DIN ISO 3408-4-2011 Ball screws - Part 4 Static axial rigidity (ISO 3408-4 2006)《滚珠丝杠副 第4部分 轴向静刚度(ISO 3408-4-2006)》.pdf

1、April 2011 Translation by DIN-Sprachendienst.English price group 11No part of this translation may be reproduced without prior permission ofDIN Deutsches Institut fr Normung e. V., Berlin. Beuth Verlag GmbH, 10772 Berlin, Germany,has the exclusive right of sale for German Standards (DIN-Normen).ICS

2、25.060.99!$nOS“1754448www.din.deDDIN ISO 3408-4Ball screws Part 4: Static axial rigidity (ISO 3408-4:2006)English translation of DIN ISO 3408-4:2011-04Kugelgewindetriebe Teil 4: Statische axiale Steifigkeit (ISO 3408-4:2006)Englische bersetzung von DIN ISO 3408-4:2011-04Vis billes Partie 4: Rigidit

3、axiale statique (ISO 3408-4:2006)Traduction anglaise de DIN ISO 3408-4:2011-04www.beuth.deDocument comprises pagesIn case of doubt, the German-language original shall be considered authoritative.2103.11 A comma is used as the decimal marker. Contents Page National foreword .3 National Annex NA (info

4、rmative) Bibliography.3 1 Scope 4 2 Normative references 4 3 Terms and definitions4 4 Symbols and subscripts .4 4.1 Symbols 4 4.2 Subscripts 5 5 Determination of static axial rigidity, R6 5.1 General6 5.2 Static axial rigidity, R .8 5.3 Static axial rigidity of ball screw, Rbs.8 5.4 Static axial rig

5、idity of ball screw shaft, Rs.8 5.4.1 General8 5.4.2 Rigid mounting of ball screw shaft at one end.8 5.4.3 Rigid mounting of ball screw shaft at both ends .9 5.5 Static axial rigidity of ball nut unit, Rnu9 5.5.1 Static axial rigidity of ball nut unit with backlash, Rnu19 5.5.2 Static axial rigidity

6、 of symmetrically preloaded ball nut unit, Rnu2,4.13 5.5.3 Correction for accuracy, far.15 Annex A (informative) Example calculation of static axial rigidity in preloaded symmetrical double nut system .17 Annex B (informative) Correction for load application, fal.20 2 DIN ISO 3408-4:2011-04 National

7、 foreword This standard has been prepared by Technical Committee ISO/TC 39 “Machine Tools”, Working Group WG 7 “Ball screws”. The responsible German body involved in its preparation was the Normenausschuss Werkzeugmaschinen (Machine Tools Standards Committee). ISO 3408 consists of the following part

8、s, under the general title Ball screws: Part 1: Vocabulary and designation Part 2: Nominal diameters and nominal leads Metric series Part 3: Acceptance conditions and acceptance tests Part 4: Static axial rigidity Part 5: Static and dynamic axial load ratings and operational life Annexes A and B are

9、 for information only. The DIN Standard corresponding to the International Standard referred to in this document is as follows: ISO 3408-1 DIN ISO 3408-1 National Annex NA (informative) Bibliography DIN ISO 3408-1, Ball screws Part 1: Vocabulary and designation 3 DIN ISO 3408-4:2011-04 1 Scope This

10、part of ISO 3408 sets forth terms and mathematical relations relevant to the determination of the static axial rigidity of the ball screw. 2 Normative references The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited ap

11、plies. For undated references, the latest edition of the referenced document (including any amendments) applies. ISO 3408-1:2006, Ball screws Part 1: Vocabulary and designation 3 Terms and definitions For the purposes of this document, the terms and definitions given in ISO 3408-1 apply. 4 Symbols a

12、nd subscripts 4.1 Symbols Symbol Description Unit Contact angle degrees, Reciprocal curvature radius mm1 Ratio of the semi-major to the semi-minor axes of the contact ellipse Lead angle degrees, l Elastic deflection m cEMaterial constant ckGeometry factor N2/3m dboDiameter of the deep hole bore mm d

13、cDiameter of load application on the ball screw shaft mm DcDiameter of load application on the ball nut mm DpwBall pitch circle diameter mm DwBall diameter mm D1Outer diameter of ball nut mm Ball screws Part 4: Static axial rigidity 4 DIN ISO 3408-4:2011-04 Symbol Description Unit E Modulus of elast

14、icity N/mm2farCorrection factor for accuracy classes (rigidity) falCorrection factor for load application frs, frnConformity (ratio of ball/balltrack radius to ball diameter) of ball screw shaft and ball nut F Axial force, load N i Number of loaded turns k Rigidity characteristic N/m3/2l Length mmls

15、Unsupported length of ball screw shaft mm m Poissons constant (e.g. for steel m = 10/3) n Rotational speed min1PhLead mm q Time percentage % R Rigidity N/msaBacklash (axial play) m Y Auxiliary value according to Hertz for the description of the elliptic integrals of the first and second kinds N2/3m4

16、/3z1Number of effectively loaded balls per turn z2Number of unloaded balls in the recirculation system, only for systems where balls will be recirculated after one turn 4.2 Subscripts Symbol Description ar refers to accuracy b refers to ball bs refers to ball screw c refers to nut body/ball screw sh

17、aft e refers to external load or the resulting deformation respectively lim refers to limit load (at this value the contact between balls and balltracks of ball screw shaft andball nut is eliminated) m refers to equivalent N refers to normal load which acts upon balls and balltracks of the ball scre

18、w shaft and ball nut inthe direction of the contact angle n refers to ball nut pr refers to preload s refers to ball screw shaft b/t refers to ball/balltrack area nu refers to ball screw within the loaded ball nut area 1 refers to ball nut 1 2 refers to ball nut 2 5 DIN ISO 3408-4:2011-04 5 Determin

19、ation of static axial rigidity, R 5.1 General The static axial rigidity of a ball screw exerts a major influence on its positioning accuracy. It is a function of the design of the ball screw, its support and bearing arrangement. For the purpose of the calculation given below support and bearing arra

20、ngement have been disregarded. The static axial rigidity of ball screws is not linear. For the purpose of the study of rigidity, a ball screw can 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 load

21、application. The deflection to be determined is caused by axial deflections of the screw shaft and the ball nut body, radial deflections of the screw shaft and the ball nut body, deflections of the balls and the thread land. The calculation of the deflections attributable to the ball contact is base

22、d on the theory related to Hertz stress. The following preconditions should be met as closely as possible: the material of the contacting partners shall be homogenous and isotropic, in addition, Hookes law applies, i.e., no plastic deformation, and in the contact area only normal stress shall be act

23、ing, i.e., a level pressure surface is generated. Moreover, the applied simplified theory of Hertz specifies identical elasticity modulus and transversal contraction parameter for the material of ball screw shaft, ball nuts and balls. When calculating axial rigidity it is important to differentiate

24、between ball nuts that have backlash and those that have none, i.e. preloaded ball nuts. It is possible to generate preload by different methods: a) Single ball nut with continuous thread. Preloading by oversize balls, resulting in four-point-ball-contact. See Figure 1. Figure 1 6 DIN ISO 3408-4:201

25、1-04 b) Single ball nut with shifted thread between the preloaded areas, achieving two-point-ball-contact. See Figure 2. Figure 2 c) Single ball nut with double start thread and shifted pitch (two-point-ball-contact). See Figure 3. Figure 3 d) Double ball nut consisting of two single ball nuts, each

26、 with continuous thread. Axial displacement of the two single ball nuts against each other. See Figure 4. Figure 4 The rigidity calculation set forth in this standard can be applied to all preloading methods described. As it is very time-consuming and hence unsuitable for practical purposes to deter

27、mine the precise axial deflection on the basis of the corresponding formulae, a reasonably simplified calculation method is outlined below so that the calculation may be effected with a pocket calculator. 7 DIN ISO 3408-4:2011-04 5.2 Static axial rigidity, R The static axial rigidity, R, constitutes

28、 the resistance to deformation and denotes the force F, in newtons, which is required to effect a component deflection l by 1 m in the axial direction of load application: FRl=(1) 5.3 Static axial rigidity of ball screw, RbsThe overall rigidity, Rbs, is arrived at by adding the pertinent rigidity va

29、lues of the components: bs s nu,ar11 1RRR=+ (2) 5.4 Static axial rigidity of ball screw shaft, Rs5.4.1 General The rigidity of the ball screw shaft follows from the elastic deflection of the ball screw shaft lscaused by an axial force F and depends on the bearing arrangement. 5.4.2 Rigid mounting of

30、 ball screw shaft at one end See Figure 5. aSee Equation (4). Figure 5 Where the rigidity is ()22cbo13s410sdd ERl =in case of a solid shaft bo0d = (3) cpwwcosdD D = (4) 8 DIN ISO 3408-4:2011-04 5.4.3 Rigid mounting of ball screw shaft at both ends See Figure 6. aSee Equation (4). Figure 6 Where the

31、rigidity is ()22cboss23ss2s2410dd ElRlll =(5) the minimum of rigidity is obtained at ss22ll = and thus is ()22cbos2,min3s10dd ERl =(6) 5.5 Static axial rigidity of ball nut unit, Rnu5.5.1 Static axial rigidity of ball nut unit with backlash, Rnu15.5.1.1 Static axial rigidity of nut body and screw sh

32、aft under resulting radial components of load Rn/sDetermination of Rn/s:n/sn/sFRl=(7) n/sn/sFlR= (8) Nut: thick-walled cylinder subjected to “internal pressure” (radial component of normal ball thrust). 9 DIN ISO 3408-4:2011-04 Screw shaft: cylinder subjected to “external pressure” (radial component

33、 of normal ball thrust). Premise: the ball screw shaft is either solid or deephole drilled; ball screw shaft and ball nut have the same Youngs modulus and Poissons ratio. The axial rigidity of the nut body and screw shaft under this type of load is 2hn/s22 2231c cbo22 221c cbo2tan10iP ERDD ddDD dd =

34、9) where cpwwcosDD D =+ (10) 5.5.1.2 Static axial rigidity in ball/balltrack area, Rb/tIn order to simplify, the ball nut body and the screw shaft deformations have been disregarded in this calculation. The same applies to uneven distribution of load on the balls and threads, machining inaccuracie

35、s, and change of contact angle. The relative displacement between ball nut and ball screw shaft due to the axial backlash has not been taken into account because it is not an elastic deflection see Figure 7 a) and b). 10 DIN ISO 3408-4:2011-04 X axial displacement between ball nut and ball screw sha

36、ft Y external axial load, FeFigure 7 The extent of the axial deflection on the ball/balltrack area is a function of load applied, nominal diameter, ball size, number of loaded balls, conformity, and angle of load application. Thus the axial deflection in the ball/balltrack area is sufficiently appro

37、ximated by the following equation: sb/t nb/tb/tcos sinlll +=(11) According to Hertz the approach of the components is calculated from: 223s,nb/t s,n E N s,nlcF=(12)11 DIN ISO 3408-4:2011-04 Where for the screw shaft balltrack/ball contact applies: swrsw pww41 2coscosDfDDD= +(13) For the nut balltrac

38、k/ball contact applies: nwrnw pww41 2coscosDfDDD= +(14)The auxiliary values Ys,ndepend upon the ratio of the semi-major to the semi-minor axes of the contact ellipse cos . The following equation makes use of sin , which can be obtained by: 2sin 1 cos = () ()1/ 4 1/ 2s,n1,282 0,154 sin 1,348 sin 0,19

39、4 sinY = + (15) cos is solely conditioned by the contour of the rolling partners. It is described as follows: rs w pw wss12coscoscosfD D D=(16) rn w pw wnn12coscoscosfD D D+=(17)0s,n 0b3Es,n0s,n 0bE11550 EcEE+=(18) with s,n,b0s,n,b2s,n,b11EEm=(19) For ball bearing steel: 5snb2,1 10EEE= snb10 / 3mmm=

40、 0s 0n 0b 0E EEE= Es En Eb E0,4643cccc= 12 DIN ISO 3408-4:2011-04 N1cos sinFFiz = (20) pw12wintegercosDzzD=(21) hpwarctanPD =(22) The rigidity characteristic k of one loaded turn of the ball screw is calculated from: 5/2 5/2133/2Eksin coszkcc=(23) and 33ks sn nc = +(24)Thus, the axial deflection due

41、 to Hertz stress exerted on a single nut can be calculated: 2/3b/tFlki=(25) ()()1/3b/t2/321dd3lF Fki= (26)The static axial rigidity of the ball/balltrack area Rb/tat the axial force F is: ()()23b/t eb/td32dFR Fikl=(27) This reveals the dependence of the spring rigidity on the load. The system rigidi

42、ty may be increased by increasing the axial force exerted on the ball screw, e.g. by a preload force Fpr. 5.5.1.3 Static axial rigidity of ball nut unit with backlash, Rnu1nu1 b/t n/s111RRR=+ (28) 5.5.2 Static axial rigidity of symmetrically preloaded ball nut unit, Rnu2,45.5.2.1 Static axial rigidi

43、ty of nut body and screw shaft under preload, Rn/s,prAs both nut bodies act like preloaded rings, the rigidity, Rn/s,pr, of a double nut is twice as high as that of a single nut: n/s,pr n/s2R R= (29) 13 DIN ISO 3408-4:2011-04 5.5.2.2 Static axial rigidity of ball/balltrack area under preload, Rn/t,p

44、r (see Figure 8) In order to obtain high rigidity in the ball/balltrack area, nut systems are preloaded. Thus the backlash in the individual nut and the relatively large ball/balltrack deflection at low load are eliminated. Key 1 ball nut 1 2 ball nut 2 3 ball screw shaft 4 straight approximation li

45、ne 5 actual curve aActual curve of the axial deflection in the ball/balltrack area of the preloaded ball nut system if an additional external load between Fc= 0 and Fc= Flimis applied. Maximum deviation between 4 and 5 is approximately 6 %. Figure 8 The preload force to be applied has to be determin

46、ed carefully, as excessive preload will reduce life. The following equation will furnish a guide value for symmetric double nuts: mpr3/22FF = (30) The equivalent load Fmis obtained from the following equation: 33mem1niiiinF Fqn=(31) 14 DIN ISO 3408-4:2011-04 The axial deflection of the ball/balltrac

47、k area due to the preload of a symmetrically preloaded nut system is calculated according to Equation (25): 2/3prb/t,prFlki=(32) For 0 Feu Flim, the rigidity Rb/tin the ball/balltrack area is determined as follows: b/t,prl for Fpris determined as for Equation (32), as there is b/t,pr b/t,limll= and 3/2lim pr2F F= (33) the approximation results in limb/tb/t,prFRl(34) ()23/23b/t pr2R Fki (35)5.5.2.3 Single or double ball nut preloaded by two-point-ball-contact, Rnu2As both nut bodies act like preloaded rings, the rigidity Rn/sof a doubl