By slinging - throwing a ball or spear or jevelin or hammer or Davidīs catapult - missiles can be thrown very far. Itīs not only the longer lever arm, that might bring that effect. The longer a catapult-arm or sling-rope, the more effectiv than it should be. Pratize however does show, also relative short lever arms do show enormous effects (like short sticks to accelerate lances).
Also, one must not turn around a sling above head many times. Thatīs but good for getting higher starting speed. Decisive however, for example at diskus-throwing or hammer-throwing, will be the īpullī at last turn. The essential effect thus must be based at that swinging-out of mass.
That problem, Dr.Habbel did put into discussion (and no physican did achieve sufficient explanations for him), in the following will be examined. I can do that but by geometric means. I will do this that kind, these conciderations about outward-throwing of mass can be transfered to rotor-systems.
A first turn will go around the body of the throwing man, here called system-axis (SA). The arm of the throwing man will be looked at to be stiff (elbow joint fixed), here called rotor beam (RT, German īRotorträgerī, like a rotating arm or disk, which can bear rotating parts). At the beginning this rotor arm will show to right side. Around the hand of the throwing man, here called rotor axis (RA), the catapult (for example a rope with a stone embedded) can do a second turning motion. This sling-unit will be called rotor (RO). The track of a point of mass (MP) at the end of will be looked at. So, this sling-unit is comparable with a mass-point at the outside (longest radius) of a rotor-disk, which is installed turnable at a rotor-beam. The rotor (RO) here shall be two third as long than the rotor beam (RT). In order to see the throuwing-out phase of mass, here the rotor does show to left side at the beginning, into direction to the system axis. The drawing here does show the positions of the rotor beam after each turn of 15 degrees around the system axis, turning clockwise. A throwing man will accelerate his arm, these sections thus will be done each in shorter time. Here, the rotor system is assumed to turn constant speed.
Sling - curve
When the rotor beam (RT) from its null-position (showing to the right) did turn 15 degrees, the mass-point (thick black point) has to move only little distance outward-forward. Corresponding to its speed now achieved, it now shows inertia into that direction. On the other hand, it next has to follow next position of the rotor axis.
The inertia-component, here ist drawn as dotted line from one position of the mass point to its next position. Same direction and same length it is extended (best to see left side up). The actual inertia will respectivly would like to transport the mass to the end of that line. On the other hand, the mass all times must again be on same radius from the rotor axis. These possible positions are marked by dotted segment of a cirle around next position of the rotor axis.
As next direction of the rotor-radius here looked at (blue thin line), here was taken the line between rotor axis and the end of the inertia-line. At this line, next position of mass-point will be.
The motion of a mass-point thus will be build, first by a movement corresponding to its inertia-direction and -value and second by a movement caused by turning of the rotor beam. The work to be done by the system, thus is the distance between end of the inertia-line and next position of the mass-point (here marked by thick red line).
That work, at the beginning is few: inside there are only little distances to walk. In addition, from that motion only the component normal (90 degrees) to the radius must be done by power. Full power thus the system has to bring, when the rotor beam did nearby 120 degrees turn. Afterwards, rotor-beam and roto-radius do show angels with more than 90 degrees, will say a power-component will pull at the system-axis, the other power-component must be done by the rotor-beam (now at a longer lever arm).
Now one may see, after 180 degrees turning of the rotor beam, the rotor did but turn about 133 degrees around its rotor-axis. Above that, next 15 degrees turn of the rotor-beam will bring highest speed of the mass (green thick line). Further turning of the rotor beam will bring no more acceleration of speed. The rotor wonīt go further ahead (like the end of a whip hanging backwards). So, here it would make sence, the throwing man would release the missile (thick green arrow).
Itīs absolutly remarkable, that here the speed of the mass is much higher than the speed of the rotor-beam (green dotted lines): while the rotor-beam will turn 15 degrees, the mass will show an angle-speed of about 20 degrees, so one third higher speed - at nearby its largest distance to the system axis!
Power
A throwing man, thus in principle, muß turn the sling-unit some more than 180 degrees (by this relation of length of rotor-beam and rotor-radius). At the beginning, the motion of missile will cost but low power, will say the man can invest his power in acceleration of his arm. Nearby 120 degree-turning, for a short time he has to bring īfull pullī. Afterwards he just has to direct the missile in direction of the aim (for example als by moving the īsystem-axisī as hammer-throwers do).
The resulting speed in degrees than is some one third higher than the speed of his hand. So, thatīs higher speed than a longer lever arm by itself could achieve.
That sectionwise detection of positions of a mass-point, in reality will show an track like smoothed curve (blue bended line). No, itīs interesting to compare that curve with a curve produced by a rotor-system with determined gear, so with fixed corelation of turning speed of rotor-beam and rotor-radius.
Here, again the rotor-beam (RT, only three positions are marked) will turn 180 degrees around the system-axis (SA). Same time, the rotor (RO) will turn around its rotor-axis 180 degrees (black thin lines). In sum, the mass-point looked at, at the end will show to left side. The track of that mass-point (thick black curve) does show the typical loop inside the rotor-beams-radius and the wide swinging outside of.
Conspicuous now will be, the track of the free pendulum (the sling-unit above, thick blue curve) is relativly flat. Opposite, the track of the rotor-system with fixed gear, will transport outside the mass much faster. This work must be done by system-power - while at the free pendulum that job is done by centrifugal force automatically. So this rotor-system is not optimal, the rotor does turn too fast.
Felix Würth by his genius of an experimentator, did clearly recognize: such rotors have to turn slower, must be a quater or a fifth ībackturningī - as he called it. The red lines here, now do show but 11 degrees turning of the rotor (while the rotor-beam turns 15 degrees). You easyly can see, that gear (red line) comes nearby the ideal sling-curve (blue curve). Here, the rotor is not forced to take it outmost position alread after 180 degree-turning of the rotor-beam, but will achieve that position later (for example after 240 degrees at 1/4 slower rotor-turning).
This will say, with such a rotor-system the advantages of sling-effect can be used: with a minimum of power-input, much higher kinetic energy will be achieved at the outside track-point. Felix Würth with his maschines takes off that surplus of energy by slowing-down the system.
This surplus of energy does result of skillful usage of inertia, which partly may go as it likes to, so an addition of pull-power and inertia-power may come true - as shown next chapter in detail.
Evert / 20.11.1999
Crop Circle Sitemap