hw2_solution_2008

hw2_solution_2008 - ME 17 Summer 2008 Homework 2 Solution...

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ME 17 Summer 2008 Homework 2 Solution TA: Gaurav Soni PROBLEM 1 Part (1) and (2): In order to determine the critical mass of the jumper, the root of the following equation was found separately by two different iterative methods: Bisection method and False Position. f(m) = sqrt(m*g/Cd)*tanh(sqrt(g*Cd/m)*t)-v = 0 where g=9.81, Cd=0.25, t=4 and v=36 (all in SI units). The successive values calculated by the two methods are shown in the following table. One can note that both of the methods converge on very similar values. Both of these values are very close to the true root of 142.7376. m Iteration # Bisection Method False Position Method 1 125.0000 176.2773 2 162.5000 162.3828 3 143.7500 154.2446 4 134.3750 149.4777 5 139.0625 146.6856 6 141.4063 145.0501 7 142.5781 144.0922 8 143.1641 143.5311 9 142.8711 143.2024 10 142.7246 143.0099 11 142.7979 142.8971 12 142.7612 142.8310 13 142.7429 142.7923 14 142.7338 142.7697 15 142.7383 142.7564 16 142.7361 142.7486 17 142.7372 142.7441 18 142.7414 19 142.7398 20 142.7389
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It is also interesting to note that the false position method takes three more iterations than the bisection method to converge. The convergence of the false position is slowed down by the fact that the value of xl (the lower bracket) tends to stay at xl=50 in all the iterations. This fact was found by printing the successive values of xl and xu. Following figure shows the successive iterated for mass by the two methods.
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C:\Documents and Settings\gsoni\My Documents\resear. ..\bungee.m Page 1 July 10, 2006 7:59:24 PM %This script file defines the equation and calls the 'bisectio' function to %find its root %In this case the equation is related to the bungee jumper's mass f=inline('sqrt(m*9.81/0.25)*tanh(sqrt(9.81*0.25/m)*4)-36'); m_bi=bisection_succesive(f,50,200) fprintf('The critical mass of the jumper calculated by bisection method is %f kg',m_bi(le ngth(m_bi))); m_fa=false_position(f,50,200) fprintf('The critical mass of the jumper calculated by false position method is %f kg',m_ fa(length(m_fa))); figure; plot(m_bi,'b.-') hold plot(m_fa,'r*-'); hold off xlabel('iteration number'); ylabel('approximate value of jumper`s mass'); legend('bisection method','false position method') title('iterative approximations for jumper`s mass');
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C:\Documents and Settings\gsoni\My Doc. ..\bisection_succesive.m Page 1 July 10, 2006 7:58:31 PM function root=bisection_succesive(f,xl,xu) %This functions finds root of an equation f=0 by bisection method %xl and xu are two initial bracketing points %check if xl and xu bracket the solution or not if f(xl)*f(xu)>0 'no bracket' return end %Bisection iter=1;
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This note was uploaded on 08/06/2008 for the course ME 17 taught by Professor Milstein during the Summer '07 term at UCSB.

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hw2_solution_2008 - ME 17 Summer 2008 Homework 2 Solution...

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