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Modeling approach for axial contact stiffness of ball screws①

更新时间:2016-07-05

0 Introduction

Ball screw is one of the most important interfaces in machine tools. The axial stiffness of ball screws has great effects on accuracy of positioning, dynamic characteristic and transmission efficiency. Many researches have been conducted on axial stiffness of ball screws. Okwudire, et al.[1,2] demonstrated theoretically and experimentally that the nut introduced a coupling between the dynamics in the axial/torsional directions and the lateral (bending) direction in ball screws. Torque-induced bending deformations were optimized to minimize the static coupling between the applied torque and lateral deformations of the screw in Ref.[3]. Uncertainty analysis of torque-induced bending deformations considering variable preload and axial stiffness was conducted in Ref.[4]. But the contact stiffness of balls was obtained by manufacturers’ catalogs in the research above. Microscopic characteristics of contact surfaces is not considered in the research.

Lin, et al.[5] established a dynamic model of balls in ball screws considering Frictional force and moment. Deformation in the contact region was also discussed. Wu, et al.[6] established a dynamic model of ball screws in machine tools, the axial and torsional stiffness was considered in the research. The simulation model for feed system considering the stiffness was also established. According to the equilibrium equation of forces and moments, Chen, et al.[7] established a mathematical model for stiffness of ball screws under complicated loads considering variable thread angle and deformation. Hertz theory was adopted to solve the contact status between balls and raceways. Hu, et al.[8] established an identifying approach for axial stiffness in ball screws. The axial stiffness of ball screws was identified based on experimental results. In our previous research[9], axial contact stiffness of ball screws was estimated by Hertz theory, in which the coupling relationship between the pressure of the raceway and the variation of the thread angle was considered.

Hertz theory was widely used to analyze contact status in bearings[10-12] and rolling guideways[13-15]. But for ball screws, contact region is asymmetrical because of helix angle. The greater helix angle is, the more asymmetrical the contact region is. Hertz theory can only be applied in symmetrical or axisymmetric contact faces. Relatively large error will occur when Hertz theory is applied.

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Therefore, a new analytical method combining the minimum excess principle with Mandelbort (MB) fractal theory is proposed to estimate axial contact stiffness of ball screws in this research. Firstly, geometrical shape of contact surfaces in ball screws is analyzed, the parameter equations about screw-raceway and nut-raceway are established. 3D contact surface of raceway is simulated and meshed based on the parameter equations. Secondly, normal contact stress is solved by the minimum excess principle. Fast Fourier transform (FFT) is adopted to solve the convolution, which is used to evaluate the effect of normal stress on deformation in the normal direction. Conjugate gradient method is employed to solve the contact model. Finally, contact stiffness of the diminutive area is obtained by the MB fractal theory. The contact stiffness of contact points is calculated by integrating contact stiffness of diminutive areas. The axial stiffness of ball screws is obtained according to the connection of contact stiffness. Microscopic fractal characteristics of joints can be considered in this research. The effectiveness of the proposed method is also validated by the self-designed experiment.

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1 Geometrical shape of contact surfaces

where ui, j is the normal deformation of node (i, j), pk, l is the normal stress subjected to the element whose center is (k, l).

Fig.1 Three coordinate system of ball-screw

ZA-axis and ZB-axis are perpendicular to the tangent plane at contact points. XA-axis and XB-axis are parallel to tangential direction of the center track of balls.

Geometrical shape of contact surfaces at the coordinate established at contact point A(AXAYAZA) and B(BXBYBZB) can be obtained by coordinate transformation.

Parameter equation of the arc in the cross section of screw-raceway at contact point A and B at Frenet-Serret coordinate system is given:

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βA(B)-βA(B)0βA(B)SβA(B)+βA(B)0

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(1)

where rS is the radius of arc in the cross section of screw-raceway, rb is the radius of balls, βA is thread angle between ball and screw-raceway, βAS is central angle corresponding to the arc in the cross section of screw-raceway, βA0 is used to define the variation of central angle βAS, rN is the radius of arc in the cross section of nut-raceway, βB is the thread angle between ball and nut raceway, βBS is the central angle corresponding to arc in the cross section of nut-raceway, βB0 is used to define the variation of central angle βBS.

According to the coordinate transformation, Eq.(1) is expressed as

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IC

(2)

(3)

where is the coordinate transformation matrix from the coordinate system OXYZ, which is attached to the screw, to Frenet-Serret coordinate OHtnb. is expressed by

(4)

The contact problem of ball/raceway can be described as

(5)

where FWS is preload of ball screws, which can be searched in manufacturers’ catalogs. nb is the number of balls on the contact status.

βA(B)-βA(B)0βA(B)sβA(B)+βA(B)0

(6)

where is the coordinate transformation matrix at θ=θC. θC is the central angle corresponding to the helix at A or B.

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In this paper, a certain type of ball screw is studied. Geometrical shape of the raceways in the ball screw can be obtained by Eq.(6), which is depicted in Fig.2.

Fig.2 Geometrical shape of the raceway

2 Normal stress analysis

Because the contact region of ball/raceway is irregular, the minimum excess principle is suitable for analyzing this problem[16,17].

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The normal contact stress in every element is approximate to be constant, and the discrete form of Eq.(7) is given by

(7)

In order to facilitate the systematic study of the kinematic rules of the ball, the Cartesian coordinate and AXAYAZA and BXBYBZB depicted in Fig.1 are established at contact point A between screw-raceway and ball and contact point B between nut-raceway and ball, respectively. A and B are contact points when contact just occurs. In this situation, there is no normal stress.

Ki, j is the influence coefficient which is defined as Ref.[18]

(8)

where S0 is the meshed region.

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From Fig.1, the coordinate transformation matrix from coordinate OHtnb to AXAYAZA and BXBYBZB can be obtained respectively.

(9)

(10)

pi, j>0, (i, j)∈IC

(11)

pi, j=0, (i, j)∉IC

(12)

(13)

where δz is the relative rigid displacement between balls and raceways, IC is the set of all nodes which is on the contact status, ds is the area of element, Fz is the total normal load.

Normal contact stress pi, jon the contact surface can be obtained by solving Eqs (9)-(13).

Generally, the contact region is unknown before the contact problem is solved, the relative rigid displacement δz is also unknown, and the total normal load Fz can be calculated by preload of ball screws according to the equilibrium equation of nuts. The expression can be given by

(14)

The equation of arc in screw-raceway and nut-raceway in coordinate system AXAYAZA and BXBYBZB can be obtained from Eqs(2)-(5).

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Detailed steps of calculation can be found in Refs[20-22].

In this work, 2D FFT is used to solve the elastic deformation problems between balls and raceways. Compared with other methods, FFT is more effective[19].

The contact problem is solved by conjugate gradient method to expedite convergent speed of iterative operation. In the process of iteration, deformations in the contact region is calculated from contact stress by fast Fourier transform(FFT).

In this study, conjugate gradient method is used to solve the contact problem, and detailed steps of calculation can be found in Ref.[23].

Normal stress on contact surface in the ball screw studied in this paper is solved by this method, which is depicted in Fig.3. The contact region can also be predicted by Fig.3.

Fig.3 Normal contact stress distribution

3 Axial contact stiffness

On the basis of contact stress obtained in Section 2, contact stiffness of every diminutive area is solved by MB fractal theory. Micro-topography of contact surfaces can be considered in MB fractal theory compared with other methods.

Contact between two contact surfaces occurs only at some discrete diminutive areas because of actual rough surfaces. The asperity on rough surfaces can be simplified to a sphere. The contact area of asperities at the critical status of elastic deformation and plastic deformation is given by

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(15)

where H is the lower hardness of two contact materials, σy is the lower yield strength of two materials, G is scale parameter of the contact surface, D is fractal dimension of the surface profile. It describes irregular patterns of the surface profile in all the scales.

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The relationship between deformation of asperities δ and its critical value δc can be expressed by

δc/δ=(a/ac)D-1 (1<D<2)

(16)

where a represents contact area of the asperity. If a<ac, according to Eq. (16), δc<δ, plastic deformation occurs on the asperity. If a>ac, δc>δ can be obtained, elastic deformation occurs on the asperity.

According to the MB fractal theory, normal contact stiffness of the diminutive area is given by Ref.[24]

Kn

(17)

where al is the maximum of contact area at contact points, al can be solved by Eq.(18).

P=

(1<D<2, D≠1.5) (18a)

(D=1.5) (18b)

where P is the normal load on the diminutive area, and it equals to pi, jΔxΔy, ΔxΔy is the area of the diminutive area.

Contact stiffness at contact point A(B) can be obtained by

(19)

In this research, the mass of balls is neglected. And all the compliance is assumed to come from contact points (A and B). Therefore, the balls are modeled as massless springs having a stiffness Kball aligned along the contact line between screw and nut[25]. According to the connection type of springs, the contact stiffness of balls is determined by

(20)

Then the axial contact stiffness is expressed as[1]

Kax=Kballnbcos2αcos2β

(21)

4 Validation

In order to validate the modeling approach for axial contact stiffness of ball screws proposed in this paper, self-designed experiment is implemented.

4.1 Experiment

A test scheme for axial stiffness is designed for validation. Axial stiffness is measured in a stretch test machine. The experimental setup is shown in Fig.4. One end of the screw is fixed on the upper chuck of the machine. The nut is connected with the part which is technically designed for the experiment. The part is connected with the flange of nut via its upper end. The lower end of the part is fixed on the lower chuck of stretch test machine. Different axial tension is provided by raising or lowering the transom of the machine. Because the test result from displacement sensors on the stretch test machine is influenced by the deformations and gaps in the machine, the result differs greatly from the actual value. In this research, axial relative displacement between nut and screw is measured by an inductance micrometer. The probe of inductance micrometer is attached to the flange of nut by a magnetic base. A cross bar is fixed on the screw, and it is perpendicular to the axial direction of the inductance micrometer in a vertical plane.

Fig.4    Flowchart for numerical simulation of the contact stiffness by MB fractal theory

The experimental result is depicted in Fig.5. From Fig.5, conclusion can be drawn that the displacement is proportional to axial load approximately. The axial contact stiffness Kaxcan be obtained by

(22)

where F is the axial load depicted in x-axis, x is the relative displacement depicted in y-axis.

4.2 Analysis and comparisons

The machining technology of balls is more mature compared with that of raceways. Therefore micro-topography of ball surface is neglected. The surface of raceway is formed by grinding. The algorithm flow chart for the numerical contact stiffness simulation by MB fractal theory is shown in Fig.4. The specifications and the experimental condition of the ball screw are shown in Table 1. According to the machining form, the fractal parameters are estimated. D=1.566 and G=0.32nm are introduced to the fractal theory[26]. Parameters of the ball screw studied in the experiment are also introduced to the theoretical method. Axial contact stiffness of the ball screw can be drawn that its value is 2.3117×108N/m. The experimental result can be approximately obtained with its value 2.8×108N/m. By comparison, the result gained by the analytical method in this research agrees well with the experimental result is shown in Fig.5. The error between theoretical result and experimental result is 17%, as shown in Fig.6. The error is acceptable in Engineering practice. The error is mainly generated by the stiffness contribution of thin oil film between balls and raceways to the experimental result, which causes that the experimental result is greater than the theoretical result.

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Fig.5 Experimental setup

Table 1 ball-screw parameters

ParametersValueUnit(I)Geometryparametersoftheballscrew  Screwpitchcycleradiusr16mmPitchL10mmContactangleβ40.2600°Helixangleα5.6833°ScrewinnerradiusofscrewrCS16.182mmNutinnerradiusofscrewrCN15.818mmRadiusofcurvatureofthescrewracewayrS3.215mmRadiusofcurvatureofthenutracewayrN3.215mmBall’sradiusrb2.9765mmPreloadFWS2562.8NNumberofballsnb206 densityoftheballmaterialρ7.8g/cm3Young’smodulusE211.1GPaPoisson’sratiov0.3 (II)Theexperimentalconditions  Roomtemperature20℃

Fig.6 Experimental results

The comparison between theoretical result and experimental result demonstrates that the analytical method combining the minimum excess principle with MB fractal theory can be implemented to estimate the axial contact stiffness of ball screws. Compared with other methods, micro-topography of contact surfaces can be considered in the analytical method proposed in this research.

5 Conclusions

It has been common practice to solve contact stiffness in ball screws by Hertz theory. But relatively large error will occur when Hertz theory is employed because of the asymmetry of contact region in the ball/raceway contact. A new analytical method combining the minimum excess principle with MB fractal theory is proposed to estimate axial contact stiffness of ball screws in this research. The minimum excess principle is employed to conduct normal stress analysis. And MB fractal theory is adopted to obtain contact stiffness in ball screws. Microscopic fractal characteristics of interfaces can be considered in this method. The effectiveness of the proposed method is validated by the self-designed experiment. The comparison between theoretical result and experimental result demonstrates that axial contact stiffness of ball screws can be obtained by the method proposed in this study.

References

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