Effect of size on the chaotic behavior of nano resonators

Present study is devoted to investigate the size effect on chaotic behavior of a micro-electro-mechanical resonator under external electrostatic excitation. Using Galerkin's decomposition method, approximating the actuation force with a new effective lumped model, and neglecting higher order terms in the Taylor-series expansion, a simplified model of the main system is developed. By utilizing the Melnikov's method and based on the new form of the electrostatic force, an expression in terms of the system parameters is developed which can be used to rapidly estimate the chaotic region of the simplified system. Based on the analysis of the simple proposed model, it is shown that the effect of size on chaotic region varies significantly depending on bias voltage. By considering the size effect, it is demonstrated that chaotic vibration initiates at much higher constant voltages than predicted by classical theories; and, in high constant voltages, it is shown that strain gradient theory predicts occurrence of chaos at much lower amplitudes.