Elsevier

Applied Mathematical Modelling

Volume 94, June 2021, Pages 737-756
Applied Mathematical Modelling

Analytical model for coupled torsional-longitudinal vibrations of marine propeller shafting system considering blade characteristics

https://doi.org/10.1016/j.apm.2021.01.042Get rights and content

Highlights

  • Main shaft and blade dynamic equations are derived using the Newton-Euler method.

  • Concentrated effects, hydrodynamic loadings and blade deformations are considered.

  • The derived PDEs are discretized by a modified Galerkin approach.

  • Numerical analysis is done on the system's natural frequencies and mode shapes.

  • Number of dominant vibration modes is obtained for each DOF of the system.

Abstract

As an attempt to investigate the torsional-longitudinal vibrations of marine propeller shafting systems, this paper develops an integrated mathematical formulation to consider different aspects of the problem as clearly as possible. The previous works in this field mostly deal with the lumped-parameter or finite element simulations of the propeller and the main shaft while this paper employs a non-FEM distributed-parameter modeling. The Newton–Euler method is used to derive dynamic equations of the cantilever blading and the rotating main shaft. The lumped effects on the main shaft such as the thrust block, the rigid coupling and the propeller loadings are considered together with the variation of cross section and pretwist angle along the blades. Galerkin method is used to discretize the equations and find the lowest number of dominant modes for each degree of freedom. The coupling effect is then explained by classifying the mode shapes into three groups. It is found that taking blade deformations into account is advantageous to the vibration analysis of the problem.

Introduction

One of the classic issues in engine-propelled watercrafts is high levels of noise and vibration. This problem not only leads to premature structural fatigue of the propulsion system but also affects the comfort of crew/passengers and may restrict the stealth performance of the vessel itself. Today, the design and manufacture of faster and lighter watercrafts have added to this problem by decreasing their stiffness and exposing them to higher vibration amplitudes and frequencies [1]. An important source of vibration in such vehicles is the rotating shaft and propeller system [2,3]. So, the knowledge of the dynamic behavior of propeller shafting system is significant for the designers, to design such systems so as not to operate within the ranges that would increase their vibration magnitudes.

Among the different types of machinery vibrations that may occur in the propeller shafting system, some require particular attention. Torsional vibrations can be the cause of severe problems, including shaft fracture, components failure, and damage to gears and couplings [4]. This type of vibrations is caused by fluctuations in rotational speed and is not easily detectable [5]. Also, axial vibrations seem to be the most probable forced vibrations to the propulsion system [6]. Both these two types of vibrations are mainly excited by the propeller's thrust and variable loads coming from the engine and power transmission mechanism [7].

To date, numerous studies have been carried out on the axial/torsional vibration of the propeller shafting system and its interaction with the engine, crankshaft, and gearbox of the whole propulsion mechanism. Pan et al. [8] conducted an experimental study on propeller and thrust bearing force through analyzing the axial shaft vibration. They investigated the effect of thrust bearing stiffness in different shaft speeds. Murawski [9] calculated the axial vibrations of a marine power transmission system using a computer program and studied the effect of couplings and boundary conditions on the crankshaft deformations. Zhang et al. [10] employed a transfer matrix model for a submarine's shafting system and took the dynamic characteristics of thrust bearing into account. A numerical study was carried out by Chen et al. [11] to investigate the vibrations of a marine propeller and its shafting system. The unsteady pressure that was considered on the propeller blades resulted in unsteady thrust transmitted to the foundation. Li et al. [12] modeled the dynamics and acoustics of a coupled propeller-shaft system by transfer matrix method, while the elasticity of propeller was considered using the equivalent reduced modeling method. A coupled longitudinal-transverse model of a marine propulsion is developed by Hamilton's principle and discretized by the Galerkin method in [13]. The model is used to investigate geometrical non-linear effects on steady-state response, resonance and stability of the system. Also, a finite element analysis model is presented in [14] to analyze the coupled longitudinal-transverse vibrations of a marine propulsion shaft. The simulations were validated against experimental measurements.

The coupled torsional-longitudinal has been studied in some research works. Parsons [15] carried out a research on the coupling effect between torsional and axial vibrations of a typical marine propulsion shafting, rising from the added mass and damping that appear when the system is working in a flow field. A research on the torsional-axial vibrations of marine propulsion shafting systems is conducted in [16] considering the crankshaft motions. Huang et al. [17] studied the coupled torsional-longitudinal dynamics of the system and compared the analytical results with numerical FEM calculations. It was shown that one must consider the coupled response in order to analyze the propulsion shafting system. A similar lumped mass model is established in [18]. The coupling effect was investigated in different rotational speeds and the numerical results were compared to experimental measurements.

In the field of vibration analysis of underwater marine propellers, extensive research is undertaken. Tsushima and Sevik [19] theoretically and experimentally analyzed the hydro-elastic effects of non-uniform velocity field generated by a ship on the dynamic response of skewed propellers. Lin and Tsai [20] studied the free vibrations of a composite marine propeller by using FEM, considering rotational effects and added mass. They compared the modal characteristics of the blade in air and water. An overview of the previous works done on the hydro-elastic analysis of propellers is provided in [21] and a detailed review of the different factors affecting the dynamic behavior of composite marine propellers is given in [22]. By improvement of computer calculation capabilities, numerical simulations became more available on the subject of dynamic marine propeller response. For instance, Li et al. [23] developed a hybrid numerical method for predicting the added mass and damping of a rotating marine propeller. The idea was extended in order to calculate the dynamic response in the wake of the submarine and to obtain dry and wet modes of propeller vibration [24]. Tian et al. [25] also provided an experimental and numerical study on the vibrations of a seven-bladed propeller in a cyclic inflow. They repeated the same set of experiments for two similar propellers with different elastic properties.

Most of the aforementioned studies lie within two general categories: the studies that investigated the vibrations of shafting and considered the propeller as a concentrated mass at the end of the propulsion shaft; and the studies that modeled the underwater propeller itself without taking the shaft deformations into account. Although all these works make understanding of individual propeller and shaft vibrations possible, it is still worth considering the mutual interaction of the shaft and flexible propeller in order to gain a better insight into the problem. This way of consideration is applied, for example, in Refs. [2,[26], [27], [28] for an elastic system of submarine propeller-shaft-hull.

To the best of the authors’ knowledge, there exists no specific paper that exclusively investigates the vibration characteristics of propulsion shaft's torsional-longitudinal motions taking propeller deformations into consideration. Furthermore, the use of numerical methods has imposed a computational burden and does not easily give room for different parametric sweeps over different properties of the system. In this regard, the current study attempts to establish a simple yet comprehensive formulation for analyzing the torsional-longitudinal vibrations of the flexible propeller and shafting system by following the basic steps towards a straightforward means of dealing with the general problem.

In this study, the modeling is based on consideration of the main propulsion shaft and the blades as continuous environments. The blades are assumed to be Euler-Bernoulli beams, which rotate with the main shaft about its axis. The position vector for an element of the shaft and a blade is written and the equations of motion are derived using the Newton-Euler method. Concentrated effects on the shaft are included using the Dirac delta function. This enables us to maintain using a single continuum for the shaft and obviates the need to break the shaft into several pieces and use compatibility conditions. Moreover, using the Newton-Euler method instead of energy methods such as Hamilton's principle removes the tedious and voluminous procedure of writing the energy equation and variations, and gives a better insight into the interactions and loadings between the different parts of the system. Next, the derived partial differential equations (PDEs) are discretized by the Galerkin method and the number of dominant modes is determined by converging the natural frequencies. The results are also compared with a derived lumped-parameter model.

Section snippets

Modeling

The main components for most marine propulsion systems are similar. Concerning the real operation of the system and regarding the literature, a model is proposed in a way that the effective elements on the torsional-longitudinal shaft vibrations are included. A schematic presentation of the model is shown in Fig. 1. The model consists of the main shaft, the thrust block, and the propeller. The main shaft is made up of the thrust shaft and propeller shaft, coupled by a rigid coupling. The

System parameters

In this section, the values of system parameters must be determined to investigate the modal properties. To this end, the required properties and parameters are sought in the literature. Physical and geometric properties of the main shaft, the rigid coupling and the thrust block are all taken from Ref. [2]. These parameters are given in Table 1.

By having the shaft diameter, its cross-sectional area and polar moment of inertia are simply calculated by As=πDs2/4 and Js=πDs4/32.

The geometric

Conclusions

A new distributed-parameter model for the coupled torsional-longitudinal vibrations of a marine shafting system was presented, based on the forces and moments acting on it. The deformable blades were also taken into consideration, modeled as non-uniform pretwisted cantilever beams. The equations of the main shaft and the blades were linked by the equations of the rigid hub. Also, the effect of rigid coupling, thrust block and hydrodynamic forces were applied on the shaft. The partial

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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