JP4491742B2 - Interacting monoplane hyperboloid structure by cylinder - Google Patents

Description

The present invention relates to a one-leaf hyperboloid structure in which cylindrical members are formed in contact with each other.

It is known that a cylinder is a member that is mechanically stronger than a square member having the same cross-sectional area. However, the cylinder is relatively rarely used as a member of a support base for supporting a heavy object because the connection between the cylinders or the connection with other members is more difficult than the square member. Although there are many examples of the cylinder used as a vertical column, it was rarely used as an oblique column used obliquely.

Many of the support bases were often made of a combination of vertical columns and horizontal girders. The connection between a vertical column and a horizontal girder has been fixed and constructed by a wooden frame, screws and nails in the case of wood, and by bolts or welding in the case of iron. However, a rectangular structure composed of vertical dwellings and horizontal girders is not a static structure, and an oblique reinforcing material such as braces or trusses is required for a static structure. In order to form a static structure with the minimum number of members, it is desirable to form a triangular structure by tilting the pillars.

There are only a few examples of the tilted pillars that support the support stand, but the whole structure is fixed with horizontal beams. The marine tower in Kobe has a single-leaf hyperboloid structure in which steel columns arranged diagonally are arranged in an arc, but the entire structure is fixed with horizontal bars. As a similar one-leaf hyperboloid structure, there is a round chair in which a large number of linear rattans are diagonally arranged and fixed with cross members. In both cases, a static structure is formed by a combination of diagonal columns and horizontal members.

The structure obtained when a line with no thickness is moved along a circle or ellipse with a constant gradient is known as a single leaf hyperboloid. The Ichiyo hyperboloid is also known as a ruled surface because it can be composed of line segments (reference: Shobayashi Kobayashi: differential geometry of curves and curved surfaces (revised version): 裳 華 房 March 20, 2002, No. 1 30 edition). When expressed in mathematical formulas,
^{x 2 / a 2 + y 2} / b 2 -z 2 / c 2 = 1
In the case of the present invention, since a single leaf hyperboloid structure that becomes a circle is handled at z = 0, a = b = r ′ (radius). FIG. 1 is a diagram showing the concept of a single leaf hyperboloid. In this case, the inclination angle θ of the straight line forming the ruled surface is expressed by the following equation when associated with the above equation.
θ = arctan (c / r ′)

The one-leaf hyperboloid structure may be able to form a static structure by obliquely arranging the cylindrical member without cutting. Such a structure has been used for a long time, with a support base that leans three cylinders diagonally, connects the gathering portions with ropes, and suspends winches, pulleys, and the like. A structure similar to this can be seen in an ornament in which three baseball bats are combined diagonally, the crossing portions are tied together with a string, and a ball is placed on it. If this is observed closely, it will be understood that the three cylinders have a one-leaf hyperboloid structure where they are in contact with each other in the cross section of z = 0.

The means for forming a one-leaf hyperboloid structure by obliquely combining three cylinders and constraining the gathering portion is an excellent means that does not require any work. There was no example in which a support base was manufactured by forming a single-leaf hyperboloid structure using a plurality of cylinders by the same means. The reason for this is that, in order to form a one-leaf hyperboloid structure in which four or more cylinders are arranged obliquely at an equal inclination angle so as to be in contact with each other, a special geometric condition is used, as will be clarified in the present invention. This is because it requires a means for satisfying and a means for maintaining it, and it is much more difficult than the triple set. In the present invention, a structure in which the cylinders are in contact with each other to form a one-leaf hyperboloid is called a mutual one-leaf hyperboloid structure.

Problems to be solved by the invention

When four or more cylinders of equal thickness are arranged obliquely to form a one-leaf hyperboloid structure in a static state, the one-leaf hyperboloid structure can be formed in a state where the cylinders are in contact with each other. . In the present invention, a portion where a plurality of cylinders are brought into contact with each other is referred to as a joint portion in the present invention. The joint portion is a cross section corresponding to z = 0 in the mathematical formula indicated by [0005]. When the cylinders are in contact with each other at an angle of inclination at the joint, the horizontal cross section of the cylinders is an ellipse, and the ellipse is in contact with a certain circle along an inscribed circle or circumscribed circle. [0004] The Kobe Marine Tower and rattan chair described in [0004] form a one-leaf hyperboloid structure, but the diagonally arranged members are not in contact with each other, and the geometrical conditions noted in the present invention are not satisfied. .

A first problem of the present invention is a one-leaf hyperboloid structure composed of N cylinders with a radius r, where N is an integer equal to or greater than 4, and the cylinders are assembled at the junction of the one-leaf hyperboloid structure. A one-leaf hyperboloid structure in which adjacent cylinders are in contact with each other along the circumference of an inscribed circle with a radius r1 and a circumscribed circle with a radius r2 that are uniquely calculated according to the values of N, r, and θ. In the structure characterized by forming, the calculation formula that defines the relationship among N, r, θ, r1, and r2, that is, the geometric condition is clarified. FIG. 2 is a diagram showing the concept of an interconnected single leaf hyperboloid structure using a cylinder having a cross section of radius r, and shows a top view of the interconnected monoleaf hyperboloid structure when N = 6. . FIG. 3 shows a state where a plurality of obliquely penetrating cylinders are in contact with each other with an elliptical cross-sectional shape on the xy projection plane at z = 0. In FIG. 3, r1 represents the radius of the inscribed circle, and r2 represents the radius of the circumscribed circle.

A second object of the present invention is to devise a means for maintaining a static state without disassembling the above-mentioned mutual one-leaf hyperboloid structure. In the present invention, as will be described later, means has been devised for preventing the disjoint one-leaf hyperboloid structure from being decomposed at the joint using an inscribed cylinder, an inscribed regular polygonal cylinder, and a circumscribed ring.

The third object of the present invention is to devise a form of utilization of an interconnected one-leaf hyperboloid structure formed by the present invention. In the present invention, as shown in the examples, it has been clarified that the present invention can be used for a flower stand, a table, a chair, a monitoring table, a polygonal pyramid roof frame, and the like.

Means for solving the problem

Inclined angle with a single-leaf hyperboloid structure composed of N cylinders with a radius r, where N is an integer greater than or equal to 4, and N cylinders arranged at equal intervals on a certain circumference Circles of an inscribed circle (radius r1) and a circumscribed circle (radius r2) that are uniquely calculated according to the values of N, r, and θ when assembled at the junction of the one-leaf hyperboloid structure with θ. In a structure characterized in that all adjacent cylinders are in contact with each other along the circumference to form an adjacent one-leaf hyperboloid structure, a calculation formula for determining the relationship among N, r, θ, r1, and r2 is as follows: It is expressed by a formula. The following formula was developed by the inventor by inducing a condition in which elliptical cross sections of cylinders having an inclination angle θ are in contact with each other in a cross section of z = 0 (see FIG. 3).
r1 = r (p-1)
r2 = r (p + 1)
Here, p is calculated by the following equation.
p = {1+ (tan φ · sin θ) ² } 1/2 / (tan φ · sin θ)
Here, φ is given by the following equation.
φ = 180 ° / N
The position of the joint that satisfies the above geometric condition may be arbitrary, and may be the tip of a cylinder or an intermediate position. Even if the lengths of the cylinders are not necessarily the same, if there is an aggregate portion that satisfies the geometrical condition, it is possible to partially form an interconnected one-leaf hyperboloid structure. In the case of using the one-leaf hyperboloid structure as a support, it is desirable that the structure is composed of cylinders of equal length.

When the radius r, the number N of cylinders, and the inscribed circle radius r1 or circumscribed circle r2 are given, the inclination angle θ of the cylinder that satisfies the geometrical condition of the connected one-leaf hyperboloid structure is given by the following equation: It is done.
p = (r1 / r) +1 or p = (r2 / r) -1
θ = arcsin {1 / ((p ² −1) 1/2 · tan φ)}

As a means for maintaining the mutual one-leaf hyperboloid structure satisfying the geometrical condition so as not to be decomposed while maintaining the z-axis vertically, a method of fixing the circumscribed circle integrally with the cylinder may be considered. Since the cylinder moves away from the center when it is disassembled, restraining the circumscribed circle is a reasonable fixing method. In the embodiment of the present invention, a means for forming an interconnected one-leaf hyperboloid structure so that the cylinders are in contact with each other in an annulus having a diameter of a circumscribed circle was devised, and its feasibility was confirmed. As shown in FIG. 4, it is possible to form a one-leaf hyperboloid structure while making a bolt hole in an annulus and a cylinder, coupling them in a freely rotating state, and applying torsional rotation. The same effect can be obtained by using a circumscribed regular polygonal ring instead of the circumscribed circle. In this case, the shape of the circumscribed regular polygon ring may be determined so that the radius of the inscribed circle is equal to the circumscribed circle r2.

When forming an interconnected one-lobe hyperboloid structure without drilling bolt holes in the cylinder, place the inscribed cylinder so that the cylinder does not fall apart, and along with the circumscribed ring, give torsional rotation little by little When the inclination angle is moved, the one-leaf hyperboloid structure is easily formed. It is also possible to use an inscribed regular polygonal cylinder instead of the inscribed cylinder. Means for joining the cylindrical bundles of the joints by using strings, chains or the like instead of the circumscribed ring can be considered, and it was confirmed that this means is effective in the examples.

A means that uses only an inscribed cylinder or an inscribed regular polygonal cylinder without using a circumscribed ring or a circumscribed regular polygon ring is also possible. As shown in FIG. 5, N bolts are fixed at equal intervals on the wall of the inscribed regular polygonal column, and the cylinder is connected to the inscribed cylinder using the bolt holes opened in the N cylinders. By tilting the cylinders until the cylinders are in contact with each other, an interdigitated one-leaf hyperboloid structure can be formed. For safety, it is desirable to restrain the outside of the joint with a string or the like.

In the one-leaf hyperboloid structure fixed by the above means, the equilibrium state is maintained by the load in the gravitational direction due to the weight of the cylinder, and when an upward force is applied to the cylinder, the equilibrium state collapses, Decompose. There are three possible means for preventing the decomposition: 1) fixing the inclination angle, 2) fixing the end of the cylinder so as not to move, and 3) fixing the cylinder so as not to cause torsional rotation. In the embodiment of the present invention, a means for preventing decomposition was devised in combination with a means for installing a support plate on the top.

As one means for preventing the disjoint one-leaf hyperboloid structure from being disassembled even when it is lifted in the air, as shown in FIG. Invented a means to fix the distance of two spheres. It was made clear in the Examples that this means is an effective means for preventing decomposition.

In order to mount and fix the top plate on the top of the one-piece hyperbolic surface structure, the cylinder is obliquely cut at an angle of (45 ° −θ / 2) according to the inclination angle θ, and the oblique cylinder is cut. It is conceivable to join them in the shape of a "letter", devise so that the end of the cylinder is perpendicular to the top plate, and fix it with screws, nails, etc. As shown in FIG. 7, if the end of the cylinder is cut obliquely, rotated by 180 degrees and joined, and the cylinder portion in contact with the top board is connected to the top board at a right angle, the work can be facilitated. It became clear in the examples.

In the present invention, as shown in FIG. 8, a circular hole larger than the diameter of the cylinder is formed in two circular support plates at the apex of the regular N polygon, and the entire cylinder is penetrated through the circular hole. It was discovered that if two disks are rotated while being twisted, an interconnected one-leaf hyperboloid structure can be formed. The diameter d of the circular hole is given by the following equation when the radius r of the cylinder, the inclination angle θ, and the thickness t of the disk are given (see FIG. 9).
d = t / tan θ + 2r / sin θ

In order to fix the mutual one-leaf hyperboloid structure formed in the shape of the cylinder penetrating into the circular hole opened in the two discs, it is necessary to fix the disc and the cylinder at the penetrating portion. Or the distance between the two disks may be fixed.

In the present invention, it has been discovered that a one-piece hyperbolic structure can be formed by a combination of one disc having a circular hole at a position corresponding to the vertex of a regular N polygon and a circumscribed ring. . This structure was found to be very stable and not decompose.

In the embodiment, four to eight wooden round bars are used as cylindrical members, and metal circles, strings, wooden circumscribed regular polygonal rings, and wooden circles for fixing inscribed circles are used as circumscribed circle fixing members for joints. Bars and wooden regular polygonal columns were used. Wooden and cardboard were used for the top board. The ball for preventing decomposition was made of wood or plastic. An adhesive and metal fittings were used as members for fixing the top plate. Four types of four, five, six and eight were used as the number of cylindrical members constituting the one-piece hyperbolic surface structure.

In the examples, the following five cases were carried out, all of which confirmed that the theory of the interconnected one-leaf hyperboloid structure described in the present invention and its maintenance means were correct, and further confirmed the feasibility.

Example 1: Model experiment: 20 cm long and 6 mm diameter wooden round bars as cylinders were changed to 4, 5, 6, and 8, and the inscribed diameters of the metallic circumscribed ring were 16 mm and 20 mm, respectively. , 20 mm and 25 mm, it was confirmed that a single-leaf hyperboloid structure having a calculated inclination angle and in which round bars were in contact with each other was formed. The inclination angle of the round bar of the one-leaf hyperboloid structure formed by the above combination has the same angle as the inclination angle obtained by the theoretical formula shown in [0013], and is as follows. It was.
Four round bars: circumscribed circle diameter 16 mm: inclination angle = 48.6 °
Five round bars: circumscribed circle diameter 20 mm: inclination angle = 39.9 °
Six round bars: circumscribed circle diameter 20 mm: inclination angle = 54.6 °
8 round bars: circumscribed circle diameter 25 mm: inclination angle = 53.3 °
Even if the joint is installed almost in the center, or even if the position of the joint is moved, as long as the joint is kept vertical, the weight of the round bar makes the one-leaf hyperboloid structure static. It was confirmed that it was stable.

Example 2: Flower stand, 45 cm in length, 5 wooden round bars with a diameter of 20 mm, 1 metal ring with an inscribed circle diameter of 64 mm, 1 wooden disc with a diameter of 24 cm and a thickness of 10 mm, diameter for stopper Using two 30 mm wooden spherical balls (with holes) and one wooden round bar with a fixed diameter of 10 mm, a flower stand with a top plate was created. In this case, the inclination angle of the round bar was 43.6 ° as in the theoretical formula. Two spherical balls are placed on the top and bottom of the joint, drilled with a circular hole with a diameter of 10 mm, fixed by pulling the spherical ball through a wooden round bar, and then joined to the top plate to connect each other. The curved body structure was prevented from being decomposed. A frame with a regular pentagonal height was attached to the back side of the top plate, and a stopper was used to prevent the end from spreading due to the weight of the round bar. As a result, a stable flower stand was completed.

Example 3: Support base: Inclination angle θ = 75 calculated by the formula [0020] using four wooden round bars with a length of 90 cm and a diameter of 30 mm, and two wooden disks with a diameter of 30 cm and a thickness of 15 mm. 4 holes of 35mm in diameter corresponding to ° are drilled at equal intervals near the edge circle of a wooden disc, and 4 discs are passed through each disc hole and 2 discs are twisted. By rotating in this way, it was confirmed that an interconnected one-leaf hyperboloid structure developed by the present invention in which round bars are in contact with each other at the joint could be formed. The length of the end portion of the round bar penetrating the two top plates can be made variable, but the vertical distance between the two top plates is kept constant. The two top plates can be used as a support base. By adjusting the length and diameter of the cylindrical member, it can be used for a work table that can be disassembled, a monitoring table for a beach, and the like.

Example 4: Chair: As shown in FIG. 7 of [0019] using four wooden round bars with a length of 46 cm and a diameter of 30 mm, a wooden disc with a diameter of 30 cm and a thickness of 15 mm, a nylon string, and a screw. The end part is obliquely cut at 5 cm so that the inclination angle corresponds to 70 °, and a round bar bent in a “shape” is created, and can be rotated horizontally at the four corners on the back side of the disk. It was confirmed that an interconnected one-leaf hyperboloid structure conforming to the geometric condition of the present invention can be formed by connecting with a screw in a state and binding with a nylon string so that the round bars are in contact with each other at the joint. The circumscribed circle diameter in this case was 73.8 mm.

Example 5: Polygonal pyramid roof frame: 6 logs with a length of 180 cm and a diameter of 80 mm, one metal ring (bicycle hub) with an inscribed circle diameter of 385 mm for fixing the circumscribed circle (radius of the circumscribed circle in the theoretical formula) r2 is 192.5mm), using six metal bolts with a diameter of 8mm, drill six circular holes with a diameter of 8mm every 60 ° in the metal ring, and at the tip of the log 150mm An interconnected one-leaf hyperboloid developed by the present invention by passing through a circular hole with a diameter of 8 mm, connecting the logs with bolts at equal intervals inside the metal annulus, and widening the ends of the logs little by little. A structure was formed. According to the theoretical formula developed in the present invention, the tilt angle is 31.6 °, which is consistent with the tilt angle in the example. The overall shape is that of a hexagonal roof, and it was judged that it could be used for a polygonal pyramid roof. It was actually experienced that even if a human with a weight of 75 kg rides on the frame formed in the above-described embodiment, it has a strength that does not frighten.

The invention's effect

The effects obtained by the present invention are as follows.
1) Given the radius, number and inclination angle of a given cylinder, and fixing the inscribed circle radius or circumscribed circle radius calculated by the theoretical formula developed in the present invention at the joint, the cylinder members are in contact with each other. It is possible to form a solid, one-leaf hyperboloid structure that is in a static state. This means can be said to be an excellent method for forming a support base on a cylinder without performing a work such as cutting.
2) As a result, the above-mentioned interleaved one-leaf hyperboloid structure can easily form a rigid support body with a minimum number of cylindrical members.
3) By fixing the inclination angle of the above-mentioned mutual one-leaf hyperboloid structure, fixing the end of the cylinder so as not to move, or fixing it so as not to cause torsional rotation, It can be maintained without disassembly and various supports can be made.
4) Applicable to structures such as flower stands, tables, chairs, monitoring tables, and polygonal pyramid roof frames.
5) Can be used for mechanics education.

Diagram showing the concept of a single leaf hyperboloid The figure which shows the concept of the mutual one-leaf hyperboloid by a cylinder (in the case of N = 6) The figure which shows the state which the cylinder contact | connects in the cross section of z = 0 (in the case of N = 6) Means for maintaining circumscribed one-leaf hyperboloid structure by connecting the circumscribed ring and cylinder with bolts Means for maintaining an interconnected monoplane hyperboloid structure using an inscribed regular polygonal cylinder Method for fixing mutually connected monoplane hyperboloid structures using spheres Joining method with top plate by oblique cutting of cylinder Method for forming an interconnected single leaf hyperboloid structure using two discs The figure explaining the relationship between the inclination angle θ, the circular hole diameter d, and the disc thickness t

Explanation of symbols

x: x-axis coordinate y: y-axis coordinate z: z-axis coordinate r ′: radius θ: inclination angle r: radius of the cylinder r1: radius of the inscribed circle r2: radius of the circumscribed circle d: diameter of the circular hole t: plate Thickness

Claims (2)

A single-leaf hyperboloid structure composed of N cylinders with a radius r, where N is an integer equal to or greater than 4, and N cylinders arranged at regular intervals on a constant circumference have a constant inclination angle θ Along the circumference of the inscribed circle of radius r1 and the circumscribed circle of radius r2, which are uniquely calculated according to the values of N, r, and θ. , A structure characterized in that all adjacent cylinders touch each other A single-leaf hyperboloid structure composed of N cylinders with a radius r, where N is an integer equal to or greater than 4, and N cylinders arranged at regular intervals on a constant circumference are connected to a constant inclination angle θ. Along the circumferences of the inscribed circle of radius r1 and the circumscribed circle of radius r2 that are uniquely calculated by the values of N, r, and θ. In the structure characterized in that all adjacent cylinders are in contact with each other, means for maintaining the structure so as not to be decomposed

Images