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Abstract

The non-uniform distribution load during machining and assembly process is crucial for spindle system, especially in complex working conditions. The conception of non-uniform preload adjustment approach was proposed and experimentally investigated in this article. Based on the mechanical equivalent principle, the non-uniform preload was theoretically transformed to the combination of uniform preload and an extra moment. Then, the non-uniform preload of rolling bearing was experimentally measured and analyzed via a spacer with 15-µm wear loss on the end face. The spindle performance factors, such as rotation accuracy, temperature rising, acceleration, and vibration, were all monitored. The rotation center of spindle was deviated in different non-uniform preload conditions. Meanwhile, the temperature and vibration performance of non-uniform preload are superior to those of uniform bearing preload.

Keywords

Rolling ball bearing non-uniform bearing preload rotational accuracy vibration

Introduction

Angular contact ball bearing is one of the most essential parts in machine tool spindles, which directly affects the machining accuracy and reliability of high-speed and ultra-precision manufacturing.^1–3 Reasonable preload exerted on bearing helps to improve the stiffness and rotational accuracy of high-speed spindles. The peak temperature and skidding can also be reduced due to different clearances.^4–7 Currently, the required performance of bearing dramatically increased with the increase of the spindle rotation accuracy, rotation speed, and load capacity. Tiny machining errors or assembly errors could affect the service performance of spindle bearing, especially in extreme machining conditions. For example, in high speed, a slight defection of bearings’ inner ring due to the dimension error of rolling elements will greatly aggravate the rotation accuracy. Some researchers have discussed the effect of preload in various working conditions and proposed a corresponding adjustment mechanism, which effectively improved bearing performance.⁸

According to the maximum contact pressure and service life, bearing preload was theoretically analyzed by Kim et al.⁹ The optimum preload, in high-speed condition, was given by GD Hagiu,¹⁰ based on the spindle reliability. Zhang et al.¹¹ developed a new nonlinear dynamic model of the rotor-bearing system considering preload and varying contact angle of the bearing. A comparative study about dynamic characteristics of high-speed spindle with respect to different preload mechanisms was discussed by Cao et al. The performance of spindle under rigid and constant force preload is investigated systematically using a mathematical model under various conditions.¹² For experimental study, Song and Shin¹³ investigated the preload in consideration of spindle speed as well as rotation accuracy.

With the development of high-speed machining, the spindle requires extraordinary stiffness at low speed and low heat at high speed, simultaneously. The traditional rigid preload or constant pressure preload cannot meet the machining requirements in the whole speed range, especially when the working conditions alternate between light cutting at high speed and heavy cutting at low speed.^14–17

To this end, much research focuses on controllable and modifiable preload technology, and several new variable preload adjustment devices were developed and fabricated.^18,19 A hydraulic variable device was developed by Jiang and Mao²⁰ to monitor the temperature rise of the bearing affected by preload. Hwang and Lee²¹ introduced an automatic variable preload unit consisting of an elastic centrifugal deformation mechanism and springs with an electromagnet. The work by Xu et al.,²² who designed a hydraulic loading system to adjust preload, shows that the axial preload of rolling bearing has a remarkable effect on the spindle performance when the rotating speed exceeds a critical value. Similar work by Chen and Chen²³ was conducted to adjust the bearing preload, in different cutting conditions, to ensure the spindle performs at high stiffness and high accuracy. In addition, bearing preload device by liquid pressure was discussed by Chio and Lee.¹⁸ The above representative work indicates that, at high rotation speed, spindle temperature is reduced by a lower preload. Meanwhile, at low rotation speed, spindle dynamic stiffness is significantly enhanced with a larger preload.

However, due to the machining error, assembly error, and external loads, neither the axis nor the center of the rolling bearing can maintain coincidence with the spindle axis. As we know, in a spindle-bearing system coupled with irrational preload, the working load on the front bearing is often heavier than that on the rear one; therefore, the heat generation on the front bearing is usually higher than that on the rear one. All these factors add up to a non-uniform distribution of axial force on either bearing’s outer or inner ring. In these conditions, the traditional uniform rigid preload, constant pressure preload, or variable preload methods cannot meet the requirements any more.

In practical application, the initial preload of the high-speed motorized spindle usually includes many springs with same stiffness coefficients, which were mounted uniformly within the bearing spacer. However, after working for a period time, the stiffness of those springs changed inconsistently due to uneven thermal conditions and stress, which will induce uneven force applied on bearing end faces.

As mentioned earlier, a new preload device is needed to eliminate the uneven force, thus an optimal bearing preload is achieved to enhance spindle service life. Principle of different preload methods is shown in Figure 1. The constant pressure or rigid preload, as shown in Figure 1(a), is a common method for high- or low-speed region. Figure 1(b) is the method of uniform variable preload, in which preload value can be changed through a specially designed component. In Figure 1(c), a non-uniform preload method is presented (F₁ = 1800 N, F₂ = F₃ = 200 N), whose value and distribution are all adjustable.

Figure 1.

Bearing preload technology: (a) constant pressure/rigid preload, (b) variable preload, and (c) non-uniform preload.

The preload exerted on the rolling ball bearing is adjustable by changing the elongation indicator of the piezoelectric actuator. Also, the preload value exerted on the bearings’ outer ring can be displayed by the force sensors. The loading points of the actuator were controlled independently. When force exerted on the bearing’s outer ring was not equal to another, the preload value presents non-uniform. Obviously, the new non-uniform method is different from the constant pressure/rigid preload or variable preload method.^16,18,20

In this article, the conception of non-uniform preload adjustment approach was proposed and investigated in the following procedure. First, the non-uniform preload equivalent transformation and non-uniform adjustment were introduced. Second, some key parameters, such as the rotation accuracy, temperature rise, and vibration of the spindle system, were experimentally monitored and discussed. Finally, the conclusion was given in the last section.

Non-uniform preload equivalent transformation and experimental platform

Equivalent transformation of non-uniform preload

In order to analyze the dynamic characteristics of rolling ball bearing–spindle system with non-uniform preload, the equivalent transformation is discussed first. As shown in Figure 2, the preload is applied on the bearing’s outer ring through piezoelectric actuators, which are uniformly distributed. When these forces are applied differently on bearing’s outer ring, the preload will present a non-uniform distribution.

Figure 2.

Point of force application on the bearing.

The relationship of bearing forces and deformations, under non-uniform preload, is a compound problem. Hence, the equivalent transformation of bearing under non-uniform preload is necessary. Based on equivalent theory, the non-uniform preload can be simplified to an axial force ΣF_ix and torque ΣM_iy acting in x-z plane and ΣM_iz acting in x-y plane, respectively. Suppose the rolling element is Z and there are n acting points uniformly distributed on the bearing’s outer ring, denoted as F₁, F₂, …, F_i…, F_n. Set the angle of F₁ as 0° and the angle of action point F_i as α_i. Assuming D_o and D_i represent external and internal diameter of the bearing’s outer ring, respectively, l_i represents the line connecting the bearing geometric center O and action point i, as shown in Figure 2.

The equivalent process is described in Figure 3.

Figure 3.

Equivalent process of non-uniform preload.

According to the mechanical equivalence principle and F = min (F₁, F₂, …, F_n), the non-uniform preload is equivalent to uniform preload and moment relative to the bearing center. The composite axial uniform preload of the bearing can be deduced as follows

$\sum F_{ix} = \sum_{i = 1}^{n} F_{i}$ (1)

where F_i is the preload exerted on the point i (N), n is the number of preload exerted on bearing’s outer ring, and F is the minimum preload exerted on the bearing’s outer ring (N).

For each component preload, it is equivalent to an axial force and moment acting on bearing center. The moment is

$M_{i} = \frac{(D_{o} + D_{i}) (F_{i} - F)}{4}, i = 1, 2, \dots, n$ (2)

Then, the decomposition moments on y-axis direction and direction can be obtained

$\sum M_{iy} = \sum_{i = 1}^{n} [\frac{(D_{o} + D_{i}) (F_{i} - F)}{4} \cos α_{i}]$ (3)

$\sum M_{iz} = \sum_{i = 1}^{n} [\frac{(D_{o} + D_{i}) (F_{i} - F)}{4} \sin α_{i}]$ (4)

In order to illustrate the equivalent transformation of non-uniform preload more clearly, an example of equivalent transformation about the experimental condition was added. The type of angular contact ball bearings used in this spindle is NSK7014C/P4.01. Setting the value of preload F₁ = 1800 N, F₂ = 200 N, and F₃ = 200 N, the moment is generated only in the direction of F₁, as shown in Figure 4. The detailed calculation processes are listed as follows.

Figure 4.

Example of equivalent transformation.

The composite axial preload of rolling ball bearing is

$\sum F_{ix} = \sum_{i = 1}^{n} F_{i} = 2200 N$ (5)

According to the NSK Super Precision Bearings Manual, the size of D_o is 90 mm and D_i is 78 mm. It can be seen from Figure 2 that α₁ is π/6 and n is 3. F = min (F₁, F₂, F₃), so F is 200 N. Substituting F₁, F₂, F₃, D_o, D_i, and α₁ in equations (3) and (4), respectively, the value of ΣM_iy and ΣM_iz can be calculated as follows

$\sum M_{y} = \frac{(90 + 78) (1800 - 200)}{4} \cos \frac{π}{6} = 58, 195 N mm$ (6)

$\sum M_{Z} = \frac{(90 + 78) (1800 - 200)}{4} \sin \frac{π}{6} = 33, 600 N mm$ (7)

where $\sum_{i}^{n} M_{iy}$ is the bending moment lined with y-axis which is caused by the non-uniform preload and $\sum_{i}^{n} M_{iz}$ is the bending moment lined with z-axis which is caused by the non-uniform preload.

Non-uniform preload adjustment system

An experimental platform was set up to investigate the mechanism of non-uniform preload. Three piezoelectric actuators separated by 120° were utilized as active load unit for preload control. The preload value was measured by force sensors and displayed in real time. The maximum speed of the experimental spindle was up to 10,000 r/min. The spindle system with non-uniform preload adjustment is given in Figure 5.

Figure 5.

Experimental platform with non-uniform preload adjustment.

Four angular contact ball bearings were used in pairs in the test spindle system. The layout of rolling ball bearings was double back to back duplex. The experimental spindle was driven by a motorized spindle. Its rotation speed was precisely controlled by a servo drive controller.

Drive mechanism of the spindle

The maximum speed of the motorized spindle is 15,000 r/min. The technical parameters of motorized spindle are shown in Table 1.

Table 1.

Technical parameters of the motorized spindle.

High-speed motorized spindle	Technical indicator
Power	11.5 kW
Maximum speed	15,000 r/min
Rated current	19 A
Static run-out of shaft end	⩽3 µm
Lubrication, cooling	Grease lubrication, water cooling
Temperature rising	Temperature rising on shell ⩽25°C
Level of dynamic balance	G1

Preload loading component

Three piezoelectric ceramic actuators were employed in this experimental spindle for their high stiffness, small size, and easy installation. Furthermore, the preload applied on the bearing’s outer ring was displayed on computer in real time via a force sensor connected with the rear actuator. The type of piezoelectric ceramic actuator was PSt150/10/20VS15. The force sensor was PACEline-CFT/5 kN produced by HBM Company, which is connected to the end of the piezoelectric actuator.

In order to obtain non-uniform preload, three piezoelectric actuators were uniformly mounted in the blind hole of sleeve. Figure 6 shows the structure of the non-uniform preload loading system. The preload was adjusted by the piezoelectric actuator voltage. When the input voltages of three piezoelectric actuators were not same, the preload on the outer ring was non-uniform.

Figure 6.

Non-uniform preload loading system.

Data acquisition system

Spindle rotation accuracy

The Spindle Error Analyzer (Lion) was used to measure spindle rotation accuracy. Four non-contact capacitive displacement sensors and seven temperature sensors were used to test the spindle rotation accuracy and thermal feature, respectively. The spindle radial rotation sensitivity, radial fixed sensitivity and the spindle drift were also measured. The data from two orthogonal capacity sensors were used to get rotational trajectory.

The arrangement of capacitive displacement sensors is shown in Figure 7(a). The displacement sensors y₁ and y₂ were mounted vertically to measure different sections of the step shaft. The sensors z₁ and z₂ were mounted in horizontal orientation. y₁ and z₁ were in a vertical plane. Similarly, y₂ and z₂ were in another vertical plane. Figure 7(b) shows the schematic diagram of those displacement sensors. The displacement signals of rotary spindle were collected by low-pass filtering to eliminate noise signal. According to the horizontal displacement signals (D_z) and vertical displacement signals (D_y), the spindle rotation orbits, the rotation center, and the spindle rotation accuracy were all monitored and analyzed.

Figure 7.

Arrangement of displacement sensors: (a) arrangement of the sensors and (b) schematic diagram of radial displacement.

Bearing temperature measurement

Bearing temperature data were monitored and sampled by YOKOGAWA-MX100 to ensure that the data are reliable. Seven Pt100 temperature sensors were employed in this experimental platform to measure bearing’s outer ring temperature because the sensor has small size to install.

To ensure the accuracy of the data measured, the deviation comparison of the two types of sensors, sensor 1 (Pt100) and sensor 2 (Lion), was conducted. As shown in Figure 8, the measuring point is located on the top of the spindle housing. The temperature rise presents an identical tendency at the same position. The temperature by Lion is slightly higher than the Pt100 by 0.2°C, so either of the sensors meets the test requirements.

Figure 8.

Deviation comparison of the two sensors.

Bearing vibration signal acquisition

A Brüel & Kjær multi-channel acquisition system was employed to sample vibration signals from the non-uniform preload spindle system. The sampling frequency is 8192 Hz, and the bandwidth is 3200 Hz. Two acceleration transducers were used to monitor the acceleration signals. Those acceleration transducers were mounted on top of the bearing house.

Results and discussion

Rotation accuracy of spindle

The spindle rotation accuracy is crucial for machine tool designing, manufacturing, and maintenance.^24,25 In this subsection, the performance of rotation accuracy was studied under different non-uniform preload conditions. According to NSK Bearing Manual, the initial preload is 250 N. Then, 11 non-uniform preload conditions (from C0 to C10) were designed, as shown in Table 2.

Table 2.

Working conditions of non-uniform preload.

Number	F₁ (N)	F₂ (N)	F₃ (N)
C0	100	100	100
C1	300	100	100
C2	600	100	100
C3	900	100	100
C4	1200	100	100
C5	1500	100	100
C6	1500	200	200
C7	1500	300	300
C8	1200	400	400
C9	1000	500	500
C10	800	600	600

The spindle rotation trajectory and the rotation center in C0, C2, C4, C6, C8, and C10 conditions of different preload ways are shown in Figure 9, at the speed of 2000, 4000, and 8000 r/min, respectively. For different rotation speeds and preload conditions, the horizontal rotation error and vertical rotation error both are about 20 ± 0.5 µm, which indicates that different preload conditions have little effect on the spindle rotation trajectory. However, the rotation center has a significant shift against preload. As shown in Figure 9, the maximum horizontal offset of the rotation center is 14.7 µm and the maximum vertical offset of the rotation center is 4.7 µm at speed 2000 r/min. When the rotation speed is 4000 r/min, the maximum horizontal offset of the rotation center is 12.91 µm and the maximum vertical offset of the rotation center is 6.67 µm. At a speed of 8000 r/min, the maximum horizontal offset of the rotation center is 13.1 µm and the maximum vertical offset of the rotation center is 3.12 µm. Therefore, it can be concluded that, in condition of different non-uniform preload rather than rotation speed, the rotation center is mainly shifted in the horizontal direction with the equivalent moment increasing (from C1 to C5), while the equivalent moment decreasing gradually (from C6 to C10) the rotation center returns to near the initial position.

Figure 9.

The trajectory of spindle rotation center: (a) n = 2000 r/min, (b) n = 4000 r/min, and (c) n = 8000 r/min.

Based on the above analysis, the rotation center of spindle is significantly changed under the non-uniform preload and the offset direction of the rotation center is relevant to the direction of equivalent moments. Furthermore, with the changing of preload condition (i.e. the magnitude of the equivalent moment is different), the rotation center offset is relevant to the equivalent moment. When the spindle speed is up to 8000 r/min, the rotation center and rotation trajectory are consistent. Therefore, the non-uniform preload has significant effects on rotation center of the tested spindle.

Temperature rise of bearing

In general, at high-speed range, due to heat generation of bearing, a lower preload is required to reduce the heat amount and lengthen the service life.^26,27 To determine the impact of non-uniform preload on spindle thermal performance, temperature rise of bearing was tested on the spindle platform. The non-uniform force on the bearing’s outer ring can be generated by installing a specially designed spacer. In order to facilitate the calculation, the end face oblique angle (θ) of outer spacer ring is denoted as

$θ = \arctan \frac{d}{L}$ (8)

where L is the outer diameter of the outer spacer (mm) and d is the interpolation between highest and lowest point on the end face of the spacer (µm).

The oblique angle of outer spacer and installation position is shown in Figure 10: Figure 10(a) shows the diagram of oblique angle, Figure 10(b) shows the highest and lowest point on the end face of the spacer. The end face is divided into 12 parts and marked with number 1 to 12, which is convenient for experimenter to discover the relative position of highest point and lowest point, and Figure 10(c) shows installation position in the spindle. In practical applications, oblique angle was usually generated on end face of the spacer due to the local wearing after using for a period of time. Based on the outer spacer non-parallelism, the temperature rise and vibration characteristic are studied to evaluate the influence of the uneven force on spindle bearing and finally to determine a proper preload. During the study, the non-parallelism of the spacer is about 15 µm, as shown in Figure 10(a).

Figure 10.

The oblique angle and installation position of spacer.

Figure 11 shows the arrangement of the temperature transducers. Six temperature transducers (Pt100) separated by 60° (sensors 1–6) were used to monitor the bearing temperature. Different preloads value and distribution are shown in Table 3 and Figure 12, respectively. The experiment was carried out in a workshop, whose temperature range is relatively constant (initial temperature range is 23–25°C). Taking the increment values Δt as a variable on each measurement point in Figure 13

$Δ t = t_{s} - t_{0}$ (9)

where t_s is the terminal temperature of the test point (°C) and t₀ is the initial temperature of the test point (°C).

Figure 11.

Arrangement of the sensors.

Table 3.

The magnitude of preload exerted on bearing.

Preload distribution	F₁ (N)	F₂ (N)	F₃ (N)	F_total (N)	M_y (N mm)	M_z (N mm)
Uniform preload	733	733	733	2200	0	0
Non-uniform (C1)	1800	200	200	2200	58,195	33,600
Non-uniform (C2)	200	1800	200	2200	−58,195	33,600
Non-uniform (C3)	200	200	1800	2200	0	67,200

Figure 12.

The diagram of preload loading.

Figure 13.

Temperature increment at different rotation speeds: (a) n = 2000 r/min, (b) n = 4000 r/min, and (c) n = 8000 r/min.

The results from Figure 13(a) show that there is little difference between uniform and non-uniform preload in condition C2 at the speed 2000 r/min. The temperature rise in conditions C1 and C3 is higher than uniform condition by 0.8°C. When the speed was increased to 4000 r/min, the temperature increment under condition C2 is 1.6°C lower than uniform preload. The temperature increment in conditions C1 and C3 is slightly higher than uniform condition by 0.3°C. However, when the spindle speed was changed to 8000 r/min, the bearing temperature in condition C2 is lower than uniform preload by 2.5°C. Moreover, the temperature in condition C3 is higher than in condition C1. It can be illustrated by the fact that the non-uniform preload C3 aggravates the internal posture more severely, which is more likely to induce more friction within the test bearing. In the non-uniform preload condition C2, the temperature is higher than uniform after a period of 30 min. During 30–40 min, the two curves are almost overlapping. After a period of 40 min, the temperature increment is less than uniform preload condition. It can be seen from the figures that the temperature increment is minimum under non-uniform preload condition C2. In other words, the heat generation of the bearing in this condition is minimal.

The results show that internal feature of the spindle bearing can be changed by non-uniform preload. Heat generation of ball bearing will decrease under appropriate non-uniform preload condition and then service performance of spindle will be improved correspondingly.

Vibration characteristic of spindle

In this subsection, the vibration characteristic affected by non-uniform preload was studied. The experimental speed ranged from 2000 to 8000 r/min. In different preload conditions, as shown in Table 4, the original signals were acquired when the spindle speed was steady.

Table 4.

Magnitude of preload exerted on bearing (total preload is 1800 N).

Preload condition	F₁ (N)	F₂ (N)	F₃ (N)	F_total (N)	M_y (N mm)	M_z (N mm)
Uniform	733	733	733	2200	0	0
C2 (non-uniform)	200	1800	200	2200	−58,195	33,600

A three-axis accelerometer was mounted on the spindle to measure the vibration signal (z-axis). Bearing vibration performance was assessed at rotation speeds of 2000, 4000, and 8000 r/min. To avoid the inertia effects, the vibration signals were measured within the steady rotation period about 30 s. The results of vibration signals are shown in Figure 14.

Figure 14.

Vibration acceleration between non-uniform C2 and uniform preload: (a) n = 2000 r/min, (b) n = 4000 r/min, and (c) n = 8000 r/min.

At rotation speed of 2000 r/min, it can be found that the vibration amplitude under uniform and non-uniform preload is almost the same. The amplitude of vibration presents a similar trend at rotation speed of 4000 r/min. The reason is that the vibration of the spindle is not very sensitive to non-uniform preload in the low-speed region. However, when the spindle rotation speed is 8000 r/min, the vibration amplitude under non-uniform preload is 45% lower than under uniform preload condition. Three repetitive tests have been done to indicate that the results presented in this article are not an accident phenomenon. All of the three results show that the vibration of the spindle is decreased under proper non-uniform preload if the spindle suffers non-uniform force. In addition, the assemblage precision of rotor-bearing system is very important which can influence the experiments seriously. We carefully check the assemblage precision as for as machining accuracy prior to the repetitive test.

In practical engineering applications, the vibration level of rolling ball bearing is assessed by vibration acceleration single peak (g). Bearing vibration acceleration reflects the size of the impact force. In low-speed region (2000–4000 r/min), the centrifugal of the bearings is relatively small, which gives small impact to the spindle. The vibration amplitudes under uniform and non-uniform preload are almost the same, which could be explained by the centrifugal of bearings induced by the oblique outer spacer. However, when the rotation speed up to a certain speed, the centrifugal will be more serious and will affect the vibration amplitude of spindle significantly. In addition, it is worth noting that the radial stiffness of bearing in the different direction is different. Rolling bearing radial stiffness has an important influence on the dynamic characteristics of high-speed rotation of the system. It affects the appearance of the critical speed, the size of the dynamic instability of the generation, and vibration response. These are the possible reasons for the vibration characteristic of the test spindle under uniform and non-uniform preload conditions.

From the test results, it seems that the spindle speed effect was a major source to the bearing vibration amplitude rise. With the speed increased, the vibration amplitude decreased under proper non-uniform preload condition more obviously. The above results indicate that the vibration amplitude is decreased when proper non-uniform preload is applied on the bearing.

Conclusion

The effect of non-uniform preload of angular contact ball bearing on the spindle performance was experimentally investigated in this article. The rotation accuracy, the vibration, and temperature performance in both uniform and non-uniform conditions were compared and discussed:

Non-uniform preload method applied on a test spindle has been experimentally investigated. The performances of the spindle focus on rotation accuracy, temperature rise, and vibration were discussed in different preload conditions.

Rotation center of the spindle is significantly changed in different non-uniform preload conditions, and the rotation center offset is relevant to the equivalent moment produced by the non-uniform preload.

In the result of the temperature measurement of the spindle, when a non-parallelism of end face about the spacer was installed in the spindle, the temperature is decreased by 2.5°C under non-uniform preload condition at the spindle speed of 8000 r/min. However, when the spindle speed is under 5000 r/min, the temperature rise between non-uniform preload method and uniform preload method has shown little difference.

In the result of the vibration measurement of the spindle, when a non-parallelism of end face about the spacer was installed in the spindle, the vibration decreased about 45% at the speed of 8000 r/min. However, vibration of the spindle is not very sensitive to non-uniform preload in the low-speed region.

For further studies, the spindle performance under non-uniform preload should be considered with external load. Furthermore, the nonlinear coupling mechanism between the non-uniform preload and performance of bearing would be conducted conveniently.

Footnotes

Academic Editor: Ling Zheng

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 work is supported,in part,by National Natural Science Foundation of China (grant nos 51105297 and 51575434) and National High Technology Research and Development Program of China (863 Program) (grant no. 2012AA040704HZ).

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Investigation of multiple spindle characteristics under non-uniform bearing preload

Abstract

Keywords

Introduction

Non-uniform preload equivalent transformation and experimental platform

Equivalent transformation of non-uniform preload

Non-uniform preload adjustment system

Drive mechanism of the spindle

Preload loading component

Data acquisition system

Spindle rotation accuracy

Bearing temperature measurement

Bearing vibration signal acquisition

Results and discussion

Rotation accuracy of spindle

Temperature rise of bearing

Vibration characteristic of spindle

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References