CH-08 (523178), страница 6
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Initially, the temperatures are assumed to be equal to zero everywhere inthe shown in Figure 13 except those on the boundary. Carry out onecomplete relaxation (starting from the top row and from left to right, andthen down to the next row and so on) cycle to upgrade the unknowntemperatures. The points A, B, and D are on a straight line.8. Rework Problem 7 if the boundary ABCD is insulated but TD remainsequal to 100°F.9. Following the same process as in Problem 8, but obtain the direct solutionof the temperature distribution for Problem 7.© 2001 by CRC Press LLCFIGURE 13.
Prtoblem 7.10. The warping of a twisted bar of uniform rectangular cross section alreadydepicted by the mesh plot shown in Figures 5 and 6 also can be observedusing the contour plotting capability of MATLAB. Apply program Warping.m and the command contour to generate a contour plot for a = 30 andb = 20 using = 100 and = 1 (Figures 14A and 14B, respectively).FIGURE 14A.
Problem 10.© 2001 by CRC Press LLCFIGURE 14B. Problem 10.FIGURE 15. Problem 4.11. Direct solution of the warping function W(X,Y) can also be obtainedfollowing the procedure described in Problem 8. For a rectangular crosssection (-a≤X≤a and -b≤Y≤b) of a twisted rod, the warping W(X,Y) needsto be found only for the upper right quadrant 0≤X≤a and 0≤Y≤b whichin general can be divided into a gridwork of (M + 1)x(N + 1).
Theantisymmetry conditions W(X = 0,Y) = W(X,Y = 0) = 0 reduces to only(M + 1)x(N + 1)-(M + N + 1) = MxN unknowns, i.e., only solving thoseW’s for X>0. That means we have to derive a system of MxN linearalgebraic equations: along the boundaries X = a and Y = b, Equations 23to 25 are to be used and for the interior grid points (0<X<a and 0<Y<b),Equation 3 is to be used.
Generate such a matrix equation and then apply© 2001 by CRC Press LLCthe program Gauss to find these MxN W’s. Compare the resulting Wdistribution with those obtained by the relaxation method shown inFigures 5 and 6 for a = 30 and b = 20.12. Same as Problem 11 except for the case a = b = 20 and for comparingwith Figure 7.13. Solve the warping problem by Mathematica.WAVEPDE1. For the string problem analyzed in the Sample Application, modify theprogram slightly so that the times required for the string to have themagnitudes of its maximum displacements reduced to 0.8, 0.6, 0.4, and0.2, and the corresponding deflected shapes can be printed.2.
Rearrange the subprogram FUNCTION F in the program WavePDE toconsider the case of an initial, upward lifting the mid-third (8≤x≤16 cm)of the string by 1 cm. Rerun the program using the same input data as inthe Sample Application.3. Consider a string which is composed of two different materials eventhough it is subjected to a uniform tension T so that the left and right onethirds (i.e., , 0≤x≤8 cm and 24≤x≤32 cm, respectively) of the string hasa wave velocity a = 80 cm/sec while its mid-third (i.e., 8≤x≤16 cm) hasa wave velocity a = 90 cm/sec.
Modify and then rerun program WavePDEusing the other input same as in the Sample Application.4. A tightened string of length L equal to 1 ft is lifted as shown in 15 andis released with a velocity distribution v = y(t = 0,x)/t = 2sinx/L inft/sec. If the constant T/m appearing in Equation 2 is equal to 8,100ft2/sec2, use a time increment t = 0.0005 sec and a space increment x =0.1L and apply Equation 7 to find the y values at t = 0.001 sec and forthe stations x2 and x3.5. In approximating Equation 1 by finite differences, we may keep the sameapproach for 2u/t2 as in deriving Equation 7 but to apply the secondcentral-difference formula for 2u/t2 not at t = ti but at t = ti + 1.
Theresulting equation, for C = (at/x)2, is:−Cu i +1, j−1 + (2 + C)u i +1, j − Cu i +1, j+1 = − u i −1, j + 2 u i, jDerive a matrix equation for solving the unknowns uj for j = 1,2,…,N–1,at t = ti + 1. Note that the boundary conditions are ui + 1,0 = ui + 1,N = 0.
Writea program WavePDE.G which uses the Gaussian Elimination method tosolve this matrix equation and run it for the Sample problem to comparethe results.6. Change the MATLAB m file WavePDE to solve Problem 4.7. Apply Mathematica to solve Problem 4.© 2001 by CRC Press LLC8.6 REFERENCES1. C. R.
Wylie, Jr., Advanced Engineering Mathematics, Chapter 9, Second Edition,McGraw-Hill, New York, 1960.2. J. P. Holman, Heat Transfer, McGraw-Hill, New York, 1963.3. S. Timoshenko and J. N. Goodier, Theory of Elasticity, Chapter 11, Second Edition,McGraw-Hill, New York, 1951.© 2001 by CRC Press LLC.
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