This tutorial models a simple harmonic oscillator with a driving force. The driving force pushes on the harmonic oscillator, which tends to increase the amplitude.
Acceleration Equations
If you have not already done so, launch the Newton Tool. As discussed in the previous tutorial, the basic equation governing the acceleration of a simple harmonic oscillator is
where c1 is the spring constant divided by the particle mass. ω, the frequency with which the system oscillates, is the square root of c1.
For a driven harmonic oscillator, the equation has an extra term--the driving force per unit mass f. This leads to the equation:
For this tutorial, we choose a sinusoidally oscillating driving force f(t) = sin(Ω t), where Ω is the frequency of f(t). We choose Ω = 2 radians/s. Open the Newton Tool, leave the number of dimensions 1, and change the expression in the acceleration equation to that below:
Once this equation is entered, click 'Next'.
Initial Conditions
Now, enter the initial conditions as shown below. These initial conditions mean we are starting the mass from rest at time = 0.
Once these initial conditions have been entered, click 'Calculate'. After the calculation finishes, click 'Next'.
Plots
Once the calculation has finished, we can make several plots. First, under 'plot on x-axis' select 'Time', and under 'plot on y-axis' select 'x position'. This will be a graph of the spring's displacement as a function of time. Because we're modeling a driven harmonic oscillator, we except to see the graph oscillate up and down but perhaps not regularly since it now has a driving force. Click 'View Plot' to see this graph.
Now return to the plot panel and generate a phase portrait. The resulting graph is quite interesting:
The motion in phase space for this driven harmonic oscillator is clearly complex.
Challenge
Resonance occurs when the driving frequency equals the natural frequency of the system. For our problem, this occurs when ω = Ω. Study this phenomenon with the Newton Tool [for example, choose the acceleration a = -4.0*x + sin(2.*t)]. Graph x vs. t and the phase portrait. You will see how the amplitude of oscillation gets larger with each cycle. Can you use this simple model to understand why soldiers break their march when crossing a bridge or why the tides are so high in the Bay of Fundy?
