Dynamical responses and stabilities of axially moving nanoscale beams with time-dependent velocity using a nonlocal stress gradient theory

A higher-order mechanical model of axially moving nanoscale beams with time-dependent velocity was developed in the framework of nonlocal stress gradient theory. Based on the correlation between effective and common nonlocal bending moments, a sixth-order partial differential equation of motion with respect to the transverse displacement was derived. Unlike some previous work which assumed the velocity of axially moving nanoscale beam to be a constant, a time-dependent axial velocity was considered for the nanoscale beams. The resonance vibration frequencies were obtained according to the governing equation of motion and corresponding boundary conditions. It was concluded a nonlocal nanoscale strengthening effect that the vibration frequencies of such axially moving nanostructure increase with stronger nonlocal effects, or a larger dimensionless nonlocal nanoscale parameter causes a higher vibration frequency. A jumping phenomenon in frequency field was observed, and the vibration frequency may decrease or increase with an increase in the axial average velocity. Critical speeds of the axially non-uniformly moving nanoscale beams were defined and determined, and the critical speed versus nonlocal nanoscale revealed step and strengthening effects. The theoretical results obtained were compared with some experimental data and good agreement was achieved. Subsequently, the steady-state and stability of such moving nanostructures including the principal parametric and combination resonances were analyzed using a multiple-scale method. Some beneficial analytical procedures and theoretical formulations at nanoscale were provided. Based on specific boundary conditions, the stability boundaries of the axially accelerating nanoscale beams were determined and the unstable regions were influenced by nonlocal nanoscale significantly.