Three-to-one internal resonances of stepped nanobeam of nonlinearity

Abstract

In this study, vibrations of stepped nanobeams were investigated according to Eringen’s nonlocal elasticity theory. Multi-time scale method, which is one of the perturbation methods, has been applied to solve dimensionless state equations. The solution is considered in two steps. First-order terms obtained from the perturbation expansion formed the linear problem in the first step. In the second step, the solution of the second order of the perturbation expansion was made and nonlinear terms emerged as corrections to the linear problem from this solution. The main issue that the study wants to emphasize is the examination of the mechanical effects of the steps, which are discontinuities encountered at the nanoscale, on the system. For this purpose, while the findings of the research were obtained, various nonlocal parameter values were obtained to capture the nano-scale effect, and frequency-response and nonlinear frequency-amplitude curves corresponding to the 1st Mode values of the beam for different step ratios and step locations were obtained to capture the step effect. One of the important features of the nonlinear system is the formation of internal resonance between the modes of the system. How this situation affects the characteristics of the system has also been examined and results have been given by graphs. The obtained data show that taking into account the nanoscale step is essential for the accuracy and sensitivity of many nanostructures such as sensors, actuators, biostructures, switches, etc. that are likely to be produced at the nanoscale in practice.