(1) % MotionGenesis file: ParticlePendulumKane.txt
   (2) % Problem: Particle pendulum with air resistance.
   (3) % Copyright (c) 2009 Motion Genesis LLC.  All rights reserved.
   (4) %------------------------------------------------------------
   (5) NewtonianFrame N             % Earth.
   (6) RigidFrame     B             % Massless cable.
   (7) Particle       Q             % Particle at end of cable.
   (8) %------------------------------------------------------------
   (9) Q.SetMass( m = 100 kg )
   (10) Constant  g = 9.8 m/s^2      % Earth's gravitational acceleration.
   (11) Constant  L = 50 m           % Cable length.
   (12) Constant  bAir = 2.0 N*s/m   % Air-resistance constant.
   (13) Variable  theta''            % Pendulum swing angle.
   (14) Variable  Tension            % Tension in cable.
   (15) %------------------------------------------------------------
   (16) %   Rotational and translational kinematics.
   (17) B.RotateZ( N, theta )
-> (18) B_N = [cos(theta), sin(theta), 0;  -sin(theta), cos(theta), 0;  0, 0, 1]
-> (19) w_B_N> = theta'*Bz>
-> (20) alf_B_N> = theta''*Bz>

   (21) Q.Translate( No,  -L*By> )
-> (22) p_No_Q> = -L*By>
-> (23) v_Q_N> = L*theta'*Bx>
-> (24) a_Q_N> = L*theta''*Bx> + L*theta'^2*By>

   (25) %------------------------------------------------------------
   (26) %   Contact and distance forces on Q.
   (27) airResistance> = -bAir * Q.GetVelocity(N)
-> (28) airResistance> = -bAir*L*theta'*Bx>

   (29) Q.AddForce( -m*g*Ny> + Tension*By> + airResistance> )
-> (30) Force_Q> = -bAir*L*theta'*Bx> + Tension*By> - m*g*Ny>

   (31) %------------------------------------------------------------
   (32) %   Form equations of motion with  Kane's method.
   (33) %   Advantage: Tension is a "constraint force", automatically eliminated by Kane's method.
   (34) SetGeneralizedSpeed( theta' )
   (35) KaneDynamics = System.GetDynamicsKane()
-> (36) KaneDynamics = [L*(m*g*sin(theta)+bAir*L*theta'+m*L*theta'')]

   (37) Solve( KaneDynamics = 0,   theta'' )
-> (38) theta'' = -(m*g*sin(theta)+bAir*L*theta')/(m*L)

   (39) %------------------------------------------------------------
   (40) %   Initial values and numerical integration parameters.
   (41) Input   theta = 60 deg,  theta' = 0 deg/sec
   (42) Input   tFinal = 120 sec,  tStep = 0.2 sec
   (43) %------------------------------------------------------------
   (44) %   List output quantities and solve ODEs.
   (45) Output  t sec,  theta degrees
   (46) ODE() ParticlePendulumKane

   (47) %------------------------------------------------------------