BS ISO 3408-4-2006 Ball screws - Static axial rigidity《滚珠丝杠 静态轴向刚性》.pdf

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1、 g49g50g3g38g50g51g60g44g49g42g3g58g44g55g43g50g56g55g3g37g54g44g3g51g40g53g48g44g54g54g44g50g49g3g40g59g38g40g51g55g3g36g54g3g51g40g53g48g44g55g55g40g39g3g37g60g3g38g50g51g60g53g44g42g43g55g3g47g36g58ICS 25.060.99Ball screws Part 4: Static axial rigidityBRITISH STANDARDBS ISO 3408-4:2006BS ISO 3408

2、-4:2006This British Standard was published under the authority of the Standards Policy and Strategy Committee on 30 November 2006 BSI 2006ISBN 0 580 49522 1Compliance with a British Standard cannot confer immunity from legal obligations.Amendments issued since publicationAmd. No. Date CommentsThis p

3、ublication does not purport to include all the necessary provisions of a contract. Users are responsible for its correct application. National forewordThis British Standard was published by BSI. It is the UK implementation of ISO 3408-4:2006. It supersedes BS 6101-4:1987 which is withdrawn.The UK pa

4、rticipation in its preparation was entrusted to Technical Committee MTE/1, Machine tools.A list of organizations represented on MTE/1 can be obtained on request to its secretary.Reference numberISO 3408-4:2006(E)INTERNATIONAL STANDARD ISO3408-4First edition2006-06-15Ball screws Part 4: Static axial

5、rigidity Vis billes Partie 4: Rigidit axiale statique BS ISO 3408-4:2006ii iiiContents Page Foreword iv 1 Scope . 1 2 Normative references . 1 3 Terms and definitions. 1 4 Symbols and subscripts 1 4.1 Symbols . 1 4.2 Subscripts . 2 5 Determination of static axial rigidity, R. 3 5.1 General. 3 5.2 St

6、atic axial rigidity, R 5 5.3 Static axial rigidity of ball screw, Rbs5 5.4 Static axial rigidity of ball screw shaft, Rs5 5.4.1 General. 5 5.4.2 Rigid mounting of ball screw shaft at one end 5 5.4.3 Rigid mounting of ball screw shaft at both ends 6 5.5 Static axial rigidity of ball nut unit, Rnu. 6

7、5.5.1 Static axial rigidity of ball nut unit with backlash, Rnu16 5.5.2 Static axial rigidity of symmetrically preloaded ball nut unit, Rnu2,4. 10 5.5.3 Correction for accuracy, far12 Annex A (informative) Example calculation of static axial rigidity in preloaded symmetrical double nut system . 14 A

8、nnex B (informative) Correction for load application, fal. 17 BS ISO 3408-4:2006iv Foreword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out throu

9、gh ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collabora

10、tes closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2. The main task of technical committees is to prepare International Standar

11、ds. Draft International Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote. Attention is drawn to the possibility that some of the elements of th

12、is document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. ISO 3408-4 was prepared by Technical Committee ISO/TC 39, Machine tools. ISO 3408 consists of the following parts, under the general title Ball screws: Part 1: Vocabulary

13、 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 BS ISO 3408-4:20061Ball screws Part 4: Static axial rigidity 1 Scope This par

14、t 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 appli

15、es. 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 and

16、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 dcDi

17、ameter 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 BS ISO 3408-4:20062 Symbol Description Unit E Modulus of elasticity N/mm2farCorrection factor for accuracy class

18、es (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 mmlsUnsupported length of ball screw shaft mm m Poisso

19、ns 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 N2/3m4/3z1Number of effectively loaded balls per turn z2

20、Number 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 shaft e refers to external load or the resulting def

21、ormation 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 screw shaft and ball nut inthe direction of the contac

22、t 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 BS ISO 3408-4:200635 Determination of static axial rigidity, R 5.1 General The static

23、 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 arrangement have been disregarded. The static axial rigidity

24、 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 application. The deflection to be determined is caused b

25、y 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 based on the theory related to Hertz stress. The following p

26、reconditions 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 acting, i.e., a level pressure surface is generated. Moreov

27、er, 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 between ball nuts that have backlash and those that have

28、 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 BS ISO 3408-4:20064 b) Single ball nut with shifted thread between the prelo

29、aded 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 with continuous thread. Axial displacement of the two single

30、 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 determine the precise axial deflection on the basis of the corresp

31、onding formulae, a reasonably simplified calculation method is outlined below so that the calculation may be effected with a pocket calculator. BS ISO 3408-4:200655.2 Static axial rigidity, R The static axial rigidity, R, constitutes the resistance to deformation and denotes the force F, in newtons,

32、 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 values of the components: bs s nu,ar11 1RRR=+ (2) 5.4 Static axial ri

33、gidity 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 ball screw shaft at one end See Figure 5. aSee Equation (4). Figur

34、e 5 Where the rigidity is ()22cbo13s410sdd ERl =in case of a solid shaft bo0d = (3) cpwwcosdD D = (4) BS ISO 3408-4:20066 5.4.3 Rigid mounting of ball screw shaft at both ends See Figure 6. aSee Equation (4). Figure 6 Where the rigidity is ()22cboss23ss2s2410dd ElRlll =(5) the minimum of rigidity is

35、 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 shaft under resulting radial components of load Rn/sDetermination of Rn/s:

36、n/sn/sFRl=(7) n/sn/sFlR= (8) Nut: thick-walled cylinder subjected to “internal pressure” (radial component of normal ball thrust). BS ISO 3408-4:20067Screw shaft: cylinder subjected to “external pressure” (radial component of normal ball thrust). Premise: the ball screw shaft is either solid or deep

37、hole 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 =+(9) where cpwwcosDD D =+ (10) 5.5.1.2 Static axial rigidity in ball/balltrack

38、 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 inaccuracies, and change of contact angle. The relative displacement between ball nut and

39、 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). BS ISO 3408-4:20068 X axial displacement between ball nut and ball screw shaft Y external axial load, FeFigure 7 The extent of the axial deflection on the ball/

40、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 approximated by the following equation: sb/t nb/tb/tcos sinlll +=(11) According to Hertz

41、the approach of the components is calculated from: 223s,nb/t s,n E N s,nlcF=(12)BS ISO 3408-4:20069Where for the screw shaft balltrack/ball contact applies: swrsw pww41 2coscosDfDDD= +(13) For the nut balltrack/ball contact applies: nwrnw pww41 2coscosDfDDD= +(14)The auxiliary values Ys,ndepend upon

42、 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 0,194 sinY = + (15) cos is solely conditioned by the contour of the rolling partners. It is des

43、cribed 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= 0s 0n 0b 0E EEE= Es En Eb E0,4643cccc= BS ISO 3408-4:200610 N1cos sinFFiz = (20) pw12winte

44、gercosDzzD=(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 to Hertz stress exerted on a single nut can be calculated: 2/3b/tFlki=(25) ()()1/3b/t2/321dd3lF

45、 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 rigidity may be increased by increasing the axial force exerted on the ball screw, e.g. by a preload f

46、orce 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 rigidity of nut body and screw shaft under preload, Rn/s,prAs both nut bodies act like preloaded rings

47、, 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) BS ISO 3408-4:2006115.5.2.2 Static axial rigidity of ball/balltrack area under preload, Rn/t,pr (see Figure 8) In order to obtain high rigidity in the ball/balltrack area, nut systems are preloade

48、d. 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 line 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 determined carefully, as excessive preload will reduce life.