# The derivation of how resonant frequency can amplify vibration in bridge

refer to here:

One of the most famous examples of a resonance disaster is the 1940 Tacoma Narrows Bridge collapse in Washington, USA. This came about not simply as a result of mechanical resonance, but also aeroelastic flutter – a process that occurs when complex, varying oscillations are caused by passing winds.

Leaving the issue of aeorelastic flutter aside, I understand that resonant frequency can somewhat increase the vibration of the bridge, and hence may result in the collapse of a bridge.

But I fail to see this, from the known physics laws. Any mathematical derivation for this from the first principles?

• Driven harmonic oscillator: en.wikipedia.org/wiki/…. The quote is correct, though, that's not enough to explain what happened at that bridge. The catastrophic events there are a lot more complex. – CuriousOne May 20 '16 at 8:12
• @count_to_10: The flutter is a complex, non-linear aerodynamic phenomenon with 3 dimensional vortex-shedding that has required an enormous amount of engineering time to understand and overcome (imagine it happens to your jet-liner's steering surfaces or wings at 600mph!), the driven harmonic oscillator is a ten line (or less?) solution to a trivial single variable linear differential equation. I don't think "just" does either problem justice. – CuriousOne May 20 '16 at 8:19
• @CuriousOne, I'm unsure how the equations in your wiki reference explains about the resonant phenomena. can elaborate it in the answer? – Graviton May 20 '16 at 8:27
• @CuriousOne Touche! And apologises for the crosspost. I watched a YouTube video of the flutter testing at near mach 1 speeds of the 747 prototype in the 60's , scary stuff, without any easy way to exit the aircraft before, potentially in that case, you were left sitting in "just" basically a large tube. – user108787 May 20 '16 at 8:28
• @count_to_10: I can only agree that some of this is indeed scary. What many of us don't realize when we are flying is how close these planes operate to some really critical performance envelopes. A few percent deviation of air-speed during landing, a few degrees climbing angle too much and catastrophe is imminent if the pilots or the control systems don't react correctly and in time. The bridge is still fascinating to watch... hard to believe that it held for as long as it did at those oscillation amplitudes. – CuriousOne May 20 '16 at 8:36