Fourier deconstruction breaks down intricate signals into their building blocks for each vibrating entity, a highly complex multi-source vibration signal disentangling approach. Dividing intricate signals into their different crucial frequencies allows for easy and unambiguous source identification. Frequency domain graphic quality determines Fourier's effectiveness, and resolution relies on signal length and windowing technique choice. Fourier excels at uncovering stable state frequency points but may be ineffective for nearby or changing frequencies. More advanced techniques such as STFT and wavelet analysis are better for separating time-varying and closely spaced frequencies. Fourier stands out for basic frequency separation and is greatly valuable in vibration analysis when combined with alternative signal processing methods.

See Courant and Hilbert, Methods of Mathematical Physics and look at the section on "separation of variables."

More references:

https://web.ma.utexas.edu/users/jmeth/DEFall19/HeatEquation.pdf

https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Alevy.pdf