This study presents nonlinear dynamic analysis and active control framework for an Euler–Bernoulli nanobeam modeled using the Nonlocal Strain Gradient Theory (NSGT). The governing sixth-order nonlinear partial differential equation is derived using Hamilton’s principle with von Kármán geometric strain assumptions and reduced to a Duffing-type single-mode model through the Galerkin method. Bifurcation analysis is conducted to examine the effects of nonlocal and strain-gradient length-scale parameters on resonance behavior, stability boundaries, and hysteresis. Two advanced nonlinear control strategies: Sliding Mode Control (SMC) and Feedback Linearization with Proportional–Derivative and Feed-Forward compensation (FBL-PD-FF) are designed and evaluated under deterministic harmonic excitation and stochastic white-noise disturbances. Numerical simulations demonstrate that SMC offers superior robustness, disturbance rejection, and overshoot suppression, while FBL-PD-FF achieves fast response but suffers from higher sensitivity to nonlinearities and noise. The results provide theoretical insight and practical guidance for designing stable, high-precision MEMS/NEMS resonators operating under nonlinear and stochastic conditions.
Babaei et al. (Fri,) studied this question.