Nonlinear dynamic modeling and analysis of ball screws considering ball recirculation process

Chang Liu^a, Jinsong Zhao^b, Weidong Li^a*, Chunyu Zhao^b

a School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China.

b School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China.

* Corresponding author: Weidong Li. E-mail address: weidongli@usst.edu.cn

Abstract

The dynamic performance of the ball screws significantly affects the positioning accuracy of the machine tools, which is mainly related to the ball-groove contact characteristics, and previous studies have ignored the effect of the sudden changes in the number of loaded balls due to the ball recirculation process. To solve this problem, this paper proposes an eight DOF dynamic model of the ball screws that includes the ball recirculation process and screw shaft’s deformation, which captures the transient impacts during the transition section when the balls enter and exit. Firstly, the geometric relationship between the ball and groove centers is determined based on rigid body kinematics and coordinate transformation matrix, and the ball-groove contact loads in the working and transition sections are derived. Then, a nonlinear dynamic model of the ball screws is established by simplifying it to a mass-spring-damping structure, and the correctness of the proposed model is verified by comparing it with the experimental results. Finally, the dynamic characteristics of the ball screws under the effects of the ball recirculation process are investigated.

$\eta =L{\text{ or }}R$

$2\eta$

$\operatorname{int} \left( {{{360i} \mathord{\left/ {\vphantom {{360i} {{\theta _b}}}} \right. \kern-0pt} {{\theta _b}}}} \right)$

$\kappa$

$\zeta$

$\xi$

${\delta _c}$

${\theta _{\text{r}}}$

${\theta _{\text{b}}}$

${\theta _{\eta i}}$

$\rho$

$\omega$

${\alpha _{\eta i}}$

${\alpha _0}$

$x_{{\text{n}}}^{{2\eta }},y_{{\text{n}}}^{{2\eta }},z_{{\text{n}}}^{{2\eta }}$

$x_{{\text{s}}}^{{2\eta }},y_{{\text{s}}}^{{2\eta }},z_{{\text{s}}}^{{2\eta }}$

${T_{\text{b}}}$

${Q_{\eta i}}$

${n_{\text{s}}}$

${L_{OW}}$

${L_{\eta {\text{b}}i}}$

${L_W}$

${L_{\text{p}}}$

${L_{\text{t}}}$

${L_{\text{S}}}$

${H_W}$

${E_{\text{S}}}$

${c_u},{c_\varepsilon }$

${A^{\prime}_{\eta i}}$

${A_{\text{S}}}$

Keywords:

Ball screws

Dynamic model

Ball recirculation process

Screw shaft’s deformation Contact characteristics

Nomenclature

A₀

Initial distance of the groove centers.

Cross-section area of the screw shaft.

Distance between the groove centers after loading.

Viscous damping coefficients caused by the ball screws.

Elasticity modulus.

${f_{ux}}$

${f_{uy}}$

${f_{uz}}$

Resultant forces of the ball screws in the x-, y-, and z-axes.

${F_x}$

${F_y}$

${F_z}$

Excitation amplitudes in the x-, y- and z-directions.

Distance between the

${\text{C}}{{\text{S}}_O}$

and the

${\text{C}}{{\text{S}}_W}$

in the x-axis.

${I_x}$

${I_y}$

Moment of inertia of the worktable around the x- and y-axes.

${k_{Sux}}$

${k_{Suy}}$

${k_{Suz}}$

Linear stiffness of the screw shaft in the x-, y-, and z-directions.

${k_{S\varepsilon x}}$

${k_{S\varepsilon y}}$

Angular stiffness of the screw shaft around the x- and y-axes.

Length of the screw shaft.

Length of the transition section.

Pitch of the screw shaft.

Distance of the worktable movement when the screw shaft rotates.

Axial displacement of ith ball center in the

${\text{C}}{{\text{S}}_S}$

Distance between the

${\text{C}}{{\text{S}}_O}$

and the

${\text{C}}{{\text{S}}_W}$

in the z-axis.

Total mass of the worktable and nut.

Rotation speed of the screw shaft.

Number of the loaded balls.

Contact load between the ball and groove.

${R_{\text{s}}}$

${R_{\text{n}}}$

${R_{\text{b}}}$

${R_{\text{m}}}$

Radius of the screw and nut grooves, ball, pitch.

Time for the ball to roll angle between adjacent balls.

${v_{\text{n}}}$

${v_{\text{s}}}$

Velocities of the contact points of the ball-nut and screw grooves.

$x_{{\eta {\text{s}}i}}^{S}$

$y_{{\eta {\text{s}}i}}^{S}$

$z_{{\eta {\text{s}}i}}^{S}$

Coordinates of the initial position for the screw’s groove center in the

${\text{C}}{{\text{S}}_S}$

$x_{{\eta {\text{n}}i}}^{W}$

$y_{{\eta {\text{n}}i}}^{W}$

$z_{{\eta {\text{n}}i}}^{W}$

Coordinates of the initial position for the nut’s groove center in the

${\text{C}}{{\text{S}}_W}$

$x_{{\eta {\text{s}}i}}^{O}$

$y_{{\eta {\text{s}}i}}^{O}$

$z_{{\eta {\text{s}}i}}^{O}$

Coordinates of the initial position for the screw’s groove center in the

${\text{C}}{{\text{S}}_O}$

$x_{{\eta {\text{n}}i}}^{O}$

$y_{{\eta {\text{n}}i}}^{O}$

$z_{{\eta {\text{n}}i}}^{O}$

Coordinates of the initial position for the nut’s groove center in the

${\text{C}}{{\text{S}}_O}$

Coordinates of the screw’s groove centers in the

${\text{C}}{{\text{S}}_{2\eta }}$

Coordinates of the nut’s groove centers in the

${\text{C}}{{\text{S}}_{2\eta }}$

Initial contact angle between the ball and groove.

Contact angle between the ball and groove after loading.

Excitation frequency.

${\omega _{\text{s}}}$

${\omega _{\text{b}}}$

Angular velocity of the screw shaft and ball center.

Pitch angle of the ball screws.

Main curvature radius of the screw groove.

Position angles of the ith ball center.

Angle between adjacent balls.

Remaining angle.

Elastic release of the ball.

${\delta _{Wx}}$

${\delta _{Wy}}$

${\delta _{Wz}}$

Linear displacements of the worktable.

${\delta _{Sx}}$

${\delta _{Sy}}$

${\delta _{Sz}}$

linear displacements of the screw shaft.

${\varepsilon _{Sx}}$

${\varepsilon _{Sy}}$

Angular displacements of the screw shaft.

Damping ratios.

Dimensionless position of the worktable.

Hertz contact constants.

Integer of

$360i$

divided by

${\theta _{\text{b}}}$

Superscripts

Coordinate systems

${\text{C}}{{\text{S}}_O}$

Coordinate systems

${\text{C}}{{\text{S}}_S}$

Coordinate systems

${\text{C}}{{\text{S}}_W}$

Coordinate systems

${\text{C}}{{\text{S}}_{2\eta }}$

Subscripts

Number of the loaded ball.

s, n

Screw and nut grooves.

x, y, z

x-, y-, and z-direction.

Left or right section of the nut.

1. Introduction

Ball screw mechanisms are widely used in CNC machine tools, industrial robots, aerospace, etc., and have the ability to convert rotary motion into linear motion efficiently, accurately and repeatably [1–3]. The dynamic performance of the ball screws in ultra-precision machining processes has become a critical factor affecting workpiece quality, particularly during the circulation process where the balls move periodically [4, 5]. The periodic entry and exit of the balls in the recirculation channel of the nut leads to sudden changes in contact load, time-varying stiffness, and impact excitation, which in turn cause nonlinear vibration and affect the motion accuracy [6, 7]. Therefore, in-depth research on the dynamic characteristics of the ball screws, taking into account the ball recirculation process is of great significance for improving the positioning accuracy of the machine tools.

Some researchers have developed dynamic models of the ball screws to analyze their dynamic characteristics, which mainly focus on determining the ball-groove contact characteristics. Xu et al. [8] developed a single degree-of-freedom (DOF) dynamic model of the ball screws that included the bearing joint, screw-nut joint, and screw shaft. Zhao et al. [9] developed a two DOF dynamic model of the ball screws considering clearance and friction to investigate the dynamic characteristics. The elastic deformation of the screw shaft affects the center position of the screw groove, which in turn affects the load distribution [10–12]. Okwudire [13] derived the stiffness matrix in the screw-nut interface, and used an optimized nut design to reduce the bending vibration caused by torque. Liu et al. [14] established a five DOF dynamic model of the ball screws, and using bifurcation diagrams, three-dimensional spectra, phase diagrams, and Poincaré sections to discuss the influences of the system parameters on the dynamic behaviors. Gao et al. [15] established a lateral equivalent dynamic model of the ball screw feed system based on the Euler-Bernoulli beam theory, incorporating the screw shaft's bending vibration, worktable displacement, and joint effects. In highly accelerated ball screws, the large inertial forces generated by the moving parts may change the actual contact state of the system's kinematic joints [16, 17]. Zhang et al. [18] developed a dynamic model of the ball screw feed system that takes into account acceleration, deriving the equivalent axial stiffness of the ball screws and bearing, and this model is also verified experimentally. On this basis, Zhou et al. [19] developed a dynamic model for the vertical ball screw feed systems, systematically investigating the effects of the spindle system position, rated dynamic load, and screw tension force on the system's natural frequency during the acceleration and deceleration. During the high-speed and heavy-duty operation of the machine tools, the vibration and friction of the rolling joints have a significant impact on the dynamic performance [20–22]. Wang et al. [23] established a friction-vibration coupling model for the feed system, and analyzed the influences of the excitation force, feed acceleration, and eccentricity on the vibration and friction characteristics. However, most of the above models only consider the nut at a specific position, and ignore the effects of the ball recirculation process. In addition, to fulfill the performance requirements of a small-mass, high-stiffness structure components and high-adjustable controller parameters of the machine tools, Wu et al. [24, 25] investigated the dynamics and control of a planar three DOF parallel manipulator with actuation redundancy, and integrated the redundant actuated parallel manipulator into a four DOF hybrid machine tool. Based on this, the mechatronics modeling and forced vibration of a parallel manipulator are investigated by using the bond graph method [26].

When the balls circulate in the working chamber, those in the loaded section of the ball screws are squeezed by the preload or external load, but the balls in the return channel are free. However, there are very few research on the ball recirculation process, Wei and Lai [27] proposed the kinematic theory of high-speed preloaded ball screws, and calculated the mechanical efficiency according to the theoretical driving torque, axial load, and ball angular velocity. Alberdi et al. [28] redefined the kinematics of the ball screws by considering the of transversal velocity component introduced by the helix angle, and demonstrated the superiority of the proposed method by comparing it with the results of Wei et al.'s model [29]. To discuss the dynamic behavior of the ball screws during the ball rolling cycle, Xie et al. [30] developed an accurate dynamic model of the ball screw feed system by considering the travel deviation and ball motion, and explained the vibration mechanism of the feed system during the ball recirculation process. Obviously, the load fluctuations generated by the balls entering and exiting the screw-nut joints, which in turn causes nonlinear dynamic behavior [31]. Xu et al. [32] established a five DOF dynamic model of the linear guideway, which considers the time-varying contact load under the periodic variation of the roller position, and discussed the effects of the load size and direction, preload and velocity on the time-varying displacement and friction of five DOF. Li et al. [33] and Xu et al. [34] proposed a time-varying dynamic model of the linear guideway with crowning considering the ball recirculation process, enabling vibration characteristic analysis under the constant loads and linear velocity. Overall, the rotation of the screw shaft and the axial and bending deformations of the screw shaft will change the ball-groove contact state. However, most studies usually assume that the worktable is in a fixed position, and ignore the effect of the sudden changes in the number of loaded balls caused by the ball recirculation process on the dynamic behaviors of the ball screws.

This study proposes a method for predicting the time-varying and nonlinear dynamic behaviors of the ball screws. The method takes into account the impact force caused by the sudden change in the number of loaded balls due to the ball recirculation process, and establishes a new eight DOF dynamic model that considers the deformations of the screw shaft. The periodic changes in the worktable displacement caused by the ball entering and exiting the transition section are accurately calculated, the effects of the external load, worktable position, initial contact angle and preload on dynamic characteristics of the worktable in the feed direction are discussed, and the results of the proposed method are compared with experimental ones for verification.

2. Theoretical model

The screw shaft’s deformations (axial, bending, and rotational deformations) and worktable movement can affect the ball-groove contact state of the ball screws, resulting in changes in the dynamic characteristics of the feed system. To reduce the impact load caused by the ball entering and exiting the screw-nut contact area when the ball screws are in operation, a transition section is added at both ends of the contact area to eliminate sudden changes in the contact load, as shown in Fig. 1. Furthermore, the screw-nut circulation channel can be divided into working, transition, and return sections. When the screw shaft is rotated at an angular velocity

${\omega _{\text{s}}}$

in a counter-clockwise direction, the balls on the left and right parts of the nut roll in a clockwise direction. In this paper, the global coordinate system

${\text{C}}{{\text{S}}_O}\left( {{x_O},{y_O},{z_O}} \right)$

is located at the cross-section center at the left end of the screw shaft, and the z-axis coincides with the screw shaft’s axis. The coordinate systems

${\text{C}}{{\text{S}}_W}\left( {{x_W},{y_W},{z_W}} \right)$

and

${\text{C}}{{\text{S}}_S}\left( {{x_S},{y_S},{z_S}} \right)$

are positioned at the geometric center of the worktable and the center of the screw axis in the screw-nut interface, respectively, and their axes are parallel to the axes of the

${\text{C}}{{\text{S}}_O}$

During the operation of the ball screws, the servo motor drives continuous rotation of the screw shaft, and the nut is fixed on the worktable by bolts and moves in a straight line. As the nut does not rotate, the velocity of the ball-nut groove contact point is zero. The velocities of the contact points (

${v_{\text{n}}}$

${v_{\text{s}}}$

) of the ball-nut and screw grooves can be expressed as

$\left\{ \begin{gathered} {v_{\text{n}}}=0 \hfill \\ {v_{\text{s}}}={\omega _{\text{s}}}\left( {{R_{\text{m}}} - {R_{\text{b}}}\cos {\alpha _0}} \right) \hfill \\ \end{gathered} \right.$

where

${\omega _{\text{s}}}$

is the angular velocity of the screw shaft;

${R_{\text{m}}}$

is the pitch radius of the ball screws;

${R_{\text{b}}}$

is the ball radius;

${\alpha _0}$

is the initial contact angle.

Fig. 1

Motion model of the ball screws in the feed system.

According to rigid body kinematics, the velocity of the ball center is

${v_{\text{b}}}=0.5{v_{\text{s}}}$

. The angular velocity

${\omega _{\text{b}}}$

at which the ball center rotates around the screw axis is expressed as

${\omega _{\text{b}}}=\frac{{{v_{\text{b}}}}}{{{R_{\text{m}}}}}=\frac{{{\text{\varvec{\uppi}}}{n_{\text{s}}}\left( {{R_{\text{m}}} - {R_{\text{b}}}\cos {\alpha _0}} \right)}}{{60{R_{\text{m}}}}}$

where

${n_{\text{s}}}$

is the rotation speed of the screw shaft.

At time t, the position angles (

${\theta _{Li}}$

${\theta _{Ri}}$

) of the ith ball center for the left and right sections of the nut can be expressed as

${\theta _i}={\theta _{Li}}={\theta _{Ri}}=\left( {i - 1} \right){\theta _{\text{b}}}+{\omega _{\text{b}}}\bmod \left( {t,{T_{\text{b}}}} \right)$

where i is the ball number;

${\theta _{\text{b}}}$

is the angle between adjacent balls;

${T_{\text{b}}}$

is the time for the ball to roll angle between adjacent balls;

$\bmod \left( {t,{T_{\text{b}}}} \right)$

is the remainder of t divided by

${T_{\text{b}}}$

2.1 Ball-groove contact load in the working section

In order to facilitate the derivation of the contact load, two coordinate systems are established: the coordinate systems

${\text{C}}{{\text{S}}_{1\eta }}({x_{1\eta }},{y_{1\eta }},{z_{1\eta }})$

and

${\text{C}}{{\text{S}}_{2\eta }}({x_{2\eta }},{y_{2\eta }},{z_{2\eta }})$

, where the subscript

$\eta =L{\text{ or }}R$

represents the left or right section of the nut, respectively. The

${\text{C}}{{\text{S}}_{1\eta }}$

is fixed at the screw axis corresponding to the ball center, its each axis is parallel to the axis of the

${\text{C}}{{\text{S}}_O}$

. The

${\text{C}}{{\text{S}}_{2\eta }}$

is fixed at the ball center, its

${y_{2\eta }}$

-axis is tangent to the trajectory of the ball center, and the angle between the

${y_{2\eta }}$

-axis and the

${z_{1\eta }}$

-axis is

$\lambda$

, as shown in Fig. 2, where the points A_η and B_η are the ball-screw and nut grooves contact points, respectively.

The coordinate transformation matrix between two coordinates

${\text{C}}{{\text{S}}_{1\eta }}$

and

${\text{C}}{{\text{S}}_{2\eta }}$

can be obtained by three transformations. Firstly, The

${\text{C}}{{\text{S}}_{1\eta }}$

is translated by

${R_{\text{m}}}\cos {\theta _i}$

and

${R_{\text{m}}}\sin {\theta _i}$

${x_{1\eta }}$

- and

${y_{1\eta }}$

-axes to coincide with the two origins of the

${\text{C}}{{\text{S}}_{1\eta }}$

and

${\text{C}}{{\text{S}}_{2\eta }}$

, which is marked as

$Trans({R_{\text{m}}}\cos {\theta _i},{R_{\text{m}}}\sin {\theta _i},0)$

. Secondly, the current-frame is rotated

$({\theta _i}+{\text{\varvec{\uppi}}}/2)^\circ$

around the

${z_{1\eta }}$

-axis, so that

${x_{1\eta }}$

- and

${y_{1\eta }}$

-axes coincide with

${x_{2\eta }}$

- and

${y_{2\eta }}$

-axes, which is marked as

$Ro{t_z}({\theta _i}+{\text{\varvec{\uppi}}}/2)$

. Thirdly, the current-frame is rotated

$(2{\text{\varvec{\uppi}}} - {\alpha _0})^\circ$

around

${y_{1\eta }}$

-axis, which is marked as

$Ro{t_y}(2{\text{\varvec{\uppi}}} - {\alpha _0})$

. The coordinate transformation matrix from

${\text{C}}{{\text{S}}_{1\eta }}$

${\text{C}}{{\text{S}}_{2\eta }}$

is marked as

${{\mathbf{T}}_{1\eta - 2\eta }}$

, which is

${{\mathbf{T}}_{1\eta - 2\eta }}=Trans({R_{\text{m}}}\cos {\theta _i},{R_{\text{m}}}\sin {\theta _i},0) \cdot Ro{t_z}({\theta _i}+{\text{\varvec{\uppi}}}/2) \cdot Ro{t_y}(2{\text{\varvec{\uppi}}} - {\alpha _0})$

Fig. 2

Relationship between different coordinate systems.

The coordinates of the groove centers for the left and right sections of the nut in the

${\text{C}}{{\text{S}}_{2\eta }}$

are expressed as

$\left\{ {\begin{array}{*{20}{c}} {\left[ {x_{{\text{s}}}^{{2L}},y_{{\text{s}}}^{{2L}},z_{{\text{s}}}^{{2L}}} \right]=\left[ {0, - ({R_{\text{s}}} - {R_{\text{b}}})\cos {\alpha _0}, - ({R_{\text{s}}} - {R_{\text{b}}})\sin {\alpha _0}} \right]} \\ {\left[ {x_{{\text{n}}}^{{2L}},y_{{\text{n}}}^{{2L}},z_{{\text{n}}}^{{2L}}} \right]=\left[ {0,({R_{\text{n}}} - {R_{\text{b}}})\cos {\alpha _0},({R_{\text{n}}} - {R_{\text{b}}})\sin {\alpha _0}} \right]} \end{array}} \right.$

$\left\{ {\begin{array}{*{20}{c}} {\left[ {x_{{\text{s}}}^{{2R}},y_{{\text{s}}}^{{2R}},z_{{\text{s}}}^{{2R}}} \right]=\left[ {0, - ({R_{\text{s}}} - {R_{\text{b}}})\cos {\alpha _0},({R_{\text{s}}} - {R_{\text{b}}})\sin {\alpha _0}} \right]} \\ {\left[ {x_{{\text{n}}}^{{2R}},y_{{\text{n}}}^{{2R}},z_{{\text{n}}}^{{2R}}} \right]=\left[ {0,({R_{\text{s}}} - {R_{\text{b}}})\cos {\alpha _0}, - ({R_{\text{s}}} - {R_{\text{b}}})\sin {\alpha _0}} \right]} \end{array}} \right.$

where

${R_{\text{s}}}$

is the radius of the screw groove; superscripts 2L and 2R are the coordinate systems

${\text{C}}{{\text{S}}_{2L}}$

and

${\text{C}}{{\text{S}}_{2R}}$

, respectively; subscripts s and n are the screw and nut grooves, respectively.

According to the transformation matrix

${{\mathbf{T}}_{1\eta - 2\eta }}$

and Eqs. (4) and (5), the initial position of the screw shaft’s groove center for the left and right sections of the nut in the

${\text{C}}{{\text{S}}_S}$

is derived as

$\left\{ {\begin{array}{*{20}{l}} {x_{{L{\text{s}}i}}^{S}=\left( {{R_s} - {R_b}} \right)\cos {\alpha _0}\cos {\theta _i}+\left( {{R_s} - {R_b}} \right)\sin {\alpha _0}\sin {\theta _i}\sin \lambda +{R_{\text{m}}}\cos {\theta _i}} \\ {y_{{L{\text{s}}i}}^{S}=\left( {{R_s} - {R_b}} \right)\cos {\alpha _0}\sin {\theta _i} - \left( {{R_s} - {R_b}} \right)\sin {\alpha _0}\cos {\theta _i}\sin \lambda +{R_{\text{m}}}\sin {\theta _i}} \\ {z_{{L{\text{s}}i}}^{S}=\left( {{R_s} - {R_b}} \right)\sin {\alpha _0}\cos \lambda +{L_{L{\text{b}}i}}} \end{array}} \right.$

$\left\{ {\begin{array}{*{20}{l}} {x_{{R{\text{s}}i}}^{S}=\left( {{R_{\text{s}}} - {R_{\text{b}}}} \right)\cos {\alpha _0}\cos {\theta _i} - \left( {{R_{\text{s}}} - {R_{\text{b}}}} \right)\sin {\alpha _0}\sin {\theta _i}\sin \lambda +{R_{\text{m}}}\cos {\theta _i}} \\ {y_{{R{\text{s}}i}}^{S}=\left( {{R_{\text{s}}} - {R_{\text{b}}}} \right)\cos {\alpha _0}\sin {\theta _i}+\left( {{R_{\text{s}}} - {R_{\text{b}}}} \right)\sin {\alpha _0}\cos {\theta _i}\sin \lambda +{R_{\text{m}}}\sin {\theta _i}} \\ {z_{{R{\text{s}}i}}^{S}= - \left( {{R_{\text{s}}} - {R_{\text{b}}}} \right)\sin {\alpha _0}\cos \lambda +{L_{R{\text{b}}i}}} \end{array}} \right.$

where superscripts S represents the coordinate system

${\text{C}}{{\text{S}}_S}$

;

${L_{\eta {\text{b}}i}}$

is the displacement of ith ball center in the z-axis direction of the

${\text{C}}{{\text{S}}_S}$

Similarly, the initial position of the nut’s groove center for the left and right sections of the nut in the

${\text{C}}{{\text{S}}_W}$

is derived as

$\left\{ {\begin{array}{*{20}{l}} {x_{{L{\text{n}}i}}^{W}= - \left( {{R_{\text{n}}} - {R_{\text{b}}}} \right)\cos {\alpha _0}\cos {\theta _i} - \left( {{R_{\text{n}}} - {R_{\text{b}}}} \right)\sin {\alpha _0}\sin {\theta _i}\sin \lambda +{R_{\text{m}}}\cos {\theta _i} - {H_W}} \\ {y_{{L{\text{n}}i}}^{W}= - \left( {{R_{\text{n}}} - {R_{\text{b}}}} \right)\cos {\alpha _0}\sin {\theta _i}+\left( {{R_{\text{n}}} - {R_{\text{b}}}} \right)\sin {\alpha _0}\cos {\theta _i}\sin \lambda +{R_{\text{m}}}\sin {\theta _i}} \\ {z_{{L{\text{n}}i}}^{W}= - \left( {{R_{\text{n}}} - {R_{\text{b}}}} \right)\sin {\alpha _0}\cos \lambda +{L_{L{\text{b}}i}}} \end{array}} \right.$

$\left\{ {\begin{array}{*{20}{l}} {x_{{R{\text{n}}i}}^{W}= - \left( {{R_{\text{n}}} - {R_{\text{b}}}} \right)\cos {\alpha _0}\cos {\theta _i}+\left( {{R_{\text{n}}} - {R_{\text{b}}}} \right)\sin {\alpha _0}\sin {\theta _i}\sin \lambda +{R_{\text{m}}}\cos {\theta _i} - {H_W}} \\ {y_{{R{\text{n}}i}}^{W}= - \left( {{R_{\text{s}}} - {R_{\text{b}}}} \right)\cos {\alpha _0}\sin {\theta _i} - \left( {{R_{\text{s}}} - {R_{\text{b}}}} \right)\sin {\alpha _0}\cos {\theta _i}\sin \lambda +{R_{\text{m}}}\sin {\theta _i}} \\ {z_{{R{\text{n}}i}}^{W}=\left( {{R_{\text{s}}} - {R_{\text{b}}}} \right)\sin {\alpha _0}\cos \lambda +{L_{R{\text{b}}i}}} \end{array}} \right.$

where

${H_W}$

is the distance in the x-direction between the

${\text{C}}{{\text{S}}_O}$

and the

${\text{C}}{{\text{S}}_W}$

The worktable posture refers to the displacements generated during the worktable movement, which can be described as three linear displacements (

${\delta _{Wx}}$

${\delta _{Wy}}$

${\delta _{Wz}}$

). Assuming that the screw shaft segment in the screw-nut interface is a rigid body [35], the screw shaft undergoes axial and bending deformations due to the contact loads, with a total of five DOF that vary with the worktable position, as shown in Fig. 3. where

${\varepsilon _{Sx}}$

and

${\varepsilon _{Sy}}$

are the angular displacements around the x- and y-axes, and

${\delta _{Sx}}$

${\delta _{Sy}}$

, and

${\delta _{Sz}}$

are the linear displacements in the x-, y-, and z-axes. The displacements of the worktable and screw shaft cause the position of the groove center to change, the coordinates of the groove centers under the displacements of the worktable and screw shaft in the

${\text{C}}{{\text{S}}_O}$

can be expressed as

$\left\{ {\begin{array}{*{20}{l}} {x_{{L{\text{n}}i}}^{O}={H_W}+{\delta _{Wx}}+x_{{L{\text{n}}i}}^{W}} \\ {y_{{L{\text{n}}i}}^{O}={\delta _{Wy}}+y_{{L{\text{n}}i}}^{W}} \\ {z_{{L{\text{n}}i}}^{O}={L_{OW}}+{\delta _{Wz}}+z_{{L{\text{n}}i}}^{W}} \end{array}} \right.$

$\left\{ {\begin{array}{*{20}{l}} {x_{{R{\text{n}}i}}^{O}={H_W}+{\delta _{Wx}}+x_{{R{\text{n}}i}}^{W}} \\ {y_{{R{\text{n}}i}}^{O}={\delta _{Wy}}+y_{{R{\text{n}}i}}^{W}} \\ {z_{{R{\text{n}}i}}^{O}={L_{OW}}+{\delta _{Wz}}+z_{{R{\text{n}}i}}^{W}} \end{array}} \right.$

$\left\{ {\begin{array}{*{20}{l}} {x_{{L{\text{s}}i}}^{O}=x_{{L{\text{s}}i}}^{S}+z_{{L{\text{s}}i}}^{S}{\varepsilon _{Sy}}+{\delta _{Sx}}} \\ {y_{{L{\text{s}}i}}^{O}=y_{{L{\text{s}}i}}^{S}+z_{{L{\text{s}}i}}^{S}{\varepsilon _{Sx}}+{\delta _{Sy}}} \\ {z_{{L{\text{s}}i}}^{O}= - x_{{L{\text{s}}i}}^{S}{\varepsilon _{Sy}}+y_{{L{\text{s}}i}}^{S}{\varepsilon _{Sx}}+{\delta _{Wz}}+z_{{L{\text{n}}i}}^{W}+{L_{OW}}} \end{array}} \right.$

$\left\{ {\begin{array}{*{20}{l}} {x_{{R{\text{s}}i}}^{O}=x_{{R{\text{s}}i}}^{S}+z_{{R{\text{s}}i}}^{S}{\varepsilon _{Sy}}+{\delta _{Sx}}} \\ {y_{{R{\text{s}}i}}^{O}=y_{{R{\text{s}}i}}^{S}+z_{{R{\text{s}}i}}^{S}{\varepsilon _{Sx}}+{\delta _{Sy}}} \\ {z_{{R{\text{s}}i}}^{O}= - x_{{R{\text{s}}i}}^{S}{\varepsilon _{Sy}}+y_{{R{\text{s}}i}}^{S}{\varepsilon _{Sx}}+{\delta _{Wz}}+z_{{R{\text{n}}i}}^{W}+{L_{OW}}} \end{array}} \right.$

where superscript O is the coordinate system

${\text{C}}{{\text{S}}_O}$

;

${L_{OW}}$

is the distance in z-axis between the

${\text{C}}{{\text{S}}_O}$

and the

${\text{C}}{{\text{S}}_W}$

Fig. 3

Axial and bending deformations of the screw shaft.

Based on the geometric relationship of the groove centers illustrated in Fig. 4 and by combining Eqs. (10)–(13), the distances between the groove centers during the movement of the worktable for the left and right sections of the nut can be derived as

${A^{\prime}_{Li}}=\sqrt {{{\left( {\left( {x_{{L{\text{s}}i}}^{O} - x_{{L{\text{n}}i}}^{O}} \right)\cos {\theta _i}+\left( {y_{{L{\text{s}}i}}^{O} - y_{{L{\text{n}}i}}^{O}} \right)\sin {\theta _i}} \right)}^2}+{{\left( {z_{{L{\text{s}}i}}^{O} - z_{{L{\text{n}}i}}^{O}} \right)}^2}}$

${A^{\prime}_{Ri}}=\sqrt {{{\left( {\left( {x_{{R{\text{s}}i}}^{O} - x_{{R{\text{n}}i}}^{O}} \right)\cos {\theta _i}+\left( {y_{{R{\text{s}}i}}^{O} - y_{{R{\text{n}}i}}^{O}} \right)\sin {\theta _i}} \right)}^2}+{{\left( {z_{{R{\text{s}}i}}^{O} - z_{{R{\text{n}}i}}^{O}} \right)}^2}}$

When the sum of the distance

${A^{\prime}_{\eta i}}$

and the contact deformation

${\delta _0}$

is greater than the initial distance

${A_0}$

, the contact load

${Q_{\eta i}}$

is zero. The contact loads on the left and right sections of the nut can be derived from the Hertz contact theory [36, 37], as follows

$\left\{ {\begin{array}{*{20}{c}} {{Q_{Li}}={{\left( {\frac{{{{A^{\prime}}_{Li}} - {A_0}+{\delta _0}}}{{2{\kappa _{{\text{sn}}}}}}} \right)}^{\frac{3}{2}}}{\text{ }}{{A^{\prime}}_{Li}}+{\delta _0}>{A_0}} \\ {{\text{ }}{Q_{Li}}=0{\text{ }}{{A^{\prime}}_{Li}}+{\delta _0} \leqslant {A_0}} \end{array}} \right.$

$\left\{ {\begin{array}{*{20}{c}} {{Q_{Ri}}={{\left( {\frac{{{{A^{\prime}}_{Ri}} - {A_0}+{\delta _0}}}{{2{\kappa _{{\text{sn}}}}}}} \right)}^{\frac{3}{2}}}{\text{ }}{{A^{\prime}}_{Ri}}+{\delta _0}>{A_0}} \\ {{\text{ }}{Q_{Ri}}=0{\text{ }}{{A^{\prime}}_{Ri}}+{\delta _0} \leqslant {A_0}} \end{array}} \right.$

where

${\delta _0}$

is the contact deformation under the preload

${Q_0}$

${\kappa _{{\text{sn}}}}$

is the Hertz contact constant.

From the geometric relationships of the groove centers, the contact angles for the left and right sections of the nut are derived as

${\alpha _{Li}}={\cos ^{ - 1}}\left( {\frac{{\left( {x_{{L{\text{s}}i}}^{O} - x_{{L{\text{n}}i}}^{O}} \right)\cos {\theta _i}+\left( {y_{{L{\text{s}}i}}^{O} - y_{{L{\text{n}}i}}^{O}} \right)\sin {\theta _i}}}{{{{A^{\prime}}_{Li}}}}} \right)$

${\alpha _{Ri}}={\cos ^{ - 1}}\left( {\frac{{\left( {x_{{R{\text{s}}i}}^{O} - x_{{R{\text{n}}i}}^{O}} \right)\cos {\theta _i}+\left( {y_{{R{\text{s}}i}}^{O} - y_{{R{\text{n}}i}}^{O}} \right)\sin {\theta _i}}}{{{{A^{\prime}}_{Ri}}}}} \right)$

Fig. 4

Geometric relationship of the groove centers. (a) Under no external load, (b) Under external load

2.2 Ball-groove contact load in the transition section

To reduce the impact load caused by the ball entering and exiting from the loaded area, the transition section is added at both ends of the contact area. Due to the difficulty in aligning the centers of the first and last balls with the two ends, assuming that the first ball is aligned with the inlet end, and there is an angle between the Nth ball and the outlet end, called the remaining angle

${\theta _{\text{r}}}$

(see Fig. 5), as follows

${\theta _{\text{r}}}=360i - \operatorname{int} \left( {\frac{{360i}}{{{\theta _{\text{b}}}}}} \right){\theta _{\text{b}}}$

where

$\operatorname{int} \left( {{{360i} \mathord{\left/ {\vphantom {{360i} {{\theta _{\text{b}}}}}} \right. \kern-0pt} {{\theta _{\text{b}}}}}} \right)$

is the integer of

$360i$

divided by

${\theta _{\text{b}}}$

Fig. 5

Ball circulation process in the ball screws.

The first and last balls are in the transition section as shown in Fig. 5, the contact deformation in the first and last transition sections increases and decreases gradually when the screw shaft rotates. The rolling process from the 0th ball to the 1th ball is divided into two stages: the return section and transition section, the elastic release

${\delta _{c01}}$

is expressed as

${\delta _{c01}}=\left\{ {\begin{array}{*{20}{c}} {{\text{ }}{\delta _{k1}}{\text{ 0}} \leqslant \bmod \left( {t,{T_{\text{b}}}} \right)<\frac{{{\theta _{\text{b}}}}}{{{\omega _{\text{b}}}}} - \frac{{{L_{\text{t}}}}}{{{{{v_{\text{b}}}} \mathord{\left/ {\vphantom {{{v_{\text{b}}}} {\cos \lambda }}} \right. \kern-0pt} {\cos \lambda }}}}} \\ {\rho - \sqrt {{\rho ^2} - {{\left( {{{{v_{\text{b}}}} \mathord{\left/ {\vphantom {{{v_{\text{b}}}} {\cos \lambda }}} \right. \kern-0pt} {\cos \lambda }}\left( {{T_{\text{b}}} - \bmod \left( {t,{T_{\text{b}}}} \right)} \right)} \right)}^2}} {\text{ }}\frac{{{\theta _{\text{b}}}}}{{{\omega _{\text{b}}}}} - \frac{{{L_{\text{t}}}}}{{{{{v_{\text{b}}}} \mathord{\left/ {\vphantom {{{v_{\text{b}}}} {\cos \lambda }}} \right. \kern-0pt} {\cos \lambda }}}} \leqslant \bmod \left( {t,{T_{\text{b}}}} \right)<\frac{{{\theta _{\text{b}}}}}{{{\omega _{\text{b}}}}}} \end{array}} \right.$

where

${\delta _{k1}}$

is the 1th ball-groove elastic deformation;

${L_{\text{t}}}$

is the length of the transition section;

$\rho$

is the main curvature radius of the screw groove;

${{{\theta _{\text{b}}}} \mathord{\left/ {\vphantom {{{\theta _{\text{b}}}} {{\omega _{\text{b}}}}}} \right. \kern-0pt} {{\omega _{\text{b}}}}}$

and

${{{L_{\text{t}}}\cos \lambda } \mathord{\left/ {\vphantom {{{L_{\text{t}}}\cos \lambda } {{v_{\text{b}}}}}} \right. \kern-0pt} {{v_{\text{b}}}}}$

are the time for the ball to pass through the angle

${\theta _{\text{b}}}$

and distance

${L_{\text{t}}}$

, respectively.

When the last ball passes through the transition section, there are two cases. Case 1: The (N + 1)th and (N + 2)th balls are in and out of the transition section, respectively. Case 2: The (N + 1)th ball is not in the transition section. For the case 1, the elastic release

${\delta _{cN(N+1)}}$

from the Nth ball to the (N + 1)th ball can be divided into two stages, which is expressed as

where

${{{\theta _{\text{r}}}} \mathord{\left/ {\vphantom {{{\theta _{\text{r}}}} {{\omega _{\text{b}}}}}} \right. \kern-0pt} {{\omega _{\text{b}}}}}$

is the time for the ball to pass through the remaining angle

${\theta _{\text{r}}}$

For the Case 2, the elastic release

${\delta _{cN(N+1)}}$

from the Nth ball to the (N + 1)th ball can be divided into three stages, which is expressed as

${\delta _{cN(N+1)}}=\left\{ {\begin{array}{*{20}{c}} {{\text{ 0 0}} \leqslant \bmod \left( {t,{T_{\text{b}}}} \right)<\frac{{{\theta _{\text{r}}}}}{{{\omega _{\text{b}}}}}} \\ {\rho - \sqrt {{\rho ^2} - {{\left( {{{{v_{\text{b}}}} \mathord{\left/ {\vphantom {{{v_{\text{b}}}} {\cos \lambda }}} \right. \kern-0pt} {\cos \lambda }}\left( {\bmod \left( {t,{T_{\text{b}}}} \right) - {T_{\text{r}}}} \right)} \right)}^2}} {\text{ }}\frac{{{\theta _{\text{r}}}}}{{{\omega _{\text{b}}}}} \leqslant \bmod \left( {t,{T_{\text{b}}}} \right)<\frac{{{L_{\text{t}}}}}{{{{{v_{\text{b}}}} \mathord{\left/ {\vphantom {{{v_{\text{b}}}} {\cos \lambda }}} \right. \kern-0pt} {\cos \lambda }}}}+\frac{{{\theta _{\text{r}}}}}{{{\omega _{\text{b}}}}}} \\ {{\text{ }}{\delta _{kN}}{\text{ }}\frac{{{L_{\text{t}}}}}{{{{{v_{\text{b}}}} \mathord{\left/ {\vphantom {{{v_{\text{b}}}} {\cos \lambda }}} \right. \kern-0pt} {\cos \lambda }}}}+\frac{{{\theta _{\text{r}}}}}{{{\omega _{\text{b}}}}} \leqslant \bmod \left( {t,{T_{\text{b}}}} \right)<\frac{{{\theta _{\text{b}}}}}{{{\omega _{\text{b}}}}}} \end{array}} \right.$

where

${\delta _{kN}}$

is the Nth ball-groove elastic deformation.

Since the (N + 1)th ball is not in the transition section, the elastic release

${\delta _{c\invalidcharacter N+1\invalidcharacter (N+2)}}$

from the (N + 1)th ball to the (N + 2)th ball for the Case 2 is zero, the elastic release

${\delta _{c\invalidcharacter N+1\invalidcharacter (N+2)}}$

for the Case 1 is expressed as

${\delta _{c\invalidcharacter N+1\invalidcharacter (N+2)}}=\left\{ {\begin{array}{*{20}{c}} {{\text{ }}\rho - \sqrt {{\rho ^2} - {{\left( {{{{v_{\text{b}}}} \mathord{\left/ {\vphantom {{{v_{\text{b}}}} {\cos \lambda }}} \right. \kern-0pt} {\cos \lambda }}\left( {\bmod \left( {t,{T_{\text{b}}}} \right)+\left( {{T_{\text{b}}} - {T_{\text{r}}}} \right)} \right)} \right)}^2}} {\text{ 0}} \leqslant \bmod \left( {t,{T_{\text{b}}}} \right)<\frac{{{L_{\text{t}}}}}{{{{{v_{\text{b}}}} \mathord{\left/ {\vphantom {{{v_{\text{b}}}} {\cos \lambda }}} \right. \kern-0pt} {\cos \lambda }}}} - \left( {\frac{{{\theta _{\text{b}}}}}{{{\omega _{\text{b}}}}} - \frac{{{\theta _{\text{r}}}}}{{{\omega _{\text{b}}}}}} \right)} \\ {{\text{ }}{\delta _{kN}}{\text{ }}\frac{{{L_{\text{t}}}}}{{{{{v_{\text{b}}}} \mathord{\left/ {\vphantom {{{v_{\text{b}}}} {\cos \lambda }}} \right. \kern-0pt} {\cos \lambda }}}} - \left( {\frac{{{\theta _{\text{b}}}}}{{{\omega _{\text{b}}}}} - \frac{{{\theta _{\text{r}}}}}{{{\omega _{\text{b}}}}}} \right) \leqslant \bmod \left( {t,{T_{\text{b}}}} \right)<\frac{{{\theta _{\text{b}}}}}{{{\omega _{\text{b}}}}}} \end{array}} \right.$

Similarly, the contact load for the left and right sections of the nut in the transition section can be expressed as

$\left\{ {\begin{array}{*{20}{c}} {{Q_{Li}}={{\left( {\frac{{{{A^{\prime}}_{Li}} - {A_0}+{\delta _0} - {\delta _{ci}}}}{{2{\kappa _{{\text{sn}}}}}}} \right)}^{\frac{3}{2}}}{\text{ }}{{A^{\prime}}_{Li}}>{A_0}+{\delta _0} - {\delta _{ck}}} \\ {{\text{ }}{Q_{Li}}=0{\text{ }}{{A^{\prime}}_{Li}} \leqslant {A_0}+{\delta _0} - {\delta _{ck}}} \end{array}} \right.{\text{ }}i=0,N+1,N+2$

$\left\{ {\begin{array}{*{20}{c}} {{Q_{Ri}}={{\left( {\frac{{{{A^{\prime}}_{Ri}} - {A_0}+{\delta _0} - {\delta _{ci}}}}{{2{\kappa _{{\text{sn}}}}}}} \right)}^{\frac{3}{2}}}{\text{ }}{{A^{\prime}}_{Ri}}>{A_0}+{\delta _0} - {\delta _{ci}}} \\ {{\text{ }}{Q_{Ri}}=0{\text{ }}{{A^{\prime}}_{Ri}} \leqslant {A_0}+{\delta _0} - {\delta _{ci}}} \end{array}} \right.{\text{ }}i=0,N+1,N+2$

2.3 Nonlinear dynamic equation of the ball screws

Based on the coordinate transformation, the contact load is divided into three orthogonal components. Taking the worktable as the object, the contact load components for the left and right sections of the nut are simplified to the origin

${O_W}$

of the

${\text{C}}{{\text{S}}_W}$

. The resultant forces (

${f_{ux}}$

${f_{uy}}$

${f_{uz}}$

) of the ball screws in x-, y-, and z-directions are derived as follows

$\left\{ {\begin{array}{*{20}{l}} {{f_{ux}}={f_{Lux}}+{f_{Rux}}=\sum\limits_{{i=0}}^{{N+2}} {\left( {{Q_{Lxi}}+{Q_{Rxi}}} \right)} } \\ {{f_{uy}}={f_{Luy}}+{f_{Ruy}}=\sum\limits_{{i=0}}^{{N+2}} {\left( {{Q_{Lyi}}+{Q_{Ryi}}} \right)} } \\ {{f_{uz}}={f_{Luz}}+{f_{Ruz}}=\sum\limits_{{i=0}}^{{N+2}} {\left( {{Q_{Lzi}}+{Q_{Rzi}}} \right)} } \\ {{f_{\varepsilon x}}={f_{L\varepsilon x}}+{f_{R\varepsilon x}}=\sum\limits_{{i=0}}^{{N+2}} {\left( {\left( { - {Q_{Lyi}}{L_{L{\text{b}}i}}+{Q_{Lzi}}{y_{L{\text{A}}i}}} \right)+\left( { - {Q_{Ryi}}{L_{R{\text{b}}i}}+{Q_{Rzi}}{y_{L{\text{A}}i}}} \right)} \right)} } \\ {{f_{\varepsilon y}}={f_{L\varepsilon y}}+{f_{R\varepsilon y}}=\sum\limits_{{i=0}}^{{N+2}} {\left( {\left( {{Q_{Lxi}}{L_{L{\text{b}}i}}+{Q_{Lzi}}\left( {{H_W} - {x_{L{\text{A}}i}}} \right)} \right)+\left( {{Q_{Rxi}}{L_{R{\text{b}}i}}+{Q_{Rzi}}\left( {{H_W} - {x_{L{\text{A}}i}}} \right)} \right)} \right)} } \end{array}} \right.$

where N is the number of the loaded balls;

${x_{L{\text{A}}i}}$

and

${y_{L{\text{A}}i}}$

are the coordinates of the ball-screw contact point in the x- and y-axes of the

${\text{C}}{{\text{S}}_S}$

;

${Q_{\eta xk}}$

${Q_{\eta yk}}$

and

${Q_{\eta zk}}$

(

$\eta =L{\text{ or }}R$

) are the components of the contact load in the x-, y-, and z-directions, which are expressed as follows

$\left\{ {\begin{array}{*{20}{l}} {{Q_{Lxi}}={Q_{Li}}\cos {\alpha _{Li}}\cos {\theta _i}+{Q_{Li}}\sin {\alpha _{Li}}\sin {\theta _i}\sin \lambda } \\ {{Q_{Lyi}}={Q_{Li}}\cos {\alpha _{Li}}\sin {\theta _i} - {Q_{Li}}\sin {\alpha _{Li}}\cos {\theta _i}\sin \lambda } \\ {{Q_{Lzi}}={Q_{Li}}\sin {\alpha _{Li}}\cos \lambda } \end{array}} \right.$

$\left\{ {\begin{array}{*{20}{l}} {{Q_{Rxi}}={Q_{Ri}}\cos {\alpha _{Ri}}\cos {\theta _i} - {Q_{Ri}}\sin {\alpha _{Ri}}\sin {\theta _i}\sin \lambda } \\ {{Q_{Ryi}}={Q_{Ri}}\cos {\alpha _{Ri}}\sin {\theta _i}+{Q_{Ri}}\sin {\alpha _{Ri}}\cos {\theta _i}\sin \lambda } \\ {{Q_{Rzi}}={Q_{Ri}}\sin {\alpha _{Ri}}\cos \lambda } \end{array}} \right.$

Due to the well-known model of the ball screw only considering axial displacement of the nut, and ignoring the coupling relationship of multiple DOF [8, 9]. In this paper, the ball screws are simplified as a mass-spring-damping model, which includes three translational spring-dampers for the screw-nut, three translational spring-dampers for the screw shaft, and two rotational spring-dampers for the screw shaft, as shown in Fig. 6. According to Eq. (27), an eight DOF nonlinear dynamic model of the ball screws with time-dependent resultant forces under the action of the external load is established, as follows

$\left\{ {\begin{array}{*{20}{l}} {{m_W}{{\ddot {u}}_{Wx}}+{c_{ux}}{{\dot {u}}_{Wx}}+{f_{ux}}={F_x}\sin \omega t} \\ {{m_W}{{\ddot {u}}_{Wy}}+{c_{uy}}{{\dot {u}}_{Wy}}+{f_{uy}}={F_y}\sin \omega t} \\ {{m_W}{{\ddot {u}}_{Wz}}+{c_{uz}}{{\dot {u}}_{Wz}}+{f_{uz}}={F_z}\sin \omega t} \\ {{m_S}{{\ddot {u}}_{Sx}}+{c_{ux}}{{\dot {u}}_{Sx}}+{k_{Sux}}{u_{Sx}} - {f_{ux}}=0} \\ {{m_S}{{\ddot {u}}_{Sy}}+{c_{uy}}{{\dot {u}}_{Sy}}+{k_{Suy}}{u_{Sy}} - {f_{uy}}=0} \\ {\zeta {m_S}{{\ddot {u}}_{Sz}}+{c_{uz}}{{\dot {u}}_{Sz}}+{k_{Suz}}{u_{Sz}} - {f_{uz}}=0} \\ {{I_x}{{\ddot {\varepsilon }}_{Sx}}+{c_{\varepsilon x}}{{\dot {\varepsilon }}_{Sx}}+{k_{S\varepsilon x}}{\varepsilon _{Sx}} - {f_{\varepsilon x}}=0} \\ {{I_y}{{\ddot {\varepsilon }}_{Sy}}+{c_{\varepsilon y}}{{\dot {\varepsilon }}_{Sy}}+{k_{S\varepsilon y}}{\varepsilon _{Sy}} - {f_{\varepsilon y}}=0} \end{array}} \right.$

where

$\omega$

is the excitation frequency;

${I_x}$

and

${I_y}$

is the moment of inertia of the worktable around the x- and y-axes;

${F_x}$

${F_y}$

and

${F_z}$

are the excitation amplitude in the x-, y- and z-directions;

${k_{Sux}}$

${k_{Suy}}$

and

${k_{Suz}}$

are the linear stiffness of the screw shaft in the screw-nut interface along the x-, y-, and z- directions,

${k_{S\varepsilon x}}$

and

${k_{S\varepsilon y}}$

are the angular stiffness of the screw shaft in the screw-nut interface around the x- and y-axes [13], which are expressed as

$\left\{ \begin{gathered} {k_{Sux}}={k_{Sux}}=\frac{{3{E_S}{I_S}}}{{L_{S}^{3}{{\left( {1 - \zeta } \right)}^2}{\zeta ^2}}} \hfill \\ {k_{Suz}}=\frac{{{E_S}{A_S}}}{{\zeta {L_S}}} \hfill \\ {k_{S\varepsilon x}}={k_{S\varepsilon y}}=\frac{{3{E_S}{I_S}}}{{{L_S}\left( {1 - 3\zeta +3{\zeta ^2}} \right)}} \hfill \\ \end{gathered} \right.$

where

${A_{\text{S}}}$

and

${L_{\text{S}}}$

is the cross-section area and length of the screw shaft;

${E_{\text{S}}}$

is the elasticity modulus;

$\zeta$

is the dimensionless position of the worktable.

The matrix form of Eq. (30) can be expressed as

${\mathbf{M\ddot {x}}}+{\mathbf{C\dot {x}}}+\left( {{\mathbf{Kx}}+{{\mathbf{F}}_{\mathbf{k}}}} \right)={\mathbf{F}}$

where

${\mathbf{x}}$

${\mathbf{\dot {x}}}$

, and

${\mathbf{\ddot {x}}}$

are the displacement, velocity, acceleration matrices, respectively; M, C, and K are the mass, damping, stiffness matrices, respectively;

${{\mathbf{F}}_{\mathbf{k}}}$

and

${\mathbf{F}}$

are the resultant force and excitation force, respectively, which are expressed as

Fig. 6

Dynamic model of the ball screws.

${\mathbf{x}}=diag\left( {{u_{Wx}},{u_{Wy}},{u_{Wz}},{u_{Sx}},{u_{Sy}},{u_{Sz}},{\varepsilon _{Sx}},{\varepsilon _{Sy}}} \right)$

${\mathbf{M}}=diag\left( {{m_W},{m_W},{m_W},{m_S},{m_S},\zeta {m_S},{I_x},{I_y}} \right)$

${\mathbf{C}}=diag\left( {{c_{ux}},{c_{uy}},{c_{uz}},{c_{ux}},{c_{uy}},{c_{uy}},{c_{\varepsilon x}},{c_{\varepsilon y}}} \right)$

${\mathbf{K}}=diag\left( {0,0,0,{k_{Sux}},{k_{Suy}},{k_{Suz}},{k_{S\varepsilon x}},{k_{S\varepsilon y}}} \right)$

${{\mathbf{F}}_{\mathbf{k}}}={\left[ {{f_{ux}},{f_{uy}},{f_{uz}}, - {f_{ux}}, - {f_{uy}}, - {f_{uz}}, - {f_{\varepsilon x}}, - {f_{\varepsilon y}}} \right]^{\text{T}}}$

${\mathbf{F}}={\left[ {{F_x},{F_y},{F_z},0,0,0,0,0} \right]^{\text{T}}} \cdot \sin \omega t$

3. Results and discussion

Due to the possible loss of the ball-groove contact, the Newmark-Newton algorithm is used to solve the proposed dynamic model [38]. The initial displacement values

${{\mathbf{x}}_0}$

of the dynamic equation (Eq. (30)) is set to

$\left[ {{u_{Wx}},{u_{Wy}},{u_{Wz}},{u_{Sx}},{u_{Sy}},{u_{Sz}},{\varepsilon _{Sx}},{\varepsilon _{Sy}}} \right]=\left[ {0,0,0,0,0,0,0,0} \right]$

, and the initial values of the velocity and acceleration can be obtained according to the Newmark algorithm. To obtain high-precision solutions, the error and time step during the numerical simulation are set to Err = 10^− 9 m and Δt = 10^− 6 s, respectively. The parameters of the ball screws are shown in Table 1.

Table 1
Parameters of the ball screws.
Parameters	Value
Total mass of worktable and nut ${m_W}$	58 kg
Pitch ${L_p}$	16 mm
Length of screw shaft ${L_S}$	1000 mm
Pitch radius ${R_{\text{m}}}$	21 mm
Radius of the ball ${R_{\text{b}}}$	3.57 mm
Radius of the grooves ${R_{\text{s}}},{R_{\text{n}}}$	3.71 mm
Circle of loaded ball	2.5
Initial contact angle ${\alpha _0}$	45°
Number of balls N	46
Preload	1690 N
Elasticity modulus ${E_S}$	206 GPa
Poisson’s ratio µ	0.3

The dynamic characteristics of the ball screws are discussed theoretically and experimentally in this section, the logical structure is as follows: Sections 3.1 to 3.4 conduct numerical simulations based on the eight DOF nonlinear dynamic model proposed in section 2.3. These simulations primarily analyze the effects of the screw shaft’s deformation, worktable position, external loads and design parameters on the dynamic characteristics of the ball screws. Section 3.5 focuses on experimental validation involving three physical experiments: the transient hammering test, the ball passage vibration test under no-load conditions, and the harmonic excitation response test. The accuracy of the proposed model will be evaluated by comparing the simulation results with experimental data.

3.1 Effect of the screw shaft’s deformation on the dynamic responses

When the worktable is located at the screw shaft's mid-position with F_z = 5000 N and F_x = F_y = 0 N, the time-history curves of the worktable in the z-direction with and without the deformation (the axial, bending, and rotation deformations) of the screw shaft are shown in Fig. 7 (a), it can be seen that the worktable displacement fluctuates twice within one cycle. First fluctuation: the last ball exits the transition section, causing a decrease in stiffness and an increase in displacement. Second fluctuation: the first ball enters the transition section, causing an increase in stiffness and a decrease in displacement. This phenomenon is caused by a decrease or increase in the resultant force of the ball screws due to the sudden change in the number of loaded balls (see Eq. (27)). To compare the influence of the screw shaft’s deformation on the axial displacement of the worktable, the ball screws are simplified as a single DOF dynamic model considering the axial displacement of the worktable (well-known model: Eq.

${m_W}{\ddot {u}_{Wz}}+{c_{uz}}{\dot {u}_{Wz}}+{f_{uz}}={F_z}\sin \omega t$

). It is clear that the worktable displacement with the screw shaft’s deformation increases significantly compared to the worktable displacement without the screw shaft’s deformation, this is because the screw shaft’s deformation leads to a decrease in the stiffness of the ball screws. By comparing the time-history curves under the action of the constant and harmonic loads, it is found that there are fluctuations in the displacement during the ball entering and exiting transition section. And the number of fluctuations in one cycle is the same as under the constant force, both fluctuating upwards and then downwards, as shown in Figs. 7 (a) and (b). Additionally, due to the low excitation frequency ω = 100 rad/s, the vibration amplitude is almost identical to the displacement under the constant load.

Fig. 7

Time-history curves with screw shaft’s deformation for axial load F_z = 5000 N and ω = 100 rad/s. (a) under the constant force, (b) under the harmonic force

Figure 8 (a) illustrates the amplitude-frequency responses for both the deformed and non-deformed screw shaft, the amplitude-frequency curves show fluctuations near the ω = 500 rad/s. As excitation frequency increases beyond the main resonance, the vibration amplitude grows significantly, exhibiting hard nonlinearity and jump phenomena. It is worth noting that the deformed shaft exhibits a lower main resonance frequency compared to the non-deformed case, which is attributed to reduced stiffness of the ball screws caused by the screw shaft’s deformation. The three-dimensional spectrum is shown in Fig. 8 (b), it can be observed that the frequency component (3f) appears in the low excitation frequency region, but the amplitude of the frequency components is much smaller than the amplitude corresponding to the main frequency, indicating that the time-history curve is close to a sinusoidal shape.

Fig. 8

Amplitude-frequency curves and 3D spectrum with screw shaft’s deformation.

3.2 Effect of the worktable position on the dynamic responses

Time-history curves of the worktable at different positions are shown in Fig. 9, it can be seen that the axial displacement of the worktable fluctuates twice in one cycle, and the displacement increases as the worktable position moves away from the angular contact ball bearing. This is because the axial stiffness of the screw shaft decreases as the worktable moves to the right, as shown in Eq. (31). When the worktable moves from position 1/4 to position 3/4 in increments of 1/4 distance, the displacement increase under the constant load is the same, with a value of 4.83 µm. As the worktable moves to the right throughout its working range, the main resonance frequency of the amplitude-frequency curve decreases significantly, as shown in Fig. 10 (a). When the worktable is located in the 3/4 position, the 3D spectrum appears the frequency component, but the amplitude of the frequency component is much smaller than the amplitude corresponding to the main resonance frequency, as shown in Fig. 10 (b).

Fig. 9

Time-history curves with different nut positions for axial load F_z = 5000 N and ω = 100 rad/s. (a) under the constant force, (b) under the harmonic force

Fig. 10

Amplitude-frequency curves and 3D spectrum with different nut positions.

3.3 Effect of the external load on the dynamic responses

Time-history curves of the worktable with the variation of axial loads are shown in Fig. 11, these curves reveal that the displacement fluctuation becomes more significant with the increase of the axial load when the ball enters and exits the transition section. As the axial load increases from

Fig. 11

Time-history curves with different axial loads for axial load F_z = 1000 N and ω = 100 rad/s. (a) under the constant force, (b) under the harmonic force

Fig. 12

Amplitude-frequency curves and 3D spectrum with different axial loads.

3000 N to 7000 N in increments of 2000 N, the displacement under the constant load all increases by 6.31 µm. The amplitude-frequency curves with different axial loads are shown in Fig. 12 (a), the nonlinearity of the worktable displacement increases and the main resonance peak all increases by 75.80 µm when the axial load increases in increments of 2000 N. The 3D spectrum for F_z=7000N is shown in Fig. 12 (b), and its variation law is similar to that of Fig. 8 (b) and Fig. 10 (b).

3.4 Effect of the design parameters on the dynamic responses

Time-history curves of the worktable with different initial contact angles and preloads are shown in Fig. 13. An increase in the initial contact angle results in a decrease in the worktable displacement, due to an increase in the resultant force of the ball screws in the z-direction caused by the increase in the initial contact angle (see Eq. (27)), as shown in Fig. 13 (a). When the preload increases proportionally from Q₀ to 3Q₀, the worktable displacement is reduced by 1.48 µm and 0.55 µm, respectively, and the higher preload leads to an increase in the displacement fluctuation when the ball enters and exits the transition section, as shown in Fig. 13 (b).

The main resonance frequency in the amplitude-frequency curves increases with increasing initial contact angle and preload for the ball screws, as shown in Fig. 14. The dynamic responses of the ball screws with the initial contact angle and preload variation are shown in Figs. 15 and 16, and it can be seen that the displacement fluctuation law in Figs. 15 (a) and 16 (a) is the same as that in Figs. 13 (a) and (b), and both fluctuate first upward and then downward. The harmonic components appear in the spectrum, as shown in Figs. 15 (b) and 16 (b). The velocity-displacement curves are not a single curve due to the displacement fluctuation of the worktable caused by the change in the number of balls, as shown in Figs. 15 (c) and 16 (c).

Fig. 13

Time-history curves with different system parameters. (a) with initial contact angle, (b) with preloads

Fig. 14

Amplitude-frequency curves with different system parameters. (a) with initial contact angle, (b) with preloads

Fig. 15

Dynamic responses with initial contact angle α = 49° for axial load F_z =5000 N and ω = 100 rad/s.

Fig. 16

Dynamic responses with preload 3Q₀ for axial load F_z = 5000 N and ω = 100 rad/s.

3.5. Dynamic test verification for the ball screws

The damping ratio of the ball screws is calculated through transient hammering tests, in which the instruments included an impact hammer (LC-04A), a charge amplifier (YE5852), and two accelerometer (CA-YD-189) with a sensitivity of 99.2 mV/g and a frequency range of 0.2–1 kHz. Two acceleration sensors are installed on the end face of the nut and the bed to eliminate the influence of bed vibration, as shown in Fig. 17. The impact force of the hammer in the z-direction of the worktable is used as input, and the acceleration is used as output to obtain the frequency spectrum curve of the worktable in the z-direction, as shown in the Fig. 18.

The damping ratio is calculated using the half-power bandwidth method, expressed as

$\xi ={{({\omega _2} - {\omega _1})} \mathord{\left/ {\vphantom {{({\omega _2} - {\omega _1})} {2{\omega _0}}}} \right. \kern-0pt} {2{\omega _0}}}$

. Where

${\omega _0}$

is the frequency corresponding to the peak of the spectrum curve, with a value of

${\omega _0}={\text{683}}{\text{.99}}$

Hz. The

${\omega _1}$

and

${\omega _2}$

are the frequencies of the half power bandwidth points (corresponding to 0.707 times the maximum amplitude), and their values are

${\omega _1}={\text{668}}{\text{.99}}$

and

${\omega _2}=700.{\text{16}}$

Hz, respectively. Furthermore, the damping ratio in the z-direction is calculated to be

${\xi _z}={\text{0}}{\text{.0228}}$

. The damping coefficient can obtained from

$c=4{\text{\varvec{\uppi}}}{\omega _0}\xi m$

, the value is

${c_{uz}}=3618{\text{\varvec{\uppi}}}$

N⋅s/m. The damping coefficients

${c_{ux}}$

and

${c_{uy}}$

can be determined by the ratio of axial to radial damping coefficients according to the Ref. [14], the values are

${c_{ux}}={c_{uy}}=1360.7{\text{\varvec{\uppi}}}$

N⋅s/m. The rotational damping coefficient is selected that is equal to 0.01 times the translational damping coefficient [34], the rotational damping coefficients can be calculated as

${c_{\varepsilon x}}={c_{\varepsilon y}}=13.607{\text{\varvec{\uppi}}}$

N⋅s/m.

Fig. 17

Hammer excitation experiment.

Fig. 18

Frequency responses of the ball screws.

To verify the accuracy of the theoretical model proposed in this paper, the ball passage vibration with zero external load and different feed speeds is analyzed using frequency-domain curves. In the experiment, a piezoelectric accelerometer (CA-YD-189) is installed on the end face of the nut. The acceleration sensor generates signal data at a sampling frequency of 5 kHz, which is transmitted to a computer via the data acquisition system (DH 5956) for analysis. The time-domain vibration signal is then converted into a frequency-domain signal using the fast Fourier transform (FFT) algorithm. At the feed speeds of 200 mm/min, 400 mm/min, 600 mm/min, and 800 mm/min, the corresponding frequency f of the ball passage vibration can be calculated to be 1.5670 Hz, 3.3332 Hz, 4.7169 Hz, and 6.5572 Hz, respectively, as shown in Fig. 19. According to the model proposed in this paper, the frequency f of the ball passage vibration through numerical simulation is 1.6967 Hz, 3.3935 Hz, 5.0902 Hz, and 6.787 Hz, respectively. When the feed speed increases from 200 mm/min to 800 mm/min in increments of 200 mm/min, the corresponding f errors for different feed speeds can be calculated to be 7.64%, 1.81%, 7.91%, and 3.50%, respectively, these errors may be caused by factors such as the ball-groove sliding and slipping.

The dynamic response test of the ball screws under the harmonic excitation is shown in the Fig. 20, where the accelerometer (CA-YD-189) and the electromagnetic shaker (JZK-50) are installed on the nut’s end face and the worktable in the z-axis direction, respectively. A sinusoidal excitation is applied in the z-direction using the shaker, and processes the acceleration signal collected by the accelerometer with a sampling frequency of 5 kHz through a data acquisition system (DH 5956). The time-domain and frequency-domain signals of the acceleration in the z-

Fig. 19

Frequency domains of test signals with different feed speeds under no-load conditions.

direction of the worktable with different excitation amplitudes (F_z=30 and 60 N) and different frequencies ω (ω = 170, 200 and 230 Hz) are compared, as shown in Figs. 21 and 22. The results indicate that the time-domain and frequency-domain signals of the acceleration in the proposed model and experiments are almost identical, and their values are greater than the results without the screw shaft’s deformation. Due to factors such as manufacturing tolerances, preload, and frame rigidity of the ball screws, the amplitude of the dynamic response in the experiment is smaller than the dynamic response in the theoretical model.

Fig. 20

Test system of the ball screws.

4. Conclusion

This study establishes a nonlinear dynamic model for the ball screws considering the ball recirculation mechanics, specifically investigating the fluctuation law of the worktable displacement when the ball enters and exits the transition section. The accuracy of the model is verified by comparing with experiments. The effects of the external load, worktable position and design parameters on the dynamic behavior of the ball screws are analyzed. The main conclusions of this study can be summarized as follows:

(1) The displacement fluctuation of the worktable is caused by the sudden changes in the number of loaded balls as a result of the balls entering and exiting the transition section, and there are two fluctuations in one cycle under the constant and excitation loads.

(2) The deformation of the screw shaft leads to a decrease in the screw-nut stiffness, resulting in a lower main resonance frequency when the screw shaft is deformed than when it is not deformed. When the worktable moves away from the angular contact ball bearing at a distance of 1/4 of the screw shaft, the displacement of the worktable increases by approximately 4.83 µm under the constant load.

(3) When the external load increases by 2000 N, the worktable displacement exhibits a corresponding increase of the same value, with values of approximately 6.31 µm and 75.80 µm under the constant and excitation loads, respectively. And the displacement fluctuation of the worktable increases with the ball entering and exiting the transition section.

(4) With the increase of the initial contact angle and preload, the worktable displacement decreases under the constant load, and the main frequency increases under the harmonic load.

Fig. 21

Acceleration comparison in the z-direction. for F_z=30 N and ω = 170, 200 and 230 Hz.

Fig. 22

Acceleration comparison in the z-direction for F_z=60 N and ω = 170, 200 and 230 Hz.

Author contributions

Chang Liu: Writing–original draft, Methodology, Investigation, Formal analysis, Conceptualization. Jinsong Zhao: Visualization, Investigation. Weidong Li: Writing– review & editing, Supervision, Methodology. Chunyu Zhao: Writing– review & editing, Conceptualization.

Funding

This research has been supported by research projects funded by the Science and Technology Commission of Shanghai Municipality (Project No. 23010503700), the National Natural Science Foundation of China (Project No. 51975444), and the Ministry of Science and Technology of China (Project No. G2022013009).

Data availability

Data will be made available on request.

Declarations

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Conflict of interest

The authors declare no competing interest

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