Numerical simulation and experimental testing of dynamic stiffness of angular contact ball bearing

Abstract

Angular contact ball bearings are widely used in the multiple rotor system, such as gear box, machine tool spindle, and aero-engine rotors. The support stiffness is very important to the vibration of shafting. In order to obtain the dynamic stiffness, a numerical simulation method for dynamic stiffness of angular contact ball bearings is presented. By means of LS-DYNA software, the displacement and stress curves of the bearing’s different components are obtained successfully, and then, the dynamic stiffness of bearing under certain working conditions also can be calculated. The maximum envelope radius of the inner ring of the measured bearing can be taken as the relative displacement change value of the inner and outer rings of the bearing. To obtain the dynamic stiffness, the test strategy for dynamic stiffness of the angular contact ball bearings is introduced. The experimental results show that the equivalent dynamic stiffness increases with the increase in the preload of measured bearing under certain conditions. The simulation result coincides with the experiments well. The research work can provide the basis for the design and dynamics analysis of the bearing rotor system and have important engineering significance to improve the service performance of angular contact ball bearings.

Keywords

Angular contact ball bearing numerical simulation experimental testing bearing dynamic stiffness

Introduction

Rolling bearing is widely used in high-speed rotors, such as machine tool spindles and aero-engines, because of their high rotation accuracy and strong bearing capacity.^1–4 The bearing stiffness is an important parameter of the bearing, which exhibits strong time-varying characteristics and nonlinear characteristics under high rotation speed, variable load, and other complex operating conditions.^4,5 Furthermore, the bearing stiffness directly affects the dynamic characteristics of the spindle system.⁵ The accuracy of bearing stiffness in rotor dynamics modeling will directly determine the reliability of analysis results. Therefore, it has very important significance to study the design of bearing stiffness parameters.^6–11

The dynamic stiffness refers to the stiffness of the bearing in all directions during operation, which is an important index to measure the anti-vibration performance of the machine tool spindle structure. The larger its value is, the greater the dynamic load required to produce a unit amplitude for the machine spindle structure. Conversely, the smaller the dynamic stiffness, the more severe the vibration of the structure. Therefore, in order to ensure the spindle system has good processing performance, the dynamic stiffness analysis of the bearing is particularly important. Huang Taiping¹² put forward the testing principle of direct and indirect methods for bearing dynamic stiffness test. Chun-jie Li¹³ designed the dynamic stiffness test bench for angular contact ball bearing, measured the dynamic stiffness of the bearing at different speeds and analyzed the influence of speed and preload on the dynamic stiffness of the bearing. And, Ali¹⁴ studied the effect of preload and rotational speed on the dynamic stiffness of the bearing through the hammering test. For rolling bearings, the difficulty in analysis lies in the contact between the rolling element and the ring body. In the no-load static condition, the contact between the rolling element and the ring body is a point. With the increase in the load and speed, the point contact gradually becomes the surface contact. Meanwhile, the position, size, shape, contact pressure, and friction distribution of the contact area are unknown before analysis. They all change with the change of external load and rotational speed, which is a boundary nonlinear problem. This article designs an experimental platform for the stiffness test of rolling bearing in view of the practical applications for testing the dynamic stiffness of the bearing. In addition, a finite element simulation method for bearing stiffness is presented. The stiffness test and finite element simulation on a certain type of bearing are carried out, respectively, to obtain the relationship among dynamic stiffness, preload, and rotational speed, which provides a basis for the design and dynamic analysis of the bearing rotor system.

Dynamics stiffness mathematical model of angular contact ball bearing

The mathematical model is based on the Jones groove control theory, and the Newton–Raphson iterative method is used for numerical calculation. The following assumptions are made before establishing the model: (1) the geometry of the bearing is ideal; (2) with outer ring fixed, inner ring rotates relative to outer ring; (3) ignore the friction between the steel ball and the inner and outer ring channels; (4) the interaction between bearing components is in accordance with Hertz contact theory; and (5) excluding the influence of internal oil film thickness and oil film resistance of the bearing.

Figure 1 shows the five possible displacements of the inner ring of an angular contact ball bearing relative to the outer ring. Figure 2 shows the position angle $φ_{k}$ of the kth ball. In Figure 1, $θ$ is the contact angle; and $δ_{x}$ , $δ_{y}$ , and $δ_{z}$ are the displacements (mm) of the inner ring relative to the outer ring in the three directions x, y, and z, respectively; while $γ_{y}$ and $γ_{z}$ are the tilt angles (rad) of the inner ring relative to the outer ring in the y and z directions, respectively.

Figure 3 shows the relative position changes of the ball center of the bearing and the center of curvature of the inner and outer ring raceways at the kth ball position before and after loading. Assuming that the outer ring is stationary, the center of curvature of the raceway is not affected by the force. When the bearing is stressed, the bearing ball and the inner ring will be relatively deformed, and the position of steel ball center and the inner ring channel’s center of curvature will also change, as well as the contact angles of the inner and outer rings. The relative distance between the center of curvature of the inner and outer ring raceway before deformation is $B_{d}$ , formulated as follows

B_{d} = B D = (f_{i} + f_{o} - 1) D

(1)

In the formula, $f_{i}$ , $f_{o}$ , and D, respectively, are the radius factor of curvature of the inner ring raceway, the radius factor of curvature of the outer ring raceway, and the ball diameter.

After the bearing is stressed, the positions of the center of balls and the center of curvature of the inner ring raceway are changed in Figure 3. At this time, the radial distance and axial distance of the center of curvature of the inner and outer raceway are as follows

U_{i k} = B D \sin θ + δ_{x} - γ_{z} r_{i c} \cos φ_{k} + γ_{y} r_{i c} \sin φ_{k}

(2)

V_{i k} = B D \cos θ + δ_{y} \cos φ_{k} + δ_{z} \sin φ_{k}

(3)

In the above formula, $r_{ic} = 0.5 D_{m} + (f_{i} - 0.5) D \cos θ$ , $D_{m}$ is the pitch circle radius of the bearing, and D is the ball diameter of the bearing. The distances from the center of the steel ball to the center of curvature of the inner and outer raceway are as follows, respectively¹⁵

Δ_{i k} = r_{i} - D / 2 + δ_{i k} = (f_{i} - 0.5) D + δ_{i k}

(4)

Δ_{o k} = r_{o} - D / 2 + δ_{o k} = (f_{o} - 0.5) D + δ_{o k}

(5)

The radial displacement and axial displacement of the center position of curvature of the inner raceway before and after the bearing stress are as follows

Δ_{i c u} = δ_{x} - Δ γ_{z} r_{i c} \cos φ_{k} + γ_{y} r_{i c} \sin φ_{k}

(6)

Δ_{i c v} = δ_{y} \cos φ_{k} + δ_{z} \sin φ_{k}

(7)

In order to facilitate the analysis of Figure 3, Jones introduces two new variables $U_{k}$ and $V_{k}$ to replace $θ_{ik}$ and $θ_{ok}$ , and the geometric relationship can be obtained as follows

{\begin{cases} \sin θ_{o k} = \frac{U_{k}}{(f_{o} - 0.5) D + δ_{o k}} \\ \cos θ_{o k} = \frac{V_{k}}{(f_{o} - 0.5) D + δ_{o k}} \\ \sin θ_{i k} = \frac{U_{i k} - U_{k}}{(f_{i} - 0.5) D + δ_{i k}} \\ \sin θ_{i k} = \frac{V_{i k} - V_{k}}{(f_{i} - 0.5) D + δ_{i k}} \end{cases}

(8)

In Figure 3, using the Pythagorean theorem, we can know that

(U_{ik} - U_{k})^{2} + (V_{ik} - V_{k})^{2} - Δ_{ik}^{2} = 0

(9)

U_{k}^{2} + V_{k}^{2} - Δ_{ok}^{2} = 0

(10)

Figure 4 shows the bearing force of the ball during operation. As the bearing works, the ball will make a rotational motion around the axis of the bearing, which will generate centrifugal inertia force. In the course of the rotation of the ball, the gyro moment is generated due to the constant change of the axis of rotation. Therefore, when analyzing the force conditions of ball, it is necessary to consider the centrifugal inertial force and the gyro moment to which the ball is subjected. The centrifugal force and the gyroscopic moment are as follows

F_{ck} = \frac{1}{2} m D_{m} ω^{2} {(\frac{ω_{E}}{ω})}^{2}

(11)

M_{gk} = J_{b} ω^{2} {(\frac{ω_{B}}{ω})}_{k} (\frac{ω_{E}}{ω}) \sin α_{k}

(12)

In the formula, m is the quality of the steel ball, $ω$ is the spindle speed, $ω_{E}$ is the revolution angle of the ball, $ω_{B}$ is the rotation angle of steel ball, $α_{k}$ is the angle between the rotation axis of the steel ball and the x-axis. To analyze the steel ball, the equilibrium equation can be obtained as

\begin{matrix} Q_{ok} \cos θ_{ok} - \frac{M_{gk}}{D} \sin θ_{ok} - Q_{ik} \cos θ_{ik} \\ + \frac{M_{gk}}{D} \sin θ_{ik} - F_{ck} = 0 \end{matrix}

(13)

Q_{ok} \sin θ_{ok} - \frac{M_{gk}}{D} \cos θ_{ok} - Q_{ik} \sin θ_{ik} + \frac{M_{gk}}{D} \cos θ_{ik} = 0

(14)

In the formula, as can be seen from Wang et al.¹⁰

Q_{ik} = K_{i} δ_{ik}^{3 / 2}

(15)

Q_{ok} = K_{o} δ_{ok}^{3 / 2}

(16)

In the formula, $K_{i}$ and $K_{o}$ are the contact deformation coefficients of the inner ring and the outer ring, respectively. $K_{i}$ , $K_{i}$ affected by the position of the ball, and $K_{i}$ , $K_{o}$ is a function of $δ_{ik}$ , $δ_{ok}$ , $U_{k}$ , and $V_{k}$ . The $δ_{ik}$ , $δ_{ok}$ , $U_{k}$ , and $V_{k}$ can be calculated numerically using the Newton–Raphson iterative method using formulas (9)–(12).

The force acting on the shaft by the bearing inner ring is

{\begin{matrix} F_{xi} = Σ_{k = 1}^{N} (Q_{ik} \sin θ_{ik} + \frac{M_{gk}}{D} \cos θ_{ik}) \\ F_{yi} = Σ_{k = 1}^{N} (Q_{ik} \cos θ_{ik} - \frac{M_{gk}}{D} \sin θ_{ik}) \cos φ_{k} \\ F_{zi} = Σ_{k = 1}^{N} (Q_{ik} \cos θ_{ik} - \frac{M_{gk}}{D} \sin θ_{ik}) \sin φ_{k} \\ M_{yi} = Σ_{k = 1}^{N} [r_{ic} (Q_{ik} \sin θ_{ik} + \frac{M_{gk}}{D} \cos θ_{ik}) - f_{i} M_{gk}] \sin φ_{k} \\ M_{zi} = - Σ_{k = 1}^{N} [r_{ic} (Q_{ik} \sin θ_{ik} + \frac{M_{gk}}{D} \cos θ_{ik}) - f_{i} M_{gk}] \cos φ_{k} \end{matrix}

(17)

Let $δ = [δ_{x}, δ_{y}, δ_{z}, γ_{y}, γ_{z}]$ , find the partial derivative of $δ$ , and the fifth-order stiffness matrix of an angular contact ball bearing is

[k] = [\begin{matrix} \frac{\partial F_{xi}}{\partial δ_{x}} & \frac{\partial F_{xi}}{\partial δ_{y}} & \frac{\partial F_{xi}}{\partial δ_{z}} & \frac{\partial F_{xi}}{\partial γ_{y}} & \frac{\partial F_{xi}}{\partial γ_{z}} \\ \frac{\partial F_{yi}}{\partial δ_{x}} & \frac{\partial F_{yi}}{\partial δ_{y}} & \frac{\partial F_{yi}}{\partial δ_{z}} & \frac{\partial F_{yi}}{\partial γ_{y}} & \frac{\partial F_{yi}}{\partial γ_{z}} \\ \frac{\partial F_{zi}}{\partial δ_{x}} & \frac{\partial F_{zi}}{\partial δ_{y}} & \frac{\partial F_{zi}}{\partial δ_{z}} & \frac{\partial F_{zi}}{\partial γ_{y}} & \frac{\partial F_{zi}}{\partial γ_{z}} \\ \frac{\partial M_{yi}}{\partial δ_{x}} & \frac{\partial M_{yi}}{\partial δ_{y}} & \frac{\partial M_{yi}}{\partial δ_{z}} & \frac{\partial M_{yi}}{\partial γ_{y}} & \frac{\partial M_{yi}}{\partial γ_{z}} \\ \frac{\partial M_{zi}}{\partial δ_{x}} & \frac{\partial M_{zi}}{\partial δ_{y}} & \frac{\partial M_{zi}}{\partial δ_{z}} & \frac{\partial M_{zi}}{\partial γ_{y}} & \frac{\partial M_{zi}}{\partial γ_{z}} \end{matrix}]

(18)

Finite element simulation of dynamic stiffness of angular contact ball bearings

Taking the 60TAC120B angular contact ball bearing produced by NSK Corporation as an example, a three-dimensional simulation model of the bearing is established in the nonlinear explicit dynamics analysis software LS-DYNA, and the dynamic stiffness of the angular contact ball bearing is simulated through the refined model processing and analysis settings under specific working conditions. The geometric parameters and material property parameters of 60TAC120B angular contact ball bearings are shown in Table 1.

Table 1.

NSK 60TAC120B bearing parameters.

No.	Geometric and material properties parameters	Value
1	Inside diameter (mm)	60
2	Outside diameter (mm)	120
3	Pitch diameter (mm)	93
4	Ball diameter (mm)	10
5	Number of balls	20
6	Ring elastic modulus (MPa)	2.06E5
7	Ferrule density (kg/m³)	7.85E3
8	Ring Poisson’s ratio	0.3
9	Ball elastic modulus (MPa)	3.2E5
10	Ball density (kg/m³)	3.2E3
11	Ball Poisson ratio	0.25
12	Contact angle (°)	60

During the three-dimensional modeling and simulation analysis of the bearing, in order to improve the simulation accuracy and save the calculation time, the following equivalent and assumption conditions were adopted:

The chamfering of bearing inner and outer ring and retainer is ignored, and hexahedral mesh is used for meshing;

The bearing clearance and the influence of the oil film are ignored in the analysis process. The materials of the bearing components in the simulation process are linear elastic materials;

It is assumed that the rigidity of the bearing mandrel and bearing seat is infinite, and the deformation of the bearing itself is not involved in the analysis process;

Among them, the inner and outer rings of the bearing and the retainer use the SOLID164 solid element, and the inner ball uses the high-order SOLID168 solid element. Considering the rigid boundary of the inner and outer ring and the solid unit of the bearing without rotational freedom, a rigid shell element is applied to the inner surface of the inner ring of the bearing to facilitate the loading and rotational speed of the bearing.

The resulting finite element model of the 60TAC120B angular contact ball bearing is shown in Figure 5. The entire model contains a total of 141,333 units and 107,153 nodes.

Based on the above-mentioned finite element model of the bearing, certain boundary conditions and load conditions are set, and the rest of operating modes can be changed accordingly:

Boundary conditions: the rigid surface of the bearing outer ring is a fixed constraint;

Load application: a preload of 2000 N and a radial load of 1500 N are applied to the center of mass of the bearing inner ring rigid surface;

Working speed: the rigid surface of the bearing inner ring applied speed of 3000 r/min;

The analysis of the dynamic stiffness of a bearing belongs to explicit dynamic analysis. The difference between explicit and implicit dynamic analyses of static stiffness is that there is no need to set a contact element in the explicit dynamic analysis. It only needs to specify the contact surface, contact type, and contact-related parameters for the model to simulate the actual contact of the bearing rotation. Angular contact ball bearings have three contact positions during the working process, which are the contact between the ball and the inner raceway, the contact between the ball and the outer raceway, and the contact between the ball and the retainer pocket. The 60TAC120B angular contact ball bearings have a total of 60 contact pairs. The contact locations are shown in Figure 6.

In addition, the static friction coefficient between the rolling elements and the inner and outer ring raceways and retainer pockets is set to be 0.3E−2, 0.3E−2, and 0.2E−2, respectively, and the dynamic friction coefficients are set to 0.15E−2, 0.15E−2, and 0.1E−2, respectively. Next, we perform analysis setup and simulation. The entire calculation time is set to 160 ms, the number of output steps is 401 steps, and the entire calculation time is about 10 h.

The movement of each component of the bearing

In the LS-PREPOST post-processing software, view the time history curves of the bearings in the post-processing software LS-PREPOST. Select the node of inner ring, the node of retainer, and the node on the ball and draw the displacement curves during the movement of the bearing, as shown in Figure 7.

From Figure 7, it can be seen that the displacement of the inner ring and the retainer of the angular contact ball bearing is sinusoidal, and the displacement of the ball shows a sinusoidal change on the basis of the displacement curve of the retainer. This is specifically described as follows:

The inner ring, balls, and retainer all show significant cycle changes during operation. The rotation period of the inner ring is 20 ms, and the revolution period of the ball is the same as the rotation period of the retainer, which is about 48 ms.

The ball slides and rolls between the raceways of the inner and outer rings of the bearing. When the ball orbits once or the retainer rotates one revolution, the inner ring of the bearing rotates approximately twice.

Because the position of the selected node is closer to the outer ring of the bearing, the maximum displacement of the inner ring of the bearing, the ball, and the node of the retainer is the diameter of the pitch circle where these nodes are located and is closer to the outer diameter of the bearing.

In the same way, the nodes of the inner ring, retainer, and ball of the angular contact ball bearing are still selected, and the speed curves during the movement of the bearing are drawn, as shown in Figure 8.

It can be seen from Figure 8 that the speeds in the Y- and Z-directions of the inner ring and the retainer are sinusoidal, and the ball speed curve still shows a sinusoidal change on the basis of the retainer speed curve. Specifically described as follows:

The velocity curves of the nodes on each moving component of the bearing are obviously periodic, and the period of the velocity curve is equal to the period of the displacement curve.

When the Y-direction velocity component of the node on the inner ring reaches the limit value of ±12.5 m/s, the Z-direction velocity component is zero. At this point, the Y-displacement of this node is zero, and the Z-direction displacement will reach the extreme value of ±65 mm.

When the Y-direction velocity component of the retainer reaches the limit value of ±8 m/s, the Z-direction velocity component is zero. At this point, the Y-displacement of this node is zero, and the Z-direction displacement will reach the extremum of ±80 mm. Due to the influence of the rotation of the ball, the contact points constantly alternate change between the inner and outer rings.

The combined speed of the nodes on the inner ring is about 12.5 m/s, the combined speed of the nodes on the retainer is about 7.5 m/s, and the combined speed of the nodes on the balls changes sinusoidally, and the range of value changes is about 2.5–13.5 m/s.

Equivalent stress of bearing components

In the post-processing simulation results, the stress nephogram of each component of the bearing is displayed as shown in Figure 9.

It can be seen from the above figure that the equivalent stress of the angular contact ball bearing is concentrated in the contact position between the ball and the inner and outer ring raceway during operation. Due to the contact angle, the maximum stress position of the ball appears near the contact point. According to the Hertzian contact theory, the contact area between the ball and the raceway is an ellipse, and the compressive stress formed between them changes along the major and minor half axes of the ellipse, and the smaller the stress value is toward the elliptical edge. It can be seen from the figure that the results of the finite element analysis are consistent with the Hertzian contact theory.

In addition, it can be seen from Figure 9(a)–(c), the maximum equivalent stress of the inner ring, outer ring, and ball at the same moment is not the same. Among them, the maximum stress of the ball is 252 MPa, followed by the inner ring of 241.8 MPa, and the lowest stress is 135.4 MPa of the outer ring. The relatively large stress position on the ball occurs in the contact area with the inner and outer rings, and the ball stress in the bearing area is greater than the ball stress in the non-bearing area. The inner ring and the outer ring are the main elements that are directly contacted and carried in the bearing. The stress distribution is closely related to the stress distribution of the ball. From the slicing diagram of the bearing in Figure 9(d), it can also be seen that the relatively large stresses in the inner and outer rings are also distributed in the contact area with the ball, and the stress in the load bearing area is greater than that in the non-load bearing area. The other parts of the ring body without apparent stress distribution. The stress distribution area mainly focuses on the contact surface and presents a distinct oval.

The time curve of the maximum equivalent stress of the bearing inner ring, outer ring, and ball during normal operation of the angular contact ball bearing is shown in Figure 10. It can be seen from the above that the equivalent stress curve of the components in the bearing changes randomly, which also indicates that the stress variation of the components during the movement of the bearing is strongly nonlinear. In addition, the maximum stress on each component at a certain moment is also different, from big to small, namely, ball, inner circle, and outer ring. This also verifies the analysis results in Figure 9.

Because the accuracy of the finite element model establishment plays a decisive role in the final analysis result, establishing an accurate and feasible finite element model is the premise of the simulation analysis. From the comparison of the above bearing stress cloud diagram and the equivalent stress time curve of each component, it can be seen that the establishment of the whole finite element simulation model satisfies the requirements, so the result of the finite element simulation analysis is credible finally.

Dynamic stiffness simulation curve of bearing

After the simulation is completed, the displacement curve between the rigid faces of the inner and outer rings of the angular contact ball bearing in the direction of radial load is shown in Figure 11.

It can be seen from the figure that there is instability in the rotation of the bearing, and the displacement curve shows obvious fluctuations. In spite of this, since the rotational speed and the external load are constant, the displacement data in the direction of the radial load can still be approximated as a linear relationship. By fitting the data of the simulation curve, it can be seen that the displacement curve becomes approximately a straight line within the time range in which the bearing operation reaches the steady state. The value of the ordinate represented by this straight line is also the value of the bearing at the speed and given operating conditions. The amount of radial displacement change is a constant value. Therefore, the radial stiffness of the bearing only needs to be removed with a radial load in this direction.

When the bearing reaches steady state, the radial displacement between the inner and outer rings is $5.38 \times 1 0^{- 4} mm$ . Under the working conditions of bearing preload of 2000 N, radial load of 1500 N, and bearing rotation speed of 3000 r/min, the radial dynamic stiffness of 60TAC120B angular contact ball bearings is $5.38 \times 10^{6} N / mm$ . This value is smaller than the theoretical calculation of the axial dynamic stiffness of the bearing, which is mainly due to the fact that the finite element simulation considers the interaction between the components of the bearing. It is more accurate than the theoretical calculation.

Test of dynamic stiffness of angular contact ball bearing

The ability of a bearing to resist deformation under an external load is called the stiffness of the bearing. The rigidity of the bearing is the ratio of the amount of change in the applied load and the relative displacement between the outer ring and the outer ring of the bearing under certain conditions.^16,17

Principle of dynamic stiffness testing

Synchronous excitation

Using a rolling bearing as a single degree of freedom system, the dynamic characteristics can be expressed as¹⁸

Z (ω) = K (ω) - m ω^{2} + iC (ω) ω

(19)

In the formula, $K (ω)$ is the equivalent stiffness of the rolling bearing; m is the equivalent parametric mass of the rolling bearing; $ω^{2}$ is the inertial term of the equivalent stiffness of the rolling bearing; $ω$ is the angular frequency of the rotor operation; and $C (ω)$ is the equivalent damping coefficient of the rolling bearing. When the rolling bearing is in operation, the radial load synchronous with the rotation speed is directly applied, and the corresponding displacement generated at the center of the bearing is measured at the same time

Z (ω) = \frac{Q \sin (ω t)}{δ \sin (ω t - ψ)}

(20)

In the formula, Q is the rotational radial force acting on the bearing inner ring synchronous with the running speed of the bearing; $δ$ is the radial displacement of the bearing center; $ψ$ is the phase difference between the displacement and the force vector.

The use of unbalanced mass excitation to apply rotational radial load in synchronism with the rotational speed is achieved by adding unbalanced mass to the experimental stage rotor system, that is, an excitation disk is mounted near the rolling bearing, and a screw constituting an eccentric mass is added to the excitation disk, so that synchronous excitation of the bearing can be realized. Then, measure the excitation displacement of the main shaft system under the action of the eccentric excitation force, and then, convert the equivalent displacement generated by the bearing center under equivalent load according to the geometric relationship. The equivalent dynamic stiffness of the tested bearing is calculated from equation (1). The layout of the experimental bench is shown in Figure 12.

The synchronously rotating radial force produced by the eccentric mass is

Q = m_{a} e ω^{2}

(21)

In the formula, $m_{a}$ is the additional unbalanced mass; e is the mass eccentricity of the additional unbalanced mass. Due to the approximate method used, the synchronous unbalanced force applied to the bearing under test is not directly located in the symmetry plane of the bearing, so the necessary data conversion is needed. The synchronous radial force acting on the tested bearing is

Q_{e} = \frac{Q L_{1}}{2 (1 + L_{1} + L_{2} + L_{3} / 2)} = \frac{m_{a} e ω^{2} L_{1}}{2 (1 + L_{1} + L_{2} + L_{3} / 2)}

(22)

$L_{1}$ is the distance between balance mass and left support bearing, $L_{2}$ is the distance between the balance mass and the tested bearing, and $L_{3}$ is the distance between tested bearing and right support bearing.

Radial displacement

As can be seen from Figure 13, the displacement sensor is located close to the bearing under test to measure the radial displacement of the inner ring of the bearing. At the same time, in order to more accurately characterize the radial displacement of the bearing, it is necessary to remove the influence of the bending deformation of the main shaft itself. The equivalent displacement is¹⁸

δ_{e} = δ_{c} - δ_{s}

(23)

δ_{s} = δ_{st} / (1 - r^{2})

(24)

δ_{st} = \frac{m_{a} e ω^{2} L_{1}^{2} {(L_{2} + L_{3} / 2)}^{2}}{3 EI (L_{1} + L_{2} + L_{3} / 2)}

(25)

In the formula, $δ_{c}$ is the vibration displacement obtained by the test, $δ_{s}$ is the dynamic deflection caused by the imbalance of the spindle, $δ_{st}$ is static deflection, and r is the ratio of operating speed to critical speed. The radial stiffness of the rolling bearings at each speed is

K_{d} = \frac{Q_{e}}{δ_{e}} = \frac{Q L_{1}}{2 (1 + L_{1} + L_{2} + L_{3} / 2) δ_{e}}

(26)

Dynamic stiffness test bench design

In order to verify the correctness of the simulation analysis results, a bearing dynamic stiffness test bench was designed. The test stand is mainly composed of the following parts: motor, accelerator, coupling, excitation disk, preload adjustment nut, preload device, precision adjustment slide, bearing assembly under test, and eddy current displacement sensor. The base of the test stand is fixed on the concrete foundation by damping rubber, which is used to reduce the system vibration caused by the unbalanced excitation of the excitation disk during the high-speed rotation of the rotor. According to the above structural design scheme, the dynamic stiffness test bench for angular contact ball bearings is shown in Figure 13.

Wherein, the motor has a speed regulating function, and the current motor rotating speed can be displayed in real time through the control panel. Since the rated speed of the motor is low, the high-speed test range of the bearing to be tested are not satisfied, and therefore, an accelerator is used to further increase the working speed of the rotor system. The exciting disk is the applicator of the radial force of the bearing to be tested. Different eccentric masses are installed on the exciting disk to generate synchronous radial forces of different sizes. The pre-tensioning device is used to apply different preloads to the tested bearing, wherein the preload adjustment nut can be screwed in and out, and the preload can be accurately converted by the pitch and rotation angle. The precision adjustment slider is a set of linear pairs used to adjust the installation accuracy of the entire test rig horizontally and vertically (including parallelism, verticality, and coaxiality). The bearing assembly to be tested is used to locate and fix the tested bearing. Its internal structure can be adapted to the installation of different types of angular contact ball bearings, which is versatile. The two eddy current displacement sensors are orthogonally distributed in the horizontal and vertical directions and are mounted near the inner ring of the tested bearing through a fixed bracket for monitoring the range of displacement in the horizontal and vertical directions during bearing rotation.

Among them, the bearing preload adjustment device is shown in Figure 14. The preload device is mainly composed of several parts, namely, adjusting nut, radial sliding bearing, sliding block, fixed support, pre-tensioned adjusting spring, angular contact ball bearing, and spindle.

Its working principle can be described as follows: when the preload adjusting nut rotates to the right side, the radial ball bearing of the thrust ball moves inwards, thereby compressing the preload adjusting spring. And, the other end of the spring also acts uniformly on the outer ring of the tested bearing through the sliding block. The inner ring of the tested bearing is positioned by the mandrel, so that the compression force of the spring acts as a preload and directly acts on the tested bearing. In order to provide different bearing preloads in different ranges, six pre-adjusting springs are used in parallel to obtain the preload applied value by giving the pitch and rotation angle of the pre-tensioned nut and calculating the compression of the spring.^19–23

The test stand adopts a servo motor and the model number is 130SJT-M150D. The rated speed of the motor is 2500 r/min, the maximum speed is 3000 r/min, the rated torque is 15 N m, and the maximum torque is 30 N m, which can meet the test speed requirement. Among them, the accelerator growth ratio is 4 to ensure that the bearing to be tested working rotating speed adjustment range of 0–10,000 r/min. Preload adjustment spring model is YB3.5×18×25, elastic coefficient k = 83.7 N/mm, and maximum compression amount is 5.85 mm. Six groups of springs are used, design maximum compression is 5.5 mm, and the maximum axial force that can be provided is 2762 N, which meets test requirements.

Dynamic stiffness test data analysis

The data analysis uses Siemens LMS high-resolution signal acquisition system. The time-domain signals collected by the eddy current displacement sensors of the tested bearing positions are shown in Figure 15(a) and (b). It can be seen that the waveforms presented by the two sensors show periodic changes with a phase difference of 90°. Due to the accuracy and nonlinear effects of the test bench, the time-domain waveforms collected by the two sensors are not smooth.

In order to more clearly show the level of displacement of the bearing being measured under a given operating condition, take a certain period out of the above time-domain waveform to draw the axis trajectory, as shown in Figure 16. In the figure, the dashed line is the true axial trajectory profile. It can be seen that the true axial trajectory curve is also approximately circular due to the error and nonlinearity of the test bench. In order to facilitate the calculation of the bearing stiffness, its circumcircle is fitted here, and the maximum envelope radius is taken as the radial displacement of the bearing to be tested under this working condition.

According to the stiffness calculation formula, the dynamic stiffness value of the tested bearing under the current working condition can be obtained. Since the actual working conditions of the bearing are different, the dynamic stiffness of the 60TAC120B angular contact ball bearing under different working conditions is obtained by changing the rotational speed of the bearing under test and adjusting the magnitude of its preload. Set the adjusting range of the bearing preload is 0–2500 N, the interval is 500 N, the changing interval of the bearing speed is 400–2400 r/min, and the interval is 400 r/min. The changing interval of the bearing speed of revolution is 400–2400 r/min, and the interval is 400 r/min. The dynamic stiffness of the tested bearing under all operating conditions is drawn into a form, as shown in Table 2.

Table 2.

Radial stiffness of bearing (10⁶ N/mm).

Preload (N)	Speed (r/min)
Preload (N)	400	800	1200	1600	2000	2400
0	2.46	5.47	7.25	8.87	12.28	13.50
500	4.14	8.21	10.40	13.31	15.07	17.47
1000	4.82	9.30	12.32	14.70	16.33	19.14
1500	5.50	10.39	13.77	16.50	17.60	21.12
2000	5.89	11.49	15.22	18.63	20.50	23.10
2500	6.85	12.58	16.67	19.44	22.23	24.57

From Table 2 above, the specific data of the radial stiffness of the bearing to be tested can be known. In order to facilitate the observation, they are drawn into corresponding graphs, with the rotation speed and the preload as the abscissa, and the radial dynamic stiffness of the bearing as the ordinate. The curves of the radial dynamic stiffness of the bearing with the change of speed of revolution and preload are shown in Figure 17(a) and (b).

From Figure 17(a), it can be known that the dynamic stiffness of the angular contact ball bearing becomes larger with the increase in the preload of the bearing, but when the preload reaches a certain degree, the change of bearing stiffness is not obvious.

From Figure 17(b), it is known that when the preload is fixed, the radial dynamic stiffness of the bearing is increased with the increase in the speed. When the rotation speed is low, the change of the dynamic stiffness of the bearing with the preload is not obvious, but when the rotation speed is high, the change range of stiffness with preload increases obviously. This is mainly because with the increase in the rotational speed, the dynamic body in the bearing to be tested will expand obviously under the action of centrifugal force. The contact area between the rolling element and the ferrule increases gradually, and the corresponding contact angle will also change, which eventually leads to the increase in the radial stiffness of the bearing.

In order to verify the consistency of the bearing stiffness of the angular contact ball bearing to be tested during loading and unloading, the change range of the bearing rotation speed is set to 0–2500 r/min, and the adjusting range of the preload is 0–2500 N, and the loading and unloading stiffness tests are carried out, respectively. When the preload of the bearing is set at 1500 N, the load and unload dynamic stiffness curves of the tested bearing with the change of rotational speed are shown in Figure 18, and its data are shown in Table 3. Similarly, when the rotation speed of the bearing is set at 1200 r/min, the load and unload dynamic stiffness curves of the tested bearing with the change of preload are shown in Figure 19, and its data are shown in Table 4.

Table 3.

The radial stiffness of loading and unloading varies with the rotation speed (10⁶ N/mm).

Condition	Speed (r/min)
Condition	400	800	1200	1600	2000	2400
Load	5.5	10.3	13.7	16.5	17.6	21.1
Unload	5.1	9.6	13.2	16.5	18.3	20.8

Table 4.

The radial stiffness of loading and unloading varies with the preload (10⁶ N/mm).

Condition	Preload (N)
	0	500	1000	1500	2000	2500
Load	7.2	10.4	12.3	13.7	15.2	16.6
Unload	6.9	10.0	11.9	13.0	15.4	16.7

It can be seen from Figure 18 and Table 3, under the condition of constant preload, the radial stiffness of angular contact ball bearings increases with the increase in the rotational speed during loading and unloading. The bearing stiffness obtained from the two processes is basically consistent. The error is less than the measured data by an order of magnitude, which can be neglected, and the consistency is good.

It can be seen from Figure 19 and Table 4 that under the condition of constant speed, the radial stiffness of the angular contact ball bearing increases with the increase in the preload during the loading process and the unloading process. The bearing stiffness obtained during the loading and unloading process is basically consistent. The error is less than an order of magnitude of the measured data and can be ignored. It also verifies the consistency of the test data and the reliability of the test bench design.

In order to verify the accuracy of the simulation results of angular contact ball bearings, the simulation analysis curve and the test curve are compared. Taking the bearing rotation speed as 1600 r/min, and the change curve of the radial dynamic stiffness of the bearing with the preload force is shown in Figure 20. Similarly, taking the preload force as 1500 N, and the change curve of the bearing radial dynamic stiffness with the rotation speed is shown in Figure 21.

Through the comparison of simulation and test curves, it can be seen that the radial dynamic stiffness of 60TAC120B angular contact ball bearings increases with the increase in the preload when the rotational speed is constant, and the variation range is 9E6–2.2E7 N/mm, as shown in Figure 20; under the condition of constant preload, the radial stiffness also increases with the increase in the rotation speed, and the range of change is 5E6–2.4E7 N/mm, as shown in Figure 21. But the test data are slightly lower than the simulation data, which is mainly due to the machining accuracy and installation error of the test bench, and the acquisition error of sensors. However, the influence of these environmental factors on stiffness is not considered in the simulation.

Conclusion

In this article, the testing method of dynamic stiffness of angular contact ball bearings is studied. The test bench for testing the dynamic stiffness of angular contact ball bearings is designed. A finite element simulation method for dynamic stiffness analysis of bearings is presented. Through simulation analysis and experimental comparison, the relationship between the equivalent dynamic stiffness of the 60TAC120B angular contact ball bearing and the preload and working speed is studied. The specific conclusions are as follows:

Comparing the dynamic stiffness simulation and experimental curves of 60TAC120B angular contact ball bearings, we can see that the dynamic stiffness of the bearing is only 1E7 N/mm when the rotation speed of the bearing is 1600 r/min and the preload is 0 N. When the preload rises to 2500 N, the dynamic stiffness of the bearing is close to 2E7 N/mm, which is twice as high as that without preload. Under certain rotational speed, the equivalent dynamic stiffness increases with the increase in the preload. When the preload reaches over 2000 N, the dynamic stiffness value changes little with the increase in the preload.

When the preload of the bearing is 1500 N and the working range of rotational speed is 400–2400 r/min, the dynamic stiffness of the bearing varies from 5E6 to 2E7 N/mm. Under certain preload, the dynamic stiffness of angular contact ball bearings increases with the increase in the rotational speed, and the speed of stiffness increase slows down with the gradual increase in the rotational speed.

Overall, the simulation analysis results are consistent with the experimental test results, but the stiffness simulation data of the local position are slightly larger than the experimental value, this is because there being the geometric gap between the ball and the ring raceway of bearing. Nevertheless, the work in this article can provide some data support for online adjustment of the pre-tightening force of angular contact ball bearings and selection of the working speed of bearings, which has important engineering significance for improving the service performance of angular contact ball bearings.

Footnotes

Handling Editor: Zengtao Chen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (grant no. 51505470) and Youth Innovation Promotion Association, Chinese Academy of Sciences (CAS).

ORCID iD

Haitao Luo

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