This paper aims at studying the vibrational behavior and dynamical stability of small-scale axially moving beams resting on the viscoelastic-Pasternak foundation in a hygro-thermal environment, according to a nonlocal strain gradient Rayleigh beam model. The Galerkin procedure is applied to determine the eigenvalues of the dynamic system of equations together with the stability regions of the system. A comparison study of the proposed method is performed, first, against the available literature. Thus, we examine the effect of the rotary inertia, flexural stiffness, boundary conditions, scale parameters, foundation conditions, and environmental loads, on the vibrational frequencies and stability boundaries of the system. Based on the numerical results, an increased flexural stiffness and strain gradient parameter enhance the vibrational frequencies of the system. It is also demonstrated that the destructive effects of hygro-thermal conditions can be alleviated by a fine-tuning of the foundation characteristics. The outcomes of the present research can represent a useful benchmark for optimization design purposes of moving nanosystems in complex environmental conditions.
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