Symmetrical angular thread ball screw mechanism
The asymmetrical thread design of the ball screw mechanism addresses excessive friction and load issues by optimizing contact areas and lubrication, ensuring high capacity and efficiency under unidirectional axial forces.
Patent Information
- Application Number
- FR2024001384
- Authority / Receiving Office
- FR · FR
- Patent Type
- Patents
- Current Assignee / Owner
- Filing Date
- 2024-02-13
- Publication Date
- 2026-01-23
- Estimated Expiration
- 2044-02-13
AI Technical Summary
Existing ball screw mechanisms experience excessive friction and are unsuitable for applications with large magnitude axial forces in a constant direction, regardless of the direction of rotation.
A ball screw mechanism with asymmetrical thread flanks and a specific geometric configuration that ensures minimal contact area and reduced friction in one direction while maintaining sufficient lubrication and space in the opposite direction, utilizing a symmetrical angular thread design with extended flanks and controlled curvature radii.
The mechanism achieves reduced friction and improved lubrication, ensuring high capacity and efficiency under unidirectional axial forces by optimizing contact areas and maintaining space for lubrication, thus enhancing performance under high loads.
Smart Images
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Abstract
Description
Title of the invention: Ball screw mechanism with symmetrical angular threads TECHNICAL FIELD OF THE INVENTION
[0001] The invention relates to the field of ball screws, and more specifically to the threads of such a screw mechanism. PREVIOUS STATE OF THE ART
[0002] Document DE102007017214 discloses a ball screw mechanism comprising a screw defining a reference axis, a nut, and at least two balls rolling on a screw thread formed on the screw and a nut thread formed on the nut. This mechanism has threads whose flanks have different radii of curvature. When the mechanism is subjected to small-amplitude forces in a given axial direction corresponding to normal operation, the balls are in contact with a flank of large radius of curvature of the screw thread and with a flank of large radius of curvature of the nut thread, in order to minimize friction.When the mechanism at rest is subjected to axial loads in a direction opposite to its normal operating direction, the balls bear against a small radius flank of the screw thread and a small radius flank of the nut thread, enabling them to withstand large static loads, albeit with increased friction. However, this mechanism is not suitable for applications where the loads exerted on the moving mechanism have a large magnitude and a constant axial direction, regardless of the mechanism's direction of rotation. Description of the invention
[0003] The invention aims to remedy the drawbacks of the prior art and to propose a ball screw mechanism that can rotate without excessive friction in both directions of rotation, while being subjected to high amplitude axial forces that are always applied in the same axial direction.
[0004] To this end, according to a first aspect of the invention, a ball screw mechanism is proposed, comprising a screw defining a reference axis, a nut, and at least two balls of radius R positioned so as to roll on a screw thread formed on the screw and a nut thread formed on the nut, the screw thread and the nut thread having, in an active portion of the screw and the nut, a helix pitch P, characterized in that, in a cutting plane containing the reference axis and passing through the center of a first ball among the at least two balls, the first ball being positioned in the active portion of the screw and the nut, any segment perpendicular to the reference axis, located at
[0005] a distance x from the center of the first ball, and having a first end which belongs to the screw thread and a second end which belongs to the nut thread, has a center which, in an orthonormal coordinate system having an x-axis coinciding with the reference axis and a y-axis which passes through the center of the first ball, has an x-coordinate equal to % and a y-coordinate y, defining a function xy = / (x) which, when x varies between 0 and P, passes through a minimum Y0 reached for an x-coordinate X0 and through a maximum Y1 reached for an x-coordinate XI, such that the following inequalities are respected: '-^R <X0<P-^R lu lu ±R <Xl<P-^R , n -ro> 0.10mm
[0006] Furthermore, any segment perpendicular to the reference axis, located at a distance x from the center of the first ball, and having a first end which belongs to the screw thread and a second end which belongs to the nut thread, has a length L defining a function x L = g(x) which, when the abscissa x of the center of the segment varies between 0 and the helix pitch P, is always greater than 0.25 mm, and preferably greater than 0.40 mm.
[0007] The screw thread is radially protruding in the area near the abscissa XI of the maximum, and in depression in an area near the abscissa X0 of the minimum. Conversely, the nut thread is radially protruding in the area near near the minimum (X0) position and in a negative pressure zone near the maximum (XI) position. This allows, when a force is applied to the nut in a preferred axial direction from X0 to X0, a large contact area between the balls and the threads of the screw and nut, thus giving the mechanism a high capacity. In the opposite axial direction, the available contact areas are smaller, which is not a disadvantage since the expected forces remain low. The minimum distance L maintained between the screw and nut threads ensures good lubrication of the mechanism. Furthermore, it prevents any contact between the screw and nut, even under high loads and deformations.
[0008] Preferably:
[0009] Kl-KO > 0.20mm
[0010] According to one embodiment: • the screw thread forms an internal helical bearing race rotated radially in the opposite direction to the reference axis and delimited by two opposing flanks of the screw thread located on either side of a thread root, • the nut thread forms a helical external bearing raceway rotated ra- diamentally towards the reference axis and delimited by two opposite flanks of the nut thread located on either side of a root of the nut thread, • One extended flank of the two threads of the nut is extended towards the reference axis relative to the other of the two thread flanks of the nut. • one extended flank of the two flanks of the screw thread is extended in a radial direction opposite to the reference axis relative to the other of the two flanks of the screw thread, and • the extended flank of the nut thread is radially opposite the other flank of the screw thread and the extended flank of the screw thread is radially opposite the other flank of the nut thread, the extended flank of the nut thread and the extended flank of the screw thread being axially turned towards each other.
[0011] In a preferred axial force direction corresponding to contact between the balls and the extended flanks of the screw and nut threads, the contact ellipse can extend over a larger area of the extended flanks of both threads, including in the areas corresponding to the flank extensions. This asymmetry increases the mechanism's capacity in this direction, while maintaining sufficient space between the screw and nut, in the area between two successive turns, for proper lubrication of the mechanism. The flank asymmetry results in a smaller available surface area on the non-extended flanks for contact with the balls, leading to a reduced mechanism capacity in the force direction opposite to the preferred direction. However, this is not a problem since, in the intended application, the forces in this direction are weak, or even negligible.
[0012] Preferably, the two flanks of the screw thread have, in a cutting plane perpendicular to the root of the screw thread, a radius of curvature at every point greater than the radius R of the balls. Similarly, the two flanks of the nut thread preferably have, in a cutting plane perpendicular to the root of the nut thread, a radius of curvature at every point greater than the radius R of the balls. This ensures good control of the quasi-point contact interface between the balls and the flanks of the screw and nut threads.
[0013] Preferably, both sides of the screw thread have a circular arc curvature in a cutting plane perpendicular to the root of the screw thread, and both sides of the nut thread have a circular arc curvature in a cutting plane perpendicular to the root of the nut thread. Preferably, one or more of the following conditions are met: • Both sides of the screw thread have the same radius of curvature; and / or • Both sides of the nut thread have the same radius of curvature; and / or • at least one of the two flanks of the screw thread has a radius of curvature identical to at least one of the two sides of the nut thread.
[0014] Thus, the coefficient of friction between balls and raceways is independent of the direction of application of the axial force (in the preferred direction or the opposite direction), for a given magnitude of the resultant axial force.
[0015] According to one embodiment, the values X0 and XI satisfy the following inequalities:
[0016] tR <X0<P-R { et \r<xkp-r
[0017] According to one embodiment, the two extremes satisfy the following double inequality, where D is the distance D between the center of the first ball and the reference axis:
[0018] fo <z><fi
[0019] According to one embodiment, D - FO - s < Y1 - D < D - FO + s; e denoting a distortion margin such that 0 < e < 0.05mm.
[0020] According to one embodiment, the function x - y = f(x) is such that, for any value z between 0 and P! 2 - R, where R is the radius of at least two balls, we observe:
[0021] (2 / (|)-f:l < / (£+s)+ / (Ç--)<2 / (£)+£l with el < 0.05mm
[0022] These inequalities reflect an approximate symmetry of the function xy = f(x), with a distortion margin el, around and near a center of symmetry located at the abscissa P / 2.
[0023] According to one embodiment:
[0024] ) <D + £1
[0025] According to one embodiment:
[0026] = £
[0027] According to one embodiment, the function .xy = f(x) is continuous.
[0028] According to one embodiment, the function xy — f(x) is strictly increasing or strictly decreasing between X0 and XI.
[0029] Near the center of the balls, the function xy = f(x) is relatively constant. Thus, where D denotes the distance between the center of the first ball and the reference axis, and e denotes a distortion margin such that 0 < e < 0.05 mm, we observe: • for 0 <x<2^,D-e< f(x) < D + e • pourp_2g <x<p,D-f< / (x)<D+ e BRIEF DESCRIPTION OF THE FIGURES
[0030] Other features and advantages of the invention will become apparent from the following description, with reference to the accompanying figures which illustrate: • [Fig.1] the [Fig.1], a ball screw mechanism in axial section, illustrating in particular balls circulating on the bearing races of a nut and a screw of the ball screw mechanism; • [Fig. 2] [Fig. 2] A detailed view of a nut thread and a screw thread ball bearings. • [Fig.3] The [Fig.3] illustrates a curve of evolution of the radial distance between the midpoint of an imaginary segment separating the two threads of the ball screw mechanism and a reference axis, as well as a curve of evolution of the length of said imaginary segment, as a function of the axial distance to the center of a ball of the mechanism.
[0031] For clarity, identical or similar elements are identified by identical reference signs throughout the figures. DETAILED description of implementation methods
[0032] Figure 1 illustrates a ball screw mechanism 1 comprising two threaded components, namely a screw 10 and a nut 12, and balls 2. The screw 10 is preferably metallic, for example steel, and has a screw thread 34 which forms an internal helical raceway 14 around a reference axis 200 of the ball screw mechanism 1 defined by the screw 10, the internal helical raceway 14 being rotated radially away from the reference axis 200. The nut 12 is preferably metallic, for example steel, and has a nut thread 134 which forms an external helical raceway 114 around the reference axis 200, and rotated radially towards the reference axis 200. One of the two threaded components, namely the screw 10 or the nut 12, further forms a channel of recirculation, not shown in the figures, which preferably spans several turns of the bearing paths 14,114 of the screw 10 and the nut 12.The balls 2 can, for example, be made of steel or ceramic, and are dimensioned and positioned to circulate in a closed circuit between the outer helical raceway 114 of the nut 12 and the inner helical raceway 14 of the screw 10, as well as in the recirculation channel, preferably without separators between the balls 2.
[0033] The internal helical raceway 14 of the screw 10 has, on at least one active part, a helical envelope of constant pitch P around the reference axis 200, with a raceway bottom 16, as well as a first flank 18 and a second flank 20 facing each other on either side of the raceway bottom 16, the first flank 18 being turned in a first axial direction 210 of the reference axis 200, the second flank 20 being turned in a second axial direction 220 of the reference axis 200.
[0034] In the remainder of the description, we will refer to Figures 1 and 2 where the ball screw mechanism 1 is shown in a sectional view. The section in [Fig. 2] is made in a cutting plane containing the datum axis 200 and passing through by the center of one of the marbles.
[0035] The flanks 18, 20 of the helical raceway 14 of the screw 10 preferably have a concave cross-section in any plane containing the reference axis 200. These concave sections have a non-constant radius of curvature in the plane of [Fig. 2]. However, viewed in any cutting plane perpendicular to a tangent to the bottom of the raceway 16, i.e. in a plane inclined at the value of the helix angle with respect to the plane of [Fig. 2], the flanks 18, 20 are preferably in the form of a circular arc, and preferably with equal radii of curvature for both flanks, so as to form an ogive extending on either side of the bottom of the raceway 16, or a continuous circular arc. The helix angle is however small enough that, in the cutting plane of [Fig.2], the variations in the radius of curvature and the center of curvature of the flanks 18, 20 are small, for example less than 0.1 mm.
[0036] The first side 18 has an end 38 opposite the bottom of the path 16. The second side 20 has an end 40 opposite the bottom of the path 16.
[0037] The end 38 of the first flank 18 of the screw 10 is radially less distant from the reference axis 200 than the end 40 of the second flank 20 of the screw 10.
[0038] The helical raceway 14 of the screw 10 comprises several successive turns separated by connecting portions 36 that project from the screw thread 34. Each connecting portion 36 links the ends 38, 40 of the flanks 18, 20 of the screw 10. The connecting portion 36 forms a straight curve 46 in the vicinity of the end 38 of the first flank 18 of the raceway 14 of the screw 10, the end 38 constituting an inflection point with two half-tangents forming an obtuse angle with each other. The connecting portion 36 has a convex curvature 45 in the vicinity of the end 40 of the second flank 20 of the bearing race 14 of the screw 10, this end 40 constituting an inflection point, the surface of the screw thread 34 crossing its tangent at this inflection point 40.
[0039] The external helical raceway 114 of the nut 12 has, on at least one active part, a helical envelope of constant pitch P around the reference axis 200, with a raceway bottom 116, as well as a first flank 118 and a second flank 120 facing each other on either side of the raceway bottom 116, the first flank 118 being rotated in the second axial direction 220 of the reference axis 200, axially opposite the first flank 18 of the raceway 14 of the screw 10 and radially opposite the second flank 20 of the raceway 14 of the screw 10, the second flank 120 being rotated in a first axial direction 210 of the reference axis 200, axially opposite the second flank 20 of the raceway 14 of the screw 10 and radially opposite the first flank 18 of the raceway bearing 14 of screw 10.
[0040] The flanks 118,120 of the helical raceway 114 of the nut 12 have A concave section is preferred in any plane containing the reference axis 200. These concave sections have a non-constant radius of curvature in the plane of [Fig. 2]. However, when viewed in any cutting plane perpendicular to a tangent to the bottom of the track 116, i.e., in a plane inclined at the value of the helix angle relative to the plane of [Fig. 2], the flanks 118, 120 are preferably circular arcs, and preferably with equal radii of curvature for both flanks, so as to form an ogive extending on either side of the bottom of the track 116, or a continuous circular arc. The helix angle is, however, sufficiently small that, in the cutting plane of [Fig. 2], the variations in the radius of curvature and the center of curvature of the flanks 118, 120 are small, for example, less than 0.1 mm.
[0041] The first flank 118 has an end 138 opposite the bottom of the track 116. The second flank 120 has an end 140 opposite the bottom of the track 116. The end 138 of the first flank 118 of the nut 12 is radially further from the reference axis 200 than the end 140 of the second flank 120 of the nut 12. The helical raceway 114 of the nut 12 has several successive turns separated by projecting connecting portions 136 of the nut thread 134. Each connecting portion 136 links the ends 138, 140 of the flanks 118, 120 of the nut 12. The connecting portion 136 forms a straight curve 146 in the vicinity of the end 138 of the first flank 18 of the raceway 114 of the nut 12, the end 138 constituting an inflection point with two half-tangents making an obtuse angle between them.The connecting portion 136 has a convex curvature 145 in the vicinity of the end 140 of the second flank 120 of the bearing race 114 of the nut 12, this end 140 constituting an inflection point, the surface of the nut thread 134 crossing its tangent at this inflection point 140. .
[0042] Thus, the second flank 120 of the nut thread 134 is extended towards the reference axis 200 relative to the first flank 118 of the nut thread 134. Similarly, the second flank 20 of the screw thread 34 is extended in the radial direction opposite to the reference axis 200 relative to the first flank 18 of the screw thread 34. The extended flank 120 of the nut thread 134 is radially opposite the unextended flank 18 of the screw thread 34, and similarly, the extended flank 20 of the screw thread 34 is radially opposite the unextended flank 118 of the nut thread 134. The extended flanks 120 and 20 are opposite each other on either side of the ball center 2.
[0043] To describe the volume delimited by the screw thread 34 and the nut thread 134 in the plane of [Fig. 1] and [Fig. 2], an orthonormal coordinate system R is defined having an abscissa axis coinciding with the reference axis 200 and a ordinate axis passing through the center Cl of the first ball 2a. An arbitrary imaginary segment 240 perpendicular to the reference axis 200 can then be observed, located at a distance x from a center Cl of a first ball 2a, the distance x varying from 0 to P, P being the value of the helix pitch of the screw thread 34 and the nut thread 134. The segment 240 has a first end belonging to the screw thread 34 as well as a second end belonging to the nut thread 134.
[0044] Segment 240 has a center C2 with coordinates x on the abscissa and y on the ordinate, defining a function x - y = π(x) whose curve F is schematically illustrated superimposed on [Fig. 3], and with reference to the scale on the left of the graph in [Fig. 3], the x-axis being graduated in units corresponding to each unit representing 1 / 5th of P. The numerical values provided are purely illustrative. In practice,
[0045] The function x -* y = f(x) passes through a minimum Y0 reached for an abscissa X0 as well as through a maximum Y1 reached for an abscissa XI, where the values X0 and XI are each greater than the radius of the balls 2. The function xy = f(x) is continuously increasing or continuously decreasing between X0 and XI, and is more generally continuous over the entire length of the screw 10.
[0046] Inside the raceways 14,114, the center of segment 240 remains at a distance from the reference axis that is close to the value D, which can be expressed as a function of an e denoting a distortion margin such that 0 < e < 0.05 mm, by: • for Q <y<^£,£> -£< f(x) < D + e • for P - — < x < P^D~e < < D + e
[0047] The center Cl of the first ball 2a is located from the reference axis 200 by a distance D corresponding to half the pitch diameter of the mechanism. The minimum Y0 is less than the distance D, while Y1 is greater than the distance D. The minimum Y0 is reached for an abscissa X0 such that
[0048] R <X0<P-R
[0049] The maximum Y1 is reached for an abscissa XI such that:
[0050] R <X1<P-R
[0051] The ball screw mechanism 1 preferably has an approximate symmetry in the cutting plane of Figures 1 and 2, such that
[0052] D-Y0-8 <Yl-D<D-Y0 + e -,
[0053] e denoting a distortion margin such that 0 < e < 0.05mm.
[0054] More generally, the central symmetry exhibited by the ball mechanism 1 at the center of symmetry located at abscissa XS=(X1+X0) / 2 or XS=P / 2, and at ordinate YS=D, can be written
[0055] <2 / (Ç)+£l
[0056] el designating a distortion margin such that 0 < wire < 0.05mm.
[0057] In these inequalities, e and el account for the inclination of the helical threads relative to the axial cutting plane of figures 1 and 2, therefore of the helix angle and the 2D pitch diameter of the mechanism.
[0058] The distortion margins e and el are related to the inclination of the helical threads relative to the axial cutting plane of Figures 1 and 2, and are a function of the helix angle and the 2D pitch diameter of the mechanism. The values of e and el are less than D / 10, and preferably less than D / 50.
[0059] In practice, the difference El - FO is significant, and respects the following inequality:
[0060] Y1 - Y0 > 0.10mm
[0061] Preferably:
[0062] El-F0 > 0.20mm
[0063] Furthermore, the length L of segment 240 varies according to the abscissa x of segment 240, defining a function xg(x) = L, whose curve G is illustrated in the graph in Figure 3, with reference to the scale on the right for the ordinates. The function x - L = Xx) passes through a minimum value Lmin, itself greater than or equal to 0.25 mm, preferably greater than 0.40 mm, for example greater than 0.45 mm or 0.50 mm, on the one hand to guarantee satisfactory grease circulation between the turns of the raceways and on the other hand to avoid impacts between the threads of the screw and the nut under high loads and high deformations, particularly in the area corresponding to the connecting portions 36,136 where this minimum value may be reached.
[0064] The axial direction 210 is a preferred direction for the forces applied by the nut 12 on the balls 2 and for the forces applied by the balls 2 and by the screw 10: when the nut applies a force on the balls 2 whose resultant has an axial component in the axial direction 210, the contact ellipse between each ball 2 and the raceway 114 of the nut 12 is formed on the second flank 120, and can extend over a significant area of the second flank, including in the area corresponding to the extension of this second flank 120. Similarly, the contact ellipse between each ball 2 and the raceway 14 of the screw 10 is formed on the second flank 20, and can extend over a significant area of the second flank 20, including in the area corresponding to the extension of this second flank 20.
[0065] This asymmetry makes it possible to increase the capacity of the mechanism in this direction, while preserving between the screw and the nut a sufficient space, in the area located radially between the connecting portions 36,136, for good lubrication of the mechanism.
[0066] The asymmetry of the flanks also results in the available surface area at the non-extended flanks 18,118 being smaller to constitute a contact interface with the balls, so that the capacity of the mechanism in the direction of forces 220 opposite to the preferred direction 210 is substantially less than the capacity in the preferred direction 210, which is not a problem however since in the envisaged application, the axial forces are unidirectional or essentially unidirectional.< / z>
Claims
Demands
1. Ball screw mechanism (1), comprising a screw (10) defining a reference axis (200), a nut (12), and at least two balls (2) of radius R positioned to roll on a screw thread (34) formed on the screw (10) and a nut thread (134) formed on the nut (12), the screw thread (34) and the nut thread (134) having, in an active portion of the screw and the nut, a helix pitch P, characterized in that, in a cutting plane containing the reference axis (200) and passing through a center (Cl) of a first ball (2a) among the at least two balls (2), the first ball being positioned in the active portion of the screw and the nut, any segment (240) perpendicular to the reference axis (200), located at a distance x from the center (Cl) of the first ball (2a), and having a first an end which belongs to the screw thread (34) and a second end which belongs to the nut thread (134), has: - a center (C2) which, in an orthonormal coordinate system having an x-axis coinciding with the reference axis (200) and an ordinate axis passing through the center (Cl) of the first ball (2a), has an x-coordinate equal to x and an ordinate y, defining a function y — f(x) which, when x varies between 0 and P, passes through a minimum Y0 reached for an x-coordinate X0 and through a maximum Y1 reached for an x-coordinate XI, such that the following inequalities are respected: l^R <XQ<P-^R lu w , Fl -F0> 0.1 0mm and - a length L defining a function x~* L = g(x) which, when the abscissa x of the center of the segment (C2) varies between 0 and the helix pitch P, is always greater than 0.25 mm, and preferably greater than 0.40 mm.
2. Ball screw mechanism (1) according to claim 1, characterized in that the screw thread (34) forms an internal bearing track (14) helical rotated radially in the opposite direction to the reference axis (200) and delimited by two opposite flanks (18, 20) of the screw thread (34) located on either side of a bottom (16) of the screw thread (34), - the nut thread (134) forms an external helical bearing race (114) rotated radially towards the reference axis (200) and delimited by two opposite flanks (118, 120) of the nut thread (134) located on either side of a bottom (116) of the nut thread (134), - an extended flank (120) among the two flanks of the nut thread (134) is extended towards the reference axis (200) relative to the other of the two flanks (118) of the nut thread (134), - an extended flank (20) among the two flanks of the screw thread (34) is extended in a radial direction opposite to the reference axis (200) relative to the other of the two flanks (18) of the screw thread (34), - the extended flank (120) of the nut thread (134) is radially opposite the other flank (18) of the screw thread (34) and the extended flank (20) of the screw thread (34) is radially opposite the other flank (118) of the nut thread (134), the extended flank (120) of the nut thread (134) and the extended flank (20) of the screw thread (34) being axially turned towards each other.
3. Ball screw mechanism (1) according to claim 2, characterized in that the two flanks (18, 20) of the screw thread (34) have a circular arc curvature in a cutting plane perpendicular to the bottom (16) of the screw thread (34) and the two flanks (118, 120) of the nut thread (134) have a circular arc curvature in a cutting plane perpendicular to the bottom (118) of the nut thread (134), and, preferably, - the two flanks (18, 20) of the screw thread (34) have an identical radius of curvature; and / or - the two flanks (118, 120) of the nut thread (134) have an identical radius of curvature; and / or - at least one of the two flanks (18, 20) of the screw thread (12) has a radius of curvature identical to at least one of the two flanks (118, 120) of the nut thread (134).
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10. Ball screw mechanism (1) according to any one of the preceding claims, characterized in that () - Y 0 - s < Y1 - D < D - YO + e ; D denoting the distance between the center of the first ball (2a) and the reference axis (200), e denoting a distortion margin such that 0 < s < 0.05mm. Ball screw mechanism (1) according to any one of the preceding claims, characterized in that the function xy = f(x) is such that, for any value z between 0 and P / 2 - R, where R is the radius of the at least two balls (2): 2 / (4) -if < / (| +;)+ / (£-;) < 2 / (£ ) +£1 avec el < 0,05mm Ball screw mechanism (1) according to any one of the preceding claims, characterized in that 2 “2 Ball screw mechanism (1) according to any one of the preceding claims, characterized in that the function xy = f(x) is continuous. Ball screw mechanism (1) according to any one of the preceding claims, characterized in that the function xy = f(x) is strictly increasing or strictly decreasing between X0 and XI. Ball screw mechanism (1) according to any one of the preceding claims, characterized in that: 71-70 > 0.20mm Ball screw mechanism (1) according to any one of the preceding claims, characterized in that: for 0 <x<-^^D-e< / W<D + e pourp__lp<x<p,£> -s <!--(x)<D + e<br--> D denoting the distance between the center of the first ball (2a) and the reference axis (200), e denoting a distortion margin such that 0 < e < 0.05mm.