Detecting both the mass and position of an accreted particle by a micro/nano-mechanical resonator sensor

In the application of a micro-/nano-mechanical resonator, the position of an accreted particle and the resonant frequencies are measured by two different physical systems. Detecting the particle position sometimes can be extremely difficult or even impossible, especially when the particle is as small as an atom or a molecule. Using the resonant frequencies to determine the mass and position of an accreted particle formulates an inverse problem. The Dirac delta function and Galerkin method are used to model and formulate an eigenvalue problem of a beam with an accreted particle. An approximate method is proposed by ignoring the off-diagonal elements of the eigenvalue matrix. Based on the approximate method, the mass and position of an accreted particle can be decoupled and uniquely determined by measuring at most three resonant frequencies. The approximate method is demonstrated to be very accurate when the particle mass is small, which is the application scenario for much of the mass sensing of micro-/nano-mechanical  resonators. By solving the inverse problem,  the position measurement becomes unnecessary, which is of some help to the mass sensing application  of a micro-/nano-mechanical resonator by reducing two measurement systems to one. How to apply the method to the general scenario of multiple accreted particles is also discussed.