Coupled longitudinal-transverse dynamics of a marine propulsion shafting under superharmonic resonances
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.
Section snippets
Mathematical model
The main structures for most of marine propulsion shafting are similar. So take one marine for example, a schematic diagram for typical marine propeller shafting is illustrated in Fig. 1. It consists of shaft, propeller, back stern bearing, front stern bearing, middle bearing and thrust bearing.
The shaft links with the motor by a flexible coupling which stiffness is small. So the shaft and the motor could be separate. The shaft is assumed to be a uniform cross-section beam; the propeller is
Multiple scales method
In this section, multiple scales method will be used to solve Eq. (34) to Eq. (36).
Based on this assumption (8) in Section 2, we know that , and the term and the term , are in the same order. So the Eqs. (35) and (36) are a cubic nonlinearity system essentially. The Eq. (34) is a quadratic nonlinearity system and quadratic and cubic nonlinear terms do not occur in one equation simultaneously.
Expanding , and in the form [39]
Numerical example
In this section, numerical examples are considered to examine the transverse superharmonic resonances under the first blade frequency excitation. Take a marine propulsion shafting in practical engineering for example. The length of the shaft is 14.5 m. The external diameter of the cross section is 240 mm and the internal diameter is 120 mm. So the slender ratio is 0.0046. Young’s modulus is 210 Gpa and the density is 7800 kg m−3. The back stern bearing stiffness, the front stern bearing stiffness,
Conclusions
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. It is assumed that strains are small but the rotation is moderate compared to the strain so that the equations of motion for the longitudinal, transverse and vertical motion are nonlinearly coupled. The shaft is assumed to be supported by some linear-elastic spring with a lumped mass at one end.
Acknowledgement
The author would like to thank State Key Laboratory of Mechanical System and Vibration in Shanghai Jiao Tong University for the support. The author also appreciates Prof. S.A.A. Hosseini for his patient replies to my questions by email when I asked for his help.
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