Question

Please do this in Matlab.

Problem8: (ME3400: Engineering Dynamics; ME5800: Combustion Engines) Root-finding techniques to solve engineering problems Consider the slider crank mechanism (commonly found in an engine crank shaft and piston assembly for most reciprocating engines). The combustion of gas generates the backward-and-forward motions of the ' slider, known as reciprocating mot converted into linear motions through the slider crank mechanism TOC ions. Th e reciprocating motions are Consider a simplified kinematics problem as shown in the figure below The position of the point P is denoted by s(t). From geometry, we have Rod where, α is the angle between the bar on the right and the horizontal. For this problem, we have0.2 m and 2 0.4 m Crank radius 400 mm 200 mm 0 The two angles are related by the law of sines sin9 sin a Differentiating equation (1) with respect to time gives the velocity of the point P ds dt15 dt From equation (2), the angular velocities are related by de dt de dt Suppose _-1 rad/s , determine the angle θ when the velocity of P, v.--0.2 m/s . Choose a suitable numerical method to obtain the answer and compare with the answer obtained from using fzero

Not sure if you need this code:

e cofunction [x, er, n] = FixedPoint(g, x1, maxtol, maxitr)

if nargin < 4, maxitr = 25; end
if nargin < 3, maxtol = 1e-3; end
k = 0 ;   
er = 1;
x = x1;   

while er >= maxtol && k < maxitr
k=k+1;
xold = x;
x=g(x);
er=abs((x-xold)/x);

fprintf('iter = %i, x = %e, er = %e ', k,x,er);
end

n=k;

if n == maxitr && er >= maxtol
fprintf(' ')
warning('Maximum number of iterations reached before convergence')
fprintf(' Solution not obtained in %d iterations. ',maxitr);
end

end

function [x,er,N]=Newtonroot(fun,funder,x1,maxtol,maxitr)
if nargin < 4, maxitr = 25; end
if nargin < 3, maxtol = 1e-3; end
k = 0 ;
er = 1;
x = x1;
g = funder;
while er >= maxtol && k < maxitr
k=k+1;
xold=x;
x=x-fun(x)/g(x);
er=abs((x-xold)/x);
fprintf('iter = %i, x = %e, er = %e ', k,x,er);

end
N=k;

if N == maxitr && er >= maxtol
fprintf(' ')
warning('Maximum number of iterations reached before convergence')
fprintf(' Solution not obtained in %d iterations. ',maxitr);
end
end

function df = derivative(f)

sym_f = sym(f);
df = matlabFunction(simplify(diff(sym_f))); % simplify command is optional
end

Answer 1

θ+2.cesa D -4 Άλθ 요 2- dt cl S dt- «A .co器. AA:8蠻.奈 im器 2 dE

%Matlab code for finding the ta value using fzero and Bisection method clear all close all %Finding root using fzero The parameterized function. 11-0.2; 12-0.4; dth-l vp-0.2; myfun (theta) vp +(11sin (theta) dth)+((112*sin theta)*cos (theta)) (sqrt(12 2- 11*2*(sin (theta))2))) dth); %Finding theta value for using fzero xx- fzero( (x) myfun (),0.001) theta_fz-rad2deg (xx); fprint f ( "The root using fzero is the ta=%f\n',the tafz) Finding theta value using Regula Falsi method %initialization of theta value ; a and b a-deg2rad ( 90); b-0; fprint f ( -\nFor initial guess of theta=%d and the ta=%d. In', rad2deg(a),b) %initialization of loop err-l c=0; xx-(atb)/2; %loop for Regula falsi iterations while err>10-12 xx (c)=(a+b )/ 2; mm=double (myfun ( xx ( c ) ) ) ; if mm>=0 a-xx(c) else b-xx (c) err-abs (myfun(xx(c))) fprintf( The root using bisection method is theta-f n',rad2deg (xx (end))) 용号号응용号号응용号号응용号号응용号号% End Of Code 융융용용융융용용융융용용융융용용융융용용융융 The root using fzero is theta-47.05859 7 For initial guess of theta-90 and theta 0. The root using bisection method is theta 47.058597 Published with MATLABE R2018a

%Matlab code for finding theta value using fzero and Bisection method
clear all
close all
%Finding root using fzero
% The parameterized function.
l1=0.2; l2=0.4; dth=1; vp=-0.2;
myfun = @(theta) vp+(l1*sin(theta)*dth)+(((l1^2*sin(theta)*cos(theta))/(sqrt(l2^2-l1^2*(sin(theta))^2)))*dth);

%Finding theta value for using fzero
xx= fzero(@(x) myfun(x),0.001);
theta_fz=rad2deg(xx);
fprintf('The root using fzero is theta=%f ',theta_fz)
%Finding theta value using Regula Falsi method

%initialization of theta value; a and b
a=deg2rad(90);
b=0;
fprintf(' For initial guess of theta=%d and theta=%d. ',rad2deg(a),b)
%initialization of loop
err=1;
c=0;
xx=(a+b)/2;

%loop for Regula falsi iterations
while err>10^-12
    c=c+1;
    xx(c)=(a+b)/2;
    mm=double(myfun(xx(c)));
    if mm>=0
        a=xx(c);
    else
        b=xx(c);
    end
    err=abs(myfun(xx(c)));
end

fprintf('The root using bisection method is theta=%f ',rad2deg(xx(end)))

%%%%%%%%%%%%%%%%%%%% End Of Code %%%%%%%%%%%%%%%%%%%%%%