**Abstract:** The stiffness of the bearing is affected by the specific structural parameters of the bearing, working conditions and other factors, and is nonlinear. In order to obtain the specific parameters of the nonlinear stiffness of a certain type of deep groove ball bearing, a measuring device was designed using the existing dynamometer and dial gauge. Aiming at the axial stiffness and radial stiffness of the bearing, two bearing assembly structures for stiffness measurement are designed respectively. Among them, when measuring the radial stiffness, the correction washer is used to realize the adjustment of the assembly pre-tightening force. Using the ANSYS finite element analysis software, the static deformation analysis of some components of the assembly structure is carried out, and the negligibility of the small measurement errors caused by the deformation of these components is verified. Using the designed measuring device, a large amount of data about the bearing stiffness of this type of deep groove ball bearing was obtained, and a relatively ideal measurement result was obtained.

**0 Preface**

A laser tracker contains a motor structure. When ANSYS finite element analysis software is used to analyze the dynamic characteristics of the laser tracker, the stiffness and stress deformation of the motor rotor bearing have a great influence on the dynamic characteristics of the laser tracker. However, the bearing is affected by its own structural parameters, operating conditions, load conditions, lubrication environment and other factors, and its stiffness is nonlinear, which cannot be effectively modeled and analyzed in ANSYS software. Certain simplification. The simplification method is to replace the ball between the inner ring and the outer ring of the model bearing with the combine7 element (ie, the three-dimensional stiffness spring) of the ANSYS software to complete the simplification of the bearing. Because the nonlinear specific parameters of bearing stiffness are essential in the simplification process, at this time, it is necessary to carry out specific and accurate measurement of the nonlinear stiffness of the bearing according to the actual working conditions of the bearing [1].

Chen Shuangqiao of Naval Engineering University and others proposed to use the boundary element method to test the dynamic stiffness and damping of bearings, and achieved good results [2]. Huang Taiping from Xiamen University College of Engineering gave a specific method to test the stiffness of dynamic bearings, which uses electronic devices such as sensors to measure, and the cost is relatively expensive [3]. Fu Wei and others from the Department of Instrument Science and Engineering, School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University designed a test system for measuring the stiffness of precision bearings, which is an automated test system with an industrial control computer as the core [4].

In this paper, based on the actual conditions and using the traditional mechanical method with lower cost, the rough stiffness of the deep groove ball bearing of the Japanese NSK6900 model is preliminarily measured. According to the radial stiffness and axial stiffness of this type of bearing, two test devices were designed to measure the deformation of the bearing under a given load. According to the load and the corresponding deformation, the specific stiffness value of the bearing can be obtained. This paper only analyzes the stiffness of the bearing under static conditions, and does not involve dynamic conditions for the time being.

**1 Measuring principle**

The stiffness K of the bearing is defined as the ratio of the load change dF to the displacement change dδ. In the formula: K is the stiffness of the bearing at a specific working point; F is the position load of the working point; δ is the deformation displacement of the working point.

As the load changes, so does the stiffness of the bearing. In practice, it is difficult to obtain sufficiently small load variation dF and displacement variation dδ, so ΔF and Δδ are used to approximate dF and dδ.

The stiffness value obtained by formula (2) is an average stiffness value at a certain working point. By designing some measuring devices, the ΔF and Δδ of different working points can be obtained, so as to obtain the bearing stiffness parameters of different working points [5] .

The existing dynamometer is used to apply axial and radial loads to the NSK 6900 deep groove ball bearings respectively, and a lever dial indicator (commonly known as a gauge head) commonly used in the machinery industry is used to measure the deformation displacement corresponding to different loads. The minimum resolution of the sub-meter is 2 μm. For this reason, this paper designs the corresponding bearing assembly structure for the axial and radial stiffness of the bearing, so as to realize the measurement of the axial and radial stiffness of the bearing.

**1.1 Axial stiffness test**

Figure 1 shows a cross-sectional view of the bearing assembly structure used for bearing axial stiffness testing. This structure uses two bearings for assembly, which can greatly reduce the measurement error caused by the radial and axial play of the bearing. When the load shown in Figure 1 is applied, the load is mainly carried by the inner ring of the upper bearing. For the upper end bearing, there will be a pair of opposite axial forces between the inner ring and the outer ring. Under the action of this pair of axial forces, there will be relative displacement between the inner ring and the outer ring. According to the magnitude of the axial force and the corresponding displacement, the axial stiffness parameters of the bearing under different load conditions can be obtained. In order to further reduce the measurement error caused by the axial movement and radial shaking of the bearing, two lever dial indicators are placed on both ends of the pressed journal to measure the deformation displacement. The specific measurement positions are shown in Figure 1, and then Calculate the average value of the deformation displacement of each point.

**1.2 Radial stiffness test**

Figure 2 shows a cross-sectional view of the bearing assembly structure used for bearing radial stiffness testing. The outside of the device is a large support frame, and the inside is a horizontally placed journal. Two bearings are assembled at both ends of the journal. . The structure is designed so that the load is loaded in the middle of the journal, and a thicker and shorter journal is used to reduce the bending deformation of the journal. When the load F is applied, the load will generate radial loads F1 and F2 on the bearings at the left and right ends respectively through the journal. Since the outer ring of the bearing is fixed relative to the support frame, it is only necessary to measure the diameter of the inner ring of the bearing. The relative displacement between the inner ring and the outer ring of the bearing can be obtained.

When measuring, it is necessary to measure the radial displacement of the inner ring of the bearing at both ends of the journal, and add them to obtain the displacement corresponding to the load. The specific measurement position is shown in Figure 2. Since the preload of the bearing assembly has a great influence on the stiffness of the bearing, when designing the assembly structure, a correction washer is added to one end of the journal, as shown in Figure 2, by changing the thickness of the correction washer, it can be The preload force of the bearing assembly is controlled, so as to realize the bearing stiffness test under different preload force assembly conditions.

**2 ANSYS finite element analysis of deformation of assembly components**

When the load is loaded on the assembly structure, the measured deformation and displacement are not all caused by the deformation of the bearing, and other components of the assembly structure will also be deformed by force, so that the measured data has a certain error. In order to minimize the error, The parts of the assembly structure are all made of 45 steel material with greater rigidity. However, it is necessary to carry out specific analysis on how much the force and deformation of these components will affect the authenticity of the measurement results.

In this paper, the static structural stress analysis module of the ANSYS finite element analysis software is used to analyze the static deformation of the two main components of the assembly structure (journal, support frame) respectively [6]. The structural static deformation analysis results of the main components are as follows: shown in Table 1. The elastic modulus of 45 steel is 2.11×1011, the Poisson’s ratio is 0.28, and the load applied by static deformation is 1000N.

It can be seen from Table 1 that the maximum deformation of the main components in the measured assembly structure is 1.12 μm. Since the bearing stiffness of the NSK6900 model is in the range of 1×107～1×108N/m, when a load of 1000N is applied, the deformation displacement of the bearing is between 10～100μm. Therefore, the deformation of the assembly member is relatively small relative to the deformation and displacement of the bearing, and the resulting error can be ignored [7].

**3 Measuring device and its operation**

Figures 3 and 4 are diagrams of measuring devices for bearing radial stiffness and axial stiffness, respectively.

The two assembly structures are respectively fixed on the platform of the dynamometer by screws, and the long handle of the dynamometer can be manipulated to apply pressure to the assembly structure. The pressure loading head of the dynamometer should just press on the concave surface specially designed for the assembly structure. , the magnitude of the applied load will be displayed on the dynamometer reading in N.

When measuring the radial stiffness of the bearing, use two lever dial gauges to measure the radial displacement between the inner ring and the outer ring of the two bearings at the left and right ends of the journal, and add the displacements read by the two gauge heads. , the radial displacement corresponding to the load displayed on the dynamometer reading can be obtained. When measuring the axial stiffness of the bearing, use two lever dial gauges to measure the displacement of the compressed journal on both sides of the upper end of the journal, and calculate the average value. The average value obtained is the load corresponding to the load on the dynamometer reading. Axial displacement and averaging can reduce the errors caused by the axial and radial sloshing of the bearing to the measurement results.

**4 Data and Analysis**

The axial load-displacement curve of the bearing is shown in Figure 5. Since the stiffness K=dF/dδ of the bearing, and the abscissa in Figure 5 is the load F, and the ordinate is the displacement δ, the stiffness K corresponding to a point on the curve in Figure 5 is the reciprocal of the slope of the curve at that point. During data calculation, it is approximately considered that the inverse of the slope of each small segment of the curve (small straight line segment) is the stiffness value corresponding to the small segment of the curve. In Figure 5, when the load increases from 200N to 1000N, the axial stiffness of the bearing is calculated to increase from 1.4×107N/m to 3×107N/m.

Under the condition of no preload, the measured radial load-displacement curve of the bearing is shown in Figure 6. Due to the influence of errors and other factors, the curve is not smooth enough. Similar to the calculation method of the axial stiffness data, when the load increases from 200N to 1000N, the radial stiffness of the bearing is roughly between 1.5×107~2.25×107N/m. As the load increases, the radial stiffness tends to increase.

Since it is difficult to measure the preload force of the bearing assembly, the control of the preload force of the bearing assembly can be realized by adjusting the thickness of the correction shim. After trimming the correction washer by 0.05mm, it is assembled, so as to add the pre-tightening force. The measured radial load-displacement curve of the bearing is shown in Figure 7. Similarly, the curve in Figure 7 is not smooth enough. When the load increases from 200N to 1000N, the radial stiffness is roughly between 2.5×107 and 3.3×107N/m, and the stiffness value has obvious increase, the load and displacement approximate a linear relationship.

Comparing Figure 5, Figure 6 and Figure 7, it can be seen that as the load increases, the radial stiffness and axial stiffness of the bearing will increase. Since there is a certain gap between the parts of the assembly structure, there is also a gap between the inner and outer rings of the bearing and the balls. In the stiffness test, this phenomenon obviously does not exist. When a load of 1000N is applied and a preload is added, the radial displacement of the bearing (90μm) is reduced by about half compared with the radial displacement of the bearing without preload (37μm). It can be seen that when the preload is increased, the radial stiffness of the bearing will increase significantly.

**5 Conclusion**

In this paper, two assembly structures are designed using a dynamometer and a lever dial indicator, respectively, and the nonlinear stiffness measurement of a certain type of deep groove ball bearing is completed. Through the actual measurement of the bearing stiffness, the specific values of the bearing stiffness under different loads are obtained, and the nonlinear relationship curve between the bearing load and the deformation displacement is obtained.