## 1 Introduction  
Nano structures have attracted a huge amount of attention from the scientific community due to their extraordinary mechanical, electrical, thermal and other physical/chemical properties. These novel properties can be for nanoelectromechanical systems (NEMS), nanoresonators, nanosensors and nanogenerators [1; 2], which are always modeled as nanobeams [3], nanorods [4], nanoribbons [5], nanoplates [6], nanoshells [7], etc to investigate the mechanical properties and other physical properties. Currently, there are three main kinds of approaches available for the study of nanomechanics, which are experimental techniques [8], molecular dynamic (MD) simulations [9] and modified continuum mechanics theories [10; 11], respectively. However, nano scale experiments are exceedingly difficult and MD simulations are costly in terms of computation. As a remedy, continuum mechanics theories are a valid selection for nanomechanics. The nonlocal elasticity theory established by [PERSON] [12; 13] is one of the typical continuum mechanics theories. It is assumed that the stress at a reference point is affected not only by the strain at that point but also by the strains at every point in the rest of the domain. Therefore, based on the nonlocal elasticity theory, a couple of researchers have applied their works to the bending, vibration and stability of nanostructures [14; 15; 16]. Moving nanosystems including NEMS, nanoresonators, nanosensors and nanogenerators are actually common in engineering applications, which can be modeled as moving continuum. Many researchers have studied the dynamical behaviors of such moving nanosystems including the transverse vibration characteristics, dynamical stabilities and dynamic response. [PERSON] et al. [17] studied dynamic stability in transverse parametric vibrations of an axially accelerating tensioned beam of a Timoshenko model on simple supports. The Galerkin method is applied to discretize the governing equation and the averaging method is done to analyze the instability phenomena. [PERSON] [18] studied the dynamic instability of an axially moving single-walled carbon nanotube (SWCNT) with simply supported ends by using nonlocal Rayleigh beam theory. The obtained results show that small-scale parameters would result in the occurrence of both divergence and flutter instabilities at lower levels of speed. [PERSON] et al. [19] presented a time/wave domain analysis for an axially moving pre-stressed nanobeam by a wavelet-based spectral element (WSE) method, which is validated to be accurate and efficient. The effects of the moving nanobeam property including velocity, pretention and nonlocal parameter on vibration and wave characteristics and dispersion curves were investigated. Also, the instability of moving nanobeams was studied considering divergence and flutter. [PERSON] et al. [20] investigated the dynamic behavior of axially moving nanobeams based on the nonlocal strain gradient theory considering the geometrical nonlinearity. The roles of small-scale parameters on the flutter critical velocity and stability were explained. [PERSON] [21] presented the transverse vibration behaviors of axially moving nanobeams considering the thermal effects and strain gradient. [PERSON] and [PERSON] [22] studied the nonlocal nonlinear vibrations of a viscoelastic Rayleigh nanobeam with small fluctuation in the axial speed by applying the nonlocal theory. [PERSON] [23] investigated the longitudinal and transverse wave propagation of stationary and axially moving carbon nanotubes (CNTs) with conveying nano-fluid. Recently, [PERSON] et al. [24] studied the transverse free vibration and stability of an axially moving nanoplate. The effects of dimensionless small-scale parameters, axial speed and boundary conditions on the natural frequencies in sub-critical region were discussed via the method of complex mode. The effects of small-scale parameters on divergence and flutter in super-critical region were investigated by the Galerkin method. [PERSON] et al.