Week 3 - Solving second order ODEs

AIM :- SOLVING SECOND ORDER ODE USING MATLAB

GROVING EQUATIONS :-

b = damping

m =mass

g = gravity

l = length

θ=θ1

dθ/dt=dθ/dt=θ2 - (1) d^2θ/dt^2+b/m dθ/dt+g/lsinθ=0

d^2θ/dt^2 = d^2θ1/dt^2 = d/dt(dθ1/dt)=d/dt(θ2) - (2) d^2θ1/dt^2+b/m dθ1/dt+g/lsinθ1

dθ2/dt +b/mθ2

dθ1/dt =θ2 dθ2/dt = -b/mθ2 -g/l sinθ1

discription :-

THE SECOND ORDER ODE IS AN FIELD AND AS ENGINERS LEARNING HOW TO WRITE A PROGRAM ABOUT SECOND ORDER ODE IN MATLAB

ODE IS EXTREMLY IMPORTENT IN ENGINARING AND DISCRIBE A LOT OF FINAMINA AND SOLVING THOSE ODE IS UNDERSTAND THE SYSTEM

BODY OF THE CONTENT :- THE SECOND ORDER ODE IS A PROGRAM WHICH IS USED IN MATLAB SOFTWARE IN THIS SOFTWARE WE CAN SOLVE THE DIFFERENT TYPES OF ODE USING THE FORMULA OF SIMPLE PENDULAM AND SHOWING HOW SECOND ORDER ODE WILL WORK IN MATLAB SOFTWARE

CONCLUSION :- THE SECOND ORDER ODE IS A PROGRAM WHICH IS USED IN MATLAB SOFTWARE IN THIS SOFTWARE WE CAN SOLVE THE DIFFERENT TYPES OF ODE USING THE FORMULA OF SIMPLE PENDULAM AND SHOWING HOW SECOND ORDER ODE WILL WORK IN MATLAB SOFTWARE THE SECOND ORDER ODE IS AN FIELD AND AS ENGINERS LEARNING HOW TO WRITE A PROGRAM ABOUT SECOND ORDER ODE IN MATLAB ODE IS EXTREMLY IMPORTENT IN ENGINARING AND DISCRIBE A LOT OF FINAMINA AND SOLVING THOSE ODE IS UNDERSTAND THE SYSTEM

CODING OF THE SECOND ORDER ODE :-

HEAR WE ARE FIND THE SECOND ORDER ODE USING THE SIMPLE SPENDILUM IN MATLAB AND EXPLING THE CODE WITH STEPS AND COMMENTS

STEP1 :- CLEAR ALL

CLOSE ALL

CLC

STEP2:- WRITE THE GIVING VALUES

b = 0.5

g = 9.81

l = 1

m = 0.1

STEP3:- GIVE INITIAL CONDITION VALUES

%initial condition

theta_0 =[0;5]

STEP4:- GIVE TIME POINTS VALUES

%time points

t_span = linespace(0,10,500)

STEP5:-GIVE SOLVING ODE VALUES

%solving ode

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

plot(t,results(:,1))

STEP6 :- GIVE THE SECOND ORDER ODE FUNCTION

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

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

clear all

close all

clc

b = 0.05;

g = 9.81;

l = 1;

m = 0.1;

% initial condition

theta_0 = [0;5];

% time points

t_span = linspace(0,10,500);

% solving ode

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

plot(t,results(:,1))

hold on

plot(t,results(:,2))

ylabel('plot')

xlabel('time')

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

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