Low-frequency Spectrum of the Sturm – Liouville Problem on a Metric Graph
DOI:
https://doi.org/10.52575/2687-0959-2026-58-2-130-136Keywords:
Metric Graph, Sturm – Liouville Problem, Laplacian, Model Order ReductionAbstract
This paper investigates the dependence of the natural oscillation frequency spectrum of a string grid on the mass distribution between the strings and nodes. The grid is modeled as a metric graph, for which a Sturm–Liouville problem is formulated. It is shown that as the grid spacing tends to zero, its dynamic properties approach those of a continuous membrane. An analysis of the equation describing the low-frequency part of the spectrum is carried out. The monotonic dependence of the eigenvalues on the linear density of the strings is proved. It is established that the spectrum of the grid is always enclosed between the spectrum of the difference Laplace operator and the spectrum of the membrane. Based on the obtained results, a justification is given for choosing the string grid as an effective reduced-order model for problems concerning membrane oscillations, which, in turn, leads to a reduction in the computational complexity of this problem.
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