A time domain three-dimensional sono-elastic method for ships’ vibration and acoustic radiation analysis in water *

Ming-song Zou , You-sheng Wu , Can Sima , Shu-xiao Liu

1. China Ship Scientific Research Center, Wuxi 214082, China

2. State Key Laboratory of Deep-sea Manned Vehicles, Wuxi 214082, China

3. National Key Laboratory on Ship Vibration and Noise, Wuxi 214082, China

Abstract: The classical three-dimensional hydroelasticity of ships is extend to include the effect of fluid compressibility, which yields the three-dimensional sono-elasticity of ships. To enable the predictions of coupled transient or nonlinear vibrations and acoustic radiations of ship structures, a time domain three-dimensional sono-elastic analysis method of acoustic responses of a floating structure is presented. The frequency domain added mass and radiation damping coefficients of the ship are first calculated by a three-dimensional frequency domain analysis method, from which a retardation function is derived and converted into the generalized time domain radiation force through a convolution integral. On this basis the generalized time domain sono-elastic equations of motion of the ship hull in water are established for calculation of the steady-state or transient-state excitation induced coupled vibrations and acoustic radiations of the ship. The generalized hydrodynamic coefficients, structural vibrations and underwater acoustic radiations of an elastic spherical shell excited by a concentrated pulsating force are illustrated and compared with analytical solutions with good agreement. The numerical results of a rectangular floating body are also presented to discuss the numerical error resultant from truncation of the upper integration limit in the Fourier integral of the frequency domain added mass coefficients for the retardation function.

Key words: Hydroelasticity, sono-elasticity, time domain, Fourier transform, vibration, acoustic radiation

Introduction

The fluid-structure interaction has been widely applied in various engineering practice, including improvement of ship motion performance and structural safety, ship vibration and noise control and enhancement of underwater acoustic stealth. In the late 70s of the twenty century, the hydroelasticity of ships was established as one new branch in the research of ship structure-fluid coupling dynamics. In early 80s, by combining the three-dimensional potential flow theory of ship motions with the three-dimensional structural dynamics, the threedimensional hydroelasticity was established by Wu[1],Bishop et al.[2]to deal with the dynamic response of arbitrarily shaped three-dimensional deformable body under internal and external loads. In the study of hydroelasticity, the water is usually treated as an incompressible potential flow[1-8]. When the structural vibration and the associated acoustic radiation of a ship are to be analyzed, the water must be treated as compressible acoustic medium. Several years ago the three-dimensional hydroelasticity theory of ships[1-2]was extended to include the effect of water compressibility, and a frequency domain sonoelasticity analysis method of acoustic radiations of an advancing floating structure responding to mechanical excitations was produced[9]. This method and the corresponding software “THAFTS-Acoustic” is feasible in predicting the steady state vibrations and underwater acoustic radiations of a ship[10-12]. The three-dimensional sono-elasticity theory is developed on the basis of the classical three-dimensional hydroelasticity theory. The sono-elastic method is closely related to the method of hydroelasticity.Generally speaking, the three-dimensional sonoelasticity theory is a branch of the hydroelasticity theory. To put it in another way, the sono-elasticity theory is also a generalized hydroelasticity theory,where the water is treated as compressible medium to include the acoustic effects. The comprehension of the three-dimensional sono-elasticity theory as well as its relationship with the three-dimensional hydroelasticity theory is described in detail in Ref. [13]. In many cases the coupled vibrations and acoustic radiations of a ship may be excited by a transient load or a nonlinear excitation caused by, for example, the interactions between a rotor shaft and the roller bearing mounted on the ship hull. The frequency domain method cannot directly deal with the transient or nonlinear problems.

To enable the coupled transient or nonlinear vibrations and acoustic radiations of a ship structure to be analyzed, a time domain sono-elasticity analysis method of floating structures is presented in this paper.This method is composed of three procedures. At first the frequency domain sono-elasticity analysis method is employed to obtain the frequency domain added mass and radiation damping coefficients of the ship in calm water. Secondly the frequency domain added mass coefficients are used to derive a retardation function by a Fourier integral, which is related to the generalized time domain radiation force by a convolution integral. Thirdly, on this basis the generalized time domain sono-elastic equations of motion of the ship hull in water are established. Their numerical solutions provide the time histories of the coupled vibrations and sound radiations of the ship excited by transient loads. The acoustic wave propagation problems with known vibration or impedance boundary conditions were solved through time domain boundary integral method (TBIM) in Refs. [14-16]. The TBIM calculation suffers the well-known exponentially diverging instability due to numerical errors. In this paper, the time domain sono-elasticity analysis method need not to solve the time domain boundary integral equation, and hence this instability problem no longer exists.

For validation of the proposed time domain sono-elasticity analysis method and the code, the generalized hydrodynamic coefficients, structural vibrations and underwater acoustic radiations of a floating elastic spherical shell excited by a concentrated pulsation force are illustrated and compared with analytical solutions. Very good agreement is achieved. The numerical results of a rectangular floating body are also presented to investigate the numerical error resultant from truncation of the Fourier integral for the retardation function. Although only transient mechanical excitations are analyzed in this paper, the presented method can also be extended to deal with acoustic responses of a ship induced by nonlinear excitations.

1. The linear sono-elasticity theory of ships in frequency domain

An equilibrium coordinate system Oxyz is introduced with the x-axis pointing towards the bow,the z-axis pointing upwards, as shown in Fig. 1. When a ship travels at a constant speed U in the x-direction in inviscid irrotational compressible fluid, vibration is excited by the on-board machineries or certain incident acoustic waves. The structural vibrations and acoustic responses as described in the equilibrium coordinate system are all small and linear. By introducing the principal coordinates ( )(=rq t r1,2, … , m) , the displacement column vector of the vibrating structure is expressed as the superposition of the principal modes of the structure in vacuum[2,9]

The total velocity potential Φ may be represented as[2,9]

Fig. 1 The equilibrium coordinate system

Assuming a time harmonic dependence in the form ofin frequency domain, the corresponding-r th radiation wave potential may be represented as

The radiation wave potentialrφ and the diffraction wave potentialin the whole fluid field can be represented by the following simple-source boundary integrals[1,17]over C in terms of the Green's function G ( r ,0r)[2,9]

The Green's functions (see Eq. (6)) which are suitable for ideal compressible fluid considering the effect of free surface are used in the sono-elastic analysis. From the perspective of acoustic analysis,there is no limit on the frequency range of the Green's functions. That is to say, the frequency range can be 0 Hz-∞ Hz. The effect of forward speed is not included in Eq. (6). As stated in Refs. [9, 13], the forward speed of a ship is much smaller than the sound speed in water. Hence in acoustic analysis the influence of forward speed on the free surface boundary condition may be neglected, the Green's function of zero forward speed is acceptable in Eq. (5).

When calculating the steady state velocity potential, the Green's function suitable for in-compressible flow in classical three-dimensional hydroelastic analysis is adopted. But the Green's function suitable for compressible acoustic medium,as shown in Eq. (6), is used when calculatingOφ ,Dφ and.

The dynamic responses of the structure induced by the mechanical excitations described by concentrative forcesor the incident waves described byOφ are governed by the generalized equations of motion in the principal coordinatesin the following matrix form

where a, b and c are the matrices of generalized modal inertia, modal stiffness of the dry structure. ξ,and G are the principal coordinate vector and the generalized excitation vector, respectively. A , B and C are respectively the matrices of generalized hydrodynamic inertia, damping and restoring coefficients

where ρ is the fluid density.

The frequency of concern is above 5 Hz for the analysis of acoustic radiation of ships in water. The excitations of machinery and acoustic waves are considered in the sono-elastic method presented in this paper. The excitation of gravitational waves is not taken into account. The component of generalized excitation vector in Eq. (7) is

The principal coordinates are obtained from Eq.(7). Then the vibration and acoustic radiation in water of ship structures can be calculated based on the modal superposition method. The acoustic pressure in water is

The sound power radiated from the wetted surface of the ship is given by[13,19]

where the superscript “*”denotes complex conjugate.

The numerical examples in Ref. [7] show that the ship speed U will influence the coupled vibration and acoustic radiation. However the influence is only limited to the low frequency range and near field, and the influence is usually negligible in engineering problems.

2. The CVIS method of eliminating irregular frequencies

In this paper, a “closed virtual impedance surface(CVIS) method” for depressing the “irregular frequencies” encountered in the numerical solution based on the simple-source distribution method is introduced.The problem of “irregular frequencies” exists because of the cavity resonances of imaginary inner fluid region occupied by the floating body. The irregular frequencies will frequently occur when using boundary integral formulations to solve exterior acoustics and water wave problems[20-22]. These frequencies do not represent any kind of physical resonance but are due to the numerical method. The numerical results will be distorted at irregular frequencies. To eliminate the irregular frequencies in the prediction of acoustic responses of a structure in ideal acoustic medium τ , it is proposed in this paper to place a closed virtual impedance surface Si2in the imaginary inner fluid domaininside the wetted surfaceas shown in Fig. 2. This impedance surface absorbs the acoustic energy and suppresses the cavity resonance in the imaginary fluid region, and hence efficiently eliminates the irregular frequencies.When the ship speed influence is eliminated,assuming φi, φ are respectively the velocity potentials defined in the inner and the outer fluid domain,the interface boundary conditions on Si2and S may be written as:

where pni2is the pressure on Si2, uni2, ZSare respectively the displacement and mechanical impedance of Si2inin direction, unis the displacement of S in n direction. The integral equations for solving the source strength σ in the outer domain and the unknown displacement uni2are as follows:

Insertion of the imaginary impedance surface inside the wetted surface of the body not only moves the oscillation frequencies of the inner fluid domain to the upper band by changing its volume, and also absorbs the oscillation energy of the inner fluid domain. This enables the elimination or removal of the irregular frequencies.

Fig. 2 The impedance surface in the imaginary inner fluid domain

3. Three-dimension sono-elasticity analysis method in time domain

3.1 Time domain equations of coupled sono-elastic response of a floating structure

In time domain, the generalized equations of motion of the structure may be written in matrix form as

where a , b and c are matrices of generalized mass, generalized damping and generalized stiffness of the dry structure. q is a column vector of the principal coordinate. If there is no incident acoustic waves, and only mechanical excitations are concerned in the present analysis of structural vibration and acoustic radiation problems, there exists generalized radiation wave force Ξ( t) and generalized mechanical exciting force G(t ) .

When the ship is stationary with zero forward speed, the radiation wave potential may be represented as

In the fluid field the acoustic radiation wave pressure is

A component of the generalized radiation wave forcecorresponding to the r-th principal mode may be written as

Substituting Eq. (15) into Eq. (17), and denoting

yields

During the process of structural vibration and acoustic radiation the fluid restoring force that the structure may sustain is rather small and negligible.The generalized equations of motion of the ship structure in time domain may then be represented in the matrix form:

where K(t -)τ is the matrix of retardation functions.Its elements are represented by Eq. (18). A component of the generalized exciting force vector G( t) corresponding to the r- th principal mode is

3.2 Retardation function

When the external excitations are sinusoidal with angular frequency ω, the principal coordinates may be expressed as

The generalized equation Eq. (20) becomes

Its counterpart, namely the generalized equation of the coupled structural vibration and fluid acoustic radiation in frequency domain has the form

where A( )ω, B( )ω are respectively the matrices of added mass and radiation damping coefficients in frequency domain. The matrix of restoring coefficients is neglected, because its effect is rather small.

The equivalence of Eqs. (23), (24) provides:

Let t= ′t -τ , it becomes

By eliminatingietωthe following relations may be obtained

In the above two equations can be replaced by t for convenience. Substitutingsin/(2i) , denoting, and employing Fourier transform, the retardation function can be represented as:

where δ( t) is the Dirac function.

Calculation of δ( t) involves an infinite integral.Its physical meaning is the Fourier transform of the acoustic reactanceIt is clear that in frequency domainwhenHence it is possible to truncate the upper integration limit with proper accuracy. This is verified in the following sections in calculation of added mass coefficients of a spherical shell and a twin-rectangular-hull barge.

3.3 Dynamic displacement and sound power

The dynamic displacement in time domain is calculated by Eq. (1) and the corresponding frequency spectrum can be obtained by using Fourier transform.

To obtain the radiated sound power spectrum, the Fourier transform of each principal coordinate is calculated at first

where Re()? denotes the real part, Brkis an element of the generalized radiation damping matrix. An appropriate window function and the method of averaged crossing periods of time variation may be used in numerical calculation.

For a non-stable signal, some time-frequency analysis methods may be used for data processing.The widely used short-time Fourier transform defined in Eq. (31) is introduced here for a non-stable signal s ( t) .

where h ( t) is the window function. s ( t) is a signal that may represent a principal coordinate, a structural acceleration, or a sound power etc. In Eq. (31), s ( t)is segmented by h ( t) .

The time-frequency spectrum of a predicted time domain signal may be obtained by Eq. (31). If the Fourier transformof each principal coordinategiven by Eq. (29) is replaced by its short-time Fourier transformrepresented by Eq. (31),the time-frequency spectrum of radiated sound power P ( t ,ω ) may be obtained.

4. Numerical example and verification of the present method

The Newmark method is used in numerical calculation of Eq. (20). The vibration and acoustic radiation of an elastic spherical shell in water excited by a concentrated force F , as shown in Fig. 3, is predicted as an example. The exciting force F is a cosine pulse with duration 1ms, as shown in Fig. 4.The shell is made of steel, with the radius 0.5 m,thickness 1mm and damping ratio 0.01.

Fig. 3 Spherical shell and the coordinate system

Fig. 4 Time variation of the exciting force

To verify the present method and the code, the predicted results of the spherical shell located far below the water surface are compared with the analytical solutions of the case of infinite fluid domain. The analytical solutions include the structural accelerations and the radiated sound power[23-24]. The latter is of the form

To investigate the influence on the numerical accuracy of the retardation function K′( t) caused by truncation of the infinite integral in Eq. (28), two methods of calculating the radiation damping coefficient B11( )ω of the rigid body translation mode are compared with each other. The first is the “Direct calculation” based on the frequency domain method.The second is called the “Calculation by retardation function”. Actually it is to use the frequency domain result of the added mass coefficient A11( )ω to calculate the retardation function K1′1( t) by Eq. (28) with the upper integration limit truncated to/2 π=3 000 Hz, and then re-calculate B11( )ω by employing Eq. (27b). The comparison of the two sets of predictions of B11( )ω is illustrated in Fig. 5 in the non-dimensional form:

Fig. 5 Non-dimensional acoustic resistance of rigid body translation mode of the spherical shell

which is actually the non-dimensional acoustic resistance ZReof the rigid body translation mode.,sA are the mass and wetted surface area of the spherical shell respectively. a11is the generalized mass of the rigid body translation mode. Figure 5 shows the two sets of results are nearly coincident,indicating that the truncation of the upper integration limitin integration of Eq. (28) is satisfactory to produce the retardation function in the frequency region

When the spherical shell is shown in Fig. 3 excited by a unit sinusoidal force F , the acceleration transfer function and the radiated sound power transfer function predicted by both the numerical method and the analytical method of Ref. [18] including Eq. (32)are illustrated and compared in Fig. 6. In the figure the source level of the radiated sound power is represented as

Fig. 6 Comparison of the present numerical predictions and the analytical solutions

The numerical predictions and the analytical results in Fig. 6 are close to each other. When the excitation force F acting on the spherical shell is a cosine pulse force shown in Fig. 4, the time variation of the acceleration at the excitation point is shown in Fig. 7. According to the short-time Fourier transform,the corresponding acceleration time-frequency spectrum and the radiated sound power time-frequency spectrum are exhibited in Fig. 8.

Fig. 7 Time variations of the acceleration at the excitation-point

Fig. 8 (Color online) The predicted time-frequency spectra of acceleration and radiated sound power

5. Analysis on a twin hull floating barge

To further verify the present method and the code,and to investigate the effect of frequency truncation of retardation functions on computational accuracy, the generalized hydrodynamic coefficients associated with two rigid body modes (surge and sway) of a twin-rectangular-hull barge floating on surface of water with infinite depth are calculated. The length,width, draught and distance between inner surfaces of the twin hull are respectively 20, 9, 5 and 15. The size of the wetted surface panel used in numerical analysis is 0.5 as shown in Fig. 9. The non-dimensional acoustic resistance of Eq. (33) is calculated by two methods which are the same as in section V. The upper integration limit of Eq. (28) is truncated tofor the second method. The comparison of the results is shown in Fig. 10.

Fig. 9 Mesh of wetted surface panels of the twin-rectangular-hull barge

Fig. 10 Non-dimensional acoustic resistance of the two rigid body translation modes

Fig. 11 The closed virtual impedance surface with impedance value ρ c abs the acoustic energy and suppresses the cavity resonances in the imaginary fluid region, and hence efficiently eliminates the irregular frequencies

In this paper, the simple-source distribution method is used to calculate the generalized hydrodynamic coefficients. The problem of “irregular frequencies” exists because of the cavity resonances of imaginary inner fluid region occupied by the floating body. Some sharp peak points in Fig. 10(a)and the peak point “a” in Fig. 10(b) correspond to the irregular frequencies. The interference of multireflection waves between the two rectangular floating bodies, so called the standing wave phenomena, make the curves of non-dimensional acoustic resistance fluctuating with regard to frequency as shown in Fig.10(b). The peak points “A”, “B”, “C” and “D”correspond to the standing wave phenomena. This is self explanatory by the fact that the frequency differences between two adjacent hills are all about 100 z, the wavelength of which equals to the distance of the twin bodies.

To eliminate the irregular frequencies in the calculation, the Closed Virtual Impedance Surface(CVIS) Method is used with a 4 m×2 m×4 m losed virtual impedance surface placed in the imaginary inner fluid region of each rectangular hull as shown in Fig. 11. The mechanical impedance ZSof the virtual impedance surface is cρ .

The new comparison of the results is shown in Fig. 12. Apparently good agreement between the results of the “Direct calculation” and the “Calculation by retardation function” is achieved, and the CVIS method efficiently eliminates the irregular frequencies.

Fig. 12 Non-dimensional acoustic resistances of the two rigid body translation modes

6. Conclusions

The basic equations and numerical methods of the three-dimensional sono-elasticity of ships are deduced in this paper. Based on the frequency domain three-dimensional sono-elasticity analysis method together with the convolution integral and Fourier transform approaches, a time domain three-dimensional sono-elasticity analysis method is established.The coupled fluid-structure acoustic responses of ship structures encountering transient excitation may then be solved in time domain.

The present method and the code are verified by the numerical tests of structural responses and acoustic radiations of a submerged elastic spherical shell excited by a concentrated transient force, and the radiation damping coefficients corresponding to two rigid body translation modes of a twin-rectangularhull barge floating on water surface. Very good agreement is exhibited between the predictions by the proposed time domain method and the results obtained by the existing analytical solutions or frequency numerical method.

The numerical error in calculation of retardation functions resultant from truncation of the Fourier integral is also discussed in the analyses of two examples.