Week 3 - Solving second order ODEs

second order differenceal equation to solve the simple pendulum simulation.

d^2theta/dt^2+b/m.dtheta/dt+g/lsin(theta) = 0

create two ODE

dtheta1/dt = theta2 and dtheta2/dtheta = -(b/m(theta2)+g/lsin(theta1))

codes for calling a function :-the function which we have define with some parameter are going to use in the other programe the function is going to solve the second order equation or we can say that it going to integrate the value

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

theta1 = theta(1);

theta2 = theta(2);

dtheta_dt = theta2;

dtheta_dt = -(b/m)*theta2-(g/l)*sin(theta1);

dtheta_dt = [dtheta1_dt;dtheta2_dt);end

code simple pendulum

ODE system are two firest order velocity and other displacement and initial condition for displacement and velocity

ode 45 inbuilt solver in matlab and octave

time theta and other parameter

b = 0.5,

g = 9.81m/s

l = 0.9 in m

m = 2kg

clear all

close all

clc

%data

b = 0.05;

g = 9.81;

l = 1;

m = 1;

x0 = 0;

y0 = 0;

ct = 1;

%initial condition

theta_0 = [0,3]

% tmie_plot

t_span = linspace(0,20,500);

%solve_ode

[t,result]=ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0);

plot(t,result(:,1))

xlable('plot')

ylable('plot')

%pendulum simulation using ode

for i = 1:length(result(i,1);

d = result(i,1);

%defining x and y coordinates

x1 = -l*sin(d);

y1 = -l*cos(d);

hold on

plot([x0 x1],[y0 y1],'linewidth',4);

plot(x1,y1,'-o','markersize',30,'markeredgecolor','g','markerfacecolor',[0.5 0.5 0.5]);

plot(x1,y1,'-o','markersize',3,'markeredecolor','g','markerfacecolor',[0.5 0.5 0.5]);

axisz9[-2 2 -2 2]);

pause(0.01);

m(ct) = getframe(gcf)

ct = ct+1;

clf()

end

%animation