Dr.Shoeb_ME212_Lec-1_numerical methods_G
- 1. Introduction to Numerical Methods Numerical Methods (ME 212) Author: Dr. Shoeb Ahmed Syed Papua New Guinea University of Technology Department of Mechanical Engineering 1
- 2. My advice If you don’t let a teacher know at what level you are by asking a question, or revealing your ignorance you will not learn or grow. You can’t pretend for long, for you will • • eventually be found out. Admission of ignorance is often the first step in our education. – Steven Covey—Seven Habits of Highly Effective People 3
- 3. Why use Numerical Methods? • To solve problems that cannot be solved exactly 2 x − − u 1 e du 2 2
- 4. Why use Numerical Methods? • To solve problems that are intractable!
- 5. Some disasters caused by numerical errors? ➢ Patriot Missile Failure ➢ Explosion of the Ariane 5 ➢ EURO page: Conversion Arithmetic's ➢ The Vancouver Stock Exchange ➢ Rounding error changes Parliament makeup
- 8. Explosion of the Ariane 5
- 9. Explosion of the Ariane 5
- 12. The Vancouver Stock Exchange
- 13. Rounding error changes Parliament makeup
- 14. Rounding error changes Parliament makeup
- 15. Steps in Solving an Engineering Problem 6
- 16. How do we solve an engineering problem? Problem Description Mathematical Model Solution of Mathematical Model Using the Solution 7
- 17. Example of Solving an Engineering Problem 8
- 20. Trunnion-Hub-Girder Assembly Procedure Step1. Step2. Step3. Trunnion immersed in dry-ice/alcohol Trunnion warm-up in hub Trunnion-Hub immersed in dry-ice/alcohol Trunnion-Hub warm-up into girder Step4. 11
- 21. Problem After Cooling, the Trunnion in Hub Got Stuck 12
- 22. Why did it get stuck? Magnitude of was 0.015” contraction needed in the trunnion or more. Did it contract enough? 13
- 23. Video of Assembly Process Unplugged Version VH1 Version 14
- 24. Consultant calculations D D = T D = 12.363" = 6.47 10−6 in / in /o F = −108 −80 = −188o F T D = (12.363)(6.47 10−6 )(−188) = −0.01504" 15
- 25. D = D T Is the formula used correct? D D = T 16 T(oF) α (μin/in/oF) -340 2.45 -300 3.07 -220 4.08 -160 4.72 -80 5.43 0 6.00 40 6.24 80 6.47
- 26. The Correct Model Would Account for Varying Thermal Expansion Coefficient Tc D (T )dT Ta D = 17
- 27. Tc D = D (T)dT Ta http://numericalmethods.eng.usf.edu Can You Roughly Estimate the Contraction? Tc = D (T )dT Ta D Ta=80oF; Tc=-108oF; D=12.363” 18
- 28. Can You Find a Better Estimate for the Contraction? Tc D (T )dT Ta D = 80oF -108oF 12.363" Ta = Tc = D = 19
- 29. Estimating Contraction Accurately Change in diameter (D) by cooling it in dry ice/alcohol is given by Tc D = D (T )dT Ta Ta = 80oF Tc = -108oF D = 12.363" = −1.227810−5 T + 6.194610−3 T 2 + 6.0150 20
- 30. So what is the solution to the problem? One solution is to immerse the trunnion in liquid nitrogen which has a boiling point of -321oF as opposed dry-ice/alcohol temperature of -108oF. to the D = −0.0244" 21
- 31. Revisiting steps to solve a problem 1) Problem Statement: Trunnion got stuck the hub. in 2) Modeling: Developed a new model Tc D = D (T )dT Ta Solution: 1) Used trapezoidal rule OR Used regression and integration. Implementation: Cool the trunnion in nitrogen. 3) b) 4) liquid 22
- 32. Mathematical Procedures Nonlinear Equations Differentiation • • • • Simultaneous Linear Curve Fitting – Interpolation – Regression Integration Equations • • • Ordinary Differential Equations Other Advanced Mathematical Procedures: – – – Partial Differential Equations Optimization Fast Fourier Transforms 25
- 33. Nonlinear Equations How much of the floating ball is under water? Diameter=0.11m Specific Gravity=0.6 10−4 x3 − 0.165x2 + 3.993 = 0 26
- 34. Nonlinear Equations How much of the floating ball is under the water? x3 − 0.165x2 + 3.99310−4 = 0 f (x) = 27
- 35. Differentiation What is the acceleration at t=7 seconds? dv dt 4 16 10 ln = a − 9.8t v(t) = 2200 4 16 10 − 5000t 28
- 36. Differentiation What is the acceleration at t=7 seconds? dv dt a = 29 Time (s) 5 8 12 V el (m/s) 106 177 600
- 37. Simultaneous Linear Equations Find the velocity profile, given v(t) = at2 + bt + c, 5 t 12 Three simultaneous linear equations 25a + 5b + c =106 64a + 8b + c = 177 144a +12b + c = 600 30 Time (s) 5 8 12 V el (m/s) 106 177 600
- 38. Interpolation What is the velocity of the rocket at t=7 seconds? 31 Time (s) 5 8 12 V el (m/s) 106 177 600
- 39. Regression Thermal expansion coefficient data for cast steel 32
- 41. Integration Finding the diametric contraction in a steel shaft when dipped in liquid nitrogen. Tfluid D Troom D = dT 34
- 42. Ordinary Differential Equations How long does it take a trunnion to cool down? d (0) room = −hA( −a ), = mc dt 35
- 44. THE END