CH-06 (523176), страница 7
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Equation (a) is for the case when both w andTx are constant throughout the cable. In fact, Equation a is the solutionof the differential equation:9122d 2 y w ds w dy ==1 + 2d x Tx dx Tx dx 5.6.7.8.9.10.where s is a variable along the length of the cable. To solve this problemby applying the Runge-Kutta method, we introduce the slope variable,= θ dy/dx and convert Equation (b) to form the system of first-orderdifferential equations dy/dx = f1(x) and d/dx = f2(x) where f1(x) = andf2(x) = w[1 + θ2]1/2/Tx. Let w = 0.12 KN/m, xA = yA = 0, xB = 200 m, andyB = 50 m and let the initial conditions be y = = 0 at x = xA = 0, iterateTx value until yB is within 99.9% of 50 m.Actually, the value of Tx in Problem 4 can be obtained by solving thetranscendental equation (a) for y = yB = 50, x = xB = 200, and w = 0.12.Select a method in the program FindRoot to find this value.How could Problem 4 be solved if xA = –100 m and yA = 25 m byapplication of the Runge-Kutta method (w remains equl to 0.12 KN/m)?Apply MATLAB to solve Problem 2.Apply MATLAB to solve Problem 4 by using an increment of x = 2 m.Apply Mathematica to solve Problem 2.Apply Mathematica to solve Problem 4 using an incremwnt of x = 2 m.ODEBVPFD1.
The deflection y(x) of the beam shown in Figure 2 can be solved fromusing the moment equation, Equation 13 instead of Equation 16, but themoment M needs to be expressed in terms of x. A similar matrix equation[C]{Y} = {R} should be derived using only two boundary conditions y =0 at x = 0 and x = 3L instead of the four boundary conditions specifiedin (17). Using the second-order, central-difference formula for d2y/dx2,derive the formulas for calculations of the elements of [C] and {R}.2.
Based on the results of Problem 1, proceed to prepare the subprogramfunctions CIJ and RI and solve for {Y}. Using the data presented in thesample application of the QuickBASIC version of program OdeBvpFD,compute {Y} and compare the two approaches.3. Following the illustrative example, run the QuickBASIC version of program OdeBvoFD for the beam problem shown in Figure 2 but for N equalto 19, 29, 59, 99, and 119.© 2001 by CRC Press LLC(b)4.
Roundoff errors in the Gaussian elimination steps begin to affect theaccuracy of the computed values of the deflection for Problem 3 whenN = 119. Change the program OdeBvpFD into double precision arithmetics and rerun the case N = 119 and compare the computed ymax to thatof analytical solution.5. Make necessary changes in the FORTRAN version of program OdeBvpFD to solve Problems 3 and 4.6. For the second sample problem (deflection of beam, Figure 2), changethe distributed loads to wm = 2 and we = 1, and the rigidities to EIm = 1and EIe = 0.5 to recalculate the maximum deflection ymax.7.
Show that for the beam deflection problem shown in Figure 2 whenEquation 13 is approximated by use of second-order, central differenceand by incorporating the boundary conditions y = 0 at x = 0 and x = 3L,it will lead to the solution of the matrix equation [C]{Y} = {R} wherethe elements of [C] and {R} denoted as ci,j and ri, respectively can becalculated by the formulas:c i ,i= −2,ci,i +1 = ci +1,i = 1,ci, j= 0,for i = 1, 2,…, Nfor i = 1, 2,…, N − 1(a)elsewhereandri = ( ∆x)8.9.10.11.2Mat x iEIwhere N is the number of stations between the two supports and x isthe stepsize equal to 3L/(N + 1).For Figure 2, if the uniformly distributed loads for the middle and endingportions are designated as wm and we, respectively, derive the expressionsfor the internal bending moments in the three portions of the beam, 0≤x≤L,L≤x≤2L, and 2L≤x≤3L.Prepare subprogram FUNCTIONS CIJ and RI for Problem 8 and find thedeflection vector {Y} by use of either FORTRAN or QuickBASICversion of the program OdeBvpFD.
Select appropriate values for thenumber of stations N so that the results obtained by this second-orderapproach can be compared to those by the fourth-order approach.Use the central finite-difference method and an increment of x = 1 tofind the y values at x = 1 and x = 2 when y is governed by the equationd2y/dx2 + 3dy/dx – y = 2x – 3 and satisfies the boundary conditions y =0 at x = 0 and x = 3.Use the central finite-difference method and an increment of x = 1 tofind the y values at x = 1 and x = 2 when y is governed by the equationd2y/dx2 + 3y = x–1 and satisfies the boundary conditions y = 0 at x = 0and x = 3.© 2001 by CRC Press LLC(b)12.
It is known that u = 0 at r = 2 and r = 5 and that for 2<r<5 u satisfies theequation d2u/dr2 – rdu/dr = –3, use central differences to approximate boththe first and second derivatives of u and an increment of r equal to 1 andthen derive two equations relating the u values at r = 3 and r = 4 and solvethem.13. Apply MATLAB to solve Problem 3.14. Apply MATLAB to solve the cable problem #4 listed under OdeBvpRKand using an increment of x = 1 m.15.
Apply MATLAB to solve Problem 10 by using an increment of x = 0.05.16. Repeat Problem 13 except by application of Mathematica.17. Repeat Problem 14 except by application of Mathematica.18. Repeat Problem 15 except by application of Mathematica.6.6 REFERENCES1. C. R. Wylie, Jr., Advanced Engineering Mathematics, McGraw-Hill, New York, 1960,Chapter 6.2. J. Water, “Methods of Numerical Integration Applied to System Having TrivialFunction Evaluation,” ACM Communication, Vol.
9, 1966, p. 293.3. A. Higdon et al., Mechanics of Materials, John Wiley & Sons, New York, 1985,Chapter 7.4. Y. C. Pao, Elements of Computed-Aided Design and Manufacturing, CAD/CAM, JohnWiley & Sons, New York, 1984.5. A. Higdon, E. H. Ohlsen, W. B. Stiles, J. A. Weese, and W. F. Riley, Mechanics ofMaterials, 4th Edition, John Wiley & Sons, New York, 1985.6. W.
Jaunzemis, Continuum Mechanics, MacMillan, New York, 1967, p. 365.7. S. Timoshenko and D. H. Young, Elements of Strength of Materials, 5th Edition, VanNostrand Reinhold Co., New York, 1968.8. W. Jaunzemis, Continuum Mechanics, MacMillan, New York, 1967, p. 365.9. J. L. Meriam and L. G. Kraige, Engineering Mechanics, Volume One: Statics, ThirdEdition, John Wiley & Sons, Inc., New York, 1992.© 2001 by CRC Press LLC.
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