### Boomerang model

Boomerang is a rotor the blades spinning linear speed of wich is low, if compared it to the translation speed of the whole boomerang. It means that the slow spin cannot produce a lift force strong enough  to keep boomerang in air like helicopter blades do. The lift energy is mainly obtained from transitional motion of boomerang. On the other hand the spin forces boomerang to act as a gyroscope which is affected by forces produced by blades. The blades shape and their orientation make the boomerang to return. Here I present a model which explains why and how boomerangs return.

Ref

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Right answer is four. According to the boomerang model (see figure above) there are two types of boomerang blades: radial and arc.

Spinning around its center of mass, a boomerang activates radial and passive blades as shown in pictures below.

The blades in active position interact with air and produce forces the angular momentum of which determines two types of Boomerang gyroscopic precessions.

First type boomerang precession
First type boomerang precession is well known to be responsible of  boomerang return (all pictures are presented for right hand boomerang).

When the boomerang is thrust outwardly in a vertical position, it spins with angular speed w and flies with speed v. Radial blades produce well-known lift force, which affects spinning boomerang at positions, when air flow gets in front of them. The lift forces have different values for forward and backward spinning radial blades, as boomerang is moving onward. The difference of lift forces has angular momentum, applied to rotation axis, and provides the first type gyroscopic precession of spinning boomerang around Z-axis with angular speed  Omega1. This first type gyroscopic precession is well known and is responsible for boomerang's turning backwards.

Second type boomerang precession
Second type boomerang procession is responsible for keeping a boomerang in the air as long as possible and determines its trajectory shape (pattern). Usually arc blades are smaller and have a little lift which by itself cannot hold a boomerang in the air. When the boomerang is thrust outwardly almost in a vertical position it has big speed, but small vertical component of lift force. While speed decreases, the vertical lift force component also decreases. It means that after some time boomerang begins to fall down. The main function of arc blades is to continuously rotate spinning plane in such way which keeps constant vertical component of the lift force (which mainly is produced by radial blades). The second type precession does the job.

The boomerang's arc blades are perpendicularly attached to the end of the radial blades. Arc blades have either negative dihedral and/or negative angle of attack or positive dihedral and/or positive angle of attack. The arc blades become active when spinning boomerang gets a position, in which radial blades are parallel to the direction of flight. The lift force generated by said arc blades results in a second type gyroscopic precession around axis X with angular speed  Omega2. Arc blades dihedral and/or angle of attack define precession  Omega2 sign and value.

 This type of precession is discussed in aeronautical engineer Alan Adler's scientific paper about the ring's flight. Actually, the flying ring can be considered as a construction, having just arc blades (stabilizers) with absent of radial blades. The paper shows that the arc blade dihedral and attack angle of stabilizer can be constructed in different ways to have either positive or  negative angular moment - nose up or nose down. When construction spins, the angular moment determines precession Omega2 direction and value. Lift of leading and trailing half-rings Vs  for a conical angled ring

OK, lets see how it really works.

First movie
shows corkscrew fly pattern of flying ring with leading edge which is orientated at negative angle of attack.

Second movie
shows the fly pattern of the same ring orientated at positive angle of attack (I just turn it in my hand 180 degrees). Note that precession direction is changed from counterclockwise to clockwise.

Returning Boomerang relation. Joining First and Second types of precessions.

 First type of precession  Omega1 Second type of precession  Omega2 Total "8" type precession + =

The flight trajectory (flight pattern) depends on "8" type boomerang precession and, thus on Omega1 and Omega2 ratio.

Both, first and second type gyroscopic precessions direct boomerang to return backwards close to its launching position. Boomerang relation  Omega2= K * Omega1 describes best condition for boomerang to return, where K range is about 1/3...1/4. The coefficient K=1/3 gives "8" type of flight pattern, the coefficient K=1/4 gives "O" type flight pattern. If K<<1/4 boomerang remains orientated vertically too long (lift force is orientated along fast spin axis) and falls down with ballistic trajectory (almost like a stone). Then it rolls on ground making short arc. If K>>1/3 boomerang flies forward with corkscrew trajectory and also does not return.
When wind speed is zero, "8" (K=1/3) type pattern has landing point in behind of launching point, while "O" (K =1/4) type pattern has landing point in front of launching point. Some in-between K values give intermediate flight patterns. Wind speed shifts the landing point. So the said boomerang has optimum return pattern for fixed speed of wind, as the coefficient K for particular boomerang is constant.

DESCRIPTION OF THE FLIGHT PATTERNS

Description of the "8" type of flight pattern.
When the boomerang is thrust outwardly in a vertical position, it will create and maintain a forward flight pattern from the point of launching; it subsequently begins an upward flight until it momentarily stalls in the air, then makes a 180 degree turn to the left, reverses its forward flight and simultaneously rotates 90 degrees from vertical, taking on a horizontal, hovering type of flight pattern, and returns closely to the point of launching, passes it and slowly flies second small loop to the opposite side. Total flight pattern looks like asymmetrical "8".

"8" type of flight pattern. Top view. See the videos:

and

Description of the circular "O" type of flight pattern.
When the boomerang is thrust outwardly in a vertical position, it will create and maintain a forward flight pattern from the point of launching; it subsequently begins a horizontal flight making a 270-360 degree turn to the left, reverses its forward flight and simultaneously rotates 90 degrees from vertical, taking on a horizontal, hovering type of flight pattern, and returns to the point of launching, passes it and slowly continues the second small loop to the same side.

Total flight pattern is helix type and looks like "O". Actually it is the same "8" pattern with large asymmetry: big first loop and very delayed small second loop of "8" which in most cases is not seen, as boomerang is landing before making it. For most cases "O" type flight pattern is preferable to "8" type one, as it has a feature of very slow hovering landing, thus making it relative safe to the thrower.

"O" type of flight pattern. Top view. See the videos:

and