Contribuições à análise não linear de microestruturas

Abstract

Transducers play a crucial role in monitoring structures, as they are responsible for converting the physical variable studied into electrical signals that can be processed, stored and analyzed computationally. The model-based approach offers the advantage of allowing generalizations with numerical instrumentation, which can be compared with normative documents to evaluate their effectiveness or ineffectiveness in relation to improving regulatory standards. From this perspective, this work presents the modeling and static and dynamic behavior of embedded Euler- Bernoulli microbeams, based on the theory of elasticity for gradient deformation. Additionally, a non-linear model was incorporated to represent the electrostatic forces generated by the displacement of the microbeam, introducing additional complexity to the problem. The partial differential equation for deformation and boundary conditions was obtained using Hamilton's variational principle. Using the variable separation method, the derivative of a sixth-order ordinary differential equation for normal modes was calculated. The analysis of the characteristic equation resulted in a categorization of the roots, providing the determination of these, as well as the normalized eigenfrequencies and eigenfunctions. Using the Galerkin method and the Gauss- Legendre Quadrature with fifteen points for the calculation of integrals, a non-linear second order differential equation was determined for the generalized coordinates due to the electrostatic force. The numerical solution was approached with the Runge-Kutta method for nonlinear systems, considering five normal modes. The section dedicated to computational experiments presents graphs of normal modes, generalized coordinates and deformation for the classical and gradient cases. Therefore, comparisons were made between the results obtained with the proposed methodology and the results from classical theory. In this way, the relationships between the stiffness and the scale of the structure are observed for both cases. Thus, it was concluded that the gradient model satisfactorily represents microelectromechanical systems in comparison to the classical model, corroborating experimental results available in the literature.