Question

4 A four-bar linkage system is shown above. The first link, a, is an input link (crank) of length 1. The second link, b, is a coupler link of length 2. The third link, c, is an output link of length 4. The forth link, d, is the fixed link (ground) of length 5. All lengths are provided in metres. The angular position of the output link (04) of a four-bar linkage corresponding to the angular position of the input link (02) can be computed using the Freudenstein's equation: 2ac The following parameters must be used for root finding: xl-120°, xu-165°, xi - 120°, pert 0.01, precision 1e-4. A. Write an m-file to find the value of 04 for 02-30° using fzero. B. Check this result using the bisection, false position and modified secant methods. For each method use fprintf to print the root and the number of iterations each method takes. Use fprintf to print a statement mentioning which method requires the least amount of iterations. C. Plot the absolute difference between the function values of the estimated root and the actual root (i.e. abs(f(root_estimated)-0)) for each method against the respective number iterations on the same plot. Remember to label the plot with a legend.

[MATLAB Coding] A four-bar linkage system is shown above. The first link, a, is an input link (crank) of length 1. The second link, b, is a coupler link of length 2. The third link, c, is an output link of length 4. The forth link, d, is the fixed link (ground) of length 5. All lengths are provided in metres. Please answer the whole question in MATLAB coding.

Answer 1

function [root,iter,absdiff] = modifiedSecantMethod(f,x0,dx,tol)

iter = 1;
xn = x0;

while true
  
Xn = xn - dx*f(xn)/(f(xn+dx)-f(xn)); %computing root
  
absdiff(iter) = abs(f(Xn)-0); % get the absolute difference
  
%termination
if absdiff(iter)<tol
root = Xn;
root = root*180/pi; % to degrees
break;
end

xn = Xn;
iter = iter+1;
end

end

------------------------------------------------------------------------------

function [root,iter,absdiff] = falsePositionMethod(f,x0,x1,tol)

a = x0;
b = x1;

if ~(f(a)*f(b)<0)
error('ERROR: f(a)*f(b) must be less than 0')
end
iter = 1;
while true
x = (a *f(b) - b*f(a))/(f(b)-f(a)); %computing the root
  
absdiff(iter) = abs(f(x)-0); %getting absolute difference
  
if absdiff(iter)<=tol %termination
break;
elseif f(a)*f(x)>=0
a = x;
elseif f(b)*f(x)>=0
b = x;
end
iter = iter+1;   
end
root = x;
root = root*180/pi; % to degrees
end

-------------------------------------------------------------------------------

function [root,iter,absdiff] = bisectionMethod(f,x_pos,x_neg,tol)

iter = 1;

while true
x_mid = (x_pos+x_neg)/2; %computing the root
if f(x_mid)<0
x_neg = x_mid;
else
x_pos = x_mid;
end

absdiff(iter) = abs(f(x_mid)-0); %getting absolute differences
  
  
if absdiff(iter)<tol %termination
break;
end
  
x_mid = (x_pos+x_neg)/2;
iter = iter+1;
end

root = x_mid;

root = root*180/pi; % to degrees
end

-----------------------------------------------------------------------------------------

%driver file

clear
clc
close all

xl = 120; xu = 165; xi = 120; %degrees
xl = xl*pi/180; xu = xu*pi/180; xi = xi*pi/180; %to radians

pert = 0.01; precision = 1e-4;
a = 1; b=2;c=4;d=5; %meters

%let x2 = theta2 and x4 = theta4

x2 = 30;%degrees
x2 = x2*pi/180; %to radians

f = @(x4) (d/a).*cos(x4) - (d/c)*cos(x2) + ((a^2-b^2+c^2+d^2)/(2*a*c)) - cos(x2-x4);

%using fzero
fprintf('\nFZERO')
root = fzero(f,[xl,xu]); %radians
root = root*180/pi; %to degrees
fprintf(['\nRoot: ' num2str(root) ' degrees'])

methods = {'modified secant','false position','bisection'};

%modified secant
fprintf('\n\nMODIFIED SECANT')
[root1,iters1,absdiff1] = modifiedSecantMethod(f,xi,pert,precision);
fprintf(['\nRoot: ' num2str(root1) ' degrees'])
fprintf(['\niterations: ' num2str(iters1)])

%false position
fprintf('\n\nFALSE POSITION')
[root2,iters2,absdiff2] = falsePositionMethod(f,xl,xu,precision);
fprintf(['\nRoot: ' num2str(root2) ' degrees'])
fprintf(['\niterations: ' num2str(iters2)])

% bisection
fprintf('\n\nBISECTION')
[root3,iters3,absdiff3] = bisectionMethod(f,xl,xu,precision);
fprintf(['\nRoot: ' num2str(root3) ' degrees'])
fprintf(['\niterations: ' num2str(iters3)])

%method requiring least amount of iterations
iters = [iters1 iters2 iters3];
minIterIndex = find(iters==min(iters));
methodLeastIters = methods{minIterIndex};
fprintf(['\n\nMethod requiring least amount of iterations: ' methodLeastIters '\n\n']);

%plotting convergence
figure
plot(1:iters1,absdiff1,'-o r')
hold on
plot(1:iters2,absdiff2,'-o g')
hold on
plot(1:iters3,absdiff3,'-o b')
title('convergence')
xlabel('iteration')
ylabel('|f(x)-0|')
legend('modified secant','false position','bisection')

%OUTPUT

FZERO Root: 144.8347 degrees MODIFIED SECANT Root: 144.836 degrees iterations: 3 FALSE POSITION Root: 144.8356 degrees iterations: 8 BISECTION Root: 144.8346 degrees iterations 13 Method requiring least amount of iterations: modified secant

convergence 0.3 _ modified secant - false position 0.25 ebisection 0.2 0.15 0.1 0.05 0 12 4 6 8 10 14 iteration

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