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 (x _o y _o z _o ) can be calculated using the Weierstrass equations in terms of a complex analytic function R(ω) as given below.

e ^ισ o(l--.ω ^δ )R(ω)dω

y=y _Q +Im [ e ^W (l+ω ² )R(ω)dω ω _Q

^ω l Z =Z _n -Re J e ^iθ (2ω)R(ω)dω ω _Q

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] e ^{1 J} +e ^{T 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) .