Solving second order ODEs

A MATLAB code was coded for a 2D O.D.E equation in which a damping equation gives the animation of the oscillation.This coding can also be done for automobile suspensions and to view its damping condition.

https://youtu.be/fsRViP_nGr8

is the link for animation

close all

clc

b=0.05;
m=1;
g=9.81;
l=1;
%initial condition
theta_0=[0;5];
%total time
t_span=linspace(0,20,700);
[t,results]=ode45(@(t,theta) ode_func(t,theta,b,m,g,l),t_span, theta_0);

subplot(5,4,[13,14,17,18]);
plot(t,results(:,1),'color','b');
hold on;
plot(t,results(:,2),'color','r');
hold off;
subplot(5,4,[15,16,19,20]);
plot(results(:,1),results(:,2));
xlabel('positive-radian');
ylabel('ang-velocity');


ct=1;
for i=1:length(results(:,1))
    x0=0;
    y0=0;
    x1=l*sin(results(i,l));
    y1=-l*cos(results(i,l));
    subplot(5,4,[1,2,3,4,5,6,7,8,9,10,11,12]);
    plot([-1,1],[0,0],'linewidth',4,'color','k');
    hold on;
    line([x0 x1],[y0 y1],'linewidth',2,'color','b');
    hold on;
    plot(x1,y1,'-o','markers',20,'markerfacecolor','r');
    axis([-1 1 -2 1])
    pause(0.001);
    hold off;
    
end



function [dtheta_dt]=ode_func(t,theta,b,m,g,l)

theta1=theta(1);
theta2=theta(2);
dtheta1_dt=theta2;
dtheta2_dt=-(b/m)*theta2-(g/l)*sin(theta1);
dtheta_dt=[dtheta1_dt;dtheta2_dt]
end

The command is expalined here,

Aim: to acquire a graph for a pendulum using the damping equation between time intervals and to solve the 2D differential equation

close all

clear all

clc

These are the input arguments for the equation

b=0.05;

m=1;

g=9.81;

l=1;

%initial condition

These represent the initial condition f zero displacement and initial velocity 5m/s

theta_0=[0;5];

%total time

The total time interval of the oscillation of pendullum

t_span=linspace(0,20,300);

[t,results]=ode45(@(t,theta)

This directs us to another code where this function is determined for theta 1 & theta 2

ode_func(t,theta,b,m,g,l),t_span, theta_0);

now subplots for the displacement ,velocity & time range are given for 2 separate graphs. They are for theta 1 & theta 2

subplot(5,4,[13,14,17,18]);

plot(t,results(:,1),'color','b');

hold on;

plot(t,results(:,2),'color','r');

hold off;

subplot(5,4,[15,16,19,20]);

plot(results(:,1),results(:,2));

xlabel('positive-radian');

ylabel('ang-velocity');

a for loop is created to represent each graph at different time levels

for i=1:length(results(:,1))

x0=0;

y0=0;

x1=l*sin(results(i,l));

y1=-l*cos(results(i,l));

subplot(5,4,[1,2,3,4,5,6,7,8,9,10,11,12]);

plot([-1,1],[0,0],'linewidth',4,'color','k');

hold on;

line([x0 x1],[y0 y1],'linewidth',2,'color','b');

hold on;

plot(x1,y1,'-o','markers',20,'markerfacecolor','r');

pause(0.001);

hold off;

end