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# N=3 2R-Virtual Wheel (Reuleaux triangle)

- Author:
- asifsound

Rounded Reuleaux triangle wheel. ------

**Alias: Trapeze Wheel/ Brachiation Wheel**, or,**Rolling quasi-triangle**, or ,**"Roly-poly toy" N=3 version**I forefeel that This is a HOME-RUN invention in 21 century. Non-friction implementation.**This is simple large radius new-type caster.**--- I think so. You only have to prepare (a) Rigid Rounded Reuleaux triangle (b) the 3 sets of pink hinge for Axis. (c) And set the hinge to triangle each vertex. ---- that's all. cf.**Simple view version**(Reuleaux triangle-Omusubi: GeoGebra). This method is the most honest implementation which matches to original Reuleaux triangle concept. (i.e. Reuleaux triangle concept is a roller. lower ground roller, and, upper/ ceiling ground roller.) This is related to next apparatus.**N=3 2R-Virtual Wheel (Roly-poly toy wheel)**(Geogebra) ---- i.e. "Roly-poly toy" N=2 version cf. Yagen (Japanese medical grinder tool for crushing herbs) is near feeling. Push & roll by hand. cf. Yagen: a Japanese medicine grinder (picture) cf. おむすびころりん 童話 動く絵本/日本の昔話 ---- typical triangle rice ball cf. THE ROLLING RICE BALL (ENGLISH) cf. Omusubi Kororin - Japanese Rice Ball Fable - 1989 Japan Cartoon with Intro --- Enjoy the picture. ----- Related cf. シロテテナガザルの大車輪 (YouTube) cf. シロテテナガザル ブラキエーション (YouTube) --- Brachiation (wikipedia) -- very fast moving !!. cf. Gibbon Brachiating (YouTube)**Features:**1. Reuleaux triangle corner is not acute angle. (many variations are there.) --- edge margin ≠ 0. (Margin = 0 case, this method logic does not work. i.e. Rounded Reuleaux Triangle, not Sharp Reuleaux Triangle.) 2. Non-friction implementation. cf.

**Reuleaux Wheeled Bicycle**(by Guan Baihua on 2009) ---- ① ----- His implementation method has

**much friction**. ---- no good at all (= not wise). & noisy. cf. My bicycle:

**N=3 2R-Virtual Wheel (Reuleaux triangle-Frame)**--- I think this is better than ①. 3. This wheel body size is r=R circle. but, it's real r=2R (i.e.

**Large radius supporting wheel**.) 4. Axis position is controlled by 6 bars (or 3 hinge sets ) linkage. (Axis is not center of wheel. along the edges.) (Here, Axis B keeps the weight/ load of cart body. axis position locates under the triangle top [= highest vertex ] corner. Top corner 's pink bars are not bent, are stretched [= 180° bent]. Pink stretch bars and seat B is

**a swing system [or pendulum system]**. ) (Pink bent bars restrict the B always to be within triangle arc space. B cannot go out the this area.) ex. If the wheel rotates clockwise, the Pink pendulum is changed from CB → EB → FB → CB →.... at 120° interval in order. 5. This is not approximation solution. Bullet B ● moves exact horizontal line. It's evident. 6. Ground tangent point lies just under the B ●. 7. This method can be applied to sector angle = 60° sector, so N=5, 7.. does not fit. Very special condition, equilateral triangle only. It comes from pink hinge switching property. 8. Axis supporting arm of chassis can be inclined. Above figure arm is vertical, but inclined arm is OK, too. From this reason, you can attach the wheel to the low top of any- angle- inclined stick. // // inclined arm is possible. // ◎

**Tip:**When Axis comes on the corner

**◎**F (blue) / C (black)/ E (orange), brown curved sector touches the ground and rounds small brown circle (radius ED=4-2√3=0.535898385), and rotates 60°, and at this time pink bars relative bent angle is not changed. After this, new pink top is set. ( ex. top C is moved to right down, E becomes new top. F moves horizontally to right. And, next new-cycle statrs.)

**■ Caster making:**Non-pedal/ motor/ engine mode, Green bars is not necessary. You only have to support 6 pink bars. ■ Which do you like? (1) 100% r = 2 circle ------ usual wheel. (2) about 80% r= 4 (Black part), 20% r= 0.5 (Brown part) circle. ----- above Reuleaux triangle wheel. (2) case, pedal mode, pedal load is heavy and light (or strong & weak) mixed. --- this is good physical exercise, isn't it ?. --- human friendly low fatigue design. (??) Or reverse of it. If you bother r= 0.5 circle acuteness, please add (1) wheel aside. i.e. (1) & (2) pair wheel. [ (2) & 60° shifted (2) pair is OK, too. ] ★

**Bicycle performance**Which is efficient for race bicycle, (1) or (2) ? Someone,

**Please investigate this problem.**I think (2) is superior to (1), because average radius is bigger (??). This method has less friction than other existing Reuleaux triangle wheel method. And further, the gibbon brachiation movement is alike this (i.e. This suggests good hint. It's the result of natural selection. quick movement. ). In general, when the radius large, its fuel cost increases perhaps, but above this method is about 2 times radius, and fuel cost is unchanged perhaps. i.e. Cost performance is good perhaps (??). ■ material cf. Inventing the wheel © 2002-2016 Mathematical Etudes |----- This video is interesting. ------ This is same logic as ①.

**Much friction occures.**not wise. ( The Triangle is rotating touching a square. It is slipping on purpose. This method's friction is very much. Therefore this is a very bad method in real world. Creater is considerable stupid. ) cf. 2.1 Non-circular wheels ( in P13 「 Miura’s Water Wheel or The Dance of the Shapes of Constant Width 」 Burkard Polster February 18, 2013) In their 2005 article Reinventing the Wheel: Non-Circular Wheels [4] Claudia Masferrer Le´on and Sebasti´an von Wuthenau Mayer, presented their beautifully simple idea for fitting non-circular wheels to a cart to achieve a smooth ride. Figure 13 shows how this works. One of the wheels consists of a Reuleaux triangle glued together with one of its parallel shapes. Then, when the triangles are restricted to rotating inside their associated squares, the top of the car will smoothly move along a horizontal without up or down movement. My comment: ----- simple, but how implementation (the triangles are restricted to rotating inside their associated squares)

**without friction**? How pedal mode (with axis height position is constant) support? ★ Less or non friction implementation of "Square frame + inner Sharp Reuleaux Triangle + outer Rounded Reuleaux Triangle" ( i.e. I' like to say this in short "Reuleaux Triangle Wheel- Window version"). is possible by using "my above method concept + exact ( or quasi-exact) straight line linkage tool". ---- This good mathematical exercise for you. Please solve it by youself. (--- This is reverse version of above invention. Ground is corresponding to upper ceilinng ) My answer is ------

**N=3 2R-Virtual Wheel (Reuleaux triangle, ver.-Window)**(GeoGebra) cf. The Reuleaux Triangle (Square Drill Bit) ----- Point C traces exact circle? --So. --- Good. Tell me why? (I can't understand.) parameter a = 0.33 is key-factor. what is this? 1/3 , 1/3 speed? 120° rotate? Angle AC rotate 180 °corresponds to AD' -60 °rotate. ------ if So, Implementation is easy by Gear or Linkage tool. I'm going to implementation by Linkage (3 times fast line symmetry) . See

**N=3 2R-Virtual Wheel (Reuleaux triangle, ver. -A)**(GeoGebra) ----- center trace is not exact circle. So, This solution is approximation.

**■ Analogy**Brown corner is corresponding to heel or toe part of foot, in human body. Black arc foot is corresponding to flat part of foot. Axis B is a puppet, real shadow axis is C, E, F. So, B is only a representative of C/E/F.

**■ Where does set the crank/ pedal?**Set it to any point on the chassis where you like. And between it and B, please insert the parallel linkage frame. Pedal/ Crank implementation can have

**much variety**. ex. inscribed circle gear at B/C , circumscribed circle gear at C (reverse rotation). etc.