Runge 4 th Order Method Industrial Engineering Majors

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Runge 4 th Order Method Industrial Engineering Majors Authors: Autar Kaw, Charlie Barker http:

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 5/25/2021 http: //numericalmethods. eng. usf. edu 1

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

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

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

Runge-Kutta 4 th Order Method For Runge Kutta 4 th order method is given by where 3 lmethods. eng. usf. edu http: //numerica

How to write Ordinary Differential Equation How does one write a first order differential

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

Example The open loop response, that is, the speed of the motor to a

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. 5 lmethods. eng. usf. edu http: //numerica

Solution Step 1: 6 For lmethods. eng. usf. edu http: //numerica

Solution Step 1: 6 For lmethods. eng. usf. edu http: //numerica

Solution Cont is the approximate speed of the motor at 7 lmethods. eng. usf.

Solution Cont is the approximate speed of the motor at 7 lmethods. eng. usf. edu http: //numerica

Solution Cont Step 2: 8 lmethods. eng. usf. edu http: //numerica

Solution Cont Step 2: 8 lmethods. eng. usf. edu http: //numerica

Solution Cont is the approximate speed of the motor at 9 lmethods. eng. usf.

Solution Cont is the approximate speed of the motor at 9 lmethods. eng. usf. edu http: //numerica

Solution Cont The exact solution of the ordinary differential equation is given by The

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 10 lmethods. eng. usf. edu http: //numerica

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

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

Effect of step size Table 1 Values of speed of the motor at 0.

Effect of step size Table 1 Values of speed of the motor at 0. 8 seconds for different step sizes Step size, 0. 8 0. 4 0. 2 0. 1 0. 05 147. 20 155. 89 51. 434 299. 54 3. 5535 1. 1724 302. 96 0. 12988 0. 042852 303. 09 0. 0062962 0. 0020773 303. 09 0. 00034702 0. 00011449 (exact) 12 lmethods. eng. usf. edu http: //numerica

Effects of step size on Runge. Kutta 4 th Order Method Figure 2. Effect

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

Comparison of Euler and Runge. Kutta Methods 14 Figure 3. Comparison of Runge-Kutta methods

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

Additional Resources For all resources on this topic such as digital audiovisual lectures, primers,

Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, Math. Cad and MAPLE, blogs, related physical problems, please visit http: //numericalmethods. eng. usf. edu/topics/runge_kutt a_4 th_method. html

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

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

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