Positioning accuracy degradation and lifetime prediction of the ball screw considering time-varying working conditions and feed modes

Abstract

Ball screw mechanisms (BSMs) are used as accuracy transmission components in a wide range of industries and are characterized by their high accuracy. More specifically, the positioning accuracy of BSM has a significant effect on the accuracy of machine tool. Based on the macro-micro multiscale method, an exponential prediction model for the BSM positioning accuracy was developed considering time-varying working conditions (load and rotational speed) and feed modes. Since the accuracy degradation is mainly caused by wear, a microscopic approach was proposed to describe the positioning accuracy retention and the microscopic wear process was investigated. The sliding contact of the asperities between the ball and raceway was analyzed, and the microscopic wear behavior of the asperities was determined. Considering the time-varying working conditions, the BSM positioning accuracy characteristics were obtained under the normal feed mode by conducting suitable tests. The exponential wear model used the wear index to describe the wear status based on the positioning accuracy measurement. The accuracy loss value and the prediction index of positioning accuracy were determined based on an exponential model, and the effective lifetime of the BSM was predicted. Finally, the exponential prediction model was used in negative/positive skew feed distribution, and the effective lifetime determined.

Keywords

Ball screw mechanism time-varying working conditions feed mode positioning accuracy retention lifetime prediction

Introduction

The ball screw mechanism (BSM) is an important functional transmission component commonly widely used in various fields, such as transportation, CNC machine tools, and other industries precision equipment. The accuracy degradation of the BSM significantly affects the transmission accuracy of the precision device. The accuracy retention indicates the ability to maintain the accuracy at a certain level under certain conditions and times. Accuracy is mainly deteriorated by wear. The studies on BSM accuracy retention mainly include accuracy degradation basic, analysis of accuracy degradation, and revealing the law of accuracy degradation.

The basics of the BSM accuracy degradation include kinematics^1–5 and dynamics,^6–13 which have been independently investigated. First, Lin et al.¹ conducted a basic study on BSM kinematics. Based on the work of Lin et al.,¹ Wei and Lin² improved the kinematics considering the effect of structural parameters such as contact angle. Based on the kinematics analysis, Lin et al.³ initially established a design method for optimizing the transmission efficiency. The contact load distribution was investigated by Lin et al.⁴ and Zhao et al.,⁵ considering the motion torque and geometric error, respectively. The research on the BSM dynamics has been mainly focused on the high-speed,⁶ high acceleration,⁷ and multi-domain integrated modeling.⁸ The working conditions that affect the BSM dynamics include the worktable position⁹ and the time-varying and piecewise-nonlinear stiffness.¹⁰ Moreover, the effect of operating temperature on the BSM dynamics was also studied.^11–13

With regard to analysis of accuracy degradation and the law of accuracy degradation, based on the basic research on friction and wear and the creep theory,^14,15 BSM wear models and wear characteristics have been established, and related experiments have been designed and conducted.^15–19 In addition, the positioning accuracy prediction²⁰ and accuracy compensation of the ball screw²¹ have also been studied. Based on the creep theory, Xu et al.¹⁵ established a BSM friction force calculation model, while Oh et al.¹⁶ improved the friction torque model of a double-nut BSM. The BSM wear characteristics have been extensively investigated by Wei et al.,¹⁷ Zhou et al.,¹⁸ and Cheng et al.¹⁹ based on the Archard theory.²² In particular, Zhou et al.¹⁸ studied the precision loss rate of screw raceway based on the wear analysis. Cheng et al.¹⁹ investigated the BSM accuracy degradation due to wear under time-varying conditions. The positioning error of the ball screw feed drive system was predicted considering different mounting conditions,²⁰ and an adaptive on-line analytical compensation model of the BSM positioning error was established.²¹

In summary, the kinematics, dynamics, and wear modeling of BSMs have been investigated in detail. However, little study revealed the macroscopic BSM wear characteristics from the microscopic wear perspective. In addition, the accuracy degradation under the action of micro-sliding wear and the lifetime prediction under specific operating conditions also need to be investigated. These problems were solved by the macro-micro multiscale method to reveal the accuracy degradation law from the microscopic wear perspective. According to the accuracy degradation characteristics, an exponential prediction model for the positioning accuracy of BSM was developed considering time-varying working conditions (load/rotational speed) and feed modes. Finally, the positioning accuracy retention was investigated under a normal feed mode, and the effective BSM lifetime was predicted.

Microscopic wear of contact asperity

Sliding contact of asperity

As shown in Figure 1, the contact surface between ball and raceway comprises many asperities. According to the GW contact principle,²³ the number of asperities $q_{bs}$ and $q_{bn}$ contacted between the ball and screw/nut raceway are represented by equations (1a) and (1b) respectively.

q_{bs} = n_{bs} \int_{h_{bs}}^{\infty} J_{bs} (z) dz

(1a)

q_{bn} = n_{bn} \int_{h_{bn}}^{\infty} J_{bn} (z) dz

(1b)

where $J_{bs} (z)$ and $J_{bn} (z)$ are the height distribution functions of asperities between the ball and screw/nut raceway, respectively, and $n_{bs}$ and $n_{bn}$ are the number of asperities between the ball and screw/nut raceway. According to Figure 1, the contact surface between the ball and raceway is equivalent to that between a rigid smooth surface and an elastic rough surface:

n_{bs} = n_{bn} = n_{a}

(2a)

h_{bs} = h_{bn} = h

(2b)

J_{bs} (z) = J_{bn} (z) = J (z)

(2c)

Subsequently, the number of asperities q between the ball and raceway were obtained as follows:

q = q_{bs} = q_{bn} = n_{a} \int_{h}^{\infty} J (z) dz

(3)

where $n_{a}$ is the number of asperities between the ball and raceway:

n_{a} = S_{c} ρ

(4)

where $ρ$ denotes the density distribution of asperities, and $S_{c}$ denotes the contact area between the ball and raceway and was determined by equation (5)²⁴:

S_{c} = \frac{1}{\sqrt{π}} {(\frac{89}{25})}^{\frac{3}{4}} {(\frac{\int_{0}^{\frac{π}{2}} (1 - ι^{2} \sin^{2} α) d α}{{(1 - ι^{2} \sin^{2} α)}^{\frac{1}{6}}})}^{\frac{3}{2}}

(5)

Then, $J (z)$ can be written as:

J (z) = \frac{1}{δ \sqrt{2 π}} e^{\frac{- (z - μ)}{2 δ^{2}}}

(6)

where $μ$ denotes the average height of contact asperities, and $δ$ denotes the standard deviation of contact asperity height. The values of $μ$ and $δ$ are listed in Table 1.

Figure 1.

Microscopic contact diagram of ball and raceway.

Table 1.

Structural parameters and surface information parameters of the BSM

Parameters	Values
Effective feed distance	1700 mm
Nominal diameter of BSM	50 mm
Radius of the ball	3.375 mm
Radius of the raceway	3.673 mm
Contact angle	45°
Helix angle	4.35°
Helical pitch	12 mm
Number of balls	136
Hardness of the screw	62 HRC
Young modulus	205 GPa
Poisson ratio	0.3
Lubrication mode	Grease
Friction coefficient	0.004
Average height of asperities	0.6332 $μ m$
Standard deviation of heightof asperities	0.7925 $μ m$
Distribution density of asperities	121.7091 ( $1 / m m^{2}$ )

Asperity wear

According to Figure 2, the number of sliding contacts is $N_{a}$ and the wear volume of a single asperity is $Δ v_{a}$ .

Figure 2.

Wear volume of asperity.

Based on Figure 2, $Δ v_{a}$ is known and can be determined by:

Δ v_{a} = \frac{π σ_{e} (3 {a_{0}}^{2} + {σ_{e}}^{2})}{6}

(7)

where $σ_{e}$ represents the normal elastic deformation of a single asperity and can be written as following²⁵:

σ_{e} = {(\frac{π cH}{2 E^{*}})}^{2} r_{a}

(8)

where c is a constant determined based on the Poisson ratio v of the material by $c = 0.454 + 0.41 v$ , H denotes the hardness of the softer materials, and $E^{*}$ denotes the equivalent modulus of elasticity and can be calculated by $1 / E^{*} = (1 - v_{1}^{2}) / E_{1} + (1 - v_{2}^{2}) / E_{2}$ .

Based on Figure 2 and equation (8), $a_{0}$ was obtained:

{a_{0}}^{2} = {r_{a}}^{2} - {(r_{a} - σ_{e})}^{2} \approx 2 r_{a} σ_{e}

(9)

According to the experimental results of Kragelsky et al.,²⁶ when the wear volume of a single asperity is $Δ v_{a}$ , the sliding contact number $N_{a}$ of the asperity is:

N_{a} = {(\frac{γ_{e}}{g τ_{e}})}^{χ}

(10)

where $γ_{e}$ is the equivalent stress value, g is a constant which, according to the strength theory, equals 3, and $χ$ represents the frictional index. The shear stress $τ_{e}$ on the contact surface was obtained by:

τ_{e} = \frac{k_{f} p_{i}}{A_{c}}

(11)

where $k_{f}$ is the friction coefficient listed in Table 1. If time-varying loads are considered,¹⁹ the contact load between the ball and raceway can be obtained by:

\begin{matrix} {p_{i}}^{2 / 3} \sin θ_{si} = p_{si - 1}^{2 / 3} \sin θ_{si - 1} \\ - \frac{1}{K_{si}} (σ_{si - 1} \sin θ_{si - 1} - σ_{si} \sin θ_{si}) \\ - \frac{1}{K_{si}} \sum_{j = 1}^{M} P_{sj} \sin θ_{sj} \cos α \end{matrix}

(12)

By combing equation (12) with equations (3), (4), and (8), the actual contact area $A_{c}$ between the ball and raceway can be determined:

A_{c} = s_{c} n_{a} \int_{h}^{\infty} J (z) dz

(13)

where $s_{c}$ is the actual contact area of a single asperity:

s_{c} = π {a_{c}}^{2} = π r_{a} σ_{e}

(14)

Macroscopic wear of BSM

Based on the microscopic wear analysis of a single contact asperity, the wear characteristics of the BSM macroscopic components were analyzed. Under time-varying rotational speed conditions, the sliding distances between the ball and raceway can be obtained¹⁹:

\begin{matrix} l_{ss - bsi} = \frac{R_{bs} ω_{s - b} r \cos α}{π (1 + R_{bs}) ν_{bs - ss} R} \\ \sqrt{R^{2} + {(\frac{L}{2 π})}^{2}} \sqrt{{(R - r \cos θ_{s})}^{2} + {(\frac{L}{2 π})}^{2}} \end{matrix}

(15a)

\begin{matrix} l_{nn - bni} = \frac{R_{bn} ω_{n - b} r \cos α}{π (1 + R_{bn}) ν_{bn - nn} R} \sqrt{R^{2} + {(\frac{L}{2 π})}^{2}} \\ \sqrt{{(R + r \cos θ_{n})}^{2} + {(\frac{L}{2 π})}^{2}} \end{matrix}

(15b)

Equations (15a) and (15b) are the sliding distances between the ball and screw/nut raceway, respectively. L is the lead of the screw and can be obtained following literature method.¹⁹ $R_{bs}$ and $R_{bn}$ are the slip-roll ratios between the ball and screw/nut raceway, respectively, and were obtained following the literature methods^2,19:

R_{bs} = \frac{2 | ω_{n} r (\sin θ_{s} + \sin θ_{n}) - ω_{b} r (\cos θ_{n} + \cos θ_{s}) - ω (\frac{R}{\cos α} + r \cos α \cos θ_{n}) |}{| ω_{n} r (\sin θ_{s} + \sin θ_{n}) - ω_{b} r (\cos θ_{n} + \cos θ_{s}) + ω R (\cos α - \frac{1}{\cos α}) - ω r \cos α (\cos θ_{s} + \cos θ_{n}) | + | ω \cos α (R - r \cos θ_{s}) |}

(16a)

R_{bn} = \frac{2 | ω_{b} r (\cos θ_{s} + \cos θ_{n}) - ω_{n} r (\sin θ_{s} + \sin θ_{n}) + ω (\frac{R}{\cos α} + r \cos α \cos θ_{n}) |}{| ω_{b} r (\cos θ_{s} + \cos θ_{n}) - ω_{n} r (\sin θ_{s} + \sin θ_{n}) | + | - ω (\frac{R}{\cos α} + r \cos α \cos θ_{n}) |}

(16b)

By combining equations (3) and (15a)–(16b), the numbers of contact asperities between the ball and raceway during sliding are represented by equations (17a) and (17b), respectively:

n_{bsi} = \frac{l_{ss - bsi}}{2 a_{bs}} q_{bs}

(17a)

n_{bni} = \frac{l_{nn - bni}}{2 a_{bn}} q_{bn}

(17b)

where $a_{bs}$ and $a_{bn}$ are the elliptical long half axes of the ball contact with screw/nut raceway,²⁷ respectively.

When the BSM is in operation the sliding contact numbers of asperities between the ball and screw/nut raceway are $N_{bs} = k_{bs} N_{a}$ and $N_{bn} = k_{bn} N_{a}$ respectively. $N_{bs} = N_{bn} \cdot l_{n} / l_{n} l_{s} l_{s}$ , $l_{n}$ is the axial length value of the nut, and $l_{s}$ is the feed value of the screw. By combining equations (3), (7), (17a), and (17b), the wear volume of single ball and screw/nut raceway were obtained:

\begin{matrix} Δ v_{bs} = k_{bs} q_{bs} Δ v_{a} n_{bsi} \\ = \frac{k_{bs} n_{a} π σ_{e} (3 {a_{0}}^{2} + {σ_{e}}^{2}) l_{ss - bsi}}{12 a_{bs}} \int_{h}^{\infty} J (z) dz \end{matrix}

(18a)

\begin{matrix} Δ v_{bn} = k_{bn} q_{bn} Δ v_{a} n_{bni} \\ = \frac{k_{bn} n_{a} π σ_{e} (3 {a_{0}}^{2} + {σ_{e}}^{2}) l_{nn - bni}}{12 a_{bn}} \int_{h}^{\infty} J (z) dz \end{matrix}

(18b)

Combining equations (18a) and (18b), the wear volume between all the balls and raceways were obtained as:

\begin{matrix} Δ v = \frac{m n_{a} π σ_{e} (3 {a_{0}}^{2} + {σ_{e}}^{2})}{12} (\frac{k_{bs} l_{ss - bsi}}{a_{bs}} + \frac{k_{bn} l_{nn - bni}}{a_{bn}}) \\ \int_{h}^{\infty} J (z) dz \end{matrix}

(19)

where m is the total number of balls.

Exponential prediction model for positioning accuracy

Figure 3 illustrates the condition of a BSM due to wear.

Figure 3.

Schematic illustration of wear in a BSM.

Figure 3 shows the $O - XYZ$ coordinate system, where the X-axis is aligned with the axis of the screw and the Y- and Z-axes are located at the left end of the screw. $r_{i}$ is the original radius of the i^th ball, $r_{i'}$ is the radius of the i^th ball after wear, $O_{i}$ is the original center of the i^th ball, $O_{i'}$ is center of the i^th ball after wear, $h_{si}$ is the contact clearance of the i^th ball and screw, and $h_{ni}$ is the contact clearances of the i^th ball and nut.

The accuracy degradation in the BSM with respect to the X, Y, and Z directions is directly caused by wear. Therefore, the arc length of each ball in the screw/nut raceway can be respectively described as:

{\begin{matrix} l_{s 1} = \int_{0}^{β s 1} \sqrt{{[X' (β)]}^{2} + {[Y' (β)]}^{2} + {[Z' (β)]}^{2}} d β \\ ⋮ \\ l_{sm} = \int_{β sm - 1}^{β sm} \sqrt{{[X' (β)]}^{2} + {[Y' (β)]}^{2} + {[Z' (β)]}^{2}} d β \end{matrix}

(20)

{\begin{matrix} l_{n 1} = \int_{0}^{β n 1} \sqrt{{[X' (β)]}^{2} + {[Y' (β)]}^{2} + {[Z' (β)]}^{2}} d β \\ ⋮ \\ l_{nm} = \int_{β nm - 1}^{β nm} \sqrt{{[X' (β)]}^{2} + {[Y' (β)]}^{2} + {[Z' (β)]}^{2}} d β \end{matrix}

(21)

where $β_{s δ}$ ( $δ = 1, 2, \dots m$ ) is the central angle of each ball in the screw raceway, $β_{n δ}$ ( $δ = 1, 2, \dots m$ ) is the central angle of each ball in the nut raceway. Furthermore, $X' (β)$ , $Y' (β)$ , and $Z' (β)$ were obtained from the first derivative of the parameter equations for the ball center, represented as $X (β)$ , $Y (β)$ , and $Z (β)$ , respectively. The parametric equation for the ball center were formulated as:

{\begin{matrix} X = \frac{L}{2 π} β \\ Y = R \cos β \\ Z = R \sin β \end{matrix}

(22)

If the average deviation of all the contact asperities of the i^th ball and screw raceway is $e_{bsi} (x_{si})$ , the average deviation of all the contact asperities of the m^th ball and screw raceway is assumed as $e_{bsm} (x_{si})$ . Both are defined relative to regression line in the spiral direction of the screw. The initial position of the ball in the nut raceway is $x_{ni}$ . The average deviation of all the contact asperities of the i^th ball and nut raceway is assumed as $e_{bni} (x_{ni})$ , while the average deviation of all the contact asperities of the m^th ball and nut raceway is $e_{bnm} (x_{ni})$ .

Equations (20) and (21) were the arc length of each ball in the screw/nut raceway, respectively. And equation (22) was the parametric equation for the ball center. Combining equations (20)–(22), the measured value of the BSM positioning accuracy $g_{fd} (x_{i})$ in the feed direction can be obtained. Subsequently, because of the accumulated wear, the measured value of the BSM positioning accuracy $g_{fd} (x_{i})$ in the feed direction can be expressed as follows:

\begin{matrix} g_{fd} (x_{i}) = \frac{(\sum_{x_{si}}^{l_{s 1} + x_{si}} e_{bs 1} (x_{si}) \sin θ_{s 1} + \sum_{l_{s 1} + x_{si}}^{l_{s 1} + l_{s 2} + x_{si}} e_{bs 2} (x_{si}) \sin θ_{s 2} + \dots + \sum_{l_{s 1} + l_{s 2} + \dots + l_{sm - 1} + x_{si}}^{l_{s 1} + l_{s 2} + \dots + l_{sm - 1} + l_{sm} + x_{si}} e_{bsm} (x_{si}) \sin θ_{sm})}{l_{s 1} + l_{s 2} + \dots + l_{sm}} \\ + \frac{(\sum_{x_{ni}}^{l_{n 1} + x_{ni}} e_{bn 1} (x_{ni}) \sin θ_{n 1} + \sum_{l_{n 1} + x_{ni}}^{l_{n 1} + l_{n 2} + x_{ni}} e_{bn 2} (x_{ni}) \sin θ_{n 2} + \dots + \sum_{l_{n 1} + l_{n 2} + \dots + l_{nm - 1} + x_{ni}}^{l_{n 1} + l_{n 2} + \dots + l_{nm - 1} + l_{nm} + x_{ni}} e_{bnm} (x_{ni}) \sin θ_{nm})}{l_{n 1} + l_{n 2} + \dots + l_{nm}} \end{matrix}

(23)

where $\sum e_{bs 1} (x_{si}) \sin θ_{s 1}$ / $\sum e_{bn 1} (x_{ni}) \sin θ_{n 1}$ is the cumulative deviation of all the contact asperities of the first ball and screw/nut raceway.

By combining equations (3), (7), (10), and (17a)–(19), the relationship between the wear volume $Δ v$ and number of sliding contacts $N_{s}$ in the BSM was derived as follows:

{\begin{matrix} Δ v_{a} ~ N_{a} = {(\frac{γ_{e}}{g τ_{e}})}^{χ} \\ Δ v = qm (k_{bs} n_{bsi} + k_{bn} n_{bni}) Δ v_{a} = C {N_{s}}^{α} \end{matrix}

(24)

where $C$ is a constant to be determined, and $α$ is the prediction index.

Based on the degradation analysis of the BSM positioning accuracy, during the stable operation period, the degradation of positioning accuracy is related to the wear as follows:

Δ g_{fd} (x_{i}) = k Δ v = kC {N_{s}}^{α}

(25)

By combining equations (23) and (25), an equation for a simplified positioning accuracy measurement point was formulated as:

s_{fd} (x_{i}) = Λ {x_{i}}^{w}

(26)

where $Λ$ represents the wear coefficient, and $w$ represents the wear ratio index.

The total number of contact asperities at different positions varies with different operation conditions and feed modes. When the total number of contacts between asperities increases continuously, the asperities at different positions exhibit different wear characteristics. According to Figure 3 and equation (26), the equation of positioning accuracy retention can be written as:

S_{fd} = S_{fd 0} (x_{i}) t^{w}

(27)

where $S_{fd 0}$ represents the initial positioning accuracy in the feed direction, $S_{fd}$ represents the real time positioning accuracy after a period of operation, t represents the operation time, and $w$ represents the real time prediction index.

BSMs play an important role in controlling the CNC machine tools (M800 made in China). Due to varying size of workpieces, the BSM requires different feed distances. The total number of feed cycles during operation is N, the minimum feed distance of the nut relative to the screw is $f_{\min}$ , the maximum feed distance is $f_{\max}$ , and the feed distance $f$ follows the normal distribution. The coefficient $ξ$ represents the feed distance, $0 \leq ξ \leq 1$ is a random variable used to measure different proportions of feed distances. According to actual operating conditions, the feed distance coefficient can follow the normal distribution (Figure 4(a)), negative skew distribution (Figure 4(b)), and positive skew distribution (Figure 4(c)).

Figure 4.

Probability density distribution function of $ξ$ : (a) the feed distance coefficient under the normal distribution, (b) the feed distance coefficient under the negative skew distribution, and (c) the feed distance coefficient under positive skew distribution.

The normal distribution shown in Figure 4(a) indicates that the proportion of smaller and larger feed distances is smaller under actual operating conditions. In Figure 4(b), the negative skew distribution indicates that the feed distance under actual operating conditions is biased toward larger values. The positive skew distribution shown in Figure 4(c) indicates that the feed distance under actual operating condition is biased toward smaller values.

Positioning accuracy retention prediction

Based on multi-functional test platform for BSM (Type: W5017-C2Z12), a suitable positioning accuracy retention test was designed and tested. On the test platform, the feed distance, rotational speed, and load can be adjusted. According to the requirements of the experimental design, the rotational speed and load conditions can be controlled through programming. The tests were designed to determine and predict positioning accuracy retention characteristics under different feed modes.

The initial parameters are listed in Table 1.

The aim of the experiment was to investigate the effect of feed mode (normal distribution) and time-varying working conditions on the positioning accuracy retention of BSM. The feed distance of the BSM obeyed the normal distribution. The time-varying load was set to $[0.5 \times \sin (π t / 180) + 1] KN$ and the rotational speed was set to $[1.5 \times \sin (π t / 180) + 2] π rad / s$ . The time interval of the dynamic load and rotational speed was set to 3.5 s. KOC301 was used as the grease-type lubrication lubricant at an ambient temperature of 23°C.

Based on Figure 4(a), the positioning accuracy retention test time was 800 h. After testing for 200 h, the positioning accuracy of the BSM was measured using an API laser interferometer at room temperature. Figure 5 shows the devices that control the required rotational speed, load, and feed modes.

Figure 5.

Devices that produce the required rotational speed, preload, and feed modes.

Figure 6 shows the settings used to test and measure the cumulative positioning accuracy. Notably, the obtained positioning accuracy results included wear-induced errors, manufacturing errors, and assembly errors.

Figure 6.

Setup to test positioning accuracy.

The results presented in Table 2 were obtained from the setup shown in Figure 6.

Table 2.

Measurement results.

Node range $x_{i}$	Positioning accuracy/ $μ m$ (200 h)	Positioning accuracy/ $μ m$ (400 h)	Positioning accuracy/ $μ m$ (600 h)	Positioning accuracy/ $μ m$ (800 h)
0–150	0.000	0.002	0.004	0.014
31–181	0.863	1.957	3.181	4.564
62–212	1.590	3.529	5.934	8.760
93–243	2.347	4.973	8.422	12.806
124–274	3.028	6.607	10.947	15.879
155–305	3.743	8.342	13.426	20.705
186–336	4.366	9.084	15.001	21.834
217–367	4.670	10.628	16.453	25.114
248–398	5.168	11.056	18.084	27.025
279–429	5.614	12.495	19.711	29.368
310–460	6.049	13.113	21.179	31.849
341–491	6.531	14.065	22.716	34.161
372–522	6.880	15.023	23.878	36.251
403–553	6.999	15.586	25.345	38.473
434–584	7.306	16.379	26.381	40.299
465–615	7.749	16.579	27.755	41.827
496–646	8.073	17.617	28.790	43.532
527–677	8.437	18.440	29.782	45.755
558–708	8.726	19.350	31.671	47.777
589–739	9.085	20.120	32.750	49.840
620–770	9.424	20.962	34.565	51.288
651–801	9.737	21.591	35.748	53.658
682–832	10.195	22.644	37.328	55.378
713–863	10.394	23.245	38.081	56.848
744–894	10.759	23.927	39.326	58.883
775–925	10.852	24.296	40.506	59.419
806–956	11.296	25.121	41.032	61.637
837–987	11.541	25.529	41.702	62.841
868–1018	11.709	26.352	42.445	64.030
899–1049	11.891	26.911	43.343	65.376
930–1080	12.095	27.374	43.950	66.300
961–1111	12.367	27.564	44.677	66.786
992–1142	12.607	27.883	45.484	67.986
1023–1173	12.785	28.185	46.127	68.947
1054–1204	12.909	28.534	46.594	69.864
1085–1235	13.025	29.105	47.471	70.949
1116–1266	13.179	29.450	48.184	71.790
1147–1297	13.331	29.789	48.738	72.537
1178–1328	13.479	30.316	48.975	73.124
1209–1359	13.618	30.330	49.475	74.096
1240–1390	13.742	31.073	49.982	75.001
1271–1421	13.862	30.976	50.420	75.658
1302–1452	13.977	31.535	50.831	75.814
1333–1483	14.067	31.461	51.208	76.924
1364–1514	14.160	31.870	51.387	77.201
1395–1545	14.245	31.852	51.857	77.898
1426–1576	14.320	32.168	52.131	78.074
1457–1607	14.385	32.278	52.366	78.425
1488–1638	14.437	32.282	52.556	78.794
1519–1669	14.471	32.675	52.682	78.984
1550–1700	14.535	32.500	52.740	79.071

The measurement results of the positioning accuracy along the feed direction are shown in Figure 7.

Figure 7.

Measured values.

Figure 7 shows that for the normal distribution of probability density, the cumulative times of contact and wear decrease gradually with increasing feed distance. According to the four measurements taken every 200 h, the growth rate of BSM positioning accuracy slowed down gradually with increasing feeding distance. The maximum positioning accuracy of BSM increases in a multiplier manner.

The research object BSM adopted in this research complies with the national standards. Thus, it can be considered that the manufacturing and assembly errors are equal at any position. In order to extract the accuracy degradation within the same feed distance, the data in Table 2 were subtracted alternately. The positioning accuracy variation was determined by calculating the difference between two rows. The midpoint of the measured asperities was selected as the coordinate center of the representative node. The positioning accuracy variation is lsited in Table 3.

Table 3.

Variation of positioning accuracy.

Representative nodecoordinate $r_{i}$	Variation of positioningaccuracy/ $μ m$ (200 h)	Variation of positioningaccuracy/ $μ m$ (400 h)	Variation of positioningaccuracy/ $μ m$ (600 h)	Variation of positioning accuracy/ $μ m$ (800 h)
31.000	1.590	3.527	5.930	8.746
62.000	1.484	3.017	5.241	8.243
93.000	1.438	3.077	5.013	7.119
124.000	1.396	3.369	5.004	7.899
155.000	1.337	2.477	4.054	5.955
186.000	0.926	2.286	3.027	4.409
217.000	0.803	1.971	3.082	5.192
248.000	0.945	1.866	3.258	4.254
279.000	0.881	2.057	3.095	4.823
310.000	0.917	1.570	3.006	4.792
341.000	0.831	1.910	2.699	4.403
372.000	0.469	1.522	2.629	4.313
403.000	0.426	1.356	2.503	4.048
434.000	0.750	0.992	2.409	3.353
465.000	0.767	1.238	2.409	3.233
496.000	0.688	1.862	2.027	3.928
527.000	0.653	1.733	2.882	4.245
558.000	0.648	1.680	2.968	4.085
589.000	0.698	1.612	2.894	3.511
620.000	0.652	1.470	2.998	3.818
651.000	0.771	1.682	2.763	4.091
682.000	0.657	1.654	2.333	3.190
713.000	0.564	1.283	1.998	3.505
744.000	0.458	1.051	2.425	2.571
775.000	0.537	1.195	1.706	2.754
806.000	0.689	1.233	1.197	3.423
837.000	0.413	1.230	1.413	2.393
868.000	0.350	1.382	1.640	2.534
899.000	0.387	1.022	1.505	2.270
930.000	0.476	0.653	1.335	1.411
961.000	0.512	0.510	1.534	1.686
992.000	0.419	0.621	1.450	2.160
1023.000	0.301	0.650	1.110	1.878
1054.000	0.239	0.920	1.344	2.003
1085.000	0.271	0.916	1.590	1.926
1116.000	0.306	0.684	1.267	1.588
1147.000	0.300	0.866	0.790	1.334
1178.000	0.288	0.541	0.737	1.558
1209.000	0.262	0.757	1.007	1.877
1240.000	0.244	0.646	0.945	1.562
1271.000	0.235	0.461	0.849	0.813
1302.000	0.205	0.484	0.788	1.266
1333.000	0.183	0.335	0.556	1.387
1364.000	0.178	0.392	0.649	0.974
1395.000	0.160	0.299	0.744	0.873
1426.000	0.140	0.426	0.509	0.527
1457.000	0.117	0.113	0.425	0.721
1488.000	0.087	0.396	0.316	0.559
1519.000	0.098	0.219	0.185	0.277

These variations occurred at different time periods and are shown in Figure 8.

Figure 8.

Positioning accuracy variation.

Figure 8 shows that for normal distribution of probability density, the variation in positioning accuracy decreases with increasing feed distance, attributing to the highest cumulative number of contacts at the starting position.

According to the Rodriguez rotation theory, the values in Table 3 can be corrected. Again, the midpoint of the measured asperities was selected as the coordinate center of the representative node and the positioning accuracy variation was corrected, and the corrected values are listed in Table 4.

Table 4.

Corrected values of the positioning accuracy.

Representative nodecoordinate $r_{i}$	Correction of positioning accuracy/ $μ m$ (200 h)	Correction of positioning accuracy/ $μ m$ (400 h)	Correction ofpositioning accuracy/ $μ m$ (600 h)	Correction ofpositioning accuracy/ $μ m$ (800 h)
31.000	1.914	3.580	6.018	8.878
62.000	1.733	3.123	5.416	8.507
93.000	1.512	3.236	5.276	7.515
124.000	1.494	3.181	5.354	7.427
155.000	1.460	2.742	4.493	6.615
186.000	1.373	2.603	3.553	6.031
217.000	0.974	2.342	3.696	6.116
248.000	1.140	2.290	3.959	5.310
279.000	1.101	2.534	3.884	6.011
310.000	1.161	2.799	3.882	6.113
341.000	1.100	2.493	3.664	5.855
372.000	0.762	2.157	3.681	5.897
403.000	0.944	3.044	3.642	5.764
434.000	1.092	2.833	3.637	5.202
465.000	1.134	2.032	3.724	5.214
496.000	1.080	2.609	3.630	6.040
527.000	1.069	2.633	3.372	6.489
558.000	1.089	2.633	3.346	6.462
589.000	1.163	2.618	3.559	6.019
620.000	1.141	2.529	3.751	6.459
651.000	1.285	2.794	3.604	6.863
682.000	1.196	2.819	4.262	6.094
713.000	1.127	2.500	4.014	6.542
744.000	1.046	2.322	4.529	5.740
775.000	1.149	2.518	3.898	6.055
806.000	1.325	2.609	3.476	6.855
837.000	1.074	2.660	3.780	5.957
868.000	1.036	2.864	4.095	6.231
899.000	1.097	2.557	4.047	6.099
930.000	1.210	2.242	3.965	5.372
961.000	1.271	2.151	4.252	5.779
992.000	1.202	2.315	4.255	6.386
1023.000	1.109	2.397	4.003	6.235
1054.000	1.072	2.720	4.325	6.492
1085.000	1.128	2.769	3.658	6.547
1116.000	1.187	2.589	4.423	6.342
1147.000	1.206	2.825	4.034	6.219
1178.000	1.218	2.553	4.068	6.576
1209.000	1.217	2.821	4.426	7.026
1240.000	1.223	2.764	4.451	6.843
1271.000	1.239	2.632	4.443	6.227
1302.000	1.233	2.708	4.470	6.811
1333.000	1.236	2.611	4.325	7.064
1364.000	1.256	2.721	4.506	6.784
1395.000	1.262	2.681	3.689	6.814
1426.000	1.266	2.861	4.541	6.600
1457.000	1.267	2.601	4.545	6.926
1488.000	1.262	2.937	4.524	6.896
1519.000	1.298	2.813	4.480	6.746

The positioning accuracy at different time periods of the wear test was corrected, and the corrected values along the feed direction are shown in Figure 9.

Figure 9.

Corrected values of positioning accuracy.

According to Figure 9, for the normal distribution of probability density, the positioning accuracy degradation is highest at the starting position.

Table 4 shows the results of linear regression processing for the corrected errors at different test periods. For the normal distribution of probability density, the prediction index for these time periods were obtained. At 200, 400, 600, and 800 h operating times, the prediction indexes were $n_{200}^{1} = 0.874023$ , $n_{400}^{1} = 0.879917$ , $n_{600}^{1} = 0.900945$ and $n_{800}^{1} = 0.896571$ , respectively. The variation between the maximum and minimum prediction index was approximately 3.08%. Since this change is small, the wear in these time periods can be considered as stable. Therefore, the average prediction indexes were assumed equal to the prediction index $n^{1} = 0.887864$ for different time periods.

At a feeding distance of 300 mm in the BSM, multiple values associated with large loss in positioning accuracy were selected, and the average value was defined as the loss of positioning accuracy. Based on equations (25)–(27) and assuming the operation time as $T_{1}$ , the positioning accuracy retention model for the normal distribution of probability density was obtained as follows:

P_{a 1} = 0.0084253 T_{1}^{0.887864}

(28)

When the loss in the positioning accuracy due to wear is equal to $25 μ m$ , the BSM is considered unsuitable to operate. According to equation (28), when the time-varying loads and rotational speed are determined according to the experimental scheme, the effective operating lifetime of the BSM was found as 8145 h.

Application of positioning accuracy prediction

According to the size of the processed specimens, the feed distance of the BSM demonstrates different distributions in the operation cycle. When the BSM feed mode presents a negative skew distribution, its actual operating condition for the feed distance is biased toward larger values, and when it presents positive skew distribution, its actual operating condition for the feed distance is biased toward smaller values. The BSM positioning accuracy characteristics were analyzed under negative/positive skew feed distribution, and the BSM lifetime was predicted.

Under the feed condition of negative/positive skew distribution, the time-varying load and rotational speed were also examined. The feed distance of the BSM and the percentage of feed distance are shown in Figure 4(b) and (c) under the negative/positive skew distribution, respectively. The experimental results shown in Figure 10(a) and (b) were recorded at different time periods under the feed condition of negative skew/positive distribution, respectively.

Figure 10.

Measured values: (a) positioning accuracy under the negative skew distribution and (b) positioning accuracy under the positive skew distribution.

Figures 7 and 10 indicate that the maximum positioning accuracy was very similar among the three tests. However, the precision loss characteristics under different feed modes were different, because of different cumulative contact times of the asperities at different positions. As shown in Figure 7, for the normal distribution of the feed mode, the cumulative times of the contact and wear decreased gradually with increasing feed distance. As shown in Figure 10(a), under the negative skew distribution of the feed mode, the BSM has a region where the asperities exhibited the same contact numbers. The positioning accuracy gradient follows essentially a linear growth trend. As shown in Figure 10(b), for the positive skew distribution of the feed mode, the trend of the positioning accuracy was similar to that shown in Figure 7. However, the variation gradient of the positioning accuracy was larger than that in Figure 7.

Again, the positioning accuracy variation was determined by calculating the difference between two acquisition points. The midpoint of the measured asperities was selected as the coordinate center of the representative node. The measured positioning accuracy variation under the feed condition of negative and positive skew distributions is shown in Figure 11(a) and (b), respectively.

Figure 11.

BSM positioning accuracy variation: (a) the measured positioning accuracy variation under the feed condition of negative skew distributions and (b) the measured positioning accuracy variation under the feed condition of positive skew distributions.

Figures 8 and 11 indicate that under the normal distribution of the feed mode, the positioning accuracy variation decreases with increasing feed distance, attributing to the highest cumulative number of contacts at the beginning of the feed. Figure 11(a) indicates that when the feed distance was less than 1050 mm, the positioning accuracy variation was almost constant. In contrast, when the feed distance exceeded 1050 mm, positioning accuracy variation decreased gradually with increasing feed distance. Figure 11(b) shows that under the positive skew distribution of the feed mode, the variation trend of the positioning accuracy was similar to that shown in Figure 8; however, the highest positioning accuracy variation was larger than that observed in Figure 8.

Similarly, the BSM positioning accuracy variation was corrected. The corrected values for the feed mode of negative and positive skew distributions are presented in Figure 12(a) and (b), respectively. According to Figure 12(a), for the negative skew distribution of probability density and feed mode, the positioning accuracy degradation was the largest at a feed distance of 1050 mm.

Figure 12.

Corrected BSM positioning accuracy: (a) the corrected values for the feed mode of negative skew distributions and (b) the corrected values for the feed mode of positive skew distributions.

For the negative skew distribution of the feed mode, the prediction index for different time periods was obtained. At 200, 400, 600, and 800 h operating times, the prediction indexes were $n_{200}^{2} = 0.825539$ , $n_{400}^{2} = 0.831331$ , $n_{600}^{2} = 0.853284$ , and $n_{800}^{2} = 0.870345$ , respectively. The relative change between the maximum and minimum prediction index was approximately 5.43%. Since this difference between wear indices was very little, the wear at the different time periods was considered as stable, and the average of the prediction indexes was assumed to be equal to the prediction index $n^{2} = 0.845124$ for different time periods.

Under the negative skew distribution, the operating time was 800 h. At a feeding distance of 300 mm, several values associated with a large loss in positioning accuracy were selected and the average value was defined as the loss of positioning accuracy. Herein, the value of $1.453 μ m$ represents the loss in BSM positioning accuracy. Based on equations (25)–(27) and assuming T₂ as the operating time, the positioning accuracy retention model for the negative skew distribution of the feed mode was obtained as following:

P_{a 2} = 0.010937 T_{2}^{0.845124}

(29)

When the loss in the positioning accuracy due to wear is $25 μ m$ , the BSM is considered unsuitable to operate. Based on equation (29), the effective lifetime of the BSM is approximately 9432 h.

According to Figure 12(b), for the negative skew distribution of the feed mode, the positioning accuracy loss was the largest at the beginning of the feed; however, the positioning accuracy loss was larger than that shown in Figure 9.

For the positive skew distribution of the feed mode, the prediction index for different time periods was obtained. At 200, 400, 600, and 800 h operating time, the prediction indexes were $n_{200}^{3} = 0.900119$ , $n_{400}^{3} = 0.909061$ , $n_{600}^{3} = 0.903647$ , and $n_{800}^{3} = 0.918011$ , respectively. The relative change between the maximum and minimum prediction index was approximately 1.99%. Consequently, the wear at the different time periods was considered as stable, and the average value of prediction indexes was assumed to be the prediction index $n^{3} = 0.907709$ for different time periods.

Again, based on equations (25)–(27) and assuming $T_{3}$ as the operating time of the BSM, the positioning accuracy retention model for the positive skew distribution of the feed mode was obtained:

P_{a 3} = 0.011025 T_{3}^{0.907709}

(30)

When the loss in positioning accuracy due to wear is $25 μ m$ , the BSM is considered to be non-functional. According to equation (30), the effective lifetime of the BSM is approximately 4974 h.

The BSM accuracy degradation values at 200, 400, 600, and 800 h were detected separately, and the wear index was obtained. Under the same accuracy level standard, the lifetime of the BSM was predicted, as listed in Table 5.

Table 5.

The service lifetime prediction.

Feed mode	The wear index					The effective lifetime/h
	Operation time/h				Average value
	200	400	600	800
The normal feed distribution	0.874023	0.879917	0.900945	0.896571	0.887864	8145
The negative skew feed distribution	0.825539	0.831331	0.853284	0.870345	0.845124	9432
The positive skew feed distribution	0.900119	0.909061	0.903647	0.918011	0.907709	4974

The comparison results in Table 5 demonstrate that different feeding methods lead to different expected lifetimes at the same accuracy level. The lifetime of the BSM under negative skew distribution condition is 9432 h, with the best accuracy retention ability. Under the positive skew distribution, the lifetime is 4974 h with the worst accuracy retention ability. Moreover, the lifetime under normally distributed feed conditions is 8145 h, and the accuracy retention ability is moderate. Under negative skew distribution, asperities have the least number of contacts. Under the normal feed conditions, the contact number of asperities increased. Under the positive skew feed, the contact number of asperities was the largest. Consequently, the wear of the BSM under the three feed modes in turn increases, and the accuracy retention ability declines. Therefore, the test results were consistent with the accuracy degradation model.

Conclusion

In conclusion, using the macro-micro multiscale method, an exponential prediction model for the positioning accuracy of the BSM was developed considering time-varying working conditions (load and rotational speed) and feed modes. Relevant positioning accuracy tests were successfully designed considering the time-varying load and rotational speed conditions. In addition, the exponential prediction model was applied to different feeding conditions, allowing correct prediction of lifetime. The main conclusions of this study can be summarized as follows:

Considering time-varying working conditions (load and rotational speed), the macro-micro multiscale method was convenient to determine the micro wear characteristics of the contact surface. The positioning accuracy retention of the BSM was investigated when the feed distance coefficient follows the normal distribution. At the operating times of 200, 400, 600, and 800 h, the prediction index were $n_{200}^{1} = 0.874023$ , $n_{400}^{1} = 0.879917$ , $n_{600}^{1} = 0.900945$ , and $n_{800}^{1} = 0.896571$ , respectively. The variation between the maximum and minimum prediction index was approximately 3.08%, and the average prediction indexes were assumed equal to the prediction index $n^{1} = 0.887864$ for the different time periods.

When the positioning accuracy prediction model was used in the negative/positive skew feed distribution, the prediction indexes are as follows: $n_{200}^{2} = 0.825539$ , $n_{400}^{2} = 0.831331$ , $n_{600}^{2} = 0.853284$ , and $n_{800}^{2} = 0.870345$ under the negative skew feed distribution. The relative change between the maximum and minimum prediction index was approximately 5.43%, and the average of the prediction indexes was $n^{2} = 0.845124$ . The prediction index was $n_{200}^{3} = 0.900119$ , $n_{400}^{3} = 0.909061$ , $n_{600}^{3} = 0.903647$ , and $n_{800}^{3} = 0.918011$ under the positive skew feed distribution. The relative change between the maximum and minimum prediction index was approximately 1.99%, and the average of the prediction index was $n^{3} = 0.907709$ . Therefore, when the ball screw mechanism was in different feed modes, the prediction index of positioning accuracy was different.

When the positioning accuracy loss value was 25 um, the lifetime of the BSM under negative skew distribution condition was 9432 h, with the best accuracy retention ability. Under the positive skew distribution, the lifetime was 4974 h with the least accuracy retention ability. Moreover, the lifetime under the normally distributed feed conditions was 8145 h, with moderate accuracy retention ability. Because the BSM was under the negative skew distribution, asperities have the least number of contacts. Under the normal feed conditions, the contact number of asperities increased. Under the positive skew feed, the contact number of asperities was the largest. The corresponding prediction indexes were $n^{2} < n^{1} < n^{3}$ , and the degradation rate of the BSM positioning accuracy under the three feed modes in turn increases, thus decreasing the accuracy retention ability. In terms of accuracy retention, the order of pros and cons under the three feed conditions was negative skew distribution, normal distribution, and positive skew distribution.

This study successfully used exponential prediction model for determining the positioning accuracy of the BSM under different feed modes, and the future study will be focused on positioning accuracy retention and life prediction of other types of ball screws. In addition, the ball screws will be tracked and detected on the machine tool to verify the accuracy degradation under time-varying working conditions and under different feed forms.

Footnotes

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: The authors sincerely acknowledge the financial supports from the National Natural Science Foundation of China (51975012), Beijing Nova Programmer Interdisciplinary Cooperation Project (Z191100001119010), National Science and Technology Major Project (2019ZX04012001-003) and Shanghai Sailing Program (19YF1418600).

ORCID iD

Zhifeng Liu

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