RattleBack 0.0.2

motion of rattle-back

RattleBack: motion of rattle-back

RattleBack(abch, ABCD, init, t, sigma=0, g=981, **kw):
abch,ABCD = tuples of four real numbers, where
  a,b,c = radii of ellipsoid (cm)
  h = OG (O = center of ellipsoid, G = center of mass) (cm)
  A,B,C = moments of inertia along a,b,c axis (cm^2)
  D = product of intertia in a-b plane (cm^2)
init = initial condition of integration (shape (6,))
t = evaluation time (1d array) (sec)
sigma = friction coefficient (cm^2/sec)
g = gravity acceleration (cm/sec^2)
kw = keyword arguments passed to scipy.integrate.odeint
return y = output of odeint (shape(len(t),6)), where
  y[:,0:3] = euler angles alpha,beta,gamma
  y[:,3:6] = angular velocities omega_i (i=1,2,3)
all variables are measured in CGS units and radians
A,B,C,D,sigma corresponds to those of
  Kane & Levinson DIVIDED BY MASS of rattleback
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reference: T. R. Kane and D. A. Levinson
  "Realistic Mathematical Modeling of the Rattleback"
   International Journal of Non-Linear Mechanics 17 (1982) 175

example code:

import numpy as np
import matplotlib.pyplot as plt
from scipy.constants import degree
from RattleBack import RattleBack

abch = (20,3,2,1)
ABCD = (2,16,17,-0.2)
sigma = 1
init = [0.5*degree, 0.5*degree, 0,0,0,-5]
t = np.linspace(0,10,500)

y = RattleBack(abch, ABCD, init, t, sigma)
alpha,beta,gamma = y.T[:3] # Euler angles
delta = np.arccos(np.cos(alpha)*np.cos(beta)) # tilt angle

plt.figure(figsize=(5,5.5))

plt.subplot(211)
plt.plot(t, gamma/degree)
plt.ylabel(r'$\gamma$  / deg')

plt.subplot(212)
plt.plot(t, delta/degree)
plt.ylabel(r'$\delta$  / deg')
plt.xlabel(r'$t$ = time  / sec')
plt.tight_layout()
plt.show()