Qingyong Niu, Tianyun Li＊, Xiang Zhuand Lu Wang

Free Transverse Vibration Analysis of an Underwater Launcher Based on Fluid-Structure Interaction

Qingyong Niu1,2, Tianyun Li1,2＊, Xiang Zhu1,2and Lu Wang1,2

1. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
2. Hubei Key Laboratory of Naval Architecture & Ocean Engineering Hydrodynamics, Huazhong University of Science and Technology, Wuhan 430074, China

A pneumatic launcher is theoretically investigated to study its natural transverse vibration in water. Considering the mass effect of the sealing cover, the launcher is simplified as a uniform cantilever beam with a top point mass. By introducing the boundary and continuity conditions into the motion equation, the natural frequency equation and the mode shape function are derived. An iterative calculation method for added mass is also presented using the velocity potential function to account for the mass effect of the fluid on the launcher. The first 2 order natural frequencies and mode shapes are discussed in external flow fields and both external and internal flow fields. The results show good agreement with both natural frequencies and mode shapes between the theoretical analysis and the FEM studies. Also, the added mass is found to decrease with the increase of the mode shape orders of the launcher. And because of the larger added mass in both the external and internal flow fields than that in only the external flow field, the corresponding natural frequencies of the former are relatively smaller.

underwater launcher; free transverse vibration; natural frequency; mode shape; added mass; fluid-structure interaction

1 Introduction1

A certain form of underwater pneumatic launcher, which uses high-pressure air as the energy source to discharge the weapon out of the launch tube, vibrates in different flow fields during and after the launching process. During the launching process, the launcher vibrates in the external flow field while after the weapon is launched out, the launcher vibrates in both the external and internal flow fields. The study of the free transverse vibration has great significance for further research on the structural response and sound radiation for these types of launchers when subjected to dynamic forces.

Because of the sealing cover at the free end, the vibration characteristics of the launcher are more difficult to study than those without top structures. Generally, the top component is simplified into a point mass (Auciello, 1996; Posiadala, 1997; White and Heppler, 1995; Wu and Chen, 2003; Salarieha and Ghorashi, 2006). For structures with top point mass which are immersed in water, there is plenty of literature concerned with their vibration. Uscilowska and Kolodziej (1998) studied the free vibration of a partially immersed column with a concentrated mass at the top. Closed form frequencies and eigenfunctions were presented. The parameters’ influence on the frequency equations was also discussed. ?z (2003) presented a non-dimensional transverse vibration equation for an Euler-Bernoulli type beam partially immersed and studied the effects of water height, tip mass and water density on the frequencies. Wu and Chen (2005) developed a theory to determine the natural frequencies and the corresponding mode shapes of an immersed and elastically fixed wedge beam. A comparison between the FEM numerical results was made and good agreement was achieved. Zhou (1997) investigated a cantilever beam with a tip mass which was elastically supported. Exact and analytical expressions for eigenfrequencies and mode shapes of a beam were derived. Wu and Hsu (2006 and 2007) studied an immersed uniform beam carrying an eccentric tip mass with elastic and fixed support. Shen (2009) used the finite difference method to investigate the nonlinear free vibration of a compliant beam partially immersed in water.

Little attention has been paid to the structures fully immersed in both external and internal flow fields. In this study, an underwater pneumatic launcher is examined by modeling it into a cantilever beam with a top mass. Not only the external flow field but also the external field and the internal flow field are studied. The natural frequency equation and mode shape function of the transverse vibration are established. An iterative calculation method of added mass is proposed, which gives more precise predictions. Vibration results of the different dimensions are analyzed as examples and the first 2 order natural frequencies and mode shapes are given. In the end, a comparison between the FEM results is made and a good consistency with the natural frequencies and mode shapes is achieved.

2 Analytical model

Fig.1 shows the sketch of a type of underwater pneumatic launcher. Considering the mass effect of the sealing cover and ignoring its volume influence, the launcher is modeled as a uniform cantilever with a top mass at the free end. Taking into account the fluid-structure interaction with the added mass, the free transverse vibration equation of the beam model may be written in the form:

whereEIis the bending stiffness,uis the transverse displacement, m′ denotes the effective mass of the beam per unit length in water. m′=m+Δm , in which m is the structure mass per unit length in air and Δm is the added mass of entrained water per unit length.

Separating the variables of z and t, the solution of Eq. (1) can be written as:

Substituting Eq. (2) into Eq. (1) yields

in which a2=EI/m′.

Fig. 1 Sketch of an underwater pneumatic launcher

Noting that the left side of Eq. (3) is a function of t only and the right side is a function of z only, the equation can be satisfied for arbitrary values of t and z only if each side of Eq. (3) is a constant. Supposing the constant is2ω-, two separate differential equations are derived:

The general solutions of Eqs. (4.a) and (4.b) can be expressed in the form:

where1A and14～BB are unknown constants, ω is the natural frequency of the transverse vibration, φ is the initial phase angle and k meets the requirements of

The transverse displacement (,)uzt in Eq. (2) must satisfy the zero displacement and zero slope boundary conditions at the fixed end and the zero moment and shear boundary conditions at the free end, i.e., the conditions:

in which Mis the mass of the sealing cover.

Substituting Eqs. (6.a) and (6.b) into Eq. (2) leads to

Writing Eqs. (7.a) ～ (7.d) in a matrix form derives:

in which14～BBform a column vector. The elements of the coefficients matrix of Eq. (8) are given by:

According to the non-zero solution condition and notingω2=k4EI/m′, an equation about k is obtained:

Considering the relationship between14～BB and solving k from Eq. (9), the mode shape function is also obtained:

Eq. (9) is a transcendental equation and has an infinite number of roots. After solving the n-th root ofnk, the n-th natural frequency is then determined:

Introducingnk into Eq. (10), the n-th mode shape of the launcher can also be determined.

3 Added mass

The mass ′m in Eqs. (1), (9) and (11) is the mass per unit length considering the effects of the added mass mΔ. It is known that mΔ is relevant to the parameters including structure geometry, material properties, mode shapes, etc. (Qian et al., 1996; Su and Huang, 2003; Ju and Zeng, 1983). In the previous study of beams with top mass, mΔwas usually calculated as the mass of the water displaced by the beam per unit length (Uscilowska and Kolodziej, 1998; Wu and Chen, 2005; Wu and Hsu, 2006 and 2007). This kind of calculation ignores the effects of the differences between mode shapes and cannot be used in the circumstances of structures immersed in both external and internal fluid.

Generally, considering the viscosity of water will make the natural vibration analysis of the structures complicated and there is no significant difference according to non-viscous water. Therefore, the additional inertia effect of the water (i.e., added mass) is mainly taken into account rather than its viscosity. As mentioned in the research done by Ju and Zeng (1983), in irrotational and non-viscous fluid, the velocity potential function ψo(hù)f a uniform hollow cylindrical beam satisfies the Laplace equation in cylindrical coordinates.

where ,,rzθ are the components in cylindrical coordinates.

At the free surface and the surface adjacent to the stationary structures, the boundary conditions must be satisfied. At the boundary surface between the beam and fluid, the continuity condition is required as follows:

One form of the solution of Eq. (12) can be obtained by assuming ψ in the following form:

in which ()fr is a function of radial coordinate r,nq˙ is the generalized velocity. Substituting Eq. (14) into Eq. (12) leads to a modified Bessel function of the first kind:

Noting the velocity continuity indicated as Eq. (13)，the function of1()frcould be expressed as follows: in which Bnsand Cnsare constants determined by the cylindrical boundary conditions, I1s(r1) and K1s(r1)are modified Bessel functions of the first kind and second kind respectively. According to the characteristics of the modified Bessel function, two series of I1s(r1) and K1s(r1) are used to express the velocity potential functions of the internal and external flow field respectively. After superposition, the unified potential function becomes:

Expanding the transverse displacement (,)uzt into a series of the product of the underwater mode shapenX and the generalized coordinatenq, one obtains

Substituting Eqs. (18) and (17) into Eq. (13), the constants ofnsB andnsCare found to be:

where I1′sand K1′sare first order derivations of the modified Bessel function of the first kind and second kind respectively, a1=D/2 and a2=d/2 are the outer and inner radius of the launch tube section respectively.

Substituting Eqs. (19.a) and (19.b) into Eq. (17), the velocity potential function ψ becomes finally

Suppose the beam vibrates under a certain frequencynω, according to the theorem of kinetic energy of fluid, one obtains

in which Tis the kinetic energy of the fluid around the beam,wρ is water density, S is the fluid boundary. Noting that2/2

TmV=Δ, in which V is the velocity, added mass mΔ can be determined as follows:

in whichnmΔ is the added mass per unit length of the n-thfree transverse vibration, ξ is the axial coordinate.Hsis expressed as follows:

Δmncould not be directly determined yet by Eq.(22) because the wet mode shape function Xnis still unknown. A feasible method to solve the problem is to substitute the wet mode shape with the dry mode shape (Ju and Zeng, 1983). It can be deduced that this type of substitution method for beams without point mass could be extended to beams with top mass. Further more, if an iterative method is used more precisely, a prediction will be reached. As for the launcher model studied in this work, first, a dry mode shape is used to give "rough" results of Δmn, ωnand Xnthrough Eqs. (22), (11) and (10) respectively. Then, with the "rough" wet mode shape Xn, a relatively more precise Δmncan be calculated, as well as ωnand Xn. By repeatedly using this iteration, a precise transverse response can be achieved. The flowchart of the iterative method is shown in Fig.2. The analytical computation mentioned above is realized with MATLAB.

Fig. 2 Flowchart of the iterative calculation

4 Results and discussion

In order to check the validation of the theoretical analysis derived in this work, a 3D FEM calculation is conducted using ANSYS. The axisymmetric plane of the FSI (fluid structure interaction) model is shown in Fig.3. The element type SOLID45 is used to model the structure and FLUID30 is used to simulate the fluid medium. Interfaces between the structure and fluid are specified as the FSI coupling surfaces in ANSYS. The fluid field in the simulation is cylinder shaped, with its radius 6 times as large as the outer diameter of the launch tube and its length 2 times as long as that of the launcher. The structure model is meshed into 5 elements in thickness direction and 100 elements in length direction. Such fluid region size and element mesh density are considered as meeting the requirements of calculation accuracy (Huang et al., 2001; Yang et al., 2009). The main parameters used in the calculation are as follows: the fluid density is 1,000 kg/m3, the sonic velocity underwater is 1,400 m/s, the structure density is 7,850 kg/m3, Yong′s modulus and Poisson ratio are 200 GPa and 0.3 respectively. Nodes at the bottom surface of the launcher are fixed in displacements. The pressure of the outer surface of the fluid is set to zero. The normal displacement of the fluid at the bottom surface is also set to zero.

Different dimensions of h, D and d are discussed as examples in ANSYS, with the sealing cover′s diameter 4 cm larger than D and the height 4 cm in length.

Fig. 3 FSI model of the launcher

4.1 External flow field

The first 2 order numerical natural frequencies are given in Table 1, and compared with the analytical results and the traditional method (i.e., added mass is regarded as the mass of water displaced by the structure per unit length).

Table 1 First 2 order natural frequencies in external flow field

It can be seen from Table 1 that the analytical frequencies have a better consistency with the FEM results, with a maximum relative error of －4.75%, less than －6.53% of the traditional results. The mode shapes of the first 2 order natural frequencies are shown in Fig.4(a)-(c). Good agreements are also achieved with the curves between the analytical and FEM results.

Fig. 4 The first 2 order mode shapes in the external flow field obtained from analytical and FEM solutions

4.2 External and internal flow field

The flow of the fluid in both external and internal cases is also considered in ANSYS by setting both the outer and inner surface of the launcher as FSI surfaces. Natural frequencies, as well as the mode shapes, are given in Table 2 and Fig. 5(a)-(c) respectively, from which good agreements can be seen. The maximum relative error rate of the first 2 order natural frequencies is about 6.81%.

Due to the relatively larger added mass of the launcher in both the external and internal flow field, the natural frequencies are generally smaller than those obtained in only the internal flow field.

Table 2 The first 2 order natural frequencies in both the external and internal flow field

Fig. 5 The first 2 order mode shapes in both the external and internal flow field obtained from analytical and FEM solutions

An examination of Fig.4 and Fig.5 indicates that the consistency of the mode shapes of the first order is better than the second order. This is probably due to the greater similarity of the first order dry mode shape to the real wet mode shape in the iteration calculation for the first time.

跌倒檢測(cè)屬于二分類問題,模型只需分辨跌倒和非跌倒運(yùn)動(dòng)兩種類別即可。二分類模型的判斷結(jié)果可能存在以下4種情況[17]:

4.3 Effects of modes on added mass

According to the proceeding derivation of added mass, Δmnis not a constant but a function determined by the orders of vibration modes and the geometry of the launcher. Table 3 gives the added mass of the first 3 modes for the launcher in different flow fields.

It can be seen from Table 3 that the added mass decreases with the increasing of the mode orders of the launcher. For the same order of vibration mode, the added mass in both the external and internal flow field is relatively larger than that of only the external flow field.

Table 3 Added mass per unit length in different flow fields of the launcher

5 Conclusions

In this research, the characteristics of the transverse vibration of an underwater launcher in two different flow fields are presented. First, the launcher is modeled as a cantilever beam with a point mass at the top by simplifying the sealing cover into a concentrated mass. The natural frequency equation and the mode shape function are then derived by taking the boundary conditions into account in the motion equation. Then, an iterative calculation method of added mass is proposed to give a more precise prediction of the vibration response. Two cases of flow field are discussed. The numerical results given by FEM are used to validate the analytical theory and results. Good agreements are obtained with the natural frequencies and mode shapes. For the flow field in both the external and internal cases, due to the relatively larger added mass, the natural frequencies are relatively smaller than with those only in the external flow field. The research involved with this work may be helpful in future studies of the dynamic response and sound radiation for underwater launchers when subjected to transient actions.

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Author biographies

Qingyong Niu is a doctoral candidate at the School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, majoring in underwater engineering.

Tianyun Li was born in 1969. He is a professor at the School of Naval Architecture and Ocean engineering, at Huazhong University of Science and Technology. His main research interests include ship and ocean structural mechanics and structure vibration and noise.

1671-9433(2014)02-0178-07

date: 2013-11-01.

Accepted date: 2014-03-31.

Supported by the National Natural Science Foundation of China (51379083) and the Specialized Research Fund for the Doctoral Program of Higher Education (20120142110051).

＊Corresponding author Email: LTYZ801@hust.edu.cn

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014