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Unformatted text preview: 05.03.1 Chapter 05.03 Newton’s Divided Difference 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 05.03.2 Chapter 05.03 Figure 1 Thermal expansion coefficient vs. temperature. Determine the value of the coefficient of thermal expansion at F 14 T using Newton’s divided difference method of interpolation and a first order polynomial. Solution For linear interpolation, the coefficient of thermal expansion is given by ) ( ) ( 1 T T b b T Since we want to find the coefficient of thermal expansion at 14 T and we are using a first order polynomial, we need to choose the two data points that are closest to 14 T that also bracket 14 T to evaluate it. The two points are T and 60 1 T . Then , T 6 10 00 . 6 ) ( T , 60 1 T 6 1 10 58 . 5 ) ( T gives ) ( T b 6 10 00 . 6 1 1 1 ) ( ) ( T T T T b 60 10 00 . 6 10 58 . 5 6 6 6 10 007 . Newton’s Divided Difference MethodMore Examples: Mechanical Engineering 05.03.3 Hence ) ( ) ( 1 T T b b T ), ( 10 007 . 10 00 . 6 6 6 T 60 T At 14 T ) 14 ( 10 007 . 10 00 . 6 ) 14 ( 6 6 F in/in/ 10 902 . 5 6 If we expand ), ( 10 007 . 10 00 . 6 ) ( 6 6 T T 60 T we get , 10 007 . 10 00 . 6 ) ( 6 6 T T 60 T This is the same expression that was obtained with the direct method. Example 2 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....
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida.
 Spring '08
 Kaw,A
 Mechanical Engineering

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