Non-planar motions due to nonlinear interactions between unstable oscillatory modes in a cantilevered pipe conveying fluid

Abstract

Cantilevered pipes conveying fluid with an end mass exhibit rich post-critical nonlinear behaviors and have attracted the interest of many researchers. In the present paper, we focus on non-planar motions due to nonlinear interactions between two unstable oscillatory modes. By applying the orthogonality condition between the eigenfunction and its adjoint function to governing equations, evolution equations governing mixed-mode non-planar motions are analytically derived. In a certain range of flow velocity, mixed-mode non-planar motion can be produced. In this region, second- and third-mode oscillations are in distinct planes, which intersect perpendicularly. In particular, three-dimensional non-contacting measurements were conducted to verify the theoretical predictions, and hidden nonlinear modal interactions were extracted from experimental data. We observed mixed-mode non-planar motion that qualitatively agrees with the theoretical predictions. Moreover, in further experiments with a long pipe for future study, a sequence of bifurcating non-planar motions was observed.