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Do you happen to know relevant references in the literature that model the mechanical vibrations of a solid object (preferably antennas) by means of complex representations as a function of time?, i.e. can we say that the radial displacement can be modeled as
$$ r(t) = \sum_{i=1}^{N}{a_i \exp\left(-j b t \cos(\phi_i)\right)}$$
where $a_i, b,$ and $\phi_i$ are constant independent of time $t$ and $j$ is the complex number such that $j^2 = -1$.
No, I didn't check but I don't think these type of equations would be in a mechanical vibrations textbook mainly because a forcing function or initial displacement is not present, nor is the distance from the base of the antenna. In addition to mechanical vibrations equations, Finite element analysis can provide radial displacement. Textbooks links are below:
