Nonlinear Random Vibration of Nanobeam with Material Nonlinear Third-Order Elastic Modulus

In the present study, a comprehensive investigation is carried out on the stochastic nonlinear vibrations of a nanobeam subjected to both free and externally driven excitations. The nanobeam is assumed to rest on a viscoelastic substrate while experiencing an external load with randomly varying intensity. In developing the theoretical model, nonlinear kinematic effects arising from von Kármán strain assumptions are considered alongside a nonlinear constitutive relation describing the material nonlinearity. The motion equations governing the system are systematically derived via an energy-based variational framework, with small-scale phenomena accounted for through the application of nonlocal continuum mechanics. A modal discretization strategy is adopted to transform the resulting system into a reduced-order formulation, which is then treated numerically. The probabilistic features of the structural response are evaluated by means of Monte Carlo-based simulations. The system’s vibrational characteristics are explored through temporal response analyses employing both single-mode and bimodal representations under scenarios with and without energy dissipation, and particular attention is devoted to assessing how material nonlinearity alters the response behavior. Moreover, time–frequency representations are utilized to clarify the impact of damping on the response evolution. The uncertainty range associated with the temporal response is quantified by determining its extremal bounds, followed by an assessment of how nonlinear material effects influence these limits. The reliability of the developed framework is ultimately confirmed through benchmarking against existing findings previously published.