Quantifying uncertainty in free vibration characteristics of nanobeam with one variable first-order shear deformation theory: an analytical investigation

This research investigates uncertainty quantification in nanobeam vibration by integrating the one-variable first-order shear deformation beam theory with Eringen’s nonlocal elasticity theory. Material uncertainties associated with mass density and Young’s modulus are represented using triangular fuzzy numbers. The governing equations for vibration are derived employing the von Kármán hypothesis and Hamilton’s principle, leading to closed-form solutions for the simply supported boundary condition through Navier’s method based on the double parametric form. A comparison of the obtained frequency parameters with those in previously published literature demonstrates strong agreement in specific cases. Additionally, a Monte Carlo Simulation Technique (MCST) based on random sampling is utilized to comprehensively assess the natural frequencies of the nanobeam amidst material uncertainties, providing insights into the variability of the system's response. In order to validate the results obtained through the uncertain model, the natural frequencies derived from Navier’s Method in terms of Lower Bound and Upper Bound are rigorously compared with those obtained through MCST. This comparative analysis underscores the efficacy, accuracy, and robustness of the proposed uncertain model in capturing the dynamic behavior of the nanobeam under varying material conditions. The computation of lower and upper bounds of frequency parameters using the double parameter, and graphical outputs demonstrating the sensitivity of the models, is plotted based on the triangular fuzzy number. This uncertainty modeling and the delineation of frequency parameter bounds offer effective tools for engineering structure design and quality optimization in nanobeam applications.