new_contents: #!/usr/bin/env python
  # -*- coding: utf-8 -*-
  """Exercises 8.12, 8.17 from Kane 1985.
  """
  
  from __future__ import division
  from sympy import simplify, solve, symbols
  from sympy.physics.mechanics import ReferenceFrame, Point
  from sympy.physics.mechanics import inertia, RigidBody
  from sympy.physics.mechanics import cross, dot, dynamicsymbols
  from util import msprint, subs, partial_velocities
  from util import generalized_active_forces, generalized_inertia_forces
  
  
  print("\nEx8.12")
  ## --- Declare symbols ---
  q1 = dynamicsymbols('q1')
  u1, u2, u3 = dynamicsymbols('u1:4')
  u_prime, R, M, g, e, f, theta = symbols('u\' R, M, g, e, f, theta')
  Q1, Q2, Q3 = symbols('Q1, Q2 Q3')
  
  # --- Reference Frames ---
  F = ReferenceFrame('F')
  P = F.orientnew('P', 'axis', [-theta, F.y])
  A = P.orientnew('A', 'axis', [q1, P.x])
  A.set_ang_vel(F, u1*A.x + u3*A.z)
  
  ## --- define points D, S*, Q on frame A and their velocities ---
  pD = Point('D')
  pD.set_vel(A, 0)
  # u3 will not change v_D_F since wheels are still assumed to roll without slip.
  pD.set_vel(F, u2 * A.y)
  
  pS_star = pD.locatenew('S*', e*A.y)
  pQ = pD.locatenew('Q', f*A.y - R*A.x)
  for p in [pS_star, pQ]:
      p.set_vel(A, 0)
      p.v2pt_theory(pD, F, A)
  
  ## --- define partial velocities ---
  partials = partial_velocities([pD, pS_star, pQ], [u1, u2, u3], F, express_frame=A)
  
  forces = [(pS_star, -M*g*F.x), (pQ, Q1*A.x + Q2*A.y + Q3*A.z)]
  torques = []
  Fr, _ = generalized_active_forces(partials, forces + torques, uaux=[u3])
  print("Generalized active forces:")
  for i, f in enumerate(Fr, 1):
      print("F{0} = {1}".format(i, msprint(f)))
  
  print("\nEx8.17")
  ### --- define new symbols ---
  a, b, mA, mB, IA, J, K, t = symbols('a b mA mB IA J K t')
  IA22, IA23, IA33 = symbols('IA22 IA23 IA33')
  q2, q3 = dynamicsymbols('q2 q3')
  q2d, q3d = dynamicsymbols('q2 q3', level=1)
  
  # define frames for wheels
  B = A.orientnew('B', 'axis', [q2, A.z])
  C = A.orientnew('C', 'axis', [q3, A.z])
  
  # masscenters of bodies A, B, C
  pA_star = pD.locatenew('A*', a*A.y)
  pB_star = pD.locatenew('B*', b*A.z)
  pC_star = pD.locatenew('C*', -b*A.z)
  for p in [pA_star, pB_star, pC_star]:
      p.set_vel(A, 0)
      p.v2pt_theory(pD, F, A)
  
  # points of B, C touching the plane P
  pB_hat = pB_star.locatenew('B^', -R*A.x)
  pC_hat = pC_star.locatenew('C^', -R*A.x)
  pB_hat.set_vel(B, 0)
  pC_hat.set_vel(C, 0)
  pB_hat.v2pt_theory(pB_star, F, B)
  pC_hat.v2pt_theory(pC_star, F, C)
  
  # the velocities of B^, C^ are zero since B, C are assumed to roll without slip
  #kde = [dot(p.vel(F), b) for b in A for p in [pB_hat, pC_hat]]
  kde = [dot(p.vel(F), A.y) for p in [pB_hat, pC_hat]]
  kde_map = solve(kde, [q2d, q3d])
  # need to add q2'', q3'' terms manually since subs does not replace 
  # Derivative(q(t), t, t) with Derivative(Derivative(q(t), t))
  for k, v in kde_map.items():
      kde_map[k.diff(t)] = v.diff(t)
  
  # inertias of bodies A, B, C
  # IA22, IA23, IA33 are not specified in the problem statement, but are
  # necessary to define an inertia object. Although the values of
  # IA22, IA23, IA33 are not known in terms of the variables given in the
  # problem statement, they do not appear in the general inertia terms.
  inertia_A = inertia(A, IA, IA22, IA33, 0, IA23, 0)
  inertia_B = inertia(B, K, K, J)
  inertia_C = inertia(C, K, K, J)
  
  # define the rigid bodies A, B, C
  rbA = RigidBody('rbA', pA_star, A, mA, (inertia_A, pA_star))
  rbB = RigidBody('rbB', pB_star, B, mB, (inertia_B, pB_star))
  rbC = RigidBody('rbC', pC_star, C, mB, (inertia_C, pC_star))
  bodies = [rbA, rbB, rbC]
  
  # create new partial velocities since we define new points A*, B*, C* and
  # new frames B, C
  system = []
  for b in bodies:
      system += [b.masscenter, b.frame]
  partials2 = partial_velocities(system, [u1, u2, u3], F, kde_map, express_frame=A)
  
  Fr_star, _ = generalized_inertia_forces(partials2, bodies, kde_map, uaux=[u3])
  print("Generalized inertia forces:")
  for i, f in enumerate(Fr_star, 1):
      print("F{0} = {1}".format(i, msprint(simplify(f))))
  
subject: Add Exercises 8.12, 8.17.