Forced Spring Systems

Part 1: An Undamped Spring with External Forcing

As we have seen in the Spring Motion module, the motion of a spring-mass system can be modeled by an initial value problem of the form

where m is the mass, c is the damping coefficient, and k is the spring constant. For convenience in this module, we assume m = 1 and (until Part 4) c = 0. That is, we have unit mass and no damping. Furthermore, to simplify notation in the solution functions, we will write the spring constant as k² instead of k. Thus, our model for an undamped and unforced spring-mass system is

Now we apply an external sinusoidal force to the moving mass with amplitude F₀ and frequency w/(2pi). Thus, the model becomes

(Note: The standard notation for the forcing frequency parameter is lower case Greek omega -- w is the closest Latin letter.)

Finally, since our forcing function has the value F₀ at t = 0, we don't need to move the mass with either an initial displacement or an initial velocity, so we take both initial values to be 0. Our model is now

In this part of the module, we assume that |w| is different from |k|, that is, the forcing frequency is different from the natural frequency of the unforced oscillation. Since the signs of k and w have no bearing on the solutions of the differential equation, we assume they are both positive.

1. Enter the differential equation with the starting parameter values, k = 5, F₀ = 1, and w = 2. Use your helper application to solve the initial value problem symbolically, and plot the solution. [Note: The symbolic form of the solution may not look familiar, because your helper application may use a trigonometric identity to write it in another form.]
2. Use your helper application to find the derivative of the solution function, and plot the trajectory in the phase plane.
3. Note that the driving frequency is quite different from the natural frequency of the system. Increase w in steps of 0.5 from 2 up to 4. For each value of w, solve the initial value problem, and plot both the solution and the trajectory in the phase plane. For each choice of w, describe in your own words what you see. Pay close attention to changing scales on the axes.