Direct and Inverse Eigenvalue Problems Towards the Design and Identification of Mass-Loaded Micro-Resonators

The spectral data i.e. eigenvalues (natural frequencies) and eigenvectors (mode-shapes), characterizes the dynamics of the system. Non-destructive vibration testing, involving advanced experimental modal analysis techniques, has a potential to obtain the spectral data of the structures. It is well known that the dynamic characteristics of a structure will change due to the change in its physical properties. In this research, such changes in spectral behavior will be exploited towards the detection of minuscule changes in the mass of microstructures such as cantilever micro-beams, micro-resonators and oscillators, by solving certain direct and inverse eigenvalue problems. Some piecewise uniform micro-cantilever beams are considered here and associated transcendental eigenvalue problems are developed. Examples relevant to the design and identification of such beams are demonstrated through systematic mathematical modeling and effective solution strategy. It is shown that spectral behavior of mass loaded piecewise uniform beams can be obtained accurately and efficiently. Moreover, location and severity of the loaded mass can be identified successfully by using finite number of eigenvalues which may be available from experiments. Such formulations can be useful for, design and optimization of microstructures (micro-cantilever beams, resonators etc.), Bio-MEMS sensor design for the detection of single/multiple microbiological cells, and structural health monitoring.