Coupled longitudinal-transverse dynamics of a marine propulsion shafting under superharmonic resonances

Highlights

• The transverse superharmonic resonance is investigated.

• We consider the longitudinal inertia effect.

• There is a jump phenomenon in transverse response.

• The nonlinearity effect is of the hardening type.

• The effect of the shaft’s parameters on the nonlinear effect is discussed.

Abstract

In this paper, the transverse superharmonic resonances of a marine propulsion shafting are investigated under the first blade frequency excitation. A coupled longitudinal-transverse dynamic model due to geometrical nonlinearity is established by Hamilton’s principle and then is discretized by Galerkin method. The method of multiple scales is applied to these equations. The steady-state response and the stabilities are analyzed. The effect of the support stiffness, load, mass of propeller, damping ratio and slender ratio on the nonlinear effect is discussed. Research shows smaller values of slender ratio, bigger values of load and smaller values of damping ratio lead to stronger nonlinear effect. The nonlinear effect is reduced by increasing the back stern bearing stiffness and increased by increasing the front stern bearing and thrust bearing stiffness and the propeller mass. While the middle bearing makes small influence to it. It is also shown that these resonance curves are of the hardening type. Results of perturbation method are agreement with numerical simulations.

Introduction

For ships, propulsion shafting is an important part in ship power installations. The dynamic analysis for the propulsion shafting is essential for design engineers. The propulsion shaft system could be considered to be a typical rotor bearing system. Early the dynamic studies of the rotating shafts are focused on the linear vibration, including the prediction of the nature frequencies, calculation of the unbalance response and so on [1], [2], [3]. For some structures, a linear dynamic analysis may be sufficient when the vibration responses are not so big. If not, the nonlinear influence must be considered and nonlinear modeling will reveal more realistic behaviors of the system, such as bifurcation and chaos phenomenon. For the propulsion shafts, on the one hand, the shaft is slender and easy to cause huge vibration response under the fluid and unbalance loading. On the other hand, the blade frequencies may be closed to the lateral nature frequencies and cause resonance. So the dynamic response may have nonlinear characteristics which need to be explored fully. Recently, lots of papers are studied in this field, including support nonlinearity, geometrical nonlinearity and so on. Ishida and coworkers in several papers considered the nonlinear vibration of a rotating shaft due to geometrical nonlinearity. They studied the nonlinear forced oscillations of a rotating shaft with quadratic nonlinearity in the restoring force [4]. Furthermore, they investigated the entrainment phenomena at the critical speeds of 1/2 order subharmonic oscillations of forward and backward whirling modes [5]. They also studied the nonstationary vibration [6] and internal resonances [7], [8] of a nonlinear rotating shaft system which the forward natural frequency and the backward natural frequency satisfied the internal resonance relation of 1:−1. Rizwan et al. [9] developed a mathematical model incorporating the higher order deformations in bending and analyzed the nonlinear dynamics of rotors. In these analyses, only transverse motion of the rotor is considered and longitudinal displacement is neglected.

Dwivedy and Kar [10], [11], [12], [13], [14] examined the nonlinear dynamics of a slender beam carrying a lumped mass. These analyses included the principal parametric resonance, combination resonance and internal resonance. Barari et al. [15] studied the nonlinear vibration of Euler−Bernoulli beams subjected to the axial loads by variational iteration and parametrized perturbation methods. These analyses relate to the plane beam structures. For the rotating shafts, there are also some studies. Khadem and coworkers made outstanding achievements about the nonlinear rotating shafts. They discussed the primary resonances of a nonlinear in-extensional rotating shaft [16] and the primary and parametric resonances of asymmetrical rotating shafts with stretching nonlinearity [17]. They also investigated two-mode combination resonances of a simply supported rotating shaft by the method of harmonic balance [18]. Hosseini and coworkers also did lots of work in this field. They investigated the primary resonances of rotating shaft with stretching nonlinearity [19]. They analyzed free vibrations of a rotating shaft with nonlinearities in curvature and inertia [20] and with stretching nonlinearity [21]. Furthermore, they studied the dynamic stability and bifurcation of a nonlinear in-extensional rotating shaft with internal damping [22]. In addition, Ishida et al. [23] discussed the forced oscillations of a vertical continuous rotor with geometric nonlinearity due to the extension of the rotor center line. Shabaneh and Zu [24] investigated the dynamic analysis of a single-rotor shaft system with nonlinear elastic bearings. Sabuncu and Evran [25] studied the dynamic stability of a rotating pre-twisted Timoshenko beam subjected to lateral parametric excitation. Shaw and Shaw [26] studied the dynamic response of an unbalanced rotating shaft with internal damping using the center manifold approach. ŁUczko [27] investigated the dynamics response of rotating shafts with internal resonance and self-excited vibration. In these models, the nonlinearities include Von Karman nonlinearity and nonlinear curvature effects. In these paper, in dealing with the coupled partial differential equations, it’s assumed that the longitudinal displacement is a function of the transverse displacement by neglecting the longitudinal inertia or in-extensional assumption [28]. To reduce the degree of freedom, then the final coupled partial differential equations were transformed into the form of integro-partial differential equations by substituting this function relation into the transverse equations.

In some papers, both longitudinal and transverse displacements are considered and the longitudinal inertia is also considered. Then the two coupled partial-differential equations are solved together. For example, Han and Benaroya [29], [30] investigated the free vibration and forced responses of a compliant tower with consideration of the coupled transverse and axial motion by the finite difference approach, considering the longitudinal inertia. Ghayesh et al. [31] studied coupled longitudinal-transverse dynamics of an axially moving beam with and without a three-to-one internal resonance between the first two transverse modes. He investigated nonlinear vibrations and stability with an intermediate spring support of an axially moving beam [32] and Timoshenko beam [33]. He analyzed the nonlinear dynamics of an axially accelerating beam [34]. These analyses both relate to the beams and the literatures on the nonlinear vibration of rotating shafts considering the longitudinal inertia are very few.

To sum up, some studies of nonlinear beam or rotating shaft vibrations are based on the in-extensional assumption. It may be reasonable for no external loads acting along longitudinal direction. And some studies are based on the simplification that the longitudinal inertia is small. It may be reasonable for the slender simple support rotors (beams) without disks because the first longitudinal frequency is much more than the first lateral frequency [35]. For propulsion shafting, it could be stretching and the longitudinal inertia is not negligible due to the existence the propeller mass (the mass may be up to tens of tons).

Therefore, in this paper, a nonlinear coupled longitudinal-transverse dynamic model of the marine propulsion shafting is established and the transverse superharmonic resonances under blade frequency excitation are investigated with consideration of longitudinal inertia. The shaft is assumed to be supported by some linear-elastic springs with a lumped mass at one end. The Euler−Bernoulli beam theory is applied with added effects such as rotary inertia, gyroscopic effect. Nonlinear partial differential equations of motion are derived by Hamilton’s principle. The first mode equations are obtained by the Galerkin method. To analyze the transverse superharmonic resonances, the method of multiple scales is applied to these mode equations. The steady-state response and the stabilities are analyzed. The effects of the support stiffness, load, mass of propeller, damping ratio and slender ratio on the nonlinear effect are also discussed.