008618266359149
Abstract: The parameterized model for angular contact ball bearings is developed based on ADAMS． With the help of impact function in ADAMS and based on the integrated analysis of the physical interpretation of impact parameters，the dynamics of angular contact ball bearings is analyzed，and the effects of the axial load，pocket clearance and guide clearance of the cages on the cage stability are investigated． The cage mass center motions are quantitative evaluated by the box － counting dimension of fractal theory and finally the best structural parameter of the cages is obtained．
Key words: angular contact ball bearing; ADAMS; dynamics analysis; fractal theory
The dynamic performance of high-speed spindle bearings directly affects the working quality of the electric spindle unit. Practical use has shown that the majority of high-speed electric spindle bearings are damaged in the form of cage damage. Therefore, it is crucial to analyze the stability of cage operation.
Fractal theory starts from the nonlinear complex system itself, focusing on the non smooth and irregular geometric shapes that appear in nature and nonlinear systems, as well as the disordered and self similar systems that widely exist in social activities [5]. It is presented in a quantitative method and has been widely applied in many fields. In practical applications, the commonly referred to fractal dimension refers to box dimension. The following text uses the box dimension method to quantitatively describe the centroid trajectory images of the cage under different structural parameters and axial loads, and calculates the box dimension value to reflect the irregularity of the trajectory, compare the operational stability of the cage, and determine the optimal cage structural parameters.
1. Definition of ADAMS collision force
ADAMS uses the nonlinear equivalent spring damping model provided by the impact function based on Hertz contact theory as the calculation model for contact force. In ADAMS, collision force is defined as
In the equation, q0 is the initial distance between two colliding objects; q is the actual distance during the collision of two objects; dq/dt is the rate of change in distance between two objects over time, i.e. velocity; K is the contact stiffness; e is the collision index; cmax is the maximum damping coefficient; d is the penetration depth, which determines when the damping reaches its maximum. When using ADAMS to analyze bearings, the input parameters that need to be determined include K, e, cmax, and d.
1.1 Contact stiffness
Given the bearing working condition and speed, a nonlinear equation system composed of the steel ball balance equation, deformation compatibility equation, and bearing balance equation can be solved using the Newton iteration method to determine the contact load and contact angle between the steel ball and the inner and outer ring channels. Given the contact load and contact angle, based on Hertz contact theory, the contact stiffness between the steel ball and the inner and outer grooves of the bearing can be obtained as
In the equation, k is the ellipticity parameter; Γ is the first type of elliptic integral; ε is the second type of elliptic integral; E' is the equivalent elastic modulus; Q is the contact load between two objects in contact; Σρ is the sum of the principal curvatures of two objects in contact. By using the Hertz contact simplified solution, k, Γ and ε can be obtained.
1.2 Collision index
From equation (1), it can be seen that the collision index e reflects the degree of nonlinearity of the material, and it is recommended to be 1.5 for metals and metal materials; The rubber material is 2, and 1.5 is taken in the text.
1.3 Maximum damping coefficient
The maximum damping coefficient cmax represents the loss of collision energy. The actual value needs to be obtained through experiments, usually ranging from 0.1% to 1% of the contact stiffness.
1.4 Penetration depth
The penetration depth d represents the depth at which the steel ball penetrates the inner and outer grooves at maximum damping. At the beginning of the collision, there is no damping force. As the invasion depth increases, the damping force increases until the maximum damping force appears. According to the collision dynamics model, damping occurs when two objects come into contact, and the damping coefficient remains constant throughout a complete collision process. Therefore, the smaller the value of d, the better. Considering the numerical convergence in ADAMS, the recommended value in ADAMS, d=0.01 mm, can generally be used.
2. Calculation of fractal dimension
With the development of information processing technology and computer technology, a large number of graphic images are obtained in the form of digital images, which can be converted into digital images. Ultimately, a two-dimensional matrix (binary graph) represented by a series of binary digits (0 or 1) can be obtained, which can be analyzed to calculate the box dimension of digital images.
The specific steps to solve the box dimension are as follows: (1) binarize the image, converting each pixel on the image to white or black, and obtaining a data file with rows and columns corresponding to the rows and columns of the binary image; (2) Divide the obtained data file into several blocks, with each block having a row and column count of k, and record the number of blocks containing (0 or 1) as Nk. Usually, k=1, 2, 4,..., 2i is taken as the boundary length for block partitioning, which is based on the dimensions of 1, 2, 4,..., 2i pixels, to obtain the number of boxes N1, N2, N4,..., N2 i.
Due to the size of pixels δ = The length of the image L/the number of pixels in a row of the image, so the edge length of a block composed of k pixels in both rows and columns is δ K=k δ (k=1, 2, 4,..., 2i). For a specific image, δ Is a constant, so the k value can be directly used instead in specific calculations δ K. Fit the data points (log) with a straight line using the least squares method in the double logarithmic coordinate plane δ k. LogNk), the negative value of the slope of the line obtained is the box dimension of the image.
3. Example analysis
Taking the high-speed internal grinder electric spindle support bearing as an example, a parameterized model established by ADAMS is used to analyze different cage structural parameters and working conditions. Figure 1 shows a parameterized model of the supporting bearing, which has added various drives, constraints, and forces to the three-dimensional model. Table 1 shows the parameters of the support bearing. Table 2 shows the contact parameters between the steel ball and the inner and outer rings applied in ADAMS calculated through Section 1. Assign contact parameters to the collision force formula in ADAMS for dynamic simulation.
Figure 1 Parameterized Model of Angular Contact Ball Bearings
Table 1 Support Bearing Parameters
Table 2 Contact parameters between steel ball and inner and outer rings
3.1 Influence of Cage Structural Parameters
The motion trajectory of the cage center of mass can reflect the stability of the cage during operation, and a stable cage center of mass trajectory should be a clear and regular circular shape. Under the conditions of bearing radial load of 20 N, axial load of 70 N, and rotational speed of 48 000 r/min, the centroid trajectory of the cage with different guide clearance values is shown in Figure 2, where Cg=0.13 mm is the original design value.
Figure 2: Center of Mass Trajectory of Cage under Different Guiding Clearances
Under the conditions of bearing radial load of 20 N, axial load of 70 N, and rotational speed of 48 000 r/min, the centroid trajectory of the cage with different pocket clearance values is shown in Figure 3, where Cs=0.274 mm is the original design value.
From Figures 2 and 3, it can be seen that during the initial start of the bearing, the steel ball and the cage collide from rest to collision, starting from the (0,0) point in the image, resulting in an irregular curve at the center of mass of the cage. Under normal operating conditions, the ideal image of the center of mass trajectory of the holder should be: the trajectory curve has a high degree of overlap, and the superimposed circular pattern is regular.
Figure 3 Center of Mass Trajectory of Cage with Different Pocket Clearances
From Figures 2 and 3, it can be seen that the centroid trajectories of the cage are relatively similar when the Cg and Cs values are different, making it difficult to distinguish the pros and cons with the naked eye. Therefore, the degree of irregularity is quantitatively described through the box dimension of fractal theory. The smaller the calculated box dimension value, the more regular the trajectory of the cage center of mass is, and the better the stability of the cage operation; On the contrary, the stability of the cage operation becomes worse. According to the principle of box dimension, a program was developed using MATLAB to process and calculate the centroid trajectory of the cage in Figures 2 and 3, and the box dimension values in Table 3 were obtained.
Table 3 Box dimension values of cage centroid trajectory under different structural parameters
From Table 3, it can be seen that as Cg gradually increases, its box dimension value first decreases, then increases, and then decreases, showing a fluctuating trend, indicating that either too large or too small a guide gap can lead to instability of the cage. As Cs gradually increase, their box dimension values also exhibit a fluctuating state.
From the simulation results, it can be seen that when the guide gap between the cage and the outer ring is 0.13 mm, and the pocket gap between the cage and the steel ball is 0.20 mm, the box dimension value is the smallest, indicating that the cage operates most stably, proving that the ADAMS simulation method is relatively reliable.
3.2 Impact of operating conditions parameters
When the radial load is 0 and the speed is 48 000 r/min, the centroid trajectory of the bearing cage under different axial load conditions is shown in Figure 4.
Figure 4 Centroid trajectory of the cage under different axial loads
From Figure 4, it can be seen that when the bearing is only subjected to axial force, the centroid trajectory of the cage is relatively regular. By calculating the box dimension values under various axial forces (Table 4), it can be seen that increasing the axial force of the bearing appropriately will reduce the instability of the cage. This is because the large axial force can ensure that the steel ball can always maintain contact with the ring, limiting the sliding of the ball, thereby reducing the collision between the steel ball and the cage, and making the operation of the cage more stable. However, the axial force cannot be too large. For example, in the simulation at 200 N, the box dimension value of the trajectory map is relatively large, and the trajectory of the holder center of mass is relatively irregular. According to the box dimension values in Table 4, the stability of the cage is better when the axial force is 150 N than when the axial force is 200 N.
Table 4 Box dimension values of cage centroid trajectory under different axial forces
4. Conclusion
(1) By using ADAMS dynamic simulation software, the dynamic characteristics of bearings can be more accurately studied. Through the post-processing module, it is easier to observe the running trajectory of the cage and the bearing force of the shaft, greatly improving the analysis efficiency.
(2)The difficulty of using ADAMS for bearing simulation lies in the setting of contact and the determination of contact parameters. Accurate operating conditions and contact parameters can make the simulation closer to reality.
(3) The gap between the pocket holes and the guide gap of the cage have a significant impact on its operational stability, and either too large or too small can cause the stability of the cage to deteriorate. By simulation, it was determined that a guide gap of 0.13 mm and a pocket gap of 0.20 mm were the optimal values.
(4) Fractal theory can quantitatively describe the irregularity of the centroid trajectory of the cage and provide a reference for determining the stability of the cage's operation.
More about KYOCM Slewing Bearing ：
Rotary bearings consist of an inner ring and an outer ring, one of which usually contains a gear. Together with the connecting holes in the two rings, they enable optimized power transmission through simple and fast connections between adjacent machine parts. Bearing raceways are designed with rolling elements, cages or gaskets to accommodate loads acting individually or in combination in any direction.
Features and advantages:
High carrying capacity
High stiffness for rigid bearing applications
Low friction
Long service life
Surface protection and corrosion resistance
Integrate other features including:
Driving mechanism
Control device
Lubrication system
Monitoring system
Sealed cassette tape
https://www.kyocm.com/products/Slewing-Bearing/745.html
2023-11-14