# Size-Dependent Free Vibration of Axially Moving Nanobeams Using [PERSON]'s Two-Phase Integral Model
## 1 Introduction  
Axially moving nanoscale beams can be used in high velocity vehicles, spacecraft antennas, flexible nanorobotic manipulators, which has been shown to exhibit outstanding mechanical and physical properties. For such nanoscale structures, as their dimensions become comparable to the microstructural characteristic lengths, experimental results of their mechanical properties have shown a significant size effect which are not present at macroscale. Thus, for the design of such engineering structures, it would be very important to investigate the size effect in the dynamic behaviors of axially moving nanoscale beams using some proper models. At a macroscale, there are many related papers which deal with nonlinear vibrations of the axially moving beams based on classical (local) elasticity theory such as in Refs. [1; 2; 3; 4; 5], etc. According to these results, the initial excitation, external load and the axially moving velocity may lead to nonlinear transverse dynamic behaviors of the beam. However, classical elasticity assumes that the stress state at a given point depends only on the strain at the same point, which apparently does not take into account the effect of the small scales of nanostructures. With the growing need in analyzing size-dependent materials, nonlocal elasticity has received much attention. Nevertheless, there are some paradoxes concerned with nonlocal differential models [6], while particular integral formulations have been shown to be consistent. Nowadays, two integral models of pure nonlocal elasticity are available in literature. The former one was proposed by [PERSON] in Ref. [7] by considering an integral convolution whose input and output fields are elastic strain and stress fields, respectively. The attenuationfunction of the convolution was selected to be the fundamental solution of a differential problem in \(R^{n}\), under the condition of vanishing at infinity. In this framework, the constitutive convolution can be advantageously replaced with a equivalent differential law and successfully exploited in nonlocal problems involving dislocations and wave propagation. Nevertheless, later on, [PERSON]'s differential nonlocal law has been improperly applied to study scale phenomena in nanostructures defined on bounded domains (see e.g., [8; 9; 10] with reference to free vibrations of beams (without axial velocity)). More recent contributions on size-dependent dynamics of axially moving nanoscale beams are described as follows. In Ref. [11], the vibrational properties of an axially moving SWCNT (single-walled carbon nanotube) with simply supported ends were studied using nonlocal Rayleigh beam theory. The roles of velocity, small-scale parameter and aspect ratio on the characteristics of longitudinal, transverse and torsional vibrations of axially moving SWCNTs were examined. [PERSON] and [PERSON] [12] studied the nonlinear vibration of axially moving nanoscale beams with time-dependent velocity. Using the multiple scales method, the small scale effect on frequency response was obtained. [PERSON] et al. [13] studied the free vibrations of axially moving nanobeams with a constant velocity. In particular, the effects of nonlocal length scale on the critical velocity and the natural frequencies were studied. [PERSON] [14] applied the nonlocal theory and Euler beam model to study the thermo-electro-mechanical coupling transverse vibrations of axially moving piezoelastic nanobeams. The effects of nonlocal parameter and axial moving on the vibration of nanobeams were discussed. [PERSON] et al. [15] investigated the nonlinear vibration of axially moving nanobeams. The resonance vibration frequencies and nonlocal effect on the vibration were obtained. In Ref. [16], [PERSON] et al. studied the free vibration of axially moving nanobeams based on nonlocal theory, and the size effect on the natural frequency was determined.