Simple Harmonic Motion

Introduction

In mechanics and physics, simple harmonic motion is a special type of periodic motion where the restoring force on the moving object is directly proportional to the magnitude of the object's displacement and acts towards the object's equilibrium position.

In this activity we will learn how to use MATLAB to model and plot approximate solutions for the simple harmonic motion of the spring.

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1. The motion of a spring-mass system

1.2 The undamped spring

A spring of (natural) length L is stretched a distance s by a weight with mass m. In equilibrium we take the position of the weight (viewed as a point-particle) to be

with the positive x-axis pointing down.

If we pull down the weight and then release it, it will start to oscillate.

If we assume the weight-spring-system moves free of any resistance (no air-resistance and no internal friction in the spring) we say the spring is undamped.

We know that the equation of motion for the undamped spring is given by

where k is the known as the spring constant, and m represents the mass. Observe that this is a homogeneous second-order linear ODE with constant coefficients, which can be expressed as the couple system of equations

Using this system, we can easily solve Initial Value Problems (IVP) in MATLAB. For example, consider the following

Problem 1: A mass of 9kg is attached to a spring with spring constant 4N/m. The spring is pulled down 1m and given an initial upward kick of -0.5 m/s. Compute and plot a numerical solution.

From the information provided we have that

1.3 Implementation in MATLAB

The following MATLAB code computes and plots an approximated solution of the IVP from Problem 1.

t0 = 0;     % Initial time
tf = 50;    % Final time, here we have chosen 50
x0 = 1;     % Initial condition x(0)=1
y0 = -0.5;  % Initial condition x'(0)=-0.5
 
k = 3;      % Spring constant
m = 5;      % Mass
 
% Define anonymous function for the system
secODEUndampedSprg = @(t,Y) [Y(2); -k/m * Y(1)];
 
[T,Y] = ode45(secODEUndampedSprg, [t0, tf], [x0; y0]);
 
% Plot solutions
plot(T, Y(:,1), 'r-', 'LineWidth', 1.5);
 
% Add extra feature to the plot
title('Motion for the undamped spring')
xlabel('t')
ylabel('x(t)')
grid on
axis([t0 tf -2 2])

2. Hands on Practice

Activity: The damped spring

For a more realistic model for the spring, we need to consider the effect of damping. This is caused by air-resistance and mechanical friction of the spring. We know that this can be represented by the equation

where β is the damping constant, k the spring coefficient and m the mass.

First, in your notebook, re-write this equation as a system of ODE. Then in MATLAB:

Compute the constant
Compute an approximated solution for each IVP defined below.
Plot the solution.
Analyse the relationship between the constant D and the solution of the IVP.

Part 1) Underdamped spring

Consider

and initial conditions

. Plot the solution on the interval

Write your code here:

Part 2) Critically damped spring

Consider

and initial conditions

. Plot the solution on the interval

Write your code here:

Part 3) Overdamped spring

Consider

and initial conditions

. Plot the solution on the interval

Write your code here: