Non-Linear Dynamics and Controls of Non-Local Strain Gradient Beam with Harmonic Balance, Sliding Mode and Feedback Linearization Methods

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.