[雙語(yǔ)翻譯]--機(jī)械外文翻譯--參數(shù)化建模滾珠絲杠主軸（原文）

1、COMPUTER AIDED ENGINEERINGParametric modeling of ball screw spindlesA. Dadalau ? M. Mottahedi ? K. Groh ?A. VerlReceived: 15 April 2010 / Accepted: 13 August 2010 / Published online: 25 August 2010 ? German Academic Soci

2、ety for Production Engineering (WGP) 2010Abstract In the product development process numerical optimization can successfully be applied in the early product design stages. In the very common case of ball screw drives, th

3、e dynamical behavior is most depending on the geometrical shape of the ball screw itself. Properties like axial and torsional stiffness, moment of inertia, max- imum velocity and acceleration are determined not only by t

4、he servo motor but also by screw diameter, slope and ball groove radius. Furthermore coupling effects between the design variables make the optimization task even more difficult. In order to capture these effects, effici

5、ent numerical (usually FEM or MBS) models are needed. In this work a new more accurate and efficient method of computing the axial and torsional stiffness of ball screw spindles is presented. We analytically derive param

6、etric equations which depicts most of the dependencies of stiffness on geometrical parameters of the screw. Further- more, we enhance the analytical model with an identified function, which increase the accuracy even mor

7、e. The presented analytical model is validated against FEM model and catalog data with the help of numerous examples.Keywords Ball screw ? Spindle ? Stiffness ? FEM ? Analytic model1 IntroductionThe axial and torsional s

8、tiffness of ball screw spindles plays an important role in the dynamic behavior of ball screw drives, since it essentially determine the first and second eigenvalues of ball screw drives. When modeling ball screw drives

9、with FEM the thread is usually ignored and some mean diameter is used to model a simplified ball screw. Therefore it is crucial to have knowledge about the best approximating mean diameter. Most of the previous work on m

10、odeling and simulating stiffness of ball screw drives concentrate on modeling the assembly between ball screw nut and ball screw spindle, which implies high accuracy modeling of contact, [1, 2]. In [3] Jarosch com- pares

11、 theoretical stiffness of different types of ball screws, but the spindle is taken into account simplified as an cyl- inder with diameter equal to the spindle outer diameter, thus ignoring the stiffness weakening due to

12、spindle thread. With knowledge about the real axial kuz and torsional kuz stiffness of a screw of unit length, a mean diameter can be computed with the help ofdm; uz¼ffiffiffiffiffiffiffiffi 4kuz pErð1Þand

13、dm; uz ¼ffiffiffiffiffiffiffiffiffiffiffi 32kuz pG4 rð2Þrespectively, E Young’s modulus and G shear modulus. The mean diameter is always less than the spindle outer diameter. For each stiffness we get two

14、different mean diameters. It depends on each application which mean diameter is the best to choose. A linear combination of the two diameters could also be done. In general ball screw manufacturers provide data for axial

15、 stiffness but not forA. Dadalau (&) ? M. Mottahedi ? K. Groh ? A. Verl Institute for Control Engineering of Machine Tools and Manufacturing Units (ISW), Stuttgart, Germany e-mail: alexandru.dadalau@isw.uni-stuttgart

16、.de URL: http://www.isw.uni-stuttgart.de123Prod. Eng. Res. Devel. (2010) 4:625–631DOI 10.1007/s11740-010-0264-z4 Analytical model of ball screwsIn our approach we intend to derive an analytical model of the ball screw dr

17、ive, capable of capturing theweakening effect of the thread. Having an accurate analytical method to compute the stiffness of ball screws also have a practical application in optimization, where efficient parameter model

18、s are preferred instead of FiniteTable 1 Different spindel geometries and corresponding stiffness (DIN, catalog, FEM)Nominal [ d0 (mm)Spindle pitch Ph (mm)Ball [ Dw (mm)Spindle [ d1 (mm)Spindle Core [ d2 (mm)Number of th

19、reads Axial Stiffness DIN ISO 3408-4 (N/lm)Axial Stiffness Catalog (N/lm)Deviation Catalog versus DINAxial Stiffness FEM (N/lm)Deviation FEM versus DINDeviation FEM versus CatalogTorsional stiffness FEM (Nm/rad)1 6.0 1.0

20、 0.800 6.0 5.3 1 4.871 5.0 2.65% 4.821 -1.02% -3.57% 6.912 6.0 2.0 0.800 6.0 5.3 2 4.871 5.0 2.65% 4.818 -1.08% -3.63% 6.923 8.0 1.0 0.800 8.0 7.3 1 9.116 9.0 -1.27% 9.049 -0.73% 0.54% 24.224 8.0 2.0 1.200 8.0 7.3 1 8.43

21、5 9.0 6.69% 9.339 10.71% 3.77% 26.315 8.0 2.5 1.588 7.5 6.3 1 7.800 8.0 2.56% 7.150 -8.33% -10.62% 15.786 12.0 2.0 1.200 11.7 10.8 1 20.510 21.0 2.39% 20.087 -2.06% -4.35% 120.837 12.0 5.0 2.000 11.4 9.9 1 18.482 18.0 -2

22、.61% 18.239 -1.31% 1.33% 103.288 12.0 10.0 2.000 11.4 9.9 2 18.482 18.0 -2.61% 18.209 -1.48% 1.16% 104.159 16.0 5.0 3.000 15.0 12.9 1 31.769 32.0 0.73% 29.939 -5.76% -6.44% 276.1610 16.0 10.0 3.000 15.0 12.9 2 31.769 32.

23、0 0.73% 29.875 -5.96% -6.64% 277.7811 16.0 16.0 3.000 15.0 12.9 2 31.769 32.0 0.73% 31.589 -0.57% -1.28% 315.4612 20.0 5.0 3.000 19.0 16.9 1 52.721 53.0 0.53% 50.372 -4.45% -4.96% 771.4913 20.0 10.0 3.000 19.0 16.9 1 52.

24、721 53.0 0.53% 53.662 1.79% 1.25% 879.9714 20.0 20.0 3.500 19.3 16.7 4 50.656 52.0 2.65% 48.887 -3.49% -5.99% 741.7115 20.0 40.0 3.500 19.0 16.4 4 50.656 52.0 2.65% 50.710 0.11% -2.48% 819.1216 25.0 5.0 3.000 24.0 21.9 1

25、 86.332 86.0 -0.38% 83.331 -3.48% -3.10% 2091.4017 25.0 10.0 3.000 24.0 21.9 1 86.332 86.0 -0.38% 87.569 1.43% 1.82% 2318.2118 25.0 25.0 3.500 24.0 21.4 4 83.684 84.0 0.38% 80.593 -3.69% -4.06% 2008.6719 32.0 5.0 3.500 3

26、1.0 28.4 1 143.778 144.0 0.15% 138.334 -3.79% -3.93% 5706.8520 32.0 10.0 3.969 31.0 27.9 1 140.566 141.0 0.31% 140.035 -0.38% -0.68% 5995.9821 32.0 20.0 3.969 31.0 27.9 2 140.566 141.0 0.31% 139.870 -0.50% -0.80% 6010.52

27、22 32.0 32.0 3.969 31.0 27.9 4 140.566 141.0 0.31% 137.205 -2.39% -2.69% 5807.8523 40.0 5.0 3.500 39.0 36.4 1 232.249 232.0 -0.11% 225.372 -2.96% -2.86% 15068.6824 40.0 10.0 6.000 38.0 33.8 1 210.882 211.0 0.06% 201.489

28、-4.45% -4.51% 12343.8825 40.0 12.0 6.000 38.0 33.8 1 210.882 211.0 0.06% 204.673 -2.94% -3.00% 12826.6226 40.0 16.0 6.000 38.0 33.8 2 210.882 211.0 0.06% 198.026 -6.10% -6.15% 11799.8727 40.0 20.0 6.000 38.0 33.8 2 210.8

29、82 211.0 0.06% 201.268 -4.56% -4.61% 12372.8328 40.0 40.0 6.000 38.0 33.8 4 210.882 211.0 0.06% 200.832 -4.77% -4.82% 12463.2529 50.0 5.0 3.500 49.0 46.4 1 372.525 373.0 0.13% 364.081 -2.27% -2.39% 39143.6230 50.0 10.0 6

30、.000 48.0 43.8 1 345.327 345.0 -0.09% 333.323 -3.48% -3.38% 33462.0931 50.0 12.0 6.000 48.0 43.8 1 345.327 345.0 -0.09% 337.559 -2.25% -2.16% 34497.5032 50.0 16.0 6.000 48.0 43.8 2 345.327 345.0 -0.09% 328.901 -4.76% -4.

31、67% 32278.6933 50.0 20.0 6.500 48.0 43.4 2 340.012 340.0 0.00% 326.939 -3.84% -3.84% 32243.1534 50.0 25.0 6.500 48.0 43.4 2 340.012 340.0 0.00% 332.081 -2.33% -2.33% 33600.0535 50.0 40.0 6.500 48.0 43.4 4 340.012 340.0 0

32、.00% 326.511 -3.97% -3.97% 32392.3036 63.0 10.0 6.000 61.0 56.8 1 569.421 569.0 -0.07% 554.196 -2.67% -2.60% 91785.0837 63.0 20.0 6.500 61.0 56.4 2 562.589 563.0 0.07% 545.848 -2.98% -3.05% 89089.1638 63.0 40.0 6.500 61.

33、0 56.4 4 562.589 563.0 0.07% 545.452 -3.05% -3.12% 89272.9839 80.0 10.0 6.500 78.0 73.3 1 937.769 938.0 0.02% 914.314 -2.50% -2.53% 248264.7340 80.0 20.0 12.700 76.0 67.0 1 831.893 832.0 0.01% 791.432 -4.86% -4.88% 19041