caused by raceway misalignment is one of the important reasons for accelerated failure of ball bearings. In order to analyze the bearing stiffness in the raceway misalignment system, it is necessary to establish a pseudo-static model of the ball bearing. Figure 1 As shown. Figure 2 In the figure, the inclined outer flow channel is represented by a solid line. to Indicates the misalignment angle of the outer ring. is the azimuth angle of the outer ring to the center axis. Figure 3 In the figure, the curvature centers of the inner raceway and the outer raceway are O i and O o , in the initial state, the center of the ball is O b . Inner ring axial movement 0.5S a After that, the ball contacts the raceway but no contact force is generated, which is called the initial contact state. In this state, the center of curvature of the inner raceway is O i ', the center of the ball is O b '. When the outer raceway is misaligned, the center of curvature of the outer raceway is O o ', the center of the ball is O b ".

[0146] In case of angular misalignment, the relative axial displacement of the center of curvature of the inner raceway B 1j It can be expressed as:

[0147]

[0148] Relative radial displacement B of the inner raceway curvature center 2j It can be expressed as:

[0149]

[0150] Among them, r po Indicates the radial distance between the bearing center and the center of curvature of the outer raceway; θ to Indicates the misalignment angle of the outer raceway; Indicates the azimuth angle of the outer ring raceway away from the center axis; is the azimuth of the jth ball; S a Indicates the axial clearance of the bearing; S r Indicates the radial clearance of the bearing;

[0151] The bearing area and the non-bearing area need to be considered separately. The conditions for judging whether the ball is in the bearing area are:

[0152] (BDsinα°+B 1j ) 2 +(BDcosα°+B 2j -X 2j ) 2 >[(f i -0.5)d b ] 2 ,

[0153] Where BD is the distance between the inner and outer raceway curvature centers in the initial contact state. α° is the initial contact angle. 2j Indicates the radial distance from the center of the ball to the center of curvature of the outer raceway. i Indicates the curvature coefficient of the inner raceway. D b The ball that satisfies the inequality is located in the load-bearing area, otherwise it is located in the non-load-bearing area.

[0154] When the ball is in the load-bearing zone, considering that too large a misalignment may cause the contact ellipse between the ball and the raceway to be intercepted by the raceway rib, see Figure 4 and Figure 5 , resulting in a decrease in the load-bearing capacity. The contact stiffness is rewritten using the contact ellipse cutoff rate, and a simple calculation expression for the contact stiffness considering the flange is given. Figure 4 and Figure 5 In, h i and h o Respectively represent the height of the inner and outer ring raceway ribs. i and r o are the curvature radii of the inner and outer ring raceways respectively.

[0155] Contact ellipse cutoff It can be expressed as:

[0156]

[0157] Where, the subscript ij / oj indicates that the jth ball is in contact with the inner or outer raceway. The subscript i / o indicates the inner or outer raceway. ij / oj It is the contact angle between the ball and the inner or outer raceway. Indicates the angle corresponding to the major semi-axis of the contact ellipse between the ball and the inner and outer ring raceways. φ i / o Indicates the angle corresponding to the inner or outer raceway rib.

[0158] when When , the ball can be considered to be in the non-load bearing area. When the contact stiffness K ij / oj can be rewritten as:

[0159]

[0160] In the formula, κ ij / oj is the eccentricity parameter of the ellipse. e Represents the equivalent elastic modulus. is the complete elliptic integral of the first kind, refers to the complete elliptic integral of the second kind, ∑ρ ij / oj is the curvature and .

[0161] Contact load Q ij / oj It can be expressed as:

[0162]

[0163] In the formula, δ ij / oj Indicates the contact deformation between the ball and the inner and outer raceways.

[0164] Therefore, the equilibrium equation of the ball in the load-bearing area is:

[0165]

[0166] In the formula, X 1j It represents the axial distance from the center of the ball to the center of curvature of the outer raceway. o Indicates the curvature coefficient of the outer raceway. F cj is the centrifugal force. gj Represents the gyroscopic torque. μ j is the average friction coefficient between the ball and the inner and outer raceways. It should be noted that the friction coefficient μ j It is not the maximum static or dynamic friction coefficient in the actual sense. It can be interpreted as the minimum friction coefficient required to keep the ball in quasi-static equilibrium.

[0167] The equilibrium equation of the ball in the non-load area can be written as:

[0168] F cj -Q oj =0.

[0169] The abnormal preload caused by angular misalignment can be expressed as:

[0170]

[0171] In the formula, F mistz Indicates abnormal preload in the axial direction due to misalignment. M mistx and M misty It is the abnormal preload torque around the x-axis and y-axis caused by misalignment. pj Indicates the radial distance between the bearing center and the inner raceway contact position.

[0172] The relative axial displacement B of the inner raceway curvature center under the combined effect of abnormal preload and support load caused by misalignment 1j It can be expressed as:

[0173]

[0174] In the formula, δ z Indicates the axial displacement of the inner ring along the z-axis. θ x and θ y Represents the angular displacement of the inner circle around the x-axis and y-axis.

[0175] Relative radial displacement B of the inner raceway curvature center 2j It can be expressed as:

[0176]

[0177] In the formula, δ x and δ y is the radial displacement of the inner ring along the x-axis and y-axis.

[0178] In the case of angular misalignment, the force balance equation for the inner ring of the bearing is:

[0179]

[0180] In the formula, F x and F y Indicates the radial support load acting on the inner ring along the x-axis and y-axis. z Refers to the axial support load along the z-axis. M x and M y Denotes the overturning moment about the x-axis and y-axis acting on the inner ring.

[0181] Considering the quasi-static model of deep groove ball bearing under the coupling of misalignment and support load, the five-dimensional stiffness matrix expression of the bearing can be obtained:

[0182]

[0183] Step 2. Calculate the mesh stiffness of the spur gear pair.

[0184] Based on the gear tooth load contact analysis method and according to the force and deformation coordination conditions of gear meshing, the gear meshing equilibrium equation is established:

[0185]

[0186] In the formula, λ c is the contact flexibility matrix. b is the overall flexibility matrix. I is the unit vector. F n is the normal contact force vector of the contact point. ste is the transmission error. ε is the initial separation gap vector. T p is the input torque applied to the pinion. bp is the base circle radius of the pinion.

[0187] Meshing stiffness k m can be calculated by the following formula:

[0188]

[0189] Where nlste is the no-load transfer error.

[0190] In order to consider the influence of the rotation direction of the active gear, the symbolic variable sgn is introduced:

[0191]

[0192] In the formula, ω p is the angular velocity of the pinion.

[0193] The spur gear pair is considered as two concentrated masses, and the meshing between them is simulated by springs and dampers. The concentrated mass model of the spur gear pair is as follows: Figure 6 As shown. Figure 6 In, o p and g are the centers of the small gear and the large gear respectively. Counterclockwise is positive and clockwise is negative. The angle ψ between the meshing line and the y-axis pg It can be expressed as:

[0194]

[0195] In the formula, α pg is the pressure angle. pg is the relative azimuth angle of the two gears.

[0196] Spur gear pair unit stiffness matrix K pgeIt can be expressed as:

[0197]

[0198] Among them, the vector α pg It can be expressed as:

[0199] α pg =[-sinψ pg cosψ pg 0 0 0 sgn×r bp sinψ pg -cosψ pg 0 0 0 sgn×r bg ].

[0200] Step 3. Calculate the equivalent stiffness of the involute spline

[0201] Based on the method of material mechanics, the involute gear teeth of the spline are simplified into a cantilever beam with variable cross-section on an elastic basis. It is generally believed that the elastic deformation of meshing gear teeth mainly includes the deformation caused by bending, shearing and compression of the cantilever beam, the additional elastic deformation of the tooth root, and the contact deformation at the meshing point of the tooth surface.

[0202] Based on the slicing method, the jth gear tooth is divided into n small pieces. The kth gear tooth is as follows: Figure 7 and Figure 8 As shown. Let the thickness of the kth sheet be t k , the corresponding tooth thickness at the cross section is s k , the section moment of inertia is I k Each small piece deforms under the action of unit force. Therefore, the flexibility of the gear cantilever beam is Bj It can be expressed as:

[0203]

[0204] In the formula, β j is the load F on the jth gear tooth j The angle with the positive direction of the y-axis. L k is the kth piece and the load F j The distance between the action points along the x direction. B is the tooth width. ν is the Poisson's ratio of the gear tooth material. E ν is the equivalent elastic modulus of the gear tooth material. When the ratio of tooth width to pitch circle tooth thickness is B / s p ≥5, E ν =E / (1-ν 2 ). When B / s p <5, E ν =E. E is the elastic modulus of the gear tooth material.

[0205] When calculating the bending, shear and compression deformation of gear teeth, it is assumed that the root of the gear tooth is a rigidly fixed cantilever beam, but in fact the root of the gear tooth is elastic, so the additional deformation of the meshing point caused by the elasticity of the tooth root must be considered.

[0206] When B / s p ≥5, the additional flexibility of the tooth root λ Mj It can be expressed as:

[0207]

[0208] Where, L f is the tooth root and load F j The distance between the points of action along the x direction. s k is the tooth thickness corresponding to the tooth root.

[0209] When B / s p When <5, the additional flexibility of the tooth root can be expressed as:

[0210]

[0211] The nonlinear contact compliance of the tooth surface meshing point can be expressed as:

[0212]

[0213] In the formula, B e is the effective tooth width.

[0214] The meshing stiffness k of the jth pair of teeth j It can be simply expressed as:

[0215]

[0216] When the external teeth are active teeth, the equivalent radial stiffness k of the spline connection sr It can be expressed as:

[0217]

[0218] In the formula, is the azimuth angle of the jth pair of teeth.

[0219] Equivalent bending stiffness k sb It can be expressed as:

[0220]

[0221] Equivalent torsional stiffness k st It can be expressed as:

[0222]

[0223] In the formula, r sbis the base circle radius of the spline.

[0224] Step 4. Establish the static model of the spline-connected gear rotor-bearing system

[0225] The finite element model of the spline-connected gear rotor-bearing system is as follows Fig. 9 As shown. The rotating shaft structure is modeled using the Timoshenko beam unit. The system mainly includes three shafts, shaft I, shaft II and shaft III. The shaft I is connected to the shaft II by a cylindrical spur gear 1, and the shaft II is connected to the shaft III by a spline 2. The shaft I is the driving shaft. The shaft I is divided into thirteen sections by fourteen nodes, the shaft II is divided into seven sections by six nodes, and the shaft III is divided into twenty-four sections by twenty-five nodes. A concentrated mass unit 3 is provided on the fourth node, the sixth node, the thirteenth node and the seventh node of the shaft I, respectively, and an elastic support element 4 is provided on the second node and the eleventh node of the shaft I, and the second node and the twenty-fourth node of the shaft III, respectively, a pinion 5 is provided at the ninth node of the shaft I, a large gear 6 is provided at the second node of the shaft II, and a deep groove ball bearing 7 is provided at the third node and the fifth node of the shaft II. The input torque is applied to the concentrated mass unit at the sixth node of the shaft I, and the load torque is applied to the twenty-fourth support element of the shaft III.

[0226] Based on the unit stiffness matrices of bearings, spur gear pairs and splines calculated in the previous article, and coupling them with the stiffness matrix of the shaft system, the total stiffness matrix K of the system can be obtained. The static equation of the system can be expressed as:

[0227] KU=F S ,

[0228] Where K is the total stiffness matrix of the system. S is the static load vector of the system. U is the displacement vector of each node of the system.

[0229] The total stiffness matrix K of the system and the static load vector F S It can be expressed as:

[0230]

[0231] In the formula, K S , K SE , K B , K Sp and K G are the stiffness matrices of the shaft, support unit, bearing, spline and spur gear pair respectively. G is the torque vector. F kn G is the restoring force vector caused by no-load static transmission error. S , G LM, G G represent the gravity vectors of the shaft, concentrated mass and gear pair respectively.

[0232] Step 5. Model solving process.

[0233] The bearing stiffness can generally be calculated based on the preload and support load. However, in the actual spline-connected gear rotor-bearing transmission system, it is difficult to directly obtain the bearing support load to calculate the bearing stiffness. In addition to the abnormal preload caused by raceway misalignment, deep groove ball bearings must also withstand the support load caused by input torque, load torque and gravity. The bearing stiffness is determined by the support load and the abnormal preload, and the support load is also affected by the bearing stiffness. Therefore, there is a bidirectional coupling effect between the bearing stiffness and the support load. To solve this problem, a method for calculating the bearing support stiffness that takes into account factors such as bearing ring misalignment, spur gear pair meshing, spline connection, and shaft flexibility is proposed. The calculation process of the bearing stiffness in the spline-connected gear rotor-bearing transmission system is as follows: Figure 1 shown.

[0234] First, the abnormal preload of the bearing can be calculated through the misalignment parameters of the raceway. Based on the initial values ​​of the abnormal preload and the support load, the support stiffness of the bearing can be estimated in advance. Then, the unit stiffness matrices of the bearing, spur gear pair, spline and shaft are combined to obtain the total stiffness matrix of the system. In addition, the deformation and support load of the bearing node can be calculated based on the static load vector and the total stiffness matrix of the system. The initial value of the support load is replaced with the calculated support load, and the above calculation process is repeated until the support load meets the tolerance condition. Finally, the support stiffness of the bearing in the system is obtained.

[0235] Example

[0236] This embodiment takes the system model established in step 4 as an example, and based on the solution process given in step 5, the stiffness of the bearing can be calculated. The bearing parameters in the system are shown in Table 1. The shaft system parameters are shown in Table 2. The concentrated mass parameters are shown in Table 3. The rotating support element adopts a diagonal stiffness matrix, and its stiffness parameters are given based on experience, as shown in Table 4.

[0237] Table 1. Parameters of deep groove ball bearings SKF6011

[0238]

[0239]

[0240] Table 2. Parameters of the shaft system

[0241]

[0242] Table 3. Parameters of concentrated mass

[0243]

[0244] Table 4. Parameters of support elements

[0245]

[0246] The azimuth angle φto of the outer ring misalignment axis is set to 0. Assume that the counterclockwise rotation of the outer ring around the misalignment axis is positive. The opposite is negative. Rolling the ball from the current position to the next ball position is regarded as the ball cycle, and the stiffness results of two ball cycles are taken for analysis.

[0247] The variation of bearing time-varying stiffness with misalignment angle is as follows: Figure 10-12 As shown. The results show that when the misalignment angle is within the range of -1.2mrad to 1.2mrad, the support stiffness does not change with the change of the misalignment angle. This is because the small misalignment does not eliminate the bearing clearance and does not produce abnormal preload. Since the azimuth angle of the misaligned axis is 0, the misalignment angle mainly causes the overturning moment M around the x-axis. mistx . M mistx K yy The effect on K xx The impact of K yy The change of K is mainly caused by abnormal preload. xx The change of K is mainly caused by the support load. At the same time, the support load is also affected by the bearing stiffness. Therefore, misalignment has a great influence on K xx The impact on K is indirect. yy The influence of K is direct. Excessive misalignment angle will cause the contact ellipse between the ball and the raceway to be truncated by the raceway ribs, thereby weakening the contact stiffness. yy In the range of -6mrad to -4.8mrad and 4.8mrad to 6mrad, it shows a downward trend. In the range of -2.4mrad to -1.2mrad and 1.2mrad to 2.4mrad, the sudden change of the supporting load caused by misalignment leads to K xx In the range of -6mrad to -1.2mrad and 1.2mrad to 6mrad, K yy and K zz It increases with the increase of misalignment angle in positive and negative directions.

[0248] When the misalignment angle θ to When the time-varying stiffness of the bearing changes with the input torque T1, the time-varying stiffness of the bearing changes with the input torque T1, respectively. Figure 13-18 As shown in Figure 2, it can be seen that in both cases, the increase in input torque leads to an increase in both the axial and radial stiffness of the bearing. In the case of misalignment, the fluctuation in bearing stiffness is more obvious.

[0249] When the misalignment angle θ to When n1 is 0 and 4 mrad, the time-varying stiffness of the bearing changes with the speed n1 as follows: Figure 19-24 As shown. It can be seen that in the healthy case (θ to =0), the increase in speed significantly reduces the radial and axial stiffness of the bearing. to =4mrad), the speed has no significant effect on the bearing stiffness. This is because the misalignment causes the balls to be pressed against the raceway. The centrifugal force and gyroscopic moment caused by the speed have far less influence on the balls than the contact force between the balls and the raceway caused by the misalignment.

[0250] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for calculating the stiffness of misaligned bearings in a spline-connected gear rotor system, characterized in that: The calculation method comprises the following steps: Establish a quasi-static model of deep groove ball bearings, consider the quasi-static model of deep groove ball bearings under the coupling effect of misalignment and support load, and establish the bearing unit stiffness matrix; Considering the influence of the rotation direction of the active gear, according to the meshing stiffness and meshing line y The angle between the shafts establishes the stiffness matrix of the spur gear pair unit; Considering the additional deformation of the meshing point caused by the elasticity of the tooth root, the spline unit stiffness matrix is ​​established; The rotating shaft structure is modeled by using Timoshenko beam elements to obtain a finite element model of a spline-connected gear rotor-bearing system, and the statics equation of the system is obtained according to the finite element model of the spline-connected gear rotor-bearing system; The quasi-static model of the deep groove ball bearing is solved according to the statics equation, the bearing unit stiffness matrix, the spur gear pair unit stiffness matrix and the spline unit stiffness matrix, and the misaligned bearing stiffness is calculated.

2. The method for calculating the bearing stiffness with misalignment in a spline-connected gear rotor system according to claim 1, characterized in that: The specific steps of establishing the quasi-static model of deep groove ball bearings are as follows: In the case of angular misalignment, the relative axial displacement equation and the relative radial displacement equation of the inner raceway curvature center are established to obtain the relative axial displacement and relative radial displacement of the inner raceway curvature center; The relative axial displacement of the center of curvature of the inner raceway is: The relative radial displacement of the center of curvature of the inner raceway is: in, r po is the radial distance between the bearing center and the center of curvature of the outer raceway; θ to is the misalignment angle of the outer raceway; φ to The azimuth angle of the outer ring raceway away from the center axis; φ j For the j The azimuth of the ball; S a is the axial clearance of the bearing; S r is the radial clearance of the bearing; The ball position is divided into a load-bearing area and a non-load-bearing area. According to the relative axial displacement and relative radial displacement of the center of curvature of the inner raceway, a load-bearing area determination formula is established to determine the ball position. The ball that satisfies the load-bearing area determination formula is located in the load-bearing area, and the ball that does not satisfy the load-bearing area determination formula is located in the non-load-bearing area. The formula for determining the bearing area is: in, BD is the distance between the centers of curvature of the inner and outer ring raceways in the initial contact state; α 0 is the initial contact angle; X 2j is the radial distance from the center of the ball to the center of curvature of the outer raceway; F i is the curvature coefficient of the inner raceway; d b is the diameter of the ball; B 2j is the relative radial displacement of the center of curvature of the inner raceway; When the ball is in the load-bearing area, considering that the contact ellipse between the ball and the raceway is truncated by the raceway rib due to excessive misalignment, resulting in a reduction in load-bearing capacity, the contact stiffness is rewritten using the contact ellipse truncation rate, and a calculation expression for the contact stiffness considering the rib is established; The contact ellipse cutoff rate is: In the formula, α ij / oj is the contact angle between the ball and the inner or outer raceway; φ aij / oj Indicates the angle corresponding to the major semi-axis of the contact ellipse between the ball and the inner and outer ring raceways; ϕ i / o Indicates the angle corresponding to the inner or outer raceway rib; The contact stiffness calculation expression is: in, κ ij / oj is the ellipse eccentricity parameter; E e is the equivalent elastic modulus; is the complete elliptic integral of the first kind; is the complete elliptic integral of the second kind; is the curvature and; The contact load is obtained according to the contact stiffness calculation expression; the contact load is: in, δ ij / oj It is the contact deformation between the ball and the inner and outer raceways; According to the contact stiffness calculation expression and the contact load, the equilibrium equation of the ball in the load-bearing area and the equilibrium equation of the ball in the non-load-bearing area are established; The balance equation of the ball in the load-bearing area is: in, X 1j is the axial distance from the center of the ball to the center of curvature of the outer raceway; f o is the curvature coefficient of the outer raceway; F cj is the centrifugal force; M gj is the gyroscopic torque; μ j is the average friction coefficient between the ball and the inner and outer raceways; Q ij For the j Contact load on inner ring of each rolling element; Q oj For the j The contact load of the outer ring of each rolling element; The balance equation of the ball in the load-bearing area is: The abnormal preload force caused by angular misalignment is obtained, and the abnormal preload force is: in, F mistz Abnormal preload in the axial direction caused by misalignment; M mistx and M misty The surrounding caused by misalignment x Axis and y Abnormal preload torque of the shaft; r pj is the radial distance between the bearing center and the inner raceway contact position; r i is the radius of curvature of the inner ring raceway; According to the abnormal preload and support load, the relative axial displacement equation and relative radial displacement equation of the inner raceway curvature center are established; The relative axial displacement equation of the inner raceway curvature center is: The relative radial displacement equation of the inner raceway curvature center is: in, δ z For the inner edge z Axial displacement of the shaft; θ x and θ y Around the inner circle x Axis and y Angular displacement of the shaft; δ x For the inner edge x Radial displacement of the shaft; δ y For the inner edge y Radial displacement of the shaft; S a is the axial clearance of the bearing; S r is the radial clearance of the bearing; According to the bearing force analysis, the force balance equation of the bearing inner ring under the condition of angular misalignment is established; the force balance equation of the bearing inner ring is: in, F x and F y The edge acting on the inner ring x Axis and y Radial support load of the shaft; F z For along z Axial support load of the shaft; M x and M y To act on the inner circle around x Axis and y The overturning moment of the shaft, N b is the number of balls.

3. The method for calculating the bearing stiffness with misalignment in a spline-connected gear rotor system according to claim 2, characterized in that: The expression of the bearing unit stiffness matrix is: 。 4. The method for calculating the bearing stiffness with misalignment in a spline-connected gear rotor system according to claim 1, characterized in that: The specific steps of establishing the stiffness matrix of the spur gear pair unit are as follows: Based on the gear tooth load contact analysis method and the force and deformation coordination conditions of gear meshing, the gear meshing equilibrium equation is established; the gear meshing equilibrium equation is: in, λ c is the contact flexibility matrix; λ b is the overall flexibility matrix; I is a unit vector; F n is the normal contact force vector of the contact point; ste is the transmission error; ε is the initial separation gap vector; T p is the input torque applied to the pinion; r bp is the base circle radius of the pinion; The meshing stiffness is obtained by solving the equilibrium equation of gear meshing; the meshing stiffness is: in, nlste is the no-load transfer error; Considering the influence of the rotation direction of the driving gear, a symbol variable is introduced; the symbol variable is: in, ω p is the angular velocity of the pinion; The spur gear pair is regarded as two concentrated masses, and the meshing between the two concentrated masses is simulated by springs and dampers. y The angle between the meshing line and the axis is defined as positive in the counterclockwise direction and negative in the clockwise direction. y The angle between the axes is: in, α pg is the pressure angle; ϕ pg is the relative azimuth angle of the two gears; Considering the influence of the rotation direction of the active gear, the stiffness matrix of the spur gear pair unit is established according to the meshing stiffness and the angle between the meshing line and the y-axis; the stiffness matrix of the spur gear pair unit is: Among them, the vector α pg for: 。 5. The method for calculating the bearing stiffness with misalignment in a spline-connected gear rotor system according to claim 1, characterized in that: The specific steps of establishing the spline unit stiffness matrix are as follows: Based on the method of material mechanics, the involute gear teeth of the spline are simplified into a cantilever beam with variable cross-section on an elastic basis. Based on the slicing method, j The gear teeth are divided into n A small piece is used to establish the flexibility equation of the gear cantilever beam; the flexibility equation of the gear cantilever beam is: in, β j For the j Load on each gear tooth F j and y The angle of the positive axis; L k For the k Sheet and load F j Between the points of action x Distance of direction; B is the gear tooth width; ν is the Poisson's ratio of the gear tooth material; E ν is the equivalent elastic modulus of the gear tooth material; when the ratio of tooth width to pitch thickness is B / s p ≥5, E ν =E / (1- ν 2 );when B / s p <5 o'clock, E ν =E , E is the elastic modulus of the gear tooth material; t k is the thickness of the kth gear tooth; I k is the section moment of inertia; Considering the additional deformation of the meshing point caused by the elasticity of the tooth root; B / s p When ≥5, the additional flexibility of the tooth root is: in, L f tooth root and load F j Between the points of action x Distance of direction; s f is the tooth thickness corresponding to the tooth root; when B / s p When <5, the additional flexibility of the tooth root is: The nonlinear contact compliance of the tooth surface meshing point is obtained, and the nonlinear contact compliance of the tooth surface meshing point is: in, B e is the effective tooth width; According to the flexibility equation of the gear cantilever beam, the nonlinear contact flexibility and the additional flexibility equation of the tooth root, the meshing stiffness equation is established; the meshing stiffness equation is: The spline unit stiffness matrix includes equivalent radial stiffness, equivalent bending stiffness and equivalent torsional stiffness, and the equivalent radial stiffness, equivalent bending stiffness and equivalent torsional stiffness are calculated respectively according to the meshing stiffness; The equivalent radial stiffness is: in, φ sj For the j The azimuth angle of the gear teeth; The equivalent bending stiffness is: The equivalent torsional stiffness is: in, r sb is the base circle radius of the spline.

6. The method for calculating the bearing stiffness with misalignment in a spline-connected gear rotor system according to claim 1, characterized in that: The finite element model of the spline-connected gear rotor-bearing system comprises: a shaft I, a shaft II and a shaft III, wherein the shaft I and the shaft II are connected by a cylindrical spur gear (1), and the shaft II and the shaft III are connected by a spline (2), the shaft I is a driving shaft, the shaft I is divided into thirteen sections by fourteen nodes, the shaft II is divided into seven sections by six nodes, and the shaft III is divided into twenty-four sections by twenty-five nodes, a concentrated mass unit (3) is respectively arranged at the fourth node, the sixth node, the thirteenth node of the shaft I and the seventh node of the shaft II, an elastic support element (4) is respectively arranged at the second node and the eleventh node of the shaft I and the second node and the twenty-fourth node of the shaft III, a small gear (5) is arranged at the ninth node of the shaft I, a large gear (6) is arranged at the second node of the shaft II, and a deep groove ball bearing (7) is respectively arranged at the third node and the fifth node of the shaft II.

7. The method for calculating the bearing stiffness with misalignment in a spline-connected gear rotor system according to claim 1, characterized in that: The specific steps of solving the quasi-static model of the deep groove ball bearing are as follows: The bearing stiffness of the bearing is estimated in advance based on the initial values ​​of the abnormal preload and the bearing load; The bearing unit stiffness matrix, the spur gear pair unit stiffness matrix and the spline unit stiffness matrix are coupled with the stiffness matrix of the system to obtain a total stiffness matrix of the system; The deformation and support load of the bearing node are calculated based on the static load vector and the total stiffness matrix of the system; The initial value of the support load is replaced by the calculated support load and the above calculation process is repeated until the support load meets the preset tolerance condition, and the support stiffness of the bearing in the system is obtained.

8. The method for calculating the bearing stiffness with misalignment in a spline-connected gear rotor system according to claim 7, characterized in that: The total stiffness matrix is: The static load vector is: in, K S is the shaft stiffness matrix; K SE is the stiffness matrix of the support element; K B is the bearing stiffness matrix; K Sp is the spline stiffness matrix; K G is the stiffness matrix of the spur gear pair; T G is the torque vector; F kn is the restoring force vector caused by no-load static transmission error; G S is the gravity vector of the axis; G LM is the gravity vector of the concentrated mass; G G is the gravity vector of the gear pair.

9. The method for calculating the bearing stiffness with misalignment in a spline-connected gear rotor system according to claim 1, characterized in that: The pseudo-static model of the deep groove ball bearing is established based on assumptions, which include: regarding the deep groove ball bearing, spline and bearing as stiffness units, ignoring the mass of the stiffness units, ignoring the influence of temperature, lubrication and sealing elements, regarding the contact between the ball and the raceway as Hertzian elastic contact between rigid bodies, and regarding the angular misalignment between the outer ring and the inner ring as an abnormal preload state.

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