Ordinary Differential Equations (advanced)
Solve an ODE by auto-generating MATLAB® or C code and pass an argument to that code.
The MotionGenesis™ command file below
auto-generates the MATLAB® file
MGProjectileMotion.m
that solves ODES() for a baseball's projectile motion.
The command OutputPlot instructs MATLAB® to plot y vs. x.
Constant m = 145 grams, g = 9.8 m/s^2, b
Variable x'' = -b*x'/m % Horizontal location.
Variable y'' = -g - b*y'/m % Vertical location
%----------------------------------------------------------
% Set initial values for variables.
Input x = 0 m, x' = 44.7*cosDegrees(30) m/s
Input y = 0 m, y' = 44.7*sinDegrees(30) m/s
%----------------------------------------------------------
% List output quantities (for ODE command and plotting).
OutputPlot x meter, y meter
%----------------------------------------------------------
% Set numerical integration parameters.
Input tFinal = 3.8 sec, tStep = 0.02 sec, absError = 1.0E-7
%----------------------------------------------------------
% Auto-generate MATLAB code (or C, Fortran, etc.).
ODE() MGProjectileMotion.m( b ) % Pass the value of b.
To run the .m file,
start MATLAB® and type MGProjectileMotion( 0.05 )
To instead auto-generate the C code MGProjectileMotion.c,
use ODE() MGProjectileMotion.c( b )
After compiling the C program, run from the operating system by typing:
MGProjectileMotion 0.05
Solve coupled 2nd-order ODEs
The MotionGenesis™ command file below
solves ODEs for a swinging-spring.
The command OutputPlot causes ODE() to
Output data and Plot θ vs. t (time).
Constant m = 1 kg, g = 9.8 m/s^2, k = 100 N/m, Ln = 0.3 m
Variable y'' % Spring stretch, 1st & 2nd derivatives.
Variable theta'' % Swing angle, 1st & 2nd derivatives.
y'' = g * cos(theta) + (Ln+y) * theta'^2 - k*y/m
theta'' = -(g*sin(theta) + 2*theta'*y') / (Ln+y )
%----------------------------------------------------------
% Set initial values for variables.
Input y = 0.2 m, y' = 0, theta = 1 deg, theta' = 0
%----------------------------------------------------------
% List output quantities (for ODE command and plotting).
OutputPlot t sec, theta degrees
%----------------------------------------------------------
% Set numerical integration parameters.
Input tFinal = 15 sec, tStep = 0.02 sec, absError = 1.0E-7
%----------------------------------------------------------
ODE() MGSwingingSpring
Optional: To instead auto-generate a MATLAB® .m file,
change the last line to ODE() MGSwingingSpring.m
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MotionGenesis™ generates fast compact symbolically-optimized codes to solve ODEs. |
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| 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™. | ||
MotionGenesis™ also does vector algebra, matrix algebra and solves sets of linear equations or nonlinear equations.
Simpler:
Check out solve ODE basics.