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6/1/2014 1 Runge 4 th Order Method Industrial Engineering Majors Authors: Autar Kaw, Charlie Barker

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Presentation on theme: "6/1/2014 1 Runge 4 th Order Method Industrial Engineering Majors Authors: Autar Kaw, Charlie Barker"— Presentation transcript:

1 6/1/ Runge 4 th Order Method Industrial Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM Undergraduates

2 Runge-Kutta 4 th Order Method

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

4 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 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 Solution Step 1: For

7 Solution Cont is the approximate speed of the motor at

8 Solution Cont Step 2:

9 Solution Cont is the approximate speed of the motor at

10 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 Comparison with exact results Figure 1. Comparison of Runge-Kutta 4th order method with exact solution

12 Step size, Effect of step size (exact) Table 1 Values of speed of the motor at 0.8 seconds for different step sizes

13 Effects of step size on Runge- Kutta 4 th Order Method Figure 2. Effect of step size in Runge-Kutta 4th order method

14 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 a_4th_method.html

16 THE END


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