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Lagrangian Interpolation

Explore the Lagrange method of interpolation in mechanical engineering, utilizing polynomials for easy evaluation, differentiation, and integration. Learn how to find values using cubic, quadratic, and linear interpolation, with examples of thermal expansion coefficients. Compare results to calculate percentage differences in diameter reduction. Access additional resources for comprehensive STEM education and practical application. Author: Autar Kaw, Jai Paul. For more details: http://numericalmethods.eng.usf.edu/topics/lagrange_method.html.

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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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