Vibration of spinning functionally graded nanotubes conveying fluid

Abstract

As a first attempt, the vibration and stability analysis of magnetically embedded spinning axially functionally graded (AFG) nanotubes conveying fluid under axial loads is performed based on the nonlocal strain gradient theory (NSGT). A detailed parametric investigation is conducted to elucidate the influence of key factors such as material distribution type and size-dependent parameters on the divergence and flutter instability borders. Also, a comparative study is conducted to evaluate the available theories in the modeling of nanofluidic systems. The material characteristics of the system are graded along the longitudinal direction based on the power-law and exponential distribution functions. To accurate model and formulate the system, the no-slip boundary condition is considered. Adopting the Laplace transform and Galerkin discretization technique, the governing size-dependent dynamical equations of the system are solved. The backward and forward natural frequencies, as well as critical fluid and spin velocities of the system, are extracted. Besides, an analytical approach is applied to identify the instability thresholds of the system. Dynamical configurations, Campbell diagrams, and stability maps are analyzed. Meanwhile, it is concluded that, in contrast to the influence of nonlocal and density gradient parameters, the increment of strain gradient and elastic modulus gradient parameters expands the stability regions and alleviate the destabilizing effect of the axial compressive load.

Appendices

Appendix 1

The normalized eigenfunctions of nonlocal beams for P–P and C–C boundary conditions are as follow, respectively [77]:

$${\varphi }_{j}\left(\xi \right)=\sqrt{2}\sin({\sigma }_{j}\xi ),$$

(52)

$${\varphi }_{j}\left(\xi \right)=\cosh\left({\sigma }_{j}\xi \right)-\cos\left({\lambda }_{j}\xi \right)-\frac{{\sigma }_{j}\sinh\left({\sigma }_{j}\right)+{\lambda }_{j}\sin\left({\lambda }_{j}\right)}{{\sigma }_{j}\left(\cosh\left({\sigma }_{j}\right)-\cos\left({\lambda }_{j}\right)\right)}\left(\sinh({\sigma }_{j}\xi )-\frac{{\sigma }_{j}}{{\lambda }_{j}}\sin({\lambda }_{j}\xi )\right),$$

(53)

where,

$$\left\{\begin{array}{c}{\sigma }_{j}\\ {\lambda }_{j}\end{array}\right\}={q}_{j}\sqrt{\frac{\sqrt{4+{\left(\mu {q}_{j}\right)}^{4}}\pm {\left(\mu {q}_{j}\right)}^{2}}{2}}.$$

(54)

The characteristic frequency equation of abovementioned conditions are given as follows, respectively [77]:

$$\sin({\sigma }_{j})=0,$$

(55)

$${\mu }^{2}{\sigma }_{j}{\lambda }_{j}\sinh\left({\sigma }_{j}\right)\sin\left({\lambda }_{j}\right)+2\cosh\left({\sigma }_{j}\right)\cos\left({\lambda }_{j}\right)=2.$$

(56)

Appendix 2

When the system is in the divergence instability condition, the minimum natural frequency of the system (the backward frequency) becomes zero. In this condition, the system loses its stiffness for the first mode. Consequently, to acquire the critical fluid (spin) velocity related to the first mode, by considering one mode (r = s = 1), Eq. (45) is reduced to the following equation:

$${\mathbf{Z}}_{11}=\left[\begin{array}{cc}{k}_{11}& {k}_{12}\\ -{k}_{12}& {k}_{11}\end{array}\right]+\left[\begin{array}{cc}{g}_{11}& {g}_{12}\\ -{g}_{12}& {g}_{11}\end{array}\right]+\left[\begin{array}{cc}{m}_{11}& 0\\ 0& {m}_{11}\end{array}\right].$$

(57)

By considering the linear variation for the material characteristics and ignoring the scale effects in the system, it can be written:

$${k}_{11}={\pi }^{4}\left({\alpha }_{E}+1\right)-{\pi }^{2}\left({U}^{2}+P\right)-\beta {\Omega }^{2}-(1-\beta ){\Omega }^{2}\left({\alpha }_{\rho }-1\right),$$

(58)

$${k}_{12}=0.$$

(59)

Based on the stability theory of the linear gyroscopic systems [78–80], when the eigenvalues of the system become zero, the determinant of the stiffness matrix equals zero. Therefore, the critical divergence fluid (spin) velocity of the system can be acquired from the following relation:

$${k}_{11}{k}_{22}-{k}_{12}{k}_{21}=0.$$

(60)