Question

Use Heun's method (RK2), choose a step size Perform the same computation as in Prob. 28.19, but rather than using a constant wind force, employ a force that varies with height according to: 200z -2:/30 f(2) 5+z Problem 28.19 The following equation can be used to model the deflection of a sailboat mast subject to a wind force: d2 (L-z) where f - wind force, E modulus of elasticity, L mast length, and I-moment of inertia. Calculate the deflection if y-0 and dyldz 0 atz 0. Use parameter values of f 60, L -30, E.25 x 108, and I - 0.05 for your computation

Use Heun's Method please! Must use MATLAB and please use ANONYMOUS function. WIll rate answer 5 stars if done correctly.

Answer 1

MATLAB Script:

close all
clear
clc

% Give: y'' = (f/(2EI))*(L-z)^2
% Letting y' = u
% u' = y'' = (f/(2EI))*(L-z)^2
% We now have a system of first-order ODEs

L = 30; E = 1.25e8; I = 0.05;

h = 1;
zi = 0; zf = L;
n = (zf - zi)/h;
z = zi:h:zf;

y(1) = 0; u(1) = 0; % Initial conditions
for i=1:n
m1 = dydz(z(i), y(i), u(i));
n1 = dudz(z(i), y(i), u(i));
  
m2 = dydz(z(i) + h, y(i) + h*m1, u(i) + h*n1);
n2 = dudz(z(i) + h, y(i) + h*m1, u(i) + h*n1);
  
y(i+1) = y(i) + 0.5*h*(m1 + m2);
u(i+1) = u(i) + 0.5*h*(n1 + n2);
end

% Exact Solution
syms y_exact(z_exact)
Dy = diff(y_exact, z_exact);
eqn = diff(y_exact, z_exact, 2) == (200*z_exact*exp(-2*z_exact/30) / (5 + z_exact)) * (L - z_exact)^2 / (2*E*I);
conds = [y_exact(0) == 0, Dy(0) == 0];
sol_y = dsolve(eqn, conds);

% Plot the solutions
figure, plot(z, subs(sol_y, z)), hold on
plot(z, y, '-o'), xlabel('z'), ylabel('y(z)')
legend('Exact Solution', 'Using Heun''s Method');

function f = dydz(~,~,u)
f = u; % y'
end

function out = dudz(z,~,~)
L = 30; E = 1.25e8; I = 0.05;
f = @(z) 200*z*exp(-2*z/30) / (5 + z);
out = (f(z)/(2*E*I))*(L-z)^2; % u'
end

Output Graph (Along with Exact Solution):

Exact 3olution 0.51 0.1F