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Nonlocal vibration of a carbon nanotube embedded in an elastic medium due to moving nanoparticle analysed by modified Timoshenko beam theory – parametric excitation and spectral response

The Timoshenko beam theory, which deals with two partial differential equations of motion in terms of deflection and rotation, is transformed into a single partial differential equation with pure bending deflection as a potential function for the determination of the total deflection, rotation angle and sectional forces. Inclusion of a nonlocal stress parameter results to the extension of the governing differential equation from the 4th to the 6th order. A simply supported nanotube is considered and the governing differential equation is decomposed into a system of ordinary differential equations by employing the modal superposition method, separation of variables and the Galerkin method. Both moving nanoparticle gravity and inertia force are consistently taken into account resulting to ordinary and parametric excitation, respectively. As a novelty, the parameters are split into a constant and a time-dependent part. The former is added to the ordinary system of equations, which is solved analytically in the frequency domain by the harmonic balance method, while the system with variable coefficients is solved in the time domain by the perturbation method. The influence of slenderness ratio, nonlocal parameter, elastic stiffness, nanoparticle gravity and inertia force, as well as velocity on the free and forced nanotube response is investigated. Special attention is paid to the influence of damping on resonance. Performed parametric analysis is physically transparent due to the obtained semi-analytical solution. Some analytical results of illustrative examples are compared with numerical ones from the relevant literature and noticed differences are discussed.

carbon nanotubes; nonlocal parametric vibration; moving gravity and inertia force; modified Timoshenko beam theory; frequency domain

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