WO2008105704A2 - A mathematical minimal surface propeller - Google Patents

A mathematical minimal surface propeller Download PDF

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Publication number
WO2008105704A2
WO2008105704A2 PCT/SE2008/000066 SE2008000066W WO2008105704A2 WO 2008105704 A2 WO2008105704 A2 WO 2008105704A2 SE 2008000066 W SE2008000066 W SE 2008000066W WO 2008105704 A2 WO2008105704 A2 WO 2008105704A2
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WO
WIPO (PCT)
Prior art keywords
propeller
minimal surface
bladed
minimal
boundaries
Prior art date
Application number
PCT/SE2008/000066
Other languages
French (fr)
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WO2008105704A3 (en
Inventor
Sten Andersson
Häkan BENJAMINSSON
Original Assignee
Sten Andersson
Benjaminsson Haekan
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
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Publication date
Application filed by Sten Andersson, Benjaminsson Haekan filed Critical Sten Andersson
Publication of WO2008105704A2 publication Critical patent/WO2008105704A2/en
Publication of WO2008105704A3 publication Critical patent/WO2008105704A3/en

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Classifications

    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
    • B64C—AEROPLANES; HELICOPTERS
    • B64C11/00—Propellers, e.g. of ducted type; Features common to propellers and rotors for rotorcraft
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B63—SHIPS OR OTHER WATERBORNE VESSELS; RELATED EQUIPMENT
    • B63H—MARINE PROPULSION OR STEERING
    • B63H1/00—Propulsive elements directly acting on water
    • B63H1/02—Propulsive elements directly acting on water of rotary type
    • B63H1/12—Propulsive elements directly acting on water of rotary type with rotation axis substantially in propulsive direction
    • B63H1/14—Propellers

Definitions

  • the present invention relates to a propeller having n blades.
  • a propeller is like a screw but with a shaft center or hub, with two or more twisted blades.
  • the pitch describes the angle between a blade and the hub.
  • a particular object is to provide a propeller that is optimal for strength and shape.
  • a propeller is provided, said propeller is common to propellers having n blades, where n is an integer larger than 0.
  • Said propeller surface is described as composed of n identical units of a mathematical minimal surface.
  • the propeller is advantageous in that it is composed of n identical units of a minimal surface. It is an object of the present invention to realize that such a surface has improved performance characteristics over conventional propellers.
  • the propeller may comprise straight and curved line boundaries defining said minimal surface area, which is advantageous in that the n identical units are defined.
  • the propeller may comprise a 2-bladed propeller which is mathematically constructed and built of two identical parts of a minimal surface, a 3-bladed propeller, which is similarly constructed and built of three identical parts of a minimal surface and a n-bladed propeller which is constructed and built of n identical parts of a minimal surface, which is advantageous in that a simple description is given to a number of different but related propellers.
  • the propeller may comprise pitch and rake properties that are defined by the minimal surface boundaries, which is advantageous in that a general mathematical description of propellers is obtained.
  • the propeller may comprise a rod as rotation axis that penetrates the inner central part of the minimal surface propeller, which can be hollow, which is advantageous in that weight is reduced and strength is increased.
  • the propeller may comprise a central surface part containing Gaussian curvature that may be obtained by a topological transformation of a hub region containing mean curvature of a classic propeller, which is advantageous in that said central part is approaching the geometry of a minimal surface.
  • the propeller may comprise a variety of materials, which is advantageous in that it works in any medium like a fluid, gas or liquid.
  • materials can be metal, metal alloys, plastic or reinforced plastic or wood.
  • Fig. (Ib) describes one of the three identical copies (6) of surface (3) from Ia that build the propeller (1) in fig (Ie) .
  • the surface (3) is built in glass fiber reinforced plastic.
  • Fig. (Ic) describes 3 identical copies that are not put together.
  • Fig. (Id) describes 3 identical copies that are put together into a 3 bladed propeller.
  • Fig. (Ie) describes 3 bladed propeller (1) built in glass fiber reinforced plastic, diameter 16 cm.
  • Fig. (If) describes different projection.
  • Fig. (2a) describes minimal surface (3) boundaries for 4 bladed propeller (1). The angle is ⁇ /2.
  • Fig. (2b) describes one of the four identical copies
  • Fig. (2c) describes 4 bladed propeller (1) built in glass fiber reinforced plastic, diameter 16 cm.
  • Fig. (2d) describes different projection.
  • Fig. (3a) describes a typical prior art fishing boat propeller, diameter 56 cm.
  • Fig. (3b) describes 3 bladed minimal surface propeller (3) with boundary (5) indicated, diameter 18 cm.
  • Fig. (4a) describes a half 4 bladed propeller (1) after eq (1) .
  • Fig. (4b) describes a half 4 bladed propeller (1) after eq (2) .
  • Fig. (4c) describes a one single calculated continues minimal surface propeller (1) from fig (4a) and (4b) .
  • Fig. (5) describes a complete 4 bladed propeller (1) as obtained by adding eqs 1 and 2 on the exponential scale in eq (3) .
  • Fig. (7a) describes a 2 bladed propeller (1) after eq (5) .
  • Fig. (7b) describes a 4 bladed propeller (1) after eq (5) .
  • Fig. (7c) describes a ⁇ bladed propeller (1) after eq (5) .
  • the invention relates to a general propeller (1) to the mathematics of minimal surfaces (4).
  • the invention is related to n bladed (2) propeller (1) built of n structure units.
  • One such unit builds one Continuous Minimal Surface (CMS) (4) which with given boundaries is designated CMS.
  • CMS Continuous Minimal Surface
  • Two identical CMS (2) build a two bladed propeller (1)
  • three identical CMS (3) build a three bladed propeller (1)
  • four identical CMS (4) build a four bladed propeller (1)
  • n identical CMS (n) build a n bladed propeller (1) . Varying boundaries give changes in pitch and rake.
  • the invention is related to a n bladed (2) propeller (1) built of n structure units.
  • One such unit builds one continuous minimal surface (4) which with given boundaries is designated CMS.
  • Two identical CMS (2) units build a two bladed propeller (1), three identical CMS (3) units build a three bladed propeller (1), four identical CMS (4) build a four bladed propeller (1), etc.
  • the geometry of a four bladed propeller (1) is obtained in exact analogy via its boundaries ( ⁇ ) as in fig (2a) to give a simple CMS (4) unit of a minimal surface (4) built in glass fiber reinforced plastic as in fig (2b) .
  • fig (3a) there is a classic propeller (CP) for use in water compared with a minimal surface (4) propeller (1) (MSP) in fig (3b) .
  • the two propellers have similar pitch. Chirality is also shifted as compared with the cases above.
  • the two propellers are very similar as indicated with a free hand drawing in fig 3b, the blades are thin and in terms of a trigonal ratio, a/c is approximately 5 for both the propellers.
  • the region around the hub in the CP case has positive mean curvature, and zero Gaussian curvature, while the corresponding region is a saddle for the MSP propeller with zero mean curvature and negative Gaussian curvature.
  • the straight-line minimal surface propeller (4) boundaries are lines of intersection and the classic hub can be said to be replaced by two singular points where these lines of intersection meet.
  • a line joining the two points is the rotation axis of the propeller (1), which can be a solid rod in reality.
  • the inner part of the minimal surface propeller (4) which can be hollow, is penetrated by this rod.
  • half the two bladed propeller (1) is the one bladed propeller that has been in use in single oar sculling.
  • the geometry of the long straight-line boundaries (6) can be used to change the pitch as seen in a comparison between fig (1) and fig (3b) .
  • the long straight lines are changed into curved lines to reinforce the similarity with a typical fishing boat propeller in fig (3a) .
  • the rake of a propeller (1) is also easily changed within the minimal surface boundaries (6) .
  • n identical CMS curved simple units made of sheets of metal put together form a n bladed propeller (1).
  • Minimal surface coordinates (x,y,z) with respect to an origin (x o y o z o ) can be calculated using the Weierstrass equations in terms of a complex analytic function R( ⁇ ) as given below.
  • R( ⁇ ) needs to be determined in order to calculate the asymmetric unit of a propeller minimal surface.
  • Equation ( 1 ) A simple saddle function describes half the four bladed propeller in equation (1) . Equation ( 1 ) :
  • Equation (2)
  • Equation (3) Equation (3)
  • Propeller surfaces (1,3) for members n odd can be obtained by bending an asymmetric part of a minimal surface like in fig (2a) into fig (Ia), only by changing the angles between the straight line boundaries (6), from ⁇ /2 to 2 ⁇ /3.
  • fig (2a) coordinates of surface of fig (1) can be arrived approximately. Corresponding points are moved until their surrounding surface (3) has a mean curvature of zero.
  • Figures (la-f) and (2a-d) are examples of how to construct and build minimal mathematical surfaces (4) .

Landscapes

  • Engineering & Computer Science (AREA)
  • Chemical & Material Sciences (AREA)
  • Combustion & Propulsion (AREA)
  • Mechanical Engineering (AREA)
  • Ocean & Marine Engineering (AREA)
  • Aviation & Aerospace Engineering (AREA)
  • Structures Of Non-Positive Displacement Pumps (AREA)
  • Turbine Rotor Nozzle Sealing (AREA)

Abstract

A propeller (1) having n blades (2) where n is an integer larger than 0. The propeller (1) having a surface (3) which is described as composed of n identical units (6). Each identical unit (6) is a mathematical minimal surface.

Description

A MATHEMATICAL MINIMAL SURFACE PROPELLER
Technical Field
The present invention relates to a propeller having n blades.
Background Art
A propeller is like a screw but with a shaft center or hub, with two or more twisted blades. The pitch describes the angle between a blade and the hub.
It is generally considered that especially boat propellers have relatively high costs of manufacturing and relatively low efficiency. Similar conclusions may be drawn from US Patent 6099256 dealing with construction of propeller/impeller blades. The conventional propellers have a mixture around the hub of two geometries of which the cylindrical (the hub) is one, which of course is bad for the flow.
Summary of the Invention
It is an object of the present invention to provide an improvement of the above techniques and prior art.
A particular object is to provide a propeller that is optimal for strength and shape.
Hence, a propeller is provided, said propeller is common to propellers having n blades, where n is an integer larger than 0. Said propeller surface is described as composed of n identical units of a mathematical minimal surface.
The propeller is advantageous in that it is composed of n identical units of a minimal surface. It is an object of the present invention to realize that such a surface has improved performance characteristics over conventional propellers. The propeller may comprise straight and curved line boundaries defining said minimal surface area, which is advantageous in that the n identical units are defined.
The propeller may comprise a 2-bladed propeller which is mathematically constructed and built of two identical parts of a minimal surface, a 3-bladed propeller, which is similarly constructed and built of three identical parts of a minimal surface and a n-bladed propeller which is constructed and built of n identical parts of a minimal surface, which is advantageous in that a simple description is given to a number of different but related propellers.
The propeller may comprise pitch and rake properties that are defined by the minimal surface boundaries, which is advantageous in that a general mathematical description of propellers is obtained.
The propeller may comprise a rod as rotation axis that penetrates the inner central part of the minimal surface propeller, which can be hollow, which is advantageous in that weight is reduced and strength is increased.
The propeller may comprise a central surface part containing Gaussian curvature that may be obtained by a topological transformation of a hub region containing mean curvature of a classic propeller, which is advantageous in that said central part is approaching the geometry of a minimal surface.
The propeller may comprise a variety of materials, which is advantageous in that it works in any medium like a fluid, gas or liquid. Such materials can be metal, metal alloys, plastic or reinforced plastic or wood. Brief description of the Drawings
Embodiments of the present invention will now be described, by way of example, with reference to the accompanying schematic drawings, in which Fig. (Ia) describes minimal surface (4) boundaries
(5) for 3 bladed propellers (1). The angle is 2π/3.
Fig. (Ib) describes one of the three identical copies (6) of surface (3) from Ia that build the propeller (1) in fig (Ie) . The surface (3) is built in glass fiber reinforced plastic.
Fig. (Ic) describes 3 identical copies that are not put together.
Fig. (Id) describes 3 identical copies that are put together into a 3 bladed propeller. Fig. (Ie) describes 3 bladed propeller (1) built in glass fiber reinforced plastic, diameter 16 cm. Fig. (If) describes different projection. Fig. (2a) describes minimal surface (3) boundaries for 4 bladed propeller (1). The angle is π/2. Fig. (2b) describes one of the four identical copies
(6) of surface (3) from (2a) that build the propeller (1) in fig. (2c) . The surface (3) is built in glass fiber reinforced plastic.
Fig. (2c) describes 4 bladed propeller (1) built in glass fiber reinforced plastic, diameter 16 cm. Fig. (2d) describes different projection. Fig. (3a) describes a typical prior art fishing boat propeller, diameter 56 cm.
Fig. (3b) describes 3 bladed minimal surface propeller (3) with boundary (5) indicated, diameter 18 cm.
Fig. (4a) describes a half 4 bladed propeller (1) after eq (1) .
Fig. (4b) describes a half 4 bladed propeller (1) after eq (2) . Fig. (4c) describes a one single calculated continues minimal surface propeller (1) from fig (4a) and (4b) .
Fig. (5) describes a complete 4 bladed propeller (1) as obtained by adding eqs 1 and 2 on the exponential scale in eq (3) .
Fig. (βa) describes a 6 bladed (n=6) half propeller.
Fig. (βb) describes a 10 bladed (n=10) half propeller. Fig. (7a) describes a 2 bladed propeller (1) after eq (5) .
Fig. (7b) describes a 4 bladed propeller (1) after eq (5) .
Fig. (7c) describes a β bladed propeller (1) after eq (5) .
Detailed Description of Embodiments
The invention relates to a general propeller (1) to the mathematics of minimal surfaces (4). The invention is related to n bladed (2) propeller (1) built of n structure units. One such unit builds one Continuous Minimal Surface (CMS) (4) which with given boundaries is designated CMS. Two identical CMS (2) build a two bladed propeller (1), three identical CMS (3) build a three bladed propeller (1), four identical CMS (4) build a four bladed propeller (1) and n identical CMS (n) build a n bladed propeller (1) . Varying boundaries give changes in pitch and rake.
The invention will become clear from the detailed description given below. Various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description.
Minimal surfaces (4) are saddle surfaces characterized by having Gaussian curvature K ≤ 0 (less than or equal to zero), and mean curvature H = O. As a minimal surface (4) can be considered as the optimal shape for strength, it is worthwhile to explore the geometrical possibilities for the use of minimal surfaces (4) in the construction of a propeller (1) .
The invention is related to a n bladed (2) propeller (1) built of n structure units. One such unit builds one continuous minimal surface (4) which with given boundaries is designated CMS. Two identical CMS (2) units build a two bladed propeller (1), three identical CMS (3) units build a three bladed propeller (1), four identical CMS (4) build a four bladed propeller (1), etc.
The invention is described directly. The straight- line boundaries (5) in fig Ia describe the extension up and down, and the slightly curved boundaries describe the perpendicular extension. Dipped in soap the simple unit of a minimal surface (4) is obtained that describes a part of a propeller surface (3,4). With a frame formed after the soap surface a model was built in glass fiber reinforced plastic and is shown in fig (Ib) . The simple unit in fig (Ib) forms a CMS (3) unit. As shown in figs (Ie) and (f) three identical CMS (3) units (6) form a three bladed propeller (1) .
The geometry of a four bladed propeller (1) is obtained in exact analogy via its boundaries (β) as in fig (2a) to give a simple CMS (4) unit of a minimal surface (4) built in glass fiber reinforced plastic as in fig (2b) .
Four of these units make a complete hollow four bladed propeller (1) as shown in figs (2c) and (2d) .
In fig (3a) there is a classic propeller (CP) for use in water compared with a minimal surface (4) propeller (1) (MSP) in fig (3b) . The two propellers have similar pitch. Chirality is also shifted as compared with the cases above. The two propellers are very similar as indicated with a free hand drawing in fig 3b, the blades are thin and in terms of a trigonal ratio, a/c is approximately 5 for both the propellers. Mathematically the region around the hub in the CP case has positive mean curvature, and zero Gaussian curvature, while the corresponding region is a saddle for the MSP propeller with zero mean curvature and negative Gaussian curvature.
The straight-line minimal surface propeller (4) boundaries are lines of intersection and the classic hub can be said to be replaced by two singular points where these lines of intersection meet. A line joining the two points is the rotation axis of the propeller (1), which can be a solid rod in reality. The inner part of the minimal surface propeller (4) which can be hollow, is penetrated by this rod.
It can be said that half the two bladed propeller (1) is the one bladed propeller that has been in use in single oar sculling.
The geometry of the long straight-line boundaries (6) can be used to change the pitch as seen in a comparison between fig (1) and fig (3b) . The long straight lines are changed into curved lines to reinforce the similarity with a typical fishing boat propeller in fig (3a) . The rake of a propeller (1) is also easily changed within the minimal surface boundaries (6) . Other advantages and usages will be apparent to those of ordinary skill in the art.
The manufacturing of a classic propeller may be described in two steps. Solid blades are fastened to a prefabricated cylindrical nave as shown in fig (3) . The thickness of the blades increases closer to the nave. The manufacturing of a minimal surface propeller (4) may be described in one step, n identical CMS curved simple units made of sheets of metal put together form a n bladed propeller (1). Mathematical description:
Minimal surface coordinates (x,y,z) with respect to an origin (xoyozo) can be calculated using the Weierstrass equations in terms of a complex analytic function R(ω) as given below.
eισo(l--.ωΔ)R(ω)dω
Figure imgf000008_0001
y=yQ +Im [ eW(l+ω2)R(ω)dω ωQ
ωl Z =Zn-Re J eiθ(2ω)R(ω)dω ωQ
R(ω) needs to be determined in order to calculate the asymmetric unit of a propeller minimal surface.
It is possible to give an approximate description to minimal surfaces (4) in terms of simple saddle mathematics as described in equations (1), (2) and (4). Simple soap water experiments confirm this. Accurate coordinates can be obtained numerically under the condition that the mean curvature is zero for a minimal surface. It is well-known that the addition of such saddle equations using exponential mathematics also conserves the curvature topology and an accurate minimal surface is obtained.
Due to the cyclic nature of the saddle mathematics only propeller surfaces (1) for members n even are described, which is shown in equations (1), (2) and (3). The way to arrive at surfaces for n odd is discussed below.
A simple saddle function describes half the four bladed propeller in equation (1) . Equation ( 1 ) :
1 i 1 2 2 1 xy cos(— πz) - cos(— πz) — (x - y ) sin(— πz) = 0
The other half is related by simple rotation as in equation (2), and the two propellers are shown in fig (4a) and (b) .
Equation (2) :
-xy cos(— 1 πz)-cos(—l πz)+—1 (x2 -y2)sin(—1 πz)=0
8 2 2 8
In order to make a four bladed propeller (1) these equations are added on the exponential scale as done in equation (3) .
Equation (3) :
- (xycosf 1 πz)- cosf J πz) — 1 (x 2 -y 2 ) sin(- 1 πz)) - (-xjcosf 1 πz)- cosf i πz) +- 1 (x 2 -y 2 ) sin(— 1 πz)) e 8 2 2 8 +e 8 2 2 8 -3 = 0
And the final complete propeller (1) is shown in fig (5) .
In Mathematica fashion half propellers - or just saddles - are done as in equation (4) :
Equation 4 :
cos(πz/8)Product[xcos(i 2% In) - ysin (i 2π In), {i,0,n/2 - 1 }] - sin(πz/8)Product[xcos( i 2π / n + π / n) - ysin (i 2π / n + π I n), { i,0,n/2 - 1 } ] - cos(πz/2)= 0 and introducing A, B and C
A = cos(πz/8)Product[xcos(i 2π/ra) -ysin (i 2π/«),{i,0,n/2- l }]
B= sin(πz/8)Product[xcos( i 2π/n + π/n)-ysin (i 2π/n + %/ή),{i,0,n/2 - 1}]
C = cos(πz/2)
there is
A-B-C=O
And the half propellers for n=β and 10 are shown in fig ( βa) and (b) . Going exponentially the general formula in equation (5) give the complete propellers (1).
Equation (5)
U-B-C] J-A+B-C] e1 J+eT J = 3
And the propellers for n=2,4 and 6 (two, four and six bladed) are given in fig (7) .
Propeller surfaces (1,3) for members n odd can be obtained by bending an asymmetric part of a minimal surface like in fig (2a) into fig (Ia), only by changing the angles between the straight line boundaries (6), from π/2 to 2π/3. As the mathematics of fig (2a) is known, coordinates of surface of fig (1) can be arrived approximately. Corresponding points are moved until their surrounding surface (3) has a mean curvature of zero.
Figures (la-f) and (2a-d) are examples of how to construct and build minimal mathematical surfaces (4) .

Claims

1. A propeller (1) having n twisted blades (2) characterized in that the outer surface (3) of said propeller (1) is described as composed of n identical units (6) of a mathematical minimal surface (4).
2. A propeller (1) according to claim 1 characterized in that straight and or curved line boundaries (5) defining said minimal surface (4) area.
3. A propeller (1) according to claim 1 and 2 characterized in that a propeller (1) is build by one single and continuous mathematical minimal surface (4) with boundaries (8 ) .
4. A propeller (1) according to claim 1, 2 and 3 characterized in that a 2-bladed propeller (1) is mathematically constructed and built of two identical parts (6) of a minimal surface (4), a 3-bladed is similarly constructed and built of three identical parts (6) of a minimal surface (4) and a n-bladed (2) is constructed and built of n identical parts (6) of a minimal surface (4) .
5. A propeller (1) according to claim 1, 2, 3 and 4 characterized in that pitch and rake properties are defined by the minimal surface (4) boundaries (5) .
6. A propeller (1) according to claim 1, 2, 3, 4, and 5 characterized in that the inner part of the propeller (1) is solid.
7. A propeller (1) according to claim 1, 2, 3, 4, 5 and 6 characterized in that a rod as rotation axis penetrates the inner central part of the minimal surface propeller (1).
8. A propeller (1) according to claim 1, 2, 3, 4, 5, 6 and 7 characterized in that the rotation is obtained through a surrounding outer force .
9. A propeller (1) according to claim 1, 2, 3, 4, 5, 6, 7 and 8 characterized in that a central surface part containing Gaussian curvature may be obtained by a topological transformation of a hub region containing mean curvature of a classic propeller.
10. A propeller (1) according to claim 1, 2, 3, 4, 5, 6, 7, 8 and 9 characterized in that said propeller (1) of minimal surface (4) geometry working in any medium like a fluid, gas or liquid.
PCT/SE2008/000066 2007-02-26 2008-01-25 A mathematical minimal surface propeller WO2008105704A2 (en)

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR20200010605A (en) * 2016-05-27 2020-01-30 섀로우 엔지니어링 엘엘씨 Propeller

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
DE2439597A1 (en) * 1974-08-17 1976-02-26 Wolfram Taeuber Generating three-dimensionally curved surfaces having min. area - using heat-shrinkable plastics film stretched over suitably shaped frames
WO1981001304A1 (en) * 1979-11-05 1981-05-14 Roehm Gmbh Rigid shaped body,biaxially drawn,of synthetic material,and process for the manufacturing thereof
US5213473A (en) * 1990-09-15 1993-05-25 Mtu Motoren-Und Turbinen-Union Munchen Gmbh Radial-flow wheel for a turbo-engine
EP1739007A1 (en) * 2005-06-30 2007-01-03 Marifin Beheer B.V. Shaftless propeller

Family Cites Families (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JPH06171593A (en) * 1992-06-12 1994-06-21 Ishizaki Masao Propeller and manufacture thereof

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
DE2439597A1 (en) * 1974-08-17 1976-02-26 Wolfram Taeuber Generating three-dimensionally curved surfaces having min. area - using heat-shrinkable plastics film stretched over suitably shaped frames
WO1981001304A1 (en) * 1979-11-05 1981-05-14 Roehm Gmbh Rigid shaped body,biaxially drawn,of synthetic material,and process for the manufacturing thereof
US5213473A (en) * 1990-09-15 1993-05-25 Mtu Motoren-Und Turbinen-Union Munchen Gmbh Radial-flow wheel for a turbo-engine
EP1739007A1 (en) * 2005-06-30 2007-01-03 Marifin Beheer B.V. Shaftless propeller

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR20200010605A (en) * 2016-05-27 2020-01-30 섀로우 엔지니어링 엘엘씨 Propeller
KR102143022B1 (en) 2016-05-27 2020-08-11 섀로우 엔지니어링 엘엘씨 Propeller

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WO2008105704A3 (en) 2011-01-20
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