The geometry of the Moineau pump

Jens Gravesen

Computer Aided Geometric Design 25 (2008) 792–800

Reporter: yangyingThursday, Nov 27, 2008

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

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.

S(ír et al., 2008Z. S(ír, 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

Introduction (1)

Moineau pump: an invention from 1931 by the French engineer René Moineau

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 .

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。

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。

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。

Introduction (5)

The geometry of the Moineau pump Two parts rotating relatively to each other in an

eccentric motion epi- and hypo-cycloids

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?

Steps:

The original design

The support function

General designs

Conclusion

About spherical curves for the design of conical Moineau pumps

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

Construct the pump

The n+1:n hypo-cycloid construction

The n+1:n epi-cycloid construction

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

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.

0l x m y n z h

( , ) 0f h N

( )h h N

Scaling Translate

Rotation

Offset

Convolution

( ) ( )h N h N

( ) ( )h N h N A N

( ) ( )h N h A N

( ) ( )h N h N r

1 2( ) ( ) ( )h N h N h N

The support function for this artical

In that case the tangent is the normal is

Distance from origin to tangent line

( )t f

( )n e

h x n

( ) ( ) ( ) '( ) ( )x h e h f

'( ) ( ( ) ''( )) ( )x h h f

cos sin cos sin( ) , ( ) , ( )

sin cos sin cose f R

Rotated

Translated

Scaled

( ) , ( )x R t x h t

, ( ) ( )x x a h a e

, ( )x c x c h

21 1

2( ) ( )

d x dt d dth h f h h e

ds ds ds d

( ) ( )ds

h hd

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,

0

General designs

The motion generated by rolling a circle of radius b inside a circle of radius a is given by

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 letn N

( )h

then

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.

examples

we consider a deformation of the 3:2 hypo-epi-cycloid design. Let c=1, =-2.

The support function

with we consider deformations

A new 3:2 design

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.

Cylindrical Moineau pumps

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).

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

gab3 52 arcsin(2sin ) 2 ( )w v v v v o v

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?

Thank you!