The Influence of Boundary Constraint Viscoelasticity on the Nonlinear Forced Vibration of Fluid-Conveying Layered Pipes ^†

Abstract

In this paper, the influence of the viscoelasticity of boundary constraints on the forced vibration of the nonlinear forced resonance of a fluid-conveying layered pipe under an external forced excitation is studied. The pipe lays on viscoelastic foundations and is simply supported at both ends, and one end is subject to a viscoelastic boundary constraint. The Kelvin–Voight model was employed to describe the viscoelasticity provided by the foundation and boundary constraint. Hamilton’s variational principle was used to obtain the governing equations, during which geometric nonlinear factors including curvature nonlinearity and inertia nonlinearity were considered. By employing a perturbation-incremental harmonic balance method (IHBM), amplitude–frequency bifurcation diagrams of the pipe were obtained. The results show that the viscoelastic constraints from the boundary and foundation have significant influence on the linear and nonlinear dynamic behavior of the pipe system.

1. Introduction

Fluid-conveying pipes are widely used in aeronautic and astronautic fields, and nonlinear vibration is one of the typical failure modes of the pipe systems [1]. Therefore, the nonlinear dynamic behaviors of pipes conveying fluid have been a topic of concern over the years [2,3,4]. Wang et al. theoretically studied the nonlinear vibrations of a fluid-conveying functionally graded cylindrical shell, which is subjected to external excitation [5]. The effect of the parametric resonance on vortex-induced vibrations of a fluid-conveying pipe was investigated by Dai et al. [6]. The nonlinear vibration of functionally graded fluid-conveying microtubes subjected to transverse excitation load was analyzed by Ma and Mu [7]. The vibration and snap-through of a graphene-reinforced sandwich pipe conveying fluid under low-velocity impact was investigated by Ren et al. [8]. Effects of the internal flow velocity and the axial base excitation on the nonlinear dynamic behavior of the fluid-conveying pipe were presented by Zhou et al. [9].

As can be seen, the pipes considered in most of the previous studies were assumed to be under an immovable boundary. However, boundaries of structures in reality are usually movable. The boundary condition was proved to have a considerable impact on the dynamic instability of the pipe system [10]. In recent years, there were several studies on the dynamic behavior of a fluid-conveying pipe with movable boundaries. Wang et al. analyzed the nonlinear dynamic instability of a functionally graded fluid-conveying pipe under movable and immovable boundary conditions [11]. Nevertheless, studies on the dynamic instability of pipe structures under movable boundaries are still limited.

In this present work, we consider a more general movable boundary condition, namely viscoelastic boundary constraint. The influence of the viscoelasticity of boundary constraints on the nonlinear forced resonance of a fluid-conveying layered pipe was studied. The nonlinear motion equation of the pipe with a movable end under a transverse excitation load was established employing Hamilton’s variational principle. During the derivation of the governing equations, nonlinear factors such as nonlinear curvature, nonlinear inertia, and nonlinear constraint force are all considered. The bifurcation diagrams of the nonlinear forced vibration of the pipe were finally obtained by using the two-step perturbation-incremental harmonic balance method (TSP-IHBM), and the effect of viscoelastic boundary constraint is discussed.

2. Theoretical Model and Solutions

Consider a cylindrical fluid-conveying pipe with length L lying on a viscoelastic foundation, as shown in Figure 1. For convenience, both the Cartesian coordinate system $o-xyz$ and the cylindrical coordinate system $o-r\alpha x$ are established, and $y=r\cos \alpha$, $z=r\sin \alpha$. The pipe is simply supported at two ends with one elastic–viscous boundary constraint. The outer and inner radii of the pipe are represented as $R_{out}$ and $R_{in}$, respectively. The pipe is composed by two coaxial layers, and the radius of the interface is $R_{int}$. The transverse excitation $F=F_0\cos \omega t$ is applied at the midspan, where $\omega$ and $t$ are the excitation frequency and time variable, respectively. The pipe is assumed to vibrate due to the transverse excitation in the x-z plane. The velocity of the inner flow is $v_f$. The fluid in the pipe is considered to be inviscid and incompressible. The viscoelastic boundary constraint and foundation are described by the Kelvin–Voight model with spring stiffnesses $k_b$, $k_{fd}$ and damping coefficients $c_b$, $c_{fd}$.

By employing the generalized beam theory [1], the displacement field within the pipe can be written as follows:

$$\left\{\begin{array}{l}{\widehat{u}}^{\left(i\right)}(x,y,z,t)=u(x,t)+z[{\phi }_1^{(i)}(x)+r^2{\phi }_2^{(i)}(x)]\\ {\widehat{v}}^{\left(i\right)}(x,y,z,t)=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}(i=1,2)\\ {\widehat{w}}^{\left(i\right)}(x,y,z,t)=w(x,t)\end{array}\right.$$

(1)

This considers the transverse shear deformation. The superscript i = 1 and i = 2, corresponding to the quantities of the outer and the inner layers, respectively. $\widehat{u}$, $\widehat{v}$, and $\widehat{w}$ are, respectively, displacements in x, y, and z directions. u and w are the longitudinal and transverse displacement components on the mid-plane, respectively. ${\phi }_1^{(i)}$ and ${\phi }_2^{(i)}$ (i = 1, 2) are unknown functions that will be determined below.

The strain–displacement relationship of the pipe is as follows:

$$\left\{\begin{array}{l}{\epsilon }_{xx}^{(i)}={\displaystyle \frac{\partial {\widehat{u}}^{(i)}}{\partial x}}={\displaystyle \frac{\partial u}{\partial x}}+z\left({\displaystyle \frac{\partial {\phi }_1^{(i)}}{\partial x}}+r^2{\displaystyle \frac{\partial {\phi }_2^{(i)}}{\partial x}}\right)\\ {\gamma }_{xy}^{(i)}={\displaystyle \frac{\partial {\widehat{u}}^{(i)}}{\partial y}}+{\displaystyle \frac{\partial {\widehat{v}}^{(i)}}{\partial x}}=2yz{\phi }_2^{(i)}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(i=1,2)\\ {\gamma }_{xz}^{(i)}={\displaystyle \frac{\partial {\widehat{u}}^{(i)}}{\partial z}}+{\displaystyle \frac{\partial {\widehat{w}}^{(i)}}{\partial x}}={\phi }_1^{(i)}+(r^2+2z^2){\phi }_2^{(i)}+{\displaystyle \frac{\partial w}{\partial x}}\end{array}\right.$$

(2)

where ${\epsilon }_{xx}^{(i)}$ (i =1, 2) and ${\gamma }_{xy}^{(i)}$, ${\gamma }_{xz}^{(i)}$ (i = 1, 2) are normal and shear strain components.

The shear strains in the cylindrical coordinate $o-r\alpha x$ can be written as follows:

$${\gamma }_{xr}^{(i)}={\gamma }_{xy}^{(i)}\cos \alpha +{\gamma }_{xz}^{(i)}\sin \alpha =z{r}^{-1}[{\phi }_1^{(i)}+{\displaystyle \frac{\partial w}{\partial x}}+3r^2{\phi }_2^{(i)}]\hspace{1em}\hspace{1em}\hspace{1em}(i=1,2)$$

(3)

The constructive equations of the pipe are as follows:

$$\left\{\begin{array}{l}{\sigma }_{xx}^{(i)}=E_i{\epsilon }_{xx}^{(i)}\\ [{\tau }_{xy}^{(i)},{\tau }_{xz}^{(i)},{\tau }_{xr}^{(i)}]=E_i[{\gamma }_{xy}^{(i)},{\gamma }_{xz}^{(i)},{\gamma }_{xr}^{(i)}]/[2(1+{\mu }_i)]\hspace{1em}(i=1,2)\end{array}\right.$$

(4)

in which E_i and ${\mu }_i$ are, respectively, Young’s modulus and Poisson’s ratio of the materials at corresponding layers of the pipe.

In the present work, interfacial slippage between the inner and outer layers of the pipe is considered, and the interfacial cohesive law is employed to describe the interfacial bounding [12]. Therefore, the following can be concluded:

$$r=R_{int}:{\tau }_{xr}^{(1)}={\tau }_{xr}^{(2)}=\kappa [{\widehat{u}}^{(1)}-{\widehat{u}}^{(2)}]$$

(5)

While the boundary condition at the outer surface can be written as follows:

$$r=R_{out}:{\tau }_{xr}^{(1)}=0$$

(6)

where $\kappa \in [0,+\infty )$ is the interfacial shear stiffness.

Considering the nonlinear curvature due to the large deflection caused by the weak boundary constraint, and the nonlinear inertia caused by the large amplitude transverse vibration, as well as the nonlinear boundary constraint force, the governing equations can be established based on Hamilton’s variational principle. Finally, the amplitude–frequency bifurcation of the fluid-conveying pipe under the transverse excitation can be obtained using Perturbation-IHBM. For details of the derivation, see reference Wang et al. [11].

3. Results and Analysis

The amplitude–frequency bifurcation diagram of the pipe under transverse harmonic excitation is studied in this section. The material properties of the two layers of the pipe are set as E₁ = 162.9 GPa, μ₁ = 0.223, ρ₁ = 2330 kg/m³, E₂ = 131.7 GPa, μ₂ = 0.208, ρ₂ = 5323 kg/m³, respectively. In addition, R_in = 9 mm, R_out = 10 mm, R_int = 9.5 mm, L = 20R_out, ρ_f = 1000 kg/m³, and V_f = 100 m/s.

Figure 2 shows the effect of the boundary string stiffness on the bifurcation diagram. Obviously, boundary stiffness has a significant influence on the nonlinear behavior of the pipe system. In addition, the bifurcation diagram changes from softening to hardening type characteristics with an increase in the boundary stiffness.

Figure 3 shows the influence of the boundary constraint damping on the nonlinear forced vibration of the pipe. As shown, the curves of the two cases are almost the same, which indicates the boundary damping hardly affects the amplitude–frequency response of the fluid-conveying pipe system.

4. Conclusions

In this present work, the nonlinear forced resonance of a fluid-conveying pipe under a viscoelastic boundary constraint is studied. The research is focused on the influence of boundary stiffness and boundary damping on the bifurcation response of the pipe system. The results show that boundary stiffness has little effect on the linear properties, but considerably changes the nonlinear bifurcation of the pipe and has a hardening nonlinearity effect on the pipe system. However, boundary damping does not affect the bifurcation response of the system.

This present research may help improve the design of fluid-conveying pipe structures under general movable boundary conditions.