|
Security Warning Do you want to run this application? ◢!◣ An unsigned application from the location below is requesting permission to run. Location: http://jars.geogebra.org http://www5d.biglobe.ne.jp Running unsigned applications like this will be blocked in a future release because it is potentially unsafe and a security risk. Select the box below, then click run to start application. ☐ I accept the risk and want to run this app. [Run] [Cancel] |
(1) Oval Chebyshev wheel - DEMOPlease drag the red bullet point ● C.cf. Explanation page. (written in English) In this circumstance, all GeoGebra function is feasible (i.e. I deleted all regulation.). (You can change now interface Language, Menu/ Options/ Language/...) You can change the parameter's value. Please change. ex. rr= 1.35 change to others, ee/2 (or exp(1)/2)--- What will happen? (i.e. Use rr slider [@left below]. or, Enter "rr=ee/2" into "Input:" box, and then, push "Enter" keyboard-key. If necessary do Trace Clear = Ctrl+F or Click corresponding to ↻ refresh mark icon at upper right [Reset construction: like a recycle-mark].) This is a intellectual TOY. Tip: • ee (or exp(1)) is 2.71828.. -- e (base of natural logarithm, e of exponential) --- 2.7 is very similar to number e, why? rr=ee/2 --- trace has refreshed, but rr=2.71828/2 ---- trace has not refreshed. ??? • Cassini oval is general this figure? --- may be. near. Anyone more research/ deepen, please. In other words, This is the new drawing apparatus for Cassini oval/ lemniscate curve. This is great invention/ discovery? Small token: I am Fumio Imai (Japan) -- self appeal --. By my test, this curve is not Cassini oval. ∵ Estimate 2 foci f1, f2, and set Point P on the curve, check the distance product Pf1×Pf2, ---- not constant. So, this is not the true Cassini. This moving figure looks like a 3D (3 dimensional) animation of rotating object .
Fumio Imai, 2013/11/06, Created with GeoGebra |
(1_a) Cassini 4 legs structure8 legs implementation is easier than 6 legs system. 4 legs system has a good property as below. So, 8 legs system is Disk-1 [for 4 legs, for 0° & 180°※ crank] = circle A right, Disk-2 [+ 90° of Disk-1, 4 legs] = circle A left.Disk-2's 4 legs are 45° differed from Disk-1 each legs angles. ※ 180° crank is replaced by below figure, so real rank doesn't exist. We can avoid the conflict of bars. Observation: If this were a wheel, Up/ down rate of the Point A or B is already so small. We can use this as the rough wheel. Don't you think so? Tip: The green rope property's simplicity may help to the real implementation. Like a Cycloidal pendulum, Please find a fence shape in this Cassini case. ( or another way: 2 L poles [= corners] locate on (2,0)/ (0,0), and Both end of Rope are the bars with 1.7 length. i.e. Left and right curve is almost 1.7 radius circle.)
Fumio Imai, 2013/11/21, Created with GeoGebra |
(1_b) Cassini 4 legs structure (Enhanced version)This is worthy of an invention.This is a Square Wheel invention. I name this invention "Cassini Square Wheel". This is simple enough, useful in real life, easily-assemble wheel, at emergency use, non-bulky/ saving storage space, easy cart, separable storage, only a bar. c. r = 4 ggg (theoretical = 3.940516825) Circular segment added edge keeps the height of Point A and B (= horizontal waist line) to be constant. ---- This theoretical curve to keep AB line horizontal is almost circular, perhaps. It is evident from my experimental check. ( No slip???, waist moves/ shifts at high speed? This circular curve has sufficient complementary ability. This slipping is the same in Reuleaux triangle case. --- Oh! This indicates the near relation Epitrochoid Rotary Engine. cf. A Rotary Engine GeoGebra DEMO © CJ Shore. What a similar shape it is! So, Can make a Cassini rotary engine? --- interesting!!!) This property is true in any other rr ≧ 1.35 value case, you only have to keep that the both ends of circular segment are square edge ends, and circular segment depth length is the gap to the ground. So, No needed rr=1.35 for this apparatus. ( In Number section, RRR is radius of Circular segment, RRRbyggg is its RRR/ggg ratio. ) cf. Circular segment (wikipedia) i.e. This easy wheel is 4 legs and longer radius partial circular tire/ shoes hybrid. c. This is a kind of Continuous track (or caterpillar track). Tip: This Cassini Oval (rr=1.35 case) has the ratio of width : height = 2.7 : 1.81 ≒ 3 : 2, and 3.25 length square edge.
Fumio Imai, 2013/11/21, Created with GeoGebra ★ If you are familiar to GeoGebra operation, please use zoom-in/ out function to see/ check precise motion. |
(1_c) Cassini Wheel (8 legs)This method has a problem which 2 legs on the ground rub the ground a bit. allowable?If supporting leg on the ground were only one, this is no problem. So, Keep AB line to be horizontal precisely. Remark: bar conflict occurs, so, green parallel control bar must be in opposite side. --- bad method it is!. (but, stride is 1.51-1.52 very stable. almost constant.) So, cut this green bar, and force C'D" = 1.51 to be constant, set such C'D" bar. I can't make/draw GeoGebra animation. why? this rough method is enough and preferable, perhaps. In the first place, 90° restriction doesn't need. It's enough to be orange frame is at rough middle position of pink frame. Follow after by constant distance. By my observation, 8 times touch the ground, each stride are, (1)just 1.51, (2)1.81-1.84, (3)1.14-1.08, (4)1.81-1.84, (5)just 1.51, (6)1.81-1.84, (7)1.14-1.08, (8)1.84-1.83, So, No problem, perhaps. c. Change the stride parameter; most right case, 1.94, most under case 1.65 ---so, mean value 1.8 force sride = 1.8 (1)just 1.8,(2) 1.51-1.58,(3)1.36-1.33,(4)1.58-1.49,(5) just 1.8,(6)1.51-1.59,(7)1.37-1.33,(8)1.58-1.51 --- gap is worse than before. c. On the perpendicular bisector of edge; (1)1.43-1.48 (2)1.47-1.43 (3)1.43-1.48 (4)1.48-1.43 (5)1.43-1.47 (6)1.48-1.43 (7)1.43-1.48 (8)1.48-1.43 --- good. but, worse than 90° method. ■ My Official answer of this problem: Keep the one orange leg to be about middle position by using a spring (or flexible wire) from both side 2 pink legs. That's all. About loosely position is OK. Do nothing to other legs. No slipping problem (∵ the distance from pink to orange leg is flexible. There's no power to rub the ground). i.e. This easy wheel is 8 legs with no tire. Its components are twigs only.
Fumio Imai, 2013/11/25, Created with GeoGebra |
(2) Chebyshev egg-crank wheelIn below Fig. Independent action is Point C ● only, other parts' motions are dependent all (ex. H •, Q • is dependent.). So, please draw C along circle A.I recommend next apparatus in real world. It looks like a check-valve ? c. I appreciate , http://www.mathematische-basteleien.de/eggcurves.htm Egg Curves Ovals c. I think this new crank mechanism is useful enough. I don't know that this crank method has industrial novelty (i.e. already the same apparatus exists or not.). So, if you use/ quote this apparatus, please attach my name (ie. Fumio Imai). I define this apparatus name for convenience sake (including next Crescent Moon shape apparatus): Circle-Ellipse hybrid curve linkage ---- [alias: Imai Linkage] (i.e. Half is circle, another half is ellipse mixed shape)
Fumio Imai, 8 November 2013, Created with GeoGebra c. In my experience, rough slide (i.e. simulate the line by large radius arc) is enough to the crescent shaped crank. |
(3) Chebyshev Crescent Moon CrankPlease consider.Try Number dd (now 1.5) to other value. ex. 1.0, 1.3, 1.7 (0.8 Peaucellier is interesting (?)) cf. related material: The Design and Optimization of a Crank-Based Leg Mechanism (pdf.) My question: Is this complex or simple? ---- I feel complicated. Please make more simple. It looks like a Ninja walking ? This Crescent Moon Crank can applied to the straight line drawer apparatus (ex. Peaucellier linkage or [Generalized] Hart's A-frame※). I inserted Peaucellier linkage sample into below Fig. ※: If you have curiosity, Please go. And consider how to make a partial circle crescent apparatus. c. This apparatus can draw a Mt. Fuji. ■Diversity/ Differs: Please check. --- Move (= drag) ▲I (2,0) to (3,0) and Set dd = 0.7, what will happen? Is it near a hopeful trace curve? (c. The word "differ" is the same family of word "diversity", by (ff-v) mapping. GRIMM's Law.)
Fumio Imai, 2013/11/09, Created with GeoGebra Full Moon : dd =0 〇Gibbous Moon 凸(convex) : 0 < dd < 0.5 Half Moon : dd = 0.5 Crescent : 0.5 < dd < 1 凹(concave) New Moon : dd = 1 Big Concave : 1 < dd Tip: In above Fig, I showed next Point F case sample. (x,y) cordinate. x(F) = x(B) - [x(E)-x(B)], y(F) = y(B) , now y(F) = y(B) = y(E) Next general variation is OK (but it becomes more complicated frame). ---- i.e. more preferable curve, we can get. x(F) = x(B) - [x(E)-x(B)], y(F) = y(B) + f [x(E)-x(B)] Please find simple function f(x). and implement it. |
(4) Chebyshev Crescent Crank Variation
Fumio Imai, 2013/11/11, Created with GeoGebra The implementation of this device is heavy (?). Orthogonal point + F is identified,F is slide point on the line EA (or on the line which is the orthogonal to CD and through C ), and the distance from the middle point of AC is half of AC. Many bars and 2 slides (i.e. D, F) are needed. If you don't mind, please implement this device. Is there any other more loose method? |
(5) Circle-Ellipse hybrid crank StandardPoint (0, kk) on x axis is ellipse most right/ left edge. kk = 0 --- just half moon, kk = +1/-1 ellipse = circle shape.
Fumio Imai, 2013/11/13, Created with GeoGebra |
(6) Unexpected foot trace (Good accident)I only have tried to make the above apparatus. GeoGebra produces an odd/ interesting result. Please investigate this result. Hysteresis (dissymmetry) shape appears instead of Mt. Fuji shape. Accuracy problem? ,, but, preferable result. Different action after critical situation happens, Please watch carefully, compare to above DEMOs. When the Point F passes over the X-axis, if we keep the Point F not to overtake the GA bar, then dis-symmetry crank moving happens. ---- So, we can get the Hysteresis foot trace. i.e. Make dis-symmetry curve on purpose, So, make dis-symmetry foot trace. (And, Consider the implementation of this symmetry sledge shaped curve. ---- This is a good exercise. Butterfly shaped frame exists. It's considerable interesting !!! I feel good omen from my intuition/ experience.) Tip: To use or not the circle C (4,0) method is an option. Point D 180° rotated point input method is better in this case. If needed, you can invisualize ■■■ colored the circle C (4,0) method point R trace curve, if you feel its disturbance. (1) Click (=reset/ set) the "● mark before R = (xx,yy)" in "Point" section in left hand "algebra view" frame. (2) execute "Ctrl + F"
Fumio Imai, 2013/11/19, Created with GeoGebra |
(7) Chebyshev Butterfly CrankThis is a desk work theory.It may be difficult to keep crossed frame shape, please try in real world. (ex. Cross keeping stopper/ restriction is needed.) Below GeoGebra was much modified to original honest setting, at first, the cross frame in y > 0 area becomes un-cross frame in y < 0 area, so I made tuning much. For example, Point G is switched to Point I. I made Japanese sword (or Samurai sword) shaped trace by Chebyshev Linkage. Perhaps, This is great, I think so. Tip: • bar length are below. so simple. AC = 1 (crank radius), DE = long, CE = AF = EG = long - 1, EF = 1 • From the result, next is found. (a) Crossed frame is near to the original Chebyshev Linkage mechanism. (b) Chebychev is, go = straight line, come = mild curve, composes the closed loop shape. now crescent moon shape is, go = circle, come = outer circle, composes the closed loop shape. 1 to 1 mapping is happened. this is natural conclusion. i.e. This is " Chebyshev Linkage in Chebyshev Linkage" or " W Chebyshev", we can say so. • This is incredible action, never be bored with watching for long time.
Fumio Imai, 2013/11/20, Created with GeoGebra |
(8) Typical Hart's A-frame
Fumio Imai, 2013/12/14, Created with GeoGebra Tip: She likes dancing? |
(9) How to Square Wheel (Rhombus)I introduce to how to make a easy square wheel. If we ignore the friction problem, It's not difficult.Below is edge length 2 square/ rhombus. This is rough/ approximate solution. Axis A ▲ keeps on almost height 1. Most wide shape is 2 equilateral triangles. The top of rhombus traces straight line or ellipse curve. Ellipse foci are H, H'1 (or H',H"). The sum of distance of point on the ellipse from 2 foci is just 2. (Here, Ellipse itself is the approximation, because the value sumRope changes to the non 2.0. ex. 1.99 etc.) 4 edge length sum is just 8. Prepare the ring of rope 8.2 length, and enclose the 4 foci and 4 top of rhombus. (凸 octagon case is 8.2. 凸 hexagon case is less than 8.2. (= 8.0-8.2)) (Please check the value test, test2. Consider its meaning.) Ring of rope is a moving regulation for making hopeful shape. Length 8.2 has backlash a bit, but is enough to real world (perhaps). Drag Point B red bullet ● -1 to 1 interval. (a) B on 0 to 0.33 case ---- octagon case; test2 = 4.1 (B'1H+HB'2+B'2H'+H'B') (b) B on 0.33 to 1 case ---- hexagon case; test = 4.1 to 4.0 (B'1H+HB'2+B'2B') --- backlash rate 0.1/4=2.5% (hexagon case, 凹 concave is replaced by 凸 convex.) (b') B on 0.4 to 1 case ---- octagon case; test3 = 4.1 to 4.08 (B'1Q+QH+HB'2+B'2B') --- backlash rate 0.02/4=0.5% ----- (b') case is Point Q 凸 adjustment is applied. 0.33 to 0.4 intarval is (b) hexagon test value applied. 0.87 to 1.0 case is hexagon. in this case, test3 value is not real, but near value (i.e. 4.08-4.1). If you could manage the concave control, please try it. It's more precise system. c. B is 1 on (1,-1) case, value test = 4.0, but rhombus shape is just square, so axis A's height is 1.0, in this case 4.0*2=8.0 is differ from 8.2, 0.2 gap exists, so, this gap can be omitted. (?) Tip: As a result, Peaucellier Linkage or Hart's A-frame apparatus mechanism added is smarter than above. No friction problem. (I pictured 2 Peaucellier elements in below Fig.. Point L and U are independent. What relation are there? U"V pink line is the tangent to A- circle.)
Fumio Imai, 2013/12/17, Created with GeoGebra Tip: Axis-A and foot top B (or B', B'1, B'2) relation is Cycloid relation. |