Resonator-based mass density identification in nanoplates

In this paper, we present a methodological and computational study of an inverse eigenvalue problem with finite data arising from the determination of a distributed mass added to a uniform rectangular nanoplate acting as a mass resonator. The nanoplate is modeled within a simplified strain-gradient theory in linear elasticity, under the Kirchhoff–Love kinematic assumptions and with simply supported boundary conditions. The identification method relies on the lowest (Formula presented) eigenvalues and is based on an iterative procedure that approximates the unknown mass density by means of a truncated generalized Fourier series computed on a specific family of test functions. The added mass is assumed to be small compared to the total mass, and its support is confined to a quarter of the rectangular domain. Numerical simulations show that the reconstruction of smooth mass distributions with connected support of diameter larger that (Formula presented) the average side of the nanoplate domain is quite accurate, even when the mass perturbation is not necessarily small. The identification of discontinuous mass densities is less accurate, and a Gibbs-like phenomenon occurs near the jumps. The accuracy of the reconstruction degrades progressively as the number of the connected components of the added mass support increases. Numerical tests show that the identification method remains stable in the presence of errors on the data, provided that such errors are, on average, significantly smaller than the natural frequency-changes induced by the added mass.