1.
Consider a spring-mass system with mass \(1\text{,}\) friction constant \(6\text{,}\) and spring constant \(25\) (with units as appropriate).
- If the mass starts at the rest position, with an initial velocity of zero, what is the solution for the position function \(x(t)\text{?}\) What is the long-term behavior? Is any work needed here?
- Now consider the case where the mass is compressed to the position \(x(0) = 1\) and released. What is the solution to this IVP for \(x(t)\text{?}\) What is the long-term behavior here?
- Let’s add a forcing function of \(f(t) = 3 \cos(5t)\) to this situation. What’s the 2nd-order ODE we are solving here? What are the general and particular solutions? Putting it all together, what is the long-term behavior of this function?
- Graph the solution to the homogeneous system, the solution to our IVP forcing problem, and the forcing function. What do you notice? It might help to only graph the forcing function and the ODE IVP solution curve to make some conclusions here.
- Now let’s contain our spring-mass system in a frictionless vacuum chamber so the friction constant becomes zero. In the absence of any forcing function, and with \(x(0)=1, x’(0)=0\) as before, what’s the new solution look like here? What’s the long-term behavior?
- We’ll put our spring-mass vacuum chamber system in the back of a truck that drives over a bumpy road. For science! The bumps of the road add a force of \(f(t)=\cos(5t)\text{.}\) With the same initial conditions, what is our solution now? What’s the long-term behavior? Describe what’s going on here in terms of the motion of the mass.
- We send the truck down a slightly different road that now adds a forcing function \(f(t) = \sin(5t)\text{.}\) Using the same initial conditions, find our new solution. Discuss the long-term behavior here in terms of our spring-mass system.
- The truck stops, and we reset the spring-mass system to rest position. Then we restart the truck – we still have the same basic equation \(x'' + 25x = \sin(5t)\text{,}\) but this time our initial conditions are \(x(0) = x’(0) = 0\text{.}\) What is the solution here? What’s the long-term behavior? How is this different from the above cases?
- If we consider sound as a complicated sinusoidal wave, what do your answers above (especially to the last three problems) mean about feedback – for example, when a microphone and an attached speaker are too close? What does this mean in terms of devices like sound-canceling headphones?
NOTE: the actual world of sounds and vibratory motion is MUCH more complicated than the models above; no perfectly undamped system can exist naturally, for example. Modeling feedback from microphones, or noise-canceling headphones, or the Tacoma Narrows Bridge collapse require techniques outside the scope of this class, but the above gives an idea of how to start thinking about resonance.
