A horizontal fluid-conveying cantilever: spatial coherent structures, beam modes and jumps in stability diagram

The proper orthogonal decomposition method is used to obtain a low-dimensional model for the planar non-linear dynamics of a horizontal fluid-conveying cantilever undergoing a limit cycle oscillation. The finite-dimensional approximation of the non-linear partial differential equation (PDE) describing the oscillation is carried out by a Galerkin projection scheme, using both the cantilever beam modes and proper orthogonal modes (coherent structures) as projection bases, which leads to a finite set of coupled ordinary differential equations. The proper orthogonal modes are obtained semi-analytically using the cantilever beam modes as a basis. A systematic study is then carried out, focusing on the jumps in the linear stability diagram of a horizontal fluid-conveying cantilever vis-à-vis the order of the finite-dimensional model obtained using either the beam modes or the proper orthogonal modes. Depending on the mass-ratio of the cantilever, while the order of the finite-dimensional model using the beam-mode basis increases steadily (up to 10), the corresponding order when the proper orthogonal modes are used to span the solution of the PDE remains unaltered (only two).