Longitudinal-Flexural Self-Sustained Vibrations of Nanotube Conveying Fluid
Beam model of geometrical nonlinear longitudinal-flexural self-sustained vibrations of nanotube conveying fluid is obtained with account of nonlocal elasticity. The Euler-Bernoulli hypotheses are the basis of this model. The geometrical nonlinear deformations are described by nonlinear relations between strains and displacements. It is assumed, that the amplitudes of the longitudinal and bending vibrations are commensurable. Using variational methods of mechanics, the system of two nonlinear partial differential equations is derived to describe the nanotube self-sustained vibrations. The Galerkin method is applied to obtain the system of nonlinear ordinary differential equations. The harmonic balanced method is used to analyze the monoharmonic vibrations. Then the analysis of the self-sustained vibrations is reduced to the system of the nonlinear algebraic equations with respect to the vibrations amplitudes. The Newton method is used to solve this system of nonlinear algebraic equations. As a result of the simulations, it is determined that the stable self-sustained vibrations originate in the Hopf bifurcation due to stability loss of the trivial equilibrium. These stable vibrations are analyzed, when the fluid velocity is changed. The results of the self- sustained vibrations analysis are shown on the bifurcation diagram. The infinite sequence of the period-doubling bifurcations of the monoharmonic vibrations is observed. The chaotic motions take place after these bifurcations. As a result of the numerical simulations it is determined, that the amplitudes of the longitudinal and flexural vibrations are commensurable.
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