Flying Principle of Boomerang .....ブーメランの飛行原理(日本語は下半分をご覧ください)


Every our Japanese Engineer who went to Sydney for their business buy the Boomerang on the gift shop on the Sydney airport. After returning to their home, they try to return flying the boomerang on his neighbor park. But everybody fails to get beautiful returning, and he feels it was only decorative gift shaped like a boomerang. But I am not such a person. I looked hundred of boomerangs on ten of gifts shops in Sydney airport and city inside. And after returning to Japan I made about 30 of boomerangs with plywood board. And after many trial and study on the homepage of Australia Boomerang Association, I got beautiful returning and 8 shaped returning with my hand made boomerangs. With my trial I got some know-how and some new idea of explaining the flight of boomerangs. So I added my homepage about it here.

1. Definition of each part of Boomerang. Notice: Some of these ward definitions are differ from them of Australian Boomerang Association.


2. The definition of the wards indicating the movement of Boomerang. Notice: They also includes some of my original definition.


Sweep: The ward,sweep indicates the changing the position of the whole boomerang. Any part of the same one boomerang sweeps for the same direction with the same speed in any one moment. So the sweeping velocity vector in any part of the boomerang is the same in any one moment. It is also understood as the movement of the Gravity center or rotating center (We can consider the gravity center and rotating center is near enough so can say same position here).

Rotation: Boomerang rotates around its rotating center anytime while it flies. Right hand boomerang rotates counter clock wise seeing from right side. In normal L shaped boomerang, the rotating velocity vector of the leading arm and the trailed arm differ 90 degrees. Rotating velocity on the rotating center is Zero. It is definition of the Rotating center.Rotating center of boomerang is not in the elbow. It exists on the fork, I looked it. I calculated and tested by hanging my boomerang, also gravity center of the boomerang No49 is on the fork.

Turning: Turning head indicates the change of the direction of the rotating arc. In this ward head indicates upper part of the rotating arc ( We call upper arc of gyro as head in Japan). Please image the skate player spinning on the ice and turning head (Innerbauer).

2. Boomerang flies by the lifting force (lift) get on the wing.

2.1 Air flows by the wing while Boomerang sweeping and rotating. It causes lift force on the wing and it make fly the boomerang.



2.1 Boomerang is made by after cutting plywood by jigsaw, scraping the both edge of the same one side. Scraped edge is called as right side. And another flat side is called as rear side. Do not call the right side as upper side, because boomerang is thrown by standing position. See fig 4. Wind flowing right side (this case upper side) flows long way than the air on rear (Lower in this case) side. So the air flowing right side is extended and it causes the decrease of the pressure of right side and cause the lift force,hanging up the boomegang. It is called as Bernoulli’s principle.


2.2 The direction of the air attacking the wing changes in every moment. See Fig5 Left(boo5)! Wind attacks the fork and it folws to elbow part of the wing. See Fig5 Right(boo6)! Wind attacks the shoulder and it flows to elbow part of the wing. Such a way the wind blows to the wing from many different directions in each second. So we should shape the wing as it can get the lift force with any direction of wind. So we should shape the wing as mountain shape.

2.3 As the wind attacks the wing with the gathered velocity vector of sweeping velocity and rotating velocity, and rotating velocity is not so strong enough as extinguish the sweeping velocity, wind flows with many direction to the wing, except two tips of each the wings. This situation is the same with Frisbee. So we can say that “Boomerang is Frisbee”.

2.4 If you are Japanese, you will associate the Bamboo-Helicopter other than Frisbee as the fling toy. If you are Foreigner, Please buy the Bamboo-helicopter in the gift shop on the Narita airport as the Japanese traditional flying toy.

In Bamboo-Helicopter case, the rotating velocity is strong enough to cancel the sweeping velocity. So it can be shaped as normal propeller having the attack angle. In Boomerang case only the two tips of each wing can be understood as rotating, so we can scrape the rear side of the wing to make undercut, and get the attach angle only on the tips of two wings. But this attack angle increase the lift force of the tip of the wings and boomerang can get unneeded turning movement. And it can have a habit to fly like screw opener indicated in Figure 30. You should know that it is difficult to use this attack angle correctly.

3. Macro-motion of Boomerang
3.1 Boomerang makes "Cone motion" in it's first 3/4 circle
3.1.1 Preparing of the Preliminary experiment

Fig9(Left): (Fig10 under)

Please tap the hook on the ceiling and connect the 1m string on the hook. bundle the center holed coin like Japanese 5 yens in another side of the string. This material is called as cone pendulum. If you can not get center holed coin, use 10mm nut! If your house is tall and you can not reach the ceiling, please attach the string on the clothes pole like fig 10! And push a coin in many different directions in many different manners. This is preliminary experiment of Boomerang motion.


3.1.2 Hold a coin to make (angle between (vertical line from hook) and the string) becomes about 60 degrees. Push a coin on the direction in the (horizontal plane including coin) and 90 degree from the string, strongly. The track of the coin becomes circle. And the remaining image of the string may be cone. This motion is the biggest and first motion of the boomerang who traces cyclic track and returns to the thrower.

3.1.3 Turning head motion of Boomerang


It is well known that flying boomerang behave like a gyro and make motion of turning head. See fig 14! When (the gyro rotating counter clock wise) begin to lean to left side of the figure, it turns the upper side of the rotating arc (head) to come to near side of this figure. To indicates this phenomena more universally, I want to explain each direction with right side hand. When head of (CCW rotating) gyro turns from thumb (p) to forefinger (a), the gyro get the magic force to be turned to middle finger (m) direction by the ghost of Coriori.

3.1.4 First 3/4 track circle of correct boomerang
Fig15: Do not only look this figure but do it your self with your fingers! Arc A is your Thumb. Arc B is your point finger.
So boomerangs rotating arc be your middle finger.

Let you grip the boomerang on your right hand as you grip the hummer. Hold it over the shoulder. Incline the right side of boomerang for outer side to your right about 30 degree and The right side should be inner to allow you to see the right side of your boomerang..Throw it front side with 20 degree upwards. The 30 degree of inclination is not so common. You should search for the best angle match to your boomerang. Then thrown boomerang get the lift force for the right side(face) of it (It be left side of sweeping direction). The vertical component of lifting force (* marked in Fig16) resist to the gravity working for the boomerang from the earth, caused by the agreement by Mr. Newton and the God of Apple on 100 years ago. The horizontal component of the lifting force (** marked in Figure 16) becomes introversive force that makes boomerang to trace the circle. With this way, boomerang trace the same way with cone motion. While making the cone motion, the boomerang turns his(her?) head to counter clock wise. By the gyroscopic precession, the boomerang turns his head to upper side from inner left side. The boomerang return to the thrower like figure 15 faceing the head to upper side and he has almost vertical lift force and lost introversive force, and continues to change his movement to next 8 letter motion.

Sorry! Before talking about 8 letter motion, I want to say about Pendulum motion.
3.2 Pendulum motion of Boomerang (This motion is not expected motion but occers ofenly)


3.2.1 Hold the coin to make the string angle about 30 degrees with vertical line. Push the coin for the direction to the vertical arc of hook strongly! Then coin climb up to another side and after lost the speed, it will falling down for the direction just to your fingers. This motion is not intended to get with boomerang. But it occurs sometimes. The boomerang return back to your side very high speed and it is danger to play with.

3.2.2 Even if the boomerang was made correctly, we can get such a track by throwing boomerang with horizontal wing (with keeping the rotating arc vertically from the ground), like Figure 11. And even if the inclination of the boomerang is correct, when thrown with the too much sweeping velocity, the boomerang get into this motion after following half of the circle motion.


3.2.3 The wrong made boomerang can not get into the circle motion and get to this pendulum motion easily. Then boomerang act as vertical jumper. I will explain in detail how we can avoid making such a “vertical jumper”. In my experiment the boomerang made with aspect rethio (rethio of length of the wing and width of the wing) within 4 or 5 must be this type. I will talk agin how to avoid this motion later.

3.3 8 letter (Eight letter) motionk (Hovering)


3.3.1 The left side figure 12 is un realistic. Coin hanged on the hook do not trace on 8 letter motion. But Boomerang can do. The boomerang on the position on the coin on Figure 12 is inclined slightly (5degrees) left side and move forward. So he get introversive force to left and change the sweeping direction to left side. So he gets gyroscopic precession and rotating arc becomes vertical when he go just vertical downside of the hook on + position on the figure 12. After he go through + position, the inertia of gyroscopic precession remains and he begin to incline to right side. So he begin to turn to right side and trace the right side half of 8 letter in figure 12. And so on.

3.3.2 While making 8 letter motion, boomerang lost the sweeping speed little by little and stop the position by rotating and get the just vertical lifting force there. This situation is called as hovering. But you can see the Hovering is one type of 8 letter motion with smallest sweep speed. See figure 18! After making cone motion (circle motion) from A to B position of Figure 18, boomerang returns back to the thrower by making S- letter motion as half of 8- letter motion, from B point to C point in Figure 18.

Fig18(left): Fig19 (lower): Please image wind come from left side of these two figures.

3.3.3 By throwing boomerang with a little less sweeping speed than last case, Boomerang ends the circle motion a little nearer then last case, and it fly by the thrower making there the cross point of the 8 letter motion. See Figure 19(boo17)! In this case the whole track of the boomerang becomes 8 letter tracks. And sometimes the same thing happens with a little bit stronger wind and also with throwing more initial rotation.

4. Rocket boomerang motion (additional explanation to 3.2.2) In this case rocket is not mean A4 but Me163,raketen fluegzeug.


4.1 Please image the boomerang was thrown the best manner indicated in figure 16! The thrown boomerang goes through the circle track as a bottom of cone motion from A point of the Figure 19 to the most far point B in the figure 19. The vertical line of the traffic (track) changes from A-O to B-O while boomerang make running half of the whole circle. By changing the angle of the arc of the rotation about 90 degree or more, the head of rotating arc is waked up about 90 degree by gyroscopic precession. So boomerangs rotating arc becomes just vertical when he comes B point. So from B point boomerang stars to move as pendulum motion in figure5 and starts vertical rise up motion indicated in figure20. This page indicates how to move out from this devil cycle.

5. How to make returning boomerang
You can make returning boomerang by reducing the effect of the gyroscopic precession. See the figure 19 once again! If (the turn of the head of the boomerang becoming vertical) delays from B point for more later point as 3/4 circle or 4/5 or more, and (on the the turning head point) the sweeping speed has reduced enough, and it is near from thrower, the circle shaped returning will be proceeded. I had prepared the two help men to delay the gyroscopic precession effect for you.

5.1 Help man No1: "Resisting force for turning head of the wing "


Hold the Japanese bamboo flap keeping the main flat part of the flap horizontally! Push it on the air with rotating the flap! The flap will fly about 10m by keeping its flat face horizontally and landing on the floor smoothly. You can be the winner of the office if you throw it with 5 degree lower side and push flap handle strong forside.Because the flap has very big flat face, this face resists to be changed its included plane and has a habit to move inside of the same one plane. Because the boomerang is also made with flat plywood, he has a habit to move inside of one plane. this effect can delay the change of the head direction caused by gyroscopic precession. With this meaning we can say that"Boomerang is bamboo flap".

5.1.2 The resisying force of the flap changes while rotating the boomerang

Fig22 Sorry please image each boomerang sweep to left direction. And (+) point means the rotating center. Horizontal line means the turning arc. Clock wise arc means the turning head motion.

Because the boomerang rotates while it flies, the shape of the boomerang anti air is changing second by second. In the timing one wing faces long to the sweeping direction (like PPP on the left figure of figure 22), the tip of the wing has big resisting effect to turn the rotating arc (head). By the other hand, like right figure, in the timing of both wings can not take long arm from the sweep direction, the most far points of the wing from the turning arc indicated with QQQ and RRR in the figure are not so far from the turning arc. So in this timing boomerang has weak resisting force for the turning head force.

5.1.3 To cover this weak point, we must make the boomerang bigger and the length of the wing should be extended. But when we make the boomerang bigger, not only resisting force but also rotating inertia itself be bigger and gyroscopic precession itself also be bigger. In order to escape from this devil circle, we should cut out the part of the wing around the arc of rotating in the timing of weakest timing of the boomerang. See lower figure! The shape of the Admirals hat has a effective plan view shape whitch has effective resistance force for head turning in weak timing. And you cam understand that to make "fat" AAA,BBB and CCC aria is also effective to get strong turn head resistance.

Fig23 Horizontal line in this figure maens turning arc.

5.1.4 Scissors angle

See the Fig22! You can understand the Boomerang can be increased its resisting force to turn head by taking more narrow scissors angle. My best boomerang No49 has 85 degree.


5.2 Help man No2 "Upper wing priority effect"

Please image movement of a tire of the car! The point touching the ground has no velocity. The center point of the wheel has the same speed of the car itself (Ve of figure 24 is same with "meter speed"). So the upper tip of the tire has double high speed of the Car (Vt=Ve*2).




The Boomerang thrown from the hand flies like a wheel tire by keeping it’s rotating arc almost horizontally for a while. So upper wing cut the air with high speed and generates bigger lift force (Lu on fig 27), while lower wing generates smaller lift force (Ll on figure 27). With this difference of the lift force, the rotating arc (Head) of the boomerang is rotated to counter clock direction and it resists the gyroscopic precession who want to turn the rotating arc to clock wise. This effect become bigger when the boomerang has long wing (high aspect ratio).



In order to get more strong this “Upper wing priority effect”, make the polyhedral undercut on the front tip of each wings by scrape the rear side of the wing. How to shape it, please see Fig 1!


5.2.4 But dont forget that to use this polyhedral undercut is very difficult. I looked 100 of boomerangs in Sydney airport but only one type had this shape and the undercut was only 0.5mm. Because the tip of each wings cut the air very fast, the attack angle of the wing generates very big lift. So boomerang with this undercut tends to have too big counter clock wise turning habit and he sometimes makes “cork screw trace” given in figure 30.

6. Advice

Like I tolled you in this home page, Boomerang can return to you when he made properly (Hard), and he was thrown properly (Soft), and he got proper wind (Lucky). So the making and throwing boomerang is very fan hobby for the middle aged Engineer like me. I can spend the time with boomerang talking with the goast of Coriori, Newton and Bernoulli.

I bought the 4mm thick plywood board in neighbour "Do it youself shop". In Japan we can not get airplane plywood cause we have never made airplane with plywood like English made mosquito.

You should not hit your boomerang to children. But you have no need to care about dogs.

You should take a clothes pole with you when you go to the park to fly the boomerang to rescue the boomerang from the tree.
Also you should take an used undershat to wipe the boomerang.

7. My best boomerang No49

Copy this shape on the “BMP” software in your computer. Print it with proper zoom to be the distance of tips of two wings be 290mm. Attach the printed paper to the tick paper with paper grew. Cut it with paper cutter. Put this “shape paper” on the 4mm plywood. Trace the shape with pencil. Cut the shape with jigsaw. With small knife, scrape same one side whole around the boomerang. You should make it sure that the all section be the shape like Fig4. Smoothen all around with #120 paper. Coat with Urethane. After coating has dried up, rub with #400 paper few times. ( Try wing length with 330 and 240mm.) (This type do not need any polyhedral undercut)


8. Two Hypothesises by JF1OZL

8.1 Hypothesis 1: Boomerang is supplied the rotating force from the air by flying.


The Trace of tip of the wing is far longer than the trace of the elbow, till the boomerang returns after thrown. So I feel rotating inertia should be reduced earlier than sweeping inertia. But the fact is not that way. When Boomerang returns to me, he continues to rotate while his sweeping speed has become almost zero. I feel that boomerang must get some supplement of rotating energy while it flies. How he gets? I asked it to my boomerang and he taught me the secret. Please see Fig50! In the instant Boomerang flies on his fork to front of sweep direction, the air attacks on the upper wing want to flow shortly over the wing surface and it is bent to upper side (flow from P to Q on the figure 50).So boomerang has given the air the upper vector force (alpha in the figure). So the boomerang get the counter force from the air, just controversies vector from alpha. So boomerang is pushed to counter clockwise direction by the air. Lower wing go through with lower speed anti air. So lower wing get weaker effect of this.

Please see Figure 51, in this timing upper wing get clockwise direction force from the air, but in this case, air escapes around the shoulder and the smaller amount of the air flows on the upper wing than later case.


********Additional description on Hypothesizes 1****2007/03/09*******

Please look the Figure boo501! This figure indicates the position 90 degree before the Boo50. In this figure, the air flow on the tip of the leading wing is twisted to upper side. So reaction of this effect gives the leading wing a down force. This down force makes the boomerang accelerate to rotate for counter clock wise. And also the wing flow over the trailed wing also assists to accelerate to rotate for counter clock wise.

Please look the figure 502! This figure indicates the position on 180 degree differ from the figure 501. On this position, the wing flow over the upper side of the boomerang is twisted to downward and upper ward. And it causes totally nothing.

As I showed you with the figure 50, 51 501 and 502, the situation of 50 and 51 have the effect to acceleration of rotation and deceleration, while only the position 501 has the effect of acceleration. So the boomerang should be accelerated of it’s rotation when the both wing be upper side.

In my experience the boomerang which has the leading wing's tip bent to inner side like the figure 503 could not make hovering when it returns to my head.

On the other hand, the boomerang which has the wings tip bent to outer side like the figure 504 could make hovering when it returns to my head. And you can understand that also the trailed wings tip bent to outer side also helps to accelerate the rotation of the boomerang. It looks like the Admirals hat, is not?

8.2 Hypothesis 2: The difference of gravity center and rotating center makes the boomerangs rotating speed change inside of each cycle. And it may cause the effect of work of the wings and helps the hypothesis 1? But I have no way to know this hypothesis is right or wrong.

And also this difference between gravity center and rotating center may cause the resistance for turning head force.

Buy a high spped camera and catch the change of the change of the rotating speed.
Rent the wind tunnel and measure the force on each direction.
When you get the Doctor award please report me! I will say congratulation and drink wine with me.


The second Hypothesis of the reason why the boomerang can continue to rotate- Description add on 3rd Feb 2008


On last Sunday when I play with my boomerang, one neighbor child rotated the boomerang around his pointing finger like Figure Boo601. While looking him, I discovered that I had forgotten the important theory of rotating. See Figure Boo602! Weight can continue to rotate by the force given by finger through the string. The string can not transpose any torque from the finger to the weight. The string can transpose only the force with the direction from the gravity center of the weight to the grapping fingers. The Force is indicated as F and it is called as introversive force. See Figure boo603! The toy hoop rotating around the waist is called as Fla-hoop in Japan. The hoop does not get rotating torque from the waist, but only get introversive force. See Figure boo604! Flying Boomerang gets the pressure on the tip of the wing. The pressure outside of the wing is indicated as Pout. And Inner pressure is indicated as Pin. The air outside of the shoulder of the boomerang can slip out from the wing. The air inside of the folk of the boomerang can not slip out anywhere. So Pin become higher then Pout. So sum upped pressure becomes introversive force F. F pushes the gravity center of the boomerang from the folk to the shoulder. Especially the Inner pressure becomes bigger when the boomerang flies with his folk for front side.


9. What No77(C-shape), No73 (Ohm-shape) and No75 (M-shape) taught me.

: In order to try how the various shaped boomerangs fly, I made these three boomerangs once a time. The result of these, tree were the same. They can trace about 3/5 of circle. But they can not do hovering. All these boomerangs have circle shaped plan view. So they have small radial wing which makes lift force in hovering. So they can not hover. I can not make how to evaluate about this point of view. But more simply I can say that the plan view shape which has big circumference part in wing (like O, C, M, Q and G) can not do hovering.

Requirement of the configuration of the boomerang


In this clause, I wrote about “How to make the boomerang.” Somebody says that “Boomerang can be made with any shapes.” But I feel “Good returning boomerang has some physical requirement on its shapes”. I wrote here about it.


1.      Thickness of the plywood board should be 3mm to 6mm. The 11mm board is too heavy to our normal adult man to throw. If you are young and can throw baseball with 140km/h speed gun, you can try 11mm plywood. 4mm thick is recommended.

2.      Length of wing (small L of Figure A1) should be from 160 to 250mm. 200mm is recommended. So whole width (L of A1) becomes from 225 to 350mm. You may feel 200mm wing length is too small for your enjoyment. But this boomerang flies over 30m far from you and returns. Is it not enough to throw in your neighbor field?

3.      Aspect ratio should be from 6 to 7. As aspect ratio 6.5 is recommended. In this home page I define the Aspect ratio as ratio wing length divided by averaged wing width. Aspect ratio=a. Wing length is small L of Figure (A2). Averaged wing width D=(D1+D2)/2…..See Fig (A2)!.....Small Aspect ratio (like 5.5) wing will be “Vertical jumper boomerang”, because it do not have enough turning resistance. And also small aspect ratio boomerang will be heavy to fly and not return to the thrower. On the other hand, the boomerang which has too big aspect ratio, like 9, can be thrown and return to the thrower. But it has tendency to be week to the wind and also week to hard landing.

4.      Board thickness ratio. Here in this homepage, Board thickness ratio(i) is defined as the number of the thickness(t) of the board divided by the averaged wing width(D). I=(t/D)*100(%)....Board thickness ratio (i) is should be smaller than 18%. As Board thickness ratio, 12 to 18 % is recommended.

5.      Wing section (aerofoil) should be simple mountain shape indicated in (A4). You should make new boomerang with such a simple shape once. And after you make trial thrown, you should add very few amount of undercut like 0.5mm on the wing tip.

6.      You can make boomerang with any wood board. But it must be smooth out sided. It must have enough strength not to be broken when it make hard landing. Plywood is recommended.

7.      Width of elbow (De of A5 figure) should be 30mm or more. This requirement is to keep the strength of hard landing.

8.      Width of the elbow (De of Figure A5) should be wider than both wings or at least the same wodth. Also not to concentrate the stress, the wing shape like (A6) or (A7) are prohibited.

9.      At least one wing should have its wing tip shape as radial direction from the gravity center of the boomerang. See Fig (A8)! The tip of the leading edge has its shape parallel with the radial line (R.L.) from the Gravity center (G.C.). When the boomerang remove from the hand, it goes away from the hand with its gravity center be far from the hand. So this shape helps to remove the Boomerang from the hand. By the other hand, the boomerang, which has two wings with circumference direction shape of the wing like figure A9, can not leave from the hand easy. It sometimes fails to be thrown. It sometimes makes hard crash on the ground by the foot of the thrower and be broken. You can throw the boomerang with holding leading wing or trailed wing (Not to say trailing wing!). So you should give this shape at least one wing of your boomerang.

10.  You should calculate the aria of the wing to have enough turn resistance for gyroscopic precession. The strength of the gyroscopic precession depends on the rotating inertia of the boomerang and it is calculated by the equation indicated in the figure A11. The strength of the turn resistance is calculated with the equation indicated in Figure A14. But to calculate these equations are not so easy to do. So I can propose easy, simple way of it. When you sketched the plan view of the boomerang, stay it on the paper on the horizontal line like figure A10. The top of the elbow is set as height, h. Let you line 3/4 h, 1/2 h and 1/4 h horizontally. The aria outer three part indicated as Ae should be more than half of the total aria. So Ae should be bigger than Ac. And if you feel your sketch is not enough, you should change the shape to be slim up the Ac aria or you can make fat on the Ae aria.

11.  Scissor angle, theta of A12 figure should be from 80 to 90 degree. As a scissor angle 85 degree is recommended.

12.  Every round on the shape should be R=10mm or more. Not to wound your hand when you catch the boomerang. Not to be the air resistance, drag too big. Not to concentrate the stress when the boomerang makes a hard landing.

13.Desnsity of the boad. Density= 0.52(g/cm3) OK.
Density = 0.67 NG. It is my experience. I do not know about between 0.52 and 0.67. Please make 10cmX10cm squere boad and measure the weit of it before you make boomerang! Do not forget to measure the thick!

You can understand the sold boomerangs also follows my proposed requirement.

back to index

Description add on 12th Feb 2007/// Proposal of Kaizen Admirals hat /// Boomerang need to be trimmed.


Please see the Figure Boo93! I down loaded this shape from the Home page of Australian Boomerang association. And I made two boomerangs with this shape. They flied very well with circle shaped trace. One of them was be broken on windy day. So I re-produced three boomerangs with the same shape. But All these re-produced boomerangs flied like Me163, indicated with Figure 20 as vertical hopper. I tried to add a paper tub to try to get better trace. Then I discovered that they can make circle trace with square paper attached like a shape (B) indicated in Figure 93. the width of paper was 30mm and the total length of the boomerang, L was 33cm. You can understand that this was caused to strengthen the resistance to be turned head effect explained in the clause 10. of The Physical Requirement.

In the process of the production of the boomerang many error is happens in the shape outline, weight of each part of plywood, cutting of the edge of wing and the smoothness of each part of the wing. So many reasons of error cause the difference of the trace of fling of the same type of boomerangs. Therefore in order to make good boomerang, it needs the trimming same as the tuning of IFT coil of radio receiver.

Please see the Figure 93 once again! I made three boomerangs with the shape of ( C ). One of them turn head too late and touch the ground on most far point. So I cut like shape (A). And it flied well. The second one indicated well trace with ( C ) line. So I stayed it. The third one could not fly up even after cut to (A) line, so I had broken it. Such a way I propose to make “The Admirals hat type boomerang with trimming tab on folk” indicated in figure 93. Please make it with 4 or 3 mm thickness plywood with the total length 33cm.


Additional description on Feb.22th 2007 This thinking came on my mind in the morning on Bank House Hotel Uttoxeter England, “The meaning of dihedral processing”.

Please see the Figure 101! Boomerang, B sweeps tracing the cone motion. The length of the hanging string is r. Center is O. The small plane including the boomerang is called as P. And P is defined as square shape. Figure 102 indicates the instant of the P has swept from P1 to P2. This boomerang traces the cone motion so the head is turning and the P1 and P2 is not in the same plane. So the total space including P1 and P2 has the thickness like D indicated on the figure 102. And the Boomerang must sweep while he mixing the air with thickness of D. Already you understood what I want to say. If the plane P is curved to upside like Figure 103, Boomerang can fly on the same curved plane, and total air resistance become very small. Yes, “Proper bend to upper side of the both wing chip reduces the air resistance while the boomerang do cone motion and it make it easier to come back of the boomerang”. With the pure geometrical calculation with the figure 105, the proper dihedral processing becomes 0.9075mm with boomerang who has the total length 33cm and turn with 30m radius. Yes it is very similar with my experience as proper dihedral processing, is not?


Description adds on 8th July 2007

 When you visit your neighbor “Do it your self shop” to buy a wood plate to make boomerang, you may be felt as miserable by looking only the remaining winded plate. You should buy that! I say you that “Make your boomerang with rear side with mountain”. Once again I want to say you “Make your boomerang with rear side with mountain”. If you make your boomerang with this material, the boomerang laid upper side right side on the desk will work as seesaw. The boomerang laid upped side read side will be bridge on the desk, and you can push the center (I say elbow) of the boomerang with the 2mm stroke. Yes it is the most important secret of the boomerang. I was looking for this secret half years. --------------- What I say here is the secret of the secret. If you get some boomerang which do not return to you, you can recover that with this process. Pour the water on your boomerang about three minutes! Dry it on the sun shine with cramping on the frame on the window frame of your house. After half hours, you can get another boomerang.

******** Underside is Japanese letter *******









2.2 ブーメランに当たる風の方向はブーメランが回って、姿勢が変わるごとに、変化します。たとえば、図5の場合は、腕に対して風は又の方向から吹いてきます。一方、図6の場合には、腕に対して風は、肩のほうから吹いてきます。すなわち、腕に対してまったく違う方向から風が吹いてくるので、腕の辺りの断面形状は、どっちから風が吹いてきても、揚力が発生する「上に凸」形状が良いことになります。

2.3 実はおおよそ、翼端以外のすべての部分は、ある意味腕と同じ風の影響をうけるので、「両方向性」のある、「上に凸」形状が良いことになります。この例がフリスビーです。フィリスビーはすべての断面が一様にすべての方向からの風を受けるので、どの方向で切っても「上に凸」形状に成っています。この意味で「ブーメランはフリスビーである。」と言うことができます。

2.4 皆さん、日本人なら、回りながら飛ぶおもちゃとしてまず、ブーメランよりも先に、「竹とんぼ」を連想するでしょう。ただし、竹とんぼは斜めうえに飛ばして戻ってくるような動きもできますが、円を描いて戻ってくるような軌道はえがけません。。。。前置きが長くなりましたが、ブーメランも翼端だけは、横滑りはするものの、逆には運動しないので、一方通行の運動ですから、プロペラ形状に、裏側を削って、向かい角が着くように削って、竹とんぼのように揚力を得ることが可能です。すなわち、図8のように、翼の裏側を削って、大きな揚力を得ることが可能です。しかし、私もずいぶん失敗しましたが、翼端揚力の多き過ぎるブーメランは、理想的な円軌道を描かずに、「逆ねじワインオープナー」のような起動を描いて、全体としては、あっちへ飛んでいってしまい帰ってこないのです。ですから、まずはこの「翼短揚力はあまり頼ってはいけないもの」としての認識が必要です。



3.1 ブーメランは円錐振り子運動をする。(正しい円軌道ブーメランのはじめの3/4周)
3.1.1 予備実験の用意



3.1.2 そして、糸がフックの垂線に対して、60度をなすような角度に5円だまを位置してから、5円玉を含む水平面上でかつ糸と直角な方向に”強く”押し出してみてください。(図11)。。5円玉は、円運動をして、糸は円錐の軌跡を描きます。これが、ブーメランが円を描いて戻ってくる第一の主要な運動です。

3.1.3 ブーメランの首振り(回頭運動)


3.1.4 正しい円軌道ブーメランのはじめの3/4周
図15(右); 図16(下)


3.2 ブーメランの振り子運動(正しくないが良く発生するうごきかた)

3.2.1 フックの垂線に対して30度の角度になるように5円玉を持っておいて、垂線に向かって強い力でブーメランを打ち出してみてください。すると、5円玉は垂線を含み地面に垂直な平面で急上昇して、やがて、初速の運動エネリギーがすべて位置エネルギーに変換されてしまうと、速度を失い、やがて重力に引き戻されて、すごいスピードで元のところに戻ってきて、指にあたるでしょう。こういう動きはブーメランにとって好ましくない運動ですし、だいたいものすごいスピードで落下してくるのであぶなくていけません。

3.2.1 正しくできたブーメランでも、投げるときに表側を垂直上側にして、図11のようにして投げると、こういう運動をします。また、最初の投げるときの角度は良くても、押し出す速度が大きすぎると、円の途中で水平姿勢になったときに、この運動モードに入ってしまいます。


3.2.2 またよくできていないブーメランは、どういうふうに投げても、この「急上昇&降下タイプ」にしか飛行できません。じつはこのようなブーメランを作るほうが、正しく戻ってくるブーメランをつくるよりもたやすいのです。これについては4章で詳細説明します。

3.3 ブーメランの8の字運動(ホバリング)

3.3.1 左の図12は正しい絵ではありません。フックにつるされた5円玉は8の字を描くような運動はできません。しかしブーメランにはできます。図の5円玉の位置にあるブーメランはやや半丸の内側(つまり進行方向左側)に傾いた、だいたい5度程度傾いた姿勢で飛んでいます。このとき、揚力の90%は重力にさからうように働き、揚力の垂直成分の10%程度がブーメランの軌道をを内側にむけて曲げる内向力として働きます。するとブーメランは図の位置からフックの垂直下の(+)点へとまがりながら向かいます。このときに、さきほど3.1.4で述べた回頭運動をするので、ブーメランの回転軸は(+)点では垂直になり、結果ブーメランの翼面は完全に地面と水平になります。この際に、回頭が(+)点ですんなり止まらずに、そのままつんのめってブーメランは逆に右側へ傾くので、図12の右側の半分を飛ぶ間今度は進行方向に対して右に傾いた状態で、右旋回をしてふたたび(+)点へ戻ってきます。あとはこの繰り返しです。

3.3.2 ホバリング

3.3.2 正しく円軌道(0軌道)で戻ってくるブーメランにおいて、最初の3/4周、すなわち図18のAからB点までは、3.1.4で述べた円錐振り子の運動で、最後のBからCまでが3.3。1で述べた8の字運動の半分のS字運動でふらふらしながら戻ってきます。

3.3.3 上記の0軌道の場合に対して、ホンノ少しブーメランの行き脚が速くて、投げた人に近いところで、円錐運動から8の字運動に移行した場合には、図19(boo17)のように、投げ上げ者を通り越してから8の字運動をするので、全体の軌跡が8の字になります。


4.1 さて、話は3.14まで戻ります。図16の理想的な投げ方で投げられたブーメランが、理想的な円軌道を描いて飛んだとしましょう。図19において、投げた人の位置A点から、最も遠い所(最遠点)ににゆくまでに、軌道の垂線A−OとB−Oはほぼ90度(かそれよりやや多い程度)かわることになりますから、そのままその回転軸の変化が回頭に影響するとなると、ブーメランはB点に来たときにすでに翼面が水平になってしまいます。すると、図19のB点のように、ブーメランを引っ張る揚力の糸は真上に来るので、そこからブーメランは3.2.1で述べた振り子運動に入ってしまい、結果として図20のような軌跡で落ちてしまいます。ブーメランを作ったひとは必ずこのようなものを作るものです。本書は「どうやってここから脱却するか」において、他者の追随を許さない考察をしています。


5.1 さきほど、4.1で述べたように、垂直上昇するブーメランの方が、むしろ当たり前のものなのです。ではどうやって「垂直上昇」せずに「帰って来る」ブーメランを作るかというと、図19でA点からB点にゆくまでに、ブーメランが回頭するのをおくらしてやって、図18や図19のくらいまで、すなわち最遠点でなく、円を3/4から7/8程度描いた後で、翼面が水平にもどるようにしてやればよいのです。そういう効果を持ったものがブーメランには2種類用意されています。ここでは仮にお助けマン1とお助けマン2と呼びましょう。次の章で詳細紹介します。

5.1 お助けマンその1「翼の回頭抗力」


5.1.2 翼の回頭抗力は姿勢で変化する



5.1.3 この図22右の不利な姿勢を改善する方策は第一に全体の翼形を大型化してやればよいようなものですが、同時にそのままでは重くなってしまい、またそのぶん回転慣性が大きくなるので、回頭抗力が増加するのと同時に回頭力自体も増加してしまうので、もとのもくあみです。そのため、大型化しておいてから、回頭抗力に寄与しない、「不利姿勢での回頭軸」に近い部分、すなわち図23の斜線部分を削り取って(回頭抗力/回頭力)比率を大きくする翼形が必要です。そうですね。図23は、はやりの「アドミラルズハット:元帥帽子」の形状に近いですね。私の49号や50号もこの系列です。

5.1.4 翼の挟み角



5.2 お助けマンその2「翼上優先効果」





この翼端効果を得るには、図29のように左右の翼の前側の先端の裏側を削って、翼の端っこにおいて向かい角(polyheadral Undercut)をあたえるのが手っ取り早いです。





下記の平面形状の翼形を型紙にして両側の翼の端面の距離が290mmに成るように、シナベニアに鉛筆でなぞって、切り出してから、小刀で図4のような翼形になるように、全体の端面を斜めに削り取ってください。その後、120番程度の紙やすりで表裏も翼端もとにかくすべての面を滑らかにした後で、2倍程度に薄めた油性のニスを塗って、ニスが乾いた後で400番程度の細かい紙やすりで4回程度軽く表面をやすってなめらかにすれば出来上がりです。(330mmで50号機、240mmで 51号機になり、それなりに飛びます):風の無いところできれいに円を描いて戻ってきますが、風が少しでもあるとあおられて落っこちたり、とんでもないところへ、飛んでいってしまう、少々神経質なブーメランです。レーゲンスブルグの森の公園の林の中で飛ばしたらとっても気持ちよかったです。





仮説2: ブーメランの回転中心と重心には結構違いがあります。となれば、重心が回転中心よりも上に来るときは回転速度はおくれるでしょうし、その逆に下に来るときは回転が加速されるはず。であれば、ブーメランはくるくると等速でまわっているのではなくて、ぎくしゃくしながら回っており、これも上の現象をたすけているのではないか?なんぞと思いますが、私には証明のしようがありません。













13. 板の比重は0.52g/cm3ならOK.。0.67g/cm3では重すぎてだめ。経験則なのでその間は判りません。比重の重いブーメランはどうゆうふに削っても途中で落っこちてしまいます。板を買ってきたら、10cm角に切り取って重さを試しに量ってみましょう。2008.2.9本項記載追加


















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