We continue our study of dynamical systems by studying systems with additional degrees of freedom.
Two masses connected by springs to fixed walls and connected to each other by a third spring have two two spatial degrees of freedom x1(t) and x2(t). Although we are very familiar with these position coordinates, other sets of coordinateness are sometimes more useful to describe the system. Normal mode coordinates, for example, are computed from the position coordinates as u1(t) = (x1(t) + x2(t))/2 and u2(t) = (x1(t) - x2(t))/2 and the inverse transformations are x1(t) = u1(t) + u2(t) and x2(t) = u1(t) - u2(t). The Coupled Oscillator Modes Model shows how these different coordinate representations evolve in time.
Try different initial conditions x1(0) and x2(0) followed by different u1(0) or u2(0). What initial conditions give symmetric motion? Asymmetric motion? Which of the two modes u1(t) or u2(t) has the higher frequency? Why? These two frequencies are the normal mode frequencies of the system.
The following differential equation models will be discussed in class.
Additional models may be be posted for self-study.
The Coupled Oscillator Modes 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/>.