**0 Introduction**

The working load of the mechanical system usually manifests as complex loads such as dynamic load and/or impact load. As the key bearing component of the mechanical system, the rolling bearing has an important influence on the working performance, life and reliability of the mechanical system. From the point of view of system dynamics, rolling bearings are essentially composed of structural elements such as rolling elements, rings and cages. It is a typical multi-body system with factors such as multiple flexible bodies and dynamic contact relationships. In order to adapt to complex working load characteristics, reduce bearing fatigue damage and optimize bearing dynamic performance, it is urgent to solve problems such as accurate rolling bearing dynamic design and dynamic analysis [1]. Wensing[2] described the elastic deformation of the outer ring of the bearing by using the generalized function, and analyzed the mechanical properties of the ball bearing by using ANSYS. Rubio et al. [3] used Algor to analyze the mechanical properties of cylindrical roller bearings. Kang Yuan [4] used ANSYS to analyze the static stiffness of ball bearings. [5] used ANSYS to study the load distribution of ball bearings. Lin Tengjiao et al. [6], Fan Li et al. [7], Li Chang et al. [8] used ANSYS/DYNA to simulate and analyze the mechanical properties of ball bearings. References [9-12] used pseudo-static or pseudo-dynamic methods to analyze the mechanical properties of ball bearings. Yao Tingqiang et al [13-14] studied the dynamic characteristics of multi-body contact in deep groove ball bearings.

The mechanical research of rolling bearing usually adopts finite element method, statics method or pseudo-dynamic method, but these methods ignore the dynamic contact relationship between rolling elements and rings, and cannot fully describe the dynamic characteristics of the bearing. The author proposes a dynamic hybrid calculation and analysis method that combines the multi-body dynamics method and the finite element method, and considers the dynamic contact relationship between the steel ball and the ferrule, the elastic deformation of the ferrule structure, radial load, axial load, Based on the function of key factors such as centrifugal force and clearance, the dynamic analysis model of flexible multi-body contact of angular contact ball bearing is established, and the dynamic characteristics of the bearing are analyzed and discussed.

**Flexible Multibody Dynamics Model of Angular Contact Ball Bearings**

1.1 Hertz contact stiffness of steel ball and ring raceway Ignoring the influence of lubrication, Hertz elastic contact theory is widely used to describe the contact stiffness of ball bearings. The geometric parameter model of the angular contact ball bearing is shown in Figure 1. The relationship between the curvature parameters and the geometric parameters is as follows:

ρb1 = ρb2 = 2/Dw

ρi1 = 2/di

ρi2 = -1/Ri

po1 = -2/De

po2 = -1/R

In the formula, ρb1, ρb2, ρi1, ρi2, ρo1, ρo2 are the first and second principal curvatures of the steel ball, inner ring raceway and outer ring raceway, respectively; di, De are the diameters of the inner and outer ring raceways, respectively ; Dw is the diameter of the steel ball; Ri, Re are the radius of curvature of the inner and outer ring raceways.

In the formula, F(ρ)i, F(ρ)o, (ρΣ)i, (ρΣ)o are the main curvature function and the sum of the main curvatures of the inner and outer raceways, respectively; d and D are the inner diameter and outer diameter of the inner ring, respectively. ring outer diameter; dm is the pitch diameter of the ball bearing; α is the contact angle. According to the Hertz contact theory, the Hertz contact force in the common normal direction of the contact point between the ball and the raceway of the ferrule is related to the elastic deformation.

The bearing steel is selected as chromium steel (GCr15), the elastic modulus is 207GPa, and the Poisson’s ratio is 0.3, then the Hertz contact stiffness between the steel ball and the ring raceway (ignoring the i or o symbol in the formula) is Kc = 2. 15×105(δ*)-3/2(ρΣ)-1/2 (5)

### 1.2 Ball bearing flexible multi-body dynamic contact mechanics model

Figure 2 shows the dynamic contact relationship between the steel ball and the raceway surface of the ferrule. The raceway surface of the elastic ring of the ball bearing is composed of quadrilateral face elements, and the element nodes of the ring finite element model on the raceway surface are the vertices of the quadrilateral face element. OXYZ is the inertial coordinate system, ooxoyozo and obxbybzb are the body coordinate systems of the flexible outer ring and the steel ball, respectively, o’sx’sy’sz’s, o’cx’cy’cz’c are the raceway surface element and the surrounding The coordinate system of the box.

Since the position of the bounding box coordinate system relative to the ferrule body coordinate system remains unchanged, the dynamic contact search and prediction calculation between the steel ball and the ferrule raceway can be converted to the bounding box coordinate system [13]. Obtain the relative position vector r”bs between the steel ball and the patch unit in the normal direction of the patch unit ns

where n is the contact force index of the ball bearing, n = 1.5; δ0 and c0 are the user-defined maximum contact depth and damping, respectively; step() is the step function; sign(δ ) is the relative velocity sign function; ” and δ are the penetration amount and penetration rate, respectively; Kc

is the contact stiffness. In the dynamic analysis model of angular contact ball bearing (Fig. 3), the rings are all flexible bodies, and the steel ball only considers the local Hertz contact elastic deformation. Considering the influence of the contact angle between the steel ball and the ferrule raceway, the relationship between the geometric parameters of the steel ball revolution angle, the rotation angle and the inner ring rotation angle, where Dm is the pitch circle diameter of the ball bearing; α is the difference between the steel ball and the ferrule raceway The average contact angle, the influence of the change of the contact angle on the angular velocity of the steel ball can be ignored, and it can usually be described by the initial contact angle; ξi and ξo are the slip ratios between the steel ball and the inner and outer ring raceways, and 0≤ξi 0≤ξo <1; θi, θbg and θbs are the rotation angle of the inner ring, the revolution angle of the steel ball and the rotation angle, respectively.

### Fig.3 Flexible multi-body dynamic simulation model of angular contact ball bearing 7010C

In essence, the motion constraint relationship between the cage, the steel ball and the ferrule is given by the formula (10).

In this paper, on the basis of ADAMS/View [15-16], the coordinate data of the surface nodes of the ferrule raceway are extracted, and the normal vector of the four-sided patch element is calculated; the penalty function method is used to apply the contact point between the steel ball and the ferrule raceway surface. force and display its data in the form of a curve. GFOSUB provided by ADAMS/Solver. f. VARSUB. f. REQSUB. User subroutines such as f, application subroutines such as SYS-FUN, SYSARY and Constraint. f. Boundingbox. f and other contact search self-defined programs, compiled and embedded in Solver in DLL file format to achieve dynamic contact collision simulation of steel balls and ferrules. In the post-processing process, through REQSUB. f The contact force is displayed in the form of a curve, and its eight parameters are defined as the components of the contact force and moment in the three directions of X, Y, and Z, and the resultant force and moment. The dynamic contact prediction algorithm of steel ball and ferrule raceway based on bounding box proposed by the author effectively combines global search and local search. . When the steel ball is in contact with a certain (several) bounding box elements, the dynamic contact calculation program only performs a local search for the contact state in these bounding box elements, and determines whether the patch elements contained in these bounding box elements are in contact with the steel ball. Contact occurs, thereby reducing the global search and improving the calculation efficiency of the contact force.

**2 Dynamic calculation example of angular contact ball bearing**

### 2.1 Computational boundary conditions

Taking the angular contact ball bearing 7010C as an example, the outer surface of the outer ring is fixed, the inner ring rotates freely, and the inner ring and the outer ring are coupled through the dynamic contact relationship between the steel ball and the raceway of the ferrule. The inner and outer ring raceway diameters of angular contact ball bearing 7010C are divided into 56.01mm and 74.037mm, the inner and outer ring raceway curvature radii are divided into 4.65mm and 4.75mm, the inner and outer ring diameters are 50mm and 80mm respectively. , the diameter and number of steel balls are 9mm and 18, respectively, the contact angle is 15°, the contact stiffness of the steel balls and the inner ring raceway and outer ring raceway are Kci = 9.31 × 105N/mm1.5, Kco = 8 .09 × 105N/mm1.5, and the maximum contact damping value c0[2] is 5 × 10-2N·s/mm.

### 2.2 Calculation results of flexible multi-body dynamics of angular contact ball bearings

Figure 4 shows the changing law of the dynamic contact force between the steel ball and the raceway of the ferrule when the rotational speed of the inner ring of the angular contact ball bearing is ni=1800r/min, and the center of the inner ring is subjected to a central radial force of 1kN and a preload of 2kN . The abscissa in Figure 4 is the direction angle of the steel ball b1 (common

rotation angle), and the ordinate is the dynamically changing contact force. It can be seen from Fig. 4 that due to the alternating change of the odd-even support mode of the bearing steel balls in the angular contact ball bearing, when the steel ball b1 revolves through the radial load area, the dynamic contact force has a reverse decrease in the direction of radial force action. . Due to the important influence of preload and contact angle (15°), and the centrifugal force of the steel ball and inner ring during operation, therefore

In the non-radial load region, there is always a contact force between the ball and the ferrule due to the dynamic contact, and the dynamic contact force in the load and non-load regions presents an approximate double-period variation law, which more realistically shows that the angular contact Load bearing characteristics of ball bearings.

### Fig. 4 Change of contact force between the steel ball and the raceway of the ferrule when the steel ball revolves once

Figures 5 and 6 show the changing law curves of dynamic contact stress at nodes Po2 and Pi2 on the inner and outer ring raceways, respectively. Since the outer ring is fixed and the inner ring is free to rotate around the axial direction, when the steel ball b1 revolves once, the dynamic contact stress at the node Po2 on the outer ring raceway will produce a contact excited by the passing frequency of the steel ball after 18 periodic changes. vibration. Since the speed of the inner ring is higher than the revolution speed of the steel ball, the node Pi2 on the inner ring raceway has a macroscopic rotation with the rotation of the inner ring, so the node Pi2 is bound to pass through the load zone and the non-load zone of the bearing, so that the contact stress of the inner ring node Pi2 The number of excitations depends on the load conditions, the speed of the inner ring and the spin-roll ratio of the steel ball.

Fig. 7 shows the dynamic continuous change law curve of the radial contact stress of the ring when the angular contact ball bearing is running. Figure 8 shows the radial and axial vibration displacement of the steel ball center and the vibration displacement response curve of the node Pi2 on the inner ring raceway when the bearing is running. Analysis of Figure 7 shows that the comprehensive stress change curve of the inner and outer ring raceways presents an approximate periodical change law, and the comprehensive stress change curve of the outer ring raceway is steeper. During operation, the steel ball and the inner ring have an approximately periodic vibration displacement response law. The root mean square value of the radial vibration displacement of the node Pi2 on the steel ball and the inner ring raceway is 1.84 × 10-2 mm and 1.98 × 10 mm, respectively. -2mm, the root mean square value of the axial vibration displacement of the steel ball is 3.72 × 10-2mm, it can be seen that under the constant pressure and preload, the axial vibration displacement response of the steel ball on the raceway of the ferrule is relatively large.

### Fig.8 Vibration displacement response of angular contact ball bearing during operation

Figure 9 shows the change law curve of the static contact angle between the steel ball and the raceway of the ferrule. The static contact angle increases with the increase of the preload, and the increasing trend decreases. The static contact angle decreases with the increase of radial force under the action of preload, and the decreasing trend decreases gradually. Because the static contact angle does not change much, the angular contact ball bearing has better supporting performance. Figure 10 shows the change law curve of the relative displacement of the center of the ferrule with the radial force under the preload of 10kN. The radial relative displacement increases with the increase of the radial force, and the axial relative displacement decreases gently with the increase of the radial force.

### Fig.10 Relative displacement of the center of the ferrule under the change of radial force

Figure 11 shows the influence law of the comprehensive contact stress distribution of the ring raceway section under different radial forces (4kN or 7kN) under 10kN preload. Figure 12 is the stress cloud diagram of the ferrule raceway. The multi-body contact between the steel ball and the ferrule raceway forms the contact ellipse with different shapes. The maximum contact stress of the raceway section of the angular contact ball bearing under the action of the preload increases with the increase of the radial force. With the increase of radial force, the contact stress distribution pattern of the outer ring raceway section gradually approaches the symmetrical position of the outer ring raceway section, while the contact stress distribution pattern of the inner ring raceway section remains at a certain level with that of the inner ring raceway. Angular offset position. Because the radial force increases, the contact angle decreases, but the fixed outer ring makes the contact position of the steel ball and the outer ring raceway change and reduces the contact angle, which makes the maximum contact stress position of the outer ring raceway section (corresponding to the joint

The point position) also changes, which causes the contact stress distribution of the outer ring raceway section to change; while the inner ring can rotate freely, so that the contact position between the steel ball and the inner ring raceway changes relatively little, and the maximum inner ring raceway section changes. The contact stress position remains basically unchanged, so the maximum contact stress distribution pattern of the inner ring raceway section also remains basically unchanged.

### Fig.12 Contour diagram of contact stress distribution of angular contact ball bearing rings

The dynamic hybrid calculation and analysis method of angular contact ball bearing proposed by the author can obtain the results of displacement, vibration acceleration, load distribution and stress distribution of the bearing at any instant under different working conditions, and can be used to analyze the transition process of motion, such as load or rotational speed. sudden changes, etc. On this basis, it is also necessary to estimate the performance parameters of the bearing through system dynamics analysis to obtain the best bearing design scheme for the considered engineering application.

** in conclusion**

(1) Under the running state of the angular contact ball bearing, the contact force, dynamic contact stress and vibration displacement of the steel ball and the raceway of the ferrule show an approximate periodic variation law with the change of the revolution angle of the steel ball. In the case of preload, the dynamic contact force exhibits a bimodal variation law in the load region and the non-load region when the radial force is small.

(2) When the radial force is large, the dynamic contact stress on the radial plane of the ring raceway and on the axial section is likely to cause contact fatigue failure of the ring raceway, thereby shortening the fatigue life of the angular contact ball bearing. In the outer ring fixed mode, when the steel ball (cage) revolves once, the excitation times of the inner and outer ring raceways by the passing frequency of the steel balls is equal to the number of steel balls, and the excitation times of the inner ring raceway by the passing frequency of the steel balls is related to the load, inner and outer ring. The ring speed is related to factors such as the spin-roll ratio of the steel ball.

(3) Due to the dynamic contact force between the steel ball and the ferrule raceway and the different curvature of the ferrule raceway, the elastic deformation area of the inner ring raceway is larger than that of the outer ring raceway, and different forms of contact are formed. oval shape. As the load increases, the contact position of the steel ball with the outer ring raceway changes, while the contact position with the inner ring raceway remains basically unchanged. The contact angle and the axial relative displacement increase with the increase of the axial load and decrease with the increase of the radial load, while the radial relative displacement increases with the increase of the radial load.

(4) For the flexible multi-body contact dynamics of rolling bearings under complex working conditions, the dynamic hybrid calculation and analysis method proposed by the author is an effective and feasible new method, which can be applied to the research of rolling bearing dynamics in the engineering field.