MGSolveNonlinearEqnContinuouslyA.html

MotionGenesis input/output:

   (1) % MotionGenesis file: MGSolveNonlinearEqnContinuouslyA.txt
   (2) % Copyright (c) 2024 Motion Genesis LLC.
   (3) % Purpose: Continuous solution to the nonlinear algebraic equation:
   (4) %          y^4 - 8*y - 0.3*t^2 - 9*sin(t) = 0
   (5) % Related: MGSolveNonlinearEqnContinuouslyA.txt
   (6) %          MGSolveNonlinearEqnsContinuouslyB.txt
   (7) %          MGSolveNonlinearEqnsContinuouslyC.txt
   (8) %          MGFigureEightLinkageInverseKinematics.txt
   (9) %---------------------------------------------------------------------------
   (10) %   For a continuous solution y(t), declare y and its 1st time-derivative.
   (11) Variable  y'
   (12) eqnToSolve = y^4 - 8*y - 0.3*t^2 - 9*sin(t)
-> (13) eqnToSolve = y^4 - 0.3*t^2 - 9*sin(t) - 8*y

   (14) %---------------------------------------------------------------------------
   (15) %   Differentiate eqnToSolve to form an equation that is _linear_ in y'.
   (16) eqnDt = Dt( eqnToSolve )
-> (17) eqnDt = 4*y^3*y' - 0.6*t - 9*cos(t) - 8*y'

   (18) %---------------------------------------------------------------------------
   (19) %   Calculate the values of y that satisfy eqnToSolve at t = 0.
   (20) yValuesAt0 = GetQuarticRoots( Evaluate(eqnToSolve, t=0) = 0,  y )
-> (21) yValuesAt0 = [-1 - 1.732051*imaginary;  -1 + 1.732051*imaginary;  -1.721328E-18;  2]

   (22) %---------------------------------------------------------------------------
   (23) %   Solve eqnDt = 0 to form an ODE (ordinary differential equation) for y'.
   (24) Solve( eqnDt = 0,   y' )
-> (25) y' = 0.15*(t+15*cos(t))/(-2+y^3)

   (26) %---------------------------------------------------------------------------
   (27) %   Set the initial value for y from one of the quadratic roots at t = 0.
   (28) Input  y = yValuesAt0[4] meters
   (29) %---------------------------------------------------------------------------
   (30) %   List output quantities (for subsequent ODE command).
   (31) Output t seconds,  y meters
   (32) %---------------------------------------------------------------------------
   (33) %   Set numerical integration parameters, solve ODEs, and plot results.
   (34) Input  tFinal = 20 seconds,  tStep = 0.02 sec,  absError = 1.0E-07
   (35) ODE() MGSolveNonlinearEqnContinuouslyA

   (36) Plot  MGSolveNonlinearEqnContinuouslyA.1
   (37) %--------------------------------------------------------------------
   (38) %   Record input together with responses.