Investigation of the non-linear vibration behaviour and 3:1 internal resonance of the multi supported nanobeam

In this present work, linear and non-linear vibration of multi-supported nanobeams, which are a fundamental part of the nano-electromechanical systems, is examined. To the best of the researchers' knowledge, there is no study performed into multi-supported nanobeam in the literature. The governing equations of the system are obtained by dint of the Hamilton principle and solved via the perturbation technique which is divided linear and non-linear parts of the main equations. The natural frequencies and mode shapes are calculated from the linear problem. The non-linear natural frequencies and amplitude-phase modulation graphs are obtained from the non-linear equation. All equations are written in generalized form, and 3, 4 and 5 supported nanobeams are investigated in detail. The nonlocal coefficient, support number and position and end condition types are focused on. The three to one internal resonance cases are also investigated. It is occurred that the clamped-end conditions shift right in the hardening behaviour graphs more than the simply supported condition. Moreover, it is shown that the supported numbers play a significant role in natural frequency.