V Computer simulation: Results and Discussion
Many graphs have been produced from the extensive amount of numerical data created with the help of the simulation program. Each plot shall be analysed and explained. Each curve was obtained by varying one and only one variable at a time. When a few curves are present on the one graph, they all have the same variable but each curve has different constants as shown on each graph.
The diagram below illustrates the angle called the "starting angle" all the way through
Figure 1
Mc=30, Mp=0.5
Lc=0.321, Lp=Variable
Sa=-46
In figure 1 we can see the variation of the maximum range of the trebuchet with the length of the projectile arm. The constants are: Mc=30, Mp=0.5, Lc=0.321 and Sa= -46. It shows us that as the length of the arm increases, the maximum range first increases steeply then decreases a little less steeply. When the arm is very short, it turns very fast but the velocity of the projectile is low since the velocity of the projectile is calculated using the following equation:
is the rotational speed in radians per second and r is the distance from the centre of rotation to the object of which the velocity is being calculated - in this case the projectile. v is the linear velocity of the projectile. When r is very small, v is small, even if w is quite large. When the projectile arm length nears 1m, the maximum firing range stays similar for quite a large range of projectile arm lengths. This shows that there is no one 'best length' but a certain range of 'best lengths'. As the graph progresses, the range falls as the length increases. This is because the projectile arm is now so long that the counterweight arm can only just lift the projectile. At a certain point, the trebuchet stops moving, represented by a range of 0. It is interesting to note that the range does not fall slowly to 0 but falls to 0 abruptly. This is because when the trebuchet can only just lift the projectile, it has some time to accelerate, firing a certain distance. When the trebuchet can no longer lift the counterweight with the starting angle specified, the range falls to 0. It is possible the trebuchet could throw the projectile with a higher starting angle as acceleration depends on angle. Another interesting note is that the range does not depend on g, gravity. With a larger g the trebuchet exerts a bigger force on the projectile which leaves with greater velocity, but it falls faster because of greater g. The fascinating part is that these two effects counter compensate each other exactly. The program was tried with gravity on the moon (g=1.6 m/s2), on earth (g=9.8 m/s2) and on Jupiter (g=24.8 m/s2). The same graph resulted each time.
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© Filip Radlinski 1996, 1997