According to common wisdom choosing the right moment for one’s action is considered to be essential in order to achieve something important. Being both too slow or too quick can result in missing an opportunity. To that end there are many proverbs about one or the other as listed for instance on Brainy Quote. The concept of the opportune moment has been around since olden times; it has even been given a name by the Ancient Greeks: Kairos, “the right, critical, or opportune moment”, a qualitative expression of time, as opposed to Chronos, is quantitative counterpart. What has the right moment to do with resonance frequencies?
In Engineering many problems can be viewed in both the time and frequency domain, in particular for repeating events, i.e. events occurring at a certain period; even a single event would have a period albeit infinity. For most engineers, shifting between the time and frequency representation of signals and events is their bread and butter. Therefore, even though recommendations about the right moment and the right frequency seem to be unrelated at a first glance, looking more closely, one realises that they talk about the same but from different points of view, in different domains: the opportune moment is linked to the opportune frequency, the resonance frequency which this story elaborates on.
Modern physics considers all matter to oscillate around a point of equilibrium, from atoms over complex mechanical structures to galaxies, each in its own natural modes and associated frequencies. The multi-modal oscillations produced by a physical body would comprise a wide range of frequencies which can be presented in a frequency spectrum, i.e. the amplitude (or energy, or power) of these modes as a function of frequency. The phenomenon of natural modes and frequences is present in many natural systems as well as human artefacts such as musical instruments. The simplest example of such a system is perhaps a vibrating string, or a children’s swing which can be modelled as a pendulum.
Let’s consider such a swing with a thin but strong rope of a certain length, mounted on one end on a robust beam, and having a person of a certain mass sitting on a board at the other end, as well as some friction in various forms, e.g. air resistance proportional to the speed of the end point of the swing. This arrangement has only one mode and one natural frequency like the pendulum used in a tall-case clock. Upon periodic excitation at a constant force amplitude, the pendulum, or the swing, responds with a periodic motion at a certain deflection amplitude depending on the excitation frequency as shown below in a so-called frequency response.
What does this frequency response plot show? Well, consider three different sample excitation frequencies, for instance when pushing somebody on a swing with a certain effort and at a frequency:
- below the natural frequency: suppose you push the swing at very low frequency, perhaps on a baby-swing. From a still stand you gently and slowly push forwards, investing energy to push the swing seat up; at its zenith, you then let it gently move back to the lowest point, absorbing all the energy already invested; then the same again backwards, up and back down to perform a full cycle and so on. It will feel tiresome as you have to maintain the swing in an “unnatural position”, pushing against gravity at all times.
- above the natural frequency: suppose you grab the swing seat and force a quick motion forwards and backwards. Apart from shaking the person on the swing beyond what can be tolerated, your arms will tire very quickly as you need to push hard against the inertia while not achieving much of an amplitude at all. Again, you would be trying to move the swing at an “unnatural speed” as it does not want to go that fast.
- at the natural frequency: suppose you push the swing at or close to its natural speed. You would have observed its motion beforehand and then only nudge it backwards and forwards. The energy invested every time you push will not be wasted but remains in the swing. The oscillation will increase in amplitude until a certain equilibrium is reached between the little energy being put in at every period and the energy dissipated in the system, e.g. through drag or other forces. Maximum gain is reached at the natural frequency.
What is the relationship between natural frequencies and opportune moments then? It is obvious for a swing: apply a gentle push forwards just after it has reached its highest point towards you and starts moving away, then repeat periodically at its natural frequency. This can be abstracted into more general terms: since everything exhibits natural frequencies, physical as well as non-physical system such as social systems, it is important to find out what the modes and their frequencies are, then act accordingly by making an effort when it is most opportune to add “energy”. The opportune moment for these dynamic systems is periodic, too, occurring at the natural frequency and within each natural oscillation period, at the right moment, at the right phase.
For many things we instinctively try to find out the natural frequencies: we tap on something and listen, we give it a bit of a push, shake it a bit, or try to get a response from a system with a controlled input, e.g. a gentle nudge of a new swing. This often tells us something about the natural frequencies and its natural modes of oscillations. Then we tend to work with the system, we don’t try to move too fast nor too slow, just at the right speed, frequency or time, all being interchangeable to a certain degree. The result is a maximum gain given a certain input, or conversely, the least resistance, if we think of how effortlessly a certain output can be achieved if the system is excited at its natural, opportune frequency.