The geometry of the Moineau pump

1. The geometry of the Moineau pump Jens Gravesen Computer Aided Geometric Design 25 (2008) 792–800 Reporter: yangying Thursday, Nov 27, 2008

2. About the author • Jens Gravesen • Techincal University of Denmark • Department of Mathematics • J.Gravesen@mat.dtu.dk Research Interests • Differential Geometry, Geometric Design • Interactive notes on Curves • Subdivision surfaces • Moineau pump

3. Previous related work • Mathematical problems for Moineau pump, Report from the 57th European Study Group with Industry, Kgs. Lyngby, Eenmark,2006, http://www2.mat.dtu.dk/ESGI/57/report. • Šír et al., 2008Z. Šír, J. Gravesen and B. Jüttler, Curves and surfaces represented by polynomial support functions, Theoret. Comput. Sci. 392 (2008), pp. 141–157. • Gravesen, Spherical curves for the design of conical Moineau pumps. SIAM Activity Group in Geometric Design, Problem Section

4. Introduction (1) Moineau pump: an invention from 1931 by the French engineer René Moineau

5. Introduction (2) 螺杆泵属于回转式容积泵，按螺杆根数可分为单螺杆泵、双螺杆泵、三螺杆泵和五螺杆泵等多种形式. 一根单头螺旋的转子和一个通常用弹性材料制造的具有双关螺旋的定子，当转子在定于型腔内绕定子的轴线作行星回转时，转定子之间形成的密闭腕就沿转子螺线产生位移.

6. Introduction (3) 螺杆(转子)具有单头螺纹，其任意截面皆为半径为R的圆，截面的中心位于螺旋线上且与螺杆的轴心线偏离一个偏心距e，绕轴旋转且沿轴向移动而形成的。

7. Introduction (4) 衬套（定子）内表面具有双头螺纹，其任意截面为一长圆，两端是半径为R（等于螺杆截面半径）的半圆，中间是长为4e的直线段。衬套的任意截面都是相同的长圆，只是彼此互相错开一个角度。

8. 螺杆(转子)装入衬套（定子）后，螺杆表面与衬套内螺纹表面之间形成一个个封闭的腔室，同时任意截面也被分成上下两个月牙形工作室。当螺杆旋转时，靠近吸入室的第一个工作室的容积逐渐增大，形成负压，在压差的作用下液体被吸入工作室。随着螺杆的继续转动，工作腔容积不断增至最大后，这个工作室封闭，并将液体沿轴向推向压出室。与此同时上下两个工作室交替循环地吸入和排出液体，因此液体被连续不断地从吸入室沿轴向推向压出室。螺杆(转子)装入衬套（定子）后，螺杆表面与衬套内螺纹表面之间形成一个个封闭的腔室，同时任意截面也被分成上下两个月牙形工作室。当螺杆旋转时，靠近吸入室的第一个工作室的容积逐渐增大，形成负压，在压差的作用下液体被吸入工作室。随着螺杆的继续转动，工作腔容积不断增至最大后，这个工作室封闭，并将液体沿轴向推向压出室。与此同时上下两个工作室交替循环地吸入和排出液体，因此液体被连续不断地从吸入室沿轴向推向压出室。

10. Introduction (5) The geometry of the Moineau pump • Two parts rotating relatively to each other in an eccentric motion • epi- and hypo-cycloids

11. Introduction (6) • Danish pump manufacturer Grundfos wanted an investigation • 57th European Study Group with Industry(2006) • Mathematical problems for Moineau pumps • Question: If it is possible to avoided the points with infinite curvature?

12. Steps: • The original design • The support function • General designs • Conclusion • About spherical curves for the design of conical Moineau pumps

13. The original design • The circle C1 rolls inside the circle C2 and the point P rolls back and forth on the diameter H2 • Offset both P and H2 with the same amount obtain the rotor and the stator

14. Construct the pump

15. The n+1:n hypo-cycloid construction

17. The n+1:n epi-cycloid construction

19. The n+1:n hypo-epi-cycloid construction

21. The support function h (by M.sabin 1974) The tangent plane T of point (x,y,z) (l,m,n) is the normal N So form is Explicit form So h is the distance from origin to the tangent line.

22. Scaling • Translate • Rotation • Offset • Convolution

23. The support function for this artical In that case the tangent is the normal is Distance from origin to tangent line

24. Rotated • Translated • Scaled

25. Theorem 1 Consider an ordinary inflection point (x0,y0) with curvature Where s is arc length and . Denote the normal direction by and let be the normal direction at the inflection point. We can introduce a parameter u such that And the support function can be written Furthermore,

26. General designs The motion generated by rolling a circle of radius b inside a circle of radius a is given by

27. Theorem 2 Let a positively curved arc of the rotor have the support function and the motion be generated by a circle of radius n rolling inside a circle of radius n+1,where let

28. then

29. Theorem 3 Consider a Moineau pump design where the horizontal sections of both the stator and rotor consists of smooth arcs with alternating strictly and negative curvature in the interior. If the support function of the positively curved arc of the rotor either has an expansion Then there are points with infinite curvature in the design.

30. examples we consider a deformation of the 3:2 hypo-epi-cycloid design. Let c=1, =-2. The support function with we consider deformations

32. A new 3:2 design

33. Conclusion • It is proved that if the rotor and stator consists of alternating arcs with positive and negative curvature then it is impossible to avoid points with infinite curvature in the design. • It is an open problem whether the inclusion of straight lines in the design allows for other curvature bounded designed than the 2:1 hypo-cycloid construction.

34. Cylindrical Moineau pumps

35. Conical Moineau pumps In order to have a periodic motion the circumference of the two circles has to be in the proportion n: n-1. If the top angles of the two cones are 2v and 2w, respectively, then (n-1)sin(w) = nsin(v).

36. If the 2:1 hypo-cycloid construction (n = 2) is intersected by the unit sphere there will be a gab of the size gab

37. The problem • Is it possible to construct a conical Moineau pump which is mathematical watertight? • If not, then design has the minimal possible gab (for a given angle v)? • Is it possible to find a representation of spherical curves that allows exact calculations similar to the planar case?

38. Thank you!