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Lagrangian Interpolation

Lagrangian Interpolation. Mechanical Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Lagrange Method of Interpolation http://numericalmethods.eng.usf.edu. What is Interpolation ?.

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Lagrangian Interpolation

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  1. Lagrangian Interpolation Mechanical 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. Lagrange 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. Lagrangian Interpolation http://numericalmethods.eng.usf.edu

  6. Example A trunnion is cooled 80°F to − 108°F. Given below is the table of the coefficient of thermal expansion vs. temperature. Determine the value of the coefficient of thermal expansion at T=−14°F using the Lagrangian method for linear interpolation. http://numericalmethods.eng.usf.edu

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

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

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

  10. Example A trunnion is cooled 80°F to − 108°F. Given below is the table of the coefficient of thermal expansion vs. temperature. Determine the value of the coefficient of thermal expansion at T=−14°F using the Lagrangian method for quadratic interpolation. http://numericalmethods.eng.usf.edu

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

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

  13. Cubic Interpolation http://numericalmethods.eng.usf.edu

  14. Example A trunnion is cooled 80°F to − 108°F. Given below is the table of the coefficient of thermal expansion vs. temperature. Determine the value of the coefficient of thermal expansion at T=−14°F using the Lagrangian method for cubic interpolation. http://numericalmethods.eng.usf.edu

  15. Cubic Interpolation (contd) http://numericalmethods.eng.usf.edu

  16. Cubic Interpolation (contd) http://numericalmethods.eng.usf.edu

  17. Comparison Table http://numericalmethods.eng.usf.edu

  18. Reduction in Diameter The actual reduction in diameter is given by where Tr = room temperature (°F) Tf= temperature of cooling medium (°F) Since Tr = 80 °F and Tr = −108 °F, Find out the percentage difference in the reduction in the diameter by the above integral formula and the result using the thermal expansion coefficient from the cubic interpolation. http://numericalmethods.eng.usf.edu

  19. Reduction in Diameter We know from interpolation that Therefore, http://numericalmethods.eng.usf.edu

  20. Reduction in diameter Using the average value for the coefficient of thermal expansion from cubic interpolation The percentage difference would be http://numericalmethods.eng.usf.edu

  21. 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/lagrange_method.html

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

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