Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints

Abstract

In this paper, the nonlinear responses of a loosely constrained cantilevered pipe conveying fluid in the context of three-dimensional (3-D) dynamics are investigated. The pipe is allowed to oscillate in two perpendicular principal planes, and hence its 3-D motions are possible. Two types of motion constraints are considered. One type of constraints is the tube support plate (TSP) which comprises a plate with drilled holes for the pipe to pass through. A second type of constraints consists of two parallel bars (TPBs). The restraining force between the pipe and motion constraints is modeled by a smoothened-trilinear spring. In the theoretical analysis, the 3-D version of nonlinear equations is discretized via Galerkin’s method, and the resulting set of equations is solved using a fourth-order Runge–Kutta integration algorithm. The dynamical behaviors of the pipe system for varying flow velocities are presented in the form of bifurcation diagrams, time traces, power spectra diagrams and phase plots. Results show that both types of motion constraints have a significant influence on the dynamic responses of the cantilevered pipe. Compared to previous work dealing with the loosely constrained pipe with motions restricted to a plane, both planar and non-planar oscillations are explored in this 3-D version of pipe system. With increasing flow velocity, it is shown that both periodic and quasi-periodic motions can occur in the system of a cantilever with TPBs constraints. For a cantilevered pipe with TSP constraints, periodic, chaotic, quasi-periodic and sticking behaviors are detected. Of particular interest of this work is that quasi-periodic motions may be induced in the pipe system with either TPBs or TSP constraints, which have not been observed in the 2-D version of the same system. The results obtained in this work highlight the importance of consideration of the non-planar oscillations in cantilevered pipes subjected to loose constraints.

Appendices

Appendix A

The various integrals appearing in Eqs. (16) and (17) can be evaluated analytically or numerically. They are given here

• (i) the linear terms:

  $$\begin{aligned} m_{ij}= & {} \int _0^1 {\varphi _i \varphi _j d\xi } \end{aligned}$$

  (A.1)
  $$\begin{aligned} c_{ij}= & {} \alpha \int _0^1 {\varphi _i {\varphi }''''_j \;\;d\xi } +2u\sqrt{\beta }\int _0^1 {\varphi _i {\varphi }'_j d\xi } \end{aligned}$$

  (A.2)
  $$\begin{aligned} k_{ij}= & {} \int _0^1 {\varphi _i {\varphi }''''_j \;\;d\xi } +u^{2}\int _0^1 {\varphi _i {\varphi }''_j d\xi } \nonumber \\&-\,\gamma \int _0^1 {\left( {1-\xi } \right) \varphi _i {\varphi }''_j d\xi }\nonumber \\&+\,\gamma \int _0^1 {\varphi _i {\varphi }'_j d\xi } \end{aligned}$$

  (A.3)

• (ii) the nonlinear terms:

  $$\begin{aligned} A_{ijkl}= & {} u^{2}\int _0^1 {\varphi _i {\varphi }'_j {\varphi }'_k {\varphi }''_l d\xi }\nonumber \\&-\,\gamma \int _0^1 {\left( {1-\xi } \right) \varphi _i {\varphi }'_j {\varphi }'_k {\varphi }''_l d\xi } \nonumber \\&-\,\frac{1}{2}\gamma \int _0^1 {\varphi _i {\varphi }'_j {\varphi }'_k {\varphi }'_l d\xi } \nonumber \\&+\,3\int _0^1 {\varphi _i {\varphi }'_j {\varphi }''_k {\varphi }'''_l d\xi } +\int _0^1 {\varphi _i {\varphi }''_j {\varphi }''_k {\varphi }''_l d\xi } \nonumber \\&-\,u^{2}\int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {{\varphi }'_k {\varphi }'''_l \;d\xi } d\xi } \nonumber \\&+\,\gamma \int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {\left( {1-\xi } \right) {\varphi }'_k {\varphi }'''_l \;d\xi } d\xi } \nonumber \\&-\,\int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {{\varphi }''_k {\varphi }''''_l d\xi } d\xi } \nonumber \\&+\,u^{2}\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {{\varphi }'_k {\varphi }'''_l \;d\xi } d\xi } d\xi } \nonumber \\&-\,\gamma \int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {\left( {1-\xi } \right) {\varphi }'_k {\varphi }'''_l \;d\xi } d\xi } d\xi } \nonumber \\&-\,\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {{\varphi }''_k {\varphi }''''_l d\xi } d\xi } d\xi } \nonumber \\&-\,u^{2}\int _0^1 {\varphi _i {\varphi }''_j \int _0^\xi {{\varphi }'_k {\varphi }''_l d\xi } d\xi } \nonumber \\&-\,\int _0^1 {\varphi _i {\varphi }''_j \int _0^\xi {{\varphi }''_k {\varphi }'''_l \;d\xi } d\xi } \nonumber \\&+\,\int _0^1 {\varphi _i {\varphi }''_j \int _0^\xi {{\varphi }'_k {\varphi }'_l d\xi } d\xi } \end{aligned}$$

  (A.4)
  $$\begin{aligned} B_{ijkl}= & {} 2u\sqrt{\beta }\int _0^1 {\varphi _i {\varphi }'_j {\varphi }'_k {\varphi }'_l d\xi } \nonumber \\&-\,2u\sqrt{\beta }\int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {{\varphi }'_k {\varphi }''_l d\xi } d\xi } \nonumber \\&+\,2u\sqrt{\beta }\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {{\varphi }'_k {\varphi }''_l d\xi } d\xi } d\xi } \nonumber \\&-\,2u\sqrt{\beta }\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {{\varphi }'_k {\varphi }'_l d\xi } d\xi } \end{aligned}$$

  (A.5)
  $$\begin{aligned} C_{ijkl}= & {} \int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {{\varphi }'_k {\varphi }'_l d\xi } d\xi } \nonumber \\&+\,\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {{\varphi }'_k {\varphi }'_l d\xi } d\xi } d\xi } \end{aligned}$$

  (A.6)
  $$\begin{aligned} D_{ijkl}= & {} u^{2}\int _0^1 {\varphi _i {\varphi }'_j {\varphi }'_k {\varphi }''_l d\xi } \nonumber \\&-\,\frac{1}{2}\gamma \int _0^1 {\left( {1-\xi } \right) \varphi _i {\varphi }''_j {\varphi }'_k {\varphi }'_l d\xi } \nonumber \\&-\,\gamma \int _0^1 {\left( {1-\xi } \right) \varphi _i {\varphi }'_j {\varphi }'_k {\varphi }''_l d\xi } \nonumber \\&-\,\frac{1}{2}\gamma \int _0^1 {\varphi _i {\varphi }'_j {\varphi }'_k {\varphi }'_l d\xi } \nonumber \\&+\,3\int _0^1 {\varphi _i {\varphi }'_j {\varphi }''_k {\varphi }'''_l d\xi } +\int _0^1 {\varphi _i {\varphi }''_j {\varphi }''_k {\varphi }''_l d\xi } \nonumber \\&-\,u^{2}\int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {{\varphi }'_k {\varphi }'''_l \;d\xi } d\xi } \nonumber \\&+\,\gamma \int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {\left( {1-\xi } \right) {\varphi }'_k {\varphi }'''_l \;d\xi } d\xi } \nonumber \\&-\,\int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {{\varphi }''_k {\varphi }''''_l \;\;d\xi } d\xi } \nonumber \\&+\,u^{2}\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {{\varphi }'_k {\varphi }'''_l \;d\xi } d\xi } d\xi } \nonumber \\&-\,\gamma \int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {\left( {1-\xi } \right) {\varphi }'_k {\varphi }'''_l \;d\xi } d\xi } d\xi } \nonumber \\&-\,\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {{\varphi }''_k {\varphi }''''_l \;\;d\xi } d\xi } d\xi } \nonumber \\&-\,u^{2}\int _0^1 {\varphi _i {\varphi }''_j \int _0^\xi {{\varphi }'_k {\varphi }''_l d\xi } d\xi } \nonumber \\&-\,\int _0^1 {\varphi _i {\varphi }''_j \int _0^\xi {{\varphi }''_k {\varphi }'''_l \;d\xi } d\xi } \nonumber \\&+\,\int _0^1 {\varphi _i {\varphi }''_j \int _0^\xi {{\varphi }'_k {\varphi }'_l d\xi } d\xi } \end{aligned}$$

  (A.7)

  Fig. 12

  

  Bifurcation diagrams for the tip displacements in the y direction of a pipe constrained by TSP constraints

  $$\begin{aligned} E_{ijkl}= & {} 2u\sqrt{\beta }\int _0^1 {\varphi _i {\varphi }'_j {\varphi }'_k {\varphi }'_l d\xi } \nonumber \\&-\,2u\sqrt{\beta }\int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {{\varphi }'_k {\varphi }''_l d\xi } d\xi } \nonumber \\&+\,2u\sqrt{\beta }\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {{\varphi }'_k {\varphi }''_l d\xi } d\xi } d\xi } \nonumber \\&-\,2u\sqrt{\beta }\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {{\varphi }'_k {\varphi }'_l d\xi } d\xi } \end{aligned}$$

  (A.8)
  $$\begin{aligned} F_{ijkl}= & {} \int _0^1 {\varphi _i {\varphi }'_j \int _0^\xi {{\varphi }'_k {\varphi }'_l d\xi } d\xi } \nonumber \\&+\,\int _0^1 {\varphi _i {\varphi }''_j \int _\xi ^1 {\int _0^\xi {{\varphi }'_k {\varphi }'_l d\xi } d\xi } d\xi } \end{aligned}$$

  (A.9)

• (iii) the nonlinear impacting force terms:

  $$\begin{aligned} f_i^1 \left( {\mathbf{q},\mathbf{p}} \right)= & {} \int _0^1 {\varphi _i f_\eta \left( {\eta ,\zeta } \right) } d\xi \nonumber \\= & {} \int _0^1 {\varphi _i f_\eta \left( {\varvec{\upvarphi }\cdot \mathbf{q},\varvec{\upvarphi }\cdot \mathbf{p}} \right) } d\xi \end{aligned}$$

  (A.10)
  $$\begin{aligned} f_i^2 \left( {\mathbf{q},\mathbf{p}} \right)= & {} \int _0^1 {\varphi _i f_\zeta \left( {\eta ,\zeta } \right) } d\xi \nonumber \\= & {} \int _0^1 {\varphi _i f_\zeta \left( {\varvec{\upvarphi }\cdot \mathbf{q},\varvec{\upvarphi }\cdot \mathbf{p}} \right) } d\xi \end{aligned}$$

  (A.11)

Appendix B

Modarres-Sadeghi [3] have used a six-mode Galerkin approximation in their calculations for predicting the 3-D dynamics of a cantilevered pipe. The purpose of this appendix is to evaluate the convergence property of the utilization of $N=6$. Three bifurcation diagrams for the tip displacement in the $\eta$ direction of a pipe with TSP constraints are presented in Fig. 12, for three different choices of truncation number. As shown in Fig. 12, the difference among the results predicted using $N=5, 6\hbox { and }7$ is not obvious. In this paper, therefore, $N=6$ will be employed in all numerical calculations.