Sage Journals: Discover world-class research

Abstract

Based on the correction factor of accuracy applied in ISO standard 3408-4:2006, this article proposed a modified method to calculate the effective ball number and basic static load rating of ball screws. Meanwhile, the basic static load rating of ball screws was obtained through a new measurement method presented in this article. The experimental results, agreeing well with the theoretical values calculated by the modified method, showed that there is a large discrepancy between the basic static load rating calculated by the traditional method and the measured result, which proves that the new modified method is valid. This study provides a more accurate method to obtain the basic static load of ball screws, which is significant for predicting the performance and service life of ball screws.

Keywords

Ball screws effective ball number basic static load stiffness permanent deformation

Introduction

As a key part of computer numerical control (CNC) machine tools, ball screws are widely used to convert rotary motion to linear motion or thrust to torque, and the reverse is the same.¹ The positioning accuracy of ball screws directly determines the machining accuracy of CNC machine tools. In recent years, there is an ever-increasing demand for high precision and high durability performance of CNC machine tools.² Because the basic static load rating of ball screws would directly affect the carrying capacity and positioning accuracy, it is worth paying more attention to research in the basic static load rating of ball screws.

In the literature, ball screws are usually simplified as thrust radial ball bearings under axial load,³ which means that the theoretical analysis of thrust radial ball bearings can be directly applied to ball screws. For ball bearings, if the permanent deformation exceeds $0.0001 D_{w}$ , the cavities formed in the raceways would cause extra vibration and noise, which leads to marginal lubrication and surface-initiated fatigue. Therefore, the basic static load rating of ball bearings is defined as the load applied to a nonrotating bearing that will result in a permanent deformation of $0.0001 D_{w}$ at the weaker of the inner or outer raceway contacts occurring at the position of the maximum-loaded rolling element in ISO standard 76:2006.⁴ Palmgren⁵ conducted indentation tests to obtain the permanent deformation of ball bearings. However, the indentation tests were dependent on the measurement devices available in the 1940s. Later, the indentation tests were repeated using modern measurement devices and the calculation formula was modified.⁶

Based on the formula for calculating the basic static load rating of ball bearings, the basic static load rating of ball screws was directly proposed in ISO standard 3408-5:2006,⁷ which is defined as the maximum static load that produces a plastic deformation no more than $0.0001 D_{w}$ at the position of the maximum-loaded rolling element. This research is mainly related to the basic dynamic load rating and the fatigue life of ball screws. The value of Weibull slope has been discussed for many years,^8–11 which is considered to be 9/10 in ISO standard 3408-5:2006. H Shimoda et al.⁸ recalculated the basic dynamic load rating of ball screws based on the calculation theory of linear bearings. However, no theoretical and experimental research on the basic static load rating of ball screws can be found in the literature.

Therefore, in this work, a more exact calculation method of the effective ball number and basic static load rating of ball screws is proposed. What is more, a new measurement method is proposed to validate the modified calculation method. The theoretical results are consistent with the experimental results, which proves that the modified formula of the basic static load rating is valid. This study contributes to the study of the service life of ball screws.

Theoretical analysis

Effective ball number in ball screws

Figure 1 shows the ball screw raceway without manufacturing error. In such a kind of condition, all balls are in good contact with the raceway and the load distribution is quite uniform. In fact, the load distribution of ball screws is not as uniform as we think due to the manufacturing error, as shown in Figure 2. This would lead to a lower effective ball number and then a lower basic static load rating and a lower axial stiffness than the designed values. For the estimation of the actual axial stiffness of ball screws, the correction factor of accuracy was applied in ISO standard 3408-4:2006¹² to eliminate the influence of manufacturing error. And the overall axial stiffness (R_bs) is obtained based on the static axial stiffness of screw shaft $(R_{s})$ and the modified static stiffness of the ball nut $(R_{nu, ar})$ , which can be expressed as

\frac{1}{R_{b s}} = \frac{1}{R_{s}} + \frac{1}{R_{n u, a r}}

(1)

where $R_{nu, ar}$ can be obtained by considering the corresponding correction factor as

R_{n u, a r} = f_{a r} R_{n u}

(2)

where $f_{ar}$ is the correction factor of accuracy and $R_{nu}$ is the static axial stiffness of the ball nut unit which can be expressed as

\frac{1}{R_{n u}} = \frac{1}{R_{b / t}} + \frac{1}{R_{n / s}}

(3)

where $R_{n / s}$ is the axial stiffness of the nut body and screw shaft, which can be written as

R_{n / s} = \frac{2 \cdot π \cdot i \cdot P_{h} \cdot E \cdot \tan^{2} α}{(\frac{D_{1}^{2} + D_{c}^{2}}{D_{1}^{2} - D_{c}^{2}} + \frac{d_{c}^{2} + d_{b 0}^{2}}{d_{c}^{2} - d_{b 0}^{2}}) \times 10^{3}}

(4)

where $P_{h}$ is the lead, $α$ is the contact angle, $D_{1}$ is the outer diameter of the ball nut, $D_{c}$ is the diameter of load application on the ball nut, $d_{c}$ is the diameter of load application on the ball screw shaft, and $d_{b 0}$ is the diameter of the deep hole bore.

Figure 1.

Ball screw raceway without manufacturing error.

Figure 2.

Ball screw raceway with manufacturing error.

According to ISO standard 3408-4:2006,¹² $D_{c}$ in equation (4) can be written as

D_{c} = D_{pw} + D_{w} \cdot \cos α

(5)

where $D_{pw}$ is the ball pitch circle diameter and $D_{w}$ is the ball diameter.

The static axial stiffness of the ball/raceway area $R_{b / t}$ can be written as

R_{b / t} = \frac{3}{2} \times \sqrt[3]{\frac{F_{e} \cdot \sin^{5} α \cdot \cos^{5} φ}{c_{E}^{6} \cdot c_{k}^{3}}} z^{2 / 3}

(6)

where z is the number of loaded balls, $F_{e}$ is the axial force exerted on the ball screw, $φ$ is the lead angle, $c_{E}$ is the material constant, and $c_{k}$ is the geometry factor that can be obtained following ISO standard 3408-4:2006.¹²

In ISO standard 3408-4:2006,¹² the correction factor of accuracy values between 0.5 and 0.6 on the basis of the standard tolerance grade, which means that the actual axial stiffness is far lower than the theoretical value. This indicates that the actual deformation (as well as the normal contact force) of the ball track is far higher than the theoretical value, which means that the effective ball number in a ball screw is far lower than the theoretical value.

According to ISO 3408-5:2006,⁷ the basic static load rating of ball screws can be written as

C_{oa} = k_{0} \cdot z \cdot \sin α \cdot D_{w}^{2} \cdot \cos φ

(7)

where $k_{0}$ is the characteristic of basic static axial load rating, which can be written as

k_{0} = \frac{27.74}{D_{w} \sqrt{(ρ_{11} + ρ_{12}) (ρ_{21} + ρ_{22})}}

(8)

ρ_{11} = ρ_{21} = \frac{2}{D_{w}}

(9)

ρ_{12} = \frac{- 1}{f_{rs} \cdot D_{w}}

(10)

ρ_{22} = \frac{\cos α}{\frac{D_{pw}}{2} - \cos α \frac{D_{w}}{2}}

(11)

where $f_{rs}$ is the conformity factor of ball screws.

It can be seen from equation (7) that the basic static load rating is proportional to the number of loaded balls. This means that if the effective ball number varies significantly due to manufacturing error, a large discrepancy between the actual basic load rating and the theoretical value would be caused.

Replacing the static axial stiffness of the ball nut unit $R_{nu}$ in equation (3) with the modified static axial stiffness of the ball nut unit $R_{nu, ar}$ , the effective ball number can be obtained. The modified static axial stiffness of the ball nut unit $R_{nu, ar}$ can be written as

\frac{1}{R_{nu, ar}} = \frac{1}{R_{b / t}^{'}} + \frac{1}{R_{n / s}}

(12)

The effective ball number can be obtained by combining equations (2), (3), (6), and (12), which can be written as

z^{'} = {[\frac{1}{f_{ar} \cdot z^{2 / 3}} + \frac{\frac{2}{3} (\frac{1}{f_{ar}} - 1) F_{e}^{1 / 3} \cdot \sin^{3 / 5} α \cdot \cos^{3 / 5} φ}{R_{n / s} \cdot c_{E}^{2} \cdot c_{k}}]}^{- 2 / 3}

(13)

Modified basic static load rating of ball screws

On the basis of the effective ball number obtained above, a modified calculation method of the basic static load rating of ball screws is proposed, which is in proportion to the effective ball number.

Replacing the number of loaded balls z in equation (7) with the effective ball number $z^{'}$ , the modified basic static load rating can be rewritten as

C_{oa}^{'} = k_{0} \cdot z^{'} \cdot \sin α \cdot D_{w}^{2} \cdot \cos φ

(14)

Experimental verification

Based on the definition of the basic static load rating of ball screws, a new measurement method of the static load rating based on a stiffness test bench is proposed. As shown in Figure 3, the stiffness test bench is composed of the load beam (1), the load cell (2), the anti-rotation unit (3), the stiffness measurement unit (4), and the fixing unit (5). When the load beam moves down to compress the ball screws, the axial load can be obtained through the load cell. Simultaneously, the axial deformation under axial load can be obtained through the displacement sensors in the stiffness measurement unit. It is worth mentioning that both loading and unloading processes are measured in the experiment. The parameters used in the test are shown in Table 1.

Figure 3.

The stiffness test bench.

Table 1.

Parameters used in the experiment.

Parameter	Value	Unit
Diameter of the ball $D_{w}$	6.35	mm
Ball pitch circle diameter $D_{pw}$	40	mm
Outer diameter of the ball nut $D_{1}$	82	mm
Helix angle $φ$	4.55	degree
Contact angle $α$	45	degree
Helical pitch $P_{h}$	10	mm
Circle’s column number i	2.5 × 2

The experimental process is shown in Figure 4. According to the theoretical results calculated through the modified method, the ratio of the modified basic static load rating to the theoretical value varies between 0.3 and 0.5. As permanent deformations of the ball screw would be produced during the experiment, the initial axial force applied on the ball screw is usually set less than 30%C_oa. However, to obtain more accurate experimental results, both the initial axial force and the loading interval are set at 10%C_oa. The experimental process is as follows:

1. The axial force is applied on the stiffness test bench. Based on the obtained axial stiffness curve of ball screws, the difference between the initial value $(l_{1})$ of the displacement sensors and the final value $(l_{2})$ after the unloading is completed can be written as

Δ l_{i} = | l_{1} - l_{2} |

(15)

The cumulative difference between the initial and final values is the plastic deformation of ball screws, which can be written as

Δ l = \sum_{t = 1}^{i} Δ l_{t}

(16)

2. Comparing the cumulative difference $Δ l$ with $0.0001 D_{w}$ , if the cumulative difference $Δ l$ is less than $0.0001 D_{w}$ , then the axial force applied on the stiffness test bench increases by $10 % C_{o a}$ .

3. Repeat the above two steps until the cumulative difference $Δ l$ is greater than $0.0001 D_{w}$ . Then the measured static load rating can be obtained as

C_{o a}^{″} = F_{a x i a l} - 5 % C_{o a}

(17)

Figure 4.

The experimental process.

Results and discussion

The measured axial stiffness curves are shown in Figures 5 –9. It can be seen from these figures that the unloading axial stiffness curve (the red solid line) is over the loading axial stiffness curve (the black solid line). The permanent difference $Δ l_{i}$ obtained through the experimental axial stiffness curves is shown in Table 2.

Figure 5.

The axial stiffness curves under the axial load of $10 % C_{o a}$ .

Figure 6.

The axial stiffness curves under the axial load of $20 % C_{o a}$ .

Figure 7.

The axial stiffness curves under the axial load of $30 % C_{o a}$ .

Figure 8.

The axial stiffness curves under the axial load of $40 % C_{o a}$ .

Figure 9.

The axial stiffness curves under the axial load of $50 % C_{o a}$ .

Table 2.

The permanent difference obtained through the experiments.

Axial load	Permanent difference (μm)	Cumulative permanent difference (μm)
$10 % C_{o a}$	0	0
$20 % C_{o a}$	0	0
$30 % C_{o a}$	0.2	0.2
$40 % C_{o a}$	0.5	0.7
$50 % C_{o a}$	2.5	3.2

As shown in Figures 5 and 6, the unloading curve can always return to the original point under the axial forces of $10 % C_{o a}$ and $20 % C_{o a}$ , which means that the deformation is elastic when the axial force is lower than 20%C_oa. As shown in Figure 7, the unloading curve cannot return to the original point anymore when the axial force reaches 30%C_oa, in which plastic deformations occurred with the permanent difference being 0.2 μm but less than 0.0001C_oa (0.635 μm). This means that the basic static load rating is greater than 30%C_oa. As shown in Figure 8, the unloading curve cannot return to the original point when the axial force reaches 40%C_oa. The corresponding permanent difference is 0.5 μm, which is still less than 0.0001D_w. However, in such a kind of condition, the cumulative permanent difference has already reached 0.7 μm (0.2 μm + 0.5 μm), which is greater than 0.0001D_w. Therefore, the actual basic static load rating is less than 40%C_oa. To make the experimental results more convincing, one more experiment under the axial load of 50%C_oa is conducted. As shown in Figure 9, the permanent difference reaches 2.5 μm, which is far greater than 0.0001D_w. Therefore, the possibility of permanent deformation caused by experimental errors is eliminated, which proves that the four experimental results above are scientific and reasonable.

As shown in Table 2, when the axial load applied on the stiffness test bench reaches 40%C_oa, the cumulative difference is $0.7 μ m$ , which is greater than 0.0001D_w. According to the definition of the static load rating of ball screws, the measured static load rating of ball screws can be obtained by

C_{o a}^{″} = 35 % C_{o a} = 48.405 kN

The static axial load calculated through the modified calculation method in this article is 51.8625 kN. The relative error of the modified calculation method is 7.14%, while the error of the traditional method (138.3 kN) is as high as 185.71%, which verifies the theoretical analysis in this work.

Conclusion

In this work, a new modified method to calculate the effective ball number and the basic static load rating of ball screws is proposed. The theoretical results show that the basic static load rating calculated through the modified method is less than half of the value calculated through the traditional method. With a new measurement method proposed in this article, the basic static load rating of ball screws was measured for the first time in the literature. The experimental results show that the relative error of the modified calculation method is 7.14%, while the error of the traditional method is 185.71%, which proves that the modified calculation method is valid. More importantly, the actual basic static load rating is only 35% of the theoretical value reported in ISO 3408-5:2006,⁶ which means that manufacturing error has a great influence on both the basic static load rating and stiffness of ball screws.

It is worth mentioning that, although the experimental results agree well with the theoretical values, the experiment is just based on a single ball screw. Therefore, in order to make the modified calculation method of basic static load rating more convincing, more number of experiments should be conducted in future research.

Footnotes

Acknowledgements

The authors are indebted to the Key Laboratory of Performance Test and Reliability Technology for CNC Machine Tool Components of Chinese Machinery Industry for providing the experiment materials.

Handling Editor: James Baldwin

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 study has been supported by the National Science and Technology Major Project of China (grant no.2017ZX04011001).

ORCID iD

Changguang-Zhou

References

Feng

Wang

et al . An automatic measuring method and system using a light curtain for the thread profile of a ball screw. Meas Sci Technol 2011; 22: 085106.

Zhou

Feng

Chen

et al . Correlation between preload and no-load drag torque of ball screws. Int J Mach Tool Manuf 2016; 102: 35–40.

Zhou

Feng

. A new model for predicting the mechanical efficiency of ball screws based on the empirical equations for the friction torque of rolling bearings. Adv Mech Eng 2018; 10: 800173.

ISO 76:2006. Rolling bearings—static load ratings.

Palmgren

. Ball and roller bearing engineering. Philadelphia, PA: SKF Industries, 1959.

Harris

. Rolling bearing analysis. Hoboken, NJ: John Wiley & Sons, 1984.

ISO 3408-5:2006. Ball screws, static and dynamic axial load ratings and operational life.

Shimoda

. Re-evaluation of basic dynamic load rating and life formula for a ball screw. Tribol Trans 2007; 50: 88–95.

Izawa

. Application technique for ball screw. Tokyo, Japan: Kogyo Chosakai, 1993.

10.

Shimoda

. Developments of a new type ball screw for long life its tests. Japan J Tribol 1994; 39: 406–413.

11.

Yamaguchi

. Vibration, noise and life of ball screw. In: Proceedings of the 96 motion engineering symposium, 1–2 March 1996. Tokyo, Japan: Japan Management Association.

12.

ISO 3408-4:2006. Ball screws—part 4: static axial rigidity.

Recalculation of the basic static load of ball screws

Abstract

Keywords

Introduction

Theoretical analysis

Effective ball number in ball screws

Modified basic static load rating of ball screws

Experimental verification

Results and discussion

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References