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mws_mec_inp_txt_direct_examples

mws_mec_inp_txt_direct_examples - 05.02.1 Chapter 05.02...

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Unformatted text preview: 05.02.1 Chapter 05.02 Direct Method of Interpolation – More Examples Mechanical Engineering Example 1 For the purpose of shrinking a trunnion into a hub, the reduction of diameter D of a trunnion shaft by cooling it through a temperature change of T is given by T D D where D original diameter in. coefficient of thermal expansion at average temperature F in/in/ The trunnion is cooled from F 80 to F 108 , giving the average temperature as F 14 . The table of the coefficient of thermal expansion vs. temperature data is given in Table 1. Table 1 Thermal expansion coefficient as a function of temperature. Temperature, F T Thermal Expansion Coefficient, F in/in/ 80 6.47 6 10 0 6.00 6 10 –60 5.58 6 10 –160 4.72 6 10 –260 3.58 6 10 –340 2.45 6 10 05.02.2 Chapter 05.02 Figure 1 Thermal expansion coefficient vs. temperature. If the coefficient of thermal expansion needs to be calculated at the average temperature of F 14 , determine the value of the coefficient of thermal expansion at F 14 T using the direct method of interpolation and a first order polynomial. Solution For first order polynomial interpolation (also called linear interpolation), we choose the coefficient of thermal expansion given by T a a T 1 Direct Method of Interpolation – More Examples: Mechanical Engineering 05.02.3 Figure 2 Linear interpolation. Since we want to find the coefficient of thermal expansion at F 14 T , and we are using a first order polynomial, we need to choose the two data points that are closest to F 14 T that also bracket F 14 T to evaluate it. The two points are F T and F 60 1 T . Then 6 10 00 . 6 , T T 6 1 1 10 58 . 5 , 60 T T gives 6 1 10 00 . 6 a a 6 1 10 58 . 5 60 60 a a Writing the equations in matrix form, we have 6 6 1 10 58 . 5 10 00 . 6 60 1 1 a a Solving the above two equations gives 6 10 00 . 6 a 6 1 10 007 . a Hence T a a T 1 60 , 10 007 . 10 00 . 6 6 6 T T At F, 14 T 14 10 007 . 10 00 . 6 14 6 6 F in/in/ 10 902 . 5 6 , y x 1 1 , y x x f 1 x y 05.02.4 Chapter 05.02 Example 2 For the purpose of shrinking a trunnion into a hub, the reduction of diameter D...
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mws_mec_inp_txt_direct_examples - 05.02.1 Chapter 05.02...

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