Identification of general added mass distribution in nanorods from two-spectra finite data

Nanomechanical resonators consisting in one-dimensional vibrating structures have remarkable performance in detecting small adherent masses. The mass sensing principle is based on the use of the resonant frequency shifts caused by unknown attached masses. In spite of its importance in applications, few studies are available on this inverse problem. Dilena et al. (2019) presented a method for reconstructing a small mass distribution by using the first N resonant frequencies of the free axial vibration of a nanorod under clamped end conditions. In order to avoid trivial non-uniqueness when spectral data belonging to a single spectrum are used, the mass variation was supposed to be supported in half of the axis interval. In this paper, we remove this a priori assumption on the mass support, and we show how to extend the method to reconstruct a general mass distribution by adding to the input data the first N lower eigenvalues of the nanorod under clamped-free end conditions. The nanobeam is modelled using the modified strain gradient theory to account for the microstructure and size effects. The reconstruction is based on an iterative procedure which takes advantage of the closed-form solution available when the mass change is small, and turns out to be convergent under this assumption. The results of an extended series of numerical simulations support the theoretical results