Modal Analysis and the collapse of Tacoma Narrows Bridge

Watch the video below before you read the rest of the post. It’s among the most famous film clips in history, important enough to have been selected for preservation by the National Film Registry of the Liberty of Congress.

The Tacoma Narrows Bridge in the U.S. state of Washington which was opened for the traffic on July 1, 1940, collapsed just after four months on November 7, 1940. The only life lost was a dog who drowned inside his owner’s car after it tumbled into the Puget Sound river. Unfortunately, the owner failed in his quest to rescue the dog from the bridge. This dramatic act has been one of the most interesting topics of engineering case studies for many years.

The question is: What made this bridge flutter so violently and what could have been done to avoid the catastrophe?

Ever tried doing a “Modal analysis” in ANSYS? If yes, you might have got the answer. For those of you who haven’t, let’s start with some basics of why we carry out modal analysis of a structure and why is it so important to do so. Eventually, you’ll be able to correlate this concept with the failure of the Tacoma Narrows bridge.

SHO

The figure above shows a Simple Harmonic Oscillator consisting of a rigid mass attached to an ideal spring. If this spring-mass system is driven (by an external source) by a sinusoidal displacement of frequency ω and peak amplitude|u|, it will produce a sinusoidal displacement of the mass M with a peak amplitude|x| at the same frequency (since the medium is the same). Now, we come across the term Transmissibility T which is given by:

T

where ω0  is the natural frequency of the spring-mass system:

omega not

When T is plotted as a function of ω/ω0 , the following graph results (couldn’t find a simpler plot than this on the net so, made my own):

capture-e1533500368264.png

Three salient characteristics are evident from the plot:

character

The second feature of the three is of paramount importance! Put simply, it says that when the frequency of the input source matches the natural frequency of the system which it is driving, the amplitude theoretically tends to infinity if the system is undamped. And this amplified amplitude of the system is what leads to catastrophic failure. But what is natural frequency?

At the sub-atomic level, the atoms in a body are in a constant state of motion having an average frequency of vibration called the natural frequency. When an external load is applied, the energy is stored in the body in the form of kinetic and potential energies and transmitted via atomic vibrations. Much of this stored energy is lost as heat due to friction. But when the frequency of the applied load is same as the natural frequency of the atoms, energy is transmitted with minimal loss. It is kind of a superposition of the applied frequency and the natural frequency which add up and result in the phenomenon of resonance.

When the input frequency interacts with the atomic vibrational frequency, one of the three responses might generate. The third response is obtained when the input frequency matches the natural frequency.

Capture3

Hence, one of the most important tasks is to calculate the natural frequencies of a system so that the frequency of the externally applied load doesn’t match with it. Additionally, proper damping of the system may be able to alleviate the resonance phenomena in case the two frequencies match somehow. Systems can behave erratically if the two frequencies near each other leading to failure and sometimes, even damping isn’t able to prevent the mishap.

And guess what? Modal analysis helps you do the same!

It is a simple way to calculate the natural frequencies of a system so you know which frequencies can be destructive for it and design the structure accordingly. You get to know the shapes and frequencies of the various modes of resonance (the number of modes can be infinite but you need just a few).

If you have read this far, you might have now made out what caused the dramatic failure of the Tacoma Narrows bridge. It is an example of forced resonance where the wind acted as an external source of frequency which matched the natural frequency of the bridge (ω ω0) leading to its fluttering and destruction.

Another area where Modal analysis is vital is when a transmission shaft is driven by a motor. The motor has a max. frequency when it operates at its max. RPM and there is a natural frequency of the shaft. The shaft is designed such that its natural frequency stays significantly above the motor’s max. frequency to avoid resonance and vibration. Modal analysis helps to determine those natural frequencies which aid in optimizing the shaft design.

I carried out the Modal analysis of a transmission shaft in ANSYS and the figure below shows the five successive resonance modes of the shaft which has a fixed support at the left end (similar to the air column in a pipe open at one end and closed at the other).

Though the collapse of Tacoma Narrows bridge was an epic engineering failure, it has served a valuable purpose for most engineering students, all attributes to Leon Moisseiff, the man who designed the bridge.

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