14
Nina del Ser - Mathematics
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The ratio of speeds is the only determining factor
At the beginning of this section it was mentioned that keeping the speed of the primary rotor fixed at 1 rad/s
would not in any way diminish the number of curves which could be generated. This is because the only
important factor is the ratio of the primary:secondary rotor angular velocities- not the velocities themselves. Β
and Ω are the angular velocities of the primary and secondary rotors respectively. Consider the following
figures:
figure 29: Β 5, Ω 35, T
figure 28: Β 1, Ω 7, T 2Π
2 Π
1
figure 30: Β , Ω 1, T 14Π
5
7
Despite travelling at different speeds, the rotors in all three illustrations share a common primary:secondary
rotor speed ratio of 1:7 and generate the same curve. This is not what we would expect from equation (6),
which suggests that the number of axes of symmetry corresponds to Ω. However, what we have not taken into
account is that the graphical display of time changes as the value of Β changes (when Β 1 the value of the
time and angle which displays it correspond exactly). Figures 31-32 explain the graphical significance of this:
figure 31: Β 1, Ω 7, T 2Π
t
figure 32: Β 2, Ω 14, T Π
2 Π
7
t 7
Π
In figure 32, from (6) there should be an axis of symmetry every
should occur every
2 Π
7
2 Π
Π
7
14
seconds, whereas in figure 31 this
seconds. However, since the seat sweeps out a circle twice as fast in figure 32, this will
“stretch” the time angle twice its size, so that the graphical representation of these axes of symmetry will be
equivalent in both cases. The value of T in figures 28-30 is the time when the full shape of the curve is
achieved (after which it starts to repeat itself).
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