Fluid-Induced Nonlinear Vibration of a Cantilevered Microtube with Symmetric Motion Constraints

Yong Guo

doi:10.1155/2020/8852357

Abstract

This paper investigates the dynamic behavior of a cantilevered microtube conveying fluid, undergoing large motions and subjected to motion-limiting constraints. Based on the modified couple stress theory and the von Kármán relationship, the strain energy of the microtube can be deduced and then the governing equation of motion is derived by using the Hamilton principle. The Galerkin method is applied to produce a set of ordinary differential equations. The effect of the internal material length scale parameter on the critical flow velocity is investigated. By using the projection method, the Hopf bifurcation is demonstrated. The results show that size effect on the vibration properties is significant.

Highlights

Zhang et al [17] investigated the nonlinear forced vibration of fluid-conveying pipes in the supercritical regime, that is, vibration about a curved equilibrium
Perhaps the first study of the dynamics of cantilevered micropipes conveying fluid was contributed by Hosseini and Bahaadini [35], who derived the linear governing equation of motion based on the modified strain gradient theory and carried out an analysis of eigenvalues with a parametric study in order to examine the effect of the length scale parameter
In another paper by Bahaadini and Hosseini [36], the effect of the fluid slip condition on the free vibration and the flutter instability of viscoelastic cantilevered carbon nanotubes (CNTs) conveying fluid was investigated. e material property of the CNT was simulated by the Kelvin–Voigt viscoelastic constitutive relationship. e equations derived by Hosseini and Bahaadini are linear

Summary

Discretization of the Governing Equation

R 1 where φr(ξ) cos hλrξ − cos λrξ − σr sin hλrξ − sin λrξ􏼁, σr sin hλr − sin λr􏼁 , cos hλr + cos λr􏼁. 􏽒1 φi d· ξ2υ−􏽰φ βj′′(􏽒φ1ξj′φφkk′′′φφ′l′′l′−dξφ−j′ 􏽒φξ0j′′φ􏽒k′1ξφ′′􏽒l dξ0ξφ−k′′φφ(lj′′4)􏽒d1ξξ dξ]dξ, φk′φ′ldξ φj′′. Φj′′ 􏽒1ξ 􏽒ξ0 φk′φ′ldξdξ)dξ,where the dots and primes represent z/zτ and z/zξ, respectively, and the indices i, j, k, and l range from 1 to N. As shown by Paidoussis and Semler [12], the dynamics in the two-mode version of the analytical model is in good qualitative agreement with the experimental observations. Since the main purpose of this paper is to investigate part of the qualitative behaviors of the present system, the two-mode expansion, that is, N 2 in (34), is adopted to discretize (33). In order to apply the available tools from dynamic system theory, we render (36) into the first-order form by introducing the following transformations: q1 x1, q2 x2,. N(X) represents the cubic-nonlinear terms of (36)

Theoretical Analysis and Numerical Computations

Conclusions