Recovering added mass in nanoresonator sensors from finite axial eigenfrequency data

Non-standard and constitutive boundary conditions in nonlocal strain gradient elasticity

“…Thai et al [4] recently published relevant examples of the application of this theory to the analysis of the mechanical behaviour of nanostructures. Moreover, it is worth to cite the works by Morassi and coworkers [5][6][7][8][9] to address problems related with nanosensors. Other widely used approaches to address the mechanical behaviour of nanomechanical systems fall into the nonlocal continuum mechanics framework.…”

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

“…The analysis is applied to the transverse vibration of a clamped-clamped nanobeam, but, however, the formulation relies on the classical elasticity theory. Using the modified strain gradient framework [9] to account for size effects, the inverse problem of determining the mass distribution of a nanorod from the knowledge of the first N resonant frequencies of the free axial vibration under clamped ends was originally addressed in [73]. Let us recall that the free axial vibration of a nanorod is governed by a differential operator with fourth order leading term, instead of a second-order operator, as it occurs for classical beams.…”

“…Assuming that the mass coefficient is a priori known on half of the nanorod, and that the added mass is a small perturbation of the total mass of the nanosensor, the reconstruction procedure produces an approximation of the unknown mass density as a generalized Fourier partial sum of order N , whose coefficients are calculated from the first N eigenvalues. The approach corresponds to a mixed formulation of the inverse eigenvalue problem with finite data, see, for example, the interesting paper by Barnes [74] and the introductory section in [73] for an overview of the main mathematical features of this class of problems.…”

“…In this paper we continue the line of research initiated in [73] and we consider the more general inverse problem of determining a mass variation not necessarily supported in half of the nanorod interval axis. More precisely, we propose a reconstruction method based on the knowledge of a finite number of lower resonant frequencies belonging to two spectra corresponding to clamped-clamped and clamped-free end conditions.…”

“…It can be shown that the recourse to a second partial spectrum is necessary in order to avoid trivial non uniqueness of the solution to the inverse problem. Roughly speaking, the information coming from one spectrum was replaced in [73] by the a priori knowledge of the mass distribution on one half of the nanobeam axis. Under the assumption that the added mass is small with respect to the global mass of the referential nanorod, we show that the first-order frequency shifts can be used to determine a set of generalized Fourier coefficients of the unknown mass variation on a suitable set of functions.…”