We continue our study of dynamical systems by studying systems with additional degrees of freedom.
The one-dimensional harmonic oscillator has one spatial degree of freedom and a two-dimensional position-velocity phase space. Two masses connected by springs to fixed walls and connected to each other by a third spring have two two spatial degrees of freedom and a four dimensional phase space. The coupling of the two degrees of freedom leads to an interesting dynamical system because each mass influences the other. The mass motions exhibit a beat pattern indicating that two frequencies are present in the system. These two frequencies are the normal mode frequencies of the system.
The model shows the time evolution of x1(t) and x2(t). Create a second graph that shows the time evolution of two different variables: u1(t) = x1(t) + x2(t) and u2(t) = x1(t) - x2(t). How do these new variables evolve in time? Try different initial conditions x1(0) and x2(0).
Create input fields that allow a user to enter initial values for u1(0) and u2(0) and use these input values to compute x1(0) and x2(0). Place the system in an initial state such that u1(0)=2 and u2(0)=0 and describe the motion. Repeat with u1(0)=0 and u2(0)=2.
The following differential equation models will be discussed in class.
Additional models may be be posted for self-study.
The Coupled Oscillator Model was created by Wolfgang Christian using the Easy Java Simulations (EJS) version 4.1 authoring and modeling tool. You can examine and modify a compiled EJS model if you run the model (double click on the model's jar file), right-click within a plot, and select "Open Ejs Model" from the pop-up menu. You must, of course, have EJS installed on your computer.
Information about Ejs is available at: <http://www.um.es/fem/Ejs/> and in the OSP comPADRE collection <http://www.compadre.org/OSP/>.