Recovering added mass in nanoresonator sensors from finite axial eigenfrequency data

Abstract

Abstract In this paper we present a method for solving a finite inverse eigenvalue problem arising in the determination of added distributed mass in nanoresonator sensors by measurements of the first N natural frequencies of the free axial vibration under clamped end conditions. The method is based on an iterative procedure that 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. To avoid trivial non-uniqueness due to the symmetry of the initial configuration of the nanorod, it is assumed that the mass variation has support contained in half of the axis interval. Moreover, the mass variation is supposed to be small with respect to the total mass of the initial nanorod. An extended series of numerical examples shows that the method is efficient and gives excellent results in case of continuous mass variations. The determination of discontinuous coefficients exhibits no negligible oscillations near the discontinuity points, and requires more spectral data to obtain good reconstruction. A proof of local convergence of the iteration algorithm is provided for a family of finite dimensional mass coefficients. Surprisingly enough, in spite of its local character, the identification method performs well even for not necessarily small mass changes. To the authors’ knowledge, this is the first quantitative study on the identification of distributed mass attached on nanostructures modelled within generalized continuum mechanics theories by using finite eigenvalue data.