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[Analog_Engine.git] / scripts / TP1 09.27 Bead on Cycloid.LACE
1 # A bead glides frictionless on a wire that has the shape of a cycloid
2 # g is the gravitational acceleration 9,81 m/s²
3 # a is the radius of the rolling circle (see Bronstein/Semendjajew p. 91)
4 # equation of motion:
5 # mu'' = - g/4a * mu, with mu = sin(phi/2) and phi a parameter of the cycloid
6
7 coefficient(1): g/4a
8 coefficient(2): 1 -> mu0'
9 coefficient(3): -1 -> -mu0
10 coefficient(4): 4ax
11 coefficient(5): 4ay # same as 4ax
12
13 iintegrate mu'' -> -mu'
14 IC: mu0'
15 iintegrate -mu' -> mu
16 IC: -mu0
17 invert mu -> -mu
18 -mu * g/4a -> -g/4a*mu
19 mu'' = -g/4a*mu
20
21 # the following is for displaying the cycloid in x-y space
22 # calculating x (NB: this includes some unacceptable approximations)
23 mu * 4ax -> 4a*mu
24 output(x): 4a*mu
25
26 # calculating y
27 multiply mu, mu -> mu^2
28 mu^2 * 4ay -> 4ay*mu^2
29 isum 4a*mu^2 -> -2a*mu^2 # just serves to devide by 2 because we need 2a instead of 4a
30 /2
31 invert -2a*mu^2 -> 2a*mu^2
32 output(y): 2a*mu^2
33
34 # display mu, so the sinus
35 output(z): mu