1. 6/1/2014 http://numericalmethods.eng.usf.edu 1 Runge 4 th Order Method Industrial Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

2. Runge-Kutta 4 th Order Method http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

3. 3 Runge-Kutta 4 th Order Method where For Runge Kutta 4 th order method is given by

4. http://numericalmethods.eng.usf.edu4 How to write Ordinary Differential Equation Example is rewritten as In this case How does one write a first order differential equation in the form of

5. http://numericalmethods.eng.usf.edu5 Example The open loop response, that is, the speed of the motor to a voltage input of 20 V, assuming a system without damping is If the initial speed is zero, and using the Runge- Kutta 4 th order method, what is the speed at t = 0.8 s? Assume a step size of h = 0.4 s.

6. http://numericalmethods.eng.usf.edu6 Solution Step 1: For

7. http://numericalmethods.eng.usf.edu7 Solution Cont is the approximate speed of the motor at

8. http://numericalmethods.eng.usf.edu8 Solution Cont Step 2:

9. http://numericalmethods.eng.usf.edu9 Solution Cont is the approximate speed of the motor at

10. http://numericalmethods.eng.usf.edu10 Solution Cont The exact solution of the ordinary differential equation is given by The solution to this nonlinear equation at t=0.8 seconds is

11. http://numericalmethods.eng.usf.edu11 Comparison with exact results Figure 1. Comparison of Runge-Kutta 4th order method with exact solution

12. Step size, 0.8 0.4 0.2 0.1 0.05 147.20 299.54 302.96 303.09 155.89 3.5535 0.12988 0.0062962 0.00034702 51.434 1.1724 0.042852 0.0020773 0.00011449 http://numericalmethods.eng.usf.edu12 Effect of step size (exact) Table 1 Values of speed of the motor at 0.8 seconds for different step sizes

13. http://numericalmethods.eng.usf.edu13 Effects of step size on Runge- Kutta 4 th Order Method Figure 2. Effect of step size in Runge-Kutta 4th order method

14. http://numericalmethods.eng.usf.edu14 Comparison of Euler and Runge- Kutta Methods Figure 3. Comparison of Runge-Kutta methods of 1st, 2nd, and 4th order.

15. Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/runge_kutt a_4th_method.html

16. THE END http://numericalmethods.eng.usf.edu