Resonant underwater radiation revisited

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4 Resonant underwater radiation revisited Christ de Jong a and Björn Petersson b a TNO TPD, P.O. Box 55, 600 AD Delft, The Netherlands, dejong@tpd.tno.nl b Institute of Technical Acoustics, Technical University of Berlin, Einsteinufer 5, D-0587 Berlin, Germany SEA can be a valuable tool for the prediction of the underwater-radiated sound of ships, provided that the sound radiation from the hull is modelled correctly. Some models are compared: radiation of the nearfield of the excitation force, edge and corner mode radiation and a novel simple statistical model for the resonant radiation from fluid-loaded, beam-stiffened plates being baffled by the adjacent structure. It is found that the statistical model yields results that capture the overall trend for the radiation, whereas the edge and corner radiation leads to large overestimation. It should be noted, however, that both models rely on baffled conditions that are not necessarily present in-situ. With a provisional correction for the presence of the free surface, the edge and corner radiation is brought to a closer agreement with the experimental data. EXPERIMENTS ON THE UNDERWATER SOUND RADIATION OF A SHIP-LIKE STRUCTURE Recent experimental studies [,] into the use of Statistical Energy Analysis (SEA) as a tool for the prediction of the underwater-radiated sound of ships, have indicated that the reliability of the predictions is limited by the fact that it is not straightforward to model the radiation from the hull correctly. The experiments are undertaken on a single and a triple section ship-like structure, see figure. These consist of cross-stiffened steel boxes (with mm thick hull plates), floating with the waterline at 00 mm from the bottom of the structure. The geometry of plate fields and stiffeners resembles that of the hull structure of a surface ship, though it is not a real scale model. The structure is excited by a shaker at several positions, in a frequency range between 50 Hz and 0 khz. The spatially averaged hull velocity and the sound power radiated into a semi-anechoic water tank are measured using the technique of Nearfield Acoustic Holography [3], see [,] for the details. MODELS FOR SOUND RADIATION FROM THE HULL COMPARED WITH MEASUREMENTS In the most relevant frequency range for the underwater sound radiation, the flexural wavelength in the ship hull is short compared with the wavelength in water. In this range, below the critical frequency, the structural waves only radiate sound from discontinuities in the wave field [4]. Point force radiation One obvious discontinuity is the excitation point. The sound power radiated from the near field of a point force F equals: P rad = k F πρc [5], where ρ, c, and k are the water density, sound velocity and wave number. This expression is valid below the critical frequency if the inertial impedance of the hull is lower than the specific impedance of the fluid. A comparison is presented in figure with the measured total radiated power from the single and triple section structures, for excitation on a bedplate (position in figure ). It is clear that the point force is not the predominant radiator. Note that the radiated power is nearly independent of the size of the structure. Radiation from resonant modes FIGURE. Sketch of the triple section ship-like structure. Sizes are in mm, numbers to 7 refer to shaker positions. The radiation from resonant modes is described by a radiation efficiency σ, the non-dimensional ratio of the radiated power and the average velocity squared v of the hull (of area S): P rad = σ ρcsv.

5 P rad / F [db re W/N ] NAH single section NAH triple section point force σ db re [ ] NAH NAH "statistical" Maidanik/Heckl M/H "dipole" k k 4k 8k frequency [Hz] FIGURE. the total radiated power per excitation force, for the single and triple section structures, compared with the model for point force radiation k k 4k 8k frequency [Hz] FIGURE 3: the radiation efficiency of the triple section structure for excitation positions and (see figure ) compared with different models for resonant radiation The Maidanik/Heckl (M/H) model [4] estimates σ for the radiation from edge and corner modes, below the critical frequency: σ = k U πk 3 f S, where k f is the flexural wavenumber (corrected for fluid loading) and U the total edge length. Figure 3 shows that this model clearly overestimates the measured efficiency. The modal density of the individual panels between the stiffeners is, however, low in a large part of the frequency range, which might invalidate the M/H model. A novel statistical model [6] describes the radiation from N individual hull panels as independent monopoles baffled by the adjacent structure. With the assumption of equipartition of energy over the panels, this model leads to an estimated radiation efficiency 5 σ = k S π N. Figure 3 shows that this expression captures the overall trend of the experimental data. It should be noted that the free water surface around the ship does not enforce the baffled condition that is presumed in the models. A monopole source at a distance z below a free surface radiates sound as a dipole, which radiates less power than a monopole below a rigid baffle by a factor ( ) kz. A provisional correction with this factor [7] would lead to an underestimation for the statistical model, but it brings the M/H model in closer agreement with the experimental data ( M/H dipole ). CONCLUDING REMARKS The experimental data show that the underwater sound radiation from the ship hull is predominated by the radiation from resonant modes. That suggests that SEA is applicable. The set of experimental results for one specific structure, however, does not enable corroboration of any of the models ( statistical and M/H dipole ) to account for the resonant contribution. REFERENCES. De Jong, C.A.F. and Termeer, M.K., Proc. ICSV7, 4-7 July 000, Garmisch-Partenkirchen, Deutschland, pp De Jong, C.A.F. and Van den Dool, T.C., NAG Journaal 53, 000, pp Maynard, J.D., Willliams, E.G. and Lee, Y., J. Acoust. Soc. Am. 78, pp (985) 4. Cremer, L., Heckl, M. and Ungar, E.E., Structure- Borne Sound, Springer Verlag, Berlin Junger, M.C. and Feit, D., Sound, sources and their interaction, MIT Press, Cambridge MA Petersson, B.A.T., Radiation from fluid-loaded plate-like structures with discontinuities, report TPD-HAG-RPT- 0008, TNO TPD, Delft, De Jong, C.A.F., Prediction of underwater sound radiation of ship hulls, report TPD-HAG-RPT-0009, TNO TPD, Delft, 000

6 Solution of Structural-Acoustic Vibration Problems by Complex Envelope Vectorization A. Carcaterra a and A. Sestieri b a INSEAN, Via di Vallerano 36, 006 Rome, Italy b Dipartimento di Meccanica e Aeronautica, Rome University, 0084 Rome, Italy This paper is focused on a new statement of the complex envelope displacement analysis (CEDA). When a discretized model of a complex structure is considered, the nodal unknowns are assembled into a single vector. The key point consists of handling this vector, i.e. the solution of the structural problem, as a sampled form of a continuous signal depending on a single dummy coordinate. In this way the problem becomes formally equivalent to the original CEDA and the same technique of can be applied to get the complex envelope displacement and related governing equations. INTRODUCTION The complex envelope displacement analysis (CEDA) is a very promising and powerful procedure to analyze high frequency problems. As other methods developed in the last decade to circumvent shortcomings typical of statistical energy analysis, CEDA is capable of providing the envelope of the physical solution by solving an appropriate differential equation whose solution is numerically much more convenient than the physical equations. CEDA is originally based on a variable transformation that maps the high frequency oscillations into a quasi-static displacement field []. The method has been successfully developed and applied to one-dimensional systems but the extension to two and three-dimensional problems met several difficulties and requires a general revision of the theory. In this paper a generalization of CEDA is proposed that transforms the higher dimensional problem into a one-dimensional problem. When a discrete model of a two or three-dimensional system is considered, whatever the used discretization, the nodal unknown displacements are assembled into a single vector, hence the name of complex envelope vectorization. The key point is just handling this vector, i.e. the solution of the structural problem as a sampled form of a continuous signal depending on a single dummy coordinate. In this way the problem of any order becomes formally equivalent to the original one-dimensional CEDA, and a similar transformation technique can be applied to obtain the envelope displacement and the related governing equations. Such procedure is presented here and some results referred to a rectangular membrane are compared with results obtained by a finite element solution. MULTI-DIMENSIONAL CEDA: SIGNAL VECTORIZATION CEDA has been developed by the use of the Hilbert transform that operates a suitable transformation of the dynamic variable: it was originally applied to onedimensional systems and the generalization presented here is related to the successful approach used for these systems. Let us consider the discrete formulation of a dynamic problem forced by a harmonic load, whose equation of motion can be written as: [ ] M + K u= f Au = f ω () where M, K are the mass and stiffness matrices, while u, f are the displacement and the external force vectors, consisting of N components. This vector can be thought as the sampled version of a continuous signal u(s) being s a dummy variable, i.e.: u i =u(s i ). By keeping in mind the idea of a discrete system, a multidimensional CEDA can be developed, leading to a direct generalization of the original method. A special variable transformation is introduced, i.e.: + u s u~ ( ξ ) ( s) = dξ, π ξ () j k0 x uˆ ( s) u( s) j u~ ( = + s), u( s) = uˆ( s) e

7 where ũ(s), û(s) and u (s) are the Hilbert transform of the displacement, the analytic signal and the envelope associated to the physical displacement, respectively. The characteristic wavenumber k 0 is related to the excitation frequency ω. It has been proven in [3] that, provided that the physical response of the system is characterized by a band limited spectrum around k 0, the envelope u (s) has a space spectrum characterized by low wavenumber components. This implies that if the equation of motion is given in terms of u(s) instead of u(s), then its numerical solution can be determined by using a coarse mesh. Thus, on the basis of both equations () and (), the governing equation of the envelope displacement must be determined. To this aim we can write: N i= a ui = f h a ( s, σ ) u ( σ ) dσ f ( s) (3) h i = I where I is a suitable interval on the s axis, and f(s) is the continuos counterpart of the vector f. The integral equation (3) is the equation of motion in terms of the signal u(s). By introducing the transform () into (3), after some mathematics one obtains: I a ( s, σ ) e u ( σ ) dσ = f ( j k 0 ( s σ ) s that is the integral equation governing the envelope displacement. f (s) is simply obtained by applying the transformation () to the physical load. By determining a correspondence between the kernels of equation () and (3), a simple way of transforming the dynamic operator of the motion equation is obtained, that provides the transformation of the physical mass and stiffness matrices and the related equation of motion: M i j = M i j e ( ω ) i ) Ki j = K M + K ) u = f j k0 l ( j j k0 l ( j i ) i j e (3) being l the characteristic size of the original mesh. The last is the governing equation of the envelope vector displacement. It must be observed that the obtained envelope matrices have size NxN, equivalent to the size of the original problem. However, when the hypothesis on the spectrum of the physical displacement is verified, the solution of the above equation does not need that the same fine mesh is used, in that the complex envelope exhibits a low wavenumber spectrum characterized by slow space oscillations. This implies that, when considering a T T suitable sub-vector partition of u = { Φ,..., ΦM}, each of the sub-vector has an informative content that can be reduced to a single value. By simple mathematics it can be proved that the matrices M, K can be reduced to size MxM (M<<N) by partitioning the original matrices into square submatrices and by replacing each sub-matrix with a single value obtained by averaging its elements. The given technique leads to new equations that can be solved by a lower computational cost with respect to that required for the original problem (). RESULTS To check the validity of the proposed procedure a square membrane ( x m) is considered, excited by a point force at a frequency of Hz. Figures and show the modulus of the physical response of the membrane and the envelope solution obtained by using a reduced number of points. FIGURE. Physical displacement (modulus) FIGURE. Envelope displacement (modulus) REFERENCES. A. Carcaterra and A. Sestieri, Complex envelope displacement analysis, JSV 0(), 997, pp

8 A confidence factor for SEA results: a comparison between two definitions based on different statistics A. Culla a, A. Carcaterra b, A. Sestieri a a Dipartimento di Meccanica e Aeronautica, Rome University La Sapienza, 0084 Rome, Italy b INSEAN, via di Vallerano 36, 006 Rome, Italy SEA provides the average of the power flow among modal groups of a system. As in any statistical approach, together with the average, some kind of variance would be required for a significant solution. This need was considered by Lyon who developed a computation to provide a confidence estimate, based on the probability distribution of modes in frequency bands. In the present work the analysis of a confidence factor, defined and used to describe the response of a system subjected to uncertainties in the physical parameters, is considered. Some results determined for the confidence factor are quite dissimilar from those obtained by Lyon. A theoretical comparative analysis between the two results is developed, together with Monte Carlo simulations that confirm the previous statement. INTRODUCTION Statistical Energy Analysis (SEA) is a procedure that determines the mean value of the mechanical power flow of complex systems, energy modal subsystem and the vibrational energy stored in each subsystem. SEA is recognized to be particularly appropriate in studying high frequency problems. Although SEA is in principle a statistical approach, it does not provide a complete statistical information. In fact, while a mean value is obtained from SEA, in general no information is obtained for the variance of the confidence level of the result [4]. Actually one must stress that this problem was in fact considered by Lyon in [3, 4] where he provides a normalized standard deviation (σ/m) for the energy of each modal system. To achieve this result he considers a population of similar systems characterized by a random distribution of normal frequencies. Based on previous works on the natural frequencies distribution [], he assumes that such distribution along the frequency axis is a Poisson distribution and computes the standard deviation by using an approach developed by Rice for the statistics of a sum of impulses. In this way he determines the confidence level as the inverse of the normalized standard deviation and shows that the SEA confidence increases with the modal overlap, with the number of modes in the excitation bandwidth and, in practice, when the frequency increases. In the present paper a theoretical and a numerical procedure to provide this missing estimate, i.e. the confidence level, is developed. by assuming that the physical parameters of the similar structures are randomly perturbed. VERSUS AN EFFECTIVE AND COMPLETE STATISTICS ON THE SEA RESULTS In this paper a different point of view from the one presented by Lyon is provided: a stochastic behaviour is imposed to the system's parameters -mass (ρ), stiffness (E) and dimensions- and the statistics of the energy to imposed random perturbations on these parameters is considered. This approach seems to be more suited in the framework of the SEA population, in that it is the stochastic nature of the physical parameters that imposes a random distribution of the natural frequencies. To present the developed approach and the obtained results, two different systems are considered and a confidence factor for the energies is defined as the inverse of the normalized standard deviation. A continuous structure with a random perturbation on the Young's modulus Consider a continuous system (e.g. a beam) excited at location by a point force F. The energy density for unit volume at location is:

9 * = ρv v = ρ M F ε () where ρ is the mass density of the structure and M the mobility between points and. Let us perturb randomly the Young's modulus of the system, E=E 0 (+ε), where ε is a random variable having a Gaussian distribution with zero mean. E 0 is the nominal Young's modulus. Note that if ε is a small perturbation (ε<<), a series expansion of the energy around ε=0 can be performed. The confidence factor of e can be computed analytically: m σ ε ε ε = ε ε= 0 ε ε= 0 σ ε () The analytical expression obtained from equation () is quite complex. However it is possible to study its asymptotic behaviour for ω and to see that the confidence factor shows a constant trend. In figure the confidence factor obtained by a Monte Carlo simulation is drawn (σ ε =0.05). The graphic shows the constant trend analytically predicted. Confidence factor of energy density Circular frequency (rad/s) FIGURE. Confidence factor Ε Two continuous structures coupled by a point mobility junction Consider two structures connected in two points ( and 3) by a massless junction presenting a random perturbed stiffness. Without loss of generality, we can refer to two beams. By considering the energy of the two systems at the junction points we can determine, as in the previous section, the confidence factors and compare them with a Monte Carlo simulation. The theoretical results yield the asymptotic values: lim f c = lim f c3 = (4) ω σ ω In figure the results obtained by the Monte Carlo simulations are presented (σ ε =0.05). Confidence factor Confidence factor 3 E+ 9E+0 8E+0 7E+0 6E+0 5E+0 4E+0 3E+0 E+0 E Circular frequency (rad/s) Circular frequency (rad/s) FIGURE. Coupled structures confidence factors CONCLUSIONS The developed study shows a rigorous analysis of the confidence factors obtained from a population of similar systems when perturbing randomly a physical parameter. These results do not provide similar solutions as those predicted by Lyon and deserve a deeper and detailed investigation. REFERENCES. R. H. Bolt, Normal Frequency Spacing Statistics, The Journal of the Acoustical Society of America, 9(), 75-90, (947).. R. H. Lyon, Statistical Analysis of Power Injection and Response in Structures and Rooms, The Journal of the Acoustical Society of America, 45(3), , (969). 3. R. H. Lyon, R. G. De Jong, Theory and Applications of Statistical Energy Analysis, Butterworth-Heinemann, (995). 4. A. Culla, A. Carcaterra, A. Sestieri, Power flow uncertainties in SEA: a confidence factor based on the theory of probability, Proceedings of International Conference NOVEM 000, , Lyon. ε

10 Vibroacoustic Modeling by Equivalent Acoustic Sources and Application to SEA T. M. Tomilina Laboratory of Structural Acoustics, Mechanical Engineering Research Institute of Russian Academy of Sciences, 0990 Moscow, Russia, A forced vibrating complex system, composed of elastic structure and fluid, is modeled by a set of elementary acoustic sources (monopole or dipole type). The strength of these equivalent acoustic sources is determined from the boundary and interface conditions. The model is the base of the effective algorithm for prediction of the acoustic and vibration fields of the system. Two aspects related to SEA are discussed: an estimation of loss factors and coupling loss factors for subsystems of corresponding SEA model and a determination of the power of vibroacoustic sources with respect to acoustic environment. INTRODUCTION SEA is widely used as a tool for predicting the mean square vibration velocities and acoustic pressures in vehicles, buildings, large machines, etc. [,]. It is based on substructuring a complex system and using the simple equations of the energy conservation law for each subsystem. However, SEA has its problems and many works are currently in progress aimed at further developing the method. One of a problem is a calculation of the subsystem parameters, loss factors and coupling loss factors (CLFs), for realistic structural configurations, especially when various subsystems are coupled through the ambient fluid. An important role of acoustic volumes for energy transmitting was shown on theoretical models in [3-5]. It has been established that geometry and mutual disposition of subsystems not only influence the coupling but also affect a great deal on acoustic power of a noise source: the resonancies both of elastic elements and of acoustic volumes could increase the total power flow (TPF) of an extended source by two orders of magnitude. The Equivalent Sources Modeling (ESM) approach was developed in [4] to study these effects quantitatively, which provides physically clear and numerically effective algorithm and it works in wide frequency range. In this paper ESM approach is applied to vibroacoustic analysis of particular system containing one active element a noise source of compact geometry modeling by vibrating sphere and one elastic thin walled element a circular clamped plate excited by this source via the medium. The main quantities under study are the power flow emitted from the sphere to the medium F s and injected to the plate F p. Physical effects caused by complex coupling between subsystems are discussed. The results may be used in SEA analysis in two ways: to obtain more precise values of the SEA power parameters (TPF of a source) and to evaluate coupling loss factors. ESM-APPROACH The geometry of the linear system under study is presented on Figure. A sphere of radius a and surface S vibrates in a fluid. v 0 (S)exp(-iωt) is the distribution of the normal velocity component specified on S. A circular clamped plate of radius b, thickness h, bending rigidity EI and material loss factor η vibrates in the vicinity (at a distance H from the center) of the sphere under the action of acoustical pressure field p(s ) produced by the sphere. The differential operator L for the bending plate vibrations in vacuum is known (the thickness h is assumed small compared with the flexural wavelength). The values of a power flow emitted from a sphere F s and injected to the plate F p are determined as following: = ) 5H S Y S 6 6 G6. 6 ) V = 5H 6 S Y G6 6 and Here S 6 H[S LωW, S 6 H[S LωW are pressure fields on the sphere surface S and on the upper (S ) and lower (S ) sides of the plate, Y6 is a lateral plate velocity. To define these quantities it is required to solve a coupled acoustoelastic problem that mathematically means to find the solution of the Helmholtz equation, which satisfies the radiation condition at the infinity, the boundary conditions on S, v(s)=v 0, and impedance conditions on S, : L Y 6 = S 6. The ESM-approach suggested for the solution of the problem is composed of three parts: () implementation of the equivalent sources(es) model, () determination of ES model parameters from the

11 boundary conditions, and then (3) evaluation of sound and vibration fields quantities required. v 0 (S) H higher order sources produce more intensive near pressure field which exciting the plate gives greater power flow. It is also seen from Figure that F p increases at eigenfrequences of the plate in fluid and depends on the losses of the plate. The power exchange between source and plate is comparable with the TPF of the source and as it was obtained from numerical results strongly depends on the distance between them. h FIGURE. Geometry and ES-model of a sphere S vibrating near elastic circular clamped plate S in a fluid. In the present problem both the vibrations and pressure fields of the system were modeled by two sets of ES with unknown amplitudes operated in a free space: N linear monopole-type sources located on internal sphere S 0, and N linear dipole-type sources located on the mid surface of a plate and oriented transversely (see Q and Q : on Figure ). The total pressure field of the problem p(r) is thus represented by the sum of the fields: SU = S 4 U _ U _ 4 + S 4 U U 4, which contains N=N +N unknown model parameters - amplitudes of the equivalent sources. Using this representation one can compute amplitudes of the radial particle velocity at M points of the sphere surface S and normal component of velocity at M points of the plate sides, and, after substituting them into the boundary conditions, obtain a set of M=M +M linear algebraic equations with N<M unknowns. The SVD-solution of this set gives the ES-amplitudes in the proposed model and thus all required quantities. In the numerical algorithm the integral form of the impedance conditions on the plate surface was used instead differential one, and the Green s function of the clamped plate in vacuum was represented in the form of expansion in normal modes as it was suggested in [5] and is described in another paper of these proceedings [6]. The accuracy of the numerical algorithm and optimal values of its parameters are also discussed in this paper. The numerical results for the vibration forms of a sphere as Y 6 = Y θ = 3 FRVθ, P Q n is a Legendre polynomial (n=0,...,5), are presented on Figure for the steel plate in water and parameters b/a=, h/a=., H/a=.5, η=.05. The power flow from the sphere to the plate depends on the frequency and that is the most important on the order of spherical harmonic, e.g. type of the source. It is seen that the form of the source vibration affects considerably the power injected into the plate and the more complex the vibration form the greater the power flow. The physical reason is that the FIGURE. Power flow (in watts) into the plate F p (dot lines) and TPF of a sphere F s vs dimensionless frequency ka for different vibration forms v 0 (S), n=0,...,5, v 0 =.0 m/s. The computed power flow from one subsystem to another through the surrounding medium can be used for obtaining the corrected values of the coupled loss factors (CLF) in the SEA. The procedure is the following. For two selected subsystems (sphere and plate in our case) the total energies and power flow are computed. Then assuming that the SEA equations are valid for the two subsystems, the CLFs are identified from comparison with the computed energy characteristics of the subsystems. ACKNOWLEDGEMENTS This research has been supported by RFBR under grant # (program: The leading scientific schools). REFERENCES. Lyon R.H., DeJong R.G. Theory and application of statistical energy analysis. Boston: Butterworth-Heinemann, Fahy F.J., Philosophical Transactions of the Royal Society, A, 346, p (994). 3. Tomilina T.M., Amplification of the total power flow from a noise source operating near elastic bounds, Proceedings of the Inter-Noise 96 Conference, Liverpool, UK, 996, 6, Yashkin V.B., Acoustical Physics, 44(5), (998). 5. Tomilina T.M., Bobrovnitskii Yu.I., Yashkin V.B., Kochkin A.A., J. of Sound and Vibration, 6(), (999). 6. Tomilina T.M. and Yashkin V.B., in this proceedings.

12 Prediction of Sound Propagation in Stiffened Shell Structures with an Energy Finite Element Method C. Cabos a and H. G. Matthies b a Germanischer Lloyd, Vorsetzen 3, 0459 Hamburg, Germany b Technical University, Braunschweig, Germany The performance of an energy finite element method is examined for the case of structure borne sound propagation in stiffened shell structures. An alternative relation between total subsystem energies and the power flow between subsystems is considered. INTRODUCTION P P When integrating noise prediction into the design process of an engineering structure, element finite element methods (EFEM) can be advantagous if the same finite element model can be used for different computations (e.g. strength, vibration, noise). This approach is taken in the method NoiseFEM [] developed at Germanischer Lloyd. NoiseFEM is used for complex stiffened shell structures, in particular ships []. A major concern in the re-use of finite element models in statistical methods for noise prediction Statistical Energy Analysis (SEA) [3] or EFEM is strong coupling. Subsystems with a common junction should be weakly coupled: the transmission coefficient τ i j, which in the wave approach to SEA denotes the ratio between the power P j leaving the junction in subsystem j to the power P i of the wave impinging on the junction in subsystem i, should be small. Here an alternative relation between total subsystem energies and the power flow between subsystems is proposed, which improves the performance of the mentioned methods for strong coupling. THE COUPLING MATRIX In SEA the coupling loss factor η i j relates the difference in total subsystem energies E i and E j to the net power flow between these subsystems P i j net ω η i j E i η ji E j () Together with the internal power loss P d i ωη i d E i within a subsystem i, a global system of equations can be assembled. Given the input power P i into each subsystem this system is then used to solve for the unknown E i. E.g. for a -subsystem model the SEA equations yield A d A c E E P d P d P c P c P P () E E E P E E E P FIGURE. Notation for deriving the coupling loss factor. where η is used to determine the coupling matrix A c : A c E E P c P c with A c ω η η η η (3) In this paper the net power P c i leaving a subsystem due to coupling and its relation to E are regarded. The reason is that equation (), not (), is used to compute the SEA response to external power sources. A c is derived on the basis of the transmission coefficient τ. The following assumptions common to SEA are made: The subsystem response can be decomposed into plane wave components propagating in different directions, and the power and energies of all waves can be summed up incoherently. Proportionality holds between the total energy of plane waves and their total power. Based on these assumptions the response of each subsystem is now split into the waves incident to the junction with total energy E and power flow P orthogonal to the junction and the waves leaving the junction with total energy E and power flow P orthogonal to the junction. Fig. shows the notation for the case of two subsystems. We consider the case of n subsystems coupled at a common junction. Upper case letters without subscripts denote vectors of the subscripted quantities. As in (3) we

13 are looking for a relation between E and P c. In matrix form, the definition of τ i j yields P j τ i j P i or P τp with τ τ i j (4) The matrix of transmission coefficients τ has the reflection coefficients on its diagonal. Therefore the sought net power loss of the subsystems is P c P P I τ P (5) where I denotes the identity matrix. If C is the diagonal matrix containing the proportionality factors between energy and power flow orthogonal to the junction, then e.g. CE P and with (4) we have CE P P I τ P (6) FIGURE. Double bottom test structure, simply supported at the outer edges. The arrow marks the excited plate. The cutting plane is drawn in gray. Combining (6) with (5) results in P c A c E with A c I τ I τ C (7) This form of the coupling matrix is used in NoiseFEM. The SEA coupling relation can be altered accordingly. A relation between η i j and τ i j can be derived from A c in special cases. E.g. in the case of two subsystems with ε : τ i j τ ji we have τ ε ε ε ε ε so A c ε Comparing this result with (3) yields ωη i j C τ i j τ i j C i. This formula is only a consequence of (7) for this case and is not used directly in implementation. Now regard the special case P 0. Then from (7) and (6) P c ε ε C E ε ε ε C E This is in agreement with [3] and [4]. It is not used in NoiseFEM, because P 0 appears to be in conflict with the assumption of a reverberant sound field. NUMERICAL EXAMPLE A plate of a typical ship double bottom structure (see Fig. ) is excited by rain on the roof out-of-plane forces. SEA and NoiseFEM predictions are compared to vibration computations with a detailed finite element model. The vibration predictions are averaged over excitation point and frequency band (00 Hz third octave). Mean square velocity response is averaged over response location within each plate. Fig. 3 shows the results of the comparison along the cut plane given in gray in Fig.. Black lines mark the FIGURE 3. Comparison between different prediction techniques. Scales are in metre and 0 db. cut model, green solid lines show the averaged finite element results, red dashed lines correspond to NoiseFEM results using (7) and blue dot-dashed lines depict the SEA τ response based on ωη i j i j C i. Good performance of the given approach is confirmed by the results from other frequency bands. REFERENCES. C. Cabos and H. Matthies. A method for the prediction of structure-borne noise propagation in ships. In Proc. of the 6th int. congress on sound and vibration. Copenhagen, C. Cabos, C. Worms, and J. Jokat. Application of an energy finite element method to the prediction of structure borne sound propagation in ships. In Proc. of Internoise, Den Haag, R. Lyon and R. DeJong. Theory and Applications of Statistical Energy Analysis. Boston, E. Sarradj. The uncertain relationship between transmission coefficient and coupling loss factor. In Proc. of NOVEM, Lyon, 000.

14 Wavelength Criteria for the Validity of the Energy Finite Element Method for High Frequency Vibrations I. Moens, D. Vandepitte and P. Sas Dept. of Mechanical Engineering, Division PMA, K.U.Leuven, Celestijnenlaan 300B, 300 Heverlee, Belgium This paper discusses wavelength criteria for the validity of the energy finite element method (EFEM) for non-dispersive and dispersive waves in rods, beams and plates. The wavelength criteria are established based on 'modal' validity criteria of SEA from literature and on experimental deduction and experience with EFEM. The results are verified with numerical calculations and linked to the basic assumptions and approximations in the derivation of the basic equations of EFEM. INTRODUCTION The energy finite element method (EFEM) [] is one of the vibrational conductivity approaches that are established as alternatives to the Statistical Energy Analysis (SEA) for modelling of high frequency dynamic behaviour of vibro-acoustic structures. The main advantages of EFEM over SEA are the use of a conventional model description, similar to classical finite element models, and the information on the spatial distribution of vibrational energy throughout the structure. Meanwhile, EFEM keeps the advantage of low computational cost at high frequencies. Over the past few years, several critical reviews are published that doubt the usefulness and general validity of the vibrational conductivity approaches, especially when extended from beams and rods to more dimensional components such as plates [,3]. THE ENERGY FINITE ELEMENT METHOD (EFEM) EFEM in a basic component (beam, plate, ) can be considered as the differential approach to SEA [,4]. The basic equation expresses the equilibrium between input power in and dissipated power diss and energy flow q out of a differential volume, all expressed in terms of (total) vibrational energy density e : cg in diss q e e () with the internal loss factor and c g the group velocity of the considered wavetype in the basic component. Since this equation is similar to the basic equation of static heat conduction, the vibrational conductivity approaches are often denoted by thermal approaches. The coupling of basic components is described based on power transmission coefficients which express the reflected and transmitted power of the different waves incident at the coupling. The full procedure with the use of a special coupling element, is discussed in [,4]. The main assumptions in the derivation of the basic EFEM equations are that potential energy equals kinetic energy, the omission of the nearfield and the interference between different waves (or spatially smoothed results), a plane wave approximation and diffuse field in D and 3D components WAVELENGTH CRITERIA FOR THE VALIDITY OF EFEM Basis for the derivation of a wavelength criterion for the validity of EFEM are modal parameters that give an indication of the validity of SEA [5] and some recent publications on EFEM that derive a wavelength criterion for the validity of EFEM from experiments or experience [6,7]. For each type of component, two wavelength criteria are established for the validity of EFEM based on a non-dimensional wavelength parameter l, that can be thought of as the number of wavelengths that are captured by the component : L l () where L is the chraracteristic dimension of the component and is the largest wavelength of the present waves in the frequency band of interest. The first wavelength criterion for the validity of EFEM is equivalent to the SEA criterion that the modal overlap factor MOF should be larger than unity [5]. For one dimensional components (rods and beams) a criterion is used that states that the non-dimensional wavenumber band kl should be larger than. The wavelength criterion states that the pameter l has a lower limit that is a function of damping. Since wavelengths decrease with frequency, the parameter l increases with frequency and this first criterion implies that the lower frequency limit of the validity region will decrease with higher damping. The second criterion for the validity of EFEM is equivalent to the SEA criterion on the mode count N. The number of modes in the considered frequency band should be high, e.g. minimum 5 modes for plates [5]. The second criterion yields an absolute lower limit of the parameter l that is calculated in case the frequency averaging is done in one-third octave bands. Table summarizes the two criteria for the different wave types, as derived in [].

15 D component D component D component D component non-dispersive wave (longitudinal, torsional) dispersive wave (flexural) non-dispersive wave (longitudinal, shear) dispersive wave (flexural) first wavelength criterion (SEA criterion MOF> or kl>) l l l l second wavelength criterion (SEA criterion on the mode count) l l Table : Summary of the different wavelength criteria for the validity of EFEM l 3 4 l VALIDATION OF THE CRITERIA BY NUMERICAL CASE STUDIES A numerical validation study is performed on EFEM for plates and coupled plates with hysteresis damping. Comparison is made between the total input power to the total dissipated power calculated with the total energy, instead of the (theoretically correct) potential energy. The comparison demonstrates that the second wavelength criterion (equivalent with the mode count) can be explained in the use of total energy in the equations for damping losses instead of potential energy. For high values of the parameter l, the kinetic and potential energy can be assumed to be equal, whereas at low values the differences can be very large. Especially in the case of coupled plates, the observations on a limit value of the parameter l are very close to results that are derived from SEA criteria (see Table.) and values that are reported in literature ([6] : l >.43 for plates). As an example, figure shows the ratio of the input power over the dissipated power for coupled plates at 50 for different damping levels. This figure clearly illustrates the correspondence to the wavelength criterion l >.47 (for flexural waves in the plates) as in Table. input power dissipated power plate dimension l flexural wavelength Figure : Energy balance of plates coupled at 50. From observations on the spatial distribution of the energy in numerical calculation on plates and couplde plates, a relation is found with the first wavelength criterion. In general, the exact results are closer to the EFEM predictions in structures with higher damping since the local variations of the exact energy density become smaller and the EFEM only predicts the spatially smoothed trend. A similar conclusion holds when comparing results at different frequencies: there is a better match at higher frequencies. In structures with low damping values and at lower frequencies, the EFEM solution provides only a mean value, similar to SEA. These observations confirm qualitatively the first wavelength criterion that was derived from the SEA criterion on the modal overlap factor. A final observation from the case studies is that, as reported in literature, the EFEM results tend to underestimate the energy density at the excitation and to overestimate the energy density away from the excitation due to the omission of the direct field in the solution. This observation applies to both the locally smoothed energy density within a plate and the distribution of energy over the coupled plates (total energy level of a plate). REFERENCES. Moens I., On the use and the validity of the energy finite element method for high frequency vibrations, PhD thesis, K.U.Leuven, (00).. Xing J.-T. and Price W.G., Proc. R. Soc. Lond. A, 455, (999). 3. Carcaterra A., Proceedings IUTAM Symposium on Statistical Energy Analysis,3-4 (997). 4. P.E.Cho, Energy Flow Analysis of coupled structures, PhD thesis, Purdue University, (993). 5. Fahy, F. J. and Mohammed A.D., Journal of Sound and Vibration, 58() (99). 6. Gur Y., Wagner D.A. and Morman K.N., Proceedings of the 999 SAE Noise and Vibration Conference, 59-67, ref (999). 7. Vlahopoulos, N. and Garza-Rios, L.O. and Mollo, C., Journal of Ship Research, 43(3) (999).

16 Component-Mode-Based Reduction Techniques for Modeling Low- to Mid-Frequency Vibration and Power Flow in Complex Structures Christophe Pierre and Matthew P. Castanier Vibrations and Acoustics Laboratory, Department of Mechanical Engineering, The University of Michigan, Ann Arbor, Michigan , USA A methodology is presented for generating low order models of vibration and power flow from a finite element model (FEM) of arbitrary size and complexity, as well as for predicting power flow statistics due to uncertainties. The modeling method is based on component mode synthesis, and utilizes a secondary modal analysis reduction technique to further decrease the number of degrees of freedom. This technique results in characteristic constraint (CC) modes, a highly-reduced order basis for capturing the motion of the interface between components and thus for formulating power flow. Next, the variation of (physical or modal) structural parameters is considered for this low-order model, and the ensemble-averaged power flow is calculated using Galerkin s method with quadrature polynomials and locally linear interpolation functions in the uncertain parameters. This statistical treatment provides efficient and accurate modeling of parameter uncertainties, which is critical for mid-frequency vibration analysis. The methodology is demonstrated for a complete finite element representation of a military vehicle structure. The investigation of low- to mid-frequency vibration transmission in complex structures requires the development of new methods for dynamic substructuring and power flow analysis, as well as the derivation of efficient and accurate approximations for statistical analysis. Finite element analysis, while accurate, becomes computationally expensive in the mid-frequency range, due to the tremendous number of degrees of freedom (DOF) needed to capture the shorter wavelengths of vibration. Therefore, in this work, the finite element model (FEM) of the full structure is divided into separate component FEMs, and component mode synthesis (CMS) is used to generate a reduced order model of the global structure. The example considered is the body structure of a military tracked vehicle, whose partitioning is illustrated in Figure. The goal is to investigate vibration transmission from the lower to the upper substructure while each road arm attachment in the lower substructure is subjected to the excitation resulting from track-terrain interaction. The Craig- Bampton method of CMS [] provides an excellent framework for predicting power flow [], since the constraint modes capture fully the motion of the interface between component structures. However, since there is necessarily one constraint mode for each FEM DOF of the interface, the cost of this CMS method is still prohibitive for a sufficiently fine mesh. Consequently, a secondary modal analysis reduction technique (SMART) is developed to further reduce the size of the CMS model. For the purpose of predicting power flow, the SMART approach may be applied to the interface between components. In particular, an eigenanalysis is performed on the constraint-mode partitions of the CMS mass and stiffness matrices. The resultant eigenvectors, which describe the characteristic motion of the interface, are called characteristic constraint (CC) modes [3]. One of the CC modes for the example vehicle structure is shown in Figure, and it is seen to capture vibration transmission in certain regions of the interface. It can be shown that power flow can be calculated accurately by a reduced order model with relatively few CC modes. Note that the concept of characteristic interface motion is motivated by the investigation of how energy is transmitted through the interface between the components. Thus, the present approach provides physical insight as well as computational efficiency for capturing the interface motion. Next, a general statistical treatment is introduced to obtain the variability and statistics of the CC-modebased power flow between the component structures. The problems of uncertainties in the context of structural-acoustic systems are addressed notably by Statistical Energy Analysis (SEA) [5] and many other variations, such as the fuzzy structure theory [6] and Power Flow Finite Element Analysis (PFFEA) [7]. It

17 is assumed in SEA that the system is represented by an ensemble a population of similar systems with statistically-described variations in properties. However, the common practice is to perform spatial and/or frequency average on the forced response of the system. The ensemble average is rarely performed due to the difficulty of evaluating effectively the resulting integration. Monte Carlo simulation is often employed to provide numerically accurate results for the multiple integrals over the uncertain parameter space. However, in general, the Monte Carlo method is computationally expensive. An alternative integration technique is Gaussian quadratures or, in multidimensional cases, cubatures. Some previous studies [8] employed weighted residual methods with those quadrature polynomials to investigate the response variability of general dynamic systems due to the uncertain system parameters. This method is exploited here to compute the ensemble-averaged power flow for the example vehicle structure in Figure, based on its reduced-order CMS model, where the component modal stiffnesses of each substructure are assumed to have uncertain variations from the nominal case. It can be seen from Figure that the results from the approximation agree extremely well with the Monte Carlo results throughout the frequency range. REFERENCES 8. W.D. Iwan and H. Jensen, ASME Journal of Applied Mechanics, 60, (993). FIGURE. The 4th CC mode shape of the upper and lower body structures of a military vehicle.. R.R. Craig and M.C.C. Bampton, AIAA Journal, 6, (968). 0 4 Monte Carlo locally based polynomials. Y.C. Tan, M.P. Castanier and C. Pierre, Characteristic- Mode-Based Component Mode Synthesis for Power Flow Analysis in Complex Structures, 4st AIAA Structures, Structural Dynamics, and Materials Conference, Atlanta, Georgia (AIAA ), M.P. Castanier, Y.C. Tan and C. Pierre, AIAA Journal, 39, 8-87 (00) 4. B.R. Mace and P.J. Shorter, Journal of Sound and Vibration, 33, (000). Power R.H. Lyon, Statistical Energy Analysis of Vibrating Systems, Cambridge: MIT Press, C. Soize, Journal of the Acoustical Society of America, 94, (993). 7. D.J. Nefske and S.H. Sung, Journal of Vibration, Acoustics, Stress and Reliability in Design,, (989) frequency (Hz) FIGURE. Ensemble-averaged power flow from the lower to the upper hull of the military vehicle, calculated using Monte Carlo simulations and the approximation with the locally based polynomials.

18 Vibro-acoustics of beam-plate configurations B.A.T. Petersson and C.A.F. de Jong Institute of Technical Acoustics, Technical University of Berlin, Einsteinufer 5, D-0587 Berlin, Germany. TNO TPD, P.O. Box 55, 600-AD Delft, The Netherlands. In two companion studies [,], the structural dynamic and sound radiation characteristics of beam-plate configurations were treated respectively. The structural configuration consists of an arbitrarily deep beam coupled to a thin plate via both normal and shear stresses, subject to point force excitation either at the free edge of the beam or on the plate at the stiffener crossing. INTRODUCTION Sound radiation from thin, isotropic, homogeneous plates has been comprehensively treated in the past. Also, the radiation from plates, stiffened by slender beams, has received substantial attention and calculation procedures have been developed, eg. [3]. In contrast, the radiation properties of built-up structures involving arbitrarily deep beams and thick plates are not equally well covered. Furthermore, the structural dynamics of combinations of such beams and plates are less amiable for interpretations than those pertaining to slender and thin structures owing to the several wave types involved in the former case and establish thus a relevant research task in themselves. Such structural configurations are found in ships, air- and spacecraft as well as in buildings. From a recent suite of studies concerning the dynamic characteristics of beams and plates subject to various excitation components [4-6], a variety of combinations can be modelled structurally. 0 0 The augmentation required to study the built-up systems radiation properties pertains to the effect of the fluid field. NUMERICAL AND EXPERIMENTAL RESULTS The structural dynamic characteristics can be analysed in terms of the point mobility. In Figure are shown the point force mobilities of a beam-plate configuration where the beam is directly excited at an edge over a length of l. The mobility is normalised with respect to that of a point excited thin plate. It is observed that the plate, of thickness H p, essentially controls the dynamic behaviour up to the dilatational resonance from where the beam, of width t and depth H b, takes over and sets the characteristics. 4 Re[Y vf ]/Y 0 0 Im[Y vf ]/Y Helmholtz number, k T l Helmholtz number, k T l FIGURE. Real (top) and imaginary (bottom) parts of the force mobility versus Helmholtz number for an experimental beamplate configuration without fluid loading, H b l = 75., H p l =, H p t =, η = ( ) measured and ( ) calculated beam-plate configuration. ( ) calculations for a single deep beam. ( ) limit of region of validity of thin plate theory.