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1、BRITISH STANDARD BS 6101-4: 1987 Incorporating Amendment No. 1 Machine tool ball screws Part 4: Method of calculation of static axial rigidity UDC 621.9:621.83.05-219.51 Licensed Copy: sheffieldun sheffieldun, na, Tue Dec 05 05:57:27 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6101-4:1987 This Bri
2、tish Standard, having been prepared under the direction of the Machine, Engineers and Hand Tools Standards Committee, was published under the authority of the Board of BSI and comes into effect on 31 July 1987 BSI 12-1999 The following BSI references relate to the work on this standard: Committee re
3、ference MTE/1 Draft for comment 85/72760 DC ISBN 0 580 15441 6 Committees responsible for this British Standard The preparation of this British Standard was entrusted by the Machine, Engineers and Hand Tools Standards Committee (MTE/-) to Technical Committee MTE/1 upon which the following bodies wer
4、e represented: Advanced Manufacturing Technology Research Institute Department of Trade and Industry (Mechanical and Electrical Engineering Division) Federation of British Engineers Tool Manufacturers GAMBICA (BEAMA Ltd.) Health and Safety Executive Institution of Production Engineers Ministry of De
5、fence Coopted members Amendments issued since publication Amd. No.Date of issueComments 6264November 1989 Indicated by a sideline in the margin Licensed Copy: sheffieldun sheffieldun, na, Tue Dec 05 05:57:27 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6101-4:1987 BSI 12-1999i Contents Page Committ
6、ees responsibleInside front cover Forewordii 0Introduction1 1Scope1 2Definitions1 3Notation1 4Method of determining rigidity2 4.1General2 4.2Overall rigidity3 4.3Rigidity of the ball screw shaft3 4.4Rigidity of nut body and screw shaft under radial component due to axial load4 4.5Rigidity in the bal
7、l/ball track area4 4.6Rigidity of a symmetrically preloaded nut system7 4.7Rigidity of the ball screw within the loaded ball nut area7 4.8Accuracy factor7 4.9Overall rigidity7 Appendix A A graphical solution to 4.5. Rigidity in the ball/ball track area8 Appendix B Worked example of calculation of ri
8、gidity16 Figure 1 Notation of the ball screw1 Figure 2 Ball screw shaft rigidly mounted at one end3 Figure 3 Ball screw shaft rigidly mounted at both ends3 Figure 4 Relative displacement between ball nut and ball screw shaft due to axial backlash5 Figure 5 Preloaded ball nuts5 Figure 6 Graphs to ill
9、ustrate calculation of rigidity9 Table 1 Accuracy factor7 Publications referred toInside back cover Licensed Copy: sheffieldun sheffieldun, na, Tue Dec 05 05:57:27 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6101-4:1987 ii BSI 12-1999 Foreword This Part of BS 6101 has been prepared under the direc
10、tion of the Machine, Engineers and Hand Tools Standards Committee. It has been produced in advance of related work being prepared in the International Organization for Standardization (ISO) in order to facilitate discussion between manufacturers and purchasers of machine tool ball screws. A British
11、Standard does not purport to include all the necessary provisions of a contract. Users of British Standards are responsible for their correct application. Compliance with a British Standard does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front co
12、ver, an inside front cover, pages i and ii, pages 1 to 18, an inside back cover and a back cover. This standard has been updated (see copyright date) and may have had amendments incorporated. This will be indicated in the amendment table on the inside front cover. Licensed Copy: sheffieldun sheffiel
13、dun, na, Tue Dec 05 05:57:27 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6101-4:1987 BSI 12-19991 0 Introduction This Part of BS 6101 describes a method of calculating static axial rigidity of ball screws due to tensile or thrust load. The rigidity of a ball screw exerts a major influence on its p
14、ositioning accuracy. It is a function of the design of the ball screw, its support and bearing arrangement. For the purpose of the calculations given in this Part of BS 6101, support and bearing arrangement have been disregarded. The rigidity of linear springs is constant, i.e. the deflection and th
15、e force are proportional to each other. This does not apply to a ball screw. For the study of rigidity, a ball screw may be conceived as a combination of several linear and non-linear spring elements. For this reason the rigidity value indicated is correct only for one value of load. The axial defle
16、ction to be determined is caused by: a) axial screw shaft and nut deflections; b) radial screw shaft and nut deflections; c) deflection of the thread land; d) Hertz stress; e) shear stress distribution. As it is very time-consuming, and hence unsuitable for practical purposes, to determine the axial
17、 deflection on the basis of precise formulae, a reasonably simplified calculation method is outlined, so that the calculation may be effected with a pocket calculator. 1 Scope This Part of BS 6101 gives a method of calculating the static axial rigidity of machine tool ball screws. For the purposes o
18、f this Part of BS 6101, where the term “rigidity” is used, it means static axial rigidity. NOTEThe titles of publications referred to are listed on the inside back cover. 2 Definitions For the purposes of this Part of BS 6101, the definitions given in BS 6101-3 apply. 3 Notation The following is the
19、 notation used in this Part of BS 6101. Figure 1 shows the notation of the ball screw. Figure 1 Notation of the ball screw Licensed Copy: sheffieldun sheffieldun, na, Tue Dec 05 05:57:27 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6101-4:1987 2 BSI 12-1999 4 Method of determining rigidity 4.1 Gene
20、ral The static axial rigidity R (in N/4m) is the resistance to deformation and denotes the force F (in N) which is required to effect a component deflection dla by 1 4m in the direction of load applications. The value of R shall be calculated from the equation: SymbolTerm cEMaterial constant (for st
21、eel cE = 0.4643) ckGeometry factor dcDiameter of load application on the ball screw shaft (in mm) DcDiameter of load application on the ball nut (in mm) dlElastic deflection (in 4m) DwBall diameter (in mm) d0Nominal diameter (in mm) of ball screw d1Ball screw shaft outer diameter (in mm) D1Ball nut
22、outer diameter (in mm) D3Ball nut internal diameter (in mm) EYoungs modulus of elasticity (in N/mm2) (for steel = 2.1 105) FaAxial force (in N) FamLimit load of anti-backlash double-nut system (in N) FNNormal force (in N) FPPreload force (in N) F1Actual axial force applied to ball nut 1 (in N) F2Act
23、ual axial force applied to ball nut 2 (in N) farAccuracy factor (rigidity) frConformity factor (ratio of ball track radius divided by ball diameter) iNumber of loaded turns kRigidity characteristic (in N/4m2/3) SymbolTerm lLength (in mm) PLead (in mm) rRadius (in mm) RRigidity (in N/4m) RtotOverall
24、rigidity (in N/4m) YAuxiliary value according to Hertz for the description of the elliptic integrals of the 1st and 2nd kind Z1Number of loaded balls per turn Lead angle (in degrees) Contact angle (in degrees) Reciprocal curvature radius (in 1/mm) Ratio of the semimajor to the semiminor axes of the
25、contact ellipse Subscript aAxial direction b/tBall and ball track area mMean nBall nut n/sNut body and screw shaft area pPreload sScrew shaft uBall screw within the loaded nut area uBall screw within the loaded nut area modified by accuracy factor (far) (1) Licensed Copy: sheffieldun sheffieldun, na
26、, Tue Dec 05 05:57:27 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6101-4:1987 BSI 12-19993 4.2 Overall rigidity The inverse of the overall rigidity Rtot (in N/4m) is the sum of the inverse rigidity values of the components and shall be calculated from the equation: Ru is the sum of Rn/s and Rb/t.
27、The methods of calculating the rigidity values of the components Rs, Rn/s, Rb/t and Ru are given in 4.3, 4.4, 4.5 and 4.7 respectively. 4.3 Rigidity of the ball screw shaft 4.3.1 General. The rigidity of the ball screw shaft Rs (in N/4m) follows from the elastic deflection of the shaft dls caused by
28、 the axial force Fa and is dependent on the bearing arrangement. The value of Rs shall be calculated as described in 4.3.1 and 4.3.2. 4.3.2 Rigid mounting of the ball screw shaft at one end. (See Figure 2.) The rigidity Rs1 (in N/4m) of the ball screw shaft rigidly mounted at one end shall be calcul
29、ated from the following equation: where 4.3.3 Rigid mounting of the ball screw shaft at both ends. (See Figure 3.) The rigidity of the ball screw shaft rigidly mounted at both ends Rs2 (in N/4m) shall be calculated from the following equation: (2) (3) dc = d0 Dw cos (4) (5) Figure 2 Ball screw shaft
30、 rigidly mounted at one end Figure 3 Ball screw shaft rigidly mounted at both ends Licensed Copy: sheffieldun sheffieldun, na, Tue Dec 05 05:57:27 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6101-4:1987 4 BSI 12-1999 The minimum of rigidity shall be calculated from the following equation: and thus
31、 4.4 Rigidity of nut body and screw shaft under radial component due to axial load The rigidity of the nut body and screw shaft under radial component due to axial load, Rn/s (in N/4m), shall be calculated from the following equation: The axial rigidity of nut body and screw Rn/s (in N/4m) under thi
32、s type of load shall be calculated from the following equation: where NOTEIt has been assumed that the ball screw shaft is solid and that the ball screw shaft and ball nut have the same Youngs modulus and Poissons ratio. As both nut bodies act like preloaded rings, the rigidity Rn/sp of a double nut
33、 is twice as high as that of a single nut, and shall be calculated from: 4.5 Rigidity in the ball/ball track area NOTE 1An alternative graphical method, suitable for approximation only, is given in Appendix A. NOTE 2The following have been disregarded in the calculation: a) the nut expansion factor
34、(common ball screws are assumed to have a sufficiently large thread cross section); b) uneven distribution of load on the balls and threads; c) machining inaccuracies; d) change of contact angle; e) relative displacement between ball nut and ball screw shaft due to the axial back lash see Figure 4(a
35、) and Figure 4(b). 4.5.1 General In order to obtain high rigidity in the ball/ball track area, nut systems are preloaded (see Figure 5). Thus the backlash in the individual nut and the relatively large ball/ball track deflection at low load are eliminated. The preload to be applied has to be determi
36、ned carefully as excessive preload will reduce life. The preload Fp for symmetrical double nuts shall be calculated from the following equation: NOTEThis equation gives an approximate value for Fp. For loads beyond Fam, the point at which the preload nut is unloaded, the double-nut system then exhib
37、its single nut behaviour. The extent of deflection in the ball/ball track area is a function of the following: a) load applied; b) number of loaded balls; c) conformity; d) contact angle; e) ball size; f) nominal diameter. The axial deflection in the ball/ball track area can be derived from Hertzian
38、 formula but shall be approximated by the following equation: Following Hertz, the approach of the components shall be calculated from: where (6) (7) (8) (9) Dc = d0 + Dw cos (10) Rn/sp = 2 Rn/s(11) (12) (13) dls,n b/t = Ys,n c2E 3(FN2 Cs,n)(14) (15) Fp 1 2.83 - Fam= Licensed Copy: sheffieldun sheff
39、ieldun, na, Tue Dec 05 05:57:27 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6101-4:1987 BSI 12-19995 Figure 4 Relative displacement between ball nut and ball screw shaft due to axial backlash Figure 5 Preloaded ball nuts Licensed Copy: sheffieldun sheffieldun, na, Tue Dec 05 05:57:27 GMT+00:00 200
40、6, Uncontrolled Copy, (c) BSI BS 6101-4:1987 6 BSI 12-1999 The auxiliary value Y depends upon the ratio of the semimajor to the semiminor axes of the contact ellipse cos . Cos is determined by the contour of the rolling element and shall be calculated from the equations: The auxiliary values Ys for
41、ball screw shafts and Yn for ball nuts shall be obtained from the following approximated equation: NOTEe is equal to 2.71828 and is the value of the exponential function E(1) used in Naperian logarithms. The material constant cE shall be calculated from the following equation: where For steel on ste
42、el contacts, i.e. screw shaft, balls and nut all of steel, E = 2.1 105 N/mm2 hence E01 = E02 and cE 0.4643 (see 4.5.2). The rigidity characteristic k of one loaded turn of the ball screw shall be calculated from the following equations: and Thus the axial deflection due to Hertz stress exerted on a
43、single nut dlb/t (in 4m) shall be calculated from the following equation: The rigidity of the ball/ball track area Rb/t at the axial force Fa shall be calculated from the following equation: This reveals the dependence of the rigidity on the load. The system rigidity may be increased by applying a p
44、reload force Fp. 4.5.2 Dissimilar materials For dissimilar materials: a) cE s,n shall be calculated for each contact, (i.e. cE s for screw to balls and cE n for balls to nut) using equations 22 and 23. b) ck s,n shall be calculated for each contact by evaluating equation 26 in separate parts as foll
45、ows: 1) cks = Ys 3Cs 2) ckn = Yn 3Cn c) ks,n shall be calculated from equation 25, applied twice using the appropriate subscripts. d) Finally equation 27 is applied twice, once for ks, again for kn and the resulting values of (dlb/t)s,n added together. (16) (17) (18) (19) (20) (21) (22) (23) (Poisso
46、ns ratio for steel)(24) (25) (26) (27) (28) m 10 3 - -= Licensed Copy: sheffieldun sheffieldun, na, Tue Dec 05 05:57:27 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6101-4:1987 BSI 12-19997 4.6 Rigidity of a symmetrically preloaded nut system NOTEOptimum preload is average axial load. Light preload
47、 is less than average axial load. Heavy preload is more than average axial load. 4.6.1 Light to optimum preload Where there is light to optimum preload, the deflection dlb/tp for one nut under preload Fp shall be calculated as in equation 27. The rigidity shall be calculated as shown in a) to d). a)
48、 dlb/tp for Fp; b) dlb/t for Fa; c) The deflection of the double-nut system dlb/ta (in 4m) shall then be calculated from the following equation: d) The rigidity Rb/t shall be calculated from the following equation: 4.6.2 Heavy preload Where there is heavy preload, the deflection dlb/tp and the rigid
49、ity Rb/t of one nut under preload Fp shall be calculated as shown in a) to c). a) dlb/tp for Fp; c) From equation 30, Rb/t shall be calculated as follows: 4.7 Rigidity of the ball screw within the loaded ball nut area The rigidity Ru of the ball screw within the loaded ball nut area shall be calculated from the following equations: Taking into account the accuracy factor far as specified in 4.8, the minimum rigidity shall be calculated from the following equation: 4.8 Accuracy factor As tolera