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#049: Eigenfunctions for Systems

In previous posts (#26 #27) I have discussed eigenvalue problems and associated eigenfunctions/ eigenvectors with the latter describing the mode shape for discrete or continuous physics setups. An eigenfunction can for example be a cosine for the pressure eigenfunction in room acoustics, or some Bessel function for the mode shape of a membrane, or it could be a more general eigenvector with values describing some modal displacement which cannot easily be expressed as a function due to the complex geometry or physics involved. However, there is another situation where the terms eigenvalue and eigenfunctions pop up, and that is when dealing with systems.

In one of the previous posts I showed how for a 0D systems eigenvalues are really just the poles of the system, and that is correct, in that the associated eigenvector describes a particular displacement (assuming the physics being structural mechanics as an example), which can be sustained indefinitely without any source applied, and the eigenvalue describes at which frequency it occurs. There can be a finite amount of these eigenvectors or an infinite amount of eigenfunctions for a continuous system, but they are all discrete, so they only exist for these characteristic modes and not at any other frequencies. Whether they are exited or not is not important for their existence, but if a modal system is excited, it will after an initial source input which is then removed, the displacement will take place as a superposition of all modes to various degrees depending on the initial excitation.

However, when looking at transfer functions, we are pretty much looking at the opposite situation, and that is that we source with some particular frequency, and after an initial time, a transient behavior will have died out, a steady-state response remains, and this is where the transfer function is defined. The transfer function is typically some rational fraction as below.

Next, the transfer function is of course describing the relationship between output and the output as

but it is important to remember that this holds for a particular class of functions; the so-called complex exponentials, such that the above can be written in terms of these phasors as

where a complex exponential on the input will result in complex exponential on the output, but scaled with a complex value for each frequency. And this is why this complex exponential is sometimes referred to as the 'eigenfunctions' of the system since one can then relate the input and output via

where the (complex) value of the transfer function then represents the 'eigenvalue' of the problem. Some textbooks explicitly mention this eigenvalue/eigenfunction relationship, while others simply mention that the input signal has to have a sinusoidal characteristic for the analysis to make sense.

What can be confusing is there now seemingly are two types of eigenfunctions; one that has to do with the modes in system at characteristic eigenfrequencies only, and one that seemingly exists for any input frequency. The former are sometimes called 'normal modes', and perhaps I should have mentioned that in the previous post, and they are as mentioned related to the system itself without any sourcing, whereas the latter are related to the steady-state behavior for a source being a complex exponential (typically pure sinusoidal for a standard frequency response), and what the former modes do here is to cause resonances in the system. So this post is simply to clear up any confusion there might be about the two view points.

Another note to the above is that while the transfer functions are defined assumed so-called Linear-Time-Invariant (LTI) systems, where the output is linear in that only the frequency seen on the output is that seen on the input, the transient behavior will for example have mode excitation such that the output does not initially follow the input, although the system is linear. This again solidifies how systems are defined via their steady-state behavior, and in fact it only makes sense to talk about frequency itself(!) under steady-state conditions with signals that have in principle be 'on' forever, although for physical systems we of course have to just wait for a short while for the transient response to die out.

Finally, it is a good idea to view this from a state-variable approach, as there are in generally two matrices of interest, where one is for the system itself, and one is for the particular input/output setup in question, but I will leave this as an exercise for the reader.

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