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