# Natural frequencies

Given the acceleration response time history of a multi-story structure, how can I find natural frequencies using time-frequency analysis techniques? If you just provide some references or articles, I truly appreciate it.

• What do you mean by "acceleration response", "multi-story", and "natural frequencies"? Commented Feb 24, 2023 at 15:41
• Is the applied acceleration known, or can one at least infer its spectrum (from, e.g., wind velocity over time, or ground acceleration over time?) If so, edit your question with this information. Commented Feb 24, 2023 at 18:47
• @OverLordGoldDragon: given that this is a signal processing group I think that "natural frequencies" should be self-explanatory. Commented Feb 24, 2023 at 18:49

In general, you need the time history of the excitation in order to interpret the response, like Dan Boschen has said.

But under certain circumstances, the excitation of the system can be assumed to be minimum entropy (or quite similarly minimum energy), which allows one to identify an estimate of the system's transfer function from the response alone. Therefore, this method does not require knowledge of the precise excitation, rather an estimate of the excitation is obtained from the method as well. This is called linear prediction.

I think the most prominent example of linear prediction is the estimation of the transfer function of the vocal tract from a voice recording. In the GSM standard this is used as a means to compress speech information, which is called Linear Predictive Coding (LPC). Only the resonances of the vocal tract have to be transmitted then (together with the tone pitch), not the individual samples of the speech. The reason why this works is that the vocal tract is excited by the vocal chords, which clap together and cause delta-peak-like excitations, which are minimum entropy. Also, if the person is whispering, the excitation is white-noise, which is also minimum entropy.

For a building, I don't know, but probably the random hits of an earthquake might serve as a minimum entropy excitation. But be aware that the total response in this case might be resulting from the building reponse and the response of the earth's crust (reflections? refractions? the original excitations is in several kilometers depth). So what enters the building foundation might already have been filtered by the earth. So don't take my word for it, that you can apply this to your use case.

If it was, you could also look at the spectrum of a single earthquake hit and see where some single most intensive resonances are, and you could even estimate the damping coefficient of a resonance (related to the width of the "spectral line"), if it is sufficiently far away from other resonances. But of course, this is only a very rough image, and it is not very rigorous. Moreover, it gets complicated if you want to find the mode shapes of the building, which requires the analysis of a lot of measurement locations in the building. This is infeasible to do by just looking at some plots, and so you would need the full math apparatus of linear prediction (again: if applicable).

The natural frequency of a system under test (in this case a building structure) cannot be determined from the response alone. One would need both the stimulus and response in order to determine the unknown characteristics of the channel. Otherwise we cannot isolate from the measurement what was due to the source or due to the system.

In the time domain, the response is the convolution of the stimulus in time with the impulse response of the channel, and have the response along with the stimulus can then allow us to determine the impulse response from which the natural frequency is readily determined using the Laplace Transform.

Similarly in the frequency domain, the spectrum of the output is the product of the spectrum of the input with the frequency response of the channel, and having the spectrum of the input and output allows us to determine the frequency response of the channel from which we can determine the natural frequency.

In either case, observe how both the input (stimulus) and output (response) would be needed and the natural frequency cannot be determined from the response alone. If the input stimulus is known, or even to an extent can be estimated, then with the resulting response the system characteristics including the natural frequency can be determined (or estimated).