Investigating External Force As you may have noticed, damping is precisely what causes a spring-mass system to eventuall
Posted: Tue Jul 12, 2022 12:46 pm
Investigating External Force
As you may have noticed, damping is precisely what causes aspring-mass system to eventually return to rest. Is this reasonableto assume about a bridge? Probably not. With the effects of trafficand natural vibrations, a bridge is typically under regularoscillation. When we add a forcing function, what we start to seechanges. Let's suppose that f(t)=sin(αt). In layman's terms,we are now imposing an oscillating external force. That forcevaries with time and is in units of Newtons (N), just like thespring force term and the damping force term. We see that theperiod is 2παseconds per cycle and the frequency of this forceis α2π cycles per second. In otherwords, α controls how rapidly the external forceoscillates.
1. Using your understanding of solving NHSOLDE's, solve the newsystem with m = k = 1and f(t)=sin(αt). 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 thelong-run. Note that your solution will have α in it.
COMMENTS: Having b=0 will do two things: a)it will simplify the mathematics, and b) it will mimic the notionthat the bridge is naturally under constant stress and vibration,thus simulating constant oscillation. (Again, all models arewrong.) Your answer will depend on α andon k. Take your time and do good book-keeping.
As a side note, the homogeneous solution, yh(t), is oftencalled the transient solution andthat yp(t) is called the steady-statesolution for these second-order systems. This makessense, because the forcing function typically contributes more tothe long-run solution of the system than does the transientsolution (the solution of the homogeneous reduction of theODE).
Final Analysis
Now that you have a solution, recallthat k=1 (the springconstant). This value causes the oscillations we see inthe homogeneous solution; this is because k isthe "springiness" of the spring (the elasticity of the bridge, inthis case). That is, without any forcing, the model bridge wouldoscillate at 12π oscillations per second (one fullup-down-up motion every 12π, or 0.16 seconds). The forcingfunction contributes a term (the nonhomogeneous part) that causesoscillations at a frequency of 1α oscillations persecond. In our general solution, we sum the homogeneous andnonhomogeneous solutions, and so the natural oscillation of thebridge and the external force's oscillation "interact". Thisobservation is very important in seeing why theTacoma Narrows began to oscillate more and more violently until thematerial could no longer withstand it (thereby crumbling).
2. Graph the solution in Desmos and create a sliderfor α on the interval from 0 to 1. Experiment with theslider. State any interesting observations. Be a detective! Mostnotably, spend some time "fine-tuning" α as it approaches1 from values close to 1 (like 0.9, 0.99, 0.999). What do younotice as α→1, that is, as α→k? What happens to theoscillations?
3. Take a peak again at your general solution equation.What happens in the denominator of one of the termsas α→1? Does this further support what you saw in thegraph in 6)?
4. You have just investigated one of the culprit's of thebridge's collapse. What can you say about what happens to thedisplacement of the bridge from rest, when the frequency ofoscillations of the external environment (sound waves, wind, etc.)approach the same frequency as that of the natural oscillations ofthe bridge?
As you may have noticed, damping is precisely what causes aspring-mass system to eventually return to rest. Is this reasonableto assume about a bridge? Probably not. With the effects of trafficand natural vibrations, a bridge is typically under regularoscillation. When we add a forcing function, what we start to seechanges. Let's suppose that f(t)=sin(αt). In layman's terms,we are now imposing an oscillating external force. That forcevaries with time and is in units of Newtons (N), just like thespring force term and the damping force term. We see that theperiod is 2παseconds per cycle and the frequency of this forceis α2π cycles per second. In otherwords, α controls how rapidly the external forceoscillates.
1. Using your understanding of solving NHSOLDE's, solve the newsystem with m = k = 1and f(t)=sin(αt). 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 thelong-run. Note that your solution will have α in it.
COMMENTS: Having b=0 will do two things: a)it will simplify the mathematics, and b) it will mimic the notionthat the bridge is naturally under constant stress and vibration,thus simulating constant oscillation. (Again, all models arewrong.) Your answer will depend on α andon k. Take your time and do good book-keeping.
As a side note, the homogeneous solution, yh(t), is oftencalled the transient solution andthat yp(t) is called the steady-statesolution for these second-order systems. This makessense, because the forcing function typically contributes more tothe long-run solution of the system than does the transientsolution (the solution of the homogeneous reduction of theODE).
Final Analysis
Now that you have a solution, recallthat k=1 (the springconstant). This value causes the oscillations we see inthe homogeneous solution; this is because k isthe "springiness" of the spring (the elasticity of the bridge, inthis case). That is, without any forcing, the model bridge wouldoscillate at 12π oscillations per second (one fullup-down-up motion every 12π, or 0.16 seconds). The forcingfunction contributes a term (the nonhomogeneous part) that causesoscillations at a frequency of 1α oscillations persecond. In our general solution, we sum the homogeneous andnonhomogeneous solutions, and so the natural oscillation of thebridge and the external force's oscillation "interact". Thisobservation is very important in seeing why theTacoma Narrows began to oscillate more and more violently until thematerial could no longer withstand it (thereby crumbling).
2. Graph the solution in Desmos and create a sliderfor α on the interval from 0 to 1. Experiment with theslider. State any interesting observations. Be a detective! Mostnotably, spend some time "fine-tuning" α as it approaches1 from values close to 1 (like 0.9, 0.99, 0.999). What do younotice as α→1, that is, as α→k? What happens to theoscillations?
3. Take a peak again at your general solution equation.What happens in the denominator of one of the termsas α→1? Does this further support what you saw in thegraph in 6)?
4. You have just investigated one of the culprit's of thebridge's collapse. What can you say about what happens to thedisplacement of the bridge from rest, when the frequency ofoscillations of the external environment (sound waves, wind, etc.)approach the same frequency as that of the natural oscillations ofthe bridge?