Pulsatile vibrations of viscoelastic microtubes conveying fluid

Abstract

In this paper, the pulsatile coupled vibrations of a viscoelastic microtube conveying pulsatile fluid is examined for the first time. The problem is grouped into the class of parametrically excited, internally damped, gyroscopic where both Coriolis and parametric forces are present in the presence of viscosity. The Kelvin–Voigt approach of the viscosity, the Euler–Bernoulli for the deformation, the modified couple stress theory for the small size, and Hamilton’s principle for deriving differential equations are used. Parametric frequency–response curves are obtained in the vicinity of the parametric resonance near the critical speed for both subcritical and supercritical regimes. The effect of the flow pulsation on the oscillations is investigated.

Appendix: Comparison study

The critical fluid speeds are compared to those available in the literature employing the classical beam theory. A closed-form solution was obtained by Li et al. (2016) for the critical speed of fluid-conveying tubes at nanoscale levels. Ignoring nonlinear terms, couple-stress effects, flow pulsation and external loading, one obtains the following equation from Eqs. (11) and (12)

$$ EI\frac{{\partial^{4} v}}{{\partial x^{4} }} + \left( {m + M} \right)\frac{{\partial^{2} v}}{{\partial t^{2} }} + 2MU\frac{{\partial^{2} v}}{\partial x\partial t} + MU^{2} \frac{{\partial^{2} v}}{{\partial x^{2} }} = 0, $$

(19)

For this special case, the transverse deflection of the tube can be written as

$$ v = \sum\limits_{k = 1}^{\infty } {V_{k} \sin \left( {\frac{k\pi x}{L}} \right)} . $$

(20)

Substituting the above relation into Eq. (19), one obtains

$$ U_{cr} = \sqrt {\frac{{k^{2} \pi^{2} EI}}{{ML^{2} }}} , $$

(21)

It is worth mentioning that the flow-profile-modification factor is neglected for comparison purposes. Equation (21) for the critical fluid speed perfectly matches that determined in the literature for macroscale tubes conveying fluid flow of a constant velocity (Li et al. 2016); size effects are ignored.