Nonlinear natural frequencies of a cantilever beam partially immersed in fluid and carrying an intermediate mass

In this study, a mathematical model for finding the nonlinear natural frequencies of a beam vertically immersed in fluid and carrying a point mass within its dry portion is developed. The model adopted the in-extensible beam theory that allows large in-plane deformations under the effects of gravity. The Lagrangian approach in conjunction with the assumed modes method is utilized in deriving the nonlinear differential equations. The resulting non-dimensional differential equations have shown both inertial and static nonlinearities and are solved using the method of harmonic balance (HB) and the time transformation technique (TT). Numerical results are presented, discussed and some trends are extracted.