Coupled longitudinal–transverse dynamics of a marine propulsion shafting under primary and internal resonances

In this study, the longitudinal primary resonance of a marine propulsion shafting is investigated with special consideration of the internal resonance between the longitudinal and transverse directions (the first longitudinal natural frequency is approximately equal to the sum of the first transverse forward and backward frequencies). A coupled longitudinal–transverse dynamic model is established by using the extended Hamilton׳s principle and discretized by Galerkin method. First, the multiple scales method to solve these discretized equations is used, and then, the steady-state response and its stability are analyzed. Our results show that the first transverse forward and backward modes could be excited if the longitudinal excitation load is larger than the critical load, even though there is no excitation in the transverse direction. A saturation phenomenon is observed with the longitudinal motion and the extra energy is transferred to the transverse modes. The energy distribution ratio between the forward and backward modes is inversely proportional to their frequency ratio. The effects of damping ratio and frequency detuning parameters on the critical load are discussed. Results obtained by using the perturbation method are validated by numerical simulations.