Both surface elasticity and surface stress can result in changes of resonant frequencies of a micro/nanostructure. There are infinite combinations of surface elasticity and surface stress that can cause the same variation for one resonant frequency. However, as shown in this study, there is only one combination resulting in the same variations for two resonant frequencies, which thus provides an efficient and practical method of determining the effects of both surface elasticity and surface stress other than an atomistic simulation. The errors caused by the different models of surface stress and mode shape change due to axial loading are also discussed.

A different set of governing equations on the large deflection of plates are derived by the principle of virtual work (PVW), which also leads to a different set of boundary conditions. Boundary conditions play an important role in determining the computation accuracy of the large deflection of plates. Our boundary conditions are shown to be more appropriate by analyzing their difference with the previous ones. The accuracy of approximate analytical solutions is important to the bulge/blister tests and the application of various sensors with the plate structure. Different approximate analytical solutions are presented and their accuracies are evaluated by comparing them with the numerical results. The error sources are also analyzed. A new approximate analytical solution is proposed and shown to have a better approximation. The approximate analytical solution offers a much simpler and more direct framework to study the plate-membrane transition behavior of deflection as compared with the previous approaches of complex numerical integration.

With the adsorption of analyte on the resonator mass sensor, the system eigenfrequencies will shift due to the changes of inertial mass and structural rigidity. How to model those changes and formulate the eigenfrequency computation is very important to the mass sensor application, which results in different accuracies and requires different amounts of computation. Different methods on the eigenfrequency computation of a beam and a plate carrying arbitrary number of concentrated mass/spring are presented and compared. The advantages and disadvantages of these methods are analyzed and discussed. A new method called finite mode transform method (FMTM) is shown to have good convergence and require much less computation for a beam carrying concentrated mass/spring. Because the previous finite sine transform method (FSTM) has only been applied to compute the eigenfrequency of the plate with four edges simply supported carrying a single concentrated mass, here a generalized FSTM is also presented for the case of the same plate carrying arbitrary number of concentrated mass and spring. When the total number of concentrated mass and spring is small, FMTM and FSTM are demonstrated to be very efficient.

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.

The mass sensing superiority of a micro-/nano-mechanical resonator sensor over conventional mass spectrometry has been, or at least is being firmly established. Because the sensing mechanism of a mechanical resonator sensor is the shifts of resonant frequencies, how to link the shifts of resonant frequencies with the material properties of an analyte formulates an inverse problem. Besides the analyte/adsorbate mass, many other factors, such as position and axial force, can also cause the shifts of resonant frequencies. The in situ measurement of the adsorbate position and axial force is extremely difficult if not impossible, especially when an adsorbate is as small as a molecule or an atom. Extra instruments are also required. In this study, an inverse problem of using three resonant frequencies to determine the mass, position and axial force is formulated and solved. The accuracy of the inverse problem solving method is demonstrated, and how the method can be used in the real application of a nanomechanical resonator is also discussed. Solving the inverse problem is helpful to the development and application of a mechanical resonator sensor for two reasons: reducing extra experimental equipment and achieving better mass sensing by considering more factors.