% MotionGenesis file:  MGMetronomePendulumEulerLagrangeKane.txt
% Problem:  Equation of motion for two-particle metronome.
% Copyright (c) 2017 Motion Genesis LLC.  All rights reserved.
%--------------------------------------------------------------------
NewtonianFrame N             % Earth with point No at metronome pivot.
RigidFrame     B             % Metronome rod.
Particle       Q1, Q2        % Particles fixed at distal ends of B.
%--------------------------------------------------------------------
Variable  theta''            % Metronome angle and 1st/2nd derivative.
Constant  L = 0.2 m          % Distance between point No and Q1.
Constant  h = 0.1456 m       % Distance between point No and Q2.
Constant  g = 9.8 m/s^2      % Earth's gravitational acceleration.
Constant  k = 2 N*m/rad      % Torsional spring constant.
Q1.SetMass( m1 = 0.1 kg )
Q2.SetMass( m2 = 0.02 kg )
%--------------------------------------------------------------------
%       Rotational and translational kinematics.
B.RotateNegativeZ( N, theta )
Q1.Translate( No, -L*By> )
Q2.Translate( No,  h*By> )
%--------------------------------------------------------------------
%       Relevant forces and torques.
System.AddForceGravity( -g*Ny> )
B.AddTorque( k*theta*Bz> )
%--------------------------------------------------------------------
%       Euler's equation of motion (angular momentum principle about No).
Euler = Dot( System.GetDynamics(No), -Bz> )
Factor( Euler,  theta'', g )
%--------------------------------------------------------------------
%       Conservation of mechanical energy.
KineticEnergy = System.GetKineticEnergy()
PotentialEnergyGravity = System.GetForceGravityPotentialEnergy( -g*Ny>, No )
PotentialEnergySpring = 1/2 * k * theta^2
PotentialEnergy = PotentialEnergyGravity + PotentialEnergySpring
MechanicalEnergy = KineticEnergy + PotentialEnergy
%--------------------------------------------------------------------
%       Lagrange's equation of motion.
SetGeneralizedCoordinate( theta )
Lagrange = System.GetDynamicsLagrange( SystemPotential = PotentialEnergy )
Factor( Lagrange,  theta'', g )
%--------------------------------------------------------------------
%       Kane's equation of motion.
SetGeneralizedSpeed( theta' )
Kane = System.GetDynamicsKane()
Factor( Kane,  theta'', g )
%--------------------------------------------------------------------
%       Save input and program responses.
Save MGMetronomePendulumEulerLagrangeKane.html
Quit