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

Abstract

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.

Analytical Solutions for Bending of Nanoscaled Bars Based on Eringen’s Nonlocal Differential Law

“…It is implied that the deflection decreases with an increase in the ratio of internal to external length scales since a larger external length scale (or a smaller ratio) leads to a higher midpoint deflection under given parameters. This is consistent with the conclusion in previous studies [9,11,13]. Hence, the validity of the present model is confirmed once again.…”

“…It seems that the horizontal compressive force should not be zero in the solution (13), but it is not a contradiction. This is because we are not studying the ordinary bending of beams (as we know, the governing equation for ordinary bending of nonlocal beams is a fourth-order differential equation), but the bending of a nanoscaled bar (as we see in Equations (5a) and (5b), it is a secondorder differential equation), and hence, P ≠ 0.…”

“…The classical solution of the midpoint deflection is recovered with e 0 = 0 or k = ffiffiffiffiffiffiffiffiffi P/EI p in Equation (13), which means that the classical counterpart can be obtained without the nonlocal effect. Assuming P = 5 N, Q = 1:5 N, q = 2:5 × 10 7 N/m, and l = 20 nm, the relationship between the deflection of the midpoint and the nonlocal small scale as well as the material rigidity is shown in Figure 3.…”

“…During the development of nonlocal theory, there were two kinds of nonlocal models in nonlocal differential constitution and both were studied extensively. When increasing the nonlocal scale parameter or when the nonlocal effect gets stronger, the increasing and decreasing trends of deformation result in different kinds of nonlocal differential models, namely, the nonlocal stiffness-weakening model (e.g., [3][4][5]) and the stiffness-strengthening model (e.g., [9,11,13]). The strengthening model declares that nanostructural stiffness is enhanced with stronger nonlocal effects, while the weakening model asserts an opposite conclusion.…”

On the Vibrations and Stability of Moving Viscoelastic Axially Functionally Graded Nanobeams

“…Their results illustrated that when the length scale parameter of the system is relatively equal to the thickness of the microbeam, the natural frequency, stability, and critical velocity of the system dramatically enhanced. Liu et al [16] surveyed the dynamical response and instability thresholds of axially moving nanobeams with time-dependent velocity. They concluded that the natural frequency of the system first decreases slightly, and then increases rapidly with an increase in the system velocity.…”