An Explanation for a Paradox in a Fluid-Discharging Cantilevered Pipe Attached With an End-Mass

For a fluid-discharging cantilevered pipe attached with an end-mass, there are two methods to account for the end-mass effect. The first is that the end-mass is considered in the boundary conditions. The second is that the end-mass is included in the equation of motion via a Dirac delta function. As the analytical solution of the linear free vibration model is not available due to the presence of Coriolis force, the eigenfunctions of a beam, which satisfy the same boundary conditions, are commonly employed in the Galerkin method. It has been found the first method is incorrect for natural frequency calculation when the internal flow velocity is nonzero. However, the intrinsic mechanism remains to be clarified. This study has demonstrated the eigenfunctions in the first method depend on the end-mass and the orthogonality relations are quite different from that of typical simple beams, based on which a new model is proposed and the prediction compares well with that in the second method. For further validation, the critical internal flow velocity, the onset flutter frequency, and the dynamic responses of suspended pipes under gravity are computed, which compare well with experimental observations. This study can provide a workbench for fluid-conveying pipes with various boundary conditions.