Theoretical and experimental analysis of drag torque for accelerated motion of ball screw mechanism

Abstract

Dynamic model of the ball in space for ball screw mechanism is established under different angular accelerations of the screw, and non-linear equations of the ball acceleration motion model were solved in accordance with ball movement characteristics during acceleration process of the ball screw mechanism. Forces and motion on the ball were analyzed taking into consideration acceleration state and the characteristics of the ball. Using the existing structural parameters of the ball screw mechanism and accelerating operating conditions of the ball, motion parameters of the ball were theoretically analyzed using simulations at different angular accelerations of the screw, along with motion parameters of the ball, which included contact angle, helix angle, the ball center locus helix path radius, normal force at contact points of the ball, the tangential velocity of ball center, relative sliding velocity of ball, and drag torque of the ball screw mechanism. A ball screw comprehensive test bench was used to experimentally measure drag torque of the ball screw mechanism under different angular accelerations of the screw. Changing regularity was found to be consistent between measured data with simulation results. As a result, acceleration dynamic model of the ball in space was demonstrated to be accurate and feasible.

Introduction

As a key component of numerical control machine tool, performance of the ball screw mechanism (BSM) has significant impact on processing quality of the workplaces. Additionally, state of accelerated motion between the ball and raceways for a BSM has been found to have significant influence on BSM performance. Running status is restricted to BSM drag torque, contact angle between balls and raceways, and movement characteristics of the ball. Previous research on BSM has focused on contact status and relative motion track between the ball and raceways. Birger and Arutunyan ¹ and Lin and colleagues^2–4 concluded that there was a nearly linear relationship between the spin angle velocity and rotational angular velocity of the balls and the rotational speed of the screw. Tsai et al. ⁵ studied a technique that can determine the onset of preload loss in a ball screw feed drive system via monitoring the change of ball pass frequency. The ball pass frequency of a ball screw with preload is smaller than that without preload due to the friction force caused by the preload. Li et al. ⁶ established the model of the dynamic contact angle of BSM combined with the geometric parameters of the ball and the raceway between nut and screw. They verified that the inner contact angles in screw raceway are always greater than the external angles in nut raceway under high-speed working conditions. Wei et al. ⁷ established a wear model that predicts the decrease in the preload and applied it to the wear estimation of the offset preloaded BSM. The model can effectively analyze the variation in the preload with the axial wear depth of the contact areas in different running strokes, rotational speeds, and ellipse ratios. They confirmed that the variations of these two contact angles with the axial load are strongly related to the critical load. Miyaguchi et al. ⁸ proposed that after the change of the motion direction of the nut, the drag torque of the double nuts ball screw will generate dynamic diversity and the transmitted torque is becoming smaller than which is in the steady state. Xu et al. ⁹ established creep analysis model more accurately to calculate the friction of a ball screw. Through the creep analysis and the influence of the creep parameters on the ball screw friction and friction distribution will be discussed. Zhou et al.^10,11 found that the error between the calculated preload using the traditional formula and measured value was significantly large. They determined total drag torque of the BSM due to friction between the ball and nut raceway, along with a linear correlation between preload and drag torque for the preloaded BSM. However, they did not study the accelerated motion characteristics of the BSM. It is important to note that research into accelerated motion characteristic for a BSM would be beneficial in improving the service lifetime and transmission precision of the BSM. ¹² Mu and Feng considered the centrifugal force and gyroscopic moment acting on the ball screw, and the dynamic model of high-speed ball screw was set up. They proposed that the effects of different working conditions, such as the running speed and the axial preload, affect the dynamic characteristic of ball screw at high speed which is greatly different from that of ball screw at low speed. ¹³ In the above cases, the motion rules of the ball were explored when the BSM is under a uniform motion and quasistatic state. However, there are unbalanced force and inadequate lubrication at two contact points of the ball of BSM in the acceleration process. The characteristics of acceleration motion and mechanical characteristics of balls of the ball screw are studied; the wear can be reduced, service life can be extended, and the transmission accuracy can be improved in the ball screw in the accelerated motion.

In this article, a dynamic model of the ball undergoing accelerated motion of the BSM in space is proposed. This model is based on the ball relative motion rule. Numerical differentiation and Newton–Raphson methods were used to solve the non-linear equations. Dynamic model of the ball in space was simulated, which allowed the changing regularity contact angle, helix angle, the ball center locus helix path radius, tangential velocity of the ball center, relative sliding velocity of the ball, and drag torque of the BSM to be obtained under different angular accelerations of the screw. Drag torque of the BSM at different screw angular accelerations were obtained experimentally using the BSM comprehensive test bench. This allowed both accuracy and feasibility of the dynamic model of the ball to be verified.

Theoretical analysis

Contact angle of BSM

In order to facilitate the systematic study of the kinematic rules of the ball, the Cartesian coordinate $A{X}_{\mathrm{A}}{Y}_{\mathrm{A}}{Z}_{\mathrm{A}}$ and $B{X}_{\mathrm{B}}{Y}_{\mathrm{B}}{Z}_{\mathrm{B}}$ depicted in Figure 1 are established at the contact point A between screw raceway and ball and the contact point B between nut raceway and ball, respectively. A and B are contact points when contact just occurs. In this situation, there is no normal stress. Z_A-axis and Z_B-axis are perpendicular to the tangent plane at contact points. X_A-axis and X_B-axis are parallel to tangential direction t of the center track of balls. Y_A-axis and Y_A-axis are determined by the right-hand rule.

Figure 1.

Coordinates of ball screws.

The contact angle of the BSM is useful to determine relative positional relationship between the ball and raceway, along with kinematic relationship between the screw, ball, and nut. Position vectors for contact point A between the ball and screw raceway and contact point B between the ball and nut raceway (Figure 2) were defined using the Frenet–Serret coordinate system $O_{\mathrm{H}}\mathrm{tnb}$

$${\mathit{P}}_{\mathrm{A}}^{\mathrm{H}}=\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{r_{\mathrm{b}}\cos {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{r_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{1}\end{array}\right]$$

(1a)

$${\mathit{P}}_{\mathrm{B}}^{\mathrm{H}}=\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{-r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}}\\ \multicolumn{1}{c}{-r_{\mathrm{b}}\sin {\beta }_{\mathrm{B}}}\\ \multicolumn{1}{c}{1}\end{array}\right]$$

(1b)

where ${\beta }_{\mathrm{A}}$ is the contact angle between the ball and screw raceway, ${\beta }_{\mathrm{B}}$ is the contact angle between the ball and nut raceway, and $r_{\mathrm{b}}$ is the radius of the ball.

Figure 2.

Contact angle.

As shown in Figure 2, $r_{\mathrm{S}}$ is the curvature radius of the screw raceway, $r_{\mathrm{N}}$ the curvature radius of the nut raceway. The arc center track of the screw raceway section intersects the normal plane of the ball center track at the point $C_{\mathrm{S}}$ . Arc center track of the nut raceway section intersects the normal plane of the ball center track at the point $C_{\mathrm{N}}$ . Ball center $O_H$ is all in the normal plane of the ball center track.

Position matrix of contact points A and B on the ball in the coordinates system $O_{\mathrm{H}}\mathrm{tnb}$ can be written as^14,15

$${\mathit{P}}_{\mathrm{bA}}^{\mathrm{H}}={\mathit{T}}_{\mathrm{H}}^{\mathrm{b}}{\mathit{P}}_{\mathrm{bA}}^{\mathrm{b}}=\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{r_{\mathrm{b}}\cos {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{r_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{1}\end{array}\right]$$

(2a)

$${\mathit{P}}_{\mathrm{bB}}^{\mathrm{H}}={\mathit{T}}_{\mathrm{H}}^{\mathrm{b}}{\mathit{P}}_{\mathrm{bB}}^{\mathrm{b}}=\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{-r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}}\\ \multicolumn{1}{c}{-r_{\mathrm{b}}\sin {\beta }_{\mathrm{B}}}\\ \multicolumn{1}{c}{1}\end{array}\right]$$

(2b)

where ${\mathit{T}}_{\mathrm{H}}^{\mathrm{b}}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ .

Ball contact point kinematics

Assume that acceleration of the ball center relative to the screw in the direction tangential to the helix path is $a_{\mathrm{bt}}^{*}$ and its angular acceleration relative to the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ is $\mathit{a}=[\begin{array}{ccc}\hfill a_{\mathrm{bt}}\hfill & \hfill a_{\mathrm{bn}}\hfill & \hfill a_{\mathrm{bb}}\hfill \end{array}{]}^{\mathrm{T}}$ . Motion accelerations (i.e. relative sliding accelerations) of contact points A and B on the ball relative to points A and B on the screw, respectively, in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ can be expressed as follows ¹⁶

$$\begin{array}{c}\left[\begin{array}{c}\hfill {\mathit{a}}_{\mathrm{AS}}^{\mathrm{H}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]=\frac{d}{\mathrm{dt}}\left(\frac{\mathrm{d}{\mathit{P}}_{\mathrm{bA}}^{\mathrm{H}}}{\mathrm{d}t}\right)=\frac{d}{\mathrm{dt}}\left(\frac{\mathrm{d}{\mathit{T}}_{\mathrm{H}}^{\mathrm{b}}}{\mathrm{d}t}\right)\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{r_{\mathrm{b}}\cos {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{r_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{1}\end{array}\right]\hfill \\ +{\mathit{T}}_{\mathrm{H}}^{\mathrm{b}}\left[\begin{array}{c}\hfill {\mathit{a}}_{\mathrm{bA}}^{\mathrm{b}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]=\left[\begin{array}{c}\hfill a_{\mathrm{bt}}^{*}+a_{\mathrm{bn}}{r}_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}-a_{\mathrm{bb}}{r}_{\mathrm{b}}\cos {\beta }_A\hfill \\ \multicolumn{1}{c}{-a_{\mathrm{bt}}{r}_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{a_{\mathrm{bt}}{r}_{\mathrm{b}}\cos {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \end{array}$$

(3)

where $P_{\mathrm{bA}}^{\mathrm{H}}=P_{\mathrm{A}}^{\mathrm{H}}$ , $(d/\mathrm{dt})(\mathrm{d}{\mathit{T}}_{\mathrm{H}}^{\mathrm{b}}/\mathrm{d}t)=(d/\mathrm{dt})\left(\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill v_{\mathrm{bt}}\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\right)=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill a_{\mathrm{bt}}^{*}\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ , and $\left[\begin{array}{c}\hfill {\mathit{a}}_{\mathrm{bA}}^{\mathrm{b}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill -a_{\mathrm{bb}}\hfill & \hfill a_{\mathrm{bn}}\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{a_{\mathrm{bb}}}& \hfill 0\hfill & \hfill -a_{\mathrm{bt}}\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{-a_{\mathrm{bn}}}& \hfill a_{\mathrm{bt}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{r_{\mathrm{b}}\cos {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{r_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{1}\end{array}\right]$

Slide-roll mixed motion at contact point A and pure rolling at contact point B on the ball exists during the accelerated motion of the screw. If relative motion between the ball and raceway of the nut is pure rolling, namely, relative sliding velocity at the contact point B is zero, then

$${\mathit{a}}_{\mathrm{BS}}^{\mathrm{H}}=\left[\begin{array}{c}\hfill a_{\mathrm{bt}}^{*}-a_{\mathrm{bn}}{r}_{\mathrm{b}}\sin {\beta }_{\mathrm{B}}+a_{\mathrm{bb}}{r}_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}+\stackrel{··}{\Omega }(\frac{r}{\cos \alpha }+r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}\cos \alpha )\hfill \\ \multicolumn{1}{c}{a_{\mathrm{bt}}{r}_{\mathrm{b}}\sin {\beta }_{\mathrm{B}}+\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \sin {\beta }_{\mathrm{B}}}\\ \multicolumn{1}{c}{-a_{\mathrm{bt}}{r}_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}-\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{B}}}\end{array}\right]=\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(4)

where $\stackrel{··}{\Omega }$ is angular acceleration of the screw, $\alpha$ is helix angle, r is radius of the helix trajectory of the ball center.

Since spinning angular acceleration of point B between the ball and nut raceway is zero, the following equation can be obtained

$$a_{\mathrm{bn}}\cos {\beta }_{\mathrm{B}}+a_{\mathrm{bb}}\sin {\beta }_{\mathrm{B}}=0$$

(5)

From equation (4), rotation angular acceleration of the ball around the axis t can be expressed as

$$a_{\mathrm{bt}}=-\stackrel{··}{\Omega }\sin \alpha$$

(6)

Then, relative sliding acceleration of the ball in the coordinates system $O_{\mathrm{H}}\mathrm{tnb}$ can be obtained as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{a}}_{\mathrm{AS}}^{\mathrm{H}}=\left[\begin{array}{c}\hfill a_{\mathrm{bt}}^{*}+a_{\mathrm{bn}}{r}_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}-a_{\mathrm{bb}}{r}_{\mathrm{b}}\cos {\beta }_{\mathrm{A}}\hfill \\ \multicolumn{1}{c}{-a_{\mathrm{bt}}{r}_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{a_{\mathrm{bt}}{r}_{\mathrm{b}}\cos {\beta }_{\mathrm{A}}}\end{array}\right]}\\ =\left[\begin{array}{c}\hfill -\stackrel{··}{\Omega }(\frac{r}{\cos \alpha }-r_{\mathrm{b}}\cos \alpha \cos {\beta }_{\mathrm{A}})\hfill \\ \multicolumn{1}{c}{\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{-\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{A}}}\end{array}\right]\hfill \end{array}$$

(7)

Based on relative acceleration of the ball (equation (4)), relative acceleration of point A in the ball, relative to screw raceway in tangential direction t can be expressed as $a_{\mathrm{ASt}}=a_{\mathrm{bt}}^{*}+{\alpha }_{\mathrm{bn}}{r}_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}-{\alpha }_{\mathrm{bb}}{r}_{\mathrm{b}}\cos {\beta }_A$ , and acceleration in the raceway section peripheral direction is $a_{\mathrm{ASc}}={\alpha }_{\mathrm{bt}}{r}_{\mathrm{b}}$ .

Assuming

$$\tan \phi =\frac{a_{\mathrm{ASt}}}{a_{\mathrm{ASc}}}=\frac{a_{\mathrm{bt}}^{*}+a_{\mathrm{bn}}{r}_{\mathrm{b}}\sin {\beta }_{\mathrm{A}}-a_{\mathrm{bb}}{r}_{\mathrm{b}}\cos {\beta }_{\mathrm{A}}}{a_{\mathrm{bt}}{r}_b}=g_{\mathrm{a}}$$

(8)

Based on equations (5)–(8), the following equations can be obtained

$$a_{\mathrm{bb}}=-\frac{g_{\mathrm{a}}{a}_{\mathrm{bt}}{r}_{\mathrm{b}}+\stackrel{··}{\Omega }(\frac{r}{\cos \alpha }+r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}\cos \alpha )}{r_{\mathrm{b}}(\frac{1}{\cos {\beta }_{\mathrm{B}}}+\cos {\beta }_{\mathrm{A}}+\sin {\beta }_{\mathrm{A}}\tan {\beta }_{\mathrm{B}})}$$

(9)

$$a_{\mathrm{bt}}^{*}=-\frac{a_{\mathrm{bb}}{r}_{\mathrm{b}}}{\cos {\beta }_{\mathrm{B}}}-\stackrel{··}{\Omega }(\frac{r}{\cos \alpha }+r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}\cos \alpha )$$

(10)

Dynamics analysis of ball contact points

When the ball screw is accelerated, it is affected by positive pressure from the raceways friction along the contact area, inertia force, oil film resistance, drag torque, inertia moment, and gyroscopic moment. The balanced force and moment equations acting on the ball in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ can be expressed as

$$\sum \mathit{F}=m{a}^{\mathrm{H}}:{\mathit{Q}}_{\mathrm{A}}+{\mathit{Q}}_{\mathrm{B}}+{\mathit{F}}_{\mathrm{SA}}+{\mathit{F}}_{\mathrm{SB}}+{\mathit{F}}_{\mathrm{IH}}+{\mathit{F}}_{\mathrm{oil}}=0$$

(11)

$$\sum \mathit{M}=0\begin{array}{cc}\hfill :\hfill & \hfill {\mathit{M}}_{\mathrm{SA}}+{\mathit{M}}_{\mathrm{SB}}+{\mathit{M}}_{\mathrm{IH}}+{\mathit{M}}_{\mathrm{g}}=0\hfill \end{array}$$

(12)

where $Q_{\mathrm{A}}$ and $Q_{\mathrm{B}}$ are the pressure of contact points A and B in the normal direction, $F_{\mathrm{SA}}$ and $F_{\mathrm{SB}}$ are the frictional force of contact points A and B, $F_{\mathrm{IH}}$ is inertia force, $F_{\mathrm{oil}}$ is the oil film resistance in the study of Mu and Feng, ¹³ $M_{\mathrm{SA}}$ and $M_{\mathrm{SB}}$ are the friction moment, $M_{\mathrm{IH}}$ is inertia moment, and $M_{\mathrm{g}}$ is gyroscopic moment.

Friction force at contact point B can be split into two components. Friction $f_{\mathrm{B}}\cos {\alpha }^{*}$ in the tangential direction of the raceway helix path t and direction of $X_{\mathrm{B}}$ are the same (Figure 1). Similarly, direction of static friction force $f_{\mathrm{B}}\sin {\alpha }^{*}$ in plane of nb and direction of $Z_{\mathrm{B}}$ are parallel, and spin torque $M_{\mathrm{BZ}}$ in plane of nb for contact point B and direction of $Z_{\mathrm{B}}$ are the same. Then, friction force and drag torque at contact point B can be obtained

$${\mathit{F}}_{\mathrm{BS}}=\left[\begin{array}{c}\hfill f_{\mathrm{B}}\sin {\alpha }^{*}\hfill \\ \multicolumn{1}{c}{-f_{\mathrm{B}}\cos {\alpha }^{*}\sin {\beta }_{\mathrm{B}}}\\ \multicolumn{1}{c}{f_{\mathrm{B}}\cos {\alpha }^{*}\cos {\beta }_{\mathrm{B}}}\end{array}\right]$$

(13)

$${\mathit{M}}_{\mathrm{BS}}=\left[\begin{array}{c}\hfill -f_{\mathrm{B}}\cos {\alpha }^{*}{r}_{\mathrm{b}}\hfill \\ \multicolumn{1}{c}{-f_{\mathrm{B}}{r}_{\mathrm{b}}\sin {\alpha }^{*}\sin {\beta }_{\mathrm{B}}+M_{\mathrm{BZ}}\cos {\beta }_{\mathrm{B}}}\\ \multicolumn{1}{c}{f_{\mathrm{B}}{r}_{\mathrm{b}}\sin {\alpha }^{*}\cos {\beta }_{\mathrm{B}}+M_{\mathrm{BZ}}\sin {\beta }_{\mathrm{B}}}\end{array}\right]$$

(14)

where ${\alpha }^{*}={\sin }^{-1}(a_{\mathrm{bt}}/\sqrt{a_{\mathrm{bt}}^2+a_{\mathrm{bn}}^2+a_{\mathrm{bb}}^2})$

Substituting equations (13) and (14) into equations (11) and (12), force and force moment balance equations of the ball can be obtained

$$\{\begin{array}{c}\hfill f_{\mathrm{B}}\sin {\alpha }^{*}+F_{\mathrm{SAt}}+F_{\mathrm{IHt}}+F_{\mathrm{oil}}=0\hfill \\ \multicolumn{1}{c}{-Q_{\mathrm{A}}\cos {\beta }_{\mathrm{A}}+Q_{\mathrm{B}}\cos {\beta }_{\mathrm{B}}+F_{\mathrm{SAn}}-f_{\mathrm{B}}\cos {\alpha }^{*}\sin {\beta }_{\mathrm{B}}+F_{\mathrm{IHn}}=0}\\ \multicolumn{1}{c}{-Q_{\mathrm{A}}\sin {\beta }_{\mathrm{A}}+Q_{\mathrm{B}}\sin {\beta }_{\mathrm{B}}+f_{\mathrm{B}}\cos {\alpha }^{*}\cos {\beta }_{\mathrm{B}}+F_{\mathrm{SAb}}+F_{\mathrm{IHb}}=0}\\ \multicolumn{1}{c}{M_{\mathrm{SAt}}-f_{\mathrm{B}}{r}_{\mathrm{b}}\cos {\alpha }^{*}+M_{\mathrm{IHt}}+M_{\mathrm{gt}}=0}\\ \multicolumn{1}{c}{M_{\mathrm{SAn}}-f_{\mathrm{B}}{r}_{\mathrm{b}}\sin {\alpha }^{*}\sin {\beta }_{\mathrm{B}}+M_{\mathrm{BZ}}\cos {\beta }_{\mathrm{B}}+M_{\mathrm{IHn}}+M_{\mathrm{gn}}=0}\\ \multicolumn{1}{c}{M_{\mathrm{SAb}}+f_{\mathrm{B}}{r}_{\mathrm{b}}\sin {\alpha }^{*}\cos {\beta }_{\mathrm{B}}+M_{\mathrm{BZ}}\sin {\beta }_{\mathrm{B}}+M_{\mathrm{IHb}}+M_{\mathrm{gb}}=0}\end{array}$$

(15)

In equation (15), $F_{\mathrm{SA}}$ , $F_{\mathrm{IH}}$ , $M_{\mathrm{SA}}$ , $M_{\mathrm{IH}}$ , and $M_{\mathrm{g}}$ are calculated based on the following analysis.

According to the relationship between total preload of BSM and contact pressure of the ball in the normal direction, pressure at contact point B in the normal direction can be obtained as

$$Q_{\mathrm{B}}=\frac{F_{\mathrm{pre}}}{n_{\mathrm{b}}\cos \alpha \sin {\beta }_{\mathrm{B}}}$$

(16)

where $F_{\mathrm{pre}}$ is the preload, $n_{\mathrm{b}}$ is the number of balls.

Non-linear equations can be solved using the Newton–Raphson method. Then, contact angles ${\beta }_{\mathrm{A}}$ and ${\beta }_{\mathrm{B}}$ , force $F_{\mathrm{B}}$ , friction coefficient ${\mu }_{\mathrm{A}}$ at the ball contact point A resistances,^16,17 spinning torque $M_{\mathrm{BZ}}$ , $g_{\mathrm{a}}$ , contact pressure $Q_{\mathrm{A}}$ of the ball at contact point A in the normal direction, and total drag torque of the BSM can be obtained by equation (17)

$$M=n_{\mathrm{b}}(F_{\mathrm{Bt}}\cos \alpha +F_{\mathrm{Bnb}}\cos {\beta }_{\mathrm{B}}\sin \alpha )(r+r_{\mathrm{b}}\sin {\beta }_{\mathrm{B}})$$

(17)

where $F_{\mathrm{Bt}}=f_{\mathrm{B}}\sin {\alpha }^{*}$ , $F_{\mathrm{Bnb}}=-f_{\mathrm{B}}\cos {\alpha }^{*}\sin {\beta }_{\mathrm{B}}$

Flowchart of numerical analyses

The flowchart exhibited in Figure 3 is presented to illustrate how all parameters in non-linear equations related to the roll-slide behavior of a ball screw are obtained in sequence by numerical analyses from the satisfaction of the convergence criterion. The specification of the BSM and the running conditions are shown in Table 1; the simulation can be obtained. By simulation of the dynamic model of the ball, the changing regularity of running parameters of the BSM can be obtained under the different angular accelerations of the screw.

Figure 3.

Flowchart for the computation of mechanical model of the BSM.

Table 1.

Ball screw parameters and running conditions.

Parameters	Value	Unit
Geometry parameters of BSM
Helix angle, α	2.767	°
Pitch, L	5	mm
Screw inner radius of screw, rCS	16.557	mm
Screw pitch cycle radius, r	16.375	mm
Nut inner radius of screw, rCN	15.818	mm
Radius of curvature of the screw raceway, rS	1.826	mm
Radius of curvature of the nut raceway, rN	1.826	mm
Ball’s radius, rb	1.588	mm
Contact angle	44.35	°
Preload, FWS	2360	N
Number of balls, nb	186
Density of the ball material, ρ	7.8	g/cm3
Young’s modulus, E	211.1	GPa
Poisson’s ratio, v	0.3
Running conditions
Viscosity of lubricant oil, SAE	30	cSt
Acceleration of screw, $\stackrel{··}{\Omega }$	78.13/52.08/26.04	rad/s2

BSM: ball screw mechanism.

Contact stress

Curvature of the ball was approximated to an ellipse and is define as

$$A=\frac{1}{2R_{\mathrm{X}}},B=\frac{1}{2R_{\mathrm{Y}}}$$

(18)

Ball contact with the screw raceway can be expressed as

$$\frac{1}{R_{\mathrm{X}}}=\frac{1}{r_{\mathrm{b}}}-\frac{1}{r_{\mathrm{S}}},\frac{1}{R_{\mathrm{Y}}}=\frac{1}{r_{\mathrm{b}}}+\frac{1}{r_{\mathrm{st}}}$$

(19)

Ball contact with the nut raceway can be expressed as

$$\frac{1}{R_{\mathrm{X}}}=\frac{1}{r_{\mathrm{b}}}-\frac{1}{r_{\mathrm{N}}},\frac{1}{R_{\mathrm{Y}}}=\frac{1}{r_{\mathrm{b}}}-\frac{1}{r_{\mathrm{nt}}}$$

(20)

where $r_{\mathrm{st}}$ is the radius of curvature corresponding to the tangential direction of the track, where center point of the ball and screw raceway are in contact, $r_{\mathrm{st}}=r_{\mathrm{CS}}/\cos {\beta }_{\mathrm{r}}-r_{\mathrm{S}}$ , $r_{\mathrm{nt}}$ is the radius of curvature corresponding to the tangential direction of the track in contact with the center point of the ball and the raceway, and $r_{\mathrm{nt}}=r_{\mathrm{CN}}/\cos {\beta }_{\mathrm{r}}+r_{\mathrm{N}}$ .

According to the Hertz contact theory, ¹⁷ boundary curve of the ball and raceway contact area can be expressed as

$${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1,a\ge b$$

(21)

where a is elliptical long axle and b is elliptical short axle.

A method of using standard elliptic integral was proposed by Thomas and Hoersch. ¹⁸ Parameters of ball and raceway contact ellipse can be expressed as

$$b=\frac{H}{B+A},a=\frac{b}{k}$$

(22)

where $H={[(3\mathrm{kE}(m)Q_n{(B+A)}^2)/\pi E\prime ]}^{1/3}$

where $E(m)$ is the second category of complete elliptical integration and k is the intermediate variable.

Effective elastic modulus between two contact surfaces can be expressed as

$$E\prime =2{(\frac{1-{v_1}^2}{E_1}+\frac{1-{v_2}^2}{E_2})}^{-1}$$

(23)

where $E_1$ and $E_2$ are the elastic modulus of the two contact bodies, $v_1$ and $v_2$ are corresponding Poisson ratio for the two contact bodies.

Normal elasticity approximation can be expressed as

$${\delta }_{\mathrm{n}}=\frac{H^{2}K(m)}{(B+A)E(m)}$$

(24)

Relationship between normal load $F_{\mathrm{n}}$ and normal elastic approximation ${\delta }_{\mathrm{n}}$ can be expressed as

$$F_{\mathrm{n}}=\frac{4}{3}E\prime R^{*1/2}{\delta }_{\mathrm{n}}^{3/2}$$

(25)

where $R^{*}$ is equivalent contact radius and $1/R^{*}=(1/a)+(1/b)$

Torsional slip friction force

Initial value for relative rigid displacement of contact bodies ${\delta }_{\mathrm{t}}$ in the tangential direction can be solved using the Johnson’s expression, ¹⁷ which is obtained based on many assumptions. However, this allows the approximation of an initial value

$${\delta }_{\mathrm{t}}={\delta }_0[1-{(1-\frac{F_{\mathrm{t}}}{{\mu }_{\mathrm{S}}}{F}_{\mathrm{n}})}^{2/3}]$$

(26)

and

$${\delta }_0=\frac{3{\mu }_{\mathrm{S}}{F}_{\mathrm{n}}}{8a_0}[\frac{(2-v_1)(1+v_1)}{E_1}+\frac{(2-v_2)(1+v_2)}{E_2}]$$

(27)

where $a_0$ is the contact radius considering normal load when a ball–plane contact occurs.

Rigid displacement is expressed as equation (26), and normal stress is calculated by Hertz contact theory. The shear stress is solved as follows ¹⁹

$$\mathit{Q}(r)=\{\begin{array}{cc}\hfill \frac{3{\mu }_{\mathrm{S}}{F}_{\mathrm{n}}}{2\pi a^3}\sqrt{a^2-r^2,}\hfill & \hfill c\le r\le a\hfill \\ \multicolumn{1}{c}{\frac{3{\mu }_{\mathrm{S}}{F}_{\mathrm{n}}}{{\pi }^2{a}^3}\sqrt{a^2-r^2}(\frac{\pi }{2}+k^{2}D(k)F(k^{*},\phi )-K(k)E(k^{*},\phi ))}& \hfill 0\le r\le c\hfill \end{array}$$

(28)

where $k^{*}=c/a$ , $k=\sqrt{1-{k^{*}}^2}$ , $\phi =\arcsin [1/k^{*}\sqrt{({k^{*}}^2-{(r/a)}^2)/(1-{(r/a)}^2)}]$ , and $D(k)=(K(k)-E(k))/k^2$

So, torsional moment and rigid angle of the ball between the ball and raceway can be expressed as

$${\varphi }_{\mathrm{z}}=\frac{3{\mu }_{\mathrm{S}}{F}_{\mathrm{n}}}{4\pi G{a}^2}{k}^{2}D(k)$$

(29)

$$\begin{array}{c}\multicolumn{1}{c}{M_{\mathrm{z}}=\frac{{\mu }_{\mathrm{S}}{F}_{\mathrm{n}}a}{4\pi }\{\frac{3{\pi }^2}{4}+k^{*}{k}^2[6K(k)+(4{k^{*}}^2-3)D(k)]-3\mathrm{kK}(k)\arcsin k^{*}}\\ -3k^2[K(k)\underset{0}{\overset{\pi /2}{\int }}\frac{\arcsin (k^{*}\sin \alpha )}{{(1-{k^{*}}^2{\sin }^2\alpha )}^{3/2}}d\alpha -D(k)\underset{0}{\overset{\pi /2}{\int }}\frac{\arcsin (k^{*}\sin \alpha )}{{(1-{k^{*}}^2{\sin }^2\alpha )}^{1/2}}d\alpha ]\}\hfill \end{array}$$

(30)

where G is the shear modulus.

Gyroscopic moment

When screw angular velocity is $\stackrel{·}{\Omega }$ $(\stackrel{··}{\Omega }t)$ in the inertial coordinate system $O_{\mathrm{W}}{X}_{\mathrm{W}}{Y}_{\mathrm{W}}{Z}_{\mathrm{W}}$ , revolution angular velocity of the ball in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ can be expressed as

$${\omega }_{\mathrm{m}}=\left[\begin{array}{c}\hfill {\omega }_{\mathrm{mt}}\hfill \\ \multicolumn{1}{c}{{\omega }_{\mathrm{mn}}}\\ \multicolumn{1}{c}{{\omega }_{\mathrm{mb}}}\\ \multicolumn{1}{c}{0}\end{array}\right]={\left(T_{\mathrm{W}}^{\mathrm{S}}{T}_{\mathrm{S}}^{\mathrm{H}}\right)}^{-1}\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{\stackrel{·}{\Omega }}\\ \multicolumn{1}{c}{0}\end{array}\right]=\left[\begin{array}{c}\hfill -\stackrel{·}{\Omega }\sin \alpha \hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{-\stackrel{·}{\Omega }\cos \alpha }\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(31)

where ${\mathit{T}}_{\mathrm{W}}^{\mathrm{S}}=\left[\begin{array}{cccc}\hfill \cos \Omega \hfill & \hfill -\sin \Omega \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{\sin \Omega }& \hfill \cos \Omega \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ , ${\mathrm{T}}_{\mathrm{S}}^{\mathrm{H}}=\left[\begin{array}{cccc}\hfill -\cos \alpha \sin \theta \hfill & \hfill -\cos \theta \hfill & \hfill \sin \alpha \sin \theta \hfill & \hfill r\cos \theta \hfill \\ \multicolumn{1}{c}{\cos \alpha \cos \theta }& \hfill -\sin \theta \hfill & \hfill -\sin \alpha \cos \theta \hfill & \hfill r\sin \theta \hfill \\ \multicolumn{1}{c}{\sin \alpha }& \hfill 0\hfill & \hfill \cos \alpha \hfill & \hfill r\theta \tan \alpha \hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ , $\Omega$ is angular displacement of the screw, $\theta$ is angular displacement of the ball center on the helix path with respect to the screw, r is the radius of the ball center locus helix path.

Gyroscopic moment of the ball during accelerated motion can be expressed as

$$M_g=-[J_{\mathrm{P}}+(J_{\mathrm{P}}+J_{\mathrm{e}})\frac{{\omega }_{\mathrm{m}}}{{\omega }_{\mathrm{b}}}\cos {\beta }^{*}]{\omega }_{\mathrm{m}}\times {\omega }_{\mathrm{b}}$$

(32)

where $J_{\mathrm{P}}$ is ball rotation inertia moment in the coordinate $O_{\mathrm{H}}\mathrm{tnb}$ and $J_{\mathrm{P}}=(1/10){\mathrm{md}}_{\mathrm{b}}^2$ , $J_{\mathrm{e}}$ is ball revolution inertia moment in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ and $J_{\mathrm{e}}=(1/4)m{d}^2$ , ${\beta }^{*}$ is angle between the spin axis of the ball and the screw axes and ${\beta }^{*}=\arctan (\sin ({\beta }_{\mathrm{B}})/(r_{\mathrm{b}}/r+\cos ({\beta }_{\mathrm{B}})))$ .

Then, rotation angular velocity of the ball in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ can be expressed as

$${\omega }_{\mathrm{b}}=\left[\begin{array}{c}\hfill {\omega }_{\mathrm{bt}}\hfill \\ \multicolumn{1}{c}{{\omega }_{\mathrm{bn}}}\\ \multicolumn{1}{c}{{\omega }_{\mathrm{bb}}}\end{array}\right]=\left[\begin{array}{c}\hfill a_{\mathrm{bt}}t\hfill \\ \multicolumn{1}{c}{a_{\mathrm{bn}}t}\\ \multicolumn{1}{c}{a_{\mathrm{bb}}t}\end{array}\right]$$

(33)

Inertia force, inertia moment

When the screw accelerates, inertial force and moment of the ball can be decomposed into two components. Inertial force and moment of the ball are caused due to acceleration of the ball. This acceleration motion of the ball is caused by the screw friction force and angular velocity of the ball.

Inertial force and inertial moment due to acceleration

Sliding acceleration of contact point A on the ball in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ can be expressed as follows

$$\left(\begin{array}{c}\hfill {\mathit{a}}_{\mathrm{Ab}}^{\mathrm{H}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)=\left(\begin{array}{c}\hfill {\mathit{a}}_{\mathrm{AS}}^{\mathrm{H}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)=\left[\begin{array}{c}\hfill {\alpha }_{\mathrm{bn}}{r}_{\mathrm{b}}(\sin {\beta }_{\mathrm{A}}+\sin {\beta }_{\mathrm{B}})-{\alpha }_{\mathrm{bb}}{r}_{\mathrm{b}}(\cos {\beta }_{\mathrm{A}}+\cos {\beta }_{\mathrm{B}})-\stackrel{··}{\Omega }(\frac{r}{\cos \alpha }+r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}\cos \alpha )\hfill \\ \multicolumn{1}{c}{\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{-\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{A}}}\end{array}\right]$$

(34)

Acceleration of contact point A on the ball is equal to acceleration at contact point A on the screw minus the sliding acceleration at contact point A on the ball. Acceleration of contact point A on the ball can be expressed as follows

$$\left(\begin{array}{c}\hfill {\mathit{a}}_{\mathrm{AbR}}^{\mathrm{H}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)=\left(\begin{array}{c}\hfill {\mathit{a}}_{\mathrm{A}}^{\mathrm{H}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)-\left(\begin{array}{c}\hfill {\mathit{a}}_{\mathrm{Ab}}^{\mathrm{H}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)$$

(35)

Acceleration of point A on the screw can be obtained as follows

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{c}\hfill a_{\mathrm{A}}^{\mathrm{H}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]=\frac{d}{\mathrm{dt}}\left(\left[\begin{array}{c}\hfill {\mathit{v}}_{\mathrm{A}}^{\mathrm{H}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]\right)={\left({\mathit{T}}_{\mathrm{W}}^{\mathrm{S}}{\mathit{T}}_{\mathrm{S}}^{\mathrm{H}}\right)}^{-1}\frac{d}{\mathrm{dt}}\left(\left[\begin{array}{c}\hfill {\mathit{v}}_{\mathrm{A}}^{\mathrm{W}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]\right)}\\ =\left[\begin{array}{c}\hfill \stackrel{··}{\Omega }(r\cos \alpha -r_{\mathrm{b}}\cos \alpha \cos {\beta }_{\mathrm{A}})\hfill \\ \multicolumn{1}{c}{-\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{\stackrel{··}{\Omega }(-r\sin \alpha +r_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{A}})}\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \end{array}$$

(36)

Equation (37) is satisfied when there is pure rolling at contact point B between the ball and nut raceway. This results in velocity of point B on the nut relative to point B on the screw to be equal to acceleration at contact point A on the ball

$${\mathit{a}}_{\mathrm{Ab}}^{\mathrm{H}}={\mathit{a}}_{\mathrm{NB}}^{\mathrm{H}}=\left[\begin{array}{c}\hfill -\stackrel{··}{\Omega }(\frac{r}{\cos \alpha }+r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}\cos \alpha )\hfill \\ \multicolumn{1}{c}{-\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \sin {\beta }_{\mathrm{B}}}\\ \multicolumn{1}{c}{\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{B}}}\end{array}\right]$$

(37)

By substituting equations (36) and (37) into equation (35), acceleration at contact point A on the ball can be expressed as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{a}}_{\mathrm{AbR}}^{\mathrm{H}}=\left[\begin{array}{c}\hfill \stackrel{··}{\Omega }(r\cos \alpha -r_{\mathrm{b}}\cos \alpha \cos {\beta }_{\mathrm{A}})\hfill \\ \multicolumn{1}{c}{-\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \sin {\beta }_{\mathrm{A}}}\\ \multicolumn{1}{c}{\stackrel{··}{\Omega }(-r\sin \alpha +r_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{A}})}\\ \multicolumn{1}{c}{0}\end{array}\right]}\\ -\left[\begin{array}{c}\hfill -\stackrel{··}{\Omega }(\frac{r}{\cos \alpha }+r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}\cos \alpha )\hfill \\ \multicolumn{1}{c}{-\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \sin {\beta }_{\mathrm{B}}}\\ \multicolumn{1}{c}{\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{B}}}\end{array}\right]\hfill \\ =\left[\begin{array}{c}\hfill \stackrel{··}{\Omega }(r\cos \alpha +\frac{r}{\cos \alpha }-r_{\mathrm{b}}\cos \alpha \cos {\beta }_{\mathrm{A}}+r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}\cos \alpha )\hfill \\ \multicolumn{1}{c}{\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha (-\sin {\beta }_{\mathrm{A}}+\sin {\beta }_{\mathrm{B}})}\\ \multicolumn{1}{c}{\stackrel{··}{\Omega }(-r\sin \alpha +r_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{A}}-r_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{B}})}\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \end{array}$$

(38)

In the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ , any particle $d\mathrm{m}$ acceleration in the ball can be expressed as

$${\stackrel{··}{x}}_{\mathrm{H}}=\stackrel{··}{\Omega }(r\cos \alpha +\frac{r}{\cos \alpha }-r_{\mathrm{b}}\cos \alpha \cos {\beta }_{\mathrm{A}}+r_{\mathrm{b}}\cos {\beta }_{\mathrm{B}}\cos \alpha )$$

(39a)

$${\stackrel{··}{y}}_{\mathrm{H}}=-\stackrel{··}{\Omega }{r}_{\mathrm{b}}\sin \alpha (\sin {\beta }_{\mathrm{A}}-\sin {\beta }_{\mathrm{B}})$$

(39b)

$${\stackrel{··}{z}}_{\mathrm{H}}=\stackrel{··}{\Omega }(-r\sin \alpha +r_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{A}}-r_{\mathrm{b}}\sin \alpha \cos {\beta }_{\mathrm{B}})$$

(39c)

The inertial force caused acceleration through integral method to point in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ . Thus, inertial force of any point in the ball can be obtained as

$$F_{\mathrm{IHt}}^{\mathrm{a}}=-\rho \underset{-r_{\mathrm{b}}}{\overset{r_{\mathrm{b}}}{\int }}\underset{0}{\overset{\sqrt{{r_b}^2-W^2}}{\int }}\underset{0}{\overset{2\pi }{\int }}{\stackrel{··}{x}}_{\mathrm{H}}{r}_{\mathrm{d}}d\phi d{r}_{\mathrm{d}}\mathrm{dW}$$

(40a)

$$F_{\mathrm{IHn}}^{\mathrm{a}}=-\rho \underset{-r_{\mathrm{b}}}{\overset{r_{\mathrm{b}}}{\int }}\underset{0}{\overset{\sqrt{{r_b}^2-W^2}}{\int }}\underset{0}{\overset{2\pi }{\int }}{\stackrel{··}{y}}_{\mathrm{H}}{r}_{\mathrm{d}}d\phi d{r}_{\mathrm{d}}\mathrm{dW}$$

(40b)

$$F_{\mathrm{IHb}}^{\mathrm{a}}=-\rho \underset{-r_{\mathrm{b}}}{\overset{r_{\mathrm{b}}}{\int }}\underset{0}{\overset{\sqrt{{r_b}^2-W^2}}{\int }}\underset{0}{\overset{2\pi }{\int }}{\stackrel{··}{z}}_{\mathrm{H}}{r}_{\mathrm{d}}d\phi d{r}_{\mathrm{d}}\mathrm{dW}$$

(40c)

By substituting equation (39) into equation (40), inertia force of the ball can be obtained as

$$\begin{array}{c}{\mathit{F}}_{\mathrm{IH}}^{\mathrm{a}}=\left[\begin{array}{c}\hfill F_{\mathrm{IHt}}^{\mathrm{a}}\hfill \\ \multicolumn{1}{c}{F_{\mathrm{IHn}}^{\mathrm{a}}}\\ \multicolumn{1}{c}{F_{\mathrm{IHb}}^{\mathrm{a}}}\end{array}\right]=-\frac{4}{3}\pi {r_{\mathrm{b}}}^3\rho \stackrel{··}{\Omega }\hfill \\ \left[\begin{array}{c}\hfill \frac{r}{r_{\mathrm{b}}}\cos \alpha +\frac{r}{r_{\mathrm{b}}\cos \alpha }-\cos \alpha \cos {\beta }_{\mathrm{A}}+\cos {\beta }_{\mathrm{B}}\cos \alpha \hfill \\ \multicolumn{1}{c}{\sin \alpha (\sin {\beta }_{\mathrm{A}}-\sin {\beta }_{\mathrm{B}})}\\ \multicolumn{1}{c}{-\frac{r}{r_{\mathrm{b}}}\sin \alpha +\sin \alpha \cos {\beta }_{\mathrm{A}}-\sin \alpha \cos {\beta }_{\mathrm{B}}}\end{array}\right]\hfill \end{array}$$

(41)

In the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ , position vector of any point in the ball can be expressed as

$$\left[\begin{array}{c}\hfill {\mathit{R}}_{\mathrm{b}}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]={\mathit{T}}_{\mathrm{b}}^{\mathrm{RV}}\left[\begin{array}{c}\hfill r_{\mathrm{d}}\cos \phi \hfill \\ \multicolumn{1}{c}{r_{\mathrm{d}}\sin \phi }\\ \multicolumn{1}{c}{W}\\ \multicolumn{1}{c}{1}\end{array}\right]$$

(42)

where ${\mathit{T}}_{\mathrm{b}}^{\mathrm{RV}}={\mathit{T}}_{\mathrm{b}}^{\mathrm{RZ}}{\mathit{T}}_{\mathrm{RZ}}^{\mathrm{RV}}=\left[\begin{array}{cccc}\hfill \cos \gamma \hfill & \hfill \sin \gamma \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{-\sin \gamma }& \hfill \cos \gamma \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill \cos {\beta }_{\mathrm{t}}\hfill & \hfill 0\hfill & \hfill \sin {\beta }_{\mathrm{t}}\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{-\sin {\beta }_{\mathrm{t}}}& \hfill 0\hfill & \hfill \cos {\beta }_{\mathrm{t}}\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ , $\phi$ is angular displacement between the X_W-axis and Y_W-axis, $\gamma$ is angular displacement of the Z_W-axis, and ${\beta }_{\mathrm{t}}$ is angular displacement of the Y_W-axis.

According to the integral method, inertia moment of ball in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ can be expressed as

$${\mathit{M}}_{\mathrm{IH}}^{\mathrm{a}}=-\rho \underset{-{\mathrm{r}}_{\mathrm{b}}}{\overset{{\mathrm{r}}_{\mathrm{b}}}{\int }}\underset{0}{\overset{\sqrt{{\mathrm{r}}_{\mathrm{b}}^2-{\mathrm{W}}^2}}{\int }}\underset{0}{\overset{2\pi }{\int }}({\mathit{R}}_{\mathrm{b}}\times {\mathit{a}}_{\mathrm{H}})r_{\mathrm{d}}d\phi d{r}_{\mathrm{d}}\mathrm{dW}$$

(43)

Inertia force and inertia moment due to angular velocity

Referring to the study of Wei et al., ³ when the screw accelerates, the inertia force which caused the angular velocity of the ball through integral method to any point in the ball can be obtained as

$$F_{\mathrm{IWx}}=-\rho \underset{-r_{\mathrm{b}}}{\overset{r_{\mathrm{b}}}{\int }}\underset{0}{\overset{\sqrt{{r_{\mathrm{b}}}^2-W^2}}{\int }}\underset{0}{\overset{2\pi }{\int }}{\stackrel{··}{x}}_{\mathrm{W}}{r}_{\mathrm{d}}d\phi d{r}_{\mathrm{d}}\mathrm{dW}$$

(44a)

$$F_{\mathrm{IWy}}=-\rho \underset{-r_{\mathrm{b}}}{\overset{r_{\mathrm{b}}}{\int }}\underset{0}{\overset{\sqrt{{r_{\mathrm{b}}}^2-W^2}}{\int }}\underset{0}{\overset{2\pi }{\int }}{\stackrel{··}{y}}_{\mathrm{W}}{r}_{\mathrm{d}}d\phi d{r}_{\mathrm{d}}\mathrm{dW}$$

(44b)

$$F_{\mathrm{IWz}}=-\rho \underset{-r_{\mathrm{b}}}{\overset{r_{\mathrm{b}}}{\int }}\underset{0}{\overset{\sqrt{{r_{\mathrm{b}}}^2-W^2}}{\int }}\underset{0}{\overset{2\pi }{\int }}{\stackrel{··}{z}}_{\mathrm{W}}{r}_{\mathrm{d}}d\phi d{r}_{\mathrm{d}}\mathrm{dW}$$

(44c)

Inertial force of the whole ball in the inertial coordinate system $O_{\mathrm{W}}{X}_{\mathrm{W}}{Y}_{\mathrm{W}}{Z}_{\mathrm{W}}$ and the Frenet–Serret coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ can be expressed as

$${\mathit{F}}_{\mathrm{IW}}=\left[\begin{array}{c}\hfill F_{\mathrm{IWx}}\hfill \\ \multicolumn{1}{c}{F_{\mathrm{IWy}}}\\ \multicolumn{1}{c}{F_{\mathrm{IWz}}}\end{array}\right]=\frac{4}{3}\pi {r_{\mathrm{b}}}^3\rho r{(\stackrel{·}{\Omega }+\frac{v_{\mathrm{bt}}}{r}\cos \alpha )}^2\left[\begin{array}{c}\hfill \cos (\Omega +\theta )\hfill \\ \multicolumn{1}{c}{\sin (\Omega +\theta )}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(45a)

$${\mathit{F}}_{\mathrm{IH}}^S={\left(T_{\mathrm{W}}^{\mathrm{S}}{T}_{\mathrm{S}}^{\mathrm{H}}\right)}^{-1}{F}_{\mathrm{IW}}=-\frac{4}{3}\pi {r_{\mathrm{b}}}^3\rho r{(\stackrel{·}{\Omega }+\frac{v_{\mathrm{bt}}}{r}\cos \alpha )}^2\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(45b)

where $\rho$ is material density of the ball.

In the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ , position vector acceleration for any point in the ball can be expressed as

$${\mathit{a}}_{\mathrm{H}}^{\mathrm{S}}=({\mathit{T}}_{\mathrm{W}}^{\mathrm{S}}{\mathit{T}}_{\mathrm{S}}^{\mathrm{H}}{)}^{-1}{\mathit{a}}_{\mathrm{W}}$$

(46)

Then, according to the integral method of the inertia moment, inertia moment of ball in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ can be expressed as

$${\mathit{M}}_{\mathrm{IH}}^S=-\rho \underset{-r_{\mathrm{b}}}{\overset{r_{\mathrm{b}}}{\int }}\underset{0}{\overset{\sqrt{r_{\mathrm{b}}^2-{\mathrm{W}}^2}}{\int }}\underset{0}{\overset{2\pi }{\int }}({\mathit{R}}_{\mathrm{b}}\times {\mathit{a}}_{\mathrm{H}})r_{\mathrm{d}}d\phi d{r}_{\mathrm{d}}\mathrm{dW}$$

(47)

Then, total inertial force and moment of the ball during acceleration in the coordinate system $O_{\mathrm{H}}\mathrm{tnb}$ can be expressed as

$${\mathit{F}}_{\mathrm{IH}}={\mathit{F}}_{\mathrm{IH}}^a+{\mathit{F}}_{\mathrm{IH}}^{\mathrm{S}}$$

(48)

$$M_{\mathrm{IH}}=M_{\mathrm{IH}}^a+M_{\mathrm{IH}}^{\mathrm{S}}$$

(49)

Sliding friction and friction torque of the ball

Contact area of the ball and the screw raceways is defined as $A_{\mathrm{cA}}$ . Pressure $P_A$ is independent of the contact area at point A resistance. ²⁰ Friction of a small element from an arbitrary position $(i,j)$ in the contact area of point A can be expressed as

$${\mathit{f}}_{\mathrm{SA}}(i,j)=-{\mu }_{\mathrm{A}}{\Delta }_{\mathrm{XA}}{\Delta }_{\mathrm{YA}}{p}_{\mathrm{A}}(i,j)\frac{{\mathit{v}}_{\mathrm{AS}}(i,j)}{|{\mathit{v}}_{\mathrm{AS}}(i,j)|}$$

(50)

where $p_A(i,j)$ is the magnitude of pressure distribution $P_A$ in the $(i,j)$ position, ${\Delta }_{\mathrm{xA}}{\Delta }_{\mathrm{yA}}$ is cell area, and $v_{\mathrm{AS}}(i,j)$ is relative sliding velocity vector of the contact element.

Total frictional force on the contact area $A_{\mathrm{cA}}$ at the contact point A can be obtained as

$${\mathit{F}}_{\mathrm{SA}}=\int \underset{A_{\mathrm{cA}}}{\int }{\mathit{f}}_{\mathrm{SA}}=\sum _{i=1}^M\sum _{j=1}^N{\mathit{f}}_{\mathrm{SA}}(i,j),(i,j)\in A_{\mathrm{cA}}$$

(51)

When macro sliding occurs at the point A, total friction moment of the corresponding contact area can be expressed as

$${\mathit{M}}_{\mathrm{SA}}=\int \underset{A_{\mathrm{cA}}}{\int }{\mathit{r}}_{\mathrm{A}}\times {\mathit{f}}_{\mathrm{SA}}=\sum _{i=1}^M\sum _{j=1}^N{\mathit{r}}_{\mathrm{A}}(i,j)\times {\mathit{f}}_{\mathrm{SA}}(i,j),(i,j)\in A_{\mathrm{cA}}$$

(52)

Experimental details

In order to verify the dynamic model of the ball in space subjected to varying angular acceleration of the screw, a test bench for BSM was set up to conduct test. A three-dimensional (3D) schematic along with the experimental setup of the test bench is shown in Figure 4. This test bench can be designed in accordance with actual working conditions. The experimental setup of this test bench has two nuts—the first nut to work, and the second nut to move with the operation platform when the screw is working. However, when the functions are reversed, that is, for the second nut to work and the first nut to move with the operation platform during drag torque test of the BSM, and the #30 lubricating oil was used to lubricate, the drag torque can be obtained at different angular accelerations. During the test, force measurements are made using a tension–compression sensor, and drag torque of BSM can be calculated by the product of the force multiplied by the arm of force.

Figure 4.

The BSM test bench.

Results and discussion

Drag torque of ball screw

Variation of the drag torque as a function of screw speed is shown in Figure 5. It can be observed that drag torque is increased with increase in the screw’s rotational velocity. Additionally, a larger angular acceleration of the screw resulted in greater friction torque of the BSM. The ball need to overcome friction force to drive nut acceleration as acceleration of the screw is increased.

Figure 5.

Variation of drag torque.

Changing regularity of the drag torque is demonstrated to be consistent between measured and simulated BSM data at varying angular acceleration of the screw. According to the dynamic model, angular acceleration was adjusted; however, drag torque of simulation can be compared with measured value.

Contact angles at screw ${\beta }_{\mathrm{A}}$ and nut ${\beta }_{\mathrm{B}}$

Variation of contact angle as a function of screw speed is shown in Figure 6. It can be seen that contact angle formed at the screw and nut is different. This difference is further increased with increase in angular acceleration of the screw. In general, contact angle formed at the screw is larger than that formed at the nut. Along with increasing speed of the screw, under same preload of the BSM, contact angle formed at the screw was increased, whereas contact angle formed at the nut decreased with increase in speed of the screw. Increase in speed results in the centrifugal force affecting the contact angle at the nut to decrease in magnitude. Difference between contact angle at the screw and nut increases with increase in angular acceleration of the screw.

Figure 6.

Variation of contact angle.

Helix angle

Different angular accelerations of the screw generate different helix angles for the BSM, even under similar running conditions. When values for the angular accelerations are substituted into initial helix angle and pitch parameters in Table 1, helix angles under the different running conditions were obtained as shown in Figure 7. Under similar running conditions, helix angle formed at lower angular acceleration always has a higher magnitude compared to larger angular acceleration. Speed and the angular acceleration of BSM are two factors that influence helix angle. In the case, where BSM speed was increased, the ball is far away from the screw axis due to the centrifugal force. This resulted in a significantly reduced helix angle at larger angular accelerations.

Figure 7.

Helix angle.

Radius of the helix trajectory at the ball center

Different angular accelerations of the screw generate different radius of the helix trajectory at the BSM ball center, in spite of similar running conditions being maintained (Figure 8). Under similar running conditions, radius of helix trajectory at the ball center always results in a higher angular acceleration in comparison to small angular acceleration. Speed and angular acceleration of BSM are two factors that influence radius of the helix trajectory at the ball center. In the case, where BSM speed was increased, the ball was far away from the screw axis due to the centrifugal force, which results in a significantly larger radius of the helix trajectory at the ball center.

Figure 8.

Radius of the helix trajectory at the ball center.

Normal force of the ball contact points

Different angular accelerations of the screw generate different normal force at contact points of the ball, in spite of similar running conditions being maintained as shown in Figure 9. In general, normal force formed at contact point A is larger than that formed at contact point B. Along with increasing speed of the screw, under same preload of the BSM, normal force formed at contact point A was decreased, whereas normal force formed at contact point B was caused by the centrifugal force. At the same time, normal force at contact points of the ball always results in a higher angular acceleration in comparison to small angular acceleration due to the inertia force. Speed and angular acceleration of BSM are two factors that influence normal force at contact points of the ball.

Figure 9.

Normal force of the ball contact points.

Motion relation of the ball contact points

As shown in Figures 10 and 11, under similar running conditions, tangential velocity of the ball center and relative slide velocity between the ball and screw at larger angular acceleration of the screw are always the obvious change than at smaller angular acceleration. Collision force between the ball and raceways is increased due to speed and angular acceleration of the screw and causes the relative slide velocity between the ball and screw, and tangential velocity of the ball center to change.

Figure 10.

Linear velocity of the ball center with respect to the raceway of the screw.

Figure 11.

Relative sliding velocity of the ball.

Friction coefficient

Different speeds of the screw generate different friction coefficients of the BSM on the same running conditions. As shown in Figure 12, the friction coefficient increases with the increase of the speed of the BSM. When the screw angular speed is small, the inertia force of the ball is smaller, and the impact force of the ball in the raceway is smaller, and the friction coefficient value is less. However, the friction coefficient increases as the speed of the BSM is larger, due to the ball appearing collision force between the ball and raceways, and it caused contact surface of the ball is larger. However it caused friction coefficient value becoming larger, with the greater angular acceleration of the screw, the larger inertial force that caused by the balls, and it caused contact surface of the ball and screw raceway becoming larger.

Figure 12.

Friction coefficient.

Conclusion

1. Drag torque was found to increase across the whole test stage, and larger angular acceleration of the screw resulted in significant change in drag torque of the BSM. Comparison of simulation and experimental results revealed that the changing regularity of the drag torque was consistent when the screw angular velocity is not more than 1800 r/min. Thus, it can be concluded that dynamic model of the ball in space was accurate and feasible.

2. Increase in velocity and angular acceleration of BSM resulted in a variable tendency of the contact angle, that is, contact angle formed at the screw was different from that at the nut. Contact angle formed at the screw and at the nut exhibited non-linear behavior.

3. Under similar conditions, helix angle formed at large angular acceleration of the screw and higher speeds of screw were always higher compared to smaller angular acceleration of the screw and lower speed. By increasing speed of BSM, helix angle can be significantly reduced.

4. Radius of the helix trajectory at the BSM ball center was likely to be affected by speed and angular acceleration of the screw. Reducing speed or angular acceleration of the screw improved radius of the helix trajectory at the BSM ball center.

5. Speed and angular acceleration of BSM are two factors that influence normal force at contact points of the ball.