In this paper, flexural vibration and power flow transmission of a ship propulsion shafting structure are analyzed via energy principle description in conjunction with Rayleigh–Ritz procedure, in which the shafting vibration displacement is constructed as a superposition of Fourier series and boundary-smoothing supplementary functions. Effect of the distributed bearing support and thrust loading of propulsion shafting system is considered in terms of potential energy of system Lagrangian. Numerical examples are presented to demonstrate the reliability and effectiveness of the established model by comparing results with those from finite element method. Results show that the current model can deal with the vibration analysis of ship propulsion shafting with thrust loading and distributed bearing very well. Influence of boundary restraints, stiffness of distributed bearings, and thrust loading on vibration characteristics of ship shafting system is studied and addressed. Numerical study on power flow analysis is also conducted to investigate the characteristics of vibrational energy transmission in such practical structure. Results show that the stiffness of spatial bearing support has significant influence on vibrational energy transmission and thrust force will greatly affect the total input power into such structure.
Propulsion shafting is an important component of the marine power system, which converts the torque generated by the main engine into the ship propulsion force. Extensive study and experimental test have been performed on its torsional vibration. Obviously, a good understanding on its dynamic characteristics will be of fundamental significance for the efficient design and operation of such complex system. Vibration characteristics have a significant effect on the energy transmission of propulsion shafting. For this reason, transverse vibration and energy transmission through the propulsion shafting system have been studied extensively for many decades [
Flexural vibration of ship shafting system can be studied by transfer matrix method, finite element method, and experimental measurement [
Although a lot of investigation has been carried out on the vibration characteristics of ship propulsion shafting, most of the bearing support is mainly simplified as point support [
The propulsion shafting will suffer extra excitation force caused by the propeller and unbalanced excitation of shafting itself which will stimulate the system vibration and energy transmission to the hull structure [
Motivated by the current limitation on these aspects in literature, we aim in this paper to establish a more general model for the vibration analysis of propulsion shafting system with multiple bearings and thrust force generated by propeller. Stepped beam model is employed to stimulate the shafting structure considering the partially distributed bearing and thrust load. In order to make the displacement expansion sufficiently smooth in the entire solving region, Fourier series supplemented with boundary smoothed auxiliary terms is employed for admissible function construction. All the unknown coefficients are determined in conjunction with energy formulation of the propulsion shafting system through Rayleigh–Ritz procedure. Numerical examples are presented to validate the proposed model through the comparison with those results calculated from other approaches. Based on the model established, influence of some important factors, including boundary restraints, bearing stiffness, and thrust loading, on the vibration characteristics of propulsion shafting structure is studied and addressed. Then, power flow analysis is conducted to investigate the vibrational energy transmission behavior in such propulsion shafting system.
For a practical shafting system shown in Figure
An illustrative model of a ship propulsion shafting system.
(a) The simplified stepped beam model of the propulsion shafting structure; (b) the coupling interface of two beam sections.
The diagram of a ship propulsion shafting.
The specific parameters of the propulsion shafting.
Category | Mass (kg) | Length (mm) | Diameter (mm) | Equivalent stiffness (N/m) |
---|---|---|---|---|
10500 | — | — | ||
① | — | 4328 | 498 | — |
② | — | 1000 | 500 | — |
③ | — | 800 | (500, 415) | — |
④ | — | 200 | 820 | — |
⑤ | — | 3075 | 390 | — |
⑥ | — | 850 | 420 | — |
⑦ | — | 2375 | 390 | — |
⑧ | — | 100 | 966 | — |
— | 1195 | 5e8 | ||
— | 500 | 1e9 | ||
— | 850 | 1e9 |
For the harmonic oscillation, transverse vibration of each beam element will be assumed as
In vibration theory, it is normal to assume the displacement into Fourier series for the vibration analysis of an elastic rod/beam structure, namely, [
We can find that the first-order differentials at both ends are always zero, which can just be used for certain classical boundary condition. For the general elastic restraints, it will not be always zero physically; namely, the standard Fourier cosine or sine series does not possess the ability for the displacement expression of propulsion shafting structure with such elastic boundary conditions. In order to overcome the differential discontinuity of classical Fourier series, the supplementary functions are introduced to remove the jump points at both elastic ends.
A modified version of Fourier series is employed to deal with this issue, in which the additional polynomials are introduced to the standard Fourier series to remove all the discontinuities associated with the spatial differentiation of the displacement field functions. Flexural vibrational displacement function of these beam elements will be expanded as follows and the detailed polynomial expressions can be obtained in [
In most of the existing studies, vibration characteristics and power flow analysis of propulsion shafting are mainly focused on the classical boundary conditions and point support conditions. In order to analyze vibration behavior of the established model in Figure
For the
For the uniform beam, the moment of inertia
Since the propulsion shafting structure is analyzed on the basis of the stepped beam model, the mechanical coupling springs between each beam section must be taken into account. The potential energy associated with the coupling springs between the handshake interfaces (Figure
For such three supporting bearings, the partial supports are considered in this work. The corresponding potential energy of these partial supports will be described as
When the thrust force is applied at the left end of this model, there will be an equal force on any of its cross sections of the propulsion shaft. Introduction of extra axial loading will affect the differential equation in equation (
Its relevant potential energy can be written as
The total kinetic energy of the
Kinetic energy of the propeller is
Once the system Lagrangian is formulated, substituting the constructed admissible function equation (
Then, the coupling stiffness matrix
Similarly, the corresponding stiffness matrix and mass matrix can be deduced from the above equations. Then, the whole stiffness matrix
The work
Finally, the system characteristic equation in a standard eigenvalue matrix form is derived
Since the constructed admissible function is sufficiently smooth in the entire solving region, the various order spatial derivations of translational displacement function can be calculated in a straightforward way, namely, in terms of term-by-term operation. Power flow can be viewed as the time average of vibrational energy transfer across any section of beam structure. Power flow through per unit cross-sectional area of such propulsion shafting model shown in Figure
As mentioned above, the supposition of the standard Fourier series and auxiliary functions equation (
In this section, numerical examples will be presented to demonstrate the reliability and effectiveness of the proposed model for analyzing flexural vibration of propulsion shaft system with thrust force and distributed bearing supports. In the current model, arbitrary distribution of partially elastic support with variable stiffness and different thrust forces can be easily solved by just changing the corresponding description coefficients. For the elastic boundary condition, when the restraining stiffness is set as a medium number (0∼ +∞), such elastic restraint can be then achieved. Any change of boundary condition, bearing support, thrust force, and variable cross section will need no much modification on the theoretical formulation and simulation code. Then, results of the original model are compared with other simplified models to illustrate the advantages of the proposed model. In the following study, thrust force will be zero if it is not mentioned specifically. Structural damping associated with the shafting system is taken into account in terms of complex Young’s modulus
Firstly, vibration characteristics of the propulsion shafting system are considered. Tabulated in Table
The first ten modal frequencies of the original model with F-F and F-C boundary conditions.
Mode order | Boundary condition | |||||
---|---|---|---|---|---|---|
F-F | F-C | |||||
Original model | COMSOL | Diff (%) | Original model | COMSOL | Diff (%) | |
1 | 15.341 | 15.342 | 0.005 | 15.381 | 15.38 | 0.003 |
2 | 16.726 | 16.740 | 0.083 | 38.491 | 38.49 | 0.006 |
3 | 39.398 | 39.442 | 0.111 | 60.027 | 60.027 | 0.001 |
4 | 60.086 | 60.081 | 0.008 | 94.812 | 94.817 | 0.006 |
5 | 92.209 | 92.20 | 0.01 | 124.975 | 124.94 | 0.028 |
6 | 110.763 | 110.82 | 0.052 | 158.209 | 158.19 | 0.012 |
7 | 152.492 | 152.51 | 0.012 | 227.067 | 227.05 | 0.007 |
8 | 218.007 | 218.03 | 0.01 | 298.02 | 297.94 | 0.027 |
9 | 285.114 | 285.15 | 0.013 | 386.86 | 386.79 | 0.018 |
10 | 365.787 | 365.81 | 0.006 | 474.613 | 474.56 | 0.011 |
The first four mode shapes under F-F and F-C boundary conditions. (a) Mode shape 1. (b) Mode shape 2. (c) Mode shape 3. (d) Mode shape 4.
Classification societies’ standard for simplified point support position of rear stern bearing.
Classification societies | Lignum vitae bearing | Lily gold bearing | Remarks |
---|---|---|---|
BV | (0.5∼0.8) | 0.5 | |
LR | (1/4∼1/3) | (1/3∼1/2) | |
NK | |||
CCS | (1/4∼1/3) | (1/7∼1/3) |
In most of the existing studies, appropriate simplification is employed in vibration analysis of propulsion shafting system such as point support instead of the bearing support. In order to illustrate the difference and advantage of the current model comparing with the simplified model, four comparative common models are introduced and used in subsequent analysis: Model A: on the basis of the original model, the variable cross section beam (3#) is changed into a uniform beam with its diameter 500 mm the same as 2# beam member. Model B: on the basis of the original model, 1#, 2#, and 3# beam sections have the same diameter of 500 mm and 5#, 6#, and 7# beam members have the same diameter of 390 mm. Model C: on the basis of the original model, point support takes place of the partial support located at the middle of all these three bearings with the same stiffness coefficients. Model D: on the basis of the original model, the point support of the stern rear bearing located at the 1/3 length of the bearing according to the standard of classification society and the other two bearings are still located at the middle of such bearing.
Vibrational characteristics of these comparative models can be easily achieved by adjusting the corresponding parameters in current modeling framework. Tables
The first ten natural frequencies of original model compared with models A and B under F-C boundary conditions.
Mode order | Original model | Model A | Model B | ||
---|---|---|---|---|---|
Value | Value | Diff (%) | Value | Diff (%) | |
1 | 15.38 | 15.38 | 0 | 15.434 | 0.345 |
2 | 38.491 | 38.007 | 1.257 | 37.710 | 2.030 |
3 | 60.027 | 60.003 | 0.039 | 59.899 | 0.213 |
4 | 94.812 | 95.937 | 1.179 | 96.929 | 2.225 |
5 | 124.975 | 125.33 | 0.286 | 126.618 | 1.317 |
6 | 158.209 | 158.825 | 0.386 | 158.959 | 0.470 |
7 | 227.067 | 229.819 | 1.203 | 229.790 | 1.190 |
8 | 298.02 | 299.124 | 0.37 | 298.099 | 0.042 |
9 | 386.861 | 392.192 | 1.377 | 390.734 | 1.000 |
10 | 474.613 | 475.743 | 0.236 | 475.303 | 0.143 s |
The first ten natural frequencies of original model compared with models C and D under F-C boundary conditions.
Mode order | Original model | Model C | Model D | ||
---|---|---|---|---|---|
Value | Value | Diff (%) | Value | Diff (%) | |
1 | 15.38 | 14.132 | 8.114 | 15.726 | 2.248 |
2 | 38.491 | 41.681 | 8.288 | 41.477 | 7.757 |
3 | 60.027 | 65.250 | 8.702 | 62.700 | 4.453 |
4 | 94.812 | 99.720 | 5.168 | 99.945 | 5.405 |
5 | 124.975 | 129.698 | 3.782 | 129.216 | 3.396 |
6 | 158.209 | 159.668 | 0.918 | 160.211 | 1.262 |
7 | 227.067 | 227.651 | 0.248 | 228.222 | 0.500 |
8 | 298.02 | 301.183 | 1.077 | 302.002 | 1.352 |
9 | 386.861 | 386.296 | 0.148 | 386.636 | 0.060 |
10 | 474.613 | 471.945 | 0.565 | 471.788 | 0.598 |
According to Table
The first ten natural frequencies of original model under various elastic boundary restraints at the right end.
Mode order | Various elastic boundary restraints (N/m) | ||||
---|---|---|---|---|---|
102 | 104 | 106 | 108 | 1010 | |
1 | 15.341 | 15.341 | 15.354 | 15.379 | 15.380 |
2 | 16.705 | 16.706 | 17.327 | 36.637 | 38.469 |
3 | 39.434 | 39.438 | 39.460 | 48.583 | 60.023 |
4 | 60.083 | 60.086 | 60.079 | 60.159 | 94.699 |
5 | 92.185 | 92.184 | 92.224 | 94.175 | 122.644 |
6 | 110.754 | 110.786 | 110.880 | 120.903 | 156.495 |
7 | 152.497 | 152.487 | 152.515 | 155.325 | 223.717 |
8 | 217.997 | 218.004 | 218.042 | 221.557 | 291.042 |
9 | 285.106 | 285.105 | 285.151 | 289.102 | 368.156 |
10 | 365.7735 | 365.7764 | 365.8292 | 370.8595 | 446.9831 |
For the propulsion shafting system, the ship navigation status can affect the inclination angle and bearing support condition. In this section, effect of various bearing support stiffnesses on the vibration characteristics of propulsion shafting system is performed. The ratio of natural frequencies calculated under various stiffnesses to those of the original model is shown in Figure
Influence of various bearing support stiffnesses on the modal frequencies under F-C boundary condition. (a) Stern rear bearing; (b) stern front bearing.
Thrust force generated by propeller is variable due to the fluctuation of shaft rotational speed and sea state. The differential governing equation of beam vibration shows that such thrust force applied on shafting structure can reduce the bending moment of the shafting, thus affecting the lateral vibration behavior of the shafting system [
The first four natural frequencies under various thrust forces with F-C boundary conditions.
For the ship shafting system, the excitation force is generated by the propeller and the frequency
Comparison of power flow analysis made by current solution and COMSOL. (a) Internal force; (b) normalized power flow.
According to Figure
Power flow analysis of the propulsion system with different stiffnesses of stern rear bearing. (a) Power flow through bearing; (b) power flow distribution across the whole structure.
Similarly, power flow with different stiffnesses of stern front bearing is also discussed in Figure
Power flow analysis of the propulsion system with different stiffnesses of stern front bearing. (a) Power flow through bearing; (b) power flow distribution across the whole structure.
In Section
Power flow analysis of propulsion shafting system with different thrust forces.
Internal force in propulsion shafting with various thrust forces. (a) Shear force; (b) bending moment.
In this paper, a general dynamic analysis model for the study of transverse vibration characteristics and power flow behavior of a propulsion shafting system with thrust force and spatial bearing support conditions is established for the first time, in which energy principle description in conjunction with Rayleigh–Ritz procedure is employed. With the aim to make admissible function sufficiently smooth in the whole beam members, a superposition of Fourier series and boundary smoothed auxiliary polynomials is constructed for the displacement function expression. Potential energy of the stepped beams with thrust force and spatial bearing supports is formulated in system Lagrangian, which allows the analysis of current model through the minimization with respect to all the unknown coefficients.
Numerical examples are then given to validate the proposed model and study the vibration characteristics and dynamic behavior of propulsion shafting with various thrust forces and spatial bearing supports. Results show that the current model can make an accurate prediction of modal characteristics of such propulsion shafting structure, and the thrust force and spatial bearing have significant influence on vibration characteristics and energy transmission in such complicated structure. Moreover, effects of thrust force and spatial bearing supports will become weaker for the higher mode. Power flow analysis shows that the stiffness of stern bearing has much greater effect for the vibrational energy transmission pattern than the front bearing, and thrust force has important effect for the shear force at power input end and the peak of bending moment. This work can shed some new lights on the systematic understanding of dynamic behavior of propulsion shafting structure with complex boundary and general bearing supports.
The data are available on request.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (Grant no. 11972125) and Fok Ying Tung Education Foundation (Grant no. 161049).