Studies on the dynamic stability of an axially moving nanobeam based on the nonlocal strain gradient theory

In this paper, we studied the parametric resonance issue of an axially moving viscoelastic nanobeam with varying velocity. Based on the nonlocal strain gradient theory, we established the transversal vibration equation of the axially moving nanobeam and the corresponding boundary condition. By applying the average method, we obtained a set of self-governing ordinary differential equations when the excitation frequency of the moving parameters is twice the intrinsic frequency or near the sum of certain second-order intrinsic frequencies. On the plane of parametric excitation frequency and excitation amplitude, we can obtain the instability region generated by the resonance, and through numerical simulation, we analyze the influence of the scale effect and system parameters on the instability region. The results indicate that the viscoelastic damping decreases the resonance instability region, and the average velocity and stiffness make the instability region move to the left- and right-hand sides. Meanwhile, the scale effect of the system is obvious. The nonlocal parameter exhibits not only the stiffness softening effect but also the damping weakening effect, while the material characteristic length parameter exhibits the stiffness hardening effect and damping reinforcement effect.