A method for analyzing dynamic characteristics of high-speed ball bearing considering deformation of race

Abstract

The application provides a high-speed ball bearing dynamic characteristic analysis method considering raceway deformation, and belongs to the technical field of bearing dynamics. The method needs to determine basic parameters such as bearing geometric parameters, material properties and operation conditions, and establishes a high-speed angular contact ball bearing dynamics model under positioning and pre-tightening based on the Hertz contact theory and by comprehensively considering the influences of centrifugal force, pre-tightening force, raceway thermal deformation and gyroscopic moment of the bearing. Then, the influences of the bearing pre-tightening force and rotating speed on the bearing dynamic characteristic parameters under different working conditions are solved by the Newton iteration method, and the relationship between the pre-tightening force, rotating speed and bearing contact angle, contact load, centrifugal force and other parameters is obtained. The method has the characteristics that the influences of thermal effect and centrifugal effect on the bearing are considered, the bearing contact angle and contact load under different cutting conditions can be accurately calculated, and the method can provide a reference for the design and use of the motorized spindle bearing.

Description

Technical Field

[0001] This invention relates to a method for analyzing the dynamic characteristics of ball bearings in high-speed electric spindles considering the deformation of the bearing races, and belongs to the field of dynamic technical analysis of electric spindle bearings in CNC machine tool spindles. Background Technology

[0002] The electric spindle unit is a core component of high-speed CNC machine tools, and its performance directly determines the machining accuracy and quality of the entire machine tool. Bearings, as key components supporting the rotation of the electric spindle, significantly impact its accuracy. Angular contact ball bearings are widely used due to their low energy consumption, low friction, and ability to simultaneously bear axial and radial forces. Throughout the operation of the electric spindle, the stiffness, load, and installation method of the angular contact ball bearing all affect its performance. Furthermore, the Hertz contact force between the bearing balls and the raceway, the bearing preload, the contact angle of the bearing balls, and centrifugal force all influence the spindle's production quality and accuracy, leading to a decrease in overall spindle performance. Therefore, developing a dynamic characteristic analysis method for high-speed ball bearings that considers ring deformation is of significant guiding importance for improving electric spindle performance. Summary of the Invention

[0003] The purpose of this invention is to provide a method for analyzing the dynamic characteristics of high-speed electric spindle ball bearings considering ring deformation. This method is based on Hertz contact theory, comprehensively considering the effects of preload, gyroscopic torque, thermal effects, and centrifugal effects on the bearing. Using Newton's iteration method, it can accurately calculate the bearing contact angle and contact load under different cutting conditions, establishing a dynamic model of a high-speed angular contact ball bearing under positioning preload, and analyzing the changes in bearing preload and rotational speed on the bearing's dynamic characteristic parameters under different operating conditions. The specific implementation steps are as follows:

[0004] Step (1) Discuss the effect of the heat generated by the spindle at high speed on the radial thermal deformation of the bearing rings, and establish its radial thermal expansion deformation model;

[0005] Step (2) Discuss the effect of the centrifugal force generated when the bearing rotates with the shaft on the centrifugal expansion of the bearing inner ring and the shaft, and establish its centrifugal expansion deformation model;

[0006] Step (3) Establish a dynamic model of the high-speed angular contact ball bearing, considering the effects of preload, gyroscopic torque, thermal effect and centrifugal effect on its deformation;

[0007] Step (4) Determine the bearing model and operating parameters, solve the bearing contact parameter set by iterative method, and explore the influence of preload and rotational speed on its dynamic characteristics.

[0008] The specific steps are described below in conjunction with the accompanying drawings. Attached Figure Description

[0009] Figure 1 Flowchart for bearing dynamics modeling

[0010] Figure 2 Geometric relationship diagram for bearing deformation compatibility

[0011] Figure 3 Bearing force diagram

[0012] Figure 4 Flowchart for bearing calculation

[0013] Figure 5a The relationship between bearing contact angle and rotational speed

[0014] Figure 5b The relationship between bearing contact angle and radial force

[0015] Figure 6a The relationship between bearing contact load and rotational speed

[0016] Figure 6b Relationship between bearing contact load and radial force

[0017] Figure 7a The relationship between bearing centrifugal force and rotational speed

[0018] Figure 7b The relationship between the centrifugal force and radial force of the bearing

[0019] Figure 8a The relationship between bearing friction torque and rotational speed

[0020] Figure 8b The relationship between bearing friction torque and radial force Detailed Implementation

[0021] A method for analyzing the dynamic characteristics of high-speed electric spindle ball bearings considering ring deformation. The implementation of this invention will be described in detail below with reference to the accompanying drawings. Figure 1 A flowchart for bearing dynamics modeling that comprehensively considers the thermal deformation and centrifugal deformation of the bearing rings.

[0022] Step (1) Establish a radial thermal expansion deformation model, including the following:

[0023] Generally, the thermal deformation of a ring-shaped part is calculated as u = aΔTd. T (1)

[0024] In the formula, u is the amount of thermal deformation, and d T Where is the diameter of the part, 'a' is the coefficient of thermal expansion, and ΔT is the temperature rise.

[0025] From equation (1), the thermal deformation of the bearing inner raceway diameter and its inner diameter are respectively...

[0026] u i '=a i ΔT i d i u i =a i ΔT i d i (2)

[0027] When the temperature increases by ΔT s At that time, the deformation of the outer diameter of the shaft is u. s =a s ΔT s (1+v s )d(3) Combining equations (2) to (3), the radial thermal deformation of the bearing inner ring under the influence of the rotating shaft is:

[0028] u i =a i ΔT i d i +[a s ΔT s (1+v s )-a i ΔT i ]d 2 / d i (4)

[0029] In the formula, a i a s These are the coefficients of thermal expansion of the inner ring and the shaft, respectively.

[0030] Similarly, when the temperature increases by ΔT h At that time, the radial thermal deformation of the bearing outer ring under the influence of the bearing housing is

[0031] u e =a h ΔT h (1+v h )D e (5)

[0032] In the formula, v h a h These are the Poisson's ratio and coefficient of thermal expansion of the bearing housing, D. e The outer diameter of the channel.

[0033] Furthermore, the thermal deformation of the bearing rolling element can be obtained as u. b =a b ΔT b D b (6)

[0034] In the formula, a b D b These are the coefficient of thermal expansion of the rolling element and the diameter of the ball, ΔT. b To increase the rolling body temperature.

[0035] Combining equations (4) to (6), the radial thermal deformation of the bearing ring is: u r =u i -u e -2u b (7)

[0036] Step (2) Establish the bearing centrifugal expansion deformation model, including the following:

[0037] When the spindle rotates at high speed, the bearing will generate centrifugal force, which causes the inner ring of the bearing to undergo centrifugal expansion deformation. According to the theory of elasticity, the amount of centrifugal expansion deformation of the shaft and the inner ring is:

[0038]

[0039]

[0040] In the formula, u cs u ci These represent the centrifugal expansion deformation of the shaft and the inner ring, respectively; ρ s v s E s and d s These are the density, Poisson's ratio, elastic modulus, and inner diameter of the shaft, respectively; ρ i v i and E i These represent the density, Poisson's ratio, and elastic modulus of the inner ring, respectively; ω is the angular velocity of the inner ring; d, d i These are the inner diameter of the inner ring and the diameter of the inner channel, respectively.

[0041] Step (3) Establish the dynamic model of the angular contact ball bearing, including the following:

[0042] 1) Bearing displacement compatibility equation

[0043] The deformation geometry of the bearing under preload positioning is as follows: Figure 2 As shown. The outer ring and bearing housing are fixed. Therefore, during high-speed operation, the center of curvature of the outer raceway remains constant, while the center of curvature of the inner raceway and the position of the ball centers change. Under the simultaneous action of axial force, radial force, and torque, the relative displacement of the inner and outer rings will change. Under load, considering the thermal and centrifugal effects of the bearing rings, the axial and radial distances A between the initial and final positions of the centers of curvature of the inner and outer raceways will change. 1j A 2j They are represented as follows:

[0044]

[0045] Where A is the distance between the centers of curvature of the inner and outer channels when unloaded, α0 is the initial contact angle, and δ a δ r And θ represents the axial, radial, and angular relative displacements of the bearing, u a u r These represent the axial and radial thermal deformation of the bearing raceway, u. ci This represents the amount of centrifugal deformation.

[0046] The displacement compatibility equation is obtained from the Pythagorean theorem and the geometric relationship of raceway contact deformation:

[0047]

[0048] Among them, X 1j X 2j These are the axial and radial distances between the outer ring curvature center and the ball center, respectively, f i f o These are the curvature coefficients of the inner and outer channels, δ ij δ oj These represent the Hertz contact deformation of the balls with the inner and outer rings under load.

[0049] 2) Force balance equations for bearings

[0050] Force analysis of bearing rolling elements as follows Figure 3 As shown. Force analysis of each bearing rolling element yields the following force balance equation:

[0051]

[0052] In the formula, Q ij Q oj These are the contact loads between the balls and the raceways of the inner and outer rings of the bearing, F. cj M gj These represent the centrifugal force and gyroscopic torque acting on the ball bearing, respectively, where the centrifugal force... Gyro torque w represents the angular velocity of the bearing.

[0053] When a bearing operates at high speed, the forces acting on it reach a state of equilibrium, and the net force is zero. Therefore, the equilibrium equation for the entire bearing is:

[0054]

[0055] In the formula, F a F r M and M represent the axial force, radial force, and torque acting on the bearing, respectively, and z represents the number of balls.

[0056] Step (4) solves for the dynamic characteristic contact parameters of the bearing, including the following:

[0057] 1) Solving nonlinear equations

[0058] Taking into account the combined effects of bearing ring thermal and centrifugal effects on bearing deformation, a modified displacement compatibility model was used. MATLAB software was employed for dynamic analysis to obtain a series of dynamic characteristic contact parameters. The calculation process is as follows: Figure 4 As shown.

[0059] First, the basic parameters of the bearing are determined. Based on the traditional model, the bearing, spindle shaft, and bearing housing are considered as a whole. Combining the spindle temperature field model, the temperature of each part of the spindle is obtained, and the radial and axial thermal deformation of the inner and outer rings of the bearing, as well as the centrifugal expansion deformation of the inner ring, are calculated. The bearing parameters are shown in Table 1.

[0060]

[0061] Then, based on the bearing parameters, the initial contact angle is set, and the nonlinear equation system consisting of the bearing coordination equation (11), the rolling body force balance equation (12), and the bearing overall balance equation (13) is solved using the Newton-Raphson iteration method. The axial displacement δ is set as the iteration variable. a Radial displacement δ r The angular displacement θ is iterated repeatedly until the variables meet the preset accuracy requirements.

[0062] The calculation results of the bearing thermal deformation and centrifugal expansion deformation were then imported into the bearing displacement deformation equation to obtain the corrected relational equation. The corrected model was iterated multiple times until the convergence condition was met, and the bearing dynamic characteristic contact parameters, including centrifugal force and gyroscopic torque, were obtained.

[0063] 2) Analysis of dynamic contact characteristics of bearings

[0064] Based on different bearing operating conditions, considering thermal and centrifugal effects, the influence of changes in axial force, radial force, and rotational speed on the bearing contact angle, contact load, centrifugal force, and gyroscopic torque is analyzed. Different contact states of the bearing balls are obtained, and their force variation trends are studied. Figures 5-8 show the trends of bearing contact angle, contact load, centrifugal force, and gyroscopic torque as a function of radial force and rotational speed.

Claims

1. A method for analyzing the dynamic characteristics of high-speed ball bearings considering ring deformation, characterized in that, By employing Hertz contact theory and comprehensively considering the effects of preload, gyroscopic torque, thermal effects, and centrifugal effects on bearings, the bearing contact angle and contact load under different cutting conditions can be accurately calculated. A dynamic model of a high-speed angular contact ball bearing under positioning preload is established, and the influence of bearing preload and rotational speed on the dynamic characteristic parameters of the bearing under different operating conditions is analyzed. The specific implementation steps are as follows: Step (1) When the spindle rotates at high speed, its heat generation affects the radial thermal deformation of the bearing rings, and a radial thermal expansion deformation model is established. Step (2) Discuss the effect of the centrifugal force generated when the bearing rotates with the shaft on the centrifugal expansion of the bearing inner ring and the shaft, and establish its centrifugal expansion deformation model; Step (3) Establish a dynamic model of the high-speed angular contact ball bearing, considering the effects of preload, gyroscopic torque, thermal effect and centrifugal effect on its deformation; Step (4) Determine the bearing model and operating parameters, solve the bearing contact parameter set by iterative method, and establish the relationship between the influence of preload and rotation speed on its dynamic characteristics; Step (1) includes the following: Calculate the thermal deformation of ring-shaped parts as follows: (1); In the formula, This is the amount of thermal deformation. For the diameter of the part, The coefficient of thermal expansion is... This refers to the temperature rise. From equation (1), the thermal deformation of the bearing inner raceway diameter and its inner diameter are respectively... ， （2）； When the temperature rises At that time, the deformation of the outer diameter of the shaft is (3); Combining equations (2) to (3), the radial thermal deformation of the bearing inner ring under the influence of the rotating shaft is: （4）； In the formula, , These are the coefficients of thermal expansion of the inner ring and the shaft, respectively. When the temperature rises At that time, the radial thermal deformation of the bearing outer ring under the influence of the bearing housing is: （5）； In the formula, , These are Poisson's ratio and coefficient of thermal expansion of the bearing housing, respectively. The outer diameter of the channel; The thermal deformation of the bearing rolling elements is (6); In the formula, , These are the coefficient of thermal expansion of the rolling element and the ball diameter, respectively. To increase body temperature; Combining equations (4) to (6), the radial thermal deformation of the bearing ring is: (7); Step (2) includes the following: When the spindle rotates at high speed, the bearing will generate centrifugal force, which causes the inner ring of the bearing to undergo centrifugal expansion deformation. According to the theory of elasticity, the amount of centrifugal expansion deformation of the shaft and the inner ring is: （8）； （9）； In the formula, , These are the centrifugal expansion deformations of the shaft and the inner ring, respectively. , , and These are the density, Poisson's ratio, elastic modulus, and inner diameter of the shaft, respectively. , and These represent the density, Poisson's ratio, and elastic modulus of the inner ring, respectively; ω is the angular velocity of the inner ring. , These are the inner diameter of the inner ring and the diameter of the inner channel, respectively. Step (3) includes the following: 1) Bearing displacement compatibility equation; Since the outer ring and bearing housing are fixed, during high-speed operation, the center of curvature of the outer raceway remains constant, while the center of curvature of the inner raceway and the position of the ball centers change. Under the simultaneous action of axial force, radial force, and torque, the relative displacement between the inner and outer rings will change. Under load, considering the thermal and centrifugal effects of the bearing rings, the axial and radial distances between the initial and final positions of the centers of curvature of the inner and outer raceways will change. , They are represented as follows: （10）； in, This is the distance between the centers of curvature of the inner and outer channels when unloaded. The initial contact angle, , and These are the axial, radial, and angular relative displacements of the bearing. , These are the axial and radial thermal deformations of the bearing raceway, respectively. This represents the amount of centrifugal deformation. The displacement compatibility equation is obtained from the Pythagorean theorem and the geometric relationship of raceway contact deformation: （11）； in, , These are the axial and radial distances between the outer ring curvature center and the ball center, respectively. , These are the curvature coefficients of the inner and outer channels, respectively. , These represent the Hertz contact deformation of the ball bearings with the inner and outer rings under load. 2) The force balance equation of the bearing; Force analysis of each rolling element of the bearing yields the following force balance equation: （12）； In the formula, , These are the contact loads between the balls and the raceways of the inner and outer rings of the bearing, respectively. , These represent the centrifugal force and gyroscopic torque acting on the ball bearing, respectively, where the centrifugal force... gyro torque , The bearing angular velocity; When a bearing operates at high speed, the forces acting on it reach a state of equilibrium, and the net force is zero. Therefore, the equilibrium equation for the entire bearing is: （13）； In the formula, , , These are the axial force, radial force, and torque acting on the bearing, respectively. This represents the number of balls.

2. The method for analyzing the dynamic characteristics of high-speed ball bearings considering ring deformation according to claim 1, characterized in that: Step (4) includes the following: 1) Solving nonlinear equation systems; Considering the combined effects of bearing ring thermal and centrifugal effects on bearing deformation, and based on the modified displacement compatibility model, MATLAB analysis software was used to perform dynamic analysis on the bearing, and a series of dynamic characteristic contact parameters were obtained. The specific solution process is as follows: First, determine the basic parameters of the bearing. Based on the traditional model, consider the bearing, the spindle shaft and the bearing housing as a whole. Combine the spindle temperature field model to obtain the temperature of each part of the spindle and calculate the radial and axial thermal deformation of the inner and outer rings of the bearing and the centrifugal expansion deformation of its inner ring. Then, based on the bearing parameters, the initial contact angle is set, and the nonlinear equation system consisting of the bearing coordination equation (11), the rolling body force balance equation (12), and the bearing overall balance equation (13) is solved using the Newton-Raphson iteration method. The axial displacement is set as the iteration variable. radial displacement and angular displacement The process is repeated multiple times until the variables meet the pre-defined precision requirements. The calculation results of the bearing thermal deformation and centrifugal expansion deformation were then imported into the bearing displacement deformation equation to obtain the corrected relational equation. The corrected model was iterated multiple times until the convergence condition was met, and the bearing dynamic characteristic contact parameters, including centrifugal force and gyroscopic torque, were obtained. 2) Analysis of bearing dynamic contact characteristics; Based on different operating conditions of the bearing, and considering thermal and centrifugal effects, the influence of changes in axial force, radial force, and rotational speed on the bearing contact angle, contact load, centrifugal force, and gyroscopic torque is analyzed to obtain the different contact states of each ball in the bearing and the trend of force changes.

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