Solve ODEs
(Ordinary Differential Equations)
Solve a 2nd-order ODE
Shown below is the MotionGenesis™ command file
to solve the ODE shown right with initial values of
x = 1 m and
x' = 0.2 m/s for
20 seconds and to
Output and Plot x(t) vs. t (with units), type:
Variable x'' = cos(2*t) + sin(x)
Input x = 1 m, x' = 0.2 m/s
Input tFinal = 20 sec, tStep = 0.1 sec
OutputPlot t seconds, x meters
ODE() MG2ndOrderODE
Note: The prime symbol ' denotes differentiation with respect to time t.
To auto-generate a MATLAB® .m file,
change ODE() MG2ndOrderODE
to ODE() MG2ndOrderODE.m
Solve coupled 1st-order ODEs
Shown below is the MotionGenesis™ command file
that solves ODEs for the angular velocity measures wx, wy, wz of a rotating rigid body.
Initial values and numerical integration parameters are entered with the Input command.
The OutputPlot command plots wx, wy, wz vs. time t.
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/s, wy = 7.0 rad/s, wz = 0.2 rad/s
Input tFinal = 4 sec, tStep = 0.02 sec
OutputPlot t sec, wx rad/sec, wy rad/sec, wz rad/sec
ODE() MGCoupledODEs
Note: The prime symbol ' denotes differentiation with respect to time t.
To auto-generate a MATLAB® .m file,
change ODE() MGCoupledODEs
to ODE() MGCoupledODEs.m
MotionGenesis™ forms and solves these dynamics equations here.
See 3D spin-stability, video, and Dzhanibekov effect here.
Solve coupled 2nd-order ODEs
Shown below is the MotionGenesis™ command file
that simulates a swinging-spring.
Initial values of y (stretch) and θ (swing angle) are entered in the Input command.
The OutputPlot command plots θ vs. t (time).
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() MGSwingingSpring
MATLAB®: To auto-generate the file MGSwingingSpring.m,
change ODE() MGSwingingSpring
to ODE() MGSwingingSpring.m
MotionGenesis™ forms and solves these dynamics equations here.
|
MotionGenesis™ generates fast compact symbolically-optimized codes to solve ODEs |
||
| Code | Command and code file | Comments |
| MATLAB® | ODE() MGSwingingSpring.m |
Modify input values in MGSwingingSpring.m
Interpreted .m codes are slower than compiled codes. |
|
C
Fortran |
ODE() MGSwingingSpring.c
ODE() MGSwingingSpring.f |
Compile and link source code.
Modify input values in MGSwingingSpring.in Compiled codes optimize for its host processor. |
| Depending on your license, these valuable versatile codes are independent of MotionGenesis™. | ||
Solve an ODE with on/off if-like statement
Shown below is the MotionGenesis™ command file
that solves the ODE: if( t < 𝛑 ) y' = sin(t) else y' = 0
isTLessThanPi = isPositive(pi-t) % true if t < 𝛑, else 0.
sintOrZero = sin(t) * isTLessThanPi % sin(t) if t < 𝛑, else 0.
Variable y' = sintOrZero
Input tFinal = 4, y = 0 % Arbitrary values.
Output t, y, y'
ODE() MGSimulateWithIfStatement
% Plot MGSimulateWithIfStatement.1[1, 2, 3]
Solve an ODE by auto-generating MATLAB® or C code and pass an argument to that code.
Shown below is the MotionGenesis™ command file
to auto-generate the MATLAB® file
MGProjectileMotion.m
that simulates projectile motion of a baseball.
The baseball's initial x (horizontal) and y (vertical) location are specified by the Input command.
The OutputPlot command instructs MATLAB® to plot y vs. x.
To run the .m file, start MATLAB® and 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™ forms dynamics equations.
To auto-generate MGProjectileMotion.c, use
ODE() MGProjectileMotion.c( b )
After compiling the C program, run from the operating system by typing:
MGProjectileMotion 0.05
Solve an ODE with various initial values.
Shown below is the MotionGenesis™ command file
to 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 ODE command).
Input y = 3 m % Set initial value for y (for ODE command).
%--------------------------------------------------------------------
% Set 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]
