BI-ORTHOGONALITY CONDITIONS FOR POWER FLOW ANALYSIS IN FLUID-LOADED ELASTIC CYLINDRICAL SHELLS: THEORY AND APPLICATIONS

Transcription

1 BI-OTHOGONALITY CONDITIONS FO POWE FLOW ANALYSIS IN FLUID-LOADED ELASTIC CYLINDICAL SHELLS: THEOY AND APPLICATIONS L. S. Ledet 1*, S. V. Sorokin 1, J. B. Larsen, M. M. Lauridsen 1 Departent of Mechanical and Manufacturing Engineering Aalborg University, Fibigerstraede 16, Aalborg, DK-9 Eail: lledet1@student.aau.dk Eail: svs@-tech.aau.dk Grundfos Holding A/S Poul Due Jensens Vej 7, Bjerringbro, DK-885 Eail: janballelarsen@grundfos.co Eail: lauridsen@grundfos.co ABSTACT The paper addresses the classical proble of tie-haronic forced vibrations of a fluid-loaded cylindrical shell considered as a ulti-odal waveguide carrying infinitely any waves. Firstly, a odal ethod for forulation of Green s atrix is derived by eans of odal decoposition. The ethod builds on the recent advances on bi-orthogonality conditions for ulti-odal waveguides, which are derived here for an elastic fluid-filled cylindrical shell. Subsequently, odal decoposition is applied to the bi-orthogonality conditions to forulate explicit algebraic equations to express the odal aplitudes independently of each other. Secondly, the ethod is verified against results available in the literature and the convergence is studied as well. Thirdly, the work conducted in the sae references is extended by eploying this ethod to assess the near field energy distribution when the coupled vibro-acoustic waveguide is subjected to separate pressure and velocity excitations. Further, it has been found and justified that the bi-orthogonality conditions can be used as a root finder to solve the dispersion equation. Finally, it is discussed how to predict the response of a fluid-filled shell when the excitation is iported fro CFD-odelling of an operating pup. 1 INTODUCTION This paper is concerned with analysis of fluid- and structure borne noise in piping systes conveying a heavy acoustic ediu, such as water or sewage. The field of acoustical and echanical

2 noise in piping systes share any coon interests e.g. by industrial partners as well as a wide range of esearch Departents around the world. However, the ain issue fro the view point of the co-authors in this paper (Grundfos Holding A/S) is the sound eission and echanical vibrations as these are directly related to cofort and fatigue durability. Consequently, it is essential to have an understanding of the acoustical and echanical vibrations and how to suppress these to accoodate with vibrational deands. This iediately entails a coupled vibro-acoustic odel to cope with the fluid-structure interaction. For the coupled vibro-acoustic waveguide we eploy the classical odels of an infinite thin elastic cylindrical shell and an inviscid copressible fluid with no ean flow. In addition to this the odel is restricted to tie-haronic forced vibrations. This is a classical proble considered in nuerous publications with [1] and [] being the ost cited references on the subject. In these references the acoustic and echanical vibrations are assessed by eans of the energy flow, fro which the doinant path(s) of energy transission can be identified. To analyse arbitrary forcing probles we eploy Green s atrix ethod which is a well-known and widely acknowledged ethod used to recover the response of a given waveguide. This ethod does indeed have a wide range of applications and in relation to the scope of this paper it has been used in [3] to predict the near field energy distribution when applied to fundaental echanical loading conditions. Siilarly, it has been used in [4] to calculate the power flow when applied to acoustical loadings with the scope of apping available CFD-data onto the shell. However, the conventional ethod for forulating Green s atrix has certain convergence and accuracy liitations in predicting the near field energy distribution of such coupled vibro-acoustic waveguides. Thus the otivation of this paper is to develop a sustainable ethod for accurately and efficiently forulating Green s atrix. The ethod developed throughout this paper is based on the recent advances on bi-orthogonality conditions for elastic ulti-odal waveguides. These conditions have been facilitated in [5] to calculate the odal aplitudes of a straight elastic layer and siilar in [6] to solve the classical ayleigh-lab proble by eans of odal decoposition. The latter reference is based on the work conducted by Achenbach in [7], fro which the work of [8] also arises. Here the concept has been adapted to siple elastic ulti-odal waveguides again with the scope of calculating the odal aplitudes by eans of odal decoposition. In the present paper the work fro the latter reference is extended to the ore coplex elastic ulti-odal waveguide of a cylindrical shell conveying heavy fluid. The bi-orthogonality conditions for this waveguide are derived here and facilitated to decopose the equation syste into a set of explicit algebraic equations for calculating odal aplitudes for, in general, five fundaental loading conditions. The paper is structured as follows: Section presents the governing equations, the orthogonality condition and the power flow equations for the fluid-filled shell. In section 3 the odal decoposition ethod is derived, starting with the derivation of the bi-orthogonality conditions and subsequently the forulation of Green s atrix for the fundaental loading conditions. Section 4 is devoted to justify of the odal decoposition ethod through a verification with respect to the work done in [3] and likewise study the convergence of the power flow for acoustical loadings. Section 5 contains an extension to the work in [3] by analysing the near field energy distribution with a converged pressure and velocity excitation applied at different locations along the radii. This section also contains a brief discussion of other useful properties of the bi-orthogonality conditions. Finally, section 6 briefly outlines the procedure for apping CFD-data onto the vibro-acoustic odel. GOVENING EQUATIONS To assess vibrations of a fluid-filled shell we eploy the standard forulation for the fluid-structure interaction proble for a thin elastic cylindrical shell loaded by an inviscid copressible fluid. The detailed derivations of the equations of otion fro the action integral can be found in e.g. [3]

3 The free vibrations of a fluid-loaded shell are described through the following syste of equations in the fraework of Novozhilov-Gol denveiser s shell theory. u x + 1 ν u 1 + ν v x ν w x + ρ str(1 ν ) E u t = 1 + ν u x 1 ν v x + v h (1 ν) 1 w + h 3 1 w 4 h ( ν) w + ρ str(1 ν ) 1 x E v x + h 1 4 v + v t = (1) ν u x + v + h 3 1 v 4 h ( ν) v 1 x + 1 w + h 4 w 1 x 4 h w + h 4 1 x 1 w 4 ρ str(1 ν ) w + ρ fl(1 ν ) φ = E t Eh t The fluid s otion is governed by the velocity potential, φ (x, r, t), through the standard wave equation presented in () with cylindrical coordinates. φ x + φ r + 1 r φ r r φ 1 φ = () c fl t and the continuity condition at the fluid-structure interface is given as φ r = w t In these equations the -spectru has been uncoupled by utilising the axial syetry of a cylindrical shell. The subscript,, represents the circuferential wave-nuber. The axial, circuferential and radial displaceents of the shell-surface are represented by u, v and w, respectively and are all axial x and tie t dependent. The echanical properties of the shell are governed by Young s odulus E, aterial density ρ str and Poisson s ratio ν, whereas the fluid properties are governed by the fluid density ρ fl and the fluid sound speed c fl. Finally, the geoetry of the shell is defined by the radius and the thickness of the shell h. To analyse the properties of free waves we derive the dispersion equation fro (1-3) by eans of the Fourier ethod and eploy a space and tie-haronic ansatz, exp(kx iωt), and in addition, a linear cobination of Bessel-functions for the velocity potential. In the ansatz; ω is the angular frequency and k the axial wave-nuber. By substitution of the ansatz into the governing equations the linear differential equations reduce to linear algebraic equations fro which the dispersion equation is found as the condition providing a non-trivial solution to the hoogeneous linear equation syste i.e. when the deterinant is zero. Confer to the chosen ansatz the equations are apped into the frequency doain and the wavenubers found at discrete frequencies as illustrated in Figure 1 for a steel-shell filled with water. The associated shell properties are given as: =, h =.35, E = 1 GPa, ν =.3, ρ str = 78 kg, ρ 3 fl = 1 kg, c 3 fl = 144 and vibrating in breathing ode, =. s For further analysis, we introduce odal coefficients defined as the ratios of aplitudes of the axial and circuferential displaceents to the aplitude of the radial displaceent. These are referred to as α and β, respectively, and are found analytically by solving any two equations of (3)

4 k I k e ω ω Propagating Decaying * Attenuating Figure 1. Dispersion of free waves for a fluid-filled shell vibrating in breathing ode, =. the linear syste. The derivation of the dispersion equation and the odal coefficients can also be found in e.g. [3]. In addition to the continuity and otion equations the action integral includes a set of individually related generalised forces and displaceents which are involved in the forulation of alternative boundary/loading conditions. The forces and displaceents are related through the following copleentary couples. Q 1 u Q v Q 3 w Q 4 w p ϑ (4) where p is the acoustic pressure, ϑ the axial fluid velocity defined fro the velocity potential as; ϑ = φ and the generalised displaceents, (u,v,w), are defined by the ansatz with their x generalised force counterparts also derived in [3] and defined as Q 1 = Eh [ u 1 ν x + ν v + ν ] w [ 1 ν Q = Eh 1 ν u + 1 ν [ Q 3 = Eh3 3 w 1(1 ν ) x [ 3 Eh 3 w Q 4 = 1(1 ν ) x p = ρ fl φ t ( ν) v x + h 1 w x ν w ν v (1 ν) v x + h 1 ] ( ν) v x ] (1 ν) ] w x Furtherore, it can be shown that the free waves are orthogonal to each other and based on any two solutions to the governing equations in (1-3) we can apply the reciprocity theore and derive the condition of orthogonality for a fluid-filled shell as shown in equation (6). (5) Q (n) 1u (j) + Q (n) v (j) + Q 3w (n) (j) + Q (n) 4w (j) + i ω Q (j) 1u (n) Q (j) v (n) Q (j) 3w (n) Q (j) 4w (n) i ω p (n) ϑ (j) rd r p (j) ϑ (n) rd r = (6)

5 In the latter equation and hereafter we introduce the index-notation of (6) where the superscript refers to the axial wave-nuber and the subscript to the circuferential wave-nuber. Furtherore 1 we introduce a scaling to the axial wave-nuber and the radial variable such that; k di = k and r di = r, however tilde is oitted in the following for brevity. Fro the reciprocity relation used in the forulation of the orthogonality condition we can generalise to the power flow forulation by following the procedure in e.g. [1 3]. The individual power flow coponents given in (7) are expressed in ters of their generalised forces and displaceents. N Axial N T orsion N Bending πω = γ πω = γ πω = γ N F luid = γ π [e(q 1 )I(u ) I(Q 1 )e(u )] [e(q )I(v ) I(Q )e(v )] [e(q 3 )I(w ) I(Q 3 )e(w ) (7) +e(q 4 )I(w ) I(Q 4 )e(w )] [e(p )I (ϑ ) I(p )e (ϑ )] rdr N T otal = N Axial + N T orsion + N Bending + N F luid where γ = =, γ = 1 and N Axial is the power flow carried in the axial coponents, N T orsion the power flow carried in the circuferential coponents etc. 3 MODAL DECOMPOSITION METHOD AND GEEN S MATIX The governing equations presented in the previous section describes free vibrations for elastic fluid-filled cylindrical shells considered as ulti-odal waveguides. This part of the analysis is a very well established subject and is carefully explored in e.g. [1, ]. In contrast to the free vibrations it is usually of higher practical interest to treat forcing probles in fluid-filled shells. The general forcing proble is forulated fro the governing equation (1) and () by introducing non-zero right-hand-sides, corresponding to echanical and acoustical forcing, respectively. In this paper the forcing probles are solved by eans of Green s atrix ethod and to ensure that any arbitrary forcing on the waveguide can be recovered, Green s atrix is forulated for five fundaental loading conditions (axial, circuferential, radial, flexural and acoustical) at each circuferential wave-nuber. The conventional ethod of forulating Green s atrix is based on solving a syste of coplex siultaneous equations. Initially, the nuber of waves included in Green s atrix is truncated to M-waves and the nuber of syetry conditions expanded accordingly to obtain M-equations associated with the M-waves. In [3] this is done by applying Galerkin s orthogonalisation procedure and orthogonalise e.g. the total pressure against individual velocity profiles. The coplexity of this equation syste does indeed affect the nuber of waves which can be retained in the analysis and in [4] only 7 are included to assess the response for acoustical excitations. Unfortunately, 7 waves ay not always provide a converged field and with a threshold in the range of 8 1 waves for this ethod, we are highly otivated for developing a new ethod providing an efficient forulation of Green s atrix. This can be achieved by eans of the recent advances on bi-orthogonality conditions and a ethod based on odal decoposition which is outlined in the following

6 3.1 Bi-orthogonality conditions The bi-orthogonality conditions provide great potential for solving probles of tie-haronic forced vibrations in ulti-odal waveguides. The bi-orthogonality conditions for a straight elastic layer has been derived in [5] and in [7] for an infinite elastic layer of unifor thickness. Fro this convenient reforulation of the conventional orthogonality condition the solution to the forced tie-haronic ayleigh-lab proble is forulated by eans of odal decoposition in [6] and the odal aplitudes can thus be found independently. The concept of bi-orthogonality is then further adapted to siple waveguides in [8] e.g. to an elastic cylindrical shell, also with the scope of calculating the odal aplitudes based on uncoupled algebraic equations rather than as a syste of siultaneous equations. The derivation of the bi-orthogonality conditions are based on discarding the case when k n = ±k j fro the conventional orthogonality condition and arrive at two equations valid only for wave-nubers of different agnitude. Thus we are able to split the conventional orthogonality condition into two equally valid conditions, which, fortunately, always holds true for ulti-odal waveguides. Further by subtraction of the two conditions we conveniently recover the original orthogonality condition of (6). Effectively the bi-orthogonality conditions can be interpreted as an advanced forulation of the reciprocity relation for ulti-odal waveguides, which accounts for the fundaental syetry properties of free waves. This is treated in detail in [7 9]. If we introduce an acoustic ediu inside the shell treated in [8] the coplexity of the equation syste increase significantly and the derivation of the bi-orthogonality conditions becoe too coprehensive to be shown here. However, the necessary atheatical anipulations are siilar to those conducted in [8] for the siple waveguides. Based on these anipulations the bi-orthogonality conditions for an elastic fluid-filled cylindrical shell are derived as in (8), to the authors knowledge, for the first tie. B1 : B : Q (n) 1u (j) + Q (n) 4w (j) + i ω n j (8) p (n) ϑ (j) rdr = Q (j) v (n) + Q (j) 3w (n) for k n ±k j (8) where the second condition is recovered siply by interchanging the indices of the first condition. 3. Forulation of Green s atrix In the forulation of Green s atrix the infinite shell is divided into two sei-infinite shells separated at the excitation point, ξ. The coplete response of both sei-infinite doains are illustrated in (9) by introducing the odule, x ξ, in the ansatz. u N (x, ξ) = v N (x, ξ) = w N (x, ξ) = φ N (x, ξ, r) = u N(j) = v N(j) = w N(j) = φ N(j) = α (j) C N(j) β (j) C N(j) ( ) k (j) exp x ξ ( ) k (j) exp x ξ ( ) (9) k C N(j) (j) exp x ξ iωc N(j) ( J κ (j) r ) [ ( dj κ (j) r ) dr r=1 ] 1 ( ) k (j) exp x ξ

7 where κ is the radial wave-nuber, M the nuber of waves included, J the Bessel-function of first kind of order and N indicates the five fundaental loading conditions, N = 1,..., Calculating odal aplitudes for echanical excitations Let us consider the third fundaental loading condition; a radial point source on the shell wall, applied at x = ξ and distributed with the wave-nuber in the circuference as cos(θ). Obviously, the sei-infinite doains ust satisfy continuity across the excitation point and a unit-jup in the real part of the excited coponent. This is ensured if the loading conditions in (1) are satisfied. u 3 (x, ξ) = w 3 (x, ξ) = ϑ 3 (x, ξ, r) = Q 3 (x, ξ) = Q 3 3(x, ξ) = 1 sgn(x ξ) for x ξ (1) Now, if we ultiply each loading condition by their copleentary generalised counterparts for a single ode, such that u 3 is ultiplied by the odal force Q 3(n) 1, w 3 by Q 3(n) 4 etc. and su the loading conditions according to the bi-orthogonality conditions we arrive at the algebraic equation of (11). Notice that the equations presented in the following are only valid at the point of excitation i.e. for x ξ, which is however oitted fro the equations for brevity. Q 3(n) 1 u 3 + Q 3(n) 4 w 3 + i ω p 3(n) ϑ 3 rdr Q 3 v 3(n) Q 3 3w 3(n) = 1 sgn(x ξ)w3(n) (11) where n indicates the n th -odal generalised force/displaceent and can be chosen arbitrarily. By substitution of (9) into (11) we get [ Q 3(n) 1 u 3(j) Q 3(j) v 3(n) + Q 3(n) 4 w 3(j) + i ω ] Q 3(j) 3 w 3(n) p 3(n) ϑ 3(j) rdr = 1 sgn(x ξ)w3(n) This equation can now be siplified to (13) by applying the bi-orthogonality conditions fro (8) as these explicitly states that for k n ±k j the suation equates to zero. Thus we swap j with n and arrive at M-uncoupled equations, n = 1,..., M. (1) Q 3(n) 1 u 3(n) + Q 3(n) 4 w 3(n) + i ω Q 3(n) v 3(n) p 3(n) ϑ 3(n) rdr Q 3(n) 3 w 3(n) = 1 sgn(x ξ)w3(n) (13) The interpretation of the latter equation is that each odal contribution is found individually and effectively the odes ust for an independent set. Hence by eploying the assuption of odal decoposition we arrive at the M algebraic equations of (13) and confer to the ansatz (using the Fourier ethod) the odal aplitudes can be expressed in the following siple for. C 3(n) (ω) = 1 sgn(x ξ) [ F (n) (ω) ] 1 (14)

8 where F (n) (ω) is given as the left-hand-side of (13) with the aplitudes sorted out of each individual ter as seen in (15). F (n) (ω) = (Q 1 u ) : [ Eh sgn(x ξ) 1 ν [ 1 ν (Q v ) : Eh 1 ν k (n) α (n) + ν β(n) + ν ] α (n) 1 (sgn(x ξ)k(n) β (n) α (n) ) + sgn(x ξ) h (1 ν) k ( (n) β (n) )] β (n) (15) [ Eh 3 1 ( (Q 3 w ) : + sgn(x ξ) ) ] k (n) 3 ( ν) k (n) ( ν) k (n) β (n) 1(1 ν ) [ (Q 4 w ) Eh 3 1 ( : + ) k (n) ν 1(1 ν ) ν ] sgn(x ξ) β(n) k (n) [ ] (p ϑ ) : sgn(x ξ)ρ fl ω k (n) dj (κ (n) r) dr J (κ (n) r) rdr r=1 As a final reark, notice that k (n) and κ (n) are iplicit functions of ω and and further that the integral in (15) is Loel s integral, which reduce to a siple algebraic expression as well. Thus we have derived a siple, purely algebraic, equation, fro which all odal aplitudes can be deterined by substitution of the angular frequency, the circuferential wave-nuber and the associated axial wave-nuber. For the reaining echanical loading conditions the procedure is copletely analogue and the governing equations for calculating the odal aplitudes becoe Q 1(n) 1 u 1(n) + Q 1(n) 4 w 1(n) + i ω Q 1(n) v 1(n) p 1(n) ϑ 1(n) rdr Q 1(n) 3 w 1(n) = 1 sgn(x ξ)u1(n) (16) 1 (16) = 1 sgn(x ξ)v(n) 1 4 (16) = 1 sgn(x 4(n) ξ)w By coparing these with (13) it is seen that only the right-hand-side change and consequently this allow us to deterine the reaining echanical odal aplitudes based on the odal aplitudes fro the radial loading condition alone. C 1(n) (ω) = α (n) C 3(n) C (n) (ω) = β (n) C 3(n) C 4(n) (ω) = (ω) (ω) (17) sgn(x ξ) k (n) C 3(n) (ω) Thus the odal aplitudes for all echanical excitations are deterined based on the single algebraic equation of (14) and in addition siple ultiplication operations. 3.. Calculating odal aplitudes for acoustical excitations For the acoustical excitations the source is introduced as a ring source profiled as the echanical sources in the circuference, concentrated at x = ξ and applied at the radius r. To evaluate the

9 difference in the near field energy distribution between applying a pressure and a velocity ring source both sources are applied as excitations in this paper. The acoustical ring source excitations are represented by the delta-function, δ(r r ), and due to the radial and axial dependence of the velocity potential the loading is introduced as in (18) in contrast to the echanical excitation (radial independent) shown in (1). p 5 (x, ξ, r) = iωρ fl 1 r δ(r r )sgn(x ξ) ϑ 6 (x, ξ, r) = 1 r δ(r r )sgn(x ξ) for x ξ (18) where the velocity excitation can be interpreted as a onopole source and as the pressure is a noral directed force the pressure excitation can be interpreted as a dipole source when the signfunction is included. By the sae analogy the governing equations for calculating the odal aplitudes for acoustical excitations are given in (19). Q 5(n) 1 u 5(n) + Q 5(n) 4 w 5(n) + i ω Q 5(n) v 5(n) p 5(n) ϑ 5(n) rdr Q 5(n) 3 w 5(n) = sgn(x ξ) ρ fl r 5 6 = sgn(x ξ) ρ fl ϑ 5(n) δ(r r )ϑ 5(n) rdr r=r (19) (19) = isgn(x ξ) ω p6(n) r=r () and by substitution of the generalised pressure and velocity the odal aplitudes are deterined as ( C 5(n) (ω) = ik (n) ωρ fl J κ (n) r ) [ ( dj κ (n) r ) ] 1 r=r C 3(n) (ω) dr r=1 ( C 6(n) (ω) = isgn(x ξ) ωρ fl J κ (n) r ) [ ( dj κ (n) r ) ] 1 (1) r=r C 3(n) (ω) dr By inspection of the odal aplitudes for the acoustical excitations it is evident that the relation between these and the odal aplitudes for the echanical excitations are once again just a siple ultiplier. In conclusion, the odal decoposition ethod allow us to express the odal aplitudes for all fundaental loading conditions explicitly and through a convenient analytic algebraic equation, dependent on the angular frequency, the circuferential wave-nuber and the associated axial wave-nuber. Based on this forulation we are able to include an arbitrarily large nuber of waves in Green s atrix provided that all wave-nubers associated with and ω are available. Furtherore, this ethod is very strong as the odal aplitudes are found independent of the nuber of waves we wish to include in the analysis, however as a coproise the syetry conditions are satisfied only as M and the convergence of this ethod is thereby related to convergence of the syetry conditions rather than convergence of the odal aplitudes. r=

10 4 VEIFICATION AND CONVEGENCE To validate the odal decoposition ethod the obvious choice would be a qualitative experiental set-up, however due to tie constraints this has not been conducted for fluid-filled pipes at this point. Nonetheless, as the conventional ethod is well established and accepted, the odal decoposition is validated against the results obtained in [3] for echanical excitations. For the acoustical excitations no paper contains, to the authors knowledge, a converged near field energy distribution for acoustical excitations and therefore these are validated through a convergence study. 4.1 Verification of the odal decoposition ethod To ensure that the odal decoposition ethod is valid for echanical excitations, it is copared to the power flow graphs presented in [3] using the sae physical paraeters (presented in section of this paper) and with waves included. Through a coparison between the graphs illustrated in [3] pp , Fig. 15 and 16, for three echanical excitations and the graphs illustrated in Figure, it is evident that the near field distributions are all identical. This effectively eans that we can obtain converged energy distributions for echanical excitations in fluid-filled shells with a low nuber of waves retained. adial - f =.3 khz adial - f = 67.8 khz Power flow - [N/s] x x Longitudinal - f = 67.8 khz Tangential - f = 67.8 khz Power flow - [N/s] x Total Axial Torsion Bending Fluid Figure : Near field energy distribution with waves included, = 3, and evaluated at f =.3 khz and f = 67.8 khz - just after the first and second cut-on frequency. The verification graphs can be found in [3] on pp , Fig. 15 and 16. In [4] the energy distribution is considered for acoustical excitations as well, however with only 7 waves included. Through the convergence study in section 4. it is clear that 7 waves does not provide a converged energy distribution and confer to the differences between the conventional ethod and the odal decoposition ethod, it can not be justified to copare the energy fields of the two, even though the sae nuber of waves are included x

11 This is siply due to the fact that the essential conditions fro the view point of the conventional ethod is syetry, whereas the essential part fro the view point of the odal decoposition ethod is each odal contribution independent of each other. Thus in the conventional ethod the syetry conditions are ensured by choosing the aplitudes artificially through a weighted averaged where the weighting is deterined by the physical paraeters of the shell. Consequently, for the non-converged case illustrated in [4] the aplitudes are scaled unintentionally corresponding to the weighting, hence the aplitudes will deviate fro those calculated by eans of the odal decoposition ethod and a coparison would be pointless. 4. Convergence study To ensure that the odal decoposition ethod is valid for acoustical excitations as well the convergence of the syetry conditions are studied in the following, however, only presented here for the acoustical velocity excitation at the excitation frequency, f = 66.7 khz. Initially, it is of interest to validate that the load converges towards the delta-function at the point of excitation. In Figure 3 the distribution of the fluid velocity, ϑ (x, ξ, r), is shown across the radii for an increasing nuber of waves at = 3 and evaluated at the point of excitation, x = ξ, with the delta-function applied at r =.5. Figure 3: Convergence of the applied excitation, ϑ(r), across the radii for an increasing nuber of waves. Noralised with respect to the intensity at the excitation point, r =.5. Based on this figure it is easily verified that for an increasing nuber of waves the source distribution resebles the delta-function. Siilarly, this also holds true for all other circuferential wave-nubers and locations of the source. With convergence of the load verified it is of interest to study the convergence of the syetry conditions. Through the derivation of the odal decoposition ethod each odal contribution is derived explicit and independent on the nuber of retained waves, hence the study of convergence is related to the syetry conditions and not the aplitudes. This iediately provides us with four distinct and readily available convergence paraeters, for which we can deterine a threshold based on desired nuerical accuracy. To verify that the odal decoposition ethod is applicable for acoustical loadings as well, the four syetry conditions for the acoustical velocity excitation located slightly fro the fluid-structure interface, at r =.95, are shown in Figure 4 for an increasing nuber of waves. Fro the figure it is seen that Q and u converges ust faster than Q 3 and w, which can easily be visualised by plotting the generalised forces and displaceents as a continuous function of the wave-nuber. The sae can be conducted in case of a pressure excitation by visualising their syetry conditions, w and Q 4, continuously along the wave-nuber. Fro the physical view point the slow convergence of these flexural coponents are governed by the distinct differences in the ebrane and flexural stiffness and effectively the flexural coponents are ore sensitive to the acoustical sources. In conclusion the convergence paraeters reduce conveniently fro four to two paraeters

12 Figure 4: Convergence of the syetry conditions for an acoustical velocity excitation located at r =.95. Furtherore, it is clear that the syetry conditions converge non-onotonic towards zero and through siilar studies it is found that the oscillating behaviour holds true independent of the source location and circuferential wave-nuber. Fortunately, the forces and displaceents will oscillate in a siilar anner causing coon intersections points as indicated in Figure 4. Thus it ay be advantageous to choose the nuber of waves for future analysis based on a convergence study of the generalised forces and displaceents as this will provide the best coproise between a low nuber of waves but converged syetry conditions. On the other hand, this will not necessarily provide a converged velocity potential. Nevertheless, for the case shown in Figure 4 with the physical paraeters of section, the excitation located at r =.95 and excited at 66.7 khz, a qualitative nuber of waves for the proceeding analysis will be M = 5. For echanical excitations the convergence study is also conducted by eans of the syetry conditions and further, for echanical excitations, it applies that the introduced load ust be recovered by suation of all odal forces in the loaded direction. Thus for echanical excitations there are five distinct convergence paraeters and dependent on the physical paraeters these are converged with 7 1 waves retained. In conclusion, the difference in the nuber of waves necessary to obtain a converged energy field for echanical and acoustical excitations are anticipated to be a consequence of the load introduction. For echanical excitations we introduce a finite-valued source but for the acoustical excitations the load is introduced as a delta-function and in effect it will excite a wider spectru of waves with a significant aplitude - including higher order odes as well. 5 ESULTS AND DISCUSSION To illustrate advantages of the proposed ethodology we present here results of soe case-studies. 5.1 Analysis of acoustical excitations of an elastic fluid-filled cylindrical shell In extension to the previous section we investigate perforance of the shell considered in [3] in case of acoustical excitations i.e. with pressure and velocity sources. In Figure 5 the near field energy distribution is shown for the velocity and pressure excitations applied at r =.95 - both noralised with their respective total energy

13 Figure 5: Near field energy distribution for velocity and pressure excitations at r =.95 with an excitation frequency of 67.8 khz at = 3 and 5 waves retained. Noralised individually with the total energy. This energy distribution reveals that with a velocity excitation 7-9% of the energy is trapped in the fluid, whereas with the pressure excitation the fluid energy escapes rapidly to the shell in ters of bending energy, which then carries 6-9% of the energy. In effect a pressure excitation close to the interface is coparable to a echanical loading and has a distribution siilar to that of a flexural excitation, however with another near field characteristic. On the other hand, as the source is located further fro the interface the energy distribution with a pressure excitation turns towards the distribution with a velocity excitation as illustrated in Figure 6. At this location approxiately 8% of the energy reains in the fluid for both excitations, when the sources are located 1 fro the fluid-structure interface. However, in the pressure excitation 4 with a slightly ore pronounced oscillating fluid energy coponent but as the source oves closer to the centre the oscillating behaviour of the fluid energy vanishes and the energy distribution fro the two sources becoe siilar. Figure 6: Near field energy distribution with velocity and pressure excitations 1 fro the fluidstructure interface, an excitation frequency of 67.8 khz at = 3 and 5 waves retained. Nor- 4 alised individually with the total energy. As a concluding reark, it is also possible to evaluate the energy distribution when the source is located in the centre, r =, as the velocity potential is defined to ensure a bounded solution at r. At this point the source transfors fro a ring source to a point sources and if the energy distribution for such an acoustical excitation is evaluated it is seen that the energy is carried

14 solely in breathing ode and further, that it is coparable to a longitudinal echanical excitation vibrating in breathing ode. The overall difference is siply that the energy travels inside the fluid rather than in the axial coponent of the shell. 5. Other useful properties of the bi-orthogonality conditions As deonstrated, the bi-orthogonality conditions are highly relevant in calculating the odal aplitudes for a set of fundaental loading conditions and effectively in the forulation of Green s atrix as well. On the other hand, through the developent of this ethod it has been discovered that the bi-orthogonality conditions also contains potential in finding the wave-nubers, which conventionally are found fro the dispersion equation. What has been detected through the work docuented in this paper is actually that certain fluid-originated roots are hidden in the dispersion equation, eaning that the convergence rate of these wave-nubers is very poor. Thus the bi-orthogonality conditions can be facilitated as an alternative forulation for finding these roots. The only requireent for one of the bi-orthogonality conditions to act as a replaceent for the dispersion equation is that an arbitrary wave-nuber, k j, which satisfies the governing equations of (1-3) is available. The idea is siply to calculate any other wave-nuber, k n, by require that it satisfies the bi-orthogonality conditions with respect to the already known wave-nuber, k j, i.e. the wave-nubers ust be bi-orthogonal so to say. Substituting the frequency and known wavenuber into equation (8) will provide us with a continuous function in k n which sees to be ore sooth than the original dispersion equation and ay therefore be convenient fro the view point of convergence rate. What is however interesting is that the characteristic of this equation change according to whether the known wave-nuber is substituted into k j or k n and siilar, through a concise investigation, it sees that specific sets of wave-nubers becoe ore pronounced in the equation depending on whether the known wave-nuber is purely iaginary, purely real or coplex. This iediately entails for a further investigation to clarify whether wave-nubers are ore pronounced in the bi-orthogonality conditions. 6 ESPONSE OF A FLUID-FILLED SHELL TO THE EXCITATION DEFINED BY DATA FOM CFD-MODELLING OF AN OPEATING PUMP The practical purpose of this paper is to reliably predict the vibro-acoustic perforance of a pup. To accoplish this, the output fro an extensive CFD analysis of an operating pup should be apped onto the vibro-acoustic odel developed in this paper. Thereby the near field energy distribution can be assessed when the waveguide is subjected to certain CFD velocity or pressure profiles. This conveniently allows for a source characterisation and thorough analysis of the transission path(s) for such pressure and/or velocity pulsations. Initially, the tie-dependent CFD-data needs to be transfored fro the tie-doain into the frequency doain, where our vibro-acoustic odel is defined. This transforation is to be conducted for each nodal value of the CFD-data and is done by eans of a (fast) Fourier transforation. The succeeding step is to adjust this frequency dependent data to the odal decoposition of the odel. The CFD-data is then decoposed in the circuferential direction at individual circuferential wave-nubers through a Fourier series. Thus we decopose the CFD-data into separate branches of data for each retained circuferential wave-nuber. Hence if the branches are suarised over the retained circuferential wave-nubers we recover the original CFD-data in the frequency doain. Then by siilar data processing for the radial distribution at each circuferential wave-nuber we are able to predict the acoustical sources in the CFD-data and evaluate the vibro-acoustic per

15 forance of an operating pup by eans of Green s atrix ethod. This is the subject of our on-going work. 7 CONCLUDING EMAKS Through this paper a highly efficient tool, the bi-orthogonality conditions, has been facilitated to decopose the governing equation syste which allows us to express the odal aplitudes explicitly and independently of each other via the excitation frequency, circuferential wave-nuber and the associated axial wave-nuber. Further, siple relations between the odal aplitudes fro different fundaental loading conditions has been detected, which effectively reduce the equations for calculating the odal aplitudes of all fundaental loading conditions to only one siple algebraic equation and siple ultiplication operations. This very strong ethod directly allows for assessing the near field energy distribution and obtain converged solutions for both echanical and acoustical excitations. To validate that the odal decoposition ethod applies, the case of echanical excitations are verified against identical loadings in [3] and for the acoustical excitations no qualitative reference odels have been found, hence these loadings are validated through a convergence study of the applied delta-function and the syetry conditions. Through the developent of this ethod it has also been discovered that the bi-orthogonality conditions ay be utilised as a root-finder in favour of the conventional dispersion equation. At this point, this is however not investigated in substantial detail and is thereby left for future works. EFEENCES [1] G. Pavić. Vibrational energy flow in elastic circular cylindrical shells. J. Sound and Vibration, 14:93 31, 199. [] C.. Fuller and F. J. Fahy. Characteristics of wave propagation and energy distributions in cylindrical elastic shells filled with fluid. J. Sound and Vibration, 81:51 518, 198. [3] S. V. Sorokin, J. B. Nielsen, and N. Olhoff. Green s atrix and the boundary integral equation ethod for the analysis of vibration and energy flow in cylindrical shells with and without internal fluid. J. Sound and Vibration, 71: , 4. [4] S. V. Sorokin and A. V. Terentiev. Flow-induced vibrations of an elastic cylindrical shell conveying a copressible fluid. J. Sound and Vibration, 96: , 6. [5] Y. I. Bobrovnitskii. Orthogonality relations for lab waves. Soviet Acoustical Physics, 18:43 433, [6] Baruch Karp. Generation of syetric lab waves by non-unifor excitations. J. Sound and Vibration, 31:195 9, 8. [7] J. D. Achenbach. eciprocity in elastodynaics. Second. Cabridge University Press, 3. [8] Sergey V. Sorokin. On the bi-orthogonality conditions for ulti-odal elastic waveguides. J. Sound and Vibration, 33: , 13. [9] D. D. Zakharov. Orthogonality of 3d guided waves in viscoelastic lainates and far field evaluation to a local acoustic source. J. Solids and Structures, 45: ,