What causes a boomerang to return?
Boomerang is a rotor the blades spinning linear speed of which 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.
How many blades has traditional Australian aboriginal boomerang?
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.
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.
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
Full article can be found on my father’s website: http://mumris.eu/