Euler’s Method and Runge-Kutta RK4
By Murray Bourne, 09 Mar 2015
I recently updated the interactive spring applet on this page (you'll see it about half way down):
Work by a variable force - spring applet
The upgrade (now javascript, rather than Flash) uses some interesting numerical solutions for a differential equation. It uses Runge-Kutta Order 4 Method. (Actually, my earlier Flash version also used Runge-Kutta method of Order 4, but it was a lot smoother than the javascript version. I'm still trying to improve it.)
The differential equation we need to solve for a spring-mass system is:
where:
x is the position of a mass at the end of the spring at time t;
k is the spring constant;
c is the amount of damping (which slows the action of the spring); and
F is some external force, varying with time t.
The code works out a good approximation for the movement of the spring. In fact, this is how a lot of calculus solutions are obtained in the real world - numerically on computers, not algebraically like in math text books.
I didn't have anything on numerical solutions for differential equations in IntMath, so I added it just now. For introduction, I also wrote a section on Euler Method. Here are the 2 pages:
Runge-Kutta (RK4) numerical solution for Differential Equations
I hope you find them useful.
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