08.06 Shooting Method for Ordinary Differential Equations

Size: px

Start display at page:

Download "08.06 Shooting Method for Ordinary Differential Equations"

Clemence Washington
5 years ago
Views:

1 8.6 Shooting Method for Ordinary Differential Eqations After reading this chapter, yo shold be able to 1. learn the shooting method algorithm to solve bondary vale problems, and. apply shooting method to solve bondary vale problems. What is the shooting method? Ordinary differential eqations are given either with initial conditions or with bondary conditions. Look at the problem below. υ x L q Figre 1 A cantilevered niformly loaded beam. To find the deflection as a fnction of location x, de to a niform load q, the ordinary differential eqation that needs to be solved is d q L x (1) dx EI where L is the length of the beam, E is the Yong s modls of the beam, and I is the second moment of area of the cross-section of the beam. Two conditions are needed to solve the problem, and those are 8.6.1

2 8.6. Chapter 8.6 d (a,b) dx as it is a cantilevered beam at x. These conditions are initial conditions as they are given at an initial point, x, so that we can find the deflection along the length of the beam. Now consider a similar beam problem, where the beam is simply spported on the two ends υ q Figre A simply spported niformly loaded beam. To find the deflection as a fnction of x de to the niform load q, the ordinary differential eqation that needs to be solved is d qx x L (3) dx EI Two conditions are needed to solve the problem, and those are L (a,b) as it is a simply spported beam at x and x L. These conditions are bondary conditions as they are given at the two bondaries, x and x L. The shooting method The shooting method ses the same methods that were sed in solving initial vale problems. This is done by assming initial vales that wold have been given if the ordinary differential eqation were an initial vale problem. The bondary vale obtained is then compared with the actal bondary vale. Using trial and error or some scientific approach, one tries to get as close to the bondary vale as possible. This method is best explained by an example. Take the case of a pressre vessel that is being tested in the laboratory to check its ability to withstand pressre. For a thick pressre vessel of inner radis a and oter radis b, the differential eqation for the radial displacement of a point along the thickness is given by d 1 d r r (5) Assme that the inner radis a 5" and the oter radis b 8", and the material of the pressre vessel is ASTM36 steel. The yield strength of this type of steel is 36 ksi. Two strain gages that are bonded tangentially at the inner and the oter radis measre the normal tangential strain in the pressre vessel as.776 t / ra x L

3 Shooting Method t / rb.386 (6ab) r a b Figre 3 Cross-sectional geometry of a pressre vessel. at the maximm needed pressre. Since the radial displacement and tangential strain are related simply by t, (7) r then '' ra rb '' (8) Starting with the ordinary differential eqation d 1 d, , r r Let d w (9) Then dw 1 w (1) r r giving s two first order differential eqations as d w, " dw w, w 5 not known (11a,b) r r Let s assme d 8 5 w Set p the initial vale problem. d w f1r,, w, "

4 8.6. Chapter 8.6 dw w fr,, w, w (1a,b) r r Using Eler s method, i 1 i f1ri, i, wi wi 1 wi f ri, i, wi (13a,b) Let s consider segments between the two bondaries, r 5" and r 8", then 8 5 h.75" i, r 5,.38731", w f1r,, w f 15,.38371,.6538 (.75) (.75).3671" w1 w f r,, w.6538 f (5,.38731,.6538) i 1, r1 r h ", ", w f1r1, 1, w f 15.75,.3671, (.75).359 w w1 f r1, 1, w f 5.75,.3671, (.75) i, r r1 h ".359", w f1r,, w.359 f 16.5,.359, (.75).3583" w3 w fr,, w f 6.5, (.75 )

5 Shooting Method (.75) i 3, r3 r h " ", w f1r3, 3, w f 17.5,.3583,.5335 (.75) (.75).363" w w3 f r3, 3, w f 7.5,.3583,.5335 (.75) (.75) At r r r3 h " we have " While the given vale of this bondary condition is 8.37" d Let s assme a new vale for 5. Based on the first assmed vale, maybe sing twice the vale of initial gess. d 8 5 w Using h. 75, and Eler s method, we get " While the given vale of this bondary condition is " Can we se the reslts obtained from the two previos iterations to get a better estimate of d the assmed initial condition of 5? One method is to se linear interpolation on the d obtained data for the two assmed vales of 5. With d , we obtained

6 8.6.6 Chapter 8.6 with 8.363", and d , we obtained " so a better starting vale of 8.377", d 5 knowing that the actal vale at we get d Using h.75", and repeating the Eler s method with w , we get " while the actal given vale of this bondary condition is 8.377". 8. If that were not the case, one In this case, this vale coincides with the actal vale of wold contine to se linear interpolation to refine the vale of till one gets close to the actal vale of 8. Note that the step size and the nmerical method sed wold inflence the accracy for the obtained vales. For the last case, the vales are as follows " " " " " See Figre for the comparison of the reslts with different initial gesses of the slope. Using h. 75 and Rnge-Ktta th order method, " " " " "

7 Shooting Method E-3 3.8E-3 d/ = Radial Displacement, (in) 3.6E-3 3.E-3 d/d r= Exact 3.E-3 d/ = E Radial Location, r (in) Figre Comparison of reslts with different initial gesses of slope Table 1 shows the comparison of the reslts obtained sing Eler s, Rnge-Ktta and exact methods. Table 1 Comparison of Eler and Rnge-Ktta reslts with exact reslts. r Exact Eler (%) Rnge-Ktta t (in) (in) (in) (in) (%) t ORDINARY DIFFERENTIAL EQUATIONS Topic Shooting method Smmary Textbook notes on the shooting method for ODE. Major General Engineering Athors Atar Kaw Last Revised December 3, 9 Web Site

Shooting Method for Ordinary Differential Equations Autar Kaw

Shooting Method for Ordinary Differential Equations Autar Kaw Shooting Method or Ordinary Dierential Eqations Atar Kaw Ater reading this chapter, yo shold be able to. learn the shooting method algorithm to solve bondary vale problems, and. apply shooting method to