RESONATOR-BASED MASS IDENTIFICATION IN NANOBEAMS

In this thesis we present a methodology for solving a finite inverse eigenvalue problem arising in the determination of added distributed mass in nanobeams by using the first lower resonant frequencies either of the axial or bending vibration under suitable sets of end conditions. the nanobeams are modelled within a generalized continuum mechanics theory called modified strain gradient theory, to take into account the size-dependent behavior. The inverse method is based on an iterative procedure that produces an approximation of the unknown mass variation as a generalized Fourier partial sum of order N. The Fourier coefficients of the added mass are evaluated from the first N resonant frequencies belonging either to a sin- gle spectrum or two different spectra, for a mass variation with support contained in half of the nanobeam axis and for a general mass variation, respectively. The initial, unperturbed nanonbeam is supposed to be uniform and the mass variation is assumed small with respect to the total mass of the nanoresonator. The reconstructive method takes advantage of a closed-form solution when the mass change is small, and a proof of local convergence of the iteration algorithm is provided for a family of finite dimensional mass coefficients. An extended series of numerical simulations, also including cases with errors on the data, shows that the method is efficient and allows for accurate reconstruction of continuous mass coefficients. The accuracy deteriorates in presence of discontinuous mass coefficients. A constrained least- squares optimization filtering shows to be very effective to reduce the spurious oscillations near the discontinuity points of the rough coefficient. Surprisingly enough, in spite of its local character, the identification method performs well even for not necessarily small mass changes.

In this thesis we present a methodology for solving a finite inverse eigenvalue problem arising in the determination of added distributed mass in nanobeams by using the first lower resonant frequencies either of the axial or bending vibration under suitable sets of end conditions. the nanobeams are modelled within a generalized continuum mechanics theory called modified strain gradient theory, to take into account the size-dependent behavior. The inverse method is based on an iterative procedure that produces an approximation of the unknown mass variation as a generalized Fourier partial sum of order N. The Fourier coefficients of the added mass are evaluated from the first N resonant frequencies belonging either to a sin- gle spectrum or two different spectra, for a mass variation with support contained in half of the nanobeam axis and for a general mass variation, respectively. The initial, unperturbed nanonbeam is supposed to be uniform and the mass variation is assumed small with respect to the total mass of the nanoresonator. The reconstructive method takes advantage of a closed-form solution when the mass change is small, and a proof of local convergence of the iteration algorithm is provided for a family of finite dimensional mass coefficients. An extended series of numerical simulations, also including cases with errors on the data, shows that the method is efficient and allows for accurate reconstruction of continuous mass coefficients. The accuracy deteriorates in presence of discontinuous mass coefficients. A constrained least- squares optimization filtering shows to be very effective to reduce the spurious oscillations near the discontinuity points of the rough coefficient. Surprisingly enough, in spite of its local character, the identification method performs well even for not necessarily small mass changes.