Nina del Ser - Mathematics
10
| octopus
Changing the angular speeds of the rotors
The same method as that used to solve the original problem (see page 2) may be generalised in order to take
into account changing the ratios of the angular speeds of the primary and secondary rotors. We will simplify
the task further by keeping the speed of the first rotor fixed at 1 rad/s anticlockwise and change only the
angular speed of the secondary rotor (just like for the ratio of lengths, this will not diminish the range of
possible ratios in any way). Therefore our generalised set of parametric equations becomes
r x cos t bcos 1 Ω t
r y cos t bsin 1 Ω t
(5)
where Ω is the angular speed, in rad/s, of the secondary rotor (the offset angle Α has been removed from the
equation because its importance has been shown to be secondary).
Simple cases
In this small interlude we will look for any simple cases, where the figure generated by changing Ω is rela-
tively obvious. The figures below display the curves produced when Ω -1 and Ω 0.
figure 7: Ω 1
figure 8: Ω 0
These shapes are easily explained simply by looking at the parametric equations which define them. The
parametric function which defines figure 7, a circle with radius 1 and centre (b,0) is r i(cost+b)+jsint, while
figure 8 is defined by r 1.5icost+1.5jsint, simply a circle centered about the origin with radius 1.5.
Integer values of Ω
In the following section we shall try to investigate and find patterns for what happens when the angular speed
of the secondary rotor is an integer number.
Both rotors spin anticlockwise
The following images are some of the curves generated by the octopus when both rotors spin in the same
direction (anticlockwise):
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