Pu Liang, Ming Hong,2* and Zheng Wang

1. School of Naval Architecture, Dalian University of Technology, Dalian 116024, China

2. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China

1 Introduction1

When a structure in contact with fluid vibrates, there is an interaction between the fluid and the structure. It is called fluid-structure interaction. Different methods should be applied to models with different levels of interaction. The structure in contact with low density fluid is a weak coupled problem, while the structure in contact with high density fluid, such as water, is a strong coupled problem. To solve the strong coupled system, a coupled method should be applied. The fluid-structure interaction level of a plate can be decided by the empirical formulation (Atalla and Bernhard, 1994).

When a structure in contact with fluid vibrates in low frequencies, the fluid is considered as irrotational ideal fluid under small linear load. If the fluid is compressible, then the governing equation of sound pressure or velocity potential is Helmholtz differential equation. In low frequency, the effect of fluid on structure is considered as an added mass effect(Wanget al., 2014). FEM/FEM and FEM/BEM are common methods for solving the vibration characteristics of fluid-structure systems.

Usually, when the structure is closed and the fluid domain is unbounded, FEM/Direct-BEM is an appropriate method(Everstine and Henderson, 1990; Everstine, 1991; Yaoet al.,2004). It is easy to use the calculated added mass matrix by Direct-BEM to build system equation, but it can only solve closed structure and the interior or exterior problem needs to be solved separately. Another difficulty in Direct–BEM is the singular integral. When the structure is open and the fluid domain is unbounded, FEM/Indirect-BEM is a widely used method (Coyette and Fyfe, 1989; Jeans and Mathews,1990; Vlahopouloset al., 1999; Weietal., 2011; Liuet al.,2014). There are two advantages to calculate the added mass matrix by Indirect-BEM. The first is that it can solve the interior and exterior problem simultaneously. The second is that the obtained matrix is symmetric so that the fluid equation and the structure equation can be coupled better.However, it costs more time to build the system matrix. And it is difficult to deal with the hypersingular integral in Indirect-BEM. FEM/FEM is usually applied to analyze the vibration characteristics of structure in bounded fluid(Gladwell, 1966; Petytetal., 1976; Volcyet al., 1979; Lu and Clough, 1982; Wanget al., 1988; Everstine, 1997; Tong and Liu, 1997; Zhenget al., 1998; Ahlem, 2004; Thompson, 2005;Yaoet al., 2006; Wu and Zhao, 2007; Zhaoet al., 2007). The advantage of calculating the added mass matrix by FEM is that the obtained fluid mass and stiffness matrices are symmetric, sparse and real matrices. It is easier to compute the matrices using FEM than Direct-BEM or Indirect-BEM.For large scale exterior problems the FEM/FEM needs lower computational cost. FEM has no singular integral. And the obtained fluid mass and stiffness matrices are irrelevant to the vibrating frequency so that they need not to be computed repeatedly. BEM doesn’t have these advantages.Experiments are often used to verify the numerical results obtained by FEM/FEM or FEM/BEM (Volcyet al., 1979;Chenet al., 1999; Chenet al., 2003; Liuet al., 2014).

This paper presents a numerical method for calculating the added mass of structure that is in bounded fluid in low frequency. After adding the added mass matrix to the structure mass matrix, a fluid-structure interaction mass matrix is obtained. Combining the matrix with structure stiffness matrix, the generalized characteristic equation of the system is attained. The vibration characteristics of the system are calculated by solving the equation. In this paper,a FORTRAN code is programmed that computes the vibration characteristics of structure in the bounded fluid domain. The cantilever plate example demonstrates that fluid changes the structure’s dynamic characteristics. A verification experiment based on a loaded ship model is performed and the results show good correlation with the numerical solutions. The derivation of numerical method for the fluid-structure has a good reference value to the research of fluid-structure dynamics and acoustic radiation.

2 Basic theories

2.1 Derivation of Helmholtz integral equation in fluid domain

As shown in Fig. 1,Vis the bounded fluid domain andWis its boundary. When the structure in contact with the fluid vibrates, the vibration has an effect on fluid, which produces a radiation sound pressure filed in the fluid domain.The sound pressure in the fluid domain also has an effect on the structure in return. For the irrotational, compressible fluid, under the small linear load, the pressurepin the fluid domain satisfies Helmholtz differential equation:

wherek=w/cis the wave number,wthe angular frequency andcthe sound velocity in fluid.

Fig. 1 Representation of fluid domain and boundary

For Eq. (1), the weighted integral form in the fluid domain is:

wherewis the weighted function.

Using integration by parts, Eq. (2) becomes:

According to the Gaussian divergence theorem, the first term of Eq. (3) becomes:

where0ris the fluid density,vis the particle velocity vector andnthe unit normal of the fluid boundary.

Substituting Eq. (4) into Eq. (3), Eq. (5) is obtained as follows:

Eq. (5) is the Helmholtz integral equation over the fluid domain and boundary. It’s the theoretical basis to solve Helmholtz differential equation.

2.2 Solving Helmholtz integral equation of the fluid domain by FEM

This paper uses 8-nodes isoparametric hexahedral element to discretize the bounded fluid domain, as shown in Fig. 2. The shape function of 8-nodes isoparametric hexahedral element is:

The corresponding coordinateis shown in Fig. 2.

Fig. 2 8-node isoparametric element

The discrete form of Eq. (5) is

wherenfis the number of fluid elements, andnthe number of fluid boundary elements.

For 8-nodes isoparametric hexahedral element,coordinates transformation using the same shape function can satisfy the compatibility demand:

The weighted function at any point in an element can be interpolated by the shape function:

The sound pressure at any point in an element can be interpolated by the shape function:

For Eq. (7), the first term on the left isaccording to Eq. (9) and Eq. (10):

where

Substituting Eq. (11) and Eq. (12) into

whereKeis a matrix of 8×8 :

Transforming Eq. (14) to the integral form under the local coordinate system:

Considering the form ofBe, it needs to be transformed from global coordinate system to local coordinate system before integration:

For Eq. (7), the second term on the left issubstituting Eq. (9) and Eq. (10) to Eq.(19):

whereNis the shape function vector andMeis a matrix of 8×8:

The integral form of Eq. (20) in local coordinate system is:

2.3 Solving the fluid-structure vibration characteristics problem by combining structure FEM and fluid FEM

The right-hand side of Eq. (7) isThe integral boundaryWcan be divided into velocity boundaryvW, acoustic impedance boundaryzWand pressure boundarypW, as shown in Fig. 3.

Fig. 3 Boundary conditions of acoustic field

Considering the derivation in Section 2.2, the governing equation of the acoustic field in FEM form is:

whereKais the stiffness matrix of the whole fluid,Mais the mass matrix of the whole fluid, ΩFais the effect of the boundary conditions of acoustic field, including the velocity boundary ΩWv, the acoustic impedance boundaryWzand the pressure boundary ΩWp.

For the fluid-structure vibration characteristics problems,the normal velocity of structure is equal to fluid velocity in the coupled interface between fluid and structure. It is shown in Fig. 4.

Fig. 4 Acoustic field boundary conditions of fluid-structure problems

Without the damping effect, the structural motion equation in the FEM form is:

whereKsis the stiffness matrix of the whole structure,the mass matrix of the whole structure,uthe displacement vector of the structure andsFthe load vector on the structure.

The sound pressurep, which is perpendicular to the fluid-structure interface, satisfies the following:

wherenseis the number of the elements at fluid-structure interface,neis the normal vector of the element,Nsis the shape function of the structure element,Nais the shape function of the fluid element,Ωseis the fluid-structure interface.

Considering the fluid-structure interaction, sound pressure acting on the structure can be seen as added normal loads.Combining Eq. (23) and Eq. (24), the structural motion equation considering fluid-structure interaction is:

whereKcis the fluid-structure coupling stiffness matrix:

On the fluid-structure interface, the velocity of structure can be seen as the boundary conditions of sound field. Using Eq. (22), the sound field equation considering fluid-structure interaction is

whereMcis the fluid-structure coupling mass matrix:

Combining Eq. (25) and Eq. (27), the fluid-structure interaction motion equations based on FEM theory is:

If fluid-structure interaction boundarysWis the only boundary (Fa= 0), thenpcan be deleted in Eq. (29):

The corresponding equation of generalized eigenvalue problem is:

The fluid-structure coupling matrixKcis related to the element normal vectoren. If there is only one fluid domain,as shown in Fig. 5(a), the normal vectorenof boundary element can either point into the fluid or point away from the fluid, and all the elements should be consistent. From the form ofMadd, it can be found that it includesKcandTherefore, for a multi-fluid domain problem, as shown in Fig. 5(b), the element normal vectorsenof each fluid domain (V1,V2,V3orV4) should be consistent.

Fig. 5 The element normal direction of single and multifluid domain

A FORTRAN program is developed to calculate the vibration characteristics of structure under fluid load and air load based on the fluid and structure finite element method theory. The program flow chart is shown in Fig. 6. Flow diagram in the dotted box is used for calculating air loaded ship model.

Fig. 6 Flow charts of the coupled fluid-structure dynamic analysis program

3 Numerical examples

For elastic cantilever plate submerged in bounded fluid domain, boundaries of the fluid domain are shown in Fig. 7.The length of the plateais 0.50 m, widthbis 0.30 m and thicknesstis 0.004 m. The Poisson ratio of the material is 0.3, the Young's modulus is 2.1×1011N/m2and the plate’s density is 7 800 kg/m3.

Fig. 7 Boundaries of the fluid domain

Assuming that the fluid is compressible, the fluid density is 1 000 kg/m3, and the sound velocity is 1 500 m/s. The upper surface of the fluid domain is free surface, so it has the boundary conditionp=0. The other five surfaces of the fluid domain are rigid surfaces, so they have the boundary conditionv=0. The vibration characteristics of cantilever plate submerged at three different depths shown in Fig. 8 are calculated. The calculated fundamental frequencies of the cantilever plates are shown in Table 1.

Fig. 8 Three depths of the submerged cantilever plate

Table 1 1st natural frequencies of the cantilever plate in air and in different submerged depths

Fig. 9 1st natural frequencies of the cantilever plate in air and in different submerged depths

It can be seen that the results shown in Table 1 and Fig. 9 are consistent with the results in Volcyet al. (1979) and Wanget al. (1988).

4 Experimental researches

To verify the derived numerical method and the validity of the FORTRAN program in this paper, modal identification experiments were conducted for a full free ship model in the water and air under different loads.

4.1 Introduction to the experimental model and instrument

The ship model’s length, width, moulded depth and steel plate thickness are 1.50, 0.30, 0.15 and 0.005 m. Fig. 10 shows the experiment that was done in the Ship Structure Vibration Laboratory and towing basin in Dalian University of Technology. The laboratory instruments included a DH5922 dynamic signal analysis instrument, ICP acceleration sensors, and so on.

Fig. 10 Representation of identification experiment for the loaded ship model in the air and water

4.2 Numerical analysis of the experiment model

In this paper, using the dynamic analysis FORTRAN program for fluid-structure interaction problems, free modes of the loaded ship model are calculated. The Poisson ratio of the material is 0.3, the Young modulus 2.1×1011N/m2, and the density 7 800 kg/m3. The loaded water on the ship model is regarded as compressible fluid. Its density is 1 000 kg/m3and the sound velocity is 1 500 m/s. The calculated frequency for the added mass matrix is 100 Hz. The meshes of the ship model and fluid are shown in Fig. 11. The boundary condition of the free surface of the fluid is set asp=0, and the boundary condition of rigid towing basin side walls and bottom is set asv=0.

Fig. 11 Representation of mesh discretization for the ship model and the fluid

4.3 The experiment model measurement

The schematic diagram of the modal identification experiment is shown in Fig. 12. The excitation on the ship model is a pulse type. The arrangement of the eleven acceleration sensors is shown in Fig. 13. Through the modal identification experiment, the ship model’s bending modes,including modal shapes and natural frequencies, can be measured.

Fig. 12 Schematic diagram of the modal identification experiment

Fig. 13 Representation of sensors’ arrangement

4.4 Comparison of results

Numerical results of the ship model under seven load conditions are compared in the experiment. Fig. 14 compares the experimental and numerical modal shapes of the full free ship model. The first two order natural bending modal shapes under different loads in the air and in the fluid are presented, where the cloud maps on the left are the numerical modal shapes and the fitted curves on the right are the corresponding experimental modal shapes.

The first two order natural frequencies of loaded ship model’s bending modes in the air are shown in Table 2 and Fig. 15. The natural frequencies in water are shown in Table 3 and Fig. 16.

Fig. 15 Representation of testing and numerical natural values comparison in air

Fig. 16 Representation of testing and numerical natural values comparison in water

Table 2 The first two-order natural frequencies of loaded ship model’s bending modes in air

Table 3 The first two-order natural frequencies of loaded ship model’s bending modes in water

From Table 2, Table 3 and Fig. 14, it can be seen that the numerical results of the vibration modes of the ship model,are well consistent with the modal identification experiment results using the method derived in this paper. The results verify the method and the program. Comparing the load conditions 2 and 3 in the air and in the water, or comparing the load conditions 4, 5, 6 and 7, it can be determined that the locations of the loads have an effect on the structure vibration characteristics when the load quantity is constant.Comparing the load condition 2 and 4 or comparing the load condition 3 and 5 in the air and in the water, it can also be seen that the change of the load quantity has little effect on natural frequencies of the structure.

5 Conclusions

Based on the Helmholtz differential equation, the added mass matrix of the structure in bounded fluid domain is derived in low frequency. Considering the fluid compressibility, the vibration characteristics of the cantilever plate submerged at different depth are calculated.The different loaded ship models’ vibration characteristics in the air and in the fluid are also calculated. Finally, a modal identification experiment of a ship model is performed.Several conclusions are drawn below.

The method of deriving the structure adding mass matrix is simple and highly accurate. It has some benefits for calculating the fluid-structure interaction vibration characteristics and acoustic radiation characteristics.

Regarding the modal identification experiment of the structure in bounded fluid domain, the identified result can be influenced by a lot of factors, such as the position and the magnitude of the excitation, load condition and the outside interference signal.

From the modal identification experiment of the ship model under different load conditions, it can be seen that different locations of loads can influence the vibration characteristics of the structure in contact with fluid and the growth of load quantity probably has not a significant influence on the vibration characteristics of the structure.

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