Hearing distributed mass in nanobeam resonators

Abstract

One-dimensional vibrating nanostructures show remarkable performance in detecting small adherent masses added to a referential configuration. The mass sensing principle is based on measuring the resonant frequency shifts caused by the unknown attached masses. In spite of its important application in several fields, few studies have been devoted to this inverse eigenvalue problem. In this paper we have developed a distributed mass reconstruction method for initially uniform nanobeams based on measurements of the first lower resonant frequencies of the free bending vibration. Two main inverse problems are addressed. In the first problem, the mass variation is determined by using the first lower eigenfrequencies of a supported nanobeam, under the a priori assumption that the mass variation has support contained in half of the axis interval. In the second problem, we show that the a priori assumption can be removed, provided that the spectral input data include an additional set of first lower eigenfrequencies belonging to a second spectrum associated to different end conditions. The nanobeam is modelled using the modified strain gradient elasticity accounting for size effects. The reconstruction is based on an iterative procedure which takes advantage of a closed-form solution when the mass change is small, and shows to be convergent under this assumption and for smooth mass variation. The accuracy of the reconstruction deteriorates in presence of discontinuous mass variation. For these cases, a constrained least-squares optimization filtering shows to be very effective to reduce the spurious oscillations around the target coefficient. Numerical simulations show that the identification method performs well even for not necessarily small mass changes and it is stable in presence of errors on the data. An experimental validation of the method has provided encouraging results, despite the fact that only the first four eigenfrequencies under cantilever end conditions were used.