Abstract
With the development of ship enlargement, the problems of coupling vibration between hull and propulsion system and vibration transmission via bearings are more and more prominent. Based on the theory of shaft vibration and the experimental system for dynamic characteristics of the shaft, an experiment plan about propulsion shaft vibration under dynamic excitations is designed in this paper. The performance of propulsion shaft vibration under hull deformation excitations applied on intermediate and stern bearings is studied. Hydraulic excitations in horizontal and vertical directions on the intermediate bearing and stern bearing of the experimental model of propulsion shaft are considered in this paper to simulate hull deformation on bearings of the ship. Vibration characteristics of the shaft under different excitations are gained and coupling effects are discussed. Moreover, the influences of amplitude and direction of excitations on bearings and the shaft rotation speed on the vibration of propulsion are studied. The results show that aiming at improving the safety and reliability of navigation, the hull deformation, especially the horizontal hull deformation excitation on the intermediate bearing, is not neglectable and should be considered during primary design. Also, rotation speed and resonant frequency are needed to be well designed with the frequencies of hull deformation excitations.
1. Introduction
With the development of ship enlargement and economization, the power and size parameters of propulsion increase apparently. Thus, the problems of coupling vibration between hull and propulsion system is more and more prominent. As the ship hull is a thin-walled structure with elasticity, its deformation due to sea wave and other environmental influencing factors is unneglectable and will significantly affect the dynamic characteristics of the propulsion shaft. Thus, the effect of hull deformation on the propulsion shaft attracts attention of ship designers and researchers. For instance, Li et al. [1] built a model for the active and accurate control of marine propulsion system coupling with the hull deformation. Yan et al. [2] studied the coupling dynamical effects between the propulsion system and hull deformation, while Xiong et al. [3] focused on interactive characteristics of ship hull and sea waves. Furthermore, a simplified model of propulsion-hull-water was built and the dynamic characteristics of the system were analysed by Xing et al. [4].
Bearings are important components which transmit the forces from ship hull to propulsion shaft, as they are connections between ship hull and propulsion shaft [5]. Tian et al. [6] took the hull deformation excitation on the bearings into account and solved the mathematical equation for the energy of propulsion shaft. Gu and Zhang [7] used the finite element method to build a three-dimensional propulsion shafting model. The effects of number, stiffness, and length of bearings are discussed. Huang et al. [8] calculated hull deformations of a very large crude carrier (VLCC) at shaft supports under different waves in a full load operation. The dynamic characteristics of the propulsion shaft are affected by changing the location of bearings. Thus, attention was paid on the relationship of the shaft line alignment and the bearing positions due to the hull deformation. Murawski [9] found that the unbalance of bearing supporting aggravates the bearing abrasion and enlarges the vibration and noise of propulsion shaft. In this way, the efficiency of the shaft is decreased. Low and Lim [10] discussed the hull deformation and shaft status by gaining the relative displacements of the bearings from the hull deformation under different loading cases and sea wave excitations, which is meaningful to improve the navigation safety. Cho et al. [11] proposed a new iterative shafting alignment calculation procedure considering the interaction between shaft deflection and bearing proposed, while in the research of Shi et al. [12], another method of shafting alignment considering ship hull is used by the finite element method. Tiwari [13] introduced the application of the finite element method (FEM) and the transfer matrix method (TMM) in condition monitoring and system identification of rotor-bearing-foundation systems. Genta [14] introduced the dynamics theory of multi-degree-of-freedom rotating systems and continuous systems. Gayen et al. [15–17] presented the finite element (FE) formulation of a functionally graded (FG) shaft having multiple cracks to study the transverse vibration of such shafts in a rotor-bearing system. Timoshenko beam was used where effects of translational and rotary inertia, transverse shear deformations, and gyroscopic moments were considered.
As the excitation and coupling effects of the propulsion shaft and other structures are complex, experimental methods are always seen as the most effective way to analyse the dynamic characteristics of the propulsion shaft system. Many research studies have been focusing on the measurement method or experimental analysis on the shaft vibration or hull deformation. Lu et al. [18] introduced the development of experimental rig and the longitudinal vibration test method for marine propulsion shafting. Huang et al. [19, 20] designed experiments to discuss the coupling effects of transverse-longitudinal vibration and torsional-longitudinal vibration of a marine propulsion shaft. Zhang et al. [21] used theoretical and experimental methods to gain load distribution for entire shaft system with high accuracy. He et al. [22] prompted a nonlinear inertial matching method for large azimuth misalignment angle for hull deformation measurement system based on ring laser gyro units.
As mentioned above, most existing research studies used the numerical method to discuss the effects of hull deformation on the vibration of shaft or used the experimental method to analyse the coupled effects of shaft vibration, bearing loads, or hull deformation, respectively. In our previous work, some primary experimental results on characteristics of the ship propulsion system under dynamic excitation on bearings are discussed in Ref. [23]. Based on these references and the previous work, in this paper, experimental tests about propulsion shaft vibration under hull deformation excitations on bearings are designed. In this study, not only the multisupported rotating shaft system is considered but also the hull deformation is applied as an excitation on the bearing, considering its influence on the vibration characteristics of the propulsion shaft and the coupling effect. And the coupling effects between different vibrations are considered. Effects of various excitations and rotational speed on vibration characteristics are taken into account. Firstly, the theory of the shaft vibration under excitations on bearings is briefly introduced. Secondly, the constitution of the experimental system and experimental scheme are introduced. Then, the vibration characteristics of the propulsion shaft under different excitations are studied. The coupling effects between different vibrations are discussed. Moreover, the influences of amplitude and direction of excitations on bearings and the shaft rotation speed on the vibration of propulsion are analysed. The vibration performances of the ship propulsion system under hull deformation excitations are discussed.
2. Theory of the Shaft Vibration under Excitations on Bearings
As shown in Figure 1, the propulsion shaft is simplified as a beam with hull excitation at the bearings, which is divided into (n + 1) segments by supports. According to the Euler–Bernoulli theory, the transverse dynamic equations of the motion of each segment can be described as follows [5]:where denotes the number of each segment with the value from 1 to n, is the time, is the mass density, is Young’s modulus, is the moment of inertia of the shaft, is the transverse displacement, and represents the location along the shaft. The solution for the above equation is assumed to bewhere is a time-dependent generalized coordinate function and is the modal function. Two ordinary differential equations can be obtained by substituting the governing equation:where . The complete solution of the ordinary differential equations can be obtained by the general solution method.where are unknown coefficients for each segment of the shaft, which can be obtained by boundary conditions and continuity conditions.
At junctions of the bearing support, which is the connection of two adjacent shaft segments, the continuity condition of displacement, slope, bending moment, and shearing force can be expressed, respectively, aswhere and denote the left and right section of the beam on the support bearing , respectively, is the stiffness of the bearing, and is the dynamic excitation on the bearing. The equations of (n + 1) segment shaft can be arranged in the matrix form of to solve the vibration responses of the shaft under excitations on the bearings.
3. Experimental System
3.1. Constitution of the Experimental System
All experimental work was carried out using the ship shaft dynamic characteristic experimental system at Wuhan University of Technology. The experimental system is composed of a motor, eccentric structure to simulate crankshaft, intermediate shaft and bearing, stern shaft and bearing, and hydraulic loading system. The experimental configuration is schematically presented in Figures 2 and 3. The parameters of main components are listed in Table 1.
3.2. Loading System
As the connections between hull and propulsion shaft, bearings transmit hull deformation excitations to the propulsion shaft. Thus, a hydraulic loading with horizontal and vertical generators is designed to simulate the hull deformation excitation on the intermediate bearing and stern bearing, as shown in Figure 2.
The hydraulic loading on the bearings is set to simulate the hull deformation excitations due to wave effect and other unbalance forces. According to the data and analysis of large ship, the frequency of the wave is always low, which is less than 10 Hz, and the frequency of hull excitation is ever lower. Meanwhile, the amplitude of excitation on the bearing is small, which is always lower than 3 mm. Moreover, it is difficult for the hydraulic loading system to exert loading with less than 1 Hz and the experiment structure cannot endure large loading. Thus, in this paper, excitation with 2 Hz is chosen, and the amplitudes of these excitations are 0.1 mm, 0.5 mm, or 1.5 mm.
Besides, several rotation speeds of the propulsion shaft are considered.
3.3. Experimental Test
In the tests, dynamic responses in horizontal, vertical, longitudinal, and rotational directions of the observed point are determined using sensors and the test system. To be specific, as shown in Figure 4, the test points of whirling (horizontal and vertical) and longitudinal are set at the end of the intermediate bearing, while that of torsional vibration is located at 25 cm from the bearing for arranging the measurement instruments reasonably. The responses of longitudinal vibration are investigated by a laser displacement sensor OPTEX CD33. The whirling vibration is reflected by the test results of displacements from eddy current sensors ZA-210803-00-05-60-02 on horizontal and vertical directions while the torsional vibration is acquired with B&K MM0071 sensor and 2523 laser torsional vibration transducer. All sensors are connected to a multichannel signal analyser PXIe-6358.
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4. Vibration Characteristics of the Ship Propulsion Shaft
The responses of ship propulsion shaft under excitation in horizontal and vertical directions are analysed in this part. Horizontal, vertical, and longitudinal directions are represented by X, Y, and Z, respectively. Horizontal and vertical are both radial directions, while longitudinal is the axial direction of the shaft. Displacements in these 3 directions are UX, UY, and UZ. The results for axis orbit and rotational angle is also presented.
4.1. Responses of Ship Propulsion System under Horizontal Hull Deformation Excitation
In this section, the horizontal loading setting at intermediate bearing is considered. The amplitude of the dynamic excitations is 0.5 mm while rotation speed is set to 120 r/min. As shown in Figure 5, the displacements in X, Y, and Z directions are all evident under horizontal excitation in X direction, which confirms the existence of coupling effects between different vibrations. Meanwhile, UX is the largest (the peak value is 0.27 mm), followed by UY (the peak value is 0.034 mm), as vibrations in X and Y are both from transversal vibration and interact with each other during shaft rotation. The peaks in all three directions appear at 2 Hz, that is, the vibration response always appears near the excitation frequency. In addition, the axis orbit is larger in the X direction than in the Y direction, since the excitation is applied in the X direction.
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4.2. Responses of Ship Propulsion System under Vertical Hull Deformation Excitation
In this section, the vertical loading setting with the amplitudes 0.5 mm at No. 2 intermediate bearing is considered. The rotation speed is also set to 120 r/min. Similarly, as shown in Figure 6, the displacements in X, Y, and Z directions are still all evident under vertical excitation in Y direction which explains the existence of coupling effects between different vibrations as well. Meanwhile, UY is the largest (the peak value is 0.27 mm) followed by UX (the peak value is 0.03 mm) in this case. It must also be noticed that UY and UZ under horizontal excitations are larger than UX and UZ under vertical excitations compared to the results in the previous section. It suggests that the coupling effect between different vibrations is larger under horizontal excitation than vertical excitation. In other words, horizontal hull deformation excitation, which may be caused by sea wave on side shell or other horizontal unbalance forces, has more influence on the coupled transverse-longitudinal vibration than the vertical hull deformation excitation, which may be caused by longitudinal sea wave or other vertical unbalance forces. The reason is that the propulsion shaft has more freedom in horizontal direction than in vertical direction as the bearing sets are in the vertical direction. The peaks of the response also appear near the excitation frequency. And the axis orbit is larger in the Y direction than in the X direction, since the excitation is applied in the Y direction.
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5. Effect of Hull Deformation Excitations on the Responses of the Ship Propulsion Shaft
The effect of amplitudes, directions, frequencies of hull deformation excitations and shaft rotation speed on the characteristics of shaft vibration are analysed in this part. The experiment cases with different amplitudes, frequencies, locations, directions of excitations, and rotation speeds are listed in Table 2.
5.1. Effect of Amplitudes of Hull Deformation Excitations on Bearings
In this section, the response displacement of the shaft is obtained for amplitudes 0.5 mm and 1.5 mm on the intermediate bearing and 0.1 mm and 0.5 mm on the stern bearing of horizontal and vertical excitations, respectively. The effect of amplitudes of excitation on bearings is discussed based on the horizontal displacement UX and vertical displacement UY in Figures 7–10.
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As shown in Figures 7 and 8, the amplitudes of response displacement increase with the amplitudes of excitation on the intermediate bearing. Besides, there are also increases for UY under horizontal excitation and the UX under vertical excitation, which confirms the existence of the coupling effects again, which means coupling effects are important for the analysis of large ship propulsion shaft and should be considered during primary design. Another interesting finding is that the amplitude of horizontal excitation on the intermediate bearing has more effects on the displacements in both directions in comparison to the case of vertical excitation on the intermediate bearing. To be specific, the vertical displacement increases more than twice as shown in Figure 7(b), but the horizontal displacement barely increases as shown in Figure 8(a). It means that horizontal hull deformation excitation, which may be caused by sea wave on side shell or other horizontal unbalance forces, has more effects than the vertical hull deformation excitation, which may be caused by longitudinal sea wave or other vertical unbalance forces, due to the structure of shaft propulsion.
Similarly, the amplitudes of displacement increase with the amplitudes of excitation on the stern bearing. As shown in Figures 9 and 10, UX increases more under horizontal excitation and UY grows more under vertical excitation when the excitations are set on the stern bearing. However, the coupling effects under excitations on the stern bearing are not as evident as that for the case of excitations on the intermediate bearing. This is supported by the results that increase of UY under horizontal excitation and UX under vertical excitation is not evident compared with the cases of excitation on the intermediate bearing. Generally, a large ship has many intermediate bearings; coupling effects for those bearings are not neglectable due to their larger effects.
5.2. Effect of Frequencies of Hull Deformation Excitations on Bearings
This part discusses the effect of frequencies of excitations on the response displacement of the shaft. The results for the case under excitations at 2 Hz and 4 Hz at intermediate and stern bearing are compared. In this part, vertical excitation is considered, and the response of the shaft is obtained for amplitudes 0.5 mm and rotation speed 120 r/min.
It can be seen from Figure 11 that under vertical excitation at the intermediate bearing, the frequencies of excitation mainly change the response result of vertical displacement. Excitation at 2 Hz excites the resonant vibration at 2 Hz, and excitation at 4 Hz excites the resonant vibration at 4 Hz. However, from Figure 12, it can be seen that peaks at 2 Hz and 4 Hz have larger amplitudes under the excitation of 4 Hz on the stern bearing. It is because the observed point is at the intermediate bearing, and the change of frequency at stern bearing enlarges the vibration at every resonant frequency.
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5.3. Effect of Directions of Hull Deformation Excitations on Bearings
In this section, the response displacement of the shaft is obtained for amplitudes 0.5 mm, excitations 2 Hz, and rotation speed 120 r/min. The effect of directions of excitations is discussed by comparing results for the case under horizontal (x) or vertical (y) excitations on the intermediate bearing or the stern bearing or simultaneously to the intermediate bearing and the stern bearing.
As shown in Figures 13 and 14, for both cases of excitations set on intermediate and stern bearing, UX is larger under horizontal excitation while UY is larger under vertical excitation. Moreover, the effect of directions of excitations on the intermediate bearings is more evident than on the stern bearings. This is supported by the results that the larger displacements are about 5 times the smaller displacements (respectively, in Figures 13(a) and 13(b)) under excitations on the intermediate bearing while under excitations on the stern bearing, they are only twice the smaller displacements (respectively, in Figures 14(a) and 14(b)). This is because the intermediate bearing is located at the middle part of the shaft and the vibration is more complex and easier to be affected by the engine force, multisupports, and other structures. Also, as discussed before, as a large ship has more intermediate bearings, the effects of directions of hull excitation should be considered. When horizontal excitation on the intermediate bearings takes the main part, the horizontal displacement should be monitored while the vertical displacement should be observed when vertical excitation on the intermediate takes the main part if largest vibration is needed to be gained for avoiding the safe boundary.
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It is shown in Figure 15 that under transversal excitations on both intermediate and stern bearing, the number of peaks of transversal and longitudinal vibration become more than when loaded separately. It is because of the coupling effects between transversal excitation at different locations. When vertical excitations are applied on intermediate and stern bearing at the same time, amplitudes of peaks are larger than when loaded separately, while the location and number of peaks do not have evident changes. It means that the coupling effects of vertical excitations at different locations are not evident.
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5.4. Effect of Shaft Rotation Speed
In this section, the response displacement of the shaft is obtained for amplitudes 0.5 mm. The effect of shaft rotation speed is discussed by comparing results observed under rotation speed at 70 r/min, 120 r/min, and 170 r/min. The cases of horizontal excitation on intermediate bearing and vertical excitation on intermediate are taken as examples.
As shown in Figure 16, response displacement of 70 r/min is largest at 2 Hz for both UX and UY under the horizontal excitation on the intermediate bearing. Meanwhile, the peaks of displacement arrive at 1.2 Hz and 2.8 Hz for 70 r/min and 170 r/min, respectively. Similarly, in Figure 17, the curves of response displacement always have peaks at 1.2 Hz for 70 r/min, 2 Hz for 120 r/min, and 2.8 Hz for 170 r/min. Besides, the highest peak always arrives at 2 Hz for 120 r/min. From these results, it can be seen the highest peak is always at 2 Hz because it is from both the excitation frequency and the rotational frequency of 70 r/min, while 1.2 Hz and 2.8 Hz are both the rotational frequencies of the shaft. Thus, the frequency of excitation should avoid the rotational resonant frequency for avoiding sharp vibration. In other words, the rotation speed and resonant should be considered with the excitations in the primary design.
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6. Conclusions
An experiment investigation about propulsion shaft vibration under hull deformation excitations on intermediate and stern bearings is conducted in this paper to study the performance of propulsion shaft vibration under hull deformation excitation. The vibration characteristics of ship propulsion shaft under different excitations are analysed. The effects of amplitudes, frequencies, and directions of these excitations and shaft rotation speed on the characteristics of shaft vibration are studied. Conclusions are gained as follows:(1)The coupling effect between different vibrations exists. The influence of horizontal hull deformation excitation on the vibration of propulsion shaft is larger than the vertical hull deformation excitation. It means that horizontal hull deformation excitation, which may be caused by sea wave on side shell or other horizontal unbalance forces, has more effects than the vertical hull deformation excitation, which may be caused by longitudinal sea wave or other vertical unbalance forces, due to the structure of shaft propulsion.(2)The effect of directions of excitations on the intermediate bearing is more evident than on the stern bearings. It means when horizontal excitation on the intermediate bearings takes the main part, the horizontal displacement should be monitored while the vertical displacement should be observed when vertical excitation on the intermediate takes the main part if largest vibration is needed to be gained for avoiding the safe boundary.(3)The highest peak always arrives at frequencies of the excitations. It means rotation speed and resonant frequency are needed to be considered with the excitations in the primary design.
From these conclusions, it can be seen that the amplitude, frequency, and direction of excitations on bearings and rotation speed all have influences on the vibration of propulsion shaft. The hull deformation, especially the horizontal hull deformation excitation on the intermediate bearing, is not neglectable and should be considered during primary design. Also, as the large ship has many intermediate bearings, coupling effects for those bearings are not neglectable due to their larger effects. Moreover, rotation speed and resonant frequency are needed to be well designed with the frequencies of hull deformation excitations, aiming at improving safety and reliability of the navigation. In the future scope, the corresponding experiments on the real large ship are needed to verify the effects of hull deformation on the vibration of propulsion shaft. Besides, based on these conclusions from the experiment, with the development of ship enlargement, the analytical and numerical methods are needed to be improved to consider effects of ship hull deformation. At the same time, coupling effects and rotational speed are also needed to be included to gain more precise vibration mechanism and results.
Data Availability
The experimental data used to support this study are available from the corresponding author upon request.
Disclosure
Part of this study was presented in the 4th International Conference on Transportation Information and Safety (ICTIS).
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This study was supported by a Program Grant of the National Natural Science Foundation of China (51609190 and 51839005).