Nonlinear analysis of L-shaped pipe conveying fluid with the aid of absolute nodal coordinate formulation

Abstract

By adopting the absolute nodal coordinate formulation, a novel and general nonlinear theoretical model, which can be applied to solve the dynamics of combined straight-curved fluid-conveying pipes with arbitrary initially configurations and any boundary conditions, is developed in the current study. Based on this established model, the nonlinear behaviors of a cantilevered L-shaped pipe conveying fluid with and without base excitations are systematically investigated. Before starting the research, the developed theoretical model is verified by performing three validation examples. Then, with the aid of this model, the static deformations, linear stability and nonlinear self-excited vibrations of the L-shaped pipe without the base excitation are determined. It is found that the cantilevered L-shaped pipe suffers from the static deformations when the flow velocity is subcritical, and will undergo the limit-cycle motions as the flow velocity exceeds the critical value. Subsequently, the nonlinear forced vibrations of the pipe with a base excitation are explored. It is indicated that period-n, quasi-periodic and chaotic responses can be detected for the L-shaped pipe, which has a strong relationship with the flow velocity, excitation amplitude and frequency.

Appendices

Appendix A

In this Appendix, the detailed derivations of Eqs. (2) and (3) will be given. As we mentioned before, since the ANC formulation was used in this study to establish the theoretical model of the considered L-shaped pipe, the equation shown in Eq. (1) should be employed to derive the nonlinear governing equation of the pipe system. Thus, each physical quantity in Eq. (1) needs to be determined.

First, the absolute velocity vector of the pipe element will be determined. Since the pipe is subjected to a base excitation in X-direction, the position vector r of an arbitrary point on the pipe element in the inertial coordinate system X₁-o₁-Y₁ consists of two parts: the position vector of this point in the local coordinate system X-o-Y, and the position vector of the local coordinate system in the inertial coordinate system. Thus, we have:

$$ {\mathbf{r}} = {\mathbf{r}}_{l} + {\mathbf{r}}_{i} = {\mathbf{Sq}} + \left[ {\begin{array}{*{20}c} {w_{b} } \\ 0 \\ \end{array} } \right] $$

(A.1)

where S and q, respectively, denote the shape function and the nodal coordinate vector of the ANCF pipe element, which have been defined in Ref. [30]. Then, according to Eq. (A.1), the absolute velocity vector of the pipe element can be easily defined:

$$ {\mathbf{v}}_{{\mathbf{P}}} = \frac{{{\text{d}}{\mathbf{r}}}}{{{\text{d}}t}} = {\dot{\mathbf{r}}} = {\mathbf{S\dot{q}}} + \left[ {\begin{array}{*{20}c} {\dot{w}_{b} } \\ 0 \\ \end{array} } \right] $$

(A.2)

According to the discussions in Ref. [30], the absolute velocity vector of the fluid element can be expressed as follows:

$$ {\mathbf{v}}_{{\mathbf{F}}} = {\mathbf{v}}_{{\mathbf{P}}} + Uf{{\varvec{\uptau}}} $$

(A.3)

where f is the longitudinal deformation gradient and τ denotes the tangential unit vector along the deformed pipe axis. Their expression can be given as follows [30]:

$$ f = \frac{{\sqrt {{\mathbf{r^{\prime}}}^{T} {\mathbf{r^{\prime}}}} }}{{\sqrt {{\mathbf{r^{\prime}}}_{0}^{T} {\mathbf{r^{\prime}}}_{0} } }},{{\varvec{\uptau}}} = \frac{{{\mathbf{r^{\prime}}}}}{{\sqrt {{\mathbf{r^{\prime}}}^{T} {\mathbf{r^{\prime}}}} }} $$

(A.4)

From the above expressions, we can quickly find that neither v_F nor v_P is a function of ${\dot{\mathbf{q}}}$, and hence, the second surface integration shown in Eq. (1) is found to be zero. Then, based on Eqs. (A.2), (A.3) and (A.4), we can obtain the following expressions of the total kinetic energy of the system and the density of kinetic energy of the fluid

$$ \begin{aligned} T = & \frac{1}{2}m\int_{0}^{l} {{\mathbf{v}}_{{\mathbf{P}}}^{T} } {\mathbf{v}}_{{\mathbf{P}}} {\text{d}}x + \frac{1}{2}M\int_{0}^{l} {{\mathbf{v}}_{{\mathbf{F}}}^{T} } {\mathbf{v}}_{{\mathbf{F}}} {\text{d}}x \\ & = \frac{1}{2}\left( {m + M} \right)\int_{0}^{l} {{\dot{\mathbf{q}}}^{T} {\mathbf{S}}^{T} {\mathbf{S\dot{q}}}{\text{d}}x} + MU\int_{0}^{l} {\frac{{{\dot{\mathbf{q}}}^{T} {\mathbf{S}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}}}{{\sqrt {{\mathbf{q}}_{0}^{T} {\mathbf{S}^{\prime}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}_{0} } }}{\text{d}}x} + \frac{1}{2}MU^{2} \int_{0}^{l} {\frac{{{\mathbf{q}}^{T} {\mathbf{S}^{\prime}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}}}{{{\mathbf{q}}_{0}^{T} {\mathbf{S}^{\prime}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}_{0} }}{\text{d}}x} \\ & \; + \left( {m + M} \right)\int_{0}^{l} {\left[ {\begin{array}{*{20}c} {\dot{w}_{b} } \\ 0 \\ \end{array} } \right]^{T} {\mathbf{S\dot{q}}}{\text{d}}x} + MU\int_{0}^{l} {\left[ {\begin{array}{*{20}c} {\dot{w}_{b} } \\ 0 \\ \end{array} } \right]^{T} \frac{{{\mathbf{S}^{\prime}\mathbf{q}}}}{{\sqrt {{\mathbf{q}}_{0}^{T} {\mathbf{S}^{\prime}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}_{0} } }}{\text{d}}x} \\ & \; + \frac{1}{2}\left( {m + M} \right)l\left( {\dot{w}_{b} } \right)^{2} \\ \end{aligned} $$

(A.5)

$$ \begin{aligned} T^{\prime} = & \frac{1}{2}\rho_{F} {\mathbf{v}}_{F}^{T} {\mathbf{v}}_{F} \\ & = \frac{1}{2}\rho_{F} {\dot{\mathbf{q}}}^{T} {\mathbf{S}}^{T} {\mathbf{S\dot{q}}} + \rho_{F} U\frac{{{\dot{\mathbf{q}}}^{T} {\mathbf{S}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}}}{{\sqrt {{\mathbf{q}}_{0}^{T} {\mathbf{S}^{\prime}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}_{0} } }} + \frac{1}{2}\rho_{F} U^{2} \frac{{{\mathbf{q}}^{T} {\mathbf{S}^{\prime}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}}}{{{\mathbf{q}}_{0}^{T} {\mathbf{S}^{\prime}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}_{0} }} \\ & \; + \rho_{F} \left[ {\begin{array}{*{20}c} {\dot{w}_{b} } \\ 0 \\ \end{array} } \right]^{T} {\mathbf{S\dot{q}}} + \rho_{F} U\left[ {\begin{array}{*{20}c} {\dot{w}_{b} } \\ 0 \\ \end{array} } \right]^{T} \frac{{{\mathbf{S}^{\prime}\mathbf{q}}}}{{\sqrt {{\mathbf{q}}_{0}^{T} {\mathbf{S}^{\prime}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}_{0} } }} + \frac{1}{2}\rho_{F} \left( {\dot{w}_{b} } \right)^{2} \\ \end{aligned} $$

(A.6)

where ρ_F is the density of the fluid, q₀ represents the nodal coordinate vector of the initial pipe element, and l is length of the pipe element. By substituting Eqs. (A.2), (A.3) and (A.6) into the first surface integration in Eq. (1), the following expression can be determined

$$ \begin{aligned} & \int_{S} {{\text{d}}{\mathbf{a}} \cdot \left( {{\mathbf{v}}_{{\mathbf{F}}} - {\mathbf{v}}_{{\mathbf{P}}} } \right)\frac{{\partial T^{\prime}}}{{\partial {\dot{\mathbf{q}}}}}} \\ & = aMU\left( {{\mathbf{S}}^{T} {\mathbf{S\dot{q}}} + U\frac{{{\mathbf{S}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}}}{{\sqrt {{\mathbf{q}}_{0}^{T} {\mathbf{S}^{\prime}}^{T} {\mathbf{S}^{\prime}\mathbf{q}}_{0} } }} + {\mathbf{S}}^{T} \left[ {\begin{array}{*{20}c} {\dot{w}_{b} } \\ 0 \\ \end{array} } \right]} \right)|_{0,l} \\ \end{aligned} $$

(A.7)

where a is a scalar and can be defined as a(x = 0) = -1 and a(x = l) = 1.

Finally, the vector of generalized forces of the system, Q, needs to be determined. Since the influence of gravity and damping is not taken into account in this article, we only need to determine the generalized elastic forces vector of the system. Recalling that the L-shaped pipe in this paper is considered to be slender, the Euler–Bernoulli beam theory is adopted. Due to this fact, the potential energy of the pipe element can be written as

$$ U_{el} = \frac{1}{2}\int_{0}^{l} {EA_{p} \varepsilon^{2} {\text{d}}x} + \frac{1}{2}\int_{0}^{l} {EI\left( {\kappa - \kappa_{0} } \right)^{2} {\text{d}}x} $$

(A.8)

where κ₀ denotes the initial curvature of the pipe element. For the straight pipe segments κ₀ = 0, and for the curved pipe segment κ₀ ≠ 0. Moreover, ε and κ are, respectively, the longitudinal strain and geometrical curvature of the deformed pipe element, which can be defined as follows [30]:

$$ \varepsilon = \frac{1}{2}\left( {f^{2} - 1} \right),\kappa = \frac{{\left| {{\mathbf{r^{\prime}}} \times {\mathbf{r^{\prime\prime}}}} \right|}}{{||{\mathbf{r^{\prime}}}||^{3} }} $$

(A.9)

According to Eq. (A.8), the vector of the generalized elastic forces is defined by

$$ {\mathbf{Q}} = - \frac{{\partial U_{el} }}{{\partial {\mathbf{q}}}} = - \int_{0}^{l} {EA_{p} \varepsilon \frac{\partial \varepsilon }{{\partial {\mathbf{q}}}}{\text{d}}x} - \frac{1}{2}\int_{0}^{l} {EI\left( {\kappa - \kappa_{0} } \right)\frac{\partial \kappa }{{\partial {\mathbf{q}}}}{\text{d}}x} $$

(A.10)

So far, all the terms in Eq. (1) have been determined. For the rest, we just need to substitute Eqs. (A.5), (A.7), (A.10) and (4) into Eq. (1) and perform some straightforward manipulations, so that nonlinear governing equations of the pipe element can be obtained.

Appendix B

It is well known that if the elements used to discretize the pipe is not enough, the numerical results will not converge, and if the number of elements is too large, the calculation cost will increase. Due to this fact, it is necessary to determine a suitable number of pipe elements. To this end, the static equilibrium configurations of the L-shaped pipe for u = 8 with four different numbers of pipe elements, including 6, 9, 12 and 15, are displayed in Fig. 15a. From this figure, it is found that the results for 12 and 15 pipe elements are almost the same, indicating that 12 pipe elements are sufficient for predicting the static equilibrium configurations of the L-shaped pipe. Furthermore, Fig. 15b shows the bifurcation diagrams of dimensionless tip-end displacements in X-direction of the pipe without the base excitation for different numbers of the pipe elements. Again, it is easy to find that the results of 12 pipe elements are almost consistent with those of 15 pipe elements. According to these two figures, therefore, it is believed that 12 pipe elements are sufficient to predict the nonlinear statics and dynamics of the considered L-shaped pipe conveying fluid.

Fig. 15

Convergence analysis on a the static equilibrium configurations of the L-shaped pipe for u = 8, and b the bifurcation diagrams of dimensionless tip-end displacements in X-direction of the pipe versus internal flow velocity