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

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Runge-Kutta 4 th Order Method

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

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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

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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.

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Solution Step 1: For

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Solution Cont is the approximate speed of the motor at

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Solution Cont Step 2:

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Solution Cont is the approximate speed of the motor at

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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

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

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

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

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Comparison of Euler and Runge- Kutta Methods Figure 3. Comparison of Runge-Kutta methods of 1st, 2nd, and 4th order.

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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

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