Problem — Natural frequency of a skyscraper

Problem

Estimate the natural frequency of horizontal oscillation of a skyscraper of height $h \approx 200\;\text{m}$, modelling it as a flexible beam analogous to an inverted pendulum. Discuss why civil engineers worry about avoiding resonance with seismic waves and wind.

Solution

A rule of thumb in structural engineering estimates the fundamental period as $T \approx N/10$ seconds, with $N$ the number of storeys. A $200\;\text{m}$ building has about $60$ storeys, hence $T \approx \frac{60}{10} = 6\;\text{s}$ A beam model gives the same order of magnitude: for geometrically similar beams the frequency scales as $f \propto 1/h$, so a tall building oscillates slowly. In any case the result falls in the band $\ev{f \sim 10^{-2}\text{–}10^{-1}\;\text{Hz}}$ that is, periods from a fraction of a second to several seconds.

Why it matters: long-period seismic waves and wind gusts have spectral content precisely in this band. If an external driving force coincides with the natural frequency, resonance occurs and the amplitudes grow until they damage the structure. This is why engineers design the building so as to shift the fundamental mode out of the dangerous bands and install tuned mass dampers (such as the large pendulum in Taipei 101), which absorb energy and reduce the amplitude.

Links

Topics: Oscillations and harmonic motion Concepts: Simple harmonic motion · Resonance · Period and frequency Skills: Fermi estimation · Small-oscillation approximation