Science

That’s Maths: Resonant vibrations from atoms to the far horizons of the cosmos

Understanding the inner workings of oscillations holds the key to comprehension of much of the modern world

Taiwan’s tallest skyscraper, the Taipei 101 building, recently survived an earthquake of 7.4 magnitude. Photograph: Sam Yeh/AFP/Getty Images
Taiwan’s tallest skyscraper, the Taipei 101 building, recently survived an earthquake of 7.4 magnitude. Photograph: Sam Yeh/AFP/Getty Images
Peter Lynch
Sat Apr 27 2024 - 12:16

Panta Rhei – everything flows – said Heraclites, describing the impermanence of the world. He might well have said “everything vibrates”.

From subatomic particles to the farthest reaches of the cosmos, we find oscillations. Vibration is key for aircraft wing, motor engine and optical system design. Ocean tides forced by the moon and seasonal variations of the atmosphere are oscillations on longer timescales. The stability of the solar system depends on orbital resonances, small-number ratios between the periods of the planets.

Many simple mechanical systems like springs and pendulums oscillate about a position of equilibrium, their motion being sustained against damping by some forcing mechanism. Thus, the balance wheel of a watch spins back and forth regularly, impelled by a spring. In a quartz watch, a crystal oscillates, with periodic injections of electrical energy to keep it going.

The particular modes of oscillation – called normal modes – characterise the system. If the force that nudges it is carefully tuned, the oscillations can reach large amplitudes. A child on a swing is a familiar example. Another is a wine glass which, if lightly tapped, will ring with a clear pitch. If subjected to a high-amplitude tone at this pitch, the glass may shatter. This strong response to an external force is called resonance.

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Resonance for good and for evil

Resonance can be beneficial if controlled. It is critical for the design and performance of most musical instruments, and essential for radio telecommunications, laser physics and many areas of engineering. But resonance can have destructive effects if precautions are not taken to prevent them. One dramatic example of catastrophic structural failure was the collapse of the Tacoma Narrows Suspension Bridge in 1940.

The Spire of Light towers up 120 metres over Dublin’s O’Connell Street. It was erected in 2003, filling the former site of Nelson’s Pillar. The Spire is simple in appearance, but its design involved some advanced engineering. If unconstrained, wind-induced oscillations of the Spire would have the potential to grow to the point of collapse. As with many mechanical structures, the Spire has normal modes of vibration. If strong winds vary with a period matching a particular mode, the oscillations can grow to destructively large levels.

Pendulums for stability

Theoretical analysis and mathematical modelling during the design phase showed the fundamental oscillation period of the Spire is about 3.6 seconds, with a second mode having a period of about 1.2 seconds. Two tuned mass dampers, each weighing about a tonne, were installed within the Spire to suppress these oscillations. They are like giant pendulums, with periods close to the first two resonant modes. The masses dissipate the energy of oscillations, like the shock-absorbers in a car. So far, they have done their work well.

The Spire is simple in appearance, but its design involved some advanced engineering

Recently, Taiwan’s tallest skyscraper, the Taipei 101 building, survived an earthquake of 7.4 magnitude. The building was designed to withstand violent typhoons with winds of up to 200km/h, and earthquakes likely within two millenniums.

The structure incorporates an enormous tuned mass damper, like that in the Spire but on a much grander scale. The damper can reduce oscillations by 50 per cent or more, averting cataclysmic failure. Hanging some 300 metres above ground level, the bob of this pendulum is a massive steel sphere weighing 660 tonnes, orders of magnitude heavier than the dampers in the Spire.

Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin. He blogs at thatsmaths.com