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Spline Interpolation Methods for Engineering Majors: Essential Concepts and Applications

This resource provides an in-depth look into spline interpolation methods, a vital tool for engineering students. It covers the basics of interpolation, with a focus on polynomial and spline methods, demonstrating how to find values at unknown points based on given data. The document illustrates practical examples, including the application of linear and quadratic splines through a rocket velocity case study. Understanding these methods will enhance numerical problem-solving skills in engineering and contribute to STEM education.

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Spline Interpolation Methods for Engineering Majors: Essential Concepts and Applications

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  1. Spline Interpolation Method Major: All Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates http://numericalmethods.eng.usf.edu

  2. Spline Method of Interpolationhttp://numericalmethods.eng.usf.edu

  3. What is Interpolation ? Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given. http://numericalmethods.eng.usf.edu

  4. Interpolants Polynomials are the most common choice of interpolants because they are easy to: • Evaluate • Differentiate, and • Integrate. http://numericalmethods.eng.usf.edu

  5. Why Splines ? http://numericalmethods.eng.usf.edu

  6. Why Splines ? Figure : Higher order polynomial interpolation is a bad idea http://numericalmethods.eng.usf.edu

  7. Linear Interpolation http://numericalmethods.eng.usf.edu

  8. Linear Interpolation (contd) http://numericalmethods.eng.usf.edu

  9. Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using linear splines. Table Velocity as a function of time Figure. Velocity vs. time data for the rocket example http://numericalmethods.eng.usf.edu

  10. Linear Interpolation http://numericalmethods.eng.usf.edu

  11. Quadratic Interpolation http://numericalmethods.eng.usf.edu

  12. Quadratic Interpolation (contd) http://numericalmethods.eng.usf.edu

  13. Quadratic Splines (contd) http://numericalmethods.eng.usf.edu

  14. Quadratic Splines (contd) http://numericalmethods.eng.usf.edu

  15. Quadratic Splines (contd) http://numericalmethods.eng.usf.edu

  16. Quadratic Spline Example The upward velocity of a rocket is given as a function of time. Using quadratic splines Find the velocity at t=16 seconds Find the acceleration at t=16 seconds Find the distance covered between t=11 and t=16 seconds Table Velocity as a function of time Figure. Velocity vs. time data for the rocket example http://numericalmethods.eng.usf.edu

  17. Solution Let us set up the equations http://numericalmethods.eng.usf.edu

  18. Each Spline Goes Through Two Consecutive Data Points http://numericalmethods.eng.usf.edu

  19. Each Spline Goes Through Two Consecutive Data Points http://numericalmethods.eng.usf.edu

  20. Derivatives are Continuous at Interior Data Points http://numericalmethods.eng.usf.edu

  21. Derivatives are continuous at Interior Data Points At t=10 At t=15 At t=20 At t=22.5 http://numericalmethods.eng.usf.edu

  22. Last Equation http://numericalmethods.eng.usf.edu

  23. Final Set of Equations http://numericalmethods.eng.usf.edu

  24. Coefficients of Spline http://numericalmethods.eng.usf.edu

  25. Quadratic Spline InterpolationPart 2 of 2http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

  26. Final Solution http://numericalmethods.eng.usf.edu

  27. Velocity at a Particular Point a) Velocity at t=16 http://numericalmethods.eng.usf.edu

  28. Acceleration from Velocity Profile b) The quadratic spline valid at t=16 is given by http://numericalmethods.eng.usf.edu

  29. Distance from Velocity Profile c) Find the distance covered by the rocket from t=11s to t=16s. http://numericalmethods.eng.usf.edu

  30. 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/spline_method.html

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

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