Resonance & Structures
Posted by Bradley Cochrane on 07.21.17
Have you ever heard the story of the opera singer breaking a wine glass with only her voice? How could that happen; does the glass shatter because of the sheer volume of her voice, or is it a trick? Although volume can play a role, the primary cause is that the singer’s voice reaches a specific frequency, the same natural frequency as the glass. When the frequency of the singer’s voice matches the natural frequency of the glass, it imparts onto the glass an oscillating force that quickly grows until the glass structure breaks into pieces.
As it turns out, this story is a perfect illustration for understanding resonance, and its affect on structures. You probably won’t find anyone wearing a helmet with horns from the opera capable of destroying a building or bridge with their voice, but wind, ground motion from earthquakes, and even synchronized footsteps certainly can and have caused structural damage through resonance. It is helpful to understand what is meant by frequency, resonance, some common examples of resonance frequencies, and a less commonly known example of resonance with a 1,000-foot-long pedestrian bridge in London.
Frequency is usually measured in hertz (Hz) which is described by one cycle per second. An example of something that has a frequency of one Hz is the second hand on your watch in that it completes one tick per second.
Fig. 1 The blue wave completes ten cycles while the red wave completes one cycle, therefore, the frequency of the blue wave is ten times higher than the red wave. (wolframalpha.com)
Resonance is defined by Oxford Dictionary as, “The condition in which an object or system is subjected to an oscillating force having a frequency close to its own natural frequency.”
What is a natural frequency? Every structure, from a simple wine glass, to a 100-story skyscraper, has a natural frequency. For example, the eyes you're using to read these words have a resonance frequency of about 50 Hz. Have you ever been to a concert and you can literally feel the music in your chest? Your chest wall has a resonance frequency range of 50-100 Hz, which coincides with the bass portion of a song. The Beach Boys may not agree but that doesn’t always inspire Good Vibrations.
Power vs. Persistence
It is safe to say that most of us have seen the destruction caused by wind and earthquakes. For example, tornados and hurricanes don’t have to sustain the resonance frequency of a building roof to damage it; usually it is the sheer power of a storm that causes destruction. This is an obvious mode of failure, and is reflected by governing codes that structural engineers use to safely design civilization’s structures. What is of special interest is the covert destroyer that doesn’t utilize a 200 mph gust of wind, but a steady 40 mph wind. After all, if a structure is safe at a high wind speed, why do we need to pay any attention to wind at lower speeds? If a pedestrian bridge in London is structurally capable of supporting 5,000 people crossing the River Thames, how come we must check possible failure modes if the bridge only has half that number of people on it? You guessed it – resonance.
London Millennium Footbridge
Fig. 2 London Millennium Footbridge crossing the River Thames (https://upload.wikimedia.org/wikipedia/commons/5/57/Mill.bridge.from.tate.modern.arp.jpg)
The London Millennium Footbridge opened to the public June 10, 2000 but was closed two days later for two years due to lateral oscillations felt as pedestrians crossed the suspension bridge. Designed for a load of 5,000 people it was somewhat surprising that it only took approximately 2,000 people for the wobbling to begin. Imagine yourself gridlocked in a crowd, trying to walk over one-thousand feet to the other side of River Thames on a beautiful new bridge. Left, right, left, right, almost like you’re marching in the military; the person in front of you barely has time to move their left foot before your left foot takes its place, and your right foot narrowly escapes the approach of the person’s behind you. Your right step now has the force of 2,000 right steps, and the same with your left step. Everyone becomes further synchronized as the bridge reacts laterally, and soon the problem is exacerbated as those steps come down with more force to keep you from falling over, which only causes the bridge to react further, and so on. Suddenly, you find yourself trying to escape the bridge while the bridge is seemingly trying to escape you.
This wouldn’t happen with just one person, but the point is that it didn’t require 5,000 people. Similar to how the volume of the opera singer’s voice does play a role in our story, but isn’t the primary factor in the glass shattering, the amount of people certainly plays a role in the oscillation of the walkway, but what really matters is the frequency of their steps. If the same crowd had slowly walked, or even sprinted in unison, the oscillation wouldn’t have occurred since the frequency of their steps would not have matched the natural frequency of the structure.
Fortunately, this was recognized early without casualties or substantial damages, and subsequently very well documented by P. Dallard and others (see References). The first thing that was attempted to limit the “wobbly bridge” from wobbling was to limit the number of people allowed to cross at any given time. As you now can imagine, this didn’t solve the problem; however, during the two-year closure a solution was found by installing dampers.
Fig. 3 One of thirty-seven fluid-viscous dampers installed to control horizontal movement (https://upload.wikimedia.org/wikipedia/commons/6/60/Bridge_vert_mode_shock.jpg)
Dampers come in many forms, and they work by dissipating energy. Two types of dampers were used in this case, fluid-viscous and tuned mass dampers. Fluid-viscous dampers help by converting kinetic energy into heat or another form of energy. Tuned mass dampers reduce the amplitude of oscillations, which effectively turns 2,000 synchronized foot steps into 20. When the bridge was re-opened after the dampers were retrofitted to the structure, the Millennium Bridge no longer wobbled and the problem posed by resonance was defeated.
We’ve learned a lot about resonance, and we are constantly learning more. Structural engineers have used the failures due to resonance that history has to offer to design new ways to protect the public in areas like London on bridges, as well as seismically active areas like Los Angeles and New Zealand using base isolation systems. The solutions are growing in number and effectiveness, leading us to a safer tomorrow. The first and most important step, as always, is to identify and understand the problem. So next time you go to the opera with your friends and they wonder why your wine is in a plastic cup, you can wait for the high note to introduce them to resonance.
Dallard, P., A. J. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. M. Ridsdill Smith, and M. Willford. “The London Millennium Footbridge.” The Structural Engineer 79.22 (2001): 17-33. Web.