CC BY
94Electronic Journal «Technical acoustics» http://webcenter.ru/~eeaa/ejta/
2 (2002) 11.1-11.10 Z.M. Xu, R.Y. Shen, H.X. Hua
State Key Laboratory of Vibration, Shock & Noise, Shanghai Jiaotong University Shanghai, 200030, China, e-mail: [email protected]
Coupling vibro-acoustic behavior from a submarine power cabin equipped with floating-raft isolating systems
Received 27.06.2002, published 10.10.2002
Prediction of the fluid-structure vibration and acoustic radiation from a submerged submarine is a key problem in the naval engineering. In this paper, for a simulated submarine power cabin with ring-stiffened double shell, the fluid-structure coupling model is built by finite element software ANSYS. The vibrating transfer function inside the cabin, vibration velocity of the outer hull and radiated acoustic power are calculated by FEM/BEM for different parameters of the floating-raft isolating systems, i. e. (1) stiffness of two-stage elastic mounts; (2) raft mass ratio; (3) raft stiffness. This computational procedure can be further used to perform acoustic design of a real submarine.
1. INTRODUCTION
It is investigated that the underwater radiation noise induced by a ship diesel via a resilient mounting system and a stiffened cylindrical hull is one of the most important radiation sources. Simulating a real ship and achieving the target of vibration transmission in the power cabin and underwater acoustic radiation are very significant. The vibration attenuation by using resilient mounts has been researched these days and the techniques of floating-raft isolating systems have been implemented more and more widely [1, 2].
For a great odd-shaped structure such as a naval ship, predicting its vibro-acoustic behavior is very complicated, since the interaction between the shell and the heavy fluid is very strong [3]. The vibrating shell produces acoustic pressure in compressible fluids. On the other hand, the pressure induces the opposite force exerted on the structural surface. Consequently, it is necessary to solve the coupling vibration equations with enormous matrices. Many researchers have carried out the coupling vibro-acoustic simulation from ribbed plates and cylindrical shells, and have obtained quite a few useful guidelines [4, 5]. However, for the structure such as a submerged ring-stiffened cylindrical double shell, the available correlative references are not frequently reported since no matter the theoretical calculation or the underwater experimental measurement is very hard to perform. So, to predict the vibration and underwater noise from a real submarine, there is still much more work to do. Owing to the fact that solving the fluid-structural problem of complicated structures is difficult, some researchers calculated the acoustic pressure by using the vibration velocity of the ship shell computed previously in vacuum [6]. However, this kind of simplified algorithms doesn’t coincide with the principle of the fluid-structure interaction essentially.
Theoretically, a common and attractive method for precisely solving vibration velocities and acoustic physical variables from the immersed structure is FEM together with BEM [7, 8]. For an arbitrary elastic structure, the ordinary method of analyzing the frequency response subjected to exterior excitation is by using the finite element method. The solutions to the reduced wave equation, in an infinite domain, are modeled using a boundary element method commonly. BEM has several technical advantages over a finite element treatment of the acoustic problem, including a reduction of the dimensionality of the problem by one, and an automatic satisfaction of the radiation condition. The drawbacks of the singular numerical integration problem and the nonuniqueness or nonexistence of solutions at the characteristic frequencies from BEM can be obviated by kinds of numerical skills.
In this article, the FEM software ANSYS is used to compute the coupling vibration level of the power cabin induced by the excitations on the diesel engine. Then the structural vibration velocity of the outer shell can be abstracted as the boundary condition and acoustic variables are computed subsequently in LMS-SYSNOISE as well.
2. FEM/BEM FORMULATION
To isolate the vibration and reduce the acoustic radiation, the modern ship is always equipped with two-stage isolating systems where an intermediate mass is attached to both the engine and the foundation. It has been proven that, in nearly all cases, a two-stage mounting system affords superior isolation effect because it can minimize the vibration transmission at high frequency and simultaneously reduce the radiated acoustic power from the hull. To reflect the thinking of underwater acoustics design, the ship components, i. e. the diesel engine, mounting system, double hull, and acoustic fluids are integrated as a whole to be modeled.
2.1 Finite element equation
For the “dry” structure within the pressure shell in the cabin, finite element method is a normal and convenient tool. For the “wet” structure attached to the fluid, the solution of the model operated by the fluid-structure interaction, is commonly implemented by discretizing the model, solving both the discrete wave equation and the structural motion equation simultaneously. At the fluid-structure surface, the surface vibration velocity produces the acoustic fluid load, and the pressure exerts force on the structural surface as well. Transforming the force from the surface to the nodes, the coupled fluid-structure problem can be written as
where [Mse ] is the structural mass matrix, [Kse ] - structural stiffness matrix, [C* ] - structural damping matrix, [Mef ] - fluid mass matrix, [Kef ] - fluid stiffness matrix, [Cf ] - acoustic damping matrix, [Ref ] - coupling matrices, p - fluid density, V- fluid volume, {pe} - nodal pressure, {ue} - nodal displacement, {Fes} - mechanical load applied on the structure.
Using Eq. (1) the displacement and pressure of the structural surface can be calculated at the same time.
(1)
2.2 BEM acoustic formulation
In a radiation problem, the Neumann boundary condition is adopted, that is, vn = vn. The
over-bar means that the variable is prescribed on the surface S, while vn is surface velocity
(normal component). The so-called direct formulation can be derived from the Helmholtz equation using Green theorems and an integral equation appears:
dS(Q); C(P) =
1 PeQ+
6/4n PeS (2)
0 PeQ-
where Q+ and Q are the exterior and interior domain respectively, d represents the outer solid angle at a point P on S, G(P, Q) = e~jkR/4nR is the free-space Green function (or
fundamental solution) for the Helmholtz equation satisfying the Sommerfeld radiation
condition, R = |P - Q defines the distance between the source point Q and observation point
P, k is the wave number, and O is the circular frequency.
From the above equation, we see that the physical variables acoustic pressure p and the normal surface velocity vn are related directly. Given a discretized structural surface with M elements, Eq. (2) readily leads itself to subdivision over the elements on the surface, such that
C(P)p(P) - X IS p(Q) 3G(P Q)dS(Q) = X I jopVnG(P, Q)dS(Q) (3)
i=1 i dn i=1 i
The numerical evaluation of Eq. (3) for each of the N nodes ultimately yields a system of algebraic equations,
[ A]{p} = [ B]{vn}, (4)
where [A], [B] are coefficient matrices. The surface pressure can be determined from the normal surface velocity on an arbitrarily shaped body. From the determined surface pressure
and the normal surface velocity, the pressure everywhere in Q+ can easily be found using
Eq. (2) with P in Q+.
With the surface and far-field solutions, a variety of other quantities of interest can be computed, including average and root-mean-square surface velocity and pressure, surface acoustic intensity, radiated power, and radiated efficiency. For instance, the normal acoustic intensity is used as an important quantity. The normal acoustic intensity I'n for a nodal i on the surface S is the product of the pressure and the component of normal velocity:
In = ^Re|?' M,)’ 1 (5)
where the asterisk denotes the complex conjugation. The total power radiated is the surface integral of acoustic intensity.
3. POWER CABIN MODEL
The case to be used in this analysis is built around a submarine power cabin with doubleshells. The power cabin is immersed in fluids of density 1000 kg/m and acoustic speed of 1500 m/s. Because the construction within the cabin is very complicated, the model should be simplified appropriately while FE modeling. Here, the shafting structure is neglected. The shape of the cabin hull looks like a thin wall of revolution, so the cabin is modeled as a cylindrical double shell with length 9 m, outer diameter 8 m and inner diameter 7.2 m. The thickness of the outer light shell and inner pressure shell is 28 mm and 15 mm, respectively. Between double shells, there are 12 ring ribs with width 16 mm and height 120 mm. The two ends of the cabin are simply supported. Inside this cabin, two diesel engines are mounted by a floating raft isolation system (in fig. 1). Engines, which are also assumed as cylindrical structures, are 6 m long, 2.5 m in diameter and 20 mm thick. Each engine is symmetrically mounted on a common steel plate, i. e. so-called a floating raft, with dimensions 6 x 6 m. Between the engines and the raft 14 vibration isolators are placed on the skew. In all 21 vibration isolators are mounted between the raft and the inner pressure hull. The vibration isolators are defined by three-DOF springs with properly damping coefficients. The initial stiffness parameters are 6.0*106 N/m for the first stage, and 2.7*106 N/m for the second stage.
Because the hull of cabin is thin, the quadrilateral element SHELL63 with Hermitian interpolation polynomial is suitable to simulate the bending and twisting deformation. We can easily adjust the thickness and the material property of the element in the commercial code ANSYS. The elastic vibration isolators are meshed by damping element COMBINE14, which is three-dimensional linear element with two nodes. To simulate a three-DOF vibration isolator, three elements should be modeled, and superposed together since one damping element can only simulate one-DOF spring.
Between the light outer shell and the pressure shell, the fluid can be modeled by threedimensional brick elements FLUID30, wherein the fluid attached to the shell should be modeled by the “contacting” elements. In view of the fact that space between the double shells is small, the meshing should be finer.
Outside the outer hull the infinite fluid exists. A primary difficulty associated with the use of finite elements for the modeling of the infinite medium stems precisely from the need to satisfy the Sommerfeld radiation condition. A typical approach for treating the difficulty consists of truncating the unbounded domain by the introduction of an absorbing boundary
at some distance from the structure. The Eq. (1) is then solved in the annular region Qf, which is bounded by the fluid-structure interface and the absorbing boundary. In order, however, for the resulting problem in Qf to be well-posed, an appropriate condition needs to be specified on . Towards this end, the second-order additional conditions are used on .
In ANSYS software, the acoustic absorbing elements perform well for low as well as high frequency excitations. Numerical experiments have determined that the placement of the absorbing elements at a distance of approximately 0.2X (X is wave length) beyond the
region occupied by the structure of vibration source can produce accurate solutions. So an absorbing spherical shell which center coincides with the geometric center of the cabin is required to be modeled. In this paper, the vibration velocity and acoustic variables are computed for the frequency range 5-200 Hz. So the virtual absorbing sphere with a radius 70 m may be adopted. Also, the mesh must be fine enough to resolve the dominant frequency for acoustic analysis. The fluid is meshed following the A/4 -criterion at 200 Hz (here max. element-length 1.875 m). From the mesh, the fluid region is represented by 4564 elements FLUID30 with fluid-structure interface capability, and further 28706 brick fluid elements to model the fluid not in contact with the structure. The absorbing spherical surface consists of 852 FLUID130 elements. The nodes on the absorbing spherical surface should be accorded with the nodes of FLUID30 elements on the surface at the radius 70 m.
Fig. 1. Floating-raft isolating system of the engines
4. VIBRO-ACOUSTIC BEHAVIOR ANALYSIS
To provide a clear insight into the vibro-acoustic behavior from a simulated submarine power cabin, the study for different parameters of the floating raft isolation system is performed. The engines are excited by harmonic vertical forces 1000 N on their top nodes, and the frequency range of analysis is defined between 5 and 200 Hz. Three vibration and acoustic levels are adopted, i.e. the vibration displacement transfer function (VR) from the engine via a resilient floating raft system to the pressure hull, the root-mean-square velocity
level (Lv) of the outer shell and the radiated acoustic power level (Lw). The reference velocity
8 12 is 5 x 10 m/s and the reference acoustic power is 10 W.
4.1 Isolator stiffness
The curve A in Fig. 2 represents the numerical results for the routine isolator stiffness of the first stage, and curve B and C illustrate the cases while changing the first stage stiffness, i. e. 5 and 0.2 times of the routine stiffness respectively. It is apparent that the increasing stiffness is harmful to the vibration transmission, outer shell vibration velocity and underwater radiation noise except for a few low frequencies. The greater is the stiffness, the less is the value of VR, Lv and Lw. As expected, choosing low stiffness vibration isolators is necessary under the premise of stability.
For the same mounting system, the frequency position of the peak values of Lv doesn’t coincide with Lw and VR completely. In other words, the reduction of the transfer function is not proportional to the reduction of vibrating velocity of the outer shell and radiated power. For instance, the transfer function at 135 Hz is bigger than that at 160 Hz, however, the vibration velocity and underwater noise is opposite. It indicates not only the vibration transmission within the cabin, but also the heavy fluid-shell coupling vibration modes contribute to vibration level of the outer shell and underwater noise level.
a)
Frequency(Hz)
b)
Frequency(Hz)
c)
Frequency (Hz)
Fig. 2. Displacement transfer function (a), root-mean-square velocity of the outer shell (b) and radiated power (c) while changing the isolator stiffness of the first stage
If the problem is now modified by changing the stiffness of the second stage, Fig. 3 illustrates the results when the stiffness is set to routine stiffness (curve A), 5 times (curve B) or 0.2 times (curve C) of the routine stiffness. It is easily found that the effect of lowering the second stage stiffness is the same as the first stage stiffness.
a) b)
Frequency(Hz) Frequency(Hz)
c)
Frequency (Hz)
Fig. 3. Displacement transfer function (a), root-mean-square velocity of the outer shell (b) and radiated power (c) while changing the isolator stiffness of the second stage
4.2 Raft mass ratio
The raft mass ratio, which is the ratio between the raft mass and the machinery mass mounted on the raft, is a major parameter of a floating raft isolation system. Fig. 4 illustrates the results when the raft mass ratio is set to routine value 0.4 (curve A), 0.8 (curve B) and 0.2 (curve C) respectively. As far as reducing vibration and noise is concerned, it is clear that the big mass ratio of the raft is a good choice. However, in view of the raft arrangement and tactical agility, the submarine mass should be light as much as possible. So the raft mass must be restricted to a certain degree as well. Many factors are required to be considered synthetically while designing the raft.
a)
b)
Frequency (Hz)
100 150
Frequency (Hz)
50
c)
Frequency (Hz)
Fig. 4. Displacement transfer function (a), root-mean-square velocity of the outer shell (b) and radiated power (c) while changing the mass ratio of the raft
4.3 Raft stiffness
In Fig.5 the curve A represents the results that the raft stiffness is routine, while B and C show the stiffness is 5 and 0.2 times of A. Curves suggest that the increasing raft stiffness still plays a role in reducing noise and vibration. Because the engines are mounted elastically on the raft, the raft should be so firm that the machinery on the raft is stable. These indicate that high stiffness of the raft is very useful.
a)
-50
-60-
-70-
-80-
-90-
CO "O -100 -
££. > -110-
-120 -
-130 -
-140-
-150-
100
Frequency (Hz)
b)
Frequency (Hz)
150
c)
100
Frequency (Hz)
Fig. 5. Displacement transfer function
(a), root-mean-square velocity of the outer shell (b) and radiated power (c) while changing the raft stiffness
200
5. CONCLUSION
The numerical prediction of the coupled vibration and acoustic radiation from a submarine power cabin with a two-stage mounting system for different parameters has been carried out using finite element method together with boundary element method. In this complex vibro-acoustic system, reducing the isolator stiffness, increasing the raft mass ratio and stiffness, will play a role in reducing vibration and noise. The results are in agreement with the vibro-acoustic theory and demonstrate the reliability of the methods. This analysis method can be further applied to acoustic prediction of a real submarine under the present hardware and software computing capability.
REFERENCES
[1] Petersson, B. and Plunt, J. On effective mobilities in the prediction of structure-bore sound transmission between a source structure and a receiving structure, Part I: theoretical background and basic experimental studies. Journal of Sound and Vibration, 82(4), 517-529, (1982)
[2] Guo, L., Zhu, Sh., and Zhu, Sh. The prediction of local vibration characteristics of naval ship structures. Journal of Ship Mechanics, 4(6), 69-83, (2000)
[3] Shen, Sh., Leng, W., and Cheng Guanyi. Research on vibration and sound radiation of multiplayer isolation system with composite structure. Shipbuilding of China (in Chinese), (3), 49-59, (1997)
[4] Shang, D. and He, Z. The numerical analysis of sound and vibration from a ring-stiffened cylindrical double-shell by FEM and BEM. ACTA ACOUSTIC A (in Chinese), 26(3), 193-201,(2001)
[5] Chen, W., and Luo, D. Sound radiation from finite ring-stiffened cylindrical shells coated with viscoelastic layer. ACTA ACOUSTICA (in Chinese), 25(1): 27-32, (2001)
[6] Zheng, H., Liu, G. R., and Tao, J. S. FEM/BEM analysis of diesel piston-slap induced ship hull vibration and underwater noise. Applied Acoustics, 62, 341-358, (2001)
[7] Everstine, G. C., and Henderson, F. M. Coupled finite element/boundary element approach for fluid-structure interaction. JASA, 87(5), 1938-1947, (1990)
[8] Seybert, A. F., Wu, T. W., and Li, W. L., A coupled FEM/BEM for fluid-structure interaction using ritz vectors and eigenvectors. Trans. ASME, Journal of Vibration and Acoustics, 115, 152-158, (1993)
