# Chebyshev Linkage Wheel

- Author:
- asifsound

- Topic:
- Square

This apparatus is a square wheel made by

**Chebyshev Linkage**(approximate straight-line mechanism). In figure, the solid-line bars are components of this apparatus. The dashed-lines are explanatory lines. Aqua-colored big bullet**● E**is a axis of this wheel.This apparatus has beautiful property.
① ABCD (= #1) is a basic component.
AB = Lower base (length 4), AD or BC = Cross-oblique sides (length 5), CD = Upper base (length 2).
(We suppose that AD and BC are always crossed. To avoid the uncrossed situation , we must attach the angle limitation stopper. i.e. Keep BP ≧ 0.)
This apparatus has 4 components (#1,#2,#3,#4). #1 and #3 are

**point-symmetry**each other. #2 and #4 are, too. ② #1 and #2 (LM, ON) position relation has next**rule**.**a) Lower base ⊥ others Upper base.**From a), ON (#2 Upper base) is always vertical. ∵ AB is fixed and horizontal. From a), LM (#2 Lower base) is rotate 0 to 180゜. ∵ CD rotates 180゜. [ In same component, Lower and Upper bases move alternately each other. i.e. When one is rotating, the other is not-rotating. ]**Rule a)**is established by various ways, but, the most simple way is used in above figure. i.e. R = the middle of BC (length 2.5 [= 5/2] ), Suppose S , RS = length 2.5, ES= length 2 [= 4/2]. From theoretical analysis, ES // AB. --- so, To make ON ⊥ AB is equivalent to to make ON ⊥ ES. (We can reduce the number of bar by this way.) At the end, Force to fix the angle between ES and ON toward 90゜. --- welding/ glue.**In real implementation, prepare short**macaroni tube==**(plural pasrts), And insert axis bar**---**into it. as**--==---==---==---==---**. ← 4 macaroni.****Axis design: ---- Implementation is easy.**( #1 is independent, others are all dependent.) #1/#3 upper-bar ① ---- == macaroni_1 (bonds with naked Axis-bar**---**) #1/#3 lower-bar ES ② ---- == macaroni_2 #2/#4 upper-bar ③ (⊥ with ②) ---- == macaroni_3 (bonds with macaroni_2) #2/#4 lower-bar EU ④ (⊥ with ①) ---- == macaroni_4 (bonds with naked Axis-bar**---**) Chassis (car_body) ---- == macaroni_5 [ i.e. The smallest configure is**------==--**style. With only one macaroni. ] I think this inventions is a home-run hit in the human being history. (This figure was made by GeoGebra Graphics View's copy-function. GeoGebra is powerful educational tool.)**★ Feeling.**Please imagine, there exists vertical real gravity ↓ [vector ON], so, ground AB is horizontal. upper base CD vector is 2nd. virtual gravity →, so, #2 Lower base MO' ⊥ CD. virtual gravity rotates, in last, it directs to real gravity direction. and, it will be baton touched/ passed.**■ Easy making (??) ---- This is a kind of instant wheel.**1. Prepare 2 sets of Chebyshev point-symmetry linkages. 2. Prepare one axis bar. 3. Insert the axis bar into 2 each symmetry points. Here, axis bar is like a BBQ skewer. 4. attach the**rule a)**cordinator 3 bars (2.5, 2.5, 2 length bars) between to 2 point-symmetry linkages. 5. Set 90° adapter (corresponds to α = 90° mechanism ).**■ About practical utility :**・ wheel for on a sand-beach, carpet (----- small loss caterpillar. ) ・ caster of travel bag ・ instant wheel ( They are only bars, usually.) ・ heavy weight lifting wheel/ roller (constant height) cf. Monster 4x4 Mud Trucks at Good Time 4x4's June 12Th 2011 (YouTube) ---- interesting vtr.**★ Variant :**Q: Please consider Hexagon (= 6 edges) wheel. 3 sets of the point symmetry Chebyshev frame. A: It's easy. ---- rule a') Lower base ∠60° & ∠120° other 2 Upper bases. (2 grounded foot width is narrow than 4 edgee case.) cf. Chebyshev Linkage Wheel2**■ Pedal driven implementation FIG.**Below**pink square**is available to make a pedal crank mechanism.