Week 3 - Solving second order ODEs

Aim:

The aim of this challenge is:

(i) to solve a 2nd order ordinary differential equation, which represernts the motion of a simple pedulum with a suspended mass

(ii) simulate the motion graphically (2D motion) for 0-20sec

Input:

(i) gravitational force, g = 9.81 m/s2

(ii) length of the pendulum, L = 1 m

(iii) mass of the ball, m = 1 kg

(iv) damping coefficient, b = 0.05

Initial condition:(@ time, t = 0)

(i) angular displacement = 0 rad

(ii) angular velocity = 3 rad/sec

Solution using MatLab:

The motion of a simple pendulum is represented by the equation:

`(d^2theta)/dt^2 + b/m. (d theta)/dt + g/L.sin theta = 0`

Codes used in the program:

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

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

(i) ode45 : predefined Matlad function/solver for ODE which is a function of time,t & Ang displacement,Θ

(ii) ode_func : custom program for solving the ODE, whcih is a function of t, Θ, b, g, L, m

Attachments:

(i) validated Matlab program - ODE.m

(ii) validated funtion program - ode_func.m

(iii) Graph generated for the executed program - Plot_Velocity_Displacementy.fig

(iv) Graph generated for Pendulum simulation - Pendulum_simulation.fig

(v) Screeenshot the successfully executed program - Screenshot_ODE.png

(vi) Videofile " Simple_Pendulum" uploaded to Youtube.

Errors/Difficulties/New commands:

1. Difficulty in tracing the length(t_span) for the Θ values in the program - resolved with the support of the TechSupport.

2. Various line commands with Marker & filling which were not covered in the Lecture Videos ,were learnt from thr TechSupport.

Conclusion:

A simple executable program,including ode45, function & for loop, is written for the given ODE with the given inputs and the desired results (i) Plots and(ii) pendulum oscillation simulation is achieved.