Week 4.1 - Solving second order ODEs

Aim: Solving a second order ODE using MATLAB.

Introduction

The motion of a pendulum is a classic problem in physics and engineering, providing insights into the dynamics of oscillatory systems. When damping is introduced, the system models real-world scenarios more accurately, such as the behavior of a pendulum clock, the motion of a child on a swing, or the suspension systems in vehicles. This report outlines the simulation of a damped pendulum's motion using MATLAB, with the aim of visualizing its behavior over time and creating an animation to demonstrate its dynamics.

Practical Applications

Understanding the motion of a damped pendulum has numerous practical applications:

1. Clocks and Timekeeping:

Pendulums are used in clocks where precise timing is crucial. Damping effects must be understood and minimized to maintain accuracy.

2. Engineering and Design:

In automotive suspension systems, the principles of a damped pendulum help in designing shock absorbers that reduce oscillations and provide a smooth ride.

3. Seismic Analysis:

Pendulum models are used to simulate and analyze the behavior of buildings and structures during earthquakes, where damping helps in mitigating vibrations.

4. Control Systems:

In robotics and aerospace, controlling the oscillations of various components is essential for stability and performance.

Steps Taken and Approach

1. Model Definition:

The mathematical model of a damped pendulum is defined by a second-order ordinary differential equation (ODE). The equation considers the effects of gravity, damping, and the pendulum's physical characteristics.

2. ODE Solver:

MATLAB's `ode45` solver is used to numerically solve the ODE over a specified time span.

3. Animation Setup:

An animation is created to visualize the pendulum's motion. This involves plotting the pendulum's position at each time step and saving the frames to a video file.

4. Plotting Results:

The angular displacement and velocity are plotted over time to analyze the system's behavior.

Matlab Code With Explanation

Function Definition (`ode_func.m`)

*****

% ode_func.m

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

% Extract the first and second components of theta

theta1 = theta(1);

theta2 = theta(2);

% Define the derivatives

dtheta1_dt = theta2;

dtheta2_dt = -(b/m) * theta2 - (g/l) * sin(theta1);

% Return the derivatives as a column vector

dtheta_dt = [dtheta1_dt; dtheta2_dt];

end

*****

This function defines the dynamics of the damped pendulum. It takes the current time 't', state vector 'theta' (containing angular displacement and velocity), and system parameters (damping coefficient 'b', gravity 'g', length 'l', and mass 'm'). It returns the derivatives of the state vector.

Main Script (`main.m`)

***

% main.m

% Parameters

b = 0.5; % damping coefficient

g = 9.81; % acceleration due to gravity

l = 1.0; % length of the pendulum

m = 1.0; % mass of the pendulum

% Initial conditions [theta(0), theta_dot(0)]

theta0 = [0; 3]; % initial angle (rad) and angular velocity (rad/s)

% Time span

tspan = [0 20];

% Solve the ODE

[t, theta] = ode45(@(t, theta) ode_func(t, theta, b, g, l, m), tspan, theta0);

% Prepare the animation

figure;

hold on;

axis equal;

grid on;

xlim([-1.2 1.2]);

ylim([-1.2 1.2]);

xlabel('X Position (m)');

ylabel('Y Position (m)');

title('Pendulum Motion Animation');

% Plot the trajectory

plot(l*sin(theta(:,1)), -l*cos(theta(:,1)), 'r--'); % Plot the pendulum path

% Initialize pendulum line and mass

pendulumLine = line([0, l*sin(theta(1,1))], [0, -l*cos(theta(1,1))], 'LineWidth', 2);

pendulumMass = plot(l*sin(theta(1,1)), -l*cos(theta(1,1)), 'ro', 'MarkerSize', 12, 'MarkerFaceColor', 'r');

% Create the animation file

videoFile = VideoWriter('pendulum_motion.avi');

open(videoFile);

for i = 1:length(t)

% Update pendulum position

set(pendulumLine, 'XData', [0, l*sin(theta(i,1))], 'YData', [0, -l*cos(theta(i,1))]);

set(pendulumMass, 'XData', l*sin(theta(i,1)), 'YData', -l*cos(theta(i,1)));

% Capture the frame

frame = getframe(gcf);

writeVideo(videoFile, frame);

pause(0.01); % Pause for a short time to create the animation effect

end

% Close the video file

close(videoFile);

% Plot the angular displacement and velocity on the same plot

figure;

yyaxis left;

plot(t, theta(:, 1), 'b-', 'LineWidth', 1.5); % Plot theta (angle) over time

xlabel('Time (s)');

ylabel('Angle (rad)');

title('Pendulum Angle and Angular Velocity over Time');

grid on;

yyaxis right;

plot(t, theta(:, 2), 'r--', 'LineWidth', 1.5); % Plot theta_dot (angular velocity) over time

ylabel('Angular Velocity (rad/s)');

legend('Angle (rad)', 'Angular Velocity (rad/s)');

*****

Results and Observations:
1. Angular Velocity and displacement as a function of time

Here we can see how at lower damping factor the pendulum keeps oscillating for a longer duration.

2. Effect of increasing damping factor from 0.05 to 0.5

As we have increased the damping factor we can see the oscillations damp quickly.

Conclusion
The simulation and animation of a damped pendulum's motion provide valuable insights into the dynamics of oscillatory systems. By solving the ODE numerically and visualizing the results, we can better understand the effects of damping and the pendulum's behavior over time. This approach is applicable in various engineering and scientific fields where understanding and controlling oscillatory motion is crucial. The MATLAB code provided effectively demonstrates how to model, simulate, and visualize such a system, offering a powerful tool for both educational and practical applications.