1 Introduction

A ball screw is a mechanical device that transforms rotational movement into linear motion. [1,2,3]. Due to its high positioning accuracy, high rigidity, and low wear, it has become a crucial component in various fields, such as CNC machine tools, equipment manufacturing, and aerospace [4,5,6]. Both the static and dynamic performance of the ball screw affects positioning accuracy and machining quality [7,8,9,10]. The load distribution has a significant effect on the ball screw service life and transmission efficiency [11, 12]. At the same time, it is also a key issue for accurately characterizing the dynamic performance of the ball screw feed system [13]. It is challenging to quantify the ball contact force directly in an experiment when it is moving because of a structural characteristic of the ball screw [14]. In order to demonstrate the static contact properties within the ball screw and the dynamic performance of the feed system, it is essential to develop an appropriate mechanical model.

In recent years, numerous academics have extensively studied the load distribution properties of ball screws. One of the most commonly used modeling methods for ball screws is the finite element method. Bertolaso et al. [15] and Abevi et al. [16] achieved finite element analysis of contact loads of balls inside the screw-nut pair. However, finite element calculations frequently need a large amount of computational power and time. Hence, more and more scholars have been paying attention to developing low-order models to simulate the contact process of balls with the raceways. Hirokazu et al. [17] presented a simplified half-pitch approach to describe the primary characteristics of ball contact load under uniaxial loading. Mei et al. [14] developed a model, which is based on axial deformation relations of the screw and nut and the assumption of all balls with the same contact angle, to analyze the effects of errors of ball screws on the contact load distribution of balls and the contact stiffness of ball screw mechanisms.

Some low-order models assume that all the balls in the same nut experience the same contact force [18, 19] or consider the Hertzian contact deformation of the balls [20,21,22]. More complex models incorporate the axial deformation of the screw when calculating load distribution [14,15,16, 23,24,25,26,27,28]. Hirokazu et al. presented a simplified half-pitch approach to describe the primary characteristics of ball contact load under uniaxial loading, considering screw deformation [17]. Mei et al. refined the unit length of the screw-nut segment, and the analysis focused on the influence of positive and negative error balls on the contact load distribution [14]. Xu et al. enhanced Mei's analytical model by incorporating changes in contact angle. According to Xu's model results, the uneven contact load curve declines from positions close to the external force. This decline becomes more noticeable when contact angle modifications are considered [23]. However, Mei's model did not account for the unloaded balls, which could be present in the raceway due to geometric flaws and external loads. Liu et al. considered the influence of unloaded balls on the ball screw contact characteristics [24]. An accurate model for calculating ball contact forces under multidirectional external loads provides significant guidance for the design of the ball screw [25]. Zhao et al. considered the influence of axial and radial loads on the contact force distribution of the feed system [26]. Additionally, Zhao et al. investigated the impact of torque caused by assembly errors on the contact loads of each ball [27]. Ni et al. calculated the influence of external cyclic loading on the fatigue life of the feed system, taking into account axial loads, radial loads, and dimensional accuracy [28]. It is significant to note that these models presume that the ball screw will not undergo transverse deformation due to axial or torsional loads. Okwudire and Altintas et al. modeled the ball screw as a Tymoshenko beam element, forecasting the relationship between the axial, torsional, and transverse dynamics of the ball screw [7]. Furthermore, based on the assumption that the screw portion within the nut is flexible, Okwudire et al. derived an interface stiffness model [29]. Okwudire et al. conducted follow-up studies to lessen the impact of torsion on the screw/nut static mode shape bending [30, 31]. These studies show that a screw can experience substantial lateral deformation in response to a single axial or torsional force. Consequently, focusing solely on the axial or torsional deformation of the screw is insufficient for accurately evaluating the contact force distribution of a ball screw.

In further numerical calculation models, Lin et al. assumed that the contact angles were equal and did not change with the load [13, 32]. Liu and Zhao et al. investigated the relationship between nut position and load distribution laws by applying the Euler–Bernoulli beam theory to the ball screw system [33, 34]. In conclusion, a lot of discussion has surrounded the ball screw contact force. However, most studies have ignored the lateral deformation of the nut and have not adequately described the influence of the position of balls within the nut on load distribution. Therefore, it is necessary to consider the impact of nut deformation on the raceway center relationship.

The axial stiffness of the ball screw is unstable when bearing the variable load [35, 36]. As a result, establishing an accurate contact stiffness model is essential. The researchers considered the Hertz contact deformation of the screw and nut when calculating the axial contact stiffness [37]. On this basis, Takafuji takes into account the impact of the contact angle [38]. In order to determine the stiffness of the ball screw, there are several ways to build the entire mechanism into a sizable spring system made of different levels of flexibility [39].

However, the approaches mentioned above assume that every ball carries the same contact load, which is inconsistent with the actual ball contact scenario. Liu and Hu et al. examined the relationship between the axial stiffness of the ball screw, the contact angle, and the normal contact force of the raceway [19, 40]. Liu and Li et al. investigated the effect of geometric errors on axial static contact stiffness [25, 41]. Zhao et al. studied the relationship between screw deformation and axial static contact stiffness [34]. As the load increases, the nut experiences more distortion, which affects the ball screw load distribution to some extent. Therefore, in developing an axial contact stiffness model for the ball screw, it is essential to consider the impact of nut deformation.

The majority of existing mechanical models focus primarily on screw deformation, often disregarding the influence of nut deformation and treating the nut as a rigid body. There is currently no research on the contact model of the screw-nut pair that incorporates nut flexibility. This paper addresses this gap by developing a ball screw static model that accounts for the flexible deformation of both the screw and the nut, allowing for a more accurate estimation of the ball screw's static properties. Additionally, the effect of the screw-nut pair stiffness on the axial and torsional vibration modes is investigated through numerical modeling.

2 Theoretical analysis

2.1 Load distribution model of ball screw

Figure 1a depicts a ball screw feed system (BSFS). The screw is connected to a servo motor through a coupling. Two pairs of contact ball bearings support the left and right sides of the screw, respectively. The entire nut is screwed through, and it is secured to the worktable supported by two linear guide rails [42].

Fig. 1
figure 1

Axial balance relationship between a screw-nut pair and ball screw drive mechanism. a Schematic diagram of ball screw driven CNC axis, b force analysis of ball screw system, c ball screw axial load balance relationship, and d contact state of adjacent balls

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The attention of this study is focused on the effects of nut flexibility on the contact load distribution of balls and the contact stiffness of ball screw mechanisms. In order to streamline the computational model and reduce the intricacy of analysis processes, it is assumed that ball screws operate at rotational speeds below 1000 r/min [43]. At such an operating state, inertia forces and friction on the contact load of balls can be negligible, as they exert insignificant influence on the distribution of contact loads [13, 33]. Under these conditions, the screw-ball and nut-ball contact loads and contact angles are equal, respectively, as shown in Fig. 1b, that is:\(Q_{{{\text{s}}i}} = Q_{{{\text{n}}i}} = Q_{i} ,\;\alpha_{{{\text{s}}i}} = \alpha_{{{\text{n}}i}} = \alpha_{i}\) .

Figure 2 establishes a set of coordinates to characterize the relative positional relationship of contact positions between screws/balls and nuts/balls. The angular contact ball bearings serve as the fixed origin (o) of the global coordinate system (CSo). The x-axis is perpendicular to the table plane, and the z-axis coincides with the ideal axis of the screw. Coordinate systems CS1n and CS1s are local contact coordinate systems with A and B as origins. A and B are contact points between balls and screw and nut raceways, respectively. Axes z1s and z1n travel through the center of the ball and point to the normal vectors of the raceway surfaces. The x1s and x1n axes coincide with the tangent directions of the contact curves of the screw and nut, respectively. The y1s and y1n axes are defined by the right-hand rule. Coordinate systems CS0s and CS0n are parallel to the coordinate system CSo, and coordinate systems CS5s and CS5n are the projects of CS0s and CS0n on z-axis, respectively. The origin of the coordinate system CS2 is the center of the ball. The x2-axis is tangent to the direction of motion of the ball. The y2-axis is perpendicular to the z-axis. The angle formed by the helix angle λ of the screw shaft is situated between the z2-axis and the z-axis. The rotating coordinate system CS3 is always parallel to CSo. The transformation relations between CS1ς (ς is n/s) and CSo can be expressed as:

$$ \begin{gathered} [x_{{\text{s}}} ,y_{{\text{s}}} ,z_{{\text{s}}} ,1]^{{\text{T}}} = {\mathbf{T}}_{o - 3} \cdot {\mathbf{T}}_{3 - 2} \cdot {\mathbf{T}}_{{2 - 1{\text{s}}}} \cdot [x,y,z,1]^{{\text{T}}} \hfill \\ [x_{{\text{n}}} ,y_{{\text{n}}} ,z_{{\text{n}}} ,1]^{{\text{T}}} = {\mathbf{T}}_{o - 3} \cdot {\mathbf{T}}_{3 - 2} \cdot {\mathbf{T}}_{{2 - 1{\text{n}}}} \cdot [x,y,z,1]^{{\text{T}}} \hfill \\ \end{gathered} $$
(1)

where To-3, T3-2, T2-1 s, and T2-1n are listed in the Appendix. The components of the contact load Qςi in three directions can be described as follows based on the transformation matrix:

$$ \left[ {Q_{\varsigma xi} ,Q_{\varsigma yi} ,Q_{\varsigma zi} ,1} \right]^{{\text{T}}} = {\text{Rot}}_{z} \left( {\theta + \frac{\pi }{2}} \right) \cdot {\text{Rot}}_{y} ( - \lambda ) \cdot {\text{Rot}}_{z} \left( { - \frac{\pi }{2}} \right) \cdot {\text{Rot}}_{y} \left( {\frac{\pi }{2} - \alpha } \right) \cdot [0,0,Q_{\varsigma i} ,1]^{{\text{T}}} $$
(2)

where θ is the angular position of ball i.

Fig. 2
figure 2

Contact coordinate system and global coordinate system

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The axial force relationship of the screw shaft balances when the system is T-C condition (nut compressed and the screw tension), as illustrated in Fig. 1c, can be expressed as:

$$ F_{a} = \sum\limits_{i = 1}^{Z} {Q_{i} \sin \alpha_{i} \cos \lambda } $$
(3)

where Fa is the axial force of the feed system, and Z is the total number of balls.

The axial internal force Fsi/Fni of the screw/nut cross-section between neighboring balls in Fig. 1c can be expressed as:

$$ F_{{{\text{s}}i}} = \left\{ {\begin{array}{*{20}l} {\sum\limits_{j = i + 1}^{Z} {Q_{j} \sin \alpha_{j} \cos \lambda, } } \hfill & {i = { 1},{ 2},\cdot\cdot\cdot,\;Z - {1}} \hfill \\ 0, \hfill & {i = Z} \hfill \\ \end{array} } \right. $$
(4)
$$ F_{{{\text{n}}i}} = \sum\limits_{j = 1}^{Z} {Q_{j} \sin \alpha_{j} \cos \lambda } ,\;i = { 1},{ 2}, \, \cdot\cdot\cdot,Z $$
(5)

At the same time, the axial deformation of screw/nut between adjacent ball contact points can be expressed as:

$$ \Delta_{\varsigma i} = \frac{{\Delta LF_{\varsigma i} }}{{E_{\varsigma } S_{\varsigma } }},\;\varsigma = {\text{n}},{\text{ s}} $$
(6)

where ΔL represents the axial distance between the contact points of adjacent balls. ς is s or n, which means screw or nut. Eς is the elastic modulus, and Sς is the effective cross-sectional area of the screw/nut [14], which can be expressed as follows:

$$ S_{{\text{s}}} = {\uppi }(R_{{\text{s}}} - r_{{\text{s}}} )^{2} $$
(7)
$$ S_{{\text{n}}} = {\uppi }(R_{{{\text{no}}}}^{2} - (R_{{{\text{ni}}}} + r_{{\text{n}}} )^{2} ) $$
(8)

in which Rs is the radius of the screw. rn/rs is the radii of the ball raceway for the nut/screw, respectively. Rno and Rni are the outer and inner radii of the nut, respectively.

The axial deformation relationship for the screw and nut between their contact locations with the neighboring balls in Fig. 1c and d can be written as:

$$ \Lambda_{i} - \Lambda_{i - 1} = \Delta_{{{\text{s}}i}} + \Delta_{{{\text{n}}i}} $$
(9)

in which Λ is the axial shift between the nut groove and screw groove centers.

Based on the theorem of translation of a force, the components of contact load in Eq. (2) can be translated to point A5i/B5i on the ideal axis of screw/nut by appending corresponding additional moments, as shown in Fig. 2. The additional moments of components Qςxi and Qςyi constitute a torsional torque of screw/nut, that is [33]:

$$ T_{\varsigma i} = Q_{\varsigma yi} x_{\varsigma 5i} - Q_{\varsigma yi} y_{\varsigma 5i} $$
(10)

The additional moment of component Qςzi can be decomposed into two bending moments, Mςxi and Mςyi, which bend the axis of screw/nut on the xz- and yz- planes, respectively, and can be expressed as:

$$ \left\{ \begin{gathered} M_{\varsigma xi} = Q_{\varsigma zi} x_{5\varsigma i} \hfill \\ M_{\varsigma yi} = Q_{\varsigma zi} y_{5\varsigma i} \hfill \\ \end{gathered} \right. $$
(11)

As illustrated in Fig. 3, the nut is treated as a cantilever beam due to its left end being securely fastened to the worktable by bolts while its right end remains free. The fixed segment of the nut boss is assumed to remain un-deformed. Thus, only the transverse deformation of the unsupported area of the nut is considered. The coordinate system CS4 is the moving one of the nut with its axes perpendicular to the corresponding ones of CSo respectively. The origin o4 of CS4 is the intersection point of the nut boss right end face with the z-axis, as shown in Fig. 3. The coordinate transformation from CS4 to CSo is expressed as the following:

$$ \left\{ {x,y,z} \right\}^{{\text{T}}} = \left\{ {x_{4} ,y_{4} ,l_{n} + z_{4} } \right\}^{{\text{T}}} = \left\{ {x_{n5i} ,y_{n5i} ,l_{n} + z_{4} } \right\}^{{\text{T}}} $$
(12)

where ln is the distance from point o4 to point o.

Fig. 3
figure 3

Mechanical model of nut cantilever beam

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When a beam undergoes bending deformation, the axis of the deformed beam forms a curve in the plane, termed the deflection curve. The ordinate of any point along the abscissa of the deflection curve is denoted by ω, representing the displacement of the centroid of the beam cross-section in the ordinate direction, known as deflection. During bending deformation, the angle β at which the cross-section of the beam deviates from its original position is termed the cross-sectional angle. When the nut is subjected to the force Qnςj at the axis point with z4n-coordinate of Lj, as shown in Fig. 3, the equations of the cross-sectional angle and the deflection curve are expressed as follows [44]:

$$ \left\{ {\begin{array}{*{20}l} {EI\beta _{{Q\zeta j}} (z_{{4{\text{n}}}} ) = \frac{1}{2}Q_{{{\text{n}}\zeta j}} (L_{j} - z_{{4{\text{n}}}} )^{2} + C} \hfill \\ {EI\omega _{{Q\zeta j}} (z_{{4{\text{n}}}} ) = - \frac{1}{6}Q_{{{\text{n}}\zeta j}} (L_{j} - z_{{4{\text{n}}}} )^{3} + Cz_{{4{\text{n}}}} + D,(0 \le z_{{{\text{4n}}}} \le L_{j} )} \hfill \\ \end{array} } \right. $$
(13)

where I is the product of inertia. Lj is the distance between the jth ball in the nut and the coordinate system CS4. The integral constant in Eq. (13) can be determined by boundary conditions: no deformation occurs at the fixed end of the nut, ω′ = 0, ω = 0, then \(C = - \frac{1}{2}Q_{{{\text{n}}\zeta i}} L_{j}^{2}\), \(D = \frac{1}{6}Q_{{{\text{n}}\zeta i}} L_{j}^{3}\). Therefore, the deflection curve equation and rotation angle equation can be expressed as:

$$ \left\{ {\begin{array}{*{20}l} {\omega _{{Q\zeta j}} (z_{{4{\text{n}}}} ) = \frac{1}{{EI_{{\text{n}}} }}\left[ { - \frac{{Q_{{{\text{n}}\zeta j}} }}{6}(L_{j} - z_{{4{\text{n}}}} )^{3} - \frac{1}{2}Q_{{{\text{n}}\zeta j}} L_{j}^{2} z_{{4{\text{n}}}} + \frac{{Q_{{{\text{n}}\zeta j}} }}{6}L_{j}^{3} } \right]} \hfill \\ {\beta _{{Q\zeta j}} (z_{{4{\text{n}}}} ) = \frac{1}{{EI_{{\text{n}}} }}\left[ {\frac{{Q_{{{\text{n}}\zeta j}} }}{2}(L_{j} - z_{{4{\text{n}}}} )^{2} - \frac{1}{2}Q_{{{\text{n}}\zeta j}} L_{j}^{2} } \right],(0 \le z_{{{\text{4n}}}} \le L_{j} )} \hfill \\ \end{array} } \right. $$
(14)

Therefore, the deflection and rotation angle of the c-section are:

$$ \left\{ {\begin{array}{*{20}l} {\omega _{{Q\zeta cj}} = - \frac{1}{3}\frac{{Q_{{{\text{n}}\zeta j}} L_{j}^{3} }}{{EI_{{\text{n}}} }}} \hfill \\ {\beta _{{Q\zeta cj}} = \frac{1}{2}\frac{{Q_{{{\text{n}}\zeta j}} L_{j}^{2} }}{{EI_{{\text{n}}} }}} \hfill \\ \end{array} } \right. $$
(15)

Since the end b of nut is free, no load is applied in the segment cb. Therefore, the deflection and rotation angle of the nut at any position on the segment cb can be expressed as:

$$ \left\{ {\begin{array}{*{20}l} {\omega _{{Q\zeta j}} (z_{4} ) = - \frac{{Q_{{{\text{n}}\zeta j}} L_{j}^{2} }}{{6EI_{{\text{n}}} }}(3z_{4} - L_{j} )} \hfill \\ {\beta _{{Q\zeta j}} (z_{4} ) = \frac{1}{2}\frac{{Q_{{{\text{n}}\varsigma j}} L_{j}^{2} }}{{EI_{{\text{n}}} }},(L_{j} \le z_{{{\text{4n}}}} \le L)} \hfill \\ \end{array} } \right. $$
(16)

where L is the total length of the nut.

Because the cross-sectional angle of the nut and the spiral radius of its raceway center are very small, it is assumed that the radial displacement of the raceway centers is equal to the lateral deformation of the nut axis. The radial displacements of the nut raceway center, which corresponds to the contact point of ball i and the raceway and are caused by the components of contact loads in x- and y-directions, can be expressed as:

$$ \omega_{{Q{\text{n}}xi}} = \sum\limits_{j = 1}^{i - 1} {\frac{{Q_{{{\text{n}}xj}} L_{i}^{2} }}{{6EI_{{\text{n}}} }}(3L_{j} - L_{i} )} + \sum\limits_{j = i}^{Z} {\frac{{Q_{{{\text{n}}xj}} L_{j}^{2} }}{{6EI_{{\text{n}}} }}(3L_{i} - L_{j} )} $$
(17)
$$ \omega_{{Q{\text{n}}yi}} = \sum\limits_{j = 1}^{i - 1} {\frac{{Q_{{{\text{n}}yj}} L_{i}^{2} }}{{6EI_{{\text{n}}} }}(3L_{j} - L_{i} )} + \sum\limits_{j = i}^{Z} {\frac{{Q_{{{\text{n}}yj}} L_{j}^{2} }}{{6EI_{{\text{n}}} }}(3L_{i} - L_{j} )} $$
(18)

Similarly, the radial displacements of the nut raceway center, caused by the moments in Eq. (11), can be expressed as:

$$ \omega_{{M{\text{n}}xi}} = \sum\limits_{j = 1}^{i - 1} {\frac{{Q_{nzj} x_{5nj} L_{j} }}{{EI_{{\text{n}}} }}\left(L_{i} - \frac{{L_{j} }}{2}\right)} + \sum\limits_{j = i}^{Z} {\frac{{Q_{nzj} x_{5nj} }}{{2EI_{{\text{n}}} }}L_{i}^{2} } $$
(19)
$$ \omega_{{M{\text{n}}yi}} = \sum\limits_{j = 1}^{i - 1} {\frac{{Q_{{n{\text{z}}j}} y_{5nj} L_{j} }}{{EI_{{\text{n}}} }}\left(L_{i} - \frac{{L_{j} }}{2}\right)} + \sum\limits_{j = i}^{Z} {\frac{{Q_{{n{\text{z}}j}} y_{5nj} }}{{2EI_{{\text{n}}} }}L_{i}^{2} } $$
(20)

Therefore, the total radial displacement of the nut raceway center corresponding to contact point of ball i can be expressed as:

$$ \delta_{{r{\text{n}}i}} = (\omega_{{M{\text{n}}xi}} + \omega_{{Q{\text{n}}xi}} )\cos \theta_{i} + (\omega_{{M{\text{n}}yi}} + \omega_{{Q{\text{n}}yi}} )\sin \theta_{i} $$
(21)

where δrni is the lateral deformation of nut under the action of contact loads.

Since the screw length is far longer than its diameter, it can be thought of as an Euler–Bernoulli beam [45]. Figure 4 shows the influence of ball contact force Qsi on screw bending deformation. The Qsζi causes lateral displacement, which can be described as follows:

$$ \begin{aligned} w_{{{\text{Qs}}xi}} = & \sum\limits_{{j = 1}}^{{i - 1}} {\frac{{Q_{{{\text{s}}xj}} (H - H_{j} )}}{{6EI_{s} H}}\left[ {\frac{H}{{H - H_{j} }}(H_{i} - H_{j} )^{3} + \left( {H^{2} - (H - H_{j} )^{2} } \right)H_{i} - H_{i}^{3} } \right]} \\ &+ \sum\limits_{{j = i}}^{Z} {\frac{{Q_{{{\text{s}}xj}} (H - H_{j} )H_{i} }}{{6EI_{s} H}}} \left( {H^{2} - H_{i}^{2} - (H - H_{j} )^{2} } \right) \\ \end{aligned} $$
(22)
$$ \begin{aligned} w_{{{\text{Qs}}yi}} = & \sum\limits_{{j = 1}}^{{i - 1}} {\frac{{Q_{{{\text{s}}yj}} (H - H_{j} )}}{{6EI_{s} H}}\left[ {\frac{H}{{H - H_{j} }}(H_{i} - H_{j} )^{3} + \left( {H^{2} - (H - H_{j} )^{2} } \right)H_{i} - H_{i}^{3} } \right]} \\ & + \sum\limits_{{j = i}}^{Z} {\frac{{Q_{{{\text{s}}yj}} (H - H_{j} )H_{i} }}{{6EI_{s} H}}} \left( {H^{2} - H_{i}^{2} - (H - H_{j} )^{2} } \right) \\ \end{aligned} $$
(23)

where H is the length of the screw.

Fig. 4
figure 4

Mechanical model of screw Euler–Bernoulli beam

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The effect of the bending moment on screw bending deformation is shown in Fig. 4c. The lateral displacements of the screw axis in the x- and y-directions at point A5i, caused by the additional moments in Eq. (11), can be expressed as:

$$ \begin{aligned} w_{{{\text{Ms}}xi}} = & \sum\limits_{j = 1}^{i - 1} {\frac{{M_{{{\text{s}}xj}} }}{{6EI_{s} H}}} \left[ { - H_{i}^{3} + 3H(H_{i} - H_{j} )^{2} + (H^{2} - 3(H - H_{j} )^{2} )H_{i} } \right] \\ & & + \sum\limits_{j = i}^{Z} {\frac{{M_{{{\text{s}}xj}} H_{i} }}{{6EI_{s} H}}\left( {H^{2} - 3(H - H_{j} )^{2} - H_{i}^{2} } \right)} \\ \end{aligned} $$
(24)
$$ \begin{aligned} w_{{{\text{Ms}}yi}} = & \sum\limits_{j = 1}^{i - 1} {\frac{{M_{{{\text{s}}yj}} }}{{6EI_{s} H}}} \left[ { - H_{i}^{3} + 3H(H_{i} - H_{j} )^{2} + (H^{2} - 3(H - H_{j} )^{2} )H_{i} } \right] \\ & + \sum\limits_{j = i}^{Z} {\frac{{M_{{{\text{s}}yj}} H_{i} }}{{6EI_{s} H}}\left( {H^{2} - 3(H - H_{j} )^{2} - H_{i}^{2} } \right)} \\ \end{aligned} $$
(25)

Therefore, the total radial displacement of the screw raceway center corresponding to contact point Ai can be expressed as:

$$ \delta_{{r{\text{s}}i}} = (\omega_{{M{\text{s}}xi}} + \omega_{{Q{\text{s}}xi}} )\cos \theta_{i} + (\omega_{{M{\text{s}}yi}} + \omega_{{Q{\text{s}}yi}} )\sin \theta_{i} $$
(26)

where δrsi is the radial deformation of screw under the action of contact loads.

Assuming that the geometrical center positions are Oni, Obi, and Osi when the ball is not loaded (as shown in Fig. 5). The relationship between them can be expressed by the following formula:

$$ A_{0} = \overline{{O_{{{\text{n}}i}} O_{{{\text{s}}i}} }} = R_{{\text{s}}} { + }R_{{\text{n}}} - 2R_{{\text{b}}} $$
(27)
$$ C_{0} = A_{0} \sin \alpha_{0} $$
(28)
$$ B_{0} = A_{0} \cos \alpha_{0} $$
(29)

in which Rb is the ball radius. A0 represents the starting distance between the two raceway centers. α0 is the initial contact, α0 = 45°. Following bearing load, the raceway center axial and lateral deformation can be expressed as follows:

$$ \Lambda_{i} = C_{i} - C_{0} $$
(30)
$$ B_{i} = B_{0} + \delta_{{{\text{rs}}i}} + \delta_{{{\text{rn}}i}} $$
(31)
Fig. 5
figure 5

Ball screw raceway center geometric relationship

Full size image

Consequently, the following is an expression of the raceway center relationship after bearing:

$$ \overline{{O^{\prime}_{{{\text{n}}i}} O^{\prime}_{{{\text{s}}i}} }} = A_{0} + \delta_{{{\text{n}}i}} + \delta_{{{\text{s}}i}} = \sqrt {B_{i}^{2} + C_{i}^{2} } $$
(32)

where δsi and δni are the Hertzian contact deformation of the screw and nut under normal contact force, which can be expressed as:

$$ \delta_{{{\text{n}}i}} = \kappa_{{\text{n}}} Q_{{{\text{n}}i}}^{2/3} $$
(33)
$$ \delta_{{{\text{s}}i}} = \kappa_{{\text{s}}} Q_{{{\text{s}}i}}^{2/3} $$
(34)

where κn/κs is the Hertzian coefficient between balls and the nut/screw raceways, respectively.

Furthermore, the contact angle equation can be written as follows:

$$ \alpha^{\prime}_{i} = \arcsin \left( {\frac{{C_{i} }}{{\sqrt {B_{i}^{2} + C_{i}^{2} } }}} \right) $$
(35)

Inserting Eq. (30) into Eq. (31) and solving it for the variable Bi, then inserting Bi into Eq. (41) yields:

$$ \Lambda^{\prime}_{i} = \sqrt {(A_{0} + \delta_{{{\text{n}}i}} + \delta_{{{\text{s}}i}} )^{2} - (B_{0} + \delta_{{{\text{rn}}i}} + \delta_{{{\text{rs}}i}} )^{2} } - C_{0} $$
(36)

Inserting Eqs. (6) and (35) into Eq. (9) yields an axial deformation compatibility equation of screw and nut between two adjacent balls as the following:

$$ \begin{gathered} \sqrt {(A_{0} + \delta_{{{\text{n}}i}} + \delta_{{{\text{s}}i}} )^{2} - (B_{0} + \delta_{{{\text{rn}}i}} + \delta_{{{\text{rs}}i}} )^{2} } - \sqrt {(A_{0} + \delta_{{{\text{n(}}i - 1)}} + \delta_{{{\text{s(}}i - 1)}} )^{2} - (B_{0} + \delta_{{{\text{rn(}}i - 1)}} + \delta_{{{\text{rs(}}i - 1)}} )^{2} } \hfill \\ = \frac{{\Delta LF_{{{\text{n}}i}} }}{{E_{{\text{n}}} S_{{\text{n}}} }} + \frac{{\Delta LF_{{{\text{s}}i}} }}{{E_{{\text{s}}} S_{{\text{s}}} }} \hfill \\ \end{gathered} $$
(37)

There are Z-1 expressions in Eq. (48), which include Z variables, i.e., Qi, i = 1, 2, …, Z. These variables also satisfy Eq. (3). So one has a group of nonlinear equations as follows:

$$ \left\{ \begin{gathered} \sum\limits_{i = 1}^{Z} {Q_{i} \sin \alpha_{i} \cos \lambda } - F_{a} = 0 \hfill \\ \sqrt {(A_{0} + \delta_{{{\text{n}}i}} + \delta_{{{\text{s}}i}} )^{2} - (B_{0} + \delta_{{{\text{rn}}i}} + \delta_{{{\text{rs}}i}} )^{2} } - \frac{{\Delta LF_{{{\text{n}}i}} }}{{E_{{\text{n}}} S_{{\text{n}}} }} - \hfill \\ \sqrt {(A_{0} + \delta_{{{\text{n(}}i - 1)}} + \delta_{{{\text{s(}}i - 1)}} )^{2} - (B_{0} + \delta_{{{\text{rn(}}i - 1)}} + \delta_{{{\text{rs(}}i - 1)}} )^{2} } - \frac{{\Delta LF_{{{\text{s}}i}} }}{{E_{{\text{s}}} S_{{\text{s}}} }} = 0, \, i = 2,3, \cdots ,Z. \hfill \\ \end{gathered} \right. $$
(38)

The axial deformation of the screw-nut pair, δa, can be expressed as:

$$ \delta_{a} = \frac{{\delta_{{{\text{Anm}}}} }}{{\sin \alpha_{{{\text{Anm}}}} \cos \lambda }} $$
(39)

where δAnm and αAnm represent the maximum deformation of the ball and its contact angle. Therefore, the axial contact stiffness kn of the screw-nut pair can be expressed as [46]:

$$ k_{n} = \frac{{dF_{a} }}{{d\delta_{a} }} $$
(40)

The Newton–Raphson iterative approach can be used to solve the nonlinear equations that make up Eq. (50):

$$ {\mathbf{x}}^{(t - 1)} {\mathbf{ = x}}^{(t)} {\mathbf{ - }}{\varvec{J}}{\mathbf{(x}}^{(t)} {\mathbf{)}}^{ - 1} {\mathbf{f(x}}^{(t)} ) $$
(41)

where x(t) is the solution of the equation in each iteration. J is the Jacobian matrix of the function.

2.2 Axial and torsional dynamic model of BSFS

Figure 6 shows the axial dynamic model of the BSFS, in which the screw-nut pair and bearings are considered spring-damping systems, respectively, and the screw is modeled as a deformable continuum. In the feed system, the guide slider mainly loads the radial load of the worktable, which does not limit the axial freedom of the worktable, so they have no effect on the axial dynamic model of the ball screw and can be ignored [12, 13].

Fig. 6
figure 6

Hybrid dynamic model of BSFS

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Lagrange’s equations are applied to set up the differential equation of motion of the BSFS [47, 48]. The expression is given as follows:

$$ \frac{d}{dt}\left( {\frac{\partial L}{{\partial \dot{q}_{i} }}} \right) - \frac{\partial L}{{\partial q_{i} }} + \frac{\partial D}{{\partial \dot{q}_{i} }} = F_{i} ,\;i = { 1}, \, \cdot\cdot\cdot,n $$
(42)

where L is a Lagrange function and L = T–V. Fi is the system generalized force associated with the ith generalized coordinate. qi is the system's generalized coordinate. The system kinetic energy is T, potential energy is V, and dissipative energy is D. This paper disregards system damping due to its minimal impact on the natural frequency in the BSFS [49, 50], that is, D = 0.

As a continuous system, the screw displacement field is a coordinate function about time and space (as shown in Fig. 6). Based on the hypothetical mode method and the screw boundary constraint conditions [51], the screw displacement field and torsion field at time t can be written as follows:

$$ u(x,t) = \sum\limits_{i = 1}^{N} {\psi_{i} (x)q_{li} (t) = } \sum\limits_{i = 1}^{N} {\sin \left( {\frac{i\pi }{{L_{s} }}x} \right)q_{li} (t)} , \, \left( {x \in \left[ {0,H} \right]} \right) $$
(43)
$$ \upsilon (x,t) = \sum\limits_{i = 1}^{N} {\varphi_{i} (x)q_{\upsilon i} (t) = } \sum\limits_{i = 1}^{N} {\sin \left( {\frac{i\pi }{{L_{s} }}x} \right)q_{\upsilon i} (t)} , \, \left( {x \in \left[ {0,H} \right]} \right) $$
(44)

in which N represents the total mode order of the screw. ψi/φi is the mode shape of the ith axial/torsional vibration of the screw. qli/qυi is the instantaneous contribution of the ith mode.

Therefore, the kinetic energy of the system can be expressed as:

$$ \begin{aligned} E & = \frac{1}{2}\left[ {m_{w} \dot{u}_{w}^{2} (t) + \int_{0}^{{H_{1} }} {\rho S_{s} \left( {\frac{{\partial u(x,t)}}{{\partial t}}} \right)^{2} {\text{d}}x + \int_{{H_{2} }}^{H} {\rho S_{s} \left( {\frac{{\partial u(x,t)}}{{\partial t}}} \right)^{2} {\text{d}}x} } } \right. \\ & \quad \left. { + \int_{0}^{{H_{1} }} {\rho I_{{ps}} \left( {\frac{{\partial \upsilon (x,t)}}{{\partial t}}} \right)^{2} {\text{d}}x} + \int_{{H_{2} }}^{H} {\rho I_{{ps}} \left( {\frac{{\partial \upsilon (x,t)}}{{\partial t}}} \right)^{2} {\text{d}}x} } \right] \\ \end{aligned} $$
(45)

where the kinetic energy of the worktable is the first term. The axial kinetic energy of the screw is the second and third term. The torsional kinetic energy of the screw is the last two terms. The worktable mass is represented by mw, its displacement function is by uw, and its density is by ρ. Ips is the cross-sectional moment of inertia of the screw.

The distances H1 and H2 represent, respectively, the separations between the left and right sides of the nut.

Equation (45) is an expression of potential energy for the BSFS, which consists of the elastic potential energy of every joint, as well as the deformation energy of the screw-nut pair.

$$ \begin{aligned} V & = \frac{1}{2}\left[ {ES_{s} \int_{0}^{{H_{1} }} {\left( {\frac{{\partial u(x,t)}}{{\partial x}}} \right)^{2} {\text{d}}x + ES_{s} \int_{{H_{2} }}^{H} {\left( {\frac{{\partial u(x,t)}}{{\partial x}}} \right)^{2} {\text{d}}x + } } GI_{{ps}} \int_{0}^{{H_{1} }} {\left( {\frac{{\partial \theta (x,t)}}{{\partial x}}} \right)^{2} {\text{d}}x} } \right. \\ & \quad + \left. {GI_{{ps}} \int_{{H_{2} }}^{H} {\left( {\frac{{\partial \theta (x,t)}}{{\partial x}}} \right)^{2} {\text{d}}x + } k_{t} u^{2} (0,t) + k_{r} u^{2} (H,t) + k_{n} \Delta _{n}^{2} } \right] \\ \end{aligned} $$
(46)

where the screw stores axial elastic potential energy in terms of the first two terms, torsional elastic potential energy in terms of the third and fourth terms, elastic potential energy stored by the bearing in terms of the fifth and sixth terms, and elastic potential energy stored by the screw-nut pair in terms of the final term. G represents the shear modulus, kt denotes the thrust bearing contact stiffness, and kr denotes the radial ball bearing contact stiffness. The axial displacement Δn of the screw nut joint surface can be expressed as:

$$ \Delta_{n} = u_{w} - v(H_{1} ,t) - \lambda \theta (H_{1} ,t) $$
(47)
$$ \lambda = \frac{p}{2\pi } $$
(48)

Inserting Eqs. (45) and (46) into Eq. (42) yields:

$$ {\mathbf{M}}{\ddot{\varvec q}} + {\mathbf{Kq}} = {\mathbf{F}} $$
(49)

where M and K are the system mass and stiffness matrices, respectively. The total order of axial mode and torsional mode is 4. In Eq. (49), q = [ql1 ql2 ql3 ql4 qw qθ1 qθ2 qθ3 qθ4]. In the static state, there is no external force in the system, that is, F = [0 0 0 0 0 0 0 0]. Therefore, M and K can be expressed as:

$$ \user2{M = }\left[ {\begin{array}{*{20}c} {{\varvec{M}}_{1} } & 0 & 0 \\ 0 & {M_{w} } & 0 \\ 0 & 0 & {{\varvec{M}}_{3} } \\ \end{array} } \right] $$
(50)
$$ \user2{K = }\left[ {\begin{array}{*{20}c} {{\varvec{K}}_{1} } & 0 & 0 \\ 0 & {K_{w} } & 0 \\ 0 & 0 & {{\varvec{K}}_{3} } \\ \end{array} } \right] $$
(51)

where M1 represents the mass matrix of the screw, Mw represents the concentrated mass on the screw, and M3 represents the moment of inertia matrix of the screw. K1, Kw and K3 represent the stiffness coefficient of the screw.

The eigenvalue wi and the corresponding eigenvector χi of the BSFS can be derived by calculating the eigenequation \(\left| {K - Mw_{i}^{2} } \right|\left\{ {\chi_{i} } \right\} = 0\) in Eq. (49). The relationship between natural frequency and characteristic value is fi = wi/2π. The following can be used to characterize the relationship between the axial natural frequency of the feed system and the stiffness of the thrust bearing, the radial ball bearing stiffness at the junction, and the screw-nut pair:

$$ f = F_{X} (k_{n} ,k_{t} ,k_{r} ) $$
(52)

Figure 7 illustrates the process of solving the dynamic equation for the feed system.

Fig. 7
figure 7

Flow chart for solving ball screw load distribution

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3 Model verification

3.1 Comparison of the calculation results

In order to verify the effectivity of the algorithm in Fig. 7, the proposed model and Lin’s model in Ref. 13 were used to calculate the contact loads of a ball screw, respectively, of which principal parameters were listed in Table 1. As described in Ref. [13], the left side of the nut is fixedly connected to the worktable to limit the freedom of the nut in six directions. The nut is positioned in the middle of the screw. The comparison of the two calculation results is shown in Fig. 8a, which demonstrates that the predicted ball load distribution in this model agrees basically with that in Lin's model. The most possible reason for the discrepancies is that the contact angle in Lin’s model is equal to 45° and no variation with loading (as shown in Fig. 8b).

Table 1 Principal ball screw parameters [13]
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Fig. 8
figure 8

Comparison of the load distribution in this work and in Reference [13]

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The ball screw return tube is located outside the nut boss. Therefore, the first ball in contact with the screw and the nut is at a certain distance from the fixed end of the nut. The contact model of the ball screw is established using ANSYS Workbench 21.1 when the initial ball distance is 10 mm from the fixed end of the nut. The purpose is to discuss the influence of the position of the ball in the nut on the load distribution of the ball screw. Its size and material parameters are identical to those of the BSFS. In this paper, the hexahedral unit mesh is used for the screw shaft and ball, and the tetrahedral mesh is used for the complex contact area between the nut and raceway. The overall unit size of the screw shaft and nut is 2 mm, and the unit size of the ball contact area is 0.2 mm. The grid division is shown in Fig. 9a. The model contains 1,204,219 elements and 2,003,070 nodes. There are 64 contact pairs, including 32 ball-screw contact pairs and 32 ball-nut contact pairs. The contact type is friction, and the friction coefficient is 0.05. The Augmented Lagrange method is used for calculation. Other settings are the program default. The ball screw bears axial load Fa = 5000 N, ignoring the geometric error of balls. Figure 9 displays the calculation flow.

Fig. 9
figure 9

Calculation flow and results of ANSYS software. a Mesh division of ball screw, b boundary conditions of ball screw, and c stress nephogram of ball screw

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Figure 10a depicts the load distribution of three models, with the first ball positioned 10 mm from the fixed end of the nut. The results reveal a high-pressure region for the first row of balls near the fixed end, accompanied by significant fluctuations in contact force, a phenomenon observed in both the proposed model and the FEM model. However, Lin's model exhibits a comparatively smoother behavior. Moving to the second row of balls, both the model described in this study and the FEM model demonstrate a similar trend, with the former displaying lower error compared to the Lin model. Further analysis delves into the influence of ball position within the nut on contact force. As depicted in Fig. 10b, it is evident that the fluctuation in ball contact force distribution increases with distance. However, the contact force behavior in Lin’s model remains consistent due to its neglect of nut deformation. Consequently, the disparate verification behavior stems from Lin’s model assumption of nut rigidity and disregard for bending deformation.

Fig. 10
figure 10

a Comparison of load distribution when the ball is 10 mm away from the fixed end of nut, and b impact of the load distribution on the ball location within the nut

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Considering the accuracy of the proposed model compared with the Lin and FEM models, it is meaningful to compare the three models in terms of computational cost. This paper uses the same parameters as Lin's paper to compare two cases (as shown in Table 2). Compared with the Lin model, the calculation time of this model is reduced by 18%. In addition, compared with FEM, our model runs about 101 times faster. Therefore, this paper is more suitable for the parameter research and optimization design of ball screws.

Table 2 Comparing the cost of calculation
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3.2 Experimental verification of load-deformation of ball screw

Figure 11a depicts the screw-nut pair rigidity test setup. A piezoelectric force sensor (Sinocera CL-YD-3210) is used to measure the axial force from the hydraulic loading device. Eddy current sensor (DH902) is used to measure the relative displacement of the screw-nut pair. Table 3 lists the model parameters for the BSFS under study. Figure 11b shows the load-deformation results from the four experiments. Its load-deformation curve varies within a certain range because of the screw-nut pair geometric error [24]. The load-deformation relationship of the flexible nut is more closely aligned with the experimental results, and the calculated results are marginally greater than the experimental values. The deviation was caused by the inaccurate pre-tightening force, inaccurate size of the screw and nut, and poor manufacturing of balls. Figure 11c illustrates the external axial force loading procedure. Since the equipment used in this paper is a hand-pressed hydraulic loading device, this will cause the increment of axial force applied to change each time. The next loading is carried out only after the previous loading has stabilized to ensure test accuracy. The whole loading process is completed in 50 s. The experimental results reveal that the disparity between the rigid nut model and experimental data widens as the axial load increases. Conversely, the model utilized in this paper exhibits closer alignment with the experimental values. This discrepancy arises from the ball bearing greater contact force under high loads, leading to noticeable bending deformation of the nut. Such deformation significantly impacts the performance and accuracy of the system, thereby rendering the results presented in this paper closer to the experimental values compared to those obtained from the rigid nut model. In practical engineering applications, nut deformation is essential to ensure optimal system performance and stability, particularly in scenarios involving high loads and precision requirements.

Fig. 11
figure 11

Test for screw-nut pair rigidity. a Stiffness test layout, b screw nut component load-deformation curves and fitted result, and c axial force loading process of ball screw system

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Table 3 Structural parameters for ball screw
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3.3 Experimental verification of dynamic equation of ball screw

Figure 12 shows the layout of the ball screw excitation experiment. Through the power amplifier (Sinocera YE 5874A), a sinusoidal excitation signal produced by the signal generator (Sinocera YE1311) is sent into the shaker (Sinocera JZK-50). The workbench is then subjected to harmonic excitation via a piezoelectric impedance sensor (CL-YD-331A) and a string. A charge amplifier (Sinocera YE5852) amplifies the excitation force signal, which is then recorded by a signal acquisition device (DH5956). Acceleration response at various screw positions is monitored using a three-directional piezoelectric accelerometer (PCB356A33) and two unidirectional piezoelectric accelerometers (PCB352C04). At a sampling frequency of 20 kHz, the acceleration response is recorded concurrently with the excitation force data. Frequency and amplitude within the excitation range of 500 N and 500 Hz are selected and applied to the worktable to simulate actual ball screw operation. The discrepancy between the calculated and experimental results in the time and frequency domains is shown in Figs. 13 and 14, respectively.

Fig. 12
figure 12

Ball screw excitation test layout

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Fig. 13
figure 13

Comparison of excitation experiments when worktable at 0.5 L

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Fig. 14
figure 14

Comparison of excitation experiments when worktable at 0.25 L

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The axial vibration response signals of the screw at different positions (0.2 L, 0.5 L, 1 L) with the worktable at 0.5 L and 0.25 L are shown in Figs. 13 and 14, respectively. The rigid and flexible nut models represent whether the deformation of the nut is considered when calculating the stiffness of the screw-nut pair (kn) in the dynamic calculation process, respectively. The stiffness of the screw-nut pair determined by two statics methods is introduced into the dynamic analysis of continuous ball screw proposed in this paper. At low loads, the small contact force between the balls causes a relatively small bending deformation of the screw and nut, which is the main reason for the divergence between the experiment and the model. In addition, the response of the system under low load is very small, and it is more susceptible to environmental factors and other variables. In the Figs. 13 and 14, there are some discrepancies in the amplitudes of 2f and 3f. However, the expected amplitude of the fundamental frequency agrees well with the measured amplitude. Compared to the simulated findings, the measured values show more frequency components. However, the measured values closely match the main frequencies of the results presented in this paper in terms of amplitude. To be more precise, the estimated values are somewhat smaller than the experimental data. The reason for this disparity between the experiment and the model is that the screw and nut only slightly bent as a result of the tiny contact force between the balls. The results indicate that when the table is at 0.25 L, the acceleration response of each position of the screw is more pronounced compared to when the screw is at 0.5 L. The simulation results corroborate this observation, affirming that the proposed flexible nut model can effectively predict the vibration response of the screw at different positions.

Figure 15a shows the axial hammering test setup for the BSFS. The impact testing equipment includes a force hammer and a piezoelectric accelerometer. The signals are collected by a data acquisition system. Hammering tests were conducted at eight positions within the travel range of 100–700 mm at intervals of 100 mm. In Fig. 15b, the first mode shape is compared. It can be observed that the axial natural frequency is related to the position of the worktable. The experimental results agree well with the simulations, with the same trend and a maximum relative error of less than 6.5%. The frequency response function with the worktable fixed at 400 mm is shown in Fig. 15c. The axial natural frequencies for the first four modes are 182 Hz, 479 Hz, 771 Hz, and 1210 Hz. The errors compared to the predictions of the model presented in this paper are 6.45%, 14.5%, 9.07%, and 8.45%, respectively (as shown in Fig. 15d). The experimental results match well with the simulations, with the discrepancies primarily attributed to the omission of the slider effects in the dynamic model.

Fig. 15
figure 15

Modal testing of BSFS. a Impact testing instrument setup, b Comparison of measured and simulation first-order modes, and c, d frequency response function and comparison between experiment and simulation when the workbench is located at 400 mm

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4 Numerical simulation

Comprehending the contact properties of the screw-nut pair is essential for the BSFS. Consideration of factors such as the structural dimensions of components, nut movement position, working conditions, and other variables is necessary to ensure system performance and reliability. This section illustrates the complex contact characteristics of the ball screw through numerical simulation. Calculations are performed to assess the effects of geometric error, nut position, pitch, raceway curvature ratio, and ball number on load distribution. Furthermore, an analysis is conducted about the impact of the screw fastening mode and nut location on the ball screw vibration mode.

4.1 Analysis of the ball screw load distribution and axial contact stiffness

Geometric errors are inevitable while ball machining is being done. The geometric error is assumed to be within 0.5 μm and 1 μm of the maximum variation range in the random distribution, respectively. Figure 16a and b illustrate the geometric errors of the balls generated randomly and the corresponding calculated load distributions, respectively. As shown in Fig. 16, the equivalent error of contact force is within the interval [− 8.6, 7.9]N when the error is ± 0.5 μm, while the geometric error is ± 1 μm, that of contact force is in the interval of [− 23.2, 18.7]N. This fact demonstrates that the bigger the geometric error of balls, the stronger non-uniformity of their contact loads, which can result in the uneven wear of the ball screw and decrease its service life.

Fig. 16
figure 16

Random error of balls and its contact force. a Geometric error curve of ball, and b Load distribution curve with geometric error

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Figure 17a illustrates the relationship between the contact force of the ball and the position of the nut. The ball screw is assumed to be subjected to an axial load of Fa = 5000 N. It is evident that the ball contact force exhibits complex nonlinear behavior when the nut position is changed. The load fluctuates more violently when the nut is close to the screw fixed end than when it is closer to the screw middle. However, the contact angle always shows the opposite trend to the contact force (as shown in Fig. 17b).

Fig. 17
figure 17

Contact characteristics of ball screw. a Relationship between nut position and contact force, and b relationship between nut position and contact angle

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Figure 18a illustrates the impact of screw pitch on ball contact force, demonstrating a positive correlation between screw pitch and maximum ball contact force. As the screw pitch increases, the system load distribution becomes more uneven. The findings indicate that smaller pitch ball screws exhibit greater bearing capacity under equivalent axial external loads. Consequently, reducing the pitch contributes to enhancing system stiffness. Figure 18b depicts the impact of raceway curvature ratio (t) on load distribution. The curvature ratio of the three raceways is like the corresponding contact force range, which is [194.5, 243.2] N. Therefore, it can be concluded that the raceway curvature ratio has little effect on ball contact force. The impact of the number of balls on the load distribution is displayed in Fig. 18c, where Z = 32, 48, and 64. That is, from the initial two rows to three or four rows. The number of balls in each row is 16. Please refer to the appendix for the corresponding ball arrangement details [33]. It is evident that as the quantity of loaded balls increases, the ball contact force reduces as well. This phenomenon is explained by the fact that the screw axis deforms less as the quantity of loaded balls increases. Therefore, when the axial load is large, increasing the number of balls contributes to improving the bearing capacity of the ball screw.

Fig. 18
figure 18

Relationship between ball load distribution and structural parameters of ball screw. a pitch, b raceway curvature ratio, and c ball numbers

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The changes in contact force and stiffness of the system as the ball screw experiences an axial load are depicted in Fig. 19. As the axial load increases, both the ball contact force and the contact stiffness of the screw-nut pair increase. The normal contact force of the ball increases roughly linearly with the increase in axial load, as shown in Fig. 19a. The flexible nut model exhibits less stiffness than the rigid nut model, as depicted in Fig. 19b, and the gap between the two widens as the external force increases. The maximal normal contact force of a ball with a geometric error is computed in Fig. 19c and e. According to the findings, a ball with a positive error has a substantially higher contact force than a ball without an error. Moreover, the contact force of the ball increases with the increase of error. This indicates that a ball screw with geometric error will experience more significant axial deformation under the same external force. The axial contact stiffness of the ball screw decreases with increasing ball geometric error, as shown in Fig. 19d. Figure 19e and f illustrate that the contact force of the ball follows the same trend when the ball has a negative error, though the increment is not significant. This is because when one ball has a negative error, the other balls share the load of the negative error ball.

Fig. 19
figure 19

Ball contact force and screw-nut pair stiffness diagram under different axial forces. a Normal contact force of ball without error, b comparison of stiffness between flexible and rigid models of nuts, c normal contact force with ball positive error, and d normal contact force with ball negative error

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4.2 Effect of screw-nut pair stiffness on axial and torsional vibration modes of screw

As shown in Figs. 13 and 14, the acceleration response at different positions of the screw is distinct when the worktable is at 0.5 L and 0.25 L. Displacement responses at three locations (0.2 L, 0.55 L, and 1 L) are calculated using the same excitation frequencies and amplitudes as shown in Figs. 13 and 14. The calculation results are shown in Figs. 20 and 21. It is found that in Fig. 20, the displacement response at the 0.2 L position is significantly higher than that at the other two positions. However, a different trend is shown in Fig. 21, and the displacement response at 0.55 L position is the largest. In addition, the displacement at the 1 L position is the smallest under both conditions. The results also indicate that the amplitude of vibration response at each position increases with increased excitation amplitude and frequency.

Fig. 20
figure 20

Displacement response of the screw when the worktable is located at 0.5 L

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Fig. 21
figure 21

Displacement response of the screw when the worktable is located at 0.25 L

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Table 4 shows the first four mode shapes of the screw when the worktable is at 0.25 L and 0.5 L positions. The relationship between the first natural frequency and the position of the worktable may be relatively obvious. When the worktable moves, the mass distribution of the system will change, which will lead to the corresponding change of the first-order natural frequency. However, higher-order natural frequencies may not be sensitive to the change of worktable position. Comparing the first mode shapes at two different positions reveals that the axial vibration of the screw increases first and then decreases from the left bearing to the worktable. The positional vibration of the worktable is slight. The axial vibration of the screw exhibits a similar trend from the worktable to the right bearing. In addition, the screw adopts fixed–fixed support methods. Therefore, there is no axial vibration on both sides. This is the same trend as the amplitude change in Figs. 20 and 21. In the second-order mode, the axial vibration in the middle of the screw is slight, and the vibration near the bearing is noticeable. The third-order and fourth-order mode shapes of the two worktable positions are similar, and the vibration changes of the worktable positions are minimal. In other positions, the screw shows strong axial vibration with a large vibration amplitude.

Table 4 Comparison of the first four axial vibration modes of screw when the worktable is located at 0.25 L and 0.5 L
Full size table

Figure 22 shows the relationship between the first-order axial mode of the screw and the position of the worktable when the stiffness of the screw-nut pair is 0.5, 1, and 2 × 108 N/m, respectively. It is found that all the modal shapes are approximately centrosymmetric, which can be attributed to the fact that the fixation methods on both sides of the screws are the same. When the screw-nut pair stiffness is 5 × 107 N/m, the mode shape is approximately saddle-shaped. However, when the stiffness is 1 × 108 N/m and 2 × 108 N/m, the surface diagrams show different trends. It can be observed that the first-order modal function corresponding to the stiffness of the screw-nut pair has a significant influence. All three modes show that the axial vibration of each part of the screw is the smallest when the worktable is located near 0.5 L. In Fig. 22a1, b1, and c1, the first-order vibration mode becomes more complex as the stiffness of the screw-nut pair increases. The maximum amplitude increased obviously from 0.57 to 0.72. At the same time, the area of the vibration groove increases. When the screw-nut pair stiffness is 5 × 107 (as shown in Fig. 22a1 and a2), the vibration shape does not change during the movement of the worktable from 0 L to 0.5 L. The maximum vibration of the screw is always at the position of 0.5 L, but the maximum amplitude of vibration gradually decreases. When the screw-nut pair stiffness is 1 × 108 N/m (as shown in Fig. 22b1 and b2), the minimum value of screw vibration appears between the left bearing and the worktable during the process of moving the worktable from 0 L to 0.5 L, and the vibration on both sides of the worktable increases first and then decreases. When the screw-nut pair stiffness is 2 × 108 N/m (as shown in Fig. 22c1 and c2), the vibration mode trend of the screw is the same as in Fig. 22b1 and b2, but the vibration of the worktable position is always at the minimum point of the vibration of each part of the screw. This demonstrates that when the stiffness of the screw-nut pair reaches a certain threshold, the vibration at the position of the worktable is minimal.

Fig. 22
figure 22

Comparison of first-order axial vibration modes of the screw. a1–a2 color mapping surface diagram and projection of the screw-nut pair with stiffness of 5 × 107 N/m, b1–b2 color mapping surface diagram and projection of the screw-nut pair with stiffness of 1 × 108 N/m, and c1–c2 color mapping surface diagram and projection of the screw-nut pair with stiffness of 2 × 108 N/m

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Figure 23 shows the first, second, third, and fourth modes of axial vibration of the screw with varying stiffness levels of the screw-nut pair. It is observed that the unstable region of the second-order mode increasingly declines as stiffness increases. The rigidity of the screw-nut pair does not significantly impact the third and fourth-order vibration modes.

Fig. 23
figure 23

Comparison of the first four modes of axial vibration of screw under different screw and nut stiffness. a Mapping diagram of the screw-nut pair stiffness 5 × 107 N/m, b mapping diagram of the screw-nut pair stiffness 1 × 108 N/m, and c mapping diagram of the screw-nut pair stiffness 2 × 108 N/m

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Figure 24 illustrates the screw torsion first-order mode at screw-nut pair stiffness values of 0.5, 1, and 2 × 108 N/m, respectively. All three mode diagrams indicate that the torsional vibration of the screw is most significant at the worktable position. With the increase in stiffness, the maximum torsional amplitude of the screw increases from 1.5 to 9.6. When the screw-nut pair stiffness is 2 × 108 N/m, and the worktable is at 0.1 L and 0.9 L positions (as shown in Fig. 24c1 and c2), the torsional vibration of the table position is maximum. As the worktable runs between 0.2 L and 0.8 L, the maximum amplitude of the screw decreases from 9.6 to 6.4. The second, third, and fourth torsional vibration modes of the screw with varying screw-nut pair stiffness are displayed in Fig. 25. It is discovered through comparison that the vibration modes of the second, third, and fourth orders with varying stiffness are identical. Thus, it can be concluded that the second, third, and fourth modes of torsional vibration of the screw are unaffected by the stiffness of the screw-nut pair. Therefore, when the stiffness of the screw-nut pair is large, it is necessary to ensure that the worktable operates within the range of 0.2–0.8 L as much as possible.

Fig. 24
figure 24

Comparison of first-order torsion vibration modes of the screw. a1–a2 Color mapping surface diagram and projection of the screw-nut pair with stiffness of 5 × 107 N/m, b1–b2 color mapping surface diagram and projection of the screw-nut pair with stiffness of 1 × 108 N/m, and c1–c2 color mapping surface diagram and projection of the screw-nut pair with stiffness of 2 × 108 N/m

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Fig. 25
figure 25

Mapping diagram Comparison of the first four modes of torsion vibration of screw under different screw-nut pair stiffness. a 5 × 107 N/m, b 1 × 108 N/m, and c 2 × 108 N/m

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4.3 Influence of screw fixing mode on axial and torsional vibration modes

There is a fixed-free support mode for the ball screw under real working situations in addition to the fixed–fixed support option. Figure 26 shows the first-order mode shapes of the screw left-side fixed and right-side freely supported under three stiffness when the worktable is located at 0.25 L and 0.5 L. When the stiffness remains unchanged, the axial vibration of the screw on the right side of the worktable remains consistent. The reason for this phenomenon is that there is no constraint on the right side of the screw. When the stiffness of the screw-nut pair is kn = 1 × 108 and kn = 2 × 108, the amplitude of the free end of the screw at the right end of the worktable is small, with no significant difference between them. In order to strengthen the distinction, the vibration modes of thread-nut pair stiffness values kn = 1 × 106 and kn = 1 × 107 are added in the Fig. 26. When the screw-nut pair stiffness is 1 × 106 N/m, the first-order vibration mode of the axial vibration of the screw increases at first and then remains unchanged. However, with the increase of stiffness, the first-order mode shape increases first, decreases, and remains stable. Moreover, the greater the stiffness, the smaller the amplitude of the worktable position. The explanation lies in the fact that as the stiffness of the screw-nut pair increases, the contact between the nut and the screw can be roughly likened to a fixed support. Therefore, we can be boldly speculated that when the stiffness is large enough, the screw on the right side of the worktable will not produce axial vibration.

Fig. 26
figure 26

The screw uses a first-order mode shape with fixed-free support. a The worktable is located at 0.25 L, and b the worktable is located at 0.5 L

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Figure 27 shows the first-order mode of the worktable at different positions when the screw is fixed-free supported. It can be observed that the first-order modal shapes corresponding to the three stiffness are similar, and the increase of stiffness has little effect on the maximum amplitude of the screw. As the stiffness of the screw-nut pair increases, the vibration area of the screw decreases. Greater stiffness in the screw-nut pair contributes to improved running stability of the fixed-free support ball screws.

Fig. 27
figure 27

Comparison of first-order axial vibration modes of screw with fixed-free support. a1–a2 Color mapping surface diagram and projection of the screw-nut pair with stiffness of 1 × 106 N/m, b1–b2 color mapping surface diagram and projection of the screw-nut pair with stiffness of 1 × 107 N/m, and c1–c2 color mapping surface diagram and projection of the screw-nut pair with stiffness of 1 × 108 N/m

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Figure 28 shows the first-order torsional mode when the screw is fixed-free supported. It is observed that the torsional vibration mode of the screw remains unaffected by the stiffness of the screw-nut pair. The first-order torsional modes corresponding to the three stiffness values exhibit a trend of increasing initially, followed by stabilization. In addition, the maximum torsional amplitude of the screw shows a downward trend during the movement of the worktable position. Therefore, to guarantee structural stability and controllability of machining performance, the actual stroke of the worktable should be confined to the middle portion of the screw. This avoids movement instability caused by proximity to the fixed-end bearing.

Fig. 28
figure 28

Comparison of first-order torsional vibration modes of screw with fixed-free support

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5 Conclusion

This study presents a contact model of a screw-nut pair, considering both the screw and nut as flexible beams. The geometric relationship between the ball and raceway centers incorporates the effects of axial, torsional, and lateral deformations of the screw and nut on ball contact forces. Additionally, the screw is modeled as a continuum, and the impact of screw-nut pair stiffness on axial and torsional vibrations is described by the feed system dynamic model. The influence of the screw shaft fixing method on vibration modes is also discussed. The main conclusions are as follows:

  1. (1)

    The contact force of the first row of balls varies significantly when the ball screw is subjected to an axial load. The nut position on the screw affects the shape of the load distribution curve. As the distance between the load end of the nut and the first ball increases, the non-uniformity of ball contact loads also increases.

  2. (2)

    The flexibility of the nut decreases the stiffness of the screw-nut pair, and this effect increases with the axial force on the ball screw. The computational results of the flexible nut model in this study align well with experimental outcomes, especially under high axial stress. Additionally, the contact stiffness of the screw-nut pair is reduced by the geometrical inaccuracy of the balls.

  3. (3)

    The lower modes of the screw axial and torsional vibrations are significantly affected by the stiffness of the screw-nut pair, while the higher modes remain largely unaffected. As the rigidity of the screw-nut pair increases, both the maximum amplitude of the screw vibration and the unstable zone grow.

  4. (4)

    As the rigidity of the screw-nut pair increases in fixed-free supported screws, the axial vibration of the worktable gradually decreases. At sufficiently high stiffness, the nut effectively acts as a fixed support. However, increasing stiffness doesn't affect the torsional vibration of fixed-free supported screws.