Euler Method Simulation - Enhanced Algorithm

close all
clear all
clc

% This script compares Euler's method of numerical integration to the analytical method.
% Euler's method
g = 9.81; %acceleration of gravity [m/s^2].
m = 1; %mass of the object [kg].
k = 1; %spring constant [N/m].
y0 = 5; %initial position in relation to equilibrium [m].
v0 = 0; %initial velocity [m/s].
dt = 0.01; %time interval between steps [s].
n = 10000; %number of steps.
t(1) = 0; %initial time condition.
y(1) = y0; %initial position condition.
v(1) = v0; %initial velocity condition.
pe(1) = .5 * k * y0^2;
ke(1) = .5 * m * v0^2;
e(1) = pe(1) + ke(1);
for i = 2:n
  t(i) = t(i-1) + dt; % adds time interval to previous time and saves the value in a vector.
  v(i) = v(i-1) - (k./m) .* y(i-1) .* dt; %Calculates new velocity and saves the value in a vector.
  y(i) = y(i-1) + v(i-1) * dt; %Calculates new position and saves the value in a vector.
  pe(i) = .5 * k * y(i)^2;
  ke(i) = .5 * m * v(i)^2;
  e(i) = pe(i) + ke(i);
end
figure 1
plot(t,v,'r',t,y,'b')
title("Euler's method of numerical integration")
legend('Velocity [m/s]','position [m]')
%----------------------------------------%
% Analytical method
v_exact = -y0 * sqrt(k ./ m) .* sin(sqrt(k ./ m) .* t);
y_exact = y0 * cos(sqrt(k ./ m) .* t);
pe_exact = .5 .* k .* y_exact .^ 2;
ke_exact = .5 .* m .* v_exact .^ 2;
e_exact = pe_exact + ke_exact;
figure 2 
plot(t,v_exact,'r',t,y_exact,'b',t,v,'r--',t,y,'b--',t,e,'g',t,e_exact,'y')
title('Euler vs exact')
legend('v exact','y exact','v Euler','y Euler','Energy Euler','Exact Energy')

close all
clear all
clc

% This script compares the Euler-cromer method of numerical integration to the analytical method.
% Euler-cromer method
g = 9.81; %acceleration of gravity [m/s^2].
m = 1; %mass of the object [kg].
k = 1; %spring constant [N/m].
y0 = 5; %initial position in relation to equilibrium [m].
v0 = 0; %initial velocity [m/s].
dt = 0.01; %time interval between steps [s].
n = 10000; %number of steps.
t(1) = 0; %initial time condition.
y(1) = y0; %initial position condition.
v(1) = v0; %initial velocity condition.
pe(1) = .5 * k * y0^2; %initial spring potential energy.
ke(1) = .5 * m * v0^2; %initial object kinetic energy.
e(1) = pe(1) + ke(1); %initial total energy.
F(1) = -k * y0; %initial force on the object.

for i = 2:n
  t(i) = t(i-1) + dt; % adds time interval to previous time and saves the value in a vector.
  F(i) = -k .* y(i-1); % calculates the force on the object.
  a(i) = F(i) / m; % calculates the accelerationon of the object.
  v(i) = v(i-1) + a(i) * dt; % calculatesthe velocity of the object.
  y(i) = y(i-1) + v(i) * dt; %calculates the position of the object.
  pe(i) = .5 * k * y(i)^2; % calculates object potential energy.
  ke(i) = .5 * m * v(i)^2; % calculates object kinetic energy.
  e(i) = pe(i) + ke(i); %calculates object total energy.
end

%----------------------------------------%
% Analytical method
v_exact = -y0 * sqrt(k ./ m) .* sin(sqrt(k ./ m) .* t);
y_exact = y0 * cos(sqrt(k ./ m) .* t);
pe_exact = .5 .* k .* y_exact .^ 2;
ke_exact = .5 .* m .* v_exact .^ 2;
e_exact = pe_exact + ke_exact;

plot(t,y_exact,'b',t,y,'r--',t,e,'g',t,e_exact,'y')
title('Euler vs exact')
legend('y exact','y Euler','Energy Euler','Exact Energy')

clc
clear all
close all

g = 9.81;
L = 2;
m = 5;
b = 3;
I = (m * L^2) / 3;
Tau = 10;
w = 2.5;

omega(1) = 0;
theta(1) = 120 * pi / 180;
alpha(1) =  ((3 * g) / (2 * L))^2 * sin(theta(1));
nsteps = 10000;
dt = .001;
t(1) = 0;
for n = 2:nsteps 
  t(n) = t(n-1) + dt;
  alpha(n) = -((3 * g) / (2 * L))^2 * sin(theta(n-1)) - (b*omega(n-1))/I + Tau * cos(w*t(n));
  omega(n) = omega(n-1) + alpha(n) * dt;
  theta(n) = theta(n-1) + omega(n) * dt;
end

plot(t,theta)
title('Euler-Cromer')
figure 2
plot(theta, omega)
title('Euler-Cromer')

clc

clear all

close all

%%% This code models and animates the motion of a rigid pendulum.

g = 9.81; %gravity

L = 2; % length

m = 5; % mass

b = 0; % resistance

I = (m * L^2) / 3; %moment of inirtia

Tau = 0; % added force

w = 2.5; % added force interval

omega(1) = 0; %initialize omega (initial velocity)

theta(1) = 30 * pi / 180; % initialize theta (initial angle)

alpha(1) = ((3 * g) / (2 * L))^2 * sin(theta(1)); %initialize alpha (initial acceleration)

nsteps = 1000; % sets number of steps

dt = .01; % sets time step

t(1) = 0; %initialize time

% Euler-Cromer method

for n = 2:nsteps

t(n) = t(n-1) + dt;

alpha(n) = -((3 * g) / (2 * L))^2 * sin(theta(n-1)) - (b*omega(n-1))/I + Tau * cos(w*t(n));

omega(n) = omega(n-1) + alpha(n) * dt;

theta(n) = theta(n-1) + omega(n) * dt;

end

plot(t,theta)

title('Euler-Cromer')

figure

plot(theta, omega)

title('Euler-Cromer')

X = -.5 * L .* sin(theta); % locates center of mass in the x direction

Y = -.5 * L .* cos(theta); % locates center of mass in the y direction

figure

% animation for the pendulum

for z = 1:length(theta)

plot ([0,2*X(z)],[0,2*Y(z)],'linewidth',2)

axis([-2,2,-2,0])

hold on

plot(X(z),Y(z),'ro')

pause(.01)

%storing frames

K(z) = getframe(gcf);

hold off

end

videofile = VideoWriter('pendulum.avi','Uncompressed AVI')

open(videofile)

writeVideo(videofile,K)

close(videofile)

clc

close all

clear all

mass_Earth = 5.972e24;

mass_object = 20000;

x(1) = 0;

y(1) = 6786100;

alt(1) = sqrt(x(1)^2 + y(1)^2);

vx(1) = 7778;

vy(1) = 0;

ax(1) = 0;

ay(1) = 0;

nsteps = 100;

dt = 60;

t(1) = 0;

theta(1) = 0;

for n = 2:nsteps

f(n) = -6.67428e-11 * (mass_Earth * mass_object) / alt(n-1)^2;

fx(n) = f(n) * sin(theta(n-1));

fy(n) = f(n) * cos(theta(n-1));

ax(n) = fx(n)/mass_object;

ay(n) = fy(n)/mass_object;

vx(n) = vx(n-1) + ax(n) * dt;

vy(n) = vy(n-1) + ay(n) * dt;

x(n) = x(n-1) + vx(n) * dt;

y(n) = y(n-1) + vy(n) * dt;

alt(n) = sqrt(x(n)^2 + y(n)^2);

v(n) = sqrt(vx(n)^2 + vy(n)^2);

theta(n) = theta(n-1) + (v(n)/(2 * pi * alt(n)))*dt;

plot(x(n),y(n),'rx','LineWidth',10)

axis([-7000000,7000000,-7000000,7000000])

pause(.1)

t(n) = t(n-1)+dt;

end