The submarine experiences longitudinal vibration in the propulsion shafting system throughout most of run. A transfer matrix model of the propulsion shafting system, in which the dynamic characteristics of oil film within thrust bearing are considered, is established to describe the dynamic behavior. Using hydrodynamic lubrication theory and small perturbation method, the axial stiffness and damping of oil film are deduced in great detail, followed by numerical estimation of the foundation stiffness with finite element method. Based upon these values of dynamic parameters, the Campbell diagram describing natural frequencies in terms of shafting rotating speeds is available, and the effect on the 1st natural frequency of considerable variations in thrust bearing stiffness is next investigated. The results indicate that the amplitude of variation of the 1st natural frequency in range of low rotating speeds is great. To reduce offresonance response without drastic changes in propulsion shafting system architecture, the measure of moving thrust bearing backward is examined. The longitudinal vibration transmission through propulsion shafting system results in subsequent axial excitation of hull; the thrust load acting on hull is particularly concerned. It is observed that the measures of structural modification are of little benefit to minimize thrust load transmitted to hull.
For marine vessels, propulsion shafting is an essential changer of engine torque and propeller thrust. Modern submarine is mostly equipped with the electric propulsion plant, realizing the mechanical separation between diesel generator and propulsion motor. Thereby, the unsteady propeller thrust constitutes the sole excitation source of longitudinal vibration in propulsion shafting system. It should be mentioned that this type of vibration is an inherent vibration attached to propeller propulsion technology. Unless the new thruster is adopted, this type of vibration cannot be eliminated thoroughly. The submarine propulsion shafting system consists of propeller, stern shaft, intermediate shaft, journal bearing, thrust bearing, and flexible coupling, as shown in Figure
Schematic diagram of submarine propulsion shafting system.
Due to existence of nonuniform flow field near the propeller caused by asymmetry in hull and protrusions of control surfaces, the axial excitation which occurs at the propeller is the result of variations in thrust when the propeller blades rotate through the nonuniform wake. The frequency of this disturbance is well known as blade passing frequency, which is equal to the shafting rotating speed multiplied by the number of blades [
The transmission path of propeller thrust in propulsion shafting system can be described as follows: propeller → shafting → thrust bearing → foundation → hull. While the thrust bearing transmits propeller thrust to hull, it also provides the transmission channel of longitudinal vibration from propulsion shafting to different regions of hull, resulting in subsequent reactions of hull. Thus, the longitudinal vibration of propulsion shafting is one of the secondary excitation sources of hull vibration relative to the primary excitation sources, such as propeller, diesel generator, and auxiliary machinery [
The study of shafting longitudinal vibration is much more complex than the torsional vibration. The reason is that in torsional vibration the vibratory system is only confined to the rotating elements [
The propulsion shafting and hull are coupled system. Compared with the hull vibration, the longitudinal vibration of propulsion shafting is local vibration. Since the mass of propulsion shafting system is far less than the hull while the shafting nature frequency is larger than the hull, the coupled effect of hull and propulsion shafting is limited. Hence, the propulsion shafting can be regarded as an isolated system away from the hull [
It should be noted that the problem of propulsion shafting longitudinal vibration occurs during the shafting rotating course. Once the shafting is still, this type of vibration will be stopped immediately. That is, the propulsion shafting is rotorbearing system, and the study of longitudinal vibration must be based on the method of rotor dynamics. The propulsion shafting can be deemed as a chained structure with a branch of thrust bearing, which is particularly suitable to use the transfer matrix method (TMM) for analysis. Since firstly proposed by Prohl [
Thrust bearing is the sole axial support equipment for propulsion shafting system. Virtually all marine vessels, both merchant and combat, use the sliding tilting pad thrust bearing for purpose of transmitting propeller thrust to hull [
Apart from the thrust bearing itself, the axial rigidity of propulsion shafting system is also contributed by the mounting foundation of thrust bearing. So an estimation of foundation stiffness is significant to predict axial rigidity of propulsion shafting system. Due to structural complexity of foundation, the numerical estimation is made of foundation stiffness using the FEM in this paper. In all, three sets of models are analyzed to identify effects of surrounding support structure. Taken as a whole, the structural stiffness of thrust bearing is resultant value of thrust bearing unit stiffness and foundation stiffness. The influence of thrust bearing structural stiffness on dynamic characteristics of propulsion shafting longitudinal vibration, especially the 1st nature frequency and vibratory response, has reached an agreement in marine engineering. However, another factor affecting the longitudinal vibration level of propulsion shafting, which is the location of thrust bearing, has drawn little attention at present. In fact, the alteration of thrust bearing location is an effective measure to raise critical speed and reduce offresonance response. This study is also conducted in the work. The ultimate objective of controlling propulsion shafting longitudinal vibration is to minimize thrust load transmitted to hull and underwater acoustic radiation. The results indicate that the measures of structural modification, such as reinforcement of thrust bearing structure and alternation of thrust bearing location, fail to reduce fluctuating thrust transmission from the propellershafting system to the hull.
With regard to longitudinal vibration, the journal bearings do not make stiffness and inertia contributions, so there is no necessity to be taken into account. Because of the chained characteristic of propulsion shafting system, the transfer matrix model has been established to describe dynamic behaviors. A modular representation of propulsion shafting system is given in Figure
Modular representation of propulsion shafting system.
The propeller is fixed at the end of stern shaft. The mass of propeller and entrained water around the propeller makes up a large proportion of the total mass of propulsion shafting system approximately 60%. Thus, the propeller can be modeled as a lumped mass. Similarly, the flexible coupling is also interpreted as a lumped mass. The transfer matrixes of propeller and flexible coupling are given by
The drive shaft is comprised of stern shaft, intermediate shaft, and thrust shaft, but the diameters of respective shaft are not consistent at all, such as the diameter of flange is much too larger than shaft neck. That is, the geometry of drive shaft is stepped. Apart from a few tiny structural defects, such as chamfer and groove, the drive shaft can be simplified as a series of uniform shafts with different diameter and length.
The transfer matrix of freefree
For
To some extent, the thrust bearing and foundation can be divided into liquid part of lubricant oil film and solid part of steel structure. The thrust bearing is independently mounted on foundation via bolts rigidly in submarines. It means that the thrust bearing and foundation provide axial support for propulsion shafting system together or the thrust bearing and foundation are two sources of axial rigidity in propulsion shafting system.
The thrust bearings of submarine are those of Michell type and Kingsbury type, of which the distinct difference is the supporting form of pivoted pads. The dynamic characteristic of thrust bearing is complicated because of its irregular configuration. Taking, for example, the Kingsbury thrust bearing, by tracing the load path, it is made up of thrust collar, pads, buttons, upper leveling links, lower leveling links, retainer ring, and house. The structural stiffness of thrust bearing unit can be obtained by computing the total flexibility. With proper simplification, the individual flexibility of various components can be calculated although the geometry of these components is irregular. Since these components are connected in series, all flexibilities are summed to get the total flexibility. This theoretical method was firstly proposed by Vassilopoulos and Hamilton [
The preceding description is confined to the stiffness generated by thrust bearing unit, but it does not incorporate the stiffness generated by foundation. Actually, the dominant factor of deciding whether thrust bearing structure is rigid or flexible is the foundation spring constant. The details of estimating foundation stiffness are described in Section
The following expression is made:
The oil film between thrust collar and tilting pads is modeled as linear spring stiffness
The hull is interpreted as the rigid boundary condition. With introduction of
Then the thrust bearing and foundation are visualized as a spring attached to thrust collar in Figure
Equivalent mechanic model of thrust bearing and foundation.
The magnitude of thrust collar axial stiffness reaches 10^{11}, which amounts to that of a rigid body. The relationship between state variables in the left and right sides of thrust collar has the following form:
Substitution of expression
For dimensional TMM, the deviation accumulates as the increase of trial frequency because of large numbers in transfer matrix, which affects accuracy of results seriously. The error can be eliminated by transforming the dimensional transfer matrix to the dimensionless form. Introducing the following transformation of state vector:
By substitution of (
By replacing
One problem in application of the above dimensionless transfer matrixes is that lumped mechanical elements, such as propeller and flexible coupling, have no geometrical parameters. To overcome this difficulty, the cross section area and length of adjacent uniform shaft are utilized as nominal cross section area and length of those lumped parts.
The relationship between state vector of the right and left ends of propulsion shafting can be written as
According to the mechanic model of propulsion shafting shown in Figure
To calculate vibratory response of propulsion shafting excited by propeller fluctuating thrust, the transfer matrix of respective subsystem still needs to be expanded. The transfer matrix equation of propeller excited by thrust load is written as
Equation (
Similarly, the expanded transfer matrixes of subsequent subsystems can be deduced with the same procedures, which are represented by
Then, the accumulated expanded transfer matrix of the complete propulsion shafting system is given by backward matrix multiplication of all subsystems:
Combined with the freefree boundary conditions, the solution of propeller response is
When the propeller reaction is determined, the vibratory response of the whole propulsion shafting system can be solved. And then the dynamic force of oil film and thrust load transmitted to hull are, respectively, given by
As one of the most important indexes of measuring propeller fluctuating thrust acting on hull through the propulsion shafting system, the force transmissibility is then easily obtained by
The values of propeller mass, structural stiffness, and lumped mass of thrust bearing, stiffness, and damping of lubricant oil film within thrust bearing are all necessary to determine characteristics of propulsion shafting longitudinal vibration. These values are easily got from producers except the dynamic parameters of oil film. The longitudinal vibration of thrust collar squeezes the oil film, resulting in the variation of film thickness at the equilibrium. Generally, this variation is so small that the small perturbation method can be used to calculate the stiffness and damping coefficients of oil film combined with hydrodynamic lubrication theory.
As is shown in Figure
Schematic diagram of thrust bearing.
The static characteristics of oil film, including distribution of thickness, viscosity, pressure, and temperature, are obtained by solving two equations of Reynolds and energy as well as two expressions of film thickness and lubricant oil viscosity temperature, simultaneously.
The steady Reynolds equation in cylindrical coordinate is expressed in dimensionless form as [
The energy equation is similarly written as [
The preceding two equations are derived from the conventional form of the equations by means of the following substitutions:
The expression of film thickness can be deduced from the geometrical shape of oil film and is given by
In the end, there is necessity to add the expression of lubricant oil viscosity temperature. Here, the Walther equation is used directly [
Note that the two equations of Reynolds and energy are coupled; there is no possibility to obtain analytic solution, but the numerical solution can be solved. With the finite difference algorithm, the oil film attached to the surface of pad is divided into a mesh pattern, as shown in Figure
Mesh pattern of oil film.
Then, the two partial differential equations of Reynolds and energy are reduced to two sets of algebraic equations. With introduction of boundary conditions of pressure and temperature, the two algebraic equations are solved by an iterative procedure until the results meet defined convergence criteria. It should be pointed out that not all convergent solutions are actual static characteristics of oil film except that they meet additional two conditions; that is, the load capacity of oil film must balance with propeller steady thrust and the moment to supporting center is equal to zero. The load capacity and moment of oil film can be solved by the following formulas. By using the Simpson formula in numerical integration, it is easy to turn the integrals into the sums:
The relationship between propeller steady thrust and shafting rotating speed can be established as
Due to a scarcity of test data of the thrust coefficient
The detailed numerical procedures are presented in the appendix.
By setting
Assuming that the magnitude of perturbation is small, a firstorder expansion of the oil film dynamic pressure can be made:
And then, the dimensionless stiffness and damping of oil film can be obtained by area integral of dynamic pressure in expression (
For dynamic analysis, the term of squeezing oil film must be added in the steady Reynolds equation (
By substitutions of pressure and film thickness by the expressions (
Note that the forms of the above two equations in the left sides are the same as the steady Reynolds equation, in which just
It can be observed that the dynamic characteristics of oil film are determined by geometry size of pad
The parameters of tilting pad thrust bearing are summarized in Table
Parameters of thrust bearing.
Parameter  Symbol  Value 

Tilting pad  
Inner radius (mm) 

165 
Outer radius (mm) 

325 
Central angle (rad) 

0.70 
Number 

8 
Supporting center  
Radial coordinate (mm) 

245 
Circumferential coordinate (rad) 

0.405 
Lubricant oil  
Density (km/m^{3}) 

890 
Specific heat capacity (J/kg·°C) 

1922.8 
Inlet temperature (°C) 

35 
Viscosity (Pa·s) 

0.051 
A program to solve dynamic characteristics of lubricant oil film has been devised on the platform of MATLAB. Since the propeller steady thrust correlates with shafting rotating speed, the dynamic characteristics of oil film must be a function of rotational speed as well. The results are given in Figure
Stiffness and damping of oil film.
The calculation of stiffness and damping of oil film is a complicated task, in which a large amount of work concentrates on the solution of static characteristics. For simplification, one approach used previously is to replace the distribution of film viscosity by an equivalent viscosity at thermal equilibrium temperature [
Here, the results at typical shafting rotating speeds based upon equivalent viscosity and variable viscosity are compared in Table
Calculated stiffness and damping of oil film based upon equivalent viscosity and variable viscosity.
Shafting rotating speed (rpm)  40  60  80  100 

Stiffness 

Variable viscosity  0.59  1.66  3.38  6.06 
Equivalent viscosity  0.63  2.09  4.51  7.80 
Damping 

Variable viscosity  1.94  3.63  5.43  7.90 
Equivalent viscosity  2.83  9.20  16.14  22.32 
To estimate the longitudinal vibration level in propulsion shafting system, it is desirable to know the axial stiffness of thrust bearing foundation in asbuilt condition. The foundation is typical welding assembly made up of plates. Briefly, the axial stiffness of foundation is the result of bend and shear deflection of plates relative to hull, particularly the reaction of shear, at the excitation of thrust load. Because of the continuity of supporting structure, the surrounding structure also contributes to foundation stiffness. For such complex structure, analytical method is not deemed to provide sufficiently accurate solution. It is therefore decided to resort to the finite element method (FEM), and the commercial finite element software ANSYS is used.
In view of acoustic design, the foundation may be made as rigid as desired. Also, to guarantee the normal operation of thrust bearing, the stiffness of foundation should be sufficient. In fact, the measure of thrust bearing independently mounted is to achieve a much more rigid system in the axial direction. From the point of extensive modifications envisioned necessary to solve the problem of insufficient foundation stiffness, there is necessity to be aware of effect in any proposed modifications to stiffen the foundation.
Since the stern hull plays major role in supporting the foundation and reacting to the propeller thrust excitation, the structural behavior of foundation can be analyzed by considering this aft portion of hull. The finite element model of stern hull including foundation and supporting structure, named Model 1, is shown in Figure
Illustration of Model 1.
Profile
Front view
Foundation and supporting structure
In order to identify contribution of various parts to foundation rigidity, additional two models are formulated, named Model 2 and Model 3, respectively. Figure
Illustrations of Model 2 and Model 3.
Model 2
Model 3
With Hooker theorem, the axial stiffness of foundation can be obtained by loading in axial direction and measuring the corresponding deformation. Since the thrust bearing is mounted on foundation with bolts, the propeller thrust is transmitted to foundation as the form of lumped force at the location of each bolt. Therefore, the static load is distributed equally over eight nodes laying four nodes each on the two surfaces of the foundation. In each case, a total axial force of 1 × 10^{5} N is applied.
The assignment of boundary conditions is crucial. For Model 1, all zero translations are located at the nodes in front of hull. For Model 2, only the axial translation is constrained to zero for the nodes lying on the bottom of plates. For Model 3, all translations are constrained to zero and all rotations are free for the nodes lying on the bottom due to constraints of hull. With these applied boundary conditions, the axial displacement of nodes at the location of bolts is measured. By taking the average of these node deformations, the axial stiffness of three models is evaluated, which is 8.233 × 10^{8} N/m, 5.046 × 10^{9} N/m, and 8.299 × 10^{8} N/m, respectively. By comparison, the rigidity of foundation itself is sufficient, but the surrounding structure connected with foundation, particularly the pedestal, is relatively flexible. The axial stiffness of marine thrust bearing unit is generally more than 1 × 10^{9} N/m; it means that the thrust bearing unit is stiffer than the mounting foundation, which should not occur in ships if possible. The results of Model 1 and Model 3 indicate that the hull makes little contribution to foundation stiffness, while the pedestal affects the rigidity of foundation greatly. That is, Model 3 is accurate enough to estimate stiffness of thrust bearing foundation.
The foundation stiffness should not be much too less than that of thrust bearing unit from viewpoint of vibration control. The foundation and pedestal are vertical structure like a hollow cantilever box. Under the action of thrust, the static deformation decreases gradually from the top to the bottom. To stiffen the foundation greatly, the measure of thickening plates is not always effective. For instance, when the plate thickness of pedestal increases 8 mm, the foundation stiffness is only 1.014 × 10^{9} N/m, or an increase of approximately 22.2%. Obviously, this improvement is limited. The effort of thickening plates does not achieve the prospective objective of reinforced foundation. It seems that it is not technically feasible to increase the foundation stiffness largely without drastic changes in structural configuration of foundation.
Thus, the efforts are directed toward modifying the internal structural configuration of pedestal. One evident shortcoming of the original internal structural configuration of pedestal is that the stiffeners are almost concentrated on the top, while the bottom seems to be relatively flexible. With reasonable supplement added in this area, the total stiffness of foundation will be improved substantially. One proposed modified internal structural configuration is demonstrated in Figure
Illustrations of the original and modified internal structural configuration of pedestal.
The original
The modified
The reinforced foundation has significant effect to improve resultant stiffness of thrust bearing. Regarding assembly structure, the total stiffness depends on individual stiffness of various components as well as the connection stiffness. So the connective quality of thrust bearing and foundation is also of importance. In all, the foundation is required to be as rigid as possible. The reinforced foundation contributes to reducing the shear deflection of the pedestal and relieving the rotation of stern hull. Once the foundation is much too flexible, the thrust collar will be misaligned with respect to the bearing gland face, which is seriously harmful to lubrication performance of thrust bearing and results in increased wear and tear of pads. Only these factors are thoroughly implemented; the problem of insufficient axial stiffness in propulsion shafting system will not occur at all.
The parameters associated with the propulsion shafting system are listed in Table
Parameters of propulsion shafting system.
Parameter  Symbol  Value 

Density (kg/m^{3}) 

7850 
Young’s modulus (GPa) 

2.0 
Propeller (includes entrained water) mass (kg) 

7150 
Flexible coupling mass (kg) 

1300 
Thrust collar mass (kg) 

445 
Thrust bearing mass (kg) 

4435 
Thrust bearing resultant stiffness (MN/m) 

1100 
The 1st and 2nd natural frequencies of propulsion shafting longitudinal vibration in terms of shafting rotating speeds are shown by the Campbell diagram in Figure
Campbell diagram.
However with an increase of rotational speed, the 1st and 2nd natural frequencies tend to be constant gradually. The variation amplitude and range of natural frequencies are related to the relative magnitude of oil film stiffness and thrust bearing stiffness. At low speeds, the oil film stiffness is less than that of thrust bearing, and then the effect of rotational speeds on natural frequencies is of increased significance. But the changeability of natural frequencies is slight once the oil film stiffness exceeds that of thrust bearing. For the study, the stiffness of oil film at shafting rotating speed 60 rpm reaches the thrust bearing stiffness, and then the 1st and 2nd natural frequencies are almost invariable in the range of subsequent rotational speeds. Generally, it is the case that the more flexible the thrust bearing is, the narrower the influence range of rotational speeds on natural frequencies is available.
The excitation frequencies of propeller fluctuating thrust are tonal at the blade passing frequency. Obviously, the critical speed corresponding to the 2nd natural frequency is above the rotational speed. Thereby the 1st frequency is of particular interest. Besides, the 1st mode is the most superior mode of propulsion shafting, which contributes to vibratory response greatly.
As far as the longitudinal vibration is concerned, the structural stiffness of thrust bearing is undoubtedly the most important effect factor. Provided that the construction of thrust bearing was reinforced, the natural frequency of propulsion shafting would rise correspondingly. However, there are rapid change region and slow change region with respect to natural frequency of propulsion shafting in the variation range of thrust bearing stiffness, which is essential to decide whether to modify thrust bearing or shafting rotor when propulsion shafting encounters severe longitudinal vibration.
Most often, the submarine sails at low shafting rotating speeds. Taking the rotational speed 100 rpm as an example, the effect of thrust bearing stiffness on the 1st natural frequency of propulsion shafting longitudinal vibration is given in Figure
Effect of thrust bearing stiffness on the 1st natural frequency.
In the case of asbuilt submarine, the mounted thrust bearing will be operational forever unless there are failures. It means that the stiffness of thrust bearing unit is established throughout the period of validity, while only the foundation stiffness is changeable. Thus, the variation of thrust bearing stiffness depends on structural modifications of foundation. If the foundation was too flexible initially, especially when the spring constants of foundation and shafting were of the same order of magnitude, a large improvement in stiffening of the foundation would lead to an increase of the 1st natural frequency. Figure
Effect of foundation stiffness on the 1st natural frequency.
Although the 1st natural frequency varies with spring constant of thrust bearing, the variation amplitude is really limited. Even though the thrust bearing stiffness is multiplied 7 times, the 1st natural frequency only adds 2.5 Hz, let alone how to achieve such large improvement of thrust bearing stiffness in practice. However, the study is very useful for estimation of the 1st natural frequency in new design period of propulsion shafting system. In the preliminary design stage, there is no necessity to determine the thrust bearing stiffness, particularly the foundation stiffness, with a high degree of accuracy. Furthermore, the stiffer the foundation is relative to shafting, the less effect an error in estimating foundation stiffness will have on the 1st natural frequency. For example, if the estimated stiffness of foundation was 1.2 × 10^{9} N/m, while the actual value was 1 × 10^{9} N/m, the calculated 1st natural frequency was 23.34 Hz compared with the exact value of 22.88 Hz. Although the error of foundation stiffness reaches up to 20%, the error of the 1st natural frequency is less than 2%.
From the viewpoint of vibration control, the measure of altering natural frequency is one of the simplest control technologies, which can be easily achieved by changes in stiffness and mass. For longitudinal vibration existing in service, the steady response of thrust bearing induced by thrust load at blade passing frequency with respect to various spring constants of thrust bearing is depicted in Figure
Steady response of thrust bearing induced by thrust load at blade passing frequency.
The longitudinal vibration transmission through the propulsion shafting system results in axial excitation of hull and generation of structureborne noise, which constitutes important source of lowfrequency acoustic signature. This type of vibration transmission is actually the propagation of stress wave. The lubricant oil film between thrust collar and tilting pads within thrust bearing serves as the key transmitter of propeller steady thrust from propeller to hull. But unfortunately, the oscillations in thrust are also passively transmitted through the oil film to thrust bearing and hull.
Concentrating at the blade passing frequency and considering the existing understanding of square functional relationship between propeller steady thrust and shafting rotating speed, the fluctuating thrust is still set as a fixed proportion 5% of steady thrust at different rotational speeds. The propeller fluctuating thrust and dynamic force of oil film in terms of shafting rotating speeds are shown in Figure
Dynamic force acting on propeller, oil film, and hull.
Further, the thrust transmission characteristics in propulsion shafting system can be described by the force transmissibility in frequency domain. The waterfall plot of force transmissibility versus frequency at different shafting rotating speeds is presented in Figure
Waterfall plot of force transmissibility.
Figure
Effect of variations in thrust bearing stiffness on force transmissibility.
Although in most of cases stiffening thrust bearing foundation is helpful to alter natural frequency and reduce vibratory response on straight course, it is not always realistic except for drastic changes in structural configuration. Otherwise, the stiffness of reinforced thrust bearing steel structure is slightly larger than that of the original. Alternatively, the change of thrust bearing location is more technically feasible in practice.
In view of space and arrangement of the stern, the thrust bearing is only suitable to move backward to the side of propeller, and the range of movement is limited. Since the stern shaft is placed between pressure hull and nonpressure hull, it is not possible to mount thrust bearing there. Hence, the range of thrust bearing movement is confined in the length of the intermediate shaft, as shown in Figure
Resultant Stiffness of shafting between thrust bearing and propeller.
Displacement (m)  0  0.5  1.0  1.5  2.0 


Stiffness ( 
2.24  2.37  2.51  2.68  2.86 
The range of thrust bearing movement.
Also, the case of shafting rotating speed 100 rpm is examined. The influence of moving thrust bearing backward by various displacements on the natural frequency and mode shape of propulsion shafting longitudinal vibration are given in Figures
Natural frequencies versus backward displacement of thrust bearing.
1st mode shape of longitudinal vibration versus backward displacement of thrust bearing.
From the standpoint of vibration mode, the decrease of mode shape amplitude means the reduction of vibratory response. Under the condition of the same thrust bearing stiffness, the calculated steady response of propeller in time domain is shown in Figure
Propeller steady response versus various backward displacements of thrust bearing at 100 rpm.
Of particular concern is the effect of minimizing thrust load transmitted to hull with the conjunction of the two measures. Figure
Effect of variations in thrust bearing stiffness and thrust bearing location on force transmissibility.
The study focuses on the problem of longitudinal vibration in submarine propulsion shafting system and belongs to the scope of rotor dynamics. In view of the chain characteristic of propulsion shafting system, the modular transfer matrix model had been established. Firstly, the propulsion shafting system was divided into several subsystems, of which dynamic characteristics are all described by transfer matrixes. To improve accuracy, the transfer matrix of individual subsystem was transformed to the dimensionless form. Then, the calculation methods of stiffness and damping of oil film and foundation stiffness were proposed. Using these obtained values of dynamic parameters, the characteristics of longitudinal vibration and thrust transmission in the propulsion shafting system were analyzed. In the end, the effect of thrust bearing location was discussed.
On the basis of the studies carried out, the following conclusions can be made.
As far as longitudinal vibration is concerned, the axial dynamic characteristics of lubricant oil film between thrust collar and tilting pads within thrust bearing should be included in mechanic model of propulsion shafting system. Combined with the hydrodynamic lubrication theory, the stiffness and damping coefficients of oil film can be solved by the small perturbation method. The thermal effect of lubricant oil affects the solutions significantly and should be taken into account for dynamic analysis of oil film. The oil film stiffness and damping keep pace with the increase of shafting rotating speed nonlinearly.
For asbuilt submarine, the foundation is the dominant factor of deciding whether the propulsion shafting system is rigid or flexible in axial direction. The FEM is capable of providing sufficient accurate estimation of the foundation axial stiffness. The pedestal makes great contribution to foundation stiffness, while the effect of hull is slight. To achieve reinforced foundation, the effort should be directed toward modifying the internal structural configuration of pedestal. One reinforced scheme is proposed, but the optimized design involving the maximum rigidity and the minimum mass addition still needs to be conducted.
The natural frequencies of propulsion shafting longitudinal vibration vary with the shafting rotating speeds, but the variation amplitude decreases gradually. The fact that the critical speed of the 1st natural frequency falls within the running range needs to attract enough attention. Although the 1st natural frequency can be altered by changes in thrust bearing stiffness, the actual effect of this measure depends on the initial spring constant of thrust bearing. Generally, the critical value 2.5 × 10^{9} N/m is proper to serve as reference. In any case, a large improvement in stiffening of thrust bearing structure contributes to reduce offresonance response of propulsion shafting longitudinal vibration to some extent.
The thrust load transmitted to hull is multiplied several times in lowfrequency band whether the propulsion shafting is in the resonance state or not. The nonlinear characteristics of oil film enable the propulsion shafting to have the force magnification effect. The measure of stiffening thrust bearing is of little benefit to minimize the transmission of propeller fluctuating thrust from the propulsion shafting system to the hull.
The benefits of altering the 1st natural frequency and reducing vibratory response validate that the measure of locating thrust bearing backward is effective. Viewed from this perspective, the measure of backward movement of thrust bearing has the same effect as reinforcement of thrust bearing structure. However, the two measures of structural modification fail to minimize thrust load transmitted to hull, and there is necessity to develop other control technologies.
With the central finite difference numerical method, the equations of Reynolds and energy are reduced to finite difference forms as follows:
Equation (
The definitions of convergence criteria are given by
The whole calculation steps can be described with the following flow chart as shown in Figure
Calculation flow chart of hydrodynamic lubrication.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors wish to appreciate the financial support from the Admiralty under Grant no. 401130301.