# Study of the Dynamic Characteristics of Ball Screw with a Load Disturbance

Authors: Zhe Du; Xiao-Lan Zhang; Tao Tao
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

The dynamic character of ball screw is the key factor that influences the machining accuracy of numerical control (NC) machine tool. To improve the dynamic characteristics of the NC machine tool, it is necessary to study the dynamic characteristics of a ball screw. In this paper, the kinematics of a ball screw mechanism (BSM) are studied to expound the dynamic process of the drive, and the load disturbance is considered to analyze the contact deformation based on the Hertzian contact theory. The velocity relationships among the ball, screw, and nut are analyzed, and the influence of the contact deformation on the dynamic characteristics is simulated and investigated experimentally. The results show that the relationships between the contact deformation, which is affected by the material characteristics, the contact angle, and the load of nut are nonlinear. The contact deformation is a factor that cannot be ignored when considering the dynamic machining error of high-speed and high-precision machine tools.


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## Body

## 1. Introduction

The numerical control (NC) machine tool is an important factor in the modern manufacturing industry, and the level of industrial development of a nation is much more dependent on the technology of NC machine tools. A ball screw is a precision transmission part of the feed unit of a machine tool, and its characteristics directly impact the precision of the machine tool. Although the linear motor is a new method to operate the feed unit, as a proven technique, the ball screw will play an important part in feed unit operations for a long time [1]. Currently, high-speed and high-precision machining techniques are thought to improve machining efficiency and precision; hence, it is necessary to intensely study the dynamic characteristics of ball screws.The dynamic model, the modal parameter identification method, and the control character have been widely studied in previous works. Vicente et al. [2] present a high-frequency finite-dimension model of a ball screw in which the ball screw is considered as a continuous subsystem. In this model, the first model of a ball screw is rigid, and the axial and angular deformations only affect the mode in the second and fourth mode shapes, respectively. Varanasi and Nayfeh [3] present a distributed inertia model to study the dynamic characteristics of a ball screw in a closed loop control system. The finite right-half plane zero in the transfer function between motor torque and carriage position is accurately predicted, and the significance of incorporating a viscoelastic damper into one of the lead-screw support bearings in the system transfer functions is noted. In a study by Okwudire [4], the deformation of a screw in a nut is considered with Timoshenko beam-shape functions for modeling the ball screw. Using this method, the identification precision of the natural frequency of the ball screw is improved. Okwudire and Altintas [5] study the lateral dynamic parameter, which is omitted in previous research, and show that the lateral dynamic parameter has an important influence on the dynamic characteristics of a ball screw. In these studies, the holistic dynamics of ball screws are studied meticulously; however, the characteristics of a combined system, including a ball, screw, and nut, are almost ignored.Lin et al. [6] studied the ball screw mechanism theoretically with the aim of finding a mathematical description of the ball, screw, and nut kinematic and contact patterns. In that study, the slide phenomenon of a ball with a screw and nut is interpreted. In another paper, Lin et al. [7] analyze the factors that influence the efficiency of a ball screw and provide the formula for efficiency based on contact angle and helical angle. Wei and Lin [8] further study the BSM based on the research result of [6]. The influential parameters, such as the friction coefficient, normal force acting on the ball, and contact angle on a ball screw’s mechanism at two contact areas, are simulated. Feng and Pan [9] use the vibrational signal of a nut (on a double nut and preload-adjustable ball screw feed drive system) to research the relationship between the preload and the vibration of a nut. Huang and Ravani [10] introduce the medial axis transform (MAT) method into the BSM analysis process. The force analysis of a ball, based on the kinematic and Hertzian contact, is provided. Cuttino et al. [11] research the nonlinear relationship between the load and deformation of a single nut ball screw contact. The instant stress state is analyzed, and the relationship of load and deformation is drawn. Mei et al. [12], taking the machining error of a ball screw into consideration, study the force distribution model of a ball based on the Hertzian contact theory.The research presented above encompasses most of the dynamic and kinematic theory. However, the influence of the contact deformation on the motion precision of a ball screw is rarely investigated. In this study, the dynamic error of a ball screw is researched under a load disturbance condition that was partly based on the theoretical analyses developed by Lin et al. [6]. In Section 2, the kinematics of a ball screw are analyzed, and the velocity deviation between the ball and the screw is given in the absolute coordinate system. The force analysis of a ball is given in Section 3. The factors that influence the contact deformation are simulated in Section 4. The load disturbance experiments are implemented in Section 5. Finally, the conclusions are summarized in Section 6.


## 2. Theoretical Analysis of BSM

### 2.1. Motion Analysis of Ball Screw

The ball screw, which is composed of balls, a screw, and a nut, can achieve the transmission of linear motion and rotation. The analysis process is limited to the transmission from rotational motion to linear motion because that is the function of a ball screw in a machine tool. In the transmission process, the rotation of the screw compels the balls to move around the screw. At the same time, the balls generate a spinning motion. The drive torque from the motor causes not only the axial and angular deformation of the screw but also the contact deformation of the balls, the screw, and the nut, according to the Hertzian contact theory. Hence, the motion precision of the table will be influenced. To study the motion error caused by the contact deformation, the relative motion of the ball, screw, and nut should be analyzed first.Three coordinate systems should be established, as shown in Figure1. The first coordinate system, X′x′,y′,z′, is fixed with its z′ axis accordant with the axis of the screw, and x′y′ plane is perpendicular to the screw axis. The second coordinate system, Xx,y,z, is used to describe the rotation of the screw rotate Ω around z′ axis. According to the rigid-body kinematics, the coordinate transformation relationship from X to X′ is (1)x′y′z′=T1xyz,where T1=CΩ-SΩ0SΩCΩ0001 and CΩ=cos⁡Ω, SΩ=sin⁡Ω, as follows.Figure 1
The position of the ball center in the chosen coordinate system.Assume that one of the balls moves an angleθ along the helical groove of the screw. So, the coordinate of the ball center in X system is (2)Pr=rmCθrmSθrmθtαT,where tα=L/2πrm, rm is the nominal radius of screw, L is the pitch of screw, and α is helical angle of the groove.InX′ system the coordinate is(3)Pw=T1rmCθrmSθrmθtαT=rmCθCΩ-rmSθSΩSΩrmCθ+CΩrmSθrmθtα.The Frenet coordinate system, Yt,n,b, is defined to describe the motion of the ball center along the groove. The origin is set on the ball center, t axis is in the tangential direction of groove, and nb plane is the cross section of the groove. The Eulerian angle between X and Y is π/2αθ. The coordinate transformation relationship is(4)xyz=P+-CαSθ-CαCθSαCθ-Sθ0SαSθSαCθCαtnb.The ball can only move relatively along t direction of Y system due to the restriction of the groove. Hence, the centers of the contact region are inevitably in nb plane (Figure 2). The contact point between the ball and nut in Y system is(5)tnbA=0rbCβArbSβAT,and the contact point of the contact point between the ball and nut in Y system is(6)tnbB=0-rbCβB-rbSβBT,where βii=A,B is the contact angle between n axis and the contact vector. rb is the radius of the ball.Figure 2
The contact point of the ball and the nut (screw),A,B, in the normal plane of the Frenet coordinates.Assume that the contact deformation has no influence on the ball center inY system, and the deformation can only affect the rotary angle of the screw and the nut position in X′ system. Taking the contact deformation, δjj=A,B, into consideration, the contact point position in Y system is(7)t¯n¯b¯A=0rb-δACβArb-δASβAT,t¯n¯b¯B=0-rb-δBCβB-rb-δBSβBT,and the contact point position in X′ system is (8)xyzA=T1P+T2YA′=rmCθCΩ-SθSΩ+rb-δACβASθSΩ-SβASαCΩ-CβACαCθCΩrmCθSΩ+SθCΩ+rb-δACβACΩSθ-CβACαCθSΩ-SβASαSΩrmθtα+rb-δACβASαCθ-SβACα,xyzB=T1P+T2YA′=rmCθCΩ+SθSΩ+rb-δbCβBCαCθCΩ+SβBSαCΩ+CβBSθSΩrmCθSΩ-SθCΩ+rb-δbCβBCαCθSΩ+SβBSαSΩ+CβBSθCΩrmθtα+rb-δbSβBCα-CβBSαCθ.By the position analysis of the ball, screw, and nut in X′ system, considering the contact deformation, the mutual affect will be analyzed in the following section.


### 2.2. Velocity Analysis of the Ball, Screw, and Nut

Mutual motion of the ball, screw, and nut, compelled by the drive torque, impacts the stability and precision of the transmission. It is necessary to study the motion process. First, assuming that the spinning angular velocity of the ball isω→=ωtωnωbT, ωi(i=t,n,b) is the ball angular velocity around the t,n,b axes of Y system. The velocity of the ball at Bb contact point in the X system is(9)XB˙=P˙+ω→×YB′=-rmSθrmCθrmtαθ˙+-rb-δBSβBωn+rb-δBCβBωbrb-δBSβBωt-rb-δBCβBωt.In X′ system, the velocity of the ball at Bb contact point is(10)VBb=T1XB˙=T1˙XB+T1XB˙=Ω˙-rmCθSΩ+SθCΩ-rb-δBCβBCαCθSΩ-SβBSαSΩ+CβBSθCΩrmCθCΩ+SθSΩ+rb-δBCβBCαCθCΩ-SβBSαCΩ+CβBSθSΩ0+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrmtαθ˙+rb-δBCβBωbCΩ-SβBωnCΩ-SβBωtSΩrb-δBCβBωbSΩ-SβBωnSΩ+SβBωtCΩ-rb-δBCβBωt.In X′ system, the angular velocity of the screw is marked as 00Ω˙T, and the velocity of the contact point on the screw, Bs, is(11)VBs=00Ω˙×XB=00Ω˙rmCθ+rb-δBCβBCαCθ-SβBSαrmSθ+rb-δBCβBSθrmθtα-rb-δBCβBSαCθ+SβBCα=-rmCθ-rb-δbCβBCαCθ+SβBSαrmSθ+rb-δbCβBSθ0Ω˙.Apparently, there are deviations between the ball velocity and the screw at the contact point B:(12)ΔVBbs=-rmCθSΩ+SθCΩ+Cθ-rb-δBCβBCαCθSΩ-1+SβBSαSΩ-1+CβBSθCΩrmCθCΩ+SθSΩ-Sθ+rb-δBCβBCαCθCΩ-SβBSαCΩ+CβBSθSΩ-10Ω˙+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrmtαθ˙+rb-δBCβBωb-SβBωnCΩ-rb-δBSβBωtSΩrb-δBCβBωb-SβBωnSΩ+rb-δBSβBωtCΩ-rb-δBCβBωt.The simulation result of [7] shows that the spinning velocity of the ball around t axis, ωt, is so small that it can be omitted. Therefore, the approximated expression of (12) is(13)ΔVBbs=-rmCθSΩ+SθCΩ+Cθ-rb-δBCβBCαCθSΩ-1+SβBSαSΩ-1+CβBSθCΩrmCθCΩ+SθSΩ-Sθ+rb-δBCβBCαCθCΩ-SβBSαCΩ+CβBSθSΩ-10Ω˙+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrmtαθ˙+rb-δBCβBωb-SβBωnCΩrb-δBCβBωb-SβBωnSΩ0.Similarly, the velocity of contact point Ab on the ball at the contact region of the ball and nut in X′ system is (14)VAb=T1XA˙=T1˙XA+T1XA˙=Ω˙-rmCθSΩ+SθCΩ+rb-δACβACαCθSΩ-SβASαSΩ+CβASθCΩrmCθCΩ+SθSΩ-rb-δACβACαCθCΩ-SβASαCΩ+CβASθSΩ0+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrmtαθ˙-rb-δACβAωbCΩ-SβAωnCΩ-SβAωtSΩrb-δACβAωbSΩ-SβAωnSΩ+SβAωtCΩ-rb-δACβAωt.In the feed unit of the machine tool, the nut is fixed to the table and can only move along the guide. So, the nut is restricted by the screw but can move in line. In X′ system, the velocity of the contact point An on the nut is(15)VAn=00rmtα-rb-δACβASαSθθ˙+00-CβASαCθ-SβACαδA˙.Because δA is a small value, the derivation of δA, with δA˙≈0, using an approximated expression of (15) is(16)VAn=00rmtα-rb-δACβASαSθθ˙.There is a difference between the nut and ball at the contact point A:(17)ΔVAbn=Ω˙-rmCθSΩ+SθCΩ+rb-δACβACαCθSΩ-SβASαSΩ+CβASθCΩrmCθCΩ+SθSΩ-rb-δACβACαCθCΩ-SβASαCΩ+CβASθSΩ0+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrb-δACβASαSθθ˙-rb-δACβAωbCΩ-SβAωnCΩ-SβAωtSΩrb-δACβAωbSΩ-SβAωnSΩ+SβAωtCΩ-rb-δACβAωt.According to the velocity analysis of the ball screw parts, there inevitably exists a relative slide in the ball, screw, and nut. The slide is one of the major factors that cause the wear of the ball screw.Based on the theory in [6], the velocity of the nut in z′ direction is (18)Vn=Ω˙L2π.In calculation, the velocity of the nut in z′ direction is(19)L2π-rb-δACβASαSθθ˙.The velocity of the nut is disturbed by the contact angle, βA, contact deformation, δA, relative position, θ, and the velocity, θ˙, of the ball around the screw axis. The variation of the nut velocity will affect the dynamic precision of the table.


## 2.1. Motion Analysis of Ball Screw

The ball screw, which is composed of balls, a screw, and a nut, can achieve the transmission of linear motion and rotation. The analysis process is limited to the transmission from rotational motion to linear motion because that is the function of a ball screw in a machine tool. In the transmission process, the rotation of the screw compels the balls to move around the screw. At the same time, the balls generate a spinning motion. The drive torque from the motor causes not only the axial and angular deformation of the screw but also the contact deformation of the balls, the screw, and the nut, according to the Hertzian contact theory. Hence, the motion precision of the table will be influenced. To study the motion error caused by the contact deformation, the relative motion of the ball, screw, and nut should be analyzed first.Three coordinate systems should be established, as shown in Figure1. The first coordinate system, X′x′,y′,z′, is fixed with its z′ axis accordant with the axis of the screw, and x′y′ plane is perpendicular to the screw axis. The second coordinate system, Xx,y,z, is used to describe the rotation of the screw rotate Ω around z′ axis. According to the rigid-body kinematics, the coordinate transformation relationship from X to X′ is (1)x′y′z′=T1xyz,where T1=CΩ-SΩ0SΩCΩ0001 and CΩ=cos⁡Ω, SΩ=sin⁡Ω, as follows.Figure 1
The position of the ball center in the chosen coordinate system.Assume that one of the balls moves an angleθ along the helical groove of the screw. So, the coordinate of the ball center in X system is (2)Pr=rmCθrmSθrmθtαT,where tα=L/2πrm, rm is the nominal radius of screw, L is the pitch of screw, and α is helical angle of the groove.InX′ system the coordinate is(3)Pw=T1rmCθrmSθrmθtαT=rmCθCΩ-rmSθSΩSΩrmCθ+CΩrmSθrmθtα.The Frenet coordinate system, Yt,n,b, is defined to describe the motion of the ball center along the groove. The origin is set on the ball center, t axis is in the tangential direction of groove, and nb plane is the cross section of the groove. The Eulerian angle between X and Y is π/2αθ. The coordinate transformation relationship is(4)xyz=P+-CαSθ-CαCθSαCθ-Sθ0SαSθSαCθCαtnb.The ball can only move relatively along t direction of Y system due to the restriction of the groove. Hence, the centers of the contact region are inevitably in nb plane (Figure 2). The contact point between the ball and nut in Y system is(5)tnbA=0rbCβArbSβAT,and the contact point of the contact point between the ball and nut in Y system is(6)tnbB=0-rbCβB-rbSβBT,where βii=A,B is the contact angle between n axis and the contact vector. rb is the radius of the ball.Figure 2
The contact point of the ball and the nut (screw),A,B, in the normal plane of the Frenet coordinates.Assume that the contact deformation has no influence on the ball center inY system, and the deformation can only affect the rotary angle of the screw and the nut position in X′ system. Taking the contact deformation, δjj=A,B, into consideration, the contact point position in Y system is(7)t¯n¯b¯A=0rb-δACβArb-δASβAT,t¯n¯b¯B=0-rb-δBCβB-rb-δBSβBT,and the contact point position in X′ system is (8)xyzA=T1P+T2YA′=rmCθCΩ-SθSΩ+rb-δACβASθSΩ-SβASαCΩ-CβACαCθCΩrmCθSΩ+SθCΩ+rb-δACβACΩSθ-CβACαCθSΩ-SβASαSΩrmθtα+rb-δACβASαCθ-SβACα,xyzB=T1P+T2YA′=rmCθCΩ+SθSΩ+rb-δbCβBCαCθCΩ+SβBSαCΩ+CβBSθSΩrmCθSΩ-SθCΩ+rb-δbCβBCαCθSΩ+SβBSαSΩ+CβBSθCΩrmθtα+rb-δbSβBCα-CβBSαCθ.By the position analysis of the ball, screw, and nut in X′ system, considering the contact deformation, the mutual affect will be analyzed in the following section.


## 2.2. Velocity Analysis of the Ball, Screw, and Nut

Mutual motion of the ball, screw, and nut, compelled by the drive torque, impacts the stability and precision of the transmission. It is necessary to study the motion process. First, assuming that the spinning angular velocity of the ball isω→=ωtωnωbT, ωi(i=t,n,b) is the ball angular velocity around the t,n,b axes of Y system. The velocity of the ball at Bb contact point in the X system is(9)XB˙=P˙+ω→×YB′=-rmSθrmCθrmtαθ˙+-rb-δBSβBωn+rb-δBCβBωbrb-δBSβBωt-rb-δBCβBωt.In X′ system, the velocity of the ball at Bb contact point is(10)VBb=T1XB˙=T1˙XB+T1XB˙=Ω˙-rmCθSΩ+SθCΩ-rb-δBCβBCαCθSΩ-SβBSαSΩ+CβBSθCΩrmCθCΩ+SθSΩ+rb-δBCβBCαCθCΩ-SβBSαCΩ+CβBSθSΩ0+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrmtαθ˙+rb-δBCβBωbCΩ-SβBωnCΩ-SβBωtSΩrb-δBCβBωbSΩ-SβBωnSΩ+SβBωtCΩ-rb-δBCβBωt.In X′ system, the angular velocity of the screw is marked as 00Ω˙T, and the velocity of the contact point on the screw, Bs, is(11)VBs=00Ω˙×XB=00Ω˙rmCθ+rb-δBCβBCαCθ-SβBSαrmSθ+rb-δBCβBSθrmθtα-rb-δBCβBSαCθ+SβBCα=-rmCθ-rb-δbCβBCαCθ+SβBSαrmSθ+rb-δbCβBSθ0Ω˙.Apparently, there are deviations between the ball velocity and the screw at the contact point B:(12)ΔVBbs=-rmCθSΩ+SθCΩ+Cθ-rb-δBCβBCαCθSΩ-1+SβBSαSΩ-1+CβBSθCΩrmCθCΩ+SθSΩ-Sθ+rb-δBCβBCαCθCΩ-SβBSαCΩ+CβBSθSΩ-10Ω˙+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrmtαθ˙+rb-δBCβBωb-SβBωnCΩ-rb-δBSβBωtSΩrb-δBCβBωb-SβBωnSΩ+rb-δBSβBωtCΩ-rb-δBCβBωt.The simulation result of [7] shows that the spinning velocity of the ball around t axis, ωt, is so small that it can be omitted. Therefore, the approximated expression of (12) is(13)ΔVBbs=-rmCθSΩ+SθCΩ+Cθ-rb-δBCβBCαCθSΩ-1+SβBSαSΩ-1+CβBSθCΩrmCθCΩ+SθSΩ-Sθ+rb-δBCβBCαCθCΩ-SβBSαCΩ+CβBSθSΩ-10Ω˙+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrmtαθ˙+rb-δBCβBωb-SβBωnCΩrb-δBCβBωb-SβBωnSΩ0.Similarly, the velocity of contact point Ab on the ball at the contact region of the ball and nut in X′ system is (14)VAb=T1XA˙=T1˙XA+T1XA˙=Ω˙-rmCθSΩ+SθCΩ+rb-δACβACαCθSΩ-SβASαSΩ+CβASθCΩrmCθCΩ+SθSΩ-rb-δACβACαCθCΩ-SβASαCΩ+CβASθSΩ0+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrmtαθ˙-rb-δACβAωbCΩ-SβAωnCΩ-SβAωtSΩrb-δACβAωbSΩ-SβAωnSΩ+SβAωtCΩ-rb-δACβAωt.In the feed unit of the machine tool, the nut is fixed to the table and can only move along the guide. So, the nut is restricted by the screw but can move in line. In X′ system, the velocity of the contact point An on the nut is(15)VAn=00rmtα-rb-δACβASαSθθ˙+00-CβASαCθ-SβACαδA˙.Because δA is a small value, the derivation of δA, with δA˙≈0, using an approximated expression of (15) is(16)VAn=00rmtα-rb-δACβASαSθθ˙.There is a difference between the nut and ball at the contact point A:(17)ΔVAbn=Ω˙-rmCθSΩ+SθCΩ+rb-δACβACαCθSΩ-SβASαSΩ+CβASθCΩrmCθCΩ+SθSΩ-rb-δACβACαCθCΩ-SβASαCΩ+CβASθSΩ0+-rmSθCΩ+CθSΩrmCθCΩ-SθSΩrb-δACβASαSθθ˙-rb-δACβAωbCΩ-SβAωnCΩ-SβAωtSΩrb-δACβAωbSΩ-SβAωnSΩ+SβAωtCΩ-rb-δACβAωt.According to the velocity analysis of the ball screw parts, there inevitably exists a relative slide in the ball, screw, and nut. The slide is one of the major factors that cause the wear of the ball screw.Based on the theory in [6], the velocity of the nut in z′ direction is (18)Vn=Ω˙L2π.In calculation, the velocity of the nut in z′ direction is(19)L2π-rb-δACβASαSθθ˙.The velocity of the nut is disturbed by the contact angle, βA, contact deformation, δA, relative position, θ, and the velocity, θ˙, of the ball around the screw axis. The variation of the nut velocity will affect the dynamic precision of the table.


## 3. Force Analysis of the Hertzian Contact

The contact problem of the ball screw is the typical rolling contact innb plane. In the process of the ball moving along the groove, the ball bears four forces: the contact force at points A,B,qA,qB and the friction force at points A,B,fA,fB, as shown in Figure 3. Assuming that the friction angles between the friction force and nb plane are φA,φB, the forces in t,n, and b directions, respectively, are(20)Ft=fAqAsin⁡φA-fBqBsin⁡φB,Fn=-qAcos⁡βA+fAqAcos⁡φAsin⁡βA+qBcos⁡βB-fBqBcos⁡φBsin⁡βA,Fb=-qAsin⁡βA-fAqAcos⁡φAcos⁡βA+qBsin⁡βB+fBqBcos⁡φBcos⁡βA.From the force analysis presented in [6],(21)Ft=mrmcos⁡αθ¨+cos2⁡α,Fn=mrmθ˙+Ω˙2,Fb=mrmsin⁡αΩ¨.Assuming an axis load, Fa, is implemented on the nut, Fa will be distributed onto every ball without considering the machining error of the groove. The force on the contact point of one ball and the nut is(22)Faz=qAsin⁡βAcos⁡α-fAqAsin⁡φAsin⁡α+fAqAcos⁡φAcos⁡βAcos⁡α,(23)qA+Fncos⁡βA+Fbsin⁡βAcos⁡βBcos⁡βA+sin⁡βBsin⁡βA=qB,where z is the number of balls in the nut.Figure 3
The force analysis of the ball in the normal plane of the Frenet coordinate system.The contact situation of the ball and screw (nut), which is similar to the bearing, can be represented by two semiellipsoids whose corresponding planes of principal curvature through the point of contact coincide and whose radii of curvature in those planes have the same values as those of the unloaded original solids at the contact point (as shown in Figure4). The relative curvatures in xz plane and yz plane are(24)1Rx=1r1x+1r2x,1Ry=1r1y+1r2y,where rix,riy are principal radii of the curvature of body 1 in xz plane and yz plane, respectively. The effective radius ratio and the ellipticity are(25)γ=RyRx,κ=ba,where a,b are the diameters of the contact ellipses parallel to xz plane and yz plane, respectively.Figure 4
The shape of the contact area between the ball and the nut (screw) in Hertzian contact theory.In the contact form of the ball and nut (screw), the curvature sum is(26)1R=1Rx-1Ry.According to the Hertzian contact theory [13], the relationship between the deformation and the normal force is(27)δ=2ς/ππ/2κ2ξ1/33PR/E′2/32R,where E′=2/1-ν1/E1+1-ν2/E2, ς=∫0π/21-1-κ-2sin2⁡θ-1/2dθ, ξ=∫0π/21-1-κ-2sin2⁡θ1/2dθ, Ei is the elastic modulus of the contact body material, and νi is the Poisson ratio.The precision of the parameter approximate formula forς,ξ is simulated in [12]. The minimum error formulas are chosen as follows:(28)κ=1.03γ0.636,(29)ς=1.5277+1.6023ln⁡γ,(30)ξ=1.0003+0.5968γ-1.By taking (22) and (28)–(30) into (27), the relationship formula changes to(31)δA=ς/π2/31/2κ2ξ1/33Fa/E′zsin⁡βAcos⁡α-fAsin⁡α2/3R1/3.Hence, the positioning error of the table caused by the contact deformation is(32)δz′=δB+δASβACα.


## 4. Simulation of the Contact Deformation

From formula (30), we know that the contact deformation is related to the helical angle (α), the contact angle (βA), the number of balls (z), the curvature sum (R), the equivalent elastic modulus (E), the effective radius ratio (γ), the friction force, and the load. The relationship between the helical angle (α), the contact angle (βA), the equivalent elastic modulus (E), the load, and the contact deformation is studied using a simulation.The simulation geometry and material parameters of the ball screw are listed in Table1. The equation, F=A1+sin⁡ωft, is used as the disturbance load, with ω=20 Hz. Firstly, the contact angle is increased from 10∘ to 90∘ and A=100 N. The variation of the contact deformation caused by the change in the contact angle is shown in Figure 5(a). Next, Young’s modulus of the nut is changed from 1.6×1011N/m2 to 2.4×1011 N/m2 by the step size of 0.1×1011 N/m2. The simulation result is shown in Figure 5(b). Finally, the relationship between the contact deformation and the load, which could be called contact stiffness, is simulated as shown in Figure 6. The load varies from 0 N to 2000 N, the contact angle is 48.5∘, and Young’s modulus values are listed in Table 1.Table 1
Geometry and material parameters of ball screw.
Parameter Value Unit Nut Screw Helical angle 2.967 Degree Pitch 6 mm Ball radius 1.5875 mm Number of balls 71 — Young’s modulus 2.10×1011 2.09×1011 N/m2 Poisson ratio 0.278 0.295 — Relative curvatureRx 1.4588 mm Relative curvatureRy 1.4588 mmFigure 5
(a) The influence of the contact angle on the deformation. (b) The influence of the elastic modal on the deformation.
(a) (b)Figure 6
The relationship between the force and the deformation.As shown in Figure5(a), the relationship between the contact angle and the contact deformation is nonlinear. When the angle is greater than 50∘, the changes in the angle have no obvious impact on the deformation. However, the deformation will increase quickly if the angle is smaller than 50∘. The relationship between Young’s modulus and the contact deformation is nonlinear as well (Figure 5(b)). Increasing the value of the equivalent elastic modulus which is decided by the smaller modulus could decrease the deformation. The so-called contact stiffness of the ball screw is nonlinear, as shown in Figure 6. The deformation will increase to μm magnitude, which will increase the machining error in the high-speed high-precision machine tool.


## 5. Experiment of the Contact Deformation with Load Disturbance

To confirm the influence of the contact deformation on the transmission precision of the ball screw, experiments are implemented on a ball screw test bed (shown in Figure7). On the test bed, a double nut, no-backlash ball screw is used as the experimental object, which has a type number of BNFN-3606-2.5. An open control card is used to drive the servomotor. An encoder is installed at the end of the screw, and a linear encoder is used in the bed to provide feedback of the position of the nut (table). An inertial loading unit is used to simulate the load disturbance. The amplitude and frequency are controlled by the speed of the motor installed on the unit. The relationship between inertia torque and speed is (33)Fn=0.143×10-2×nr2,where nr is the speed of the motor.Figure 7
The ball screw drive set-up.The encoder mounted at the end of the screw, which has a precision of 0.58μm in the axial direction, and the linear encoder, whose precision is 0.1 μm, are used to test the deformation of the contact region. The data acquisition instrument used here was developed by the research group based on the IK220 counter card. The deformation of the screw rotation, which is restricted under the static condition, is tested first. The speed of the inertial loading unit increases from 30 r/min to 330 r/min by a step of 10 r/min. In other words, the peak of the simulated load disturbance increases from 1.287 N to 155.72 N. The signals of the end encoder and the linear encoder are sampled synchronously with a 1000 Hz sampling frequency, and part of the sampling signal, when tested at 200 r/min, is shown in Figure 8(a). The amplitude of the contact deformation is on μm scale under 200 r/min conditions, and there exists a phase delay between the encoder and the linear encoder. The signal frequency of the tested signals is the same as the rotation frequency of the inertia unit. Both of the amplitudes of the contact deformation tested in each speed are used to determine the relationship between the load disturbance and the contact deformation, as shown in Figure 8(b). The contact stiffness curves that were tested at the table and at the screw are generally consistent with each other when the load is greater than 65 N. In the range of 0 N to 65 N, the curves have similar trends; however, the amplitude deviation is comparatively large. The amplitude of the tested stiffness curve is a little bigger than the simulation result.Figure 8
(a) The position signals of the linear encoder and the encoder in a static test (r=200 r/min). (b) The test result of the relationship between force and deformation.
(a) (b)Another test is implemented to test the dynamic precision of the ball screw with the load disturbance. The straight reciprocating motion of the ball screw with a speed of 10 mm/s is tested. In the process, the speed inertia is set to 0 r/min, 200 r/min, and 300 r/min, respectively, and the data acquisition equipment is the same as that used for the static test. The test results are shown in Figure9. The fluctuation frequency, which is the rotational frequency of the inertia unit of the table position, is 3.33 Hz under the 200 r/min condition and 5.003 Hz in 300 r/min. The fluctuation frequencies clearly show the influence of the load disturbance on the position precision. However, the amplitude is approximately ten times that tested in the static condition. This result is due to the other sources of error, such as the pitch error, and the axial deformation of the screw may be included in the data.Figure 9
The tested signals of the linear encoder when moving under the applied force.


## 6. Conclusions

High-speed and high-precision NC machine tools are critical for modern industry. To improve the dynamic machining precision of NC machine tool, the transmission process of the ball screw is studied in this paper. Rigid-body kinematics are used to analyze the static part of the ball screw by considering contact deformation. Additionally, the Hertzian contact theory is used to analyze the influence factor on contact deformation. The relationships among the load disturbance, the deformation, and the influence factor are studied by simulation and experiment. The results show that the contact deformation indeed partially impacts the motion precision. The relationship between the load disturbance and the deformation is nonlinear, which leads to so-called contact stiffness. The relationship is influenced by the contact angle and the material characteristics, whereas the helical angle has little influence. When the amplitude of the load disturbance is close to 1000 N, the contact deformation will not be ignored in the feed system of a high-speed and high-precision machine tool.



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*Source: 8208241-2016-06-16.xml*