Applications: Longitudinal oscillating chains; PDEs

[Analog_Engine.git] / scripts / TP1 09.27 Bead on Cycloid.LACE

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+# A bead glides frictionless on a wire that has the shape of a cycloid 
+# g is the gravitational acceleration 9,81 m/s²
+# a is the radius of the rolling circle (see Bronstein/Semendjajew p. 91)
+# equation of motion:
+#   mu'' = - g/4a * mu, with mu = sin(phi/2) and phi a parameter of the cycloid 
+
+coefficient(1): g/4a
+coefficient(2): 1 -> mu0'
+coefficient(3): -1 -> -mu0
+coefficient(4): 4ax
+coefficient(5): 4ay # same as 4ax
+
+iintegrate mu'' -> -mu'
+   IC: mu0'
+iintegrate -mu' -> mu
+   IC: -mu0
+invert mu -> -mu
+-mu * g/4a -> -g/4a*mu
+mu'' = -g/4a*mu
+
+# the following is for displaying the cycloid in x-y space
+# calculating x (NB: this includes some unacceptable approximations)
+mu * 4ax -> 4a*mu
+output(x): 4a*mu
+
+# calculating y
+multiply mu, mu -> mu^2
+mu^2 * 4ay -> 4ay*mu^2
+isum 4a*mu^2 -> -2a*mu^2     # just serves to devide by 2 because we need 2a instead of 4a
+  /2
+invert -2a*mu^2 -> 2a*mu^2 
+output(y): 2a*mu^2
+
+# display mu, so the sinus
+output(z): mu