What is Vibration and Resonance? The Basic Concept Behind and A Simple Mathematical Explanation

If you say that you and a friend or anyone close to you are in the same wavelength or you have the same vibration, it means both of you resonate with each other, or both of you are in sync. And the thing is, it is good. But in structural engineering, if the frequency of the excitation and the structure matched, it’s not that good at all. It will result in a “bad vibration”!

Boss Albert discussed this on our subject earthquake engineering under the topic of vibration and resonance due to harmonic and periodic excitations. His story about the actual application of the mathematics of vibration is very simple yet very compact that even a young engineer can grasp. And so I’ll relate his story with much gusto in my hope to illuminate the minds who seek a very simple explanation on the subject. I hope I can relay the very heart of which with clarity like how Boss Albert explained it to us:

Boss Albert had this steel bridge project which lies between Boac and Santa Cruz in Marinduque. The owners were alarmed one day when they allegedly experienced a violent vibration in the structure. And because of this, they called his immediate attention.

Along with a colleague who is also an expert in the field, they went to the site to investigate. They attached motion sensors to the bridge and rented a pick-up truck to simulate traffic which hopefully would cause the bridge to vibrate.

The pickup truck went back and forth along the bridge at various speeds. Nothing happened. It vibrated yes but the vibration is within acceptable limits. The behaviour of the bridge is normal.

This puzzled them and so for the whole day they repeated the process with the pick-up truck but to no avail.

When the day was almost over, exhausted and the problem still unsolved, they sat on the sides pondering what to do next.

Then came the answer to their dilemma.

People coming from Boac going to Sta. Cruz were flocking to cross the bridge. An approach is still under construction and thus unpassable for vehicles. And so the passengers have to pass through the bridge on foot going to the waiting jeepneys on the other side. They were in a hurry for a better seat inside the jeepneys because those who cannot make it inside will have to seat on top (which is typical in the provinces.)

It was a steady rhythm of 2 beats per second. Consistent. Periodic.

Just then, the bridge started to vibrate in a disturbing manner. So this was it.

He never told us how they solved the vibration problem which was beyond acceptable limits but it’s enough to point out the concept of resonance.

So what just happened?

The frequency of the external excitation (people walking) matched that, or came very near the natural frequency of the bridge and thus the violent vibration.

How do you explain it mathematically?

According to structural dynamics books on the topic regarding harmonic and periodic excitations (the snapshot of the equation above was screen-grabbed from Anil K. Chopra’s Dynamics of Structures, Theory and Applications to Earthquake Engineering) the ratio of the static displacement to the dynamic displacement Rd will be at its largest if the frequency of the excitation will match that of the natural frequency of a structure or a bridge which is the example above. In the first term of the denominator, if the frequency of the excitation w(omega) = w(omega,n) natural frequency, it will cancel or will yield zero, leaving only the second term. Without punching in the digits, we can see that the divisor will be less and thus will have a greater quotient and thus a greater Rd.

So let’s say we have a steel bridge with a damping ratio of 0.02, if w(omega) = w(omega,n) the resulting Rd factor would be 10. It just means that the static deflection will be magnified tenfold and it will vibrate back and forth violently. It would be lucky if it would only happen within a second or two but prolonged vibration of this magnitude would snap something somewhere and may result to the eventual failure of the entire bridge.

And so the main reason why they put tuned mass dampers on bridges and other similar devices on buildings is to prevent any external excitations to vibrate the structure equal to its natural frequency.