J. Cent. South Univ. (2021) 28: 1357-1376

DOI: https://doi.org/10.1007/s11771-020-4537-1

Precision loss of ball screw mechanism under sliding-rolling mixed motion behavior

QI Bao-bao(齐宝宝)^{1, 2}, CHENG Qiang(程强)^{1, 2}, LI Shun-lei(李顺磊)^{1, 3},LIU Zhi-feng(刘志峰)^{1, 3}, YANG Cong-bin(杨聪彬)^{1, 3}

1. Institute of Advanced Manufacturing and Intelligent Technology, Beijing University of Technology, Beijing 100124, China;

2. Beijing Key Laboratory of Advanced Manufacturing Technology, Beijing University of Technology, Beijing 100124, China;

3. Mechanical Industry Key Laboratory of Heavy Machine Tool Digital Design and Testing,Beijing University of Technology, Beijing 100124, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: The sliding-rolling mixed motion behavior degrades the ball screw’s precision at different levels. Based on the sliding-rolling mixed motion between ball and screw/nut raceway, the ball screw’s precision loss considering different given axial loading and rotational speed working conditions was investigated. Since creep and lubrication relate to sliding and rolling motion wear, the creep and lubrication characteristics are analyzed under different working conditions. Besides, the precision loss was calculated considering the sole influence of sliding behavior between ball and screw and compared with the results from other current models. Finally, research on precision loss owing to the sliding-rolling mixed motion behavior was realized under given working conditions, and suitable wear tests were carried out. The analytical results of precision loss are in good agreement with the experimental test conclusions, which is conducive to better predicting the law of precision loss in stable wear period.

Key words: ball screw mechanism; precision loss; sliding-rolling mixed motion; creep; lubrication

Cite this article as: QI Bao-bao, CHENG Qiang, LI Shun-lei, LIU Zhi-feng, YANG Cong-bin. Precision loss of ball screw mechanism under sliding-rolling mixed motion behavior [J]. Journal of Central South University, 2021, 28(5): 1357-1376. DOI: https://doi.org/10.1007/s11771-020-4537-1.

1 Introduction

As driving functional components, ball screw mechanisms (BSMs) are among the most used components for precision equipment in various fields, such as machine tools [1]. The decline of the BSM precision level can restrict the accuracy level of precision equipment. Precision retention of ball screw refers to the ability to maintain precision at a certain precision level under specified operating conditions and times. However, the sliding-rolling mixed motion behavior influences the ball screw’s precision degradation at different degrees. Studies on BSM precision loss include precision degradation basics and analysis of precision degradation, revealing the law of precision degradation.

The basics of the BSM precision degradation include statics, kinematics [2-4], and dynamics [1, 5]. LIN et al [2] proposed the basic kinematic analysis method for ball screws. Further, WEI et al [3] improved the ball screws kinematic model based on LIN’s model by considering the effect of contact angle and elastic contact deformation on kinematics. The kinematics model [2] was applied to the optimization analysis of BSM transmission efficiency by LIN et al [4], and WEI et al [6] extended the transmission efficiency optimization model of the BSM based on LIN’s work. By considering the influence of preload and lubrication on contact kinematics, WEI et al [7] selected the preload-adjustable BSM as the research object. YANG et al [1] and LIU et al [5], respectively, studied the BSM feed dynamics control and dynamic performance optimization. Moreover, CHEN et al [8] established BMS’s contact stiffness considering the impact of the contact angle and helix’s rise angle on dynamics.

Besides, ZHAO et al [9], ZHEN et al [10], and LIN et al [11] studied the contact load distribution by considering the motion torque and geometric error. KANG [12] designed a series of experiments to detect the BSM friction torque models. Through creep analysis, XU et al [13] proposed a novel ball screw friction torque model. By considering the impact of preload on friction torque, ZHOU et al [14] selected the preload-adjustable BSM as the object of their experimental investigation. Finally, WEI et al [15] and KAMALZADEH et al [16] revealed the effects of preload and elastic deformation on the BSM wear rate.

At present, studies on BSM precision degradation are mainly performed from the perspective of sliding wear. According to the ball screw kinematics analysis results, there are two types of motion in BSM, rolling and sliding [14, 17, 18]. Based on ARCHARD’s [19] model, WEI et al [15], ZHOU et al [20], LIU et al [21], CHENG et al [22], and ZHOU et al [23] studied the BMS wear. ZHOU et al [20] and LIU et al [21] studied the screw precision degradation rate in line with wear model, and the BSM precision degradation due to wear result under time-varying working conditions was studied by CHENG et al [22]. The BSMs wear coefficient was obtained by ZHOU et al [23] through design experiments. LIU et al [24] developed a compensation method for thermal error. By detecting the motor current and the vibration signal of BSM, NGUYEN et al [25] realized the preload monitoring. CHENG et al [26] established the attenuation model of machine tool accuracy, considering the decline of transmission components accuracy. Then, CAI et al [27] and CHENG et al [28, 29] described the coordinate transformation relationship between two adjacent bodies in a multi-body system and established an accuracy prediction model for the vertical machining center [30, 31]. Moreover, the research results can be applied to the accuracy prediction of various machine tools.

Despite the progress, the degradation law of the ball screw’s positioning precision considering the effects of the sliding-rolling mixed motion behavior has not been studied, as well as the BSM precision degradation and its precision loss rate due to sliding-rolling mixed motion behavior. In order to update the study of precision loss, the BSM sliding-rolling mixed motion behavior considering given axial loading and rotational speed working conditions was studied in this paper. Furthermore, the BSM precision degradation and BSM precision loss rate were also gained in the light of the sliding-rolling mixed motion behavior in this study.

2 Ball screw mechanisms creep rate

Creep analysis is the basis for resolving the problem of the BSM rolling contact, including the BSM rolling friction [13] and the BSM rolling contact wear [32, 33]. This paper uses the BSM coordinate system shown in Figure 1 [13, 22].

O-XYZ is the global coordinate system, and O_b-TNB is the free coordinate system. Both S-x_sy_sz_s and N-x_ny_nz_n are local coordinate systems. For the meaning of other parameters in the figure, refer to the Nomenclature.

The creep rate of ball and nut along the x_n and y_n direction, i.e., _xn and _yn, according to KALKER’s linear creep [31], is expressed as follows:

                 (1a)

                 (1b)

where and are the components of ball spin angular velocity ω_br in the x_n and y_n directions, respectively. ω_br was obtained [34, 35]:

                (2)

Therefore, ω_b-xn and ω_b-yn can be written as:

Figure 1 BSM global-contact coordinate system and transformation relationship between coordinates O-XYZ, O_b-TNB, S-x_sy_sz_s and N-x_ny_nz_n

         (3a)

                       (3b)

The components ω_N-xn and ω_N-yn of nut’s angular velocity ω_N in the x_n and y_n directions, respectively, are as follows:

            (4a)

                              (4b)

The rolling speed between the ball and nut, ω_b-N is:

   (5)

In the equations above, δ_ni is the contact deformation between the i-th ball and the nut, expressed as follows:

                  (6)

where χ_GSni (see Table 1) is the correction factor in the Hertz contact equation.

Table 1 Initial parameters of BSM

ω_n, ω_b and ω_t are given as:

                          (7a)

                      (7b)

                        (7c)

ω_N is nut’s angular velocity, expressed as:

               (8)

_xn and _yn are expressed as:

            　 (9a)

             (9b)

Similarly, based on Ref. [33], the creep rates between the ball and screw in the x_s and y_s directions, _xs and _ys, are expressed as:

                 (10a)

                 (10b)

where ω_b-S is the rolling speed between the ball and screw, given by

(11)

The angular velocity components of ball or screw in the x_s and y_s directions, namely ω_b-xs, ω_b-ys or ω_S-xs, ω_S-ys, can be calculated as:

         (12a)

                       (12b)

               (13a)

                              (13b)

where δ_si is the contact deformation between the i-th ball and the screw, expressed as:

                  (14)

where r_s is is the track curvature radii of the screw.

Finally, _xs and _ys are expressed as:

    (15a)

 (15b)

3 Ball screw mechanisms lubrication analysis

The lubricant film thickness and contact surface roughness have diverse degrees of influence on the wear coefficient [36]. Figure 2 shows the BSM lubrication analysis diagram.

According to Ref. [7], the lubricant film thickness between the ball and the raceway h_b-k can be written as follows:

 (16)

where h_cb-k (k=s or n) is the contact center lubricant thickness between the ball and screw or the nut that can be obtained from Ref. [35]. R₁ or R₂ is the curvature sum of ball and screw or ball and nut, and e_b-k (i_k, j_k) is the elastic deformation of ball and screw and that of ball and nut [37].

 (17a)

 (17b)

                         (18)

                          (19)

where U_b-s and U_b-n are the dimensionless velocity between the ball and screw and that between the ball and nut [34]; G_b-s and G_b-n are the BSM material parameters and relate to the pressure-viscosity coefficient of lubricant, γ; W_b-s and W_b-n are dimensionless contact loading; D_s and D_n are the inner diameter of screw and that of the nut, respectively; E^* is the equivalent elastic modulus of the ball and the raceway.

Figure 2 Lubricated contact status between ball and raceway

                          (20a)

                        (20b)

                        (21)

                           (22a)

                          (22b)

                    (23)

where η_vs and η_vn are the lubricant’s dynamic viscosity. The relationship between viscosity, pressure, and temperature is as follows[38]:

     (24a)

   (24b)

where η_vs,0 is the lubricant’s viscosity at temperature T_vs,0; η_vn,0 is the lubricant’s viscosity at temperature T_vn,0; γ/λ is the pressure-viscosity coefficient of lubricant (Table 1); λ is the temperature-viscosity coefficient of lubricant (Table 1); F_si and F_ni are contact loadings between the i-th ball and screw and nut, respectively.

Moreover, u_b-s and u_b-n are the average sliding velocity in S-x_sy_sz_s and N-x_ny_nz_n, respectively:

     (25a)

    (25b)

where v_b-xs and v_b-ys are the components of the ball velocity in the x_s and y_s direction, respectively; v_s-xs and v_s-ys are the components of the screw speed in the x_s and y_s directions, respectively; v_b-xn and v_b-yn are the components of the ball velocity in the x_n and y_n directions, respectively; v_n-xn and v_n-yn are the components of the nut speed in the x_n and y_n directions, respectively. These terms are expressed as follows:

           (26a)

                       (26b)

           (27a)

                        (27b)

 (28a)

 (28b)

 (29a)

 (29b)

In the above equations, the term v_b-s is ball’s rotational speed relative to screw in the helical direction [3], given by

                (30)

By combining Eqs. (16a)-(18b), the minimum lubricant film thickness h_minb-s and h_minb-n can be written as:

                         (31a)

                        (31b)

In line with the minimum lubricant thickness and the mean square roughness of contact surface, this study introduced the parameter Φ to describe the BSM lubrication state as follows:

                          (32)

where σ_b and σ_l are the mean-square roughness values of the ball and the raceway, respectively. When h_min=h_minb-s and σ_l=σ_s, Eq. (32) expresses the lubrication parameter of ball and screw; when h_min=h_minb-n and σ_l=σ_n, Eq. (32) is the lubrication parameter of the ball and nut.

The roughness morphology and profile heights of the raceways and balls can be obtained through three-dimensional surface topography technique using a three-dimensional topography signal instrument. Figure 3 shows the measured micro-morphology signal of the screw raceway, and Table 1 summarizes the mean-square value of the roughness. This paper assumes that the screw raceway and nut raceway have the same roughness value. Similarly, the mean square roughness value of the ball was also obtained.

Figure 3 Three-dimensional micro-morphology signal of screw raceway:

The relationship between the BSM wear coefficient k_d and lubrication parameter Φ is as follows [37]:

              (33)

where k_bl is the BSM wear coefficient under the boundary lubrication state (see Table 1).

4 Precision degradation calculation

According to the mixed motion analysis, this study calculates the BSM precision degradation for the sliding and rolling motion separately.

4.1 Precision degradation due to sliding motion wear

According to the Archard model [19], the sliding wear amount can be expressed as follows:

           (34)

where C_b-s and C_b-n are the total cycles of the ball and screw, and the ball and nut, respectively; V_b-s(n) and V_b-n(n) are the wear amount of the ball and screw, and the ball and nut, respectively, in one cycle.

 (35a)

 (35b)

where V¹_b-s(t) and V¹_b-n(t) are the wear amount of the first ball and screw and that of the first ball and nut during one cycle.

                         (36a)

                         (36b)

whereare the wear amounts of ball and screw from t_ss1 to t_ss2, from t_ssi to t_ssi+1 and from t_ssM to t_ss1 in Eq. (37a); are the wear amounts of ball and nut from t_ns1 to t_ns2, from t_nsi to t_nsi+1 and from t_nsM to t_ns1 in Eq. (37b):

          (37a)

         (37b)

where are the wear amounts of ball and screw/nut at t_ss1, …t_ssi …t_ssM/t_ns1, …t_nsi…t_nsM in Eqs. (38a) and (38b). And F_s1, …, F_si…, F_sM/F_n1, …F_ni …F_nM are the contact loadings of each ball and screw/nut.

             (38a)

           (38b)

whereand…, are the screw and nut sliding distances relative to the 1st, 2nd, …, M-th ball. By taking the i-th ball as an example, one obtains:

              (39a)

              (39b)

where R_bsi and R_bni are the sliding-to-rolling ratios of the i-th ball and screw, and that of the i-th ball and nut, respectively. Based on Ref. [22], R_bsi and R_bni can be calculated:

 (40)

 (41)

By combining Eqs. (37a)-(41), one can calculate …,…,and …, …, Moreover, the sliding wear amount is obtained by bringing Eqs. (36a) and (36b) into Eqs. (35a), (35b) and (34). Based on Eqs. (34) and (36b), the sliding wear depth of ball and screw/nut is given as:

                (42a)

               (42b)

where h_sbs(n) and h_sbn(n) are the screw raceway length and that of nut raceway; a_bs is the contact surface long half axes of ball and screw, and b_bs is the contact surface short half axes of ball and screw; a_bn is the contact surface long half axes of ball and nut; b_bn is the contact surface short half axes of ball and nut. The length of the raceway of the screw is and the length of the raceway of the nut is

According to Refs. [21, 39], the contact area can be established:

          (43)

When k=s, Eq. (43) gives the contact area of ball and screw; when k=n, Eq. (43) gives the contact area of the ball and nut.

The calculation of contact loading was demonstrated. MEI et al [40] established the calculation model for the BSM contact loading. ZHAO et al [9], ZHEN et al [10], and LIN et al [11] studied the contact load distribution by considering the motion torque and geometric error. By considering the preload, WEI et al [7] pointed out the contact state between ball and raceway under different preloads; furthermore, based on the contact state, LI et al [41] conceptualized the correlation among contact loading, axial loading, and preload. Based on Refs. [40], [7] and [41], this paper calculates the contact loading between the ball and screw/nut considering the preload attenuation.

According to Refs. [7, 41], the contact state between ball and raceway can be represented as shown in Figure 4. In Figure 4(a), the axial loading is higher than 2.83 times of preload; in Figure 4(b), the axial loading is lower than 2.83 times of preload. When the axial loading almost equals the preload, BSM is not suitable for precision transmission [7]. Therefore, the study does not discuss this situation.

In Figure 4(a), F_1a is the total contact loading among balls, screw, and nut 1 raceway; F_2a is the total contact loading among balls, screw, and nut 2　 raceway, and they can be written as follows:

 (44)

Similarly, F_1a and F_2a in Figure 4(b) are:

     (45)

According to Ref. [40], F_si and F_ni (i=1, …, M) can be obtained as:

Figure 4 Contact state:

                (46a)

               (46b)

where F_p(t) can be obtained from Ref. [42]. In contact state in Figure 4(a), when the ball contacts nut 1, when the ball contacts nut 2,In contact state in Figure 4(b), when the ball contacts nut 1, when the ball contacts with nut 2, For the detailed calculations, refer to Refs. [40, 7, 41, 42].

4.2 Precision degradation due to rolling motion wear

In the rolling motion contact area of ball and raceway, there are partial sliding and partial adhesion phenomena [32, 33]. Figure 5 shows a schematic of rolling motion contact of ball and screw/nut raceways.

Based on Ref. [32], the rolling wear depth of single ball and screw is deduced:

                     (47)

where k_bs is the dimension wear coefficient, k_bs=k_d/H_b-s; according to Eq. (44), A()≈a_kb_k/9; v_rbs is the creep velocity.

At time t_i, the wear depth is given owing to rolling motion as:

   (48)

where S_bs is the creep value in rolling motion contact.

By combining Eqs. (47) and (48), the accumulative wear depth owing to rolling motion behavior in the x_s direction can be obtained:

   (49)

where is the sliding velocity between ball and screw in the x_s direction given by:

                            (50)

In Eq. (50), is the average rolling velocity of ball and screw in the x_s direction calculated by combining Eqs. (12a) and (13a).

Figure 5 Rolling motion contact of ball and screw/nut raceway: Partial slip and partial stick regions

            (51)

By combining Eqs. (49)-(51), Eq. (48) was deduced as:

         (52)

In Figure 5, at contact point A_xs, and at contact point C_xs,

By considering the rolling cycles number, is expressed as:

 (53)

Similarly, the rolling wear depth in the y_s direction can be obtained:

 (54)

The term is the sliding velocity between ball and screw in the y_s direction, given by

                            (55)

where is the average rolling velocity of ball and screw in the y_s direction calculated by combining Eqs. (12b) and (13b) as:

                       (56)

Similarly, the cumulative rolling wear depth of ball and nut in the x_n and y_n direction is obtained:

 (57a)

(57b)

The sliding velocities of ball and nut in the x_n and y_n direction, i.e., and in Eqs. (57a) and (57b), can be calculated:

                           (58a)

                          (58b)

where _xn and _yn are the creep rate of ball and screw in the x_n and y_n directions given by Eqs. (9a) and (9b), respectively. The terms and are the average rolling velocity of ball and nut in the x_n and y_n directions calculated according to Eqs. (3a), (4a), and Eqs. (3b) and (4b):

          (59a)

                      (59b)

Considering the rolling cycles number and combining Eqs. (57a) and (57b), and are given by Eqs. (60a) and (60b), respectively:

 (60a)

 (60b)

where and are the rolling wear depths of ball and nut in the x_n and y_n directions, respectively, when the number of the ball rolling cycles is (n-1). In Figure 5, at contact point A_xn, at contact point C_xn, At contact point A_yn, and at contact point C_yn,

5 Analysis of BSM precision loss

5.1 Working conditions on precision loss

Figure 6 shows the flowchart of the whole calculation process. When the ball is at different phase angles, the creep rate between ball and screw/nuts varies with the operating conditions (i.e., axial loading and rotational speed). The BSM lead and raceway curvature ratios were obtained from Table 1. The variation of the creep rate with the axial loading can be investigated based on the creep calculation presented in Section 2; the rotational speed of the screw was set to 60 r/min, and the initial preload was set to 2500 N. Besides, the law of the BSM creep rate can be obtained considering the different axial loading conditions in Figures 7(a)-(d). Similarly, the BSM lead was 20 mm, and the raceway curvature ratio was 0.2725 based on Table 1. When the axial loading was set to 2000 N, and the initial preload was 2500 N, the creep rate was obtained considering the different rotational speeds in Figures 7(e)-(h).

Figure 6 Flowchart of calculation process for BSM precision loss and precision loss rate

In Figures 7(a) and (b), the changing characteristics of the creep rate between the ball and the screw raceway are consistent with those of the contact angle. That is, the creep rate between the ball and the screw raceway decreases with reducing contact angle. Also, the creep rate decreases with the increase of the axial load. However, the contact angle θ_n between the ball and the nut raceway increases as the axial load increases. Therefore, in Figures 7(c) and (d), the changing characteristics of the creep rate between the ball and the nut raceway are also consistent with the changing characteristics of the contact angle. That is, the absolute value of the creep rate between the ball and the nut increases as the contact angle increases, and it became more significant as the axial load increases.

In Figures 7(e)-(h), because of the ball spin angular velocity, the rolling speed between ball and screw/nut relates to the contact angle. When the rotational speed of the screw increases, the contact angle θ_s becomes larger. Therefore, in Figures 7(e) and (f), the changing characteristics of the creep rate between the ball and the screw raceway are consistent with the changing characteristics of the contact angle. That is, the creep rate between the ball and the screw enlarges as the contact angle increases. The performance result was that the creep rate increased with the increase of the screw rotation speed. However, the contact angle θ_n becomes small as the screw rotation speed increases. Therefore, in Figures 7(g) and (h), the changing characteristics of the creep rate between the ball and the nut raceway are also consistent with those of the contact angle. That is, the absolute value of the creep rate between the ball and the screw becomes smaller as the contact angle decreases. Also, as the screw rotation speed increases, the creep rate reduces.

Using KOC301 lubricant, the lubricant viscosities η_vs and η_vn can be calculated from Eqs. 24(a) and (b). According to Figures 8(a) and (b), both the rotational speed and the axial loading have certain influences on the minimum lubricant film thickness between ball and screw/nut. However, the influence of the rotational speed is more significant than that of the axial loading. Also, the minimum film thickness between the ball and nut is slightly larger than that between ball and screw, because with the increase of the rotational speed and centrifugal force, the minimum lubricant thickness between ball and nut was increased more than that between ball and screw. As the axial loading increases, the contact loading of the ball and screw/nut also increases. Therefore, the minimum lubricant film thickness of the ball and screw/nut reduces.

Figure 7 BSM creep rates under rolling contact considering different axial loading conditions (a, b, c, d) and rotational speed conditions (e, f, g, h):

Figure 8 BSM minimum lubricant film thickness under different rotational speed and axial loading working conditions:

In this study, when the ball moved from the ith ball position to the (i+1)th ball position, the sliding length of screw/nut relative to the original contact point is called the sliding distance. Axial loading and rotational speed have different degrees of influence on the sliding distance. When the initial preload was 2500 N, by combining Eqs. (39a)-(41), the sliding distance of ball and screw/nut is obtained (see Figure 9) considering the effects of axial loading and rotational speed.

According to Figure 9, both the axial loading and rotational speed of the screw have different degrees of influence on the BSM sliding distance. However, compared to the effect of axial loading on the sliding distance, the effect of the rotational speed is more significant. Also, the sliding distance on the screw is greater than that on the nut. The reason for this phenomenon is the transmission principle of BSM; that is, the movement of the screw drives the ball, and in turn, drives the nut to move. When a sliding movement exists between the ball and the screw raceway, it directly affects the movement between the ball and nut. It causes sliding-to-rolling ratio R_bs between the ball and the screw greater than the R_bn between ball and nut. Therefore, the sliding distance between the ball and screw is more significant than that between the ball and nut.

Figure 9 BSM sliding distance at different rotational speed and axial loading working conditions:

5.2 Experimental tests for evaluating precision loss

According to Refs. [6, 20-22], the cumulative wear depth in the stroke direction corresponds to the BSM precision loss value. By combining Eqs. (42a) and (43), the BSM precision degradation owing to the accumulated sliding motion wear depth of ball and screw/nut can be obtained:

            (61a)

           (61b)

Compared with WEI’s [6], ZHOU’s [20], and LIU’s [21] models, the BSM precision loss due to sliding motion wear of ball and screw was obtained. Based on Eq. 61(b), the precision loss due to the accumulated sliding motion wear between ball and nut in the stroke direction can also be gained. Figure 10 shows the BSM precision loss due to sliding motion wear.

Figure 10 BSM precision loss due to sliding motion wear:

By combining Eqs. (9), (15) and (22), the BSM creep rate can be calculated considering the operating conditions. The BSM creep characteristics were obtained at different axial loadings (see Figures 7(a)-(d)) and at different rotational speeds (Figures 7(e)-(h)). Based on Eqs. (47)-(52), the rolling motion wear depth of single ball and screw was obtained. Then, according to Eq. (53), the accumulated rolling motion wear depth of all effective transmission balls and screw in the x_s direction is given by Eq. (62).

           (62)

Similarly, according to Eq. (54), the accumulated rolling motion wear depth of all effective transmission balls and screw in the y_s direction was also obtained:

           (63)

Figure 11(a) shows the variation trend of BSM cumulative rolling motion wear depth obtained by combining Eqs. (62) and (63). Similarly, according to Eqs. (60a) and (60b), the cumulated rolling motion wear depth of all effective transmission balls and nut in the x_n and y_n directions can be written by Eq. (64), and Figure 11(b) shows its variation law.

Figure 11 BSM precision loss due to rolling motion wear:

Similarly, according to Eqs. (61), the accumulated wear depth of all effective transmission balls and nut in x_n/y_n direction can be written as follows:

         (64a)

         (64b)

In order to obtain the amount of precision loss when the BSM runs in a stable operating state, the BSM after fully running-in was selected as the experimental object. The positioning error after running was taken as the initial precision value. The experiments were performed using a test platform and a laser interferometer (XD Dual Frequency Laser Interferometer).

The positioning precision of the BSM after the wear test was measured. The difference between the test value and the initial value was the precision loss due to sliding-rolling mixed motion behavior. The following experimental conditions were established: axial loading of 2000 N, rotational speed of 60 r/min, grease-type lubrication using KOC301 lubricant, ambient temperature of 23 °C, test overall duration of 160 h. By combining Eq. (61a) with Eq. (64b), the test process and test results due to sliding-rolling mixed motion were obtained (see Figure 12).

Figure 12 BSM precision loss due to sliding-rolling mixed motion behavior

ZHOU et al [20] and LIU et al [21] established the BSM precision loss model due to screw sliding motion, mainly focusing on screw precision degradation. However, since the screw, nut and ball have different degrees of precision degradation, this study focuses on the BSM precision degradation due to sliding-rolling mixed motion behavior. According to Refs. [20, 21] and combining Eqs. (61a)-(64b), the BSM precision degradation, and precision loss rate were obtained in Eq. (65). Figure 13 shows the total precision loss rate for the BSM.

Figure 13 Total precision loss rate of BSM due to sliding-rolling mixed motion behavior

 (65)

In Figure 13, both the axial loading and rotational speed have different degrees of impact on the BSM precision loss rate. Therefore, a quantitative analysis of the influence degree of axial load and rotational speed on the BSM precision loss rate is required. At a given screw speed, this study uses the variation of the precision loss rate and the variation rate of the precision loss rate to analyze the influence degree of axial load conditions on the precision loss rate. Table 2 summarizes the calculation results. In order to facilitate the quantitative analysis, the accuracy loss rate under an axial load of 500 N was used as a reference.

When analyzing the influence degree of the rotation speed on the accuracy loss rate, it was more reasonable to use the variation rate of the precision loss rate as the quantitative analysis index since the increase rate of the screw rotation speed was different. The variation of the precision loss rate was an intermediate value for calculating the variation rate of the precision loss rate, and it was also essential. In order to facilitate the quantitative analysis, the precision loss rate at a rotational speed of 20 r/min was used as a reference point. Note that due to the different increase in the screw rotation speed, that is, when the increment of the rotation speed was 40 or 60 r/min, it was necessary to divide by the multiple (2 or 3) of the rotation speed increment when calculating the variation rate of the precision loss rate. After that, the criteria for the unified quantitative analysis can be obtained (Table 3).

From Figure 13 and Table 2, at a given rotational speed, the precision loss rate increases with the increase of axial load. However, when the axial load increases by the same amplitude, the variation (increased amount) of the precision loss rate gradually decreases, and the variation rate of precision loss rate gradually decreases. The reason is that as the axial load increases, the contact angle between the ball and the screw decreases. As a result, the contact load between the ball and screw enlarges, and so does the precision loss rate between the ball and screw. However, the contact angle between the ball and the nut increases so that the contact load between the ball and the nut becomes smaller, resulting in the precision loss rate between the ball and nut also becoming smaller. Therefore, when the axial load increases, although the precision loss rate enlarges, the change value (increased amount) of the total precision loss rate of the ball screw decreases, and the variation rate of the precision loss rate gradually decreases.

At a given screw rotation speed, the increase in the precision loss rate is the maximum when the axial load changes from 500 to 1000 N. Moreover, the variation rate (increase rate) of the accuracy loss rate also reaches the maximum at 1000 N axial load. As the axial load increases, the amplitudes of attenuation of the variation value and the variation rate of the BSM precision loss are different. In particular, through the variation rate of precision loss rate, it is more convenient to analyze the change law of precision loss rate quantitatively.

According to the analysis of Figure 13 and Table 3, at a given axial load, the precision loss rate enlarges as the screw rotation speed increases. However, as the rotation speed gradually increases, the increase rate of the precision loss rate gradually decreases. On the one hand, due to the increase in rotation speed, the sliding motion velocity becomes more significant and so does the sliding motion distance (Eq. (39) and Figure 9), resulting in the accelerated sliding motion wear, and the increased precision loss and precision loss rate. On the other hand, as the rotational speed of the screw increases, the contact angle between the ball and the nut decreases. The contact load between the ball and the nut increases, increasing the precision loss rate between the ball and nut. However, the contact angle between ball and screw increases, so that the contact load between ball and screw reduces. Also, the precision loss rate between the ball and screw reduces. Therefore, when the rotation speed increases, although the precision loss rate becomes larger, the variation rate (increase rate) of the total precision loss rate of the BSM decreases.

Table 2 Influence degree of axial load conditions on BSM precision loss rate

Table 3 Influence degree of rotational speed conditions on BSM precision loss rate

At a given axial load, the variation rate of the precision loss rate reaches its maximum when the screw rotation speed changes from 20 to 40 r/min. When the speed is 40 r/min, as the axial load increases, the increase rate in the precision loss rate gradually becomes larger. Furthermore, at the 4000 N axial load, the increase rate of the precision loss rate is the maximum. As the screw rotation speed increases, the variation value and variation rate of the BSM precision loss rate have different amplitudes of attenuation. Especially through the variation rate of the precision loss rate, it was more convenient to quantitatively analyze the influence degree of the rotation speed on the BSM precision loss rate.

6 Conclusions

1) Under the action of sliding motion, the precision loss value of the screw is greater than that of nut, because the sliding distance between the ball and the screw is greater than that between the ball and the nut. The sliding motion wear depth between ball and screw contributes a large proportion to the BSM precision loss.

2) The precision loss value under rolling motion occupies a certain proportion of the precision loss value under sliding motion, and the precision loss value under rolling motion is a part of the total precision loss of the ball screw. The precision loss value due to rolling motion has a certain level of influence on the BSM total precision loss.

3) As the rotation speed gradually increases, the increase rate of the precision loss rate gradually decreases. The rotation speed had a certain effect on the contact angle and could increase the sliding velocity, resulting in the increase of precision loss rate.

Contributors

QI Bao-bao provided the data curation, methodology and writing-original draft. CHENG Qiang provided the funding acquisition and methodology. LI Shun-lei replied to reviewers’ comments and revised the final version. LIU Zhi-feng conducted the formal analysis and funding acquisition. YANG Cong-bin provided the software.

Conflict of interest

QI Bao-bao, CHENG Qiang, LI Shun-lei, LIU Zhi-feng and YANG Cong-bin declare that they have no conflict of interest.

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(Edited by FANG Jing-hua)

中文导读

滑-滚混合运动行为作用下的滚珠丝杠副精度损失

摘要：滑-滚混合运动行为在不同程度上会退化滚珠丝杠副的精度。基于滚珠与丝杠/螺母滚道之间的滑-滚混合运动分析，研究了滚珠丝杠副在不同轴向负载及旋转速度工况下的精度损失。由于滑动、滚动磨损与蠕滑、润滑紧密相关，研究了不同运行工况下蠕滑及润滑特性。此外，当探究滚珠与滚道之间的滑动运动行为对精度损失的影响时，计算了滚珠丝杠副的精度损失并对比了其他模型。最后，在给定运行工况下，研究了滑-滚混合运动下的精度损失，设计相关实验，对精度衰退模型进行验证。精度损失的分析结果与试验测试具有很好的吻合性，表明该模型能够很好地预测处于稳定磨损时期内的精度损失。

关键词：滚珠丝杠副；精度损失；滑-滚混合运动；蠕滑；润滑

Foundation item: Project(51975012) supported by the National Natural Science Foundation of China; Project(Z1511000003150138) supported by the Beijing Nova Program, China; Project(Z191100001119010) supported by the Shanghai Sailing Program, China; Project(2018ZX04033001-003) supported by the National Science and Technology Major Project, China

Received date: 2020-04-15; Accepted date: 2020-09-07

Corresponding author: CHENG Qiang, PhD, Professor; Tel: +86-15901009350; E-mail: chengqiang@bjut.edu.cn; ORCID: https://orcid. org/0000-0001-5446-3658