The mortars and concretes used in civil construction are materials with a great many valuable properties, but they almost always have a low tensile strength. This is shown by the cracks seen on almost all buildings of a certain age. For this reason modern structural engineering found the remedy of reinforcing structures with steel bars (a material that is, instead, very resistant to this kind of stress). Reinforced concrete was born, with the result that weak or moderate earthquakes cause very little serious damage to modern skyscrapers, while they cause terrible devastation in less developed countries, where houses are built of stones or bricks bound with primitive mortars.

Here we want to illustrate briefly how the study of the resonances of a building, given the properties of the materials it is made of, can provide useful indications for designing buildings that better withstand dynamic stresses and, in particular, earthquakes.

A simplified dynamic analysis

From the point of view of mechanical stresses, it is useful to consider every building as a mechanical system made up of massive but virtually inelastic parts, and light, elastic connecting parts. If at this point "masses and springs" and "linear approximation" mean nothing to you, try looking at the sections on the superposition principle, normal modes and impedance.

Buildings are not always easily described by simple linear models, but the study of linear models is the first step towards an understanding, albeit approximate, of their dynamics.

Indeed, we know that a linear model, however complex, can always be broken down, with suitable mathematical manipulations, into a set of simple, independent oscillators. Each normal mode corresponds to a resonance of the system, that is, to a natural frequency at which the system is particularly vulnerable. Indeed, an external force, even of moderate intensity but of the right frequency, can induce in the system oscillations that can grow until they exceed the maximum stresses the system can tolerate, and break it to pieces.

physical model	dynamical model	dynamic simulation

The simulation shown above, a simple "masses and springs" model, shows the response of a relatively simple system to a sinusoidal excitation at the base (like an earthquake). The colour scale indicates the compression or extension of the elements. On this page of the Structural Control and Earthquake Engineering Laboratory of Washington University you will find an excellent series of videos showing scale models of buildings subjected to various excitations.

What we must do, then, is study the response of the system to excitations and, if necessary, design auxiliary oscillating systems to be introduced into the building so that its resonances at the critical frequencies are suitably damped. One example of this technique leads to the construction of seismic dampers, and of other systems for damping self-induced oscillations.

Seismic dampers

The page on resonance in engines shows how a system subject to vibration can be isolated from the base on which it rests by turning it into an oscillator and suitably modifying its mass, stiffness and friction.

But what can be done if the source of the oscillations is the base itself?

This is precisely the case of interest for buildings subject to seismic forces: the building itself can be considered a complex oscillating system (see the previous section), and the base (that is, the ground on which it rests) oscillates irregularly. Can we exploit the resonances of the building to our advantage? Or: can we modify them to our advantage?

The good news is that we already know the response function of the system. Indeed, the amplitude response, that is, the maximum amplitude of oscillation of the system as a function of the maximum amplitude of oscillation of the base, is identical to the force response of the system with a fixed base, that is, the maximum force transmitted to the base as a function of the maximum force applied to the system. We do not prove this statement, but we invite you to look at the behaviour of the response function on the page resonance in engines.

We already have three solutions: changing the mass of the building, changing its elasticity, increasing its friction. However, it is not easy to change the mass, because the building is designed to bear a certain static load, and increasing it is not a bright idea. As for the other two possibilities, we can think of mounting the whole construction on an elastic base and incorporating viscous friction dampers into it. Incredible as it may seem, this solution is possible, and has been put into practice, for example, for the historic Los Angeles City Hall building shown alongside. Seismic damping is obtained by means of 416 elastic rubber bands and 64 viscous dampers.^[1]

There is, however, another possibility that we have not yet considered. Again, to grasp the principle, let us study a simplified system, once again a harmonic oscillator with one degree of freedom.

Vibration absorption

Suppose we have a vibrating mass m₁. What happens if, instead of changing the parameters of the existing oscillator, we connect it to another oscillator of mass m₂? The page coupled oscillators gives the full theory of the free oscillations of two harmonic oscillators connected by a spring. Here the case is very similar: we want to study the forced oscillations of the system. The differences are that the external force is transmitted by the base, and that we have only two "springs" instead of three.

In any case a general principle holds (already stated on the introductory page on resonance): the overall system of the two oscillators has two degrees of freedom. We therefore expect it to have two resonances instead of one.

But then haven't we made the situation worse?

It depends: given that any mechanical system has natural frequencies and resonances at those frequencies, what we can aim for is not the elimination of all resonances, but the damping of some. Which resonances should be damped? Simple: those at which the system is least robust (that is, those with the largest amplitude), or those that affect it most often because they correspond to frequencies always present in the driving force. For example, it makes little sense to try to eliminate high-frequency oscillations in a building. Usually the dangerous frequencies are those corresponding to the first normal modes.

In our "toy" model there is only one resonance, so our goal becomes: can we damp the only existing resonance? Where are the two new resonance frequencies of the system made up of the two oscillators?

Without going into the details of the calculation of the response, we describe what happens in the following animations. In all cases we see that the response at the main resonance of the system is strongly damped, while two "side" maxima appear, corresponding to the two new normal modes of the system. Of course, we obtain satisfactory isolation only within a certain frequency range around f₀.

For ease of visualization, we start from the situation in which the main oscillator and the absorbing one are equal, m₁=m₂ and k₁=k₂. Obviously this is not the case for buildings, where in general ${\displaystyle m_1\gg m_2}$.

	Increasing the mass of the absorbing oscillator lowers the lower resonance frequency, while it has little influence on the upper one. The band in which the system does not respond to the external excitation widens towards low frequencies.
	Increasing the stiffness of the spring between the two oscillators favours the in-phase oscillation of the two masses. The lower frequency, in the limit, tends to ${\displaystyle \frac{1}{2\pi }\sqrt{\frac{k_1}{m_1+m_2}}}$, while the higher one becomes larger and larger. In this case too, the band of strongly attenuated frequencies widens as k increases.
	For simplicity, we consider only the friction of the absorbing oscillator, because that of the whole building is generally negligible. We see that now, with the position of the two resonances fixed, the effect of damping is to attenuate the response, particularly at high frequencies.

Real examples

The absorption mechanism illustrated in the previous section has been applied to real constructions such as bridges and skyscrapers. In skyscrapers and towers, adding a large oscillating mass on the top floors helps to stabilize the building when it is set into oscillation. If the source of the oscillations is seismic events, base isolation is preferred, like that illustrated on the page resonance in engines and in the section on seismic dampers, but there are also other sources of external forces. In particular (as also happens for bridges and suspended structures) one must consider forces of hydrodynamic origin, produced by the wind striking the exposed surface of the construction.

Although not very intense at any instant, these forces, even though they are not periodic, can give rise to self-induced oscillations (see the case of the Tacoma Narrows Bridge). Installing a damper can limit the damage to the structure caused by the oscillation. Below is a photograph of the largest damping mass in the world: a 662-tonne pendulum placed at a height of 380 m between the 88th and the 92nd floor of the Taipei 101 tower in Taipei, Taiwan (photograph on the left). Next to it, the principle diagram of the damper, and a technical drawing of the top floors of the tower, showing precisely where the oscillating mass has been placed.

Finally, we only mention the possibility of obtaining better control over the oscillatory response of complex systems by using active dampers. These are real actuators that act so as to actively counter the unwanted forces acting on a structure by developing counteracting forces, that is, by negative feedback. Unfortunately, going deeper into this interesting field of engineering is beyond the scope of this page.

References and external links

• On the site of the Structural Control and Earthquake Engineering Laboratory of Washington University you will find an excellent series of videos showing scale models of buildings subjected to various excitations.
• A list of large structures fitted with dampers of various types on the site of the National Information Service for Earthquake Engineering.