Professor MathTheBeautiful, thank you for an exceptional video/lecture on the Classical Resonance and its powerful impact on everyday life on planet earth. This is an error free video/lecture on UA-cam.
Super Amazing explanation ... I ever had being a PhD phy Student... Thanks for series professor... students around you are really lucky.. how you are making mathematics colourful is astonishing... please accept by best regards...
Prof. Grinfeld, regarding the Tacoma Bridge, how come the wind has a frequency? If yes, how do I calculate it? I think of the wind as an equalization of temperatures, kind of a one way street, but no frequency involved! Thanks!
That's a good question. I think of it as air traveling through a trumpet. It passes through (like wind through a valley) but there's also a standing pressure wave whose length is determined by the dimensions of the cavity (valley).
Daniel Volinski If you look at the bridge in elevation view, it's possible that the wind load is vibrating if there's a spiral path out of page (normal to elevation). This is a fluid mechanics problem, but the design of the solids can create a path for wind to do this (in other words, the solids would act as boundary conditions for a flow problem). Not sure how you'd determine that, but, in structural engineering, we use precedent. USGS site provides sufficient data... on the other hand, I'm sure there's research on this. But USGS is the standard for using territory data for design such as eq and wind loading. To mitigate resonant failures, you basically increase damping and stiffness. But you could also add innovative dampers that would be more effective in catastrophic events. Precendent helps because it gives us a reasonable way to "exaggerate" stiffness and damping values in designing. One last thing I'll add is to keep in mind the reality of wind loads. Wind loads are typically loads that do not last for too long. So, even with resonance, you'll want to look at collapse mechanisms so as to account for a reasonably long lasting wind (maybe 5 minutes). That can also give a insight on how to brace for example. This idea of "short" time for loading seems contradictory to what this professor said about the t*cos () dominating, but it's all relative; 5 minutes means infinte in that sense and yet when wind is gone, the terms that vanished come back (sorry if that was poorly worded).
The reason why just the sine or just the cosine didn't work is because sines and cosines come in pairs - they are both two parts of the same circle, or in mathematical terms, of the same complex exponential. When you were dealing with just the exponentials, it worked, because the exponentials are complete. But when you have sines or cosines, you have to use both to complete the picture.
One thing I am still confused about: in the "real world," you would expect to have a very small error, and your equation might actually be u''+9u = 2sin((3+e)t) for a very small number e. But in this case, the solution is just the no resonance case with no t in front of the particular solution. However, intuitively, you would expect that if e is very small, resonance should still happen, even if it doesn't happen "as well." How is this problem solved? P.S. Thank you so much for this series, it's amazing!
Professor MathTheBeautiful, thank you for an exceptional video/lecture on the Classical Resonance and its powerful impact on everyday life on planet earth. This is an error free video/lecture on UA-cam.
Super Amazing explanation ... I ever had being a PhD phy Student... Thanks for series professor... students around you are really lucky.. how you are making mathematics colourful is astonishing... please accept by best regards...
You're great, please keep them coming. :)
Prof. Grinfeld, regarding the Tacoma Bridge, how come the wind has a frequency? If yes, how do I calculate it? I think of the wind as an equalization of temperatures, kind of a one way street, but no frequency involved! Thanks!
That's a good question. I think of it as air traveling through a trumpet. It passes through (like wind through a valley) but there's also a standing pressure wave whose length is determined by the dimensions of the cavity (valley).
Daniel Volinski If you look at the bridge in elevation view, it's possible that the wind load is vibrating if there's a spiral path out of page (normal to elevation). This is a fluid mechanics problem, but the design of the solids can create a path for wind to do this (in other words, the solids would act as boundary conditions for a flow problem). Not sure how you'd determine that, but, in structural engineering, we use precedent. USGS site provides sufficient data... on the other hand, I'm sure there's research on this. But USGS is the standard for using territory data for design such as eq and wind loading.
To mitigate resonant failures, you basically increase damping and stiffness. But you could also add innovative dampers that would be more effective in catastrophic events. Precendent helps because it gives us a reasonable way to "exaggerate" stiffness and damping values in designing.
One last thing I'll add is to keep in mind the reality of wind loads. Wind loads are typically loads that do not last for too long. So, even with resonance, you'll want to look at collapse mechanisms so as to account for a reasonably long lasting wind (maybe 5 minutes). That can also give a insight on how to brace for example. This idea of "short" time for loading seems contradictory to what this professor said about the t*cos () dominating, but it's all relative; 5 minutes means infinte in that sense and yet when wind is gone, the terms that vanished come back (sorry if that was poorly worded).
It's not that the wind that is vibrating but the bridge is vibrating on some (non vibrating load), its called flutter, pls look it up
The reason why just the sine or just the cosine didn't work is because sines and cosines come in pairs - they are both two parts of the same circle, or in mathematical terms, of the same complex exponential. When you were dealing with just the exponentials, it worked, because the exponentials are complete. But when you have sines or cosines, you have to use both to complete the picture.
Please why did he choose the cosine for Up instead of sine?
One thing I am still confused about: in the "real world," you would expect to have a very small error, and your equation might actually be u''+9u = 2sin((3+e)t) for a very small number e. But in this case, the solution is just the no resonance case with no t in front of the particular solution. However, intuitively, you would expect that if e is very small, resonance should still happen, even if it doesn't happen "as well." How is this problem solved? P.S. Thank you so much for this series, it's amazing!
Oh, this is just the conundrum of near resonance you talk about in a few more videos
That's right! Well done!
love the pizza tower !
The jokes are strong in this lecture series.
Your students don't realize how lucky they are!
Thanks for your comment. We have pretty amazing students at Drexel that make these videos possible.