Investigating External Force As you may have noticed, damping is precisely what causes a spring-mass system to eventuall

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Investigating External Force As You May Have Noticed Damping Is Precisely What Causes A Spring Mass System To Eventuall 1 (181.34 KiB) Viewed 31 times

Investigating External Force As you may have noticed, damping is precisely what causes a spring-mass system to eventually return to rest. Is this reasonable to assume about a bridge? Probably not. With the effects of traffic and natural vibrations, a bridge is typically under regular oscillation. When we add a forcing function, what we start to see changes. Let's suppose that f(t) = sin(at). In layman's terms, we are now imposing an oscillating external force. That force varies with time and is in units of Newtons (N), just like the spring force term and the damping force term. We see that the period is 2 seconds per cycle and the frequency of this force is cycles per second. In other words, a controls how rapidly the external force oscillates. 1. Using your understanding of solving NHSOLDE's, solve the new system with m = k = 1 and f(t) = sin(at). However, assume that bridge is undamped (b = 0). In the homogeneous solution, assume k₁=k₂ = 1. This just takes off some of our workload! Initial conditions are not important in what happens in the long-run. Note that your solution will have a in it. COMMENTS: Having b=0 will do two things: a) it will simplify the mathematics, and b) it will mimic the notion that the bridge is naturally under constant stress and vibration, thus simulating constant oscillation. (Again, all models are wrong.) Your answer will depend on a and on k. Take your time and do good book-keeping. As a side note, the homogeneous solution, yn (t), is often called the transient solution and that yp (t) is called the steady-state solution for these second- order systems. This makes sense, because the forcing function typically contributes more to the long-run solution of the system than does the transient solution (the solution of the homogeneous reduction of the ODE). Final Analysis Now that you have a solution, recall that k=1 (the spring constant). This value causes the oscillations we see in the homogeneous solution; this is because k is the "springiness" of the spring (the elasticity of the bridge, in this case). That is, without any forcing, the model bridge would oscillate at oscillations per second (one full up-down-up motion every, or 0.16 seconds). The forcing function contributes a term (the nonhomogeneous part) that causes oscillations at a frequency of oscillations per second. In our general solution, we sum the homogeneous and nonhomogeneous solutions, and so the natural oscillation of the bridge and the external force's oscillation "interact". This observation is very important in seeing why the Tacoma Narrows began to oscillate more and more violently until the material could no longer withstand it (thereby crumbling). 2. Graph the solution in Desmos and create a slider for a on the interval from 0 to 1. Experiment with the slider. State any interesting observations. Be a detective! Most notably, spend some time "fine-tuning" a as it approaches 1 from values close to 1 (like 0.9, 0.99, 0.999). What do you notice as a → 1, that is, as a → k? What happens to the oscillations? 3. Take a peak again at your general solution equation. What happens in the denominator of one of the terms as a → 1? Does this further support what you saw in the graph in 6)?
4. You have just investigated one of the culprit's of the bridge's collapse. What can you say about what happens to the displacement of the bridge from rest, when the frequency of oscillations of the external environment (sound waves, wind, etc.) approach the same frequency as that of the natural oscillations of the bridge?