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Fractional calculus is a branch of mathematics which in recent decades has been of great interest to scientists in various disciplines, including engineering. One of the applications of this branch in engineering, is in modeling the viscoelastic materials using fractional differentiation. In this article, by inserting fractional calculus as a viscoelastic material compatibility equations in nonlocal beam theory, a viscoelastic Euler-Bernoulli nano-beam with different boundary conditions at two ends, has been modeled. Material properties of a carbon nanotube is considered and two methods, pure numerical and numerical-analytical have been used for solving obtained equations in time domain. Main method is completely numerical and operator matrices used in it to discrete equations in time and spatial domain. Second method is introduced for validation of pervious method’s answers. In this method equation of system reduced to an ordinary differential equation using Galerkin and obtained equation solved using a numerical direct integrator method. Finally, in a case study, the effects of fractional order, viscoelasticity coefficient and nanlocal theory coefficient on the time response of the viscoelastic Euler-Bernoulli nano-beam with different boundary conditions have been studied.

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Inimitable properties of graphene sheets enable a variety of applications such as axially moving nanodevices. Axial velocity affects dynamical response of systems. In this study linear vibration of an axially moving two-layer graphene nonoribbon with interlayer shear effect is proposed using nonlocal elasticity theory. Based on this theory stress at a point is a function of strain at all other points of the body. Euler-Bernoulli theory is used to model the system due to nanoribbon thickness and length. It is assumed that the layers have the same transverse displacement and curvature and there is no transverse separation between layers surfaces. A shear modulus is imported in the potential energy expression in order to consider the interlayer shear effect due to weak Van der Waals forces. Governing equations are obtained using Hamilton’s principle and are solved by Galerkin approach. Results for clamped-free boundary conditions are presented and compared to other available studies. Results for pinned-pinned boundary conditions are presented and it is observed that increasing axial velocity causes divergence and flutter instabilities in the system. Effects of different shear modulus and nonlocal parameter on critical speeds are also proposed.

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This study aims to investigate the transverse vibration of single- and double-layered graphene sheets embedded in an elastic medium based on the third-order shear deformation theory considering the axial force effect within the framework of Eringen’s nonlocal elasticity theory, where the governing equations of motion are obtained using Hamilton’s principle. The superiority of the studied non-local continuum model to its local counterpart is to consider the effect of size on the mechanical behavior of the structure. The results from a natural frequency analysis are obtained for different conditions such as the effect of size and aspect ratio, axial force, nonlocal coefficient, and change in the stiffness properties of the surrounding elastic medium by using the Navier-type solution for simply supported boundary conditions. Given that in a double-layered graphene sheet, the system has an in-phase vibrational mode and anti-phase vibrational mode with 180-degrees phase difference, the effect of van der Waals force on both vibrational modes is attempted to be investigated and it is shown that the van der Waals force has no effect on in-phase vibrational mode and by increasing it, the anti-phase frequency increases. It is also demonstrated that the nonlocal parameter is not a constant parameter but its value depends on the size and atomic structure, like chiral and zigzag configurations, and even on the type of boundary conditions.

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In this paper, an atomic force microscope is modeled based on non-classical nonlocal theory and nonlinear vibration of the system is analyzed and controlled. In this modeling, the Hamilton principle is used to derive the governing equation of Euler-Bernoulli nanocantilever based on the Eringen nonlocal elasticity theory considering Von-Karman geometric non-linearity. In the next step, using the Galerkin method, the governing dynamics differential equation of the atomic force microscope is obtained in the presence of attractive and repulsive van der Waals forces. The governing nonlinear equation is solved by employing multiple time scales method, and primary and secondary resonance of the atomic force microscope is studied. In this regard, the frequency response and excitation amplitude curves of primary, superharmonic and subharmonic resonances are plotted for different values ​​of the nonlocal parameter. Accordingly, it is shown that primary, superharmonic and subharmonic resonances of atomic force microscope are significantly affected by the nonlocal parameter. The results show that the use of nonlocal theory is a fundamental necessity for analyzing nonlinear vibrations of the atomic force microscope. Then, in addition to dynamic analysis, the chaotic vibrations are completely controlled and removed in the nonlocal model of the atomic force microscope by designing and implementing the robust adaptive fuzzy controller. For this task, the robust adaptive fuzzy controller which is considered as a powerful method of chaos controlling is used in the nonlocal model of atomic force microscope. The obtained results are used in the design and control process of the atomic force microscope.