Vibration analysis of a submarine elastic propeller-shaft-hull system using FRF-based substructuring method

Accepted Manuscript Vibration Analysis of a Submarine Elastic Propeller-shaft-hull System using FRFbased Substructuring Method

Feng Chen, Yong Chen, Hongxing Hua PII:

S0022-460X(18)30817-4

DOI:

10.1016/j.jsv.2018.11.053

Reference:

YJSVI 14536

To appear in:

Journal of Sound and Vibration

Received Date:

11 June 2018

Accepted Date:

30 November 2018

Please cite this article as: Feng Chen, Yong Chen, Hongxing Hua, Vibration Analysis of a Submarine Elastic Propeller-shaft-hull System using FRF-based Substructuring Method, Journal of Sound and Vibration (2018), doi: 10.1016/j.jsv.2018.11.053

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ACCEPTED MANUSCRIPT Vibration Analysis of a Submarine Elastic Propeller-shaft-hull System using FRF-based Substructuring Method Feng Chen*1,2, Yong Chen1,2, Hongxing Hua1,2 1,2 State Key Laboratory of Mechanical System and Vibration, Laboratory of Vibration, Shock & Noise, Shanghai Jiao Tong University, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, China, 200240 * Corresponding author: [email protected]

Abstract: A dynamic model to study the coupled longitudinal and transverse vibrations of a submarine elastic propeller-shaft-hull system is developed using the FRF-based substructuring method (FBSM). The total system is firstly modeled as two substructures: the elastic propeller-shaft subsystem and the hull shell. For the former substructure, the elastic propeller is modeled by using harmonic blade array elements and the shafts are assumed to be Timoshenko beams, while the latter one is modeled using traditional finite element method. After that, the two substructures are synthesized using FBSM. The modes, the natural frequencies and the coupled longitudinal and transverse vibration characteristics of the propeller-shaft subsystem, the hull shell, and the total system are analyzed. An experiment studying the dynamic characteristics of a large-scale submarine experimental setup is processed and compared with the numerical results, which shows great consistency. Finally, a further discussion is carried out focused on how the bearing stiffness affects the coupled vibration characteristics of the total system.

Keywords: Propeller-shaft-hull system; FRF-based substructuring method (FBSM); Coupled vibration characteristics; Harmonic blade array element

1. Introduction The low-frequency vibration of the submarine can generate significant radiated noise levels, which highly restricts the submarine stealth performance. An important excitation source is the rotating propeller-shaft system. The excitation forces result from the residual unbalance of the propeller, misalignment of the shafts and fluid-induced forces acting on the propeller caused by rotation of the propeller in a non-stationary wake. These forces will transmit to the submarine hull through the shaft,

ACCEPTED MANUSCRIPT bearings and elastic foundations, leading to corresponding vibration and significant sound radiation of the hull. Thus, detailed research on the vibration responses of the coupled propeller-shaft-hull system is of great important significance to guide the practical design of a stealth submarine. Owing to the complexity of the elastic propeller-shaft-hull system, a relatively simpler approach is to analyze the vibration of the submarine propeller, shafts, and hull shell separately. The elastic property of the propeller, the interaction of vibration between the propeller-shaft system and the hull shell are obviously ignored. At lower frequencies, the submarine hull shell can be simplified as a long, thin stiffened cylindrical shell. The difficulty of studying the vibration characteristics of the hull is how to model the bulkheads, stiffeners and distributed concentrated mass. These studies were starting from the infinite cylindrical shell by Rayleigh [1], after that Forsberg [2] discussed the free vibration of a finite cylindrical shell with different boundary conditions. Bernblit [3] compared the dynamic response of a hull with different types of spacing stiffeners using finite element method. Wilken [4] used the admittance method to analyze the modal characteristics of a stiffened cylindrical hull and the stiffeners were considered as substructures. Laulagnet [5] described the rib force resultant moment with impedance matrix, then the ribs and the shell was connected through the boundary conditions, and the solution was derived by complex equations. Caresta [6] utilized an analytical method, a wave approach, and a power series solution, to calculate the cylindrical shell and the conical shell of the submarine hull respectively. Qu [7, 8] developed a semianalytical method for linear vibration analyses of functionally graded bodies of revolution with arbitrary boundary conditions to simulate the submarine hull with either bulkheads, stiffeners or distributed concentrated mass. Analytical method, transfer matrix method and finite element method are the three common tools adapted to study the dynamic characteristics of shafts. The coupled torsional, longitudinal and transverse vibration and the mutual influences between them had been extensively studied, such as Parsons [9], Schwibinger [10] and Kane [11]. Some simplifications to simulate the propeller, water lubricated and thrust bearings were implemented by Pan [12] and Schwanecke [13]. As for the propeller, in elementary rotordynamics, it was assumed to be a rigid body or simplified as a modal mass and stiffness system attached to the shaft [14]. Now, the common approach to study the dynamics of an array of blades is using commercial FEM codes, while it is still difficult when gyroscopic effect caused by rotation, the entrained water of the propeller and the coupling effect between propeller-

ACCEPTED MANUSCRIPT shaft and hull system are considered [15]. Carrera [16, 17] defined a unified formulation to perform freevibrational analyses of the rotating structures, which offers a procedure to obtain refined structural theories that account for variable kinematic description. Genta [18, 19] used a complex coordinates approach to derive the equation of motion of zero and first order disc and array of blades elements that take into account the blades' stagger angle. Genta and Chen [20, 21] developed a second and higher order disc and blade array element to study the dynamic characteristics of a ship propeller. There were also some studies conducted on coupled vibration between the shafting system and the hull shell, the numerical and experimental methods were often implemented instead of analytical research. Dylejko and Kessissoglou [22, 23] established a bar model for the shafting system and a cabin hull model to study admittance results under the action of the thrust bearing. Merz [24] established the coupling model of the propeller-shaft system and the submarine hull. He found the low-frequency vibrational modes of the hull and propeller/shafting system may result in a high level of radiated noise. Cao [25] took the finite element method to investigate the longitudinal and transverse vibrations of a shaft-hull structure excited by propeller force. Zou [26, 27, 28] proposed a series work to developed a three-dimensional sonoelastic method in the frequency domain to analyze the fluid-structure interactions, acoustic radiation and acoustic propagation of a ship. The hydro-elastic analysis and sono-elastic analysis methods are incorporated with the Green’s function of the Pekeris ocean hydro-acoustic waveguide model. The method was validated by calculating the acoustic radiation of a floating elastic spherical shell and an experiment of the underwater acoustic radiation of two ring-stiffened cylindrical shell models. Qi [29] employed a three-dimensional sono-elasticity method to explore the acoustic and vibrational characteristics of a propeller-shaft-hull coupled system. The acoustic field was solved by introducing Green’s function together with the Price–Wu generalized fluid–structure interface boundary conditions. Numerical models for hull structures with a shaft and without a shaft are designated. The correlations of the line spectra of acoustic radiation and the corresponding vibration modes of the hull are identified. Recent years, a powerful method called the transfer function synthesis method or the FRF-based substructuring method (FBSM) [30, 31] was developed for the analysis of the response of a complex builtup structure with high modal density. Its superiority comes primarily from the ability to incorporate experimental FRFs into the formulation. It can also predict the responses of the total structure from the FRFs of the substructures and the substructures can be connected through linear spring or the interface impedances. There were several applications introduced FBSM in studying the shaft-hull system. Missaoui

ACCEPTED MANUSCRIPT [32] simulated the coupling between the hull and the shaft by introducing artificial springs via variation equation. Guo [33] used displacement admittance theory to deal with the force and moment acting on the plate and the shell, and the influences of the internal structure on the vibration of the structure were studied. Liu and Ewins [34] introduced the elastic boundary to consider the connection between two substructures, the degrees of freedom of the two substructures were no need to be consistent and it also allowed to modify the total system dynamic characteristics by changing the parameters of the connecting points. To sum up, all the above studies include four ways of implementation. First, ignore the coupling among the propeller-shaft-hull system, but to study the dynamics of the elastic propeller, the shafts, and the hull shell themselves. Second, use theoretical formula with many simplifications to model the coupled shaft-hull system considering the propeller as a rigid mass or a lumped mass-spring system. Third, use finite element method to study the coupled vibrations of the elastic propeller, the shaft and the hull shell with low efficiency and high adjustment complexity. The last, use FBSM method to study the coupled dynamics of the propeller-shaft-hull system, but the elastic property of the propeller is still idealized, and the supportive experimental data are inadequate. If the hydrodynamic characteristics and the hydro-elastic phenomena of the propeller in either open water flow or wake flow and the complex Fluid Structure Interaction (FSI) need to be considered, the most used numerical procedure is to develop a full 3D FE/BE model, which is a very time-consuming and complex work. This is mainly because the geometry structure of the propeller and the Fluid Structure Interaction between the whole system and water are very complicated. If we develop a model, the propeller is modeled using simple harmonic elements, which can consider the elasticity of the propeller and the FSI between propeller and hull, meanwhile the FSI between hull structure and water is implemented using FEM/BEM as common, the hull substructure and the propeller-shaft are able to connect using FBSM. Then it is obvious that this model will take great advantage to a full 3D FEM/BEM procedure when used to analyze the low-frequency vibrations and acoustic radiations of the propeller-shaft-hull system excited by unsteady propeller bearing force. This is because the complex structure and FSI of the propeller and water can be avoided to take into consideration. Thus in the present article, the chief goal is to develop a dynamic model to study the low frequency longitudinal and transverse vibrations of the elastic propeller-shaft-hull system in open air using the FRFbased substructuring method (FBSM). The elastic property of the propeller is considered using harmonic

ACCEPTED MANUSCRIPT elements instead of using full 3D finite elements. This model can be used as the foundation to analyze the vibrations and acoustic radiations of the propeller-shaft-hull system excited by unsteady propeller bearing force in open water flow or wake flow. The paper is arranged as follows: in section 2, the research object is briefly introduced.

In section 3, the elastic propeller-shaft system and the hull shell are firstly modeled

separately, the elastic propeller uses harmonic blade array elements, the shafts use Timoshenko beam elements and the hull shell uses shell elements, then the substructures are synthesized using FBSM. In section 4, the modes, the natural frequencies, and the coupled longitudinal and transverse vibration characteristics of the propeller-shaft substructure, the hull shell, and the total system are simulated. In section 5, an experiment studying the vibration characteristics on a large scale experimental setup is carried out and compared with the numerical results. In section 6, a further discussion is carried out focused on how the stiffness of the bearings affect the dynamic characteristics of the total system. Section 7 concludes the paper and gives the readers a blueprint of our future work.

2. Submarine propeller-shaft-hull system The research object is a submarine propeller shaft-hull system, which shows in Fig 1. It consists of two substructures: a hull shell and an elastic propeller-shaft system.

Figure 1. Sketch of the submarine propeller shaft-hull system.

The hull shell as shown in Fig 2 is mainly composed of four parts: the cabin shell, the bulkhead, the bearing houses (or seats) and the motor seat. The cabin shell is the combination of six cylindrical and one conical segments. The diameter of the hull is 1.85m. The length of the hull, of the conical segment and of each cylindrical segment are 18.58m, 5.08m and 2.25m. The thickness of the conical and cylindrical shell are 5 mm and 6 mm. A stiffening ring is used between every two cylinder segments to reinforce the structure, the span between the two stiffening rings is 2.25m. In each segment, there are also several stiffeners, the distances between them are 0.15m.

Figure 2. The hull shell subsystem.

Fig 3 shows the sketch of the elastic propeller-shaft system. It is assembled by a propulsion motor and

ACCEPTED MANUSCRIPT its base, two rotors and an elastic propeller. The total length of the propeller-shaft system is 4.09 m, the diameter of the solid rotors is 0.07 m, the distances between the propeller, the bearings, and the couplings are listed in Table 1. The propeller contains five blades, the diameter of the propeller is 0.458 m. A flexible coupling is used to connect the motor to the thrust shaft and one rigid coupling connects the two shafts. Two water lubricated bearings (the stern bearing and the intermediate bearing) and an oil-lubricated bearing (the thrust bearing) are fixed on the hull shell to support the shafts. The driven elastic coupling and the thrust plate are simplified as concentrated mass and their inertias. The drive elastic coupling at one end is connected to the motor, while at the other end is connected to the driven coupling through the longitudinal spring. Since the coupling stiffness is very low compared to the shaft, the drive elastic coupling and the motor can be assumed as a lumped mass-spring system. All these property parameters are listed in Table 1.

Figure 3. The elastic propeller-shaft subsystem.

Table 1. Key physical parameters of the elastic propeller shaft system.

Fig 4 shows a photograph of the experimental setup. Compared to the numerical model, the boundary condition of the experimental setup is elastic because of the air springs, which is used to simulate the submarine floating environment at a natural frequency about 2.5Hz. Since this natural frequency is much lower than the natural frequencies of the propeller-shaft-hull system, the influence to the modal characteristics of the system itself is negligible.

Figure 4. Elastic propeller-shaft-hull experimental setup.

In the next chapters, the numerical simulations and experimental comparisons will be carried out on the above research object. The procedure is as follows: first, the dynamic model for the propeller-shaft subsystem and the hull shell will be established, and the two substructures will be synthesized using the equivalent stiffness of the bearing and its base. Second, the natural frequencies, modes and the forced responses of the dynamic model will be studied and compared with the experimental results. Last, the dynamic characteristics varying the bearing stiffness will be further discussed.

ACCEPTED MANUSCRIPT 3. The dynamic model of the submarine propeller-shaft-hull system In this section, the submarine propeller-shaft system and the hull shell is first modeled separately, the elastic propeller is modeled by using harmonic blade array elements, the shafts are assumed to be Timoshenko beams, and the hull shell is modeled using finite element method. Then the total system is established using an FRF-based substructuring method, the modeling procedure is as follows.

3.1 The elastic propeller-shaft subsystem The elastic propeller-shaft subsystem is assumed to be a flexible beam and an array of propeller blades. Since the shaft can be simply considered as a Timoshenko beam which can be easily found in any FEM textbook, only the modeling of the propeller blades and the connection between the shaft and the blades are briefly introduced, detailed derivation processes can be found in reference [18-21]. 3.1.1 Array of blades The propeller blades are modeled as Euler-Bernoulli beams and assumed to be equal, aligned along the radial direction and their sheer center coincides with the center of mass of each section. A typical cross-section of a blade perpendicular to the radial direction is shown in Fig 5. G is the section mass center, u1, u2, u3 are principal inertial axes while axes u, v and w lie along the radial, tangential and axial directions. The twist angle ψ is varied along the blade radius r. All the array properties concentrated in the mid-plane of the blade.

Figure 5. Cross section of each blade at a certain radius.

Assuming uj, vj, and wj are the radial, tangential and axial displacement components of the section of the jth blade taken at a radius r. Define an inertial frame whose origin is O, Pj is the coordinates in an inertial reference of point P, which can be expressed as 4

(P j - O)  (C - O)   R k ({r 0 0}T  {u j k 1

vj

w j }T ) ,

(1)

where C and O are the coordinates of the shaft-blade attachment point and the inertial frame origin O, respectively. P j - O is the displacement between the generic point P and point O and C - O is the displacement between the shaft center C and point O, Rk is the rotation matrices as a function of the angle

ACCEPTED MANUSCRIPT of the rigid body motion as reported in reference [20]. To describe the shape functions approximating the deformations of the propeller array of

blades,

define the following parameters: χ is a non-dimensional radius of each blade; A, I2, and I3 are the area of the cross-section of each blade and their area moments of inertia about the principal inertia axis (u2, u3 in Fig 5) of the cross-section. The displacement vector [uj, vj, wj] is then approximated by means of an Nth order truncated Fourier’s series in the angular coordinate θj

u

N

v w j   u v wic cos  i j   u v wis sin  i j   ,

(2)

i 1

where the coefficient of the various harmonics displacement uic,s and vic,s refer to the in-plane displacement while wic,s are related to the out-of-plane displacement, which can be approximated by shape functions nu, nv and nw, leading to

uic (  , t )  nu (  )qux (t ),

uis  , t )  nu (  )quy (t ),

vis (  , t )  n v (  )q vx (t ),

wic (  , t )  n w (  )q wx (t ),

vic (  , t )  n v (  )q vy (t ), wis (  , t )  n w (  )q wy (t ).

(3)

Three nodes are defined on this type of element: node 0 lies on the rotation axis and is common to all elements of a given array of blades, while nodes 1 and 2 are located at the inner radius and the outer radius of the element, respectively. The displacements of node 0 describe the rigid body motion of the array of blades, and the displacements of node 1 and 2, describe the deflections of the element from its rigid configurations. Since the coordinates and shape functions are derived, the kinetic energy of the array of blades is r

1 N 0 Ti     A(r )P jT,i P j ,i dr , 2 i 1 r

(4)

i

where ρ is the density of the blades, A(r) is the cross-section of the blades at radius r and Pj,i denoting the displacement of the center of mass of the jth blade at the radius r relative to the inertial reference. In differentiating with respect to time, angle θj must be considered as a function of time. Owing to the orthogonally of the harmonic functions, the kinetic energy can be split into in-plane and out-of-plane contributions

Ti  Tinp ,i  Toutp ,i . Therefore the in-plane and out-of-plane contributions to the kinetic energy is

(5)

ACCEPTED MANUSCRIPT 1 T T T T T Tinp ,i  [ 2 (qux m inp1,i qux  quy m inp1,i quy  q vx m inp 2,i q vx  q Tvy m inp 2,i q vy  2qux m inp 3,i q vx  2quy m inp 3,i q vy ) 2 T T T T T   (quy m inp1,i q ux  q Tvy m inp 3,i q ux  qux m inp1,i q uy  q vx m inp 3,i q uy  quy m inp1,i q vx  q vy m inp 2,i q vx 1 T T T  q m inp 3,i q vy  q m inp 2,i q vy )  (q ux m inp1,i q ux  q uy m inp1,i q uy  q vx m inp 3,i q vx  q Tvy m inp 3,i q vy )]. 2 1 2 T T Toutp ,i  [ (q wx m outp ,i q wx  q wy m outp ,i q wy )   (2q Twy m outp ,i q wx  2q Twx m outp ,i q wy )  q Twx m outp ,i q wx  q Twy m outp ,i q wy ]. 2 T ux

(6)

T vx

Thus matrices minp,i and moutp,i are given by integrals r

m inp1,i  m inp 3,i 

r

N 0  AnuT nu dr , 2 ri

m inp 2,i 

r0

N  AnuT n v dr , 2 ri

m outp ,i 

N 0  An Tv n v dr , 2 ri

(7)

r0

N  An Twn wdr. 2 ri

The potential energy is contributed to the elastic strain-stress natural of the material (Ue,i) and to the geometric effect (Ug,i), which is

U i  U e ,i  U g ,i .

(8)

The elastic energy is related to the radial extension and flexural deflections r

U e ,i 

2 2 2  10  A 1 E  2  s1, i   4  I 2  s2, i   I 3  s3, i   dr   2 ri  r r 

(9)

The prime indicates the partial derivative relative to the radial coordinates r and E is Young's modulus. The displacements s along the inertial axes is linked to the axial, tangential and radial directions by angle ψ, which is a function of the radial coordinate as

s1,i  u j ,i ,

s2,i  v j ,i cos  w j ,i sin ,

s3,i  w j ,i cos  v j ,i sin .

(10)

Then, the components of the elastic potential energy are expressed in terms of element generalized coordinates as 1 T T T U einp ,i  (qux k einp1,i qux  quy k einp1,i quy  q vx k einp 2,i q vx  q Tvy k einp 2,i q vy ), 2 1 U eoutp ,i  (q Twx k eoutp ,i q wx  q Twy k eoutp ,i q wy ). 2

(11)

The stiffness matrices are obtained from the shape functions by the following integrals r

k einp1,i

r

r

N 0 N 0 N 0 ' T ' '' T ''  EAnu nu dr , k einp 2,i  EI wn v n v dr , k eoutp ,i  EI v n ''wT n ''wdr , 2  4  4  2r ri 2r ri 2r ri

(12)

where Iv and Iw are the area moments of inertia of the cross-section in circumferential and axial direction v and w in Fig 5. The geometric potential energy is caused by the centrifugal forces Fr,i, which can be expressed as

ACCEPTED MANUSCRIPT Fr ,i  

r0 2

  Ardr 

2

Prw (r ).

(13)

r

And the geometric contribution to the potential energy can also be split into two independent contributions and expressed as

U g ,i  U ginp ,i  U goutp ,i 

1 2r 2

N r0

F i 1 ri

r ,i

(r )[v 'j ,i 2  w'j ,i 2 ]dr.

(14)

Integrating the equations, the geometric potential energy is

1 U ginp ,i  (q Tvx k ginp ,i q vx  q Tvy k ginp ,i q vy ), 2

1 U goutp ,i  (q Twx k goutp ,i q wx  q Twy k goutp ,i q wy ). 2

(15)

The stiffness matrices are given by the integrals of the shape functions r

k ginp ,i

N 0  2  Pr n 'v T n 'v dr , r ri

r

k goutp ,i

N 0  2  Pr n 'wT n 'wdr. r ri

(16)

If no external force acts on the element, the equations of motion for the array of blades will be 2 2   iG Q  M inp ,i Q inp ,i inp ,i inp ,i  (K inp ,i   K inp ,i   M niinp ,i )Q inp ,i  0, 2 2   M outp ,i Q outp ,i  iG outp ,i Q outp ,i  (K outp ,i   K  outp ,i   M nioutp ,i )Q outp ,i  0,

(17)

The in-plane and out-of-plane coordinates can be assembled in vectors as

Qinp ,i

 X 0  iY0     qux  iquy   q  iq  vy   vx

(71)

 y 0  ix 0  , Q outp ,i    . q wx  iq wy (51)

(18)

Note that X0+iY0 and Φy0-iΦx0 are the rigid body motion of the array of blades which indicates the displacements of the rotor center. The element mass, in-plane and out-of-plane gyroscopic, centrifugal and thermal stiffening and stiffness matrices are obtained by using Lagrange’s equations

M inp ,i

0 0   0 m inp1,i 0 0 

M niinp ,i

0   2i 0 0 

K inp ,i

0 0   0 0 0 0 

2

0 0  0 0  0  0  0 0    0  , M outp ,i   , G inp ,i  2i 0 m inp1,i m inp 3,i  , G outp ,i  2i  ,  0 m outp ,i  0 m outp ,i       m inp 2,i  0 m inp 3,i m inp 2,i  0 0 0  0 0  0 0  0  (19) 0    2  m inp1,i m inp 3,i  , M nioutp ,i  i  0  , K outp ,i    , K inp ,i  0 k einp1,i , 0 m outp ,i  0 k eoutp ,i  0  m inp 3,i m inp 2,i  0 k einp 2,i   0  0  0  0  , K outp ,i   . 0 k goutp ,i    k ginp ,i 

3.1.2 Beam-blade transition element

ACCEPTED MANUSCRIPT Beam and blade array element can't be directly linked for the compatibility of the displacement fields at the shaft-blade interface are not insured. A suitable transition element has thus been developed. This element is provided with two nodes, the first located on the shaft rotation axis, and the second located at the outer radius of the shaft (at the interface with the subsequent blade array element). The matrices for the shaft-blade transition element have been obtained from those described above for the blade element by just deleting all rows and columns linked with the degrees of freedom at node 1. This corresponds to constraining the displacements of the point at the inner radius of the element as a rigid body motion.

3.2 The hull shell subsystem The hull shell subsystem is modeled by using the commercial FEM software ABAQUS, its main components are shown in Fig 6. As mentioned in section 2, the hull shell subsystem is composed by the cabin shell, the bulkhead, the bearing seats (or bearing houses) and the motor seat. The cabin shell is modeled using shell element S4R, the thickness of the conical and cylindrical segments are 5 mm and 6 mm. The stiffening rings between every two cylinder segments are also modeled using shell element S4R with the thickness of 6mm, while the stiffeners are modeled using beam element B31. The motor seat and thrust bearing seat are modeled using shell element with a uniform thickness of 5mm. Since the stiffness of the thrust bearing house, intermediate bearing house, and the stern bearing house is far larger than that of the bearing seat and hull shell, they can be modeled as a rigid shell with a mass and inertia property. The element type and the number of the hull shell subsystem are listed in Table 2. The boundary condition of the hull shell is set free.

Figure 6. FEM model of the hull shell subsystem and its main components.

Table 2. Element type and the number of elements of the hull shell subsystem.

3.3 Connecting the two systems using FBSM 3.3.1 Frequency response functions of each substructure After implementing the propeller-shaft subsystem and the hull shell subsystem into finite element models, the dynamic equation for these two subsystems can both be presented as

  t  +Cu  t   Ku  t  =Q  t  , Mu

(20)

ACCEPTED MANUSCRIPT (t ) u (t ) and u(t ) where, M, K, and C are mass, stiffness and damping matrices for each substructure, u represents the acceleration, velocity, and displacement vectors of the substructure degree of freedoms, and Q(t) is the external load acting on each substructure. The first n natural frequencies and their regularized corresponding modes are related to the following equation

ΦiT MΦi  I,

 i  1,

2, ..., n  .

(21)

Assume the ith longitudinal natural frequencies and their regularized corresponding modes are  z ,i

Wi ( z ) . The vibration frequency response function (FRF) of each substructure can be obtained by former P order modal superposition, namely:

Wn ( za )Wn ( zb ) ,   2  2 j z ,n r n 0  P

H ( za , zb )  

2 z ,n

(22)

where n is the natural frequency of the substructure，  is the excitation frequency，  r is the system modal damping ratio and j 

1 .

If S nodes of the substructure are studied, the FRF of the substructure can be represented as

 H (u1 , u1 )  H (u1 , uS )  . H        H (uS , u1 )  H (uS , uS ) 

(23)

3.3.2 Connection of the two substructures Define the propeller-shaft subsystem and hull shell subsystem are substructures A and B, they are connected through the stern, intermediate, thrust bearing and their houses (or seats), and in this paper, they will be presented as equivalent stiffness and damping as shown in Fig 7.

Figure 7. The symbolic coupling of two substructures. n: number of connectors, for the present system, n=3; f i :reaction force of substructure A of ith connector; x i : non-interface displacement of substructure A of ith connector; fi : reaction force of substructure B of ith connector; xi : noninterface displacement of substructure B of ith connector; f ci : force of substructure A at ith interface; xci : displacement of substructure A at ith interface; f ci : force of substructure B at ith

ACCEPTED MANUSCRIPT interface; xci : displacement of substructure B at its interface.

The force and displacement relationship at the connection interface is

f  Zx , where f   f











f , f  f1 , f 2 , f 3 , f  f1 , f2 , f3 ,

(24)

xc   xc 1 , xc 2 , xc 3  , and xc   xc1 , xc 2 , xc 3  , Z is T

T

the impedance matrix (or equivalent stiffness) of the elastic connections. To resolve the force and displacement relationship at the connection interface, a simple implementation is using inner force balancing before and after the connecting of two substructures following Appendix A. Thus the FRF equation of the propeller-shaft-hull system is the FRF combination of the propeller-shaft system (substructure A) and the hull shell (substructure B), which is

 X IA   H IIA H IIAB  B  H IIB XI     A  XC    X CB   sym

H ICAA H ICBA A H CC

H ICAB   FI A    H ICBB   FIB  . AB   A  H CC F C   B  B H CC   FC 

(25)

Apparently, using the above procedure, the dynamic characteristics of the two subsystems and the total system are analyzed in the frequency domain. There are several advantages in studying the current system using FBSM than using commercial FEM codes, which are: (1) Under daily operation, the thrust bearing stiffness is varied in a quite wide range with the changing of the shaft rotating speed. It is more convenient to study the changes of the system dynamic characteristics if the connection stiffness is widely changed. (2) Since the frequency response function of the substructures can be obtained using not only FEM, but also analytical, semi-analytical, or the actual tests. It is possible to model and study the total system by combining the FRF of the substructures from different procedures. (3) Since the bearings are regarded as connecting units of the substructures, it is convenient to study and compare the relations and differences of the dynamic characteristics of the propeller-shaft subsystem, the hull shell, and the total system.

4. Dynamic characteristics of the propeller-shaft-hull system In this section, the modes, natural frequencies and the coupled longitudinal and the transverse vibration characteristics of the propeller-shaft subsystem, the hull shell, and the total system are analyzed

ACCEPTED MANUSCRIPT separately following the models given in the above section.

4.1 Dynamic characteristics of the elastic propeller-shaft subsystem As mentioned in the above sections, a new approach is introduced to model the propeller blades using harmonic blade array elements in this paper (hereafter called the harmonic model). A traditional finite element procedure will also be carried out to compare and verify the correctness of the results. From Table 1, the steel shaft is 4.09m long with a diameter of 0.07m. The density is 7800 kg/m3, the elastic modulus E is 210GPa and the Poisson's ratio is 0.3. A five-bladed Ni-Al-Bronze Cu3 alloy propeller is studied here, the density is 7590 kg/m3, the elastic modulus E is 186GPa and the Poisson’s ratio is 0.3. The key parameters of the propeller are listed in Table 3. The finite element model of the propeller-shaft subsystem is modeled with ABAQUS, the propeller is meshed with 17926 10-node quadratic tetrahedron elements (C3D10) and the shaft is meshed with 9276 8node linear brick elements (C3D8R) as shown in Fig 8. The middle coupling, the thrust plate and other additional components that attached to the shaft are simplified with associated concentrated mass and centralized inertia as given in Table 1.

Table 3. Key parameters of the propeller.

Figure 8. Elastic propeller-shaft system model. (a):Isometric view; (b):Side view; (c):blade model; (d):cross sections and splines of a single blade. Silver: steel shaft; Brown: Ni-Al-Bronze Cu3 alloy propeller. Note: due to page constraints, only part of the shaft structure is displayed.

For the harmonic model, since the geometry of the blades are the same, the array is axisymmetric and ten harmonic blade array elements are used to model the propeller array of blades. For each blade array element, the geometric center, the moment of inertia and the twist angle ψ of all the blade cross sections are needed as shown in Fig 8(d). The shaft is modeled using 17 Timoshenko beam element and one bladeshaft transition element is used to connect the interface between the blades and the shaft. Some of the cross sections along the non-dimensional radius χ are extracted and plotted in Fig 9(a)-Fig 9(d), while the propeller-shaft system model is shown in Fig 9(e).

ACCEPTED MANUSCRIPT Figure 9. The propeller-shaft harmonic model, shaft: Timoshenko beam, propeller: harmonic blade elements. (a)-(d): some of the cross sections along the radius (non-dimensional radii χ=0.15, 0.25, 0.5, 0.9); (e): whole system model. Note: Due to page constraints, only two shaft elements are displayed. Fig 10 and Fig 11 show the comparison of the first longitudinal and first three bending modes of propeller shaft subsystem and the first three local modes of the propeller blades at standstill using ABAQUS model and the harmonic model, respectively. Table 4 lists the comparison of the ABAQUS model and the harmonic model of the natural frequencies and their errors of the above modes. It is very clear that the listed modes and their natural frequencies of the propeller-shaft subsystem of ABAQUS and harmonic model are in great agreement. The maximum error of the given natural frequencies is no more than 6%. It is also noticed that for the harmonic model, only 10 blade array elements and 17 Timoshenko beam element are used, while for ABAQUS model, there are 27202 solid elements in total.

Table 4. Natural frequencies of propeller-shaft subsystem: comparison of the first three bending, first longitudinal and first three propeller modes using harmonic and ABAQUS model at standstill (ω=0).

Figure 10. The first longitudinal and first three bending modes of propeller shaft subsystem at standstill (ω=0). Left column (a)-(d): current harmonic models, solid black line with o: un-deformed system; solid red line with o: deformed system; right column (e)-(h): ABAQUS finite element model.

Figure 11. The first three local modes of the propeller at standstill (ω=0). Upper row (a)-(c): first-, secondand third- order modes of harmonic model, solid black line with *: un-deformed propeller blades, solid red line with >: deformed propeller blades; middle row (d)-(f): first-, second- and third- order modes of ABAQUS model; bottom row (g)-(i): comparison of the non-deformed and deformed shapes at three modes of one single blade using the two methods, solid black line with *: an un-deformed propeller blade, solid red line with >: a deformed propeller blade using harmonic model, solid blue line with o: a deformed propeller blade using FE model.

When considering rotation, the harmonic model still agrees with the ABAQUS model if the gyroscopic effect is neglected. This means that in both models only centrifugal stiffening is taken into

ACCEPTED MANUSCRIPT consideration. If all the contributions in the BAQUS model are accounted for, it underestimates the whirl frequency of forwarding modes, and this error increases with the increasing rotating speed. The comparison of the 1# natural frequency of the propeller blades at different rotating speed is plotted in Fig 12. But as for the ship propeller, generally the rotating speed is working at a range around 1-3Hz (60200rpm), which means the centrifugal stiffening and gyroscopic effect barely affect the rotating natural frequency of the propeller, thus in the following chapters, the rotation effect is neglected.

Figure 12. First forward and backward frequencies as functions of the speed for the propeller array of blades using the harmonic and ABAQUS model.

Using equation (20)-(23), the frequency response function (FRF) of the propeller-shaft subsystem can be derived as shown in Fig 13. The excitations are applied to the propeller hub center in longitudinal and lateral directions. The longitudinal responses at the stern bearing, the intermediate bearing and the thrust bearing node are listed in Fig 13(a)-Fig 13(c), where the studying frequency range is 10-1000Hz. The frequencies at which peaks in the response amplitude occur (hereafter called the peak response frequencies) under longitudinal excitation are marked with red dashed lines and tagged as a-f, while a, c, f denotes the first three longitudinal modes of shaft and b, d, e are the first three local modes of propeller blades. The transverse responses at the stern bearing, the intermediate bearing and the thrust bearing node are listed in Fig 13(d)-Fig 13(f), where the studying frequency range is 10-225Hz. The peak response frequencies under lateral excitation are marked with red dashed lines and tagged as g-l, while g-l are all the bending modes of the shaft.

Figure 13. Frequency response function of the propeller-shaft subsystem under longitudinal and lateral excitation apply to the propeller hub center. Left column: longitudinal response at the stern bearing, intermediate bearing and thrust bearing node; Right column: transverse response at the stern bearing, intermediate bearing and thrust bearing node. In figure (c): red dashed line: peak response frequencies under longitudinal excitation, a-f: peak response tagging; in figure (d): red dashed line: peak response frequencies under lateral excitation, g-l: peak response tagging.

Comparing to the natural frequencies listed in Table 4, we can notice:

ACCEPTED MANUSCRIPT (1) The peak response frequencies are all match to the natural frequencies of the propeller-shaft subsystem. (2) The responses at longitudinal and lateral directions are uncoupled. (3) The vibrations caused by the blade local modes do not exist in the transverse responses. This is because the studied blade modes are all in the out-of-plane direction, which are perpendicular to the lateral direction. (4) The longitudinal response at the thrust bearing node is larger than that at the other two bearing nodes, while the transverse response at stern bearing node is much larger, which means thrust bearing is the main longitudinal load transfer path from the propeller to the hull shell, while stern bearing is the main lateral load transfer path.

4.2 Dynamic characteristics of the hull shell subsystem The natural frequencies and their corresponding modes of hull shell subsystem are simulated using the finite element model illustrated in section 3.2. Fig 14 shows the global modes of the hull shell subsystem in frequency range 10-200Hz, while starting from 10Hz is to eliminate the rigid body motions of the hull. Table 5 lists the natural frequencies and their characterized vibration modes.

Figure 14. Nine global modes of the hull shell subsystem in frequency range 10-200Hz. (a):1st bending mode at 24.8 Hz; (b):2nd bending mode at 56.4Hz; (c):1st torsional mode at 78.2Hz; (d):3rd bending mode at 95.4Hz; (e):1st longitudinal mode at 105.1Hz; (f):2nd longitudinal mode at 134.4Hz; (g):4th bending mode at 140.2Hz; (h):3rd longitudinal mode at 153.4Hz; (i):2nd torsional mode at 156.9Hz; (j):5th bending mode at 185.8Hz.

Table 5. Natural frequencies and their global modes of hull shell subsystem in frequency range 10200Hz. B for bending; L for longitudinal; T for torsional.

According to the above natural frequencies and modes, we can notice that the global modes are all bending, torsional and longitudinal vibrations of the hull shell. But as the hull is a complicate stiffened thin shell with additional structures (motor seat, bearing seat, etc.), and it is not totally axisymmetric, the modes are represented as the combination of bending, torsional and longitudinal vibrations. It also needs

ACCEPTED MANUSCRIPT to be clear that there are many cabin section modes of the hull shell existed in the studying frequency range. But since their natural frequencies and modes are lack of regularity and they do not give many contributions to the interaction between the propeller-shaft and the hull shell subsystem, they will not discuss in this paper. Like in section 4.1, the frequency response function (FRF) of the hull shell subsystem is derived. As shown in Fig 15, a harmonic longitudinal excitation (Fz) is applied to a reference point of the thrust bearing center, and the harmonic lateral excitation (Fx) is applied to a reference of the stern bearing center.

Figure 15. Longitudinal and lateral excitations apply to the hull shell subsystem.

The longitudinal and transverse responses at thrust bearing node under longitudinal excitation are plotted in Fig 16(a), where the studying frequency range is 10-200Hz. The peak response frequencies of longitudinal response are marked with red dashed lines and tagged as e-h, while the peak response frequencies of transverse response are marked with blue dashed lines and tagged as a-j. It is clear that: (1) under longitudinal excitation, all the longitudinal modes are excited (marked as red e, f, h refer to Fig 14 and Table 5); (2) due to the asymmetry of the hull shell structure, both the longitudinal and transverse responses are excited under longitudinal excitation, but the amplitude at lateral direction is much lower. The longitudinal responses at the stern bearing, the intermediate bearing, and the thrust bearing are compared in Fig 16(b). We can notice that the longitudinal response at thrust bearing node is much larger than the other two, which indicates that thrust bearing is the main transfer path for the longitudinal load.

Figure 16. Frequency response function of hull shell subsystem under longitudinal excitation apply to the reference point of the thrust bearing center. (a): longitudinal and transverse response at thrust bearing node; red dashed line: peak response frequencies of longitudinal response; red e-h: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue a-j: peak response tagging of longitudinal response. (b): Compare of the longitudinal response at the stern bearing, intermediate bearing, and thrust bearing.

The longitudinal and transverse responses at stern bearing node under lateral excitation is plotted in Fig 17(a), where the studying frequency range is also 10-200Hz. The peak response frequencies of

ACCEPTED MANUSCRIPT longitudinal response are marked with red dashed lines and tagged as b-j, while the peak response frequencies of transverse response are marked with blue dashed lines and tagged as a-j. The dynamic characteristics under lateral excitation are quite similar to that under longitudinal excitation. Both the longitudinal and transverse responses are excited and the amplitude at longitudinal direction is much lower. But the peak numbers are more abundant in longitudinal direction compared to that under longitudinal excitation. It indicates that the global modes are easier to bring out under lateral excitation. The transverse responses at the stern bearing, the intermediate bearing, and the thrust bearing are compared in Fig 17(b). We can observe that the transverse response amplitude at stern bearing is much larger than the other two, which shows that the stern bearing is the main transfer path for the lateral load.

Figure 17. Frequency response function of hull shell subsystem under lateral excitation apply to the reference point of the stern bearing center. (a): longitudinal and transverse response at the stern bearing node; red dashed line: peak response frequencies of longitudinal response; red b-j: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue a-j: peak response tagging of longitudinal response. (b): Compare of the transverse response at the stern bearing, intermediate bearing, and thrust bearing.

4.3 Dynamic characteristics of the propeller-shaft-hull system It is easy to study the frequency response characteristics of the propeller-shaft-hull system by using FBSM, but the modes of the system cannot be obtained. Thus to analyze the peak response frequencies of the system and to verify the accuracy of the method, a FEM approach is first used to achieve the modes and natural frequencies of the propeller-shaft-hull system. The finite element model of the propeller-shaft-hull system is shown in Fig 18. The shaft is connected to the hull via a stern bearing, an intermediate bearing, and a thrust bearing. The stern and intermediate bearings are simplified as horizontal and vertical springs and damping, while the thrust bearing is modeled as tri-dimensional springs and damping. All the lumped mass, stiffness and damping properties are listed in Table 1. The boundary condition of the system is set free.

Figure 18. The finite element model of the propeller-shaft-hull system. Note: due to page constraints, only part of the propeller-shaft-hull structure is displayed.

ACCEPTED MANUSCRIPT Fig 19 shows the global modes of the propeller-shaft-hull system in the studying frequency range 10250Hz, while starting from 10Hz is to eliminate the rigid body motions of the system. It can be observed that most of the global modes are the combination of hull shell and propeller-shaft subsystem, while among them, most are dominated by the hull shell. It also needs to be clear that the first propeller blade mode is an out of plane local mode as shown in Fig 19(n), thus its natural frequency is the same as that of the propeller-shaft subsystem as listed in Table 4.

Figure 19. Twelve global modes of the propeller-shaft-hull system in frequency range 10-250Hz. (a):1st bending mode of hull at 24.6Hz; (b):2nd bending mode of hull plus 1st bending mode of shaft at 54.9Hz; (c):2nd bending mode of shaft at 59.3Hz; (d):3rd bending mode of shaft at 74.4Hz; (e):1st torsional mode of hull at 77.9Hz; (f):3rd bending mode of hull at 98.0Hz; (g):1st longitudinal mode of hull at 104.6Hz; (h):2nd bending mode of hull at 132.6Hz; (i):4th bending mode hull at 140.2Hz; (j):1st longitudinal mode of shaft at 140.4Hz; (k):3rd longitudinal mode of hull at 152.6Hz; (l):2nd torsional mode of hull at 156.9Hz; (m):5th bending mode of hull at 183.1Hz; (n):1st local mode of propeller at 210.6Hz.

The longitudinal and transverse responses at thrust bearing node under longitudinal excitation apply to the propeller hub center are plotted in Fig 20(a), where the studying frequency range is 10-250Hz. The peak response frequencies of longitudinal response are marked with red dashed lines and tagged as g-n, while for transverse response are marked with blue dashed lines and tagged as b-n. It is clear that: (1) under longitudinal excitation, all the longitudinal modes are excited (marked as red g, h, j refers to Fig 19) and the out of plane local mode of propeller blade is also excited as Fig 19(n); (2) due to the asymmetry of the hull structure, both the longitudinal and transverse responses are excited under longitudinal excitation, but the amplitude at lateral direction is much lower. The longitudinal responses at the shaft, the stern bearing, the intermediate bearing and the thrust bearing are compared in Fig 20(b). We can notice that since the longitudinal excitation is directly applied on the shaft, the response at the shaft node is largest. During the load transmitting, the amplitude of longitudinal response at thrust bearing node is much larger than that at stern bearing and intermediate bearing node, which indicates that thrust bearing is the main transfer path for longitudinal propeller

ACCEPTED MANUSCRIPT bearing force.

Figure 20. Frequency response function of the propeller-shaft-hull system under longitudinal excitation apply to the propeller hub center. (a): longitudinal and transverse response at thrust bearing node; red dashed line: peak response frequencies of longitudinal response; red g-n: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue b-n: peak response tagging of longitudinal response. (b): Compare of the longitudinal response at shaft, at the stern bearing, at the intermediate bearing and at the thrust bearing node.

The longitudinal and transverse responses at stern bearing node under lateral excitation apply to propeller hub center are plotted in Fig 21(a) with the studying frequency range of 10-250Hz. The peak response frequencies of longitudinal response are marked with red dashed lines and tagged as b-n, while for transverse response are marked with blue dashed lines and tagged as a-n. Similar to the responses under longitudinal excitation, all the longitudinal modes (marked as red g, h, j refers to Fig 19), the local mode of propeller blade ( marked as n refer to Fig 19) and most of the low frequency lateral modes are excited, but the torsional modes of the hull shell are not evident. Also, the longitudinal and transverse response is coupled under lateral excitation, but the amplitude at the longitudinal direction is much lower. The transverse response at the shaft, the stern bearing, the intermediate bearing and the thrust bearing are compared in Fig 21(b). Since the lateral excitation is directly applied to the shaft, the response at the shaft node is largest. During the load transmitting, the transverse response at the stern bearing node is much larger than that at the intermediate and thrust bearing node, which indicates that stern bearing is the main transfer path for lateral propeller bearing force.

Figure 21. Frequency response functions of hull shell subsystem under lateral excitation apply to the propeller hub center. (a): longitudinal and transverse response at the stern bearing node; red dashed line: peak response frequencies of longitudinal response; red b-n: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue a-n: peak response tagging of longitudinal response. (b): Compare of the transverse response at the shaft, at the stern bearing, at the intermediate bearing, and a thrust bearing.

ACCEPTED MANUSCRIPT The natural frequencies and their corresponding global modes of the propeller-shaft-hull system using ABAQUS and FBSM are compared in Table 6. The natural frequencies using both procedures have excellent consistency with the maximum error less than 7%, which proves the validity of the method. Associated with section 4.1 and 4.2, the dynamic characteristics of the propeller-shaft-hull system and the two substructures are interrelated follow the rules: (1) The natural frequencies of the total system are one-to-one correspondence to the natural frequencies of the hull shell substructure or the propeller-shaft substructure. (2) The modes of the total system are the combination of the two substructures. When the natural frequency of the total system is close to that of the propeller-shaft subsystem, the mode of the total system is dominated by this subsystem mode, and that’s the same for hull shell dominated natural frequencies and modes. When the natural frequency of the total system is close to both natural frequencies of the two substructure, the mode expressed as the coupled vibration of the two (such as mode b in Fig 19). (3) When the propeller-shaft substructure is connected to the hull shell, the natural frequencies of the total system dominated by the hull shell are slimly decreased compare to that of the hull shell substructure, indicates that the shaft only contributes little effect to the dynamic characteristics of the total system.

Table 6. Comparison of the natural frequencies and their global modes of the propeller-shaft-hull system using ABAQUS and FSBM in frequency range 10-250Hz. B for bending; L for longitudinal.

5. Experiment results and comparison In this section, the natural frequencies and modes of the propeller-shaft-hull system and the frequency response functions under longitudinal and lateral white noise excitation apply to the propeller hub center are tested and compared to the numerical results analyzed in section 4. Fig 22 illustrates the loading and excitation devices, data acquisition system and the installation locations of the sensors. A longitudinal loading device is used to apply static thrust to the propeller shaft as shown in Fig 22(a). Air springs are used to apply the static load, and the stiffness of the air spring is very low, which has little effect on the boundary condition of the system. A vibration exciter is used to apply excitation force on the propeller hub center as Fig 22(b). Accelerometers are stuck on the hull shell and shaft. The testing acceleration data are collected using an LMS data acquisition system as Fig 22(c). Fig 22(d) gives the location of sensors mounted on the hull, section 1-8 are on cylindrical segments and 8

ACCEPTED MANUSCRIPT equally spaced three-dimensional accelerometers are used at each section, while section 9-14 are on conical segment and 16 equally spaced three-dimensional accelerometers are used at each section, the LMS TestLab geometry of the hull shell is also plotted. H1-H11 in Fig 22(e) are 11 three-dimensional accelerometers mounted on the shaft. Fig 22(f) shows the installation method and locations of the sensors on propeller blade. Since all the blades are axisymmetric, analysis has proceeded on one blade. In addition, impact hammer is used in testing the propeller modes.

Figure 22. The loading and excitation devices, data acquisition system and the location of the sensors.

Fig 23 shows the comparison of the global modes of the propeller-shaft-hull system between numerical simulations and experiment results. The first part is the modes of the propeller-shaft-hull system, the first three bending, the first torsional and the first longitudinal modes of the system are plotted. The second part is the first three bending and first longitudinal modes of the propeller-shaft substructure. The last part is the first local mode of a propeller blade, the experiment result is given in top, front and isotropic view. The modes are all corresponding to the modes given in Fig 19 and Table 6. Each mode of numerical and experiment results keeps good consistency, most of the global modes presented as the combination of hull shell and propeller-shaft subsystem, except for the first mode of the propeller, which is the local mode of the propeller blades itself.

Figure 23. Global modes of the propeller-shaft-hull system, the propeller-shaft substructure and 1st order local mode of the propeller blade. For each mode, left figure: numerical simulation; right figure: experiment result.

Fig 24 and Fig 25 shows the frequency response function of longitudinal and transverse responses under longitudinal and lateral excitation force apply to the propeller hub center, respectively. The listed measuring point results are chosen randomly from the hull conical segment, the cylindrical segment, the shaft, and the blade. The excitation force is a white noise signal with a frequency range 10-250Hz. In each figure, the peak response frequencies are tagged using the same alphabet compared to Fig 19 and Table 6 given in section 4.3. It is clear that: (1) the FRFs are quite similar at each measuring point; (2) for a

ACCEPTED MANUSCRIPT certain point, the amplitude at longitudinal direction is much higher than that at the other two directions under longitudinal excitation force, while the amplitude of transverse response is higher under lateral excitation; (3) most of the global modes are excited whether under longitudinal or lateral excitation, which shows the coupling of the longitudinal and transverse dynamic responses.

Figure 24. The longitudinal and transverse response under longitudinal excitation acting on propeller hub center. Black dashed line: peak response frequencies; black b-h: peak response tagging.

Figure 25. The longitudinal and transverse response under lateral excitation acting on propeller hub center. Black dashed line: peak response frequencies; black a-h: peak response tagging.

Table 7 compares the natural frequency and mode results of the propeller-shaft-hull system with FBSM and experiment results, the conclusions can be derived as: (1) The frequencies at which peaks in the response amplitude occur using FBSM are very close to that of experiment results, the maximum error for each mode is less than 10%, which verifies the correctness of the modeling procedure. (2) Most of the global modes are excited whether under longitudinal or lateral excitation, which reveals the coupling of the longitudinal and transverse dynamic responses.

Table 7. Comparison of the natural frequencies and their global modes of the propeller-shaft-hull system using experiment and FSBM in frequency range 10-250Hz. B for bending; L for longitudinal.

6. Further discussion As discussed in section 4, for the current studying system, the thrust bearing is the main vibration transfer path for longitudinal excitation, while the stern bearing is the main vibration transfer path for lateral excitation. In this section, we will focus on how the stiffness of these two bearings affect the dynamic characteristics of the propeller-shaft-hull system. The procedure is quite simple using FBSM by adjusting the stiffness of the two bearings when using the FRF equations of the shaft-hull system as in equation (25).

ACCEPTED MANUSCRIPT 6.1 Dynamic characteristics under longitudinal excitation To simplify the analysis, only the longitudinal stiffness of the thrust bearing is changed. Three longitudinal stiffness values of the thrust bearing are chosen, which are ktx1=1107N/m, ktx2=1108N/m, and ktx3=1109N/m. The comparison of longitudinal responses at thrust bearing node under longitudinal excitation apply to propeller hub center are plotted in Fig 26 with the studying frequency range of 10-250Hz. The peak response frequencies are marked with a dashed line and tagged as g-n. We can notice that: (1) All the longitudinal modes are excited under longitudinal excitation, e.g. g, h, k are the first three longitudinal modes of the hull shell, j is the first longitudinal mode of shaft and n is the first local mode of the propeller blade. (2) With the increase of the thrust bearing stiffness, the first longitudinal natural frequency of the shaft changed enormously (from j1:65Hz to j3:146Hz), but the hull shell mode dominated natural frequencies are barely changed (g1 to g3, h1 to h3 and k1 to k3 in Fig 26). It indicates that the change of the longitudinal stiffness of thrust bearing mainly affects the longitudinal dynamics of the shaft, while has little influence on the dynamic characteristics of the hull.

Figure 26. Comparison of frequency response function of the propeller-shaft-hull system under longitudinal excitation apply to the propeller hub center. Dashed line: peak response frequencies of longitudinal response; g-n: peak response tagging of longitudinal response; subscript 1: peaks for stiffness ktx1=1107N/m; subscript 2: peaks for stiffness; ktx2=1108N/m; subscript 3: peaks for stiffness ktx3=1109N/m.

6.2 Dynamic characteristics under lateral excitation Like section 6.1, the dynamic characteristics under lateral vibration is only studied by changing the lateral stiffness of the stern bearing. Also, three lateral stiffness values of stern bearing are chosen, which are ksy1=1106N/m, ksy2=1107N/m, and ksy3=1108N/m. The comparison of transverse responses at the stern bearing node under lateral excitation acting on propeller hub center are plotted in Fig 27 with the frequency range of 10-250Hz. The peak response frequencies are marked with a dashed line and tagged as a-n.

ACCEPTED MANUSCRIPT Similar to that under longitudinal excitation, we can notice that: (1) All the bending modes are excited under lateral excitation, e.g. a, f, m are the bending modes of the hull, c, d are the first two bending modes of the shaft. The longitudinal modes of the system are also excited, such as g, k for the hull, j for the shaft and n for the propeller blade. (2) With the increase of the stiffness of stern bearing, the bending natural frequencies related to the shaft are highly changed (the 2nd bending from c1:43Hz to c3:57Hz, the 3rd bending from d1:62Hz to d3:78Hz), but the longitudinal modes of shaft, and the hull mode dominated natural frequencies are barely changed (j1 to j3 for the shaft, g1 to g3, h1 to h3 and k1 to k3 for hull shell in Fig 27). It reveals two things, one is the coupling of the transverse and longitudinal response caused by the change of the bearing stiffness is weak, and two is the change of the lateral stiffness of stern bearing contributes little to the dynamic characteristics of the hull.

Figure 27. Comparison of frequency responses function of the propeller-shaft-hull system under lateral excitation apply to the propeller hub center. Dashed line: peak response frequencies of transverse response; a-n: peak response tagging of transverse response; subscript 1: peaks for stiffness ksy1=1106N/m; subscript 2: peaks for stiffness; ksy2=1107N/m; subscript 3: peaks for stiffness ksy3=1108N/m.

7. Conclusion This paper develops a dynamic model to study the coupled longitudinal and transverse vibrations of a submarine elastic propeller-shaft-hull system using the FRF-based sub-structuring method (FBSM). The dynamic characteristics of the elastic propeller-shaft system, the hull, and the total system are analyzed and compared between FBSM, FEM and experiment results. Several conclusions can be achieved: First, the frequencies at which peaks in the response amplitude occur of the propeller-shaft, the hull shell and the propeller-shaft-hull system using FBSM are very close to that using FEM and the experiment results. The maximum error for each mode is less than 10%, and the modes using either method are in great agreement, which verifies the correctness of the developed FBSM modelling procedure. Second, due to the asymmetry of the hull structure, most of the global modes are excited whether under longitudinal or lateral excitation, which reveals the coupling of the longitudinal and transverse dynamic responses. Third, the modes of the propeller-shaft-hull system are the coupled vibrations of the hull, the shaft, and the

ACCEPTED MANUSCRIPT propeller. Some of them are dominated by the hull shell, while others are dominated by the shaft or the propeller. Fourth, the thrust bearing is the main transfer path for longitudinal propeller bearing force from the propeller to the hull, while the stern bearing is the main transfer path for transverse force. Last, with the increase of the stiffness of the stern bearing or the thrust bearing, only the bending or the longitudinal natural frequencies related to the shaft have sharply changed. It indicates that the change of the bearing stiffness mainly affects the dynamic responses of the shaft, while has little influence on the dynamic characteristics of the hull. For future work, our goal is to develop a model to analyze the vibrations and acoustic radiations of the propeller-shaft-hull system excited by propeller bearing force, which is a four step work: first, develop an elastic propeller-shaft-hull system model using FBSM instead of using full 3D model as established in this paper; then, consider the FSI between propeller-shaft-hull system and water; after that, derive the unsteady thrust of propeller induced by inflow turbulence and add to the propeller; last, use the developed FBSM method to analyze the low-frequency vibrations and acoustic radiations of the system excited by propeller bearing force.

9. Acknowledgments Authors may acknowledge financial support by the Foundation for Innovative Research Groups of National Natural Science Foundation of China (Grant No. 51405292).

Appendix A: Inner force balancing procedure For the propeller-shaft subsystem (substructure A) and hull shell subsystem (substructure B), the force and displacement relationship at the connection interface is

f  Zx , where: f   f











f , f  f1 , f 2 , f 3 , f  f1 , f2 , f3 , xc

(A.1)

  xc 1 , xc 2 , xc 3  ,and xc   xc1 , xc 2 , xc 3  , Z is the T

T

 Z cc   Z cc

impedance matrix (or equivalent stiffness) of the elastic connections，where Z  

Z cc  , Z cc  Z ccT  Z cc  

T T , Z cc  Z cc and Z cc    Z cc   . Since no inner point exists at the connection interface, there are no non

interface coordinates.

ACCEPTED MANUSCRIPT At the connection interface, the displacement balancing equation is satisfied, which is：

 xci   xi       , (i  1, 2,3) .  xci   xi 

(A.2)

Also as the force balancing equation：

 f ci   FCi   fi           , (i  1, 2,3) ,  f ci   FCi   fi 

(A.3)

where Fci and Fci are the external forces acting on the ith interface of substructure A and B. Before synthesizing substructure A and B, the relationship between force and displacement of each substructure can be represented using frequency response functions as

 xi   H ii     xc    H ci  x  H  c   ci

H ic H cc H cc

H ic   fi    H cc   f c  . H cc    f c 

(A.4)

We can derive xc and xc from the above equation and replace them in equation (A.2).

Z cc ( H ci fi  H cc f c  H cc f c )  Z cc ( H ci fi  H cc f c  H cc  f c )  f  0 ,

(A.5)

Z cc ( H ci fi  H cc f c  H cc f c )  Z cc  ( H ci fi  H cc f c  H cc  f c )  f  0 .

(A.6)

Using the relationship in equation (A.1) and replace in equation (A.5-A.6)，while noticed that the force acting on the interface point will not change before and after connection, which is f i  FI , then

Z cc  H ci fi  H cc f c  H cc f c   Z cc  H ci fi  H cc f c  H cc  f c   f  0 ,

(A.7)

Z cc ( H ci FI  H cc f c  H cc f c )  Z cc  ( H ci FI  H cc f c  H cc  f c )  FC  f c  0 .

(A.8)

Which also can be rearranged as

a1 f c  a2 f c  FC  a3 FI ,

(A.9)

b1 f c  b2 f c  FC  b3 FI .

(A.10)

The reaction force of the system before synthesizing the two substructures can be represented by using the external force after synthesizing:

f c  B 1  b3  b2 a21a3  FI  b2 a21 FC  FC  ,

(A.11)

f c  B 1  a3  a1b11b3  FI  FC  a1b11 FC  ,

(A.12)

ACCEPTED MANUSCRIPT where:

a1  Z cc H cc  Z cc H cc  I ， a2  Z cc  H cc  Z cc H cc  ， a3  Z cc H ci  Z cc H ci b1  Z cc H cc  Z cc H cc ， b2  Z cc H cc  Z cc H cc  I ， b3  Z cc H ci  Z cc H ci B  b2 a21a1  b1 , B  a1b11b2  a2 .

(A.13)

Compare equation (A.10), (A.11) and (A.3), and using the equilibrium equation, achieve

X I  xi , X C  xc , X C  xc .

(A.14)

Then

 X I   H II     X C    H CI  X  H  C   CI

H IC H CC H CC 

H IC   FI    H CC   FC  ,   H CC     FC 

(A.15)

where: T T T , H CI  H IC , H CC H CI  H IC   H CC ,

H II  H ii  H ic D  H a  H ic D T H b ， H IC  H ic D  H c  H ic D T H cc ,

H IC  H ic D T H d  H ic D  H cc  ， H CC  H cc D  H c , T H CC   H cc D  H cc  ， H CC    H cc  D Hd ,  H a  ( H cc  Z cc  H cc  Z cc  Z cc Z cc  Z cc Z cc ) H ci  H ci ， H c  ( H cc  Z cc   I ) Z cc ,

H b  ( H cc Z cc  H cc Z cc Z cc Z cc   Z cc Z cc  ) H ci  H ci , H d  ( H cc Z cc  I ) Z cc . Finally, the frequency domain response function for the propeller-shaft-hull system is

 X IA   H IIA H IIAB  B  H IIB XI    A   XC    X CB   sym

H ICAA H ICBA A H CC

H ICAB   FI A    H ICBB   FIB  . AB   A  H CC FC   B  B H CC   FC 

(A.17)

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ACCEPTED MANUSCRIPT

Collected Table captions Table 1. Key physical parameters of the elastic propeller shaft system. Table 2. Element type and the number of elements of the hull shell subsystem. Table 3. Key parameters of the propeller. Table 4. Natural frequencies of propeller-shaft subsystem: comparison of the first three bending, first longitudinal and first three propeller modes using harmonic and ABAQUS model at standstill (ω=0). Table 5. Natural frequencies and their global modes of hull shell subsystem in frequency range 10-200Hz. B for bending; L for longitudinal; T for torsional. Table 6. Comparison of the natural frequencies and their global modes of the propeller-shaft-hull system using ABAQUS and FSBM in frequency range 10-250Hz. B for bending; L for longitudinal. Table 7. Comparison of the natural frequencies and their global modes of the propeller-shaft-hull system using experiment and FSBM in frequency range 10-250Hz. B for bending; L for longitudinal.

ACCEPTED MANUSCRIPT

Table 1. Key physical parameters of the elastic propeller shaft system. No

Parameter

Symbol

Value

1

Shaft and propeller density

ρ

7800 kg/m3, 7590 kg/m3

2

Poisson's ratio

υ

0.3

3

Modulus of elasticity of shaft and propeller

E

2.06×1011Pa, 1.86×1011Pa

4

Length of shaft

L

4.09m

5

Diameter of the rotors

D

0.07m

6

Distance between propeller to stern bearing

L1

0.33m

L2

2.10m

L3

1.01m

Distance between stern to intermediate 7 bearing Distance between intermediate to thrust 8 bearing 9

Motor and drive coupling mass

m0

85kg

10

Rigid coupling mass

m1

5.51kg

11

Rigid coupling inertia

J1x，J1y，J1z

0.067, 0.033, 0.033 kg·m2

12

Thrust plate mass

m2

11.2kg

13

Thrust plate inertia

J2x，J2y，J2z

0.183, 0.092, 0.092 kg·m2

14

Driven elastic coupling mass

m3

34kg

15

Driven elastic coupling inertia

J3x，J3y，J3z

0.25,0.12, 0.12 kg·m2

16

Coupling stiffness

k1x，k1y，k1z

k1x=3×105N/m，k1y=k1z=3.2×105N/m

17

Motor spring stiffness

k2x，k2y，k2z

k2x=k2z=3.3×106N/m， k2y=5.5×106N/m

ACCEPTED MANUSCRIPT 18

Stiffness of stern bearing

ksy，ksz

ksy=ksz=1.1×107N/m

19

Stiffness of intermediate bearing

kiy，kiz

kiy=kiz=8.2×106N/m

20

Stiffness of thrust bearing

ktx，kty，ktz

kty=ktz=2.5×107N/m，ktx=1.2×108N/m

Table 2. Element type and the number of elements of the hull shell subsystem. Element

Type

No. of elements

Position Hull shell, stiffening rings, motor, and bearing

Shell

S4R

15624

seat

Rigid Shell

R3D4

982

Thrust, intermediate and stern bearing House

Beam

B31

6568

stiffeners

3

Thrust, intermediate and stern bearing House

Concentrated mass

ACCEPTED MANUSCRIPT

Table 3. Key parameters of the propeller. No. Blades

5

Inner Radius

0.07m

Outer Radius

0.279 m

Expanded Area Ratio

0.72

Pre-twist Angle

20°

Mean Spacing

0.221 m

Blade Mass

37.8kg

Blade Moment(Jx, Jy, Jz)

0.257, 0.152, 0.143kg.m3

ACCEPTED MANUSCRIPT

Table 4. Natural frequencies of propeller-shaft subsystem: comparison of the first three bending, first longitudinal and first three propeller modes using harmonic and ABAQUS model at standstill (ω=0). Modes

Harmonic

ABAQUS

Error (%)

Modes

Harmonic

ABAQUS

Error (%)

#1 Bending

49.6

48.1

3.1

#1 Blade

217.9

210.6

3.0

#2 Bending

55.9

53.3

4.8

#2 Blade

426.1

404.2

5.4

#3 Bending

75.6

71.7

5.5

#3 Blade

608.4

575.7

5.7

#1 Longitudinal

139.9

132.8

5.4

ACCEPTED MANUSCRIPT

Table 5. Natural frequencies and their global modes of hull shell subsystem in frequency range 10-200Hz. B for bending; L for longitudinal; T for torsional. No.

Frequency(Hz)

Mode

No.

Frequency(Hz)

Mode

No.

Frequency(Hz)

Mode

a)

24.8

#1 B

b)

56.4

#2 B

c)

78.2

#1 T

d)

95.4

#3 B

e)

105.1

#1 L

f)

134.4

#2 L

g)

140.2

#4 B

h)

153.4

#3 L

i)

156.9

#2 T

j)

186.8

#5 B

ACCEPTED MANUSCRIPT

Table 6. Comparison of the natural frequencies and their global modes of the propeller-shaft-hull system using ABAQUS and FSBM in frequency range 10-250Hz. B for bending; L for longitudinal. Propeller-shaft-hull system No

FEM method

Propeller-shaft

FBSM

Error

Hull shell

Mode

Freq. (Hz)

Mode

Freq. (Hz)

2.4

---

---

#1 B

24.8

58.4

2.6

#1 B

49.6

#2 B

56.4

59.7

64.5

2.8

#2 B

55.9

---

---

#3 B Shaft

74.4

76.3

2.3

#3 B

75.6

---

---

f

#3 B Hull

98.0

91.7

-6.4

---

---

#3 B

95.4

g

#1 L Hull

104.6

108.6

3.8

---

---

#1 L

105.1

h

#2 L Hull

128.6

131.1

1.9

---

---

#2 L

134.4

i

#4 B Hull

139.7

139.9

0.1

---

---

#4 B

140.2

j

#1 L Shaft

140.4

142.2

1.3

#1 L

139.9

---

---

k

#3 L Hull

152.6

159.8

4.7

---

---

#3 L

153.4

m

#5 B Hull

183.1

186.3

1.7

---

---

#5 B

186.8

n

#1 Propeller

210.6

217.9

3.5

#1 Propeller

217.9

---

---

Mode

Freq. (Hz)

(Hz)

(%)

a

#1 B Hull

24.6

25.2

b

#2 B Hull+#1 B Shaft

54.9

c

#2 B Shaft

d

ACCEPTED MANUSCRIPT

Table 7. Comparison of the natural frequencies and their global modes of the propeller-shaft-hull system using experiment and FSBM in frequency range 10-250Hz. B for bending; L for longitudinal. Experiment No

FBSM (Frequency: Hz)

Error (%)

23.5

25.2

7.2

#2 B Hull+#1 B Shaft

54.7

58.4

6.8

c

#2 B Shaft

60.8

64.5

6.1

d

#3 B Shaft

71.8

76.3

6.3

f

#3 B Hull

95.6

91.7

-4.1

g

#1 L Hull

100.9

108.6

7.6

h

#2 L Hull

120.2

131.1

9.1

i

#4 B Hull

134.6

139.9

3.9

j

#1 L Shaft

138.3

142.2

2.8

k

#3 L Hull

145.9

159.8

9.5

m

#5 B Hull

182.2

186.3

2.3

n

#1 Propeller

222.5

217.9

-2.1

Mode

Frequency (Hz)

a

#1 B Hull

b

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Collected figure captions Figure 1. Sketch of the submarine propeller shaft-hull system. Figure 2. The hull shell subsystem. Figure 3. The elastic propeller-shaft subsystem. Figure 4. Elastic propeller-shaft-hull experimental setup. Figure 5. Cross section of each blade at a certain radius. Figure 6. FEM model of the hull shell subsystem and its main components. Figure 7. The symbolic coupling of two substructures. n: number of connectors, for the present system, n=3; f i :reaction force of substructure A of ith connector; x i : non-interface displacement of substructure A of ith connector; fi : reaction force of substructure B of ith connector; xi : non-interface displacement of substructure B of ith connector; f ci : force of substructure A at ith interface; xci : displacement of substructure A at ith interface; f ci : force of substructure B at ith interface; xci : displacement of substructure B at ith interface. Figure 8. Elastic propeller-shaft system model. (a):Isometric view; (b):Side view; (c):blade model; (d):cross sections and splines of a single blade. Silver: steel shaft; Brown: Ni-Al-Bronze Cu3 alloy propeller. Note: due to page constraints, only part of the shaft structure is displayed. Figure 9. The propeller-shaft harmonic model, shaft: Timoshenko beam, propeller: harmonic blade elements. (a)(d): some of the cross sections along the radius (non-dimensional radii χ=0.15, 0.25, 0.5, 0.9); (e): whole system model. Note: Due to page constraints, only two shaft elements are displayed.

ACCEPTED MANUSCRIPT Figure 10. The first longitudinal and first three bending modes of propeller shaft subsystem at standstill (ω=0). Left column (a)-(d): current harmonic models, solid black line with o: un-deformed system; solid red line with o: deformed system; right column (e)-(h): ABAQUS finite element model. Figure 11. The first three local modes of the propeller at standstill (ω=0). Upper row (a)-(c): first-, second- and third- order modes of harmonic model, solid black line with *: un-deformed propeller blades, solid red line with >: deformed propeller blades; middle row (d)-(f): first-, second- and third- order modes of ABAQUS model; bottom row (g)-(i): comparison of the non-deformed and deformed shapes at three modes of one single blade using the two methods, solid black line with *: an un-deformed propeller blade, solid red line with >: a deformed propeller blade using harmonic model, solid blue line with o: a deformed propeller blade using FE model. Figure 12. First forward and backward frequencies as functions of the speed for the propeller array of blades using the harmonic and ABAQUS model. Figure 13. Frequency response function of propeller-shaft subsystem under longitudinal and lateral excitation apply to the propeller hub center. Left column: longitudinal response at stern bearing, intermediate bearing and thrust bearing node; Right column: transverse response at stern bearing, intermediate bearing and thrust bearing node; in figure (c): red dashed line: peak response frequencies under longitudinal excitation, a-f: peak response tagging; in figure (d): red dashed line: peak response frequencies under lateral excitation, g-l: peak response tagging. Figure 14. Nine global modes of the hull shell subsystem in frequency range 10-200Hz. (a):1st bending mode at 24.8 Hz; (b):2nd bending mode at 56.4Hz; (c):1st torsional mode at 78.2Hz; (d):3rd bending mode at 95.4Hz; (e):1st longitudinal mode at 105.1Hz; (f):2nd longitudinal mode at 134.4Hz; (g):4th bending mode at 140.2Hz; (h):3rd longitudinal mode at 153.4Hz; (i):2nd torsional mode at 156.9Hz; (j):5th bending mode at 185.8Hz. Figure 15. Longitudinal and lateral excitations apply to the hull shell subsystem. Figure 16. Frequency response function of hull shell subsystem under longitudinal excitation apply to the reference point of the thrust bearing center. (a): longitudinal and transverse response at thrust bearing node; red dashed line: peak response frequencies of longitudinal response; red e-h: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue a-j: peak response tagging of longitudinal response. (b): Compare of the longitudinal response at the stern bearing, intermediate bearing, and thrust bearing. Figure 17. Frequency response function of hull shell subsystem under lateral excitation apply to the reference point of the stern bearing center. (a): longitudinal and transverse responses at the stern bearing node; red dashed

ACCEPTED MANUSCRIPT line: peak response frequencies of longitudinal response; red b-j: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue a-j: peak response tagging of longitudinal response. (b): Compare of the transverse response at the stern bearing, intermediate bearing, and thrust bearing. Figure 18. The finite element model of the propeller-shaft-hull system. Note: due to page constraints, only part of the propeller-shaft-hull structure is displayed. Figure 19. Twelve global modes of the propeller-shaft-hull system in frequency range 10-250Hz. (a):1st bending mode of hull at 24.6Hz; (b):2nd bending mode of hull plus 1st bending mode of shaft at 54.9Hz; (c):2nd bending mode of shaft at 59.3Hz; (d):3rd bending mode of shaft at 74.4Hz; (e):1st torsional mode of hull at 77.9Hz; f):3rd bending mode of hull at 98.0Hz; (g): 1st longitudinal mode of hull at 104.6Hz; (h): 2nd bending mode of hull at 132.6Hz; (i): 4th bending mode hull at 140.2Hz; (j): 1st longitudinal mode of shaft at 140.4Hz; (k):3rd longitudinal mode of hull at 152.6Hz; (l):2nd torsional mode of hull at 156.9Hz; (m):5th bending mode of hull at 183.1Hz; (n):1st local mode of propeller at 210.6Hz. Figure 20. Frequency response function of the propeller-shaft-hull system under longitudinal excitation apply to the propeller hub center. (a): longitudinal and transverse responses at thrust bearing node; red dashed line: peak response frequencies of longitudinal response; red g-n: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue b-n: peak response tagging of longitudinal response. (b): Compare of the longitudinal response at the shaft node, at the stern bearing, at the intermediate bearing and at the thrust bearing. Figure 21. Frequency response function of hull shell subsystem under lateral excitation apply to the propeller hub center. (a): longitudinal and transverse response at the stern bearing node; red dashed line: peak response frequencies of longitudinal response; red b-n: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue a-n: peak response tagging of longitudinal response. (b): Compare of the transverse response at the shaft, at the stern bearing, at the intermediate bearing, and at the thrust bearing. Figure 22. The loading and excitation devices, data acquisition system and the location of the sensors. Figure 23. Global modes of the propeller-shaft-hull system, the propeller-shaft substructure and 1st order local mode of the propeller blade. For each mode, left figure: numerical simulation; right figure: experiment result. Figure 24. The longitudinal and transverse response under longitudinal excitation acting on propeller hub center. Black dashed line: peak response frequencies; black b-h: peak response tagging.

ACCEPTED MANUSCRIPT Figure 25. The longitudinal and transverse response under lateral excitation acting on propeller hub center. Black dashed line: peak response frequencies; black a-h: peak response tagging. Figure 26. Comparison of frequency response function of the propeller-shaft-hull system under longitudinal excitation apply to the propeller hub center. Dashed line: peak response frequencies of longitudinal response; g-n: peak response tagging of longitudinal response; subscript 1: peaks for stiffness ktx1=1107N/m; subscript 2: peaks for stiffness; ktx2=1108N/m; subscript 3: peaks for stiffness ktx3=1109N/m. Figure 27. Comparison of frequency response function of the propeller-shaft-hull system under lateral excitation apply to the propeller hub center. Dashed line: peak response frequencies of transverse response; a-n: peak response tagging of transverse response; subscript 1: peaks for stiffness ksy1=1106N/m; subscript 2: peaks for stiffness; ksy2=1107N/m; subscript 3: peaks for stiffness ksy3=1108N/m. Stiffening ring Conical shell

Stern bearing

Intermediate bearing

Thrust bearing

Motor

Coupling Propeller

Base

Base

Figure 1. Sketch of the submarine propeller shaft-hull system.

cylindrical shell

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Figure 2. The hull shell subsystem.

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Figure 3. The elastic propeller-shaft subsystem.

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Figure 4. Elastic propeller-shaft-hull experimental setup.

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Figure 5. Cross section of each blade at a certain radius.

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Figure 6. FEM model of the hull shell subsystem and its main components.

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Figure 7. The symbolic coupling of two substructures. n: number of connectors, for the present system, n=3; f i :reaction force of substructure A of ith connector; x i : non-interface displacement of substructure A of ith connector; fi : reaction force of substructure B of ith connector; xi : non-interface displacement of substructure B of ith connector; f ci : force of substructure A at ith interface; xci : displacement of substructure A at ith interface; f ci : force of substructure B at ith interface; xci : displacement of substructure B at ith interface.

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Figure 8. Elastic propeller-shaft system model. (a):Isometric view; (b):Side view; (c):blade model; (d):cross sections and splines of a single blade. Silver: steel shaft; Brown: Ni-Al-Bronze Cu3 alloy propeller. Note: due to page constraints, only part of the shaft structure is displayed.

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Figure 9. The propeller-shaft harmonic model, shaft: Timoshenko beam, propeller: harmonic blade elements. (a)(d): some of the cross sections along the radius (non-dimensional radii χ=0.15, 0.25, 0.5, 0.9); (e): whole system model. Note: Due to page constraints, only two shaft elements are displayed.

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Figure 10. The first longitudinal and first three bending modes of propeller shaft subsystem at standstill (ω=0). Left column (a)-(d): current harmonic models, solid black line with o: un-deformed system; solid red line with o: deformed system; right column (e)-(h): ABAQUS finite element model.

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Figure 11. The first three local modes of the propeller at standstill (ω=0). Upper row (a)-(c): first-, second- and third- order modes of harmonic model, solid black line with *: un-deformed propeller blades, solid red line with >: deformed propeller blades; middle row (d)-(f): first-, second- and third- order modes of ABAQUS model; bottom row (g)-(i): comparison of the non-deformed and deformed shapes at three modes of one single blade using the two methods, solid black line with *: an un-deformed propeller blade, solid red line with >: a deformed propeller blade using harmonic model, solid blue line with o: a deformed propeller blade using FE model.

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Figure 12. First forward and backward frequencies as functions of the speed for the propeller array of blades using the harmonic and ABAQUS model.

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Figure 13. Frequency response function of propeller-shaft subsystem under longitudinal and lateral excitation apply to the propeller hub center. Left column: longitudinal response at stern bearing, intermediate bearing and thrust bearing node; Right column: transverse response at stern bearing, intermediate bearing and thrust bearing node; in figure (c): red dashed line: peak response frequencies under longitudinal excitation, a-f: peak response tagging; in figure (d): red dashed line: peak response frequencies under lateral excitation, g-l: peak response tagging.

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Figure 14. Nine global modes of the hull shell subsystem in frequency range 10-200Hz. (a):1st bending mode at 24.8 Hz; (b):2nd bending mode at 56.4Hz; (c):1st torsional mode at 78.2Hz; (d):3rd bending mode at 95.4Hz; (e):1st longitudinal mode at 105.1Hz; (f):2nd longitudinal mode at 134.4Hz; (g):4th bending mode at 140.2Hz; (h):3rd longitudinal mode at 153.4Hz; (i):2nd torsional mode at 156.9Hz; (j):5th bending mode at 185.8Hz.

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Figure 15. Longitudinal and lateral excitations apply to the hull shell subsystem.

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Figure 16. Frequency response function of hull shell subsystem under longitudinal excitation apply to the reference point of the thrust bearing center. (a): longitudinal and transverse response at thrust bearing node; red dashed line: peak response frequencies of longitudinal response; red e-h: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue a-j: peak response tagging of longitudinal response. (b): Compare of the longitudinal response at the stern bearing, intermediate bearing, and thrust bearing.

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Figure 17. Frequency response function of hull shell subsystem under lateral excitation apply to the reference point of the stern bearing center. (a): longitudinal and transverse response at the stern bearing node; red dashed line: peak response frequencies of longitudinal response; red b-j: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue a-j: peak response tagging of longitudinal response. (b): Compare of the transverse response at the stern bearing, intermediate bearing, and thrust bearing.

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Figure 18. The finite element model of the propeller-shaft-hull system. Note: due to page constraints, only part of the propeller-shaft-hull structure is displayed.

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Figure 19. Twelve global modes of the propeller-shaft-hull system in frequency range 10-250Hz. (a):1st bending mode of hull at 24.6Hz; (b):2nd bending mode of hull plus 1st bending mode of shaft at 54.9Hz; (c):2nd bending mode of shaft at 59.3Hz; (d):3rd bending mode of shaft at 74.4Hz; (e):1st torsional mode of hull at 77.9Hz; f):3rd bending mode of hull at 98.0Hz; (g): 1st longitudinal mode of hull at 104.6Hz; (h): 2nd bending mode of hull at 132.6Hz; (i): 4th bending mode hull at 140.2Hz; (j): 1st longitudinal mode of shaft at 140.4Hz; (k):3rd longitudinal mode of hull at 152.6Hz; (l):2nd torsional mode of hull at 156.9Hz; (m):5th bending mode of hull at 183.1Hz; (n):1st local mode of propeller at 210.6Hz.

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Figure 20. Frequency response function of the propeller-shaft-hull system under longitudinal excitation apply to the propeller hub center. (a): longitudinal and transverse response at thrust bearing node; red dashed line: peak response frequencies of longitudinal response; red g-n: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue b-n: peak response tagging of longitudinal response. (b): Compare of the longitudinal response at the shaft node, at the stern bearing, at the intermediate bearing and at the thrust bearing.

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Figure 21. Frequency response function of hull shell subsystem under lateral excitation apply to the propeller hub center. (a): longitudinal and transverse response at the stern bearing node; red dashed line: peak response frequencies of longitudinal response; red b-n: peak response tagging of longitudinal response; blue dashed line: peak response frequencies of transverse response; blue a-n: peak response tagging of longitudinal response. (b): Compare of the transverse response at the shaft, at the stern bearing, at the intermediate bearing, and at the thrust bearing.

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Figure 22. The loading and excitation devices, data acquisition system and the location of the sensors.

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Figure 23. Global modes of the propeller-shaft-hull system, the propeller-shaft substructure and 1st order local mode of the propeller blade. For each mode, left figure: numerical simulation; right figure: experiment result.

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Figure 24. The longitudinal and transverse response under longitudinal excitation apply to the propeller hub center. Black dashed line: peak response frequencies; black b-h: peak response tagging.

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Figure 25. The longitudinal and transverse response under lateral excitation apply to the propeller hub center. Black dashed line: peak response frequencies; black a-h: peak response tagging.

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Figure 26. Comparison of frequency response function of the propeller-shaft-hull system under longitudinal excitation apply to the propeller hub center. Dashed line: peak response frequencies of longitudinal response; g-n: peak response tagging of longitudinal response; subscript 1: peaks for stiffness ktx1=1107N/m; subscript 2: peaks for stiffness; ktx2=1108N/m; subscript 3: peaks for stiffness ktx3=1109N/m.

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Figure 27. Comparison of frequency response function of the propeller-shaft-hull system under lateral excitation apply to the propeller hub center. Dashed line: peak response frequencies of transverse response; a-n: peak response tagging of transverse response; subscript 1: peaks for stiffness ksy1=1106N/m; subscript 2: peaks for stiffness; ksy2=1107N/m; subscript 3: peaks for stiffness ksy3=1108N/m.

ACCEPTED MANUSCRIPT Highlights: The dynamic model of an elastic propeller-shaft-hull system is developed using FBSM. The elastic propeller is modelled by using harmonic blade array elements. The vibration behaviors of the substructures and total system are simulated and compared. Experiment is performed to validate the coupled vibrations of the substructures and total system. Coupled vibration analysis using FBSM is of reasonable accuracy and high efficiency.