A simplified-nonlocal model for transverse vibration of nanotubes acted upon by a moving nanoparticle
This study provides a simpliﬁed solution forestimating the dynamic response of a single-walled carbonnanotube when excited by a moving nanoparticle. At ﬁrst,the strong form of the equation of motion for a nonlocalRayleigh nanotube is deduced, and the inertia effect of amoving nanoparticle along a nanobeam is then considered.For obtaining a weak form of the above nonlocal model,we use the Galerkin method, where the test functions are aset of orthogonal polynomials generated from a polynomialsatisfying given boundary conditions. This process leads toa second-order differential equation which for a movingload the matrix coefﬁcients are time dependent. In thestate-space formulation, the forced response depends upona transition matrix that can be locally approximated by thematrix exponential by assuming that the coefﬁcients arelocally constant. The normalized frequencies for a movingforce are calculated and compared to those obtained inprevious studies, and good agreement between them wasobserved. After acquiring the dynamic responses of ananotube for a wide range of velocities and weights ofmoving nanoparticles, as well as for the nonlocal effects ona nanobeam, a nonlinear regression analysis is adapted toestimate the response of a nanobeam according to ananalogous classical Rayleigh beam. These equivalentresults in three multipliers (a,b, and c) are functions ofkinetic parameters and nonlocal effects. Due to the nor-malization of the variables, these multipliers can be usedfor various types of beam-like structures in both the nano-and macro-domains. The accuracy of these coefﬁcients isevaluated using the results gained by the analytical solu-tion. This paper offers a remedy for a time-consumingprocess by means of some simple substitutions.