Stochastic investigation of the input uncertainty effects on the dynamic responses of constrained pipelines conveying fluids

Abstract

The dynamic response of a cantilevered pipe conveying fluid is investigated when several input parameters of the system are introduced to uncertainty. After the nonlinear equations of motions are derived, five input parameters are subject to a $\pm \,5\%$ uncertainty with a uniform distribution. First, a parametric study is performed by varying each parameter individually. Then, the Pearson correlation coefficients are calculated and discussed before a full Monte Carlo simulation is performed, and the histograms of results are investigated. It is evident that the outer diameter of the pipe has the largest effect on the maximum displacement of the pipe in the post-flutter regime. Chaotic behavior is exhibited when motion-limiting constraints are present in the system, so the system is tested with motion-limiting constraints as well. Monte Carlo simulation is performed, and bivariate diagrams are plotted to investigate how uncertainty affects the maximum displacement and periodicity of the oscillations together. Again, the outer diameter of the pipe is seen to be the most sensitive parameter to uncertainty when motion-limiting constraints are present. However, the parameters besides the outer diameter exhibit more sensitivity at high flow speeds. The results indicate that it is necessary to control the uncertainty introduced in the outer diameter to achieve expected dynamical responses at low flow speeds, but the uncertainty in all parameters must be controlled at higher flow speeds when motion-limiting constraints are present to achieve the expected behavior and chaotic responses.

Appendix

$$ \beta = \sqrt[4]{{\frac{m + M}{{EI}}\omega^{2} }} $$

(A1)

$$ C_{ij} = EI\eta \int_{0}^{L} {\beta^{4} \phi_{j} \phi_{i} {\text{d}}S = } \delta_{ij} \eta \omega^{2} $$

(A2)

$$ \left( {Cu} \right)_{ij} = 2M\int_{0}^{L} {\phi^{\prime}_{j} \phi_{i} } $$

(A3)

$$ K_{ij} = \int_{0}^{L} {\left[ {EI\beta^{4} \phi_{j} - \left( {m + M} \right)g\left( {L - S} \right)\phi^{\prime\prime}_{j} + \left( {m + M} \right)\phi^{\prime}_{j} } \right]\phi_{i} {\text{d}}S} $$

(A4)

$$ \left( {Ku} \right)_{ij} = M\int\limits_{0}^{1} {\phi^{\prime\prime}_{j} \phi_{i} {\text{d}}s} $$

(A5)

$$ M_{ijkl} = \int_{0}^{L} {\left( {\phi^{\prime}_{l} \int_{0}^{s} {\left( {m + M} \right)\phi^{\prime}_{k} \phi^{\prime}_{j} ds} - \phi^{\prime\prime}_{l} \int_{s}^{L} {\int_{0}^{s} {\left( {m + M} \right)\phi^{\prime}_{k} \phi^{\prime}_{j} {\text{d}}s} {\text{d}}s} } \right)\phi_{i} {\text{d}}s} $$

(A6)

$$ N_{ijkl} = 2M\int_{0}^{L} {\left( {\phi^{\prime}_{l} \phi^{\prime}_{k} \phi^{\prime}_{j} + \phi^{\prime}_{l} \int_{0}^{s} {\phi^{\prime\prime}_{k} \phi^{\prime}_{j} {\text{d}}s} - \phi^{\prime\prime}_{l} \int_{s}^{L} {\left( {\phi^{\prime\prime}_{k} \phi^{\prime}_{j} {\text{d}}s - \int_{0}^{s} {\phi^{\prime\prime}_{k} \phi^{\prime}_{j} {\text{d}}s} } \right){\text{d}}s} } \right)\phi_{i} {\text{d}}s} $$

(A7)

$$ \begin{aligned} P_{ijkl} & = \int_{0}^{L} {\left[ { - \frac{3}{2}\left( {m + M} \right)g\left( {L - s} \right)\phi^{\prime\prime}_{l} \phi^{\prime}_{k} \phi^{\prime}_{j} - \frac{1}{2}\left( {m + M} \right)g\phi^{\prime}_{l} \phi^{\prime}_{k} \phi^{\prime}_{j} + 3EI\phi^{\prime\prime\prime}_{l} \phi^{\prime\prime}_{k} \phi^{\prime}_{j} + EI\phi^{\prime\prime}\phi^{\prime\prime}\phi^{\prime\prime}} \right.} \\ & \quad + \phi^{\prime}_{l} \int_{0}^{s} {\left( {\left( {m + M} \right)g\left( {L - s} \right)\phi^{\prime\prime\prime}_{k} \phi^{\prime}_{j} + EI\phi^{\prime\prime\prime}_{k} \phi^{\prime\prime}_{j} } \right){\text{d}}s} - \phi^{\prime\prime}_{l} \int_{s}^{L} {\left( {\left( { - \left( {m + M} \right)g\phi^{\prime}_{k} \phi^{\prime}_{j} + EI\phi_{k}^{iv} \phi^{\prime\prime}_{j} } \right){\text{d}}s} \right.} \\ & \quad \left. {\left. { + \int_{0}^{s} {\left( {\left( {m + M} \right)g\left( {L - s} \right)\phi^{\prime\prime\prime}_{k} \phi^{\prime}_{j} + EI\phi_{k}^{iv} \phi^{\prime\prime}_{j} } \right)_{j} {\text{d}}s} } \right){\text{d}}s} \right]\phi_{i} {\text{d}}s \\ \end{aligned} $$

(A8)

$$ \left( {Pu} \right)_{ijkl} = M\int_{0}^{L} {\left( {\phi^{\prime\prime}_{l} \phi^{\prime}_{k} \phi^{\prime}_{j} - \phi^{\prime}_{l} \int_{0}^{s} {\phi^{\prime\prime\prime}_{k} \phi^{\prime}_{j} {\text{d}}s} - \phi^{\prime\prime}_{l} \int_{s}^{L} {\left( {\phi^{\prime\prime}_{k} \phi^{\prime}_{j} {\text{d}}s - \int_{0}^{s} {\phi^{\prime\prime\prime}_{k} \phi^{\prime}_{j} {\text{d}}s} } \right){\text{d}}s} } \right)\phi_{i} {\text{d}}s} $$

(A9)