Uncertainty analysis of the free vibration of nanobeams with random non-ideal supports

This study investigates the free vibration behavior of nanobeams with uncertain non-ideal boundary conditions within the framework of the modified couple stress theory. To represent manufacturing induced variability in nanoscale supports, the non-ideal boundary parameters are modeled as bounded Beta-distributed random variables. Their mean values are assigned according to the support configuration, while the concentration parameter controls the uncertainty level. The main novelty of this study is the stochastic modeling of non-ideal support parameters in a size-dependent nanobeam formulation based on the modified couple stress theory. The resulting eigenvalue problem is solved by combining the analytical characteristic equation with Monte Carlo simulation. For each support configuration, 4000 Monte Carlo samples are generated, and the first three natural frequencies are statistically evaluated using probability density functions, box plots, and 5th-95th percentile bounds. The results show that boundary condition uncertainty significantly affects both the natural frequency levels and their statistical spread. The clamped-clamped configuration exhibits the highest frequencies and widest uncertainty range, whereas the simply supported configuration produces the lowest frequencies and narrowest spread. The proposed formulation may support the evaluation of support uncertainty effects in nanoscale resonator, sensor, and actuator applications.