MotionGenesis: ODE

MotionGenesis: F=ma Software, textbooks, training, consulting.

Solve ODEs (Ordinary Differential Equations)

Solve a nonlinear 2^nd-order ODE

To solve the ODE shown right with initial values of

x = 1 m and x' = 0.2 m/s, for t = 20 seconds, and to

Output and Plot various quantities (with units), type:

Variable x'' = cos(2*t) + sin(x)
Input    x = 1 m,   x' = 0.2 m/s,  tFinal = 20 sec     
OutputPlot  t sec,  x m,  x' m/s,  x'' km/s^2
ODE()  SecondOrderODE

Note: The prime symbol ' denotes differentiation with respect to t.
Commands may be entered from a text file, e.g., SecondOrderODE.txt.
Change SecondOrderODE to SecondOrderODE.m to auto-generate a MATLAB® .m file.

x'' = cos(2 * t) + sin(x)

Plot with MotionGenesis™ or MATLAB®.

The output file SecondOrderODE.1 stores x(t), etc.

Solve coupled nonlinear 1^st-order ODEs

Shown below are ODEs for the angular velocity measures wx, wy, wz
for the spinning-book (shown right) along with the commands to
Input initial values and numerical integration parameters and to
Output and Plot wx, wy, wz via the ODE command.

Constant  Ixx = 1 kg*m^2,  Iyy = 2 kg*m^2,  Izz = 3 kg*m^2
Variable  wx',  wy',  wz'
wx' = (Iyy-Izz) * wy * wz / Ixx
wy' = (Izz-Ixx) * wx * wz / Iyy
wz' = (Ixx-Iyy) * wx * wy / Izz
Input  wx = 0.2 rad/sec,   wy = 7.0 rad/sec,  wz = 0.2 rad/sec
Input  tFinal = 4 sec,   tStep = 0.02 sec
OutputPlot  t sec,  wx rad/sec,  wy rad/sec,  wz rad/sec
ODE() CoupledODESFirstOrder

Note: The prime symbol ' denotes differentiation with respect to t.
Note: Commands may be entered from a text file, e.g., CoupledODEs.txt.
Note: Change CoupledODEs to CoupledODEs.m to auto-generate a MATLAB® .m file.
Note: MotionGenesis™ forms and solves these dynamics equations here.
Note: Video and information about 3D spin-stability and Dzhanibekov effect here.

MotionGenesis: Solving coupled nonlinear first-order ODEs

Plot with MotionGenesis™ or MATLAB®.

The output file CoupledODEs.1 stores t, wx, wy, wz.

Solve coupled 2^nd-order ODEs

Shown below are the ODEs for y and theta that govern motions of the
swinging-spring (shown right) along with the MotionGenesis™ commands to
Input initial values and numerical integration parameters and to
Output and Plot theta vs. t via the ODE command.

Constant  m = 1 kg,  g = 9.8 m/s^2,  k = 100 N/m,  Ln = 0.3 m
Variable  y'',  theta''
y'' = g * cos(theta)  +  (Ln+y) * theta'^2  -  k*y/m
theta'' = -(g*sin(theta) + 2*theta'*y') / (Ln+y )
Input  y = 0.2 m,  y' = 0,  theta = 1 deg,  theta' = 0
Input  tFinal = 15 sec,  tStep = 0.02 sec
OutputPlot  t sec,  theta degrees 
ODE() SwingingSpring

Note: Commands may be entered from a text file, e.g., SwingingSpring.txt.
Note: Change SwingingSpring to SwingingSpring.m to auto-generate a MATLAB® .m file.
Note: MotionGenesis™ forms these dynamics equations here.

MotionGenesis: Solving coupled second-order nonlinear ODEs

Plot with MotionGenesis™ or MATLAB®.

The output file SwingingSpring.1 stores the solution.

MotionGenesis™ generates fast compact symbolically-optimized code to solve ODEs
Code	Command and code file	Comments
MATLAB®	ODE() SwingingSpring.m	Modify input values in SwingingSpring.m Note: Interpreted .m codes may be slower than compiled codes.
C Fortran	ODE() SwingingSpring.c ODE() SwingingSpring.f	Compile and link source code. Modify input values in SwingingSpring.in Note: Compiled codes can optimize for its host processor.

Depending on your license, these valuable versatile codes are independent of MotionGenesis™.

Solve an ODE with an on/off if-like statement

MotionGenesis™ commands to solve the ODE: if( t < 𝛑 ) y' = sin(t) else y' = 0

isTLessThanPi = isPositive(pi-t)      % true   if t < pi, otherwise 0 (false).
sintOrZero = sin(t) * isTLessThanPi   % sin(t) if t < pi, otherwise 0 (false).
Variable y' = sintOrZero
Input  tFinal = 4,  y = 0             % Arbitrary input values.
Output  t,  y,  y'                   
ODE()  simulateWithIfStatement
% Plot simulateWithIfStatement.1[1, 2, 3]

Solve ODE by auto-generating MATLAB® or C code and pass an argument to that program.

Shown below are the MotionGenesis™ commands to solve the
ODEs for x and y that govern projectile motion of a baseball.
Initial values are specified in the input command and the OutputPlot
command plots y vs. x after the ODE command is run.
These commands auto-generate the MATLAB® file MGProjectileMotion.m
To run from the MATLAB® prompt, type: MGProjectileMotion( 0.05 )

Constant  m = 145 grams,  g = 9.8 m/s^2,  b
Variable  x'' = -b*x'/m
Variable  y'' = -g - b*y'/m
%-------------------------------------------
Input  x = 0 m,  x' = 44.7*cosDegrees(30) m/s
Input  y = 0 m,  y' = 44.7*sinDegrees(30) m/s
Input  tFinal = 3.8 s
OutputPlot  x meter,  y meter
ODE() MGProjectileMotion.m(  b  )

MotionGenesis Projectile motion with air-resistance

Plot with MotionGenesis™ or MATLAB®.

Commands may be entered from a text file.
MotionGenesis™ forms these dynamics equations here.


Note:	To pass b, to the C-program MGProjectileMotion.c, type:
	ODE() MGProjectileMotion.c( b )

	After compiling, run from the operating system prompt by typing:
	MGProjectileMotion 0.05

Solve an ODE with various initial values

Variable  y' = cos(t) + sin(y)  % Arbitrary ODE.
Output  t sec,  y m,  y' m/s    % List output quantities  (for subsequent ODE command).
Input   y = 3 m                 % Set initial value for y (for subsequent ODE command).
%--------------------------------------------------------------------
%   Set numerical integration parameters and solve the ODE with initial value y = 3.
Input  tFinal = 5 sec,  tStep = 0.02 sec,  absError = 1.0E-07
ODE()  ODEWithInitialValueA     % Solve the ODE and output results.
%--------------------------------------------------------------------
Input  y := 0 m                 % Modify the initial value for y.
ODE()  ODEWithInitialValueB     % Solve the ODE and output results.
%--------------------------------------------------------------------
%   Optional: Plot results
Plot ODEWithInitialValueA.1 [1,2]  ODEWithInitialValueB.1 [1,2]

Commands may be entered from a text file, e.g.,
MGSolveODEWithVariousInitialValues.txt.