mws_mec_dif_discrete_examples

mws_mec_dif_discrete_examples - Chapter 02.03...

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Unformatted text preview: Chapter 02.03 Differentiation of Discrete Functions-More Examples Mechanical Engineering Example 1 To find the contraction of a steel cylinder immersed in a bath of liquid nitrogen, one needs to know the thermal expansion coefficient data as a function of temperature. This data is given for steel in Table 1. Table 1 Coefficient of thermal expansion as a function of temperature. Temperature, T ( ) F Coefficient of thermal expansion, (in/in/ ) F 80 6 10 47 . 6 40 6 10 24 . 6 40 6 10 72 . 5 120 6 10 09 . 5 200 6 10 30 . 4 280 6 10 33 . 3 340 6 10 45 . 2 (a) Is the rate of change of the coefficient of thermal expansion with respect to temperature more at F 80 T than at F 340 T ? (b) The data given in Table 1 can be regressed to 2 2 to get 2 11 . Compare the results with part (a) if you used the regression curve to find the rate of change of the coefficient of thermal expansion with respect to temperature at F 80 1 T a T a a 10 T 9 6 1.2215 10 6.2790 10 6.0216 T T and at F 340 . T Solution (a) Using the forward divided difference approximation method at F 80 T , T T T dT T d i i i 1 ) ( 80 i T 40 T T T T i i 1 40 40 80 02.03.1 02.03.2 Chapter 02.03 40 80 40 ) 80 ( dT d 40 10 47 . 6 10 24 . 6 6 6 2 9 F in/in/ 10 75 . 5 Using the backward divided difference approximation method at F 340 T , T T T dT T d i i i 1 340 i T 60 T T T T i i 1 280 60 340 60 280 340 340 dT d 60 10 33 . 3 10 45 . 2 6 6 2 7 F in/in/ 10 14667 . From the above two results it is clear that the rate of change of the coefficient of thermal expansion is more at than F 80 T F 340 T . b) Given: 2 11 9 6 10 1.2215 10 6.2790 10 6.0216 T T T dT d α 11 9 10 2.443 10 6.279 80 10 2.443 10 6.279 80 11 9 dT d 2 9 F in/in/ 10 3246 . 4 340 10 2.443 10 6.279 340 11 9 dT d 2 7 F in/in/ 10 14585 . Table 2 Summary of change in coefficient of thermal expansion using different approximations. Change in Coefficient of Thermal Expansion, i T dT d Temperature, T i Divided Difference Approximation 2 nd Order Polynomial Regression 80 2 9 F in/in/ 10 75 . 5 2 9 F in/in/ 10 3246 . 4 340 2 7 F in/in/ 10 14667 . 2 7 F in/in/ 10 14585 . Differentiation of Discrete Functions...
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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 - Tampa.

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mws_mec_dif_discrete_examples - Chapter 02.03...

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