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The aim of the investigation was to establish relations between length of head and tail of an unfinned arrow (for a crossbow or electromagnetic accelerator), and density of their materials. Lets look at this problem in more details. As it is known (for example, see the distance from nose to AFC must be at least twice the distance from nose to MC. Hereinafter we will consider an arrow with constant diameter along its length – this is the only type of arrow which can be used in coilguns (although crossbow bolts often have the tails which thicken to ends). In such situation MC is situated exactly in the middle of arrow. Suggesting the length of head to be l and the length of tail - k·l (i.e. the tail is k times longer than the head), and distance from MC to rear end of the head to be ∆, we will get following equation:
This is illustrated on picture below (hereinafter the tail and the head are simple cylinders). Besides, linear densities of materials of the head and tail are depicted as
To go further we must understand how to determine exact position of MC. Let us imagine some prolonged body, balancing on a point pivot (fig. 2). Variation of linear density of the body is here achieved by variation of its diameter (not its material), but it doesn’t change the concept.
Remember the school physics: the pivot will be in MC, if the summary force momentum from the body is zero. Mathematically it is written as: (2) Considering our mixed-materials case, this equation will be: (3a)
When MC is inside the head (i.e. (3b) Dividing on (4)
So, we have correlation which defines the position of MC in relation to back end of the head through the head’s length, and ratios of densities ( I.e., MC is inside the tail ( Now, substituting (4) to (1), we have following square inequation:
m-1)+1 ≤ 0 (5)As we know from the school mathematics, the condition of its real solutions is nonnegative discriminant, i.e. (1-3 This leads to (7) (the second solution m>1 is incidental, because the material of the tail is less dense than one of the head according to preconditions of our problem). Thus, we have an important conclusion: Solution of inequation (5) looks like range of values of arbitrary tail length: (8)
The origin of this range is intuitively clear – too short arrow (small The solution above is visualized in figure below:
Dashed line Thinking about application of these results in electromagnetic accelerators, we should obviously focus on the case of minimal length of arrow, as it ensures minimal «parasitic» mass of tail and, hence, maximum acceleration efficiency. Table below contains minimum length of stabilizer (in a form of The table provides us with some interesting and useful conclusions. First, it becomes clear that all sorts of wood can be utilized as solid cylindrical stabilizers (even as dense as beech). On the contrary, none of plastics can be used as solid stabilizer. Second, considering tubular stabilizers we should keep in mind their inside-to-outside diameter ratio (i.e. thickness of a tube). This ratio must be more than 0.8 for aluminum, which corresponds to standard 8/10 mm tubes widely spread in stores. At the same time, the ratio is worse (less) for thinner tubes, so we can conclude that standard aluminum tubes are not appropriate for stabilizing projectiles of less than 10-mm caliber. On can also note that archery and crossbow arrows use special thin-wall aluminum tubes (even finned one). As for plastic tubes, the restrictions on their thickness are quite softer. At last, as one can see from the table, | |

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