L intensité de structure

Transcription

1 L intensité de structure Jean-Claude Pascal LAUM, ENSIM

2 Plan Epression de l intensité dans les poutres Formulation approchée La mesure

3 Epression de l intensité de structure Epression de l intensité de structure L intensité structurale ou vibratoire correspond à la densité de flu de puissance [en W/m ] transporté par les ondes vibratoires densité de force (tenseur des contraintes) vecteur vitesse L intensité moenne dans le temps i i ( t) = σ ( t) ( t) i, j =,, j ij j I i = i i ( ) { } t = Re σ j ij j ( ) = { } ( t) = Re{ } avec les grandeurs complees j ω σ t Re σ e t et ij ij j j

4 Epression de l intensité des ondes de fleion Définition de l intensité des ondes de fleion L intensité structurale des ondes de fleion dans une poutre se réduit à I { } ( ) = Re Q( ) ( ) + M ( ) θ ( ) La théorie d Euler Bernouilli permet d eprimer toutes les quantités à partir du déplacement déplacement angulaire ( ) ( ) θ = moment de fleion ( ) ( ) M = EI effort tranchant ( ) ( ) Q = EI I ( ) = EI Im ω ( ) ( ) ( ) ( ) [W] avec = jω 4

5 5 ENSIM A Utilisation des différences finies pour eprimer les dérivées spatiales Intensité approchée par différence finie ( ) ( ) ( ) { } 4 4 Im 4 EI I ω jω = ( ) { } 4 Im 4 4 EI I ω Epression de l intensité des ondes de fleion

6 Epression de l intensité des ondes de fleion Approimation de champ lointain Epression générale du déplacement jk k jk k ( ) = A e + A e + A e A e + 4 avec le nombre d onde de fleion k = ω 4 ρ A EI En champ lointain quand il n a pas d ondes évanescentes I = jω k k EI ρ A { } EI ρ A Im I Im I EI ρ A Im ω { } 6

7 CHARACTERISATION OF A DISSIPATIVE ASSEMBLY BY STRUCTURAL INTENSITY A. Ho to calculate energetic quantities from laser vibrometer measurements B. Analsis of assembl plate using energetic quantities C. Transformation of D model to D junction model D. Use energ conservation lo to compute joint dissipation 7

8 8 ENSIM A MEASURED ENERGETIC QUANTITIES Force distribution Divergence of the structural intensit Potential energ densit Kinetic energ densit Structural intensit { } = v v B 4 Im ω I ( ) ( ) ( ) ( ) v k v j B F B,,, 4 4 = ω 4 v T ρh = ( ) = Re 4 v v v v B V ν ω ( ) ( ) = v v v v v v B Im υ ω I

9 MEASURED ENERGETIC QUANTITIES : ADVANCED METHODS OF WAVENUMBER PROCESSING Use of SFT for calculation of spatial derivatives SFT v, V k, k ) ( ) V k, k ) is the Spatial Fourier Transform v ( of, ) ( m n + m v (, ) n TF a ( jk ) m ( jk ) n k V ( k ( The derivatives of vibrating velocit are easil calculated b, k ) k SFT on truncated signals amplifies the components of high avenumbers, bringing large contributions of the high avenumber components to the results, especiall in the case of the high-order derivatives of the velocit. 9

10 MEASURED ENERGETIC QUANTITIES : ADVANCED METHODS OF WAVENUMBER PROCESSING Mirror methods used to reduce errors caused b operation of SFT The idea of the mirror method is to build a continuous and periodic signal (the resulting signal) from the signal to be processed b SFT (the original signal) Signa l Mirror s ignal + Origina l s ignal Origina l s ignal m pl u d e it A dis ta nce (m) 0

11 EXPERIMENTAL CONFIGURATION free edge 0 mm 90 mm clamped edge 850 mm A scanning vibrometer use a OFV 00 optical head To galvo-driven mirrors direct the laser beam horizontall and verticall measurement points Test assembl consists of to steel plates of thickness mm The to opposite edges are clamped. The to other edges are free A normal point force is acting on the plate

12 ASSEMBLY PLATE ANALYSIS : STRUCTURAL INTENSITY I = I + IA = φ + A φ Standard structural intensit Wavenumber processing as used Irrotational structural intensit Data integrated over a frequenc band of 55 to 000 Hz

13 ASSEMBLY PLATE ANALYSIS : INJECTED OR DISSIPATED POWER AND FORCE DISTRIBUTION Divergence of the structural intensit Force distribution At frequenc 668 Hz. these to quantities sho that the dissipation produced b the joint is maimum at the positions here the joint is constrained b the tightening of the bolts

14 ASSEMBLY PLATE ANALYSIS : FORCE DISTRIBUTIONS AVERAGED IN FREQUENCY BAND [549,600] Hz [60,649] Hz [650,699] Hz [700,750] Hz [75,800] Hz [80,850] Hz [85,900] Hz [90,950] Hz The dominating zones of dissipation correspond to the points of maimum constraints introduced b the bolts ensuring the contact of the to parts of the plate on the joint. Hoever this behaviour of the joint ill depend largel on the frequenc 4

15 EQUIVALENT D MODEL : AVERAGED POWER FLOW ON PLATE 0 ( ) = I(, ) join t forc e Plate Total poer flo in - direction Total poer flo in - L Plate direction φ 0 d φ L ( ) = I (, ) 0 d 5

16 EQUIVALENT D MODEL : ONE-DIMENTIONAL BEAM-LIKE MODEL The evolution of the averaging poer flo over one direction of a plate reveals the similar behaviour to that of a onedimensional sstem like a beam in the other direction of the plate F Beam 6

17 EQUIVALENT D MODEL : ONE-DIMENTIONAL BEAM-LIKE MODEL The basic idea is thus to represent the joint like a node of junction of one-dimensional elements + P = a a + a a a a + a + P = a a P = a plate junction plate The third branch comprising onl an outgoing ave is used to epress the poer dissipated b the joint. Thus the poer of each branch entering in the junction element can respectivel be ritten b 7

18 EQUIVALENT D MODEL : ONE-DIMENTIONAL BEAM-LIKE MODEL Far-field scattering matri a a a a td td Epression of the conservation of flo entering in the junction r, t P r = t t r + P + P = 0 a + + the reflection and transmission coefficients P P P P plate junction plate An approimation of the dissipated poer c g t d e + e td cg e, e the dissipation coefficient the group velocit for the fleural aves the densities of total energ on both sides of the junction 8

19 USE ENERGY CONSERVATION LAW TO COMPUTE JOINT DISSIPATION From the eact conservation la of energ I (, ) + Wdis (, ) = W (, ) I(, ) is the measured structural intensit ( ) W dis, W (, ) is represented b a simplified model for dissipation proposed b Nefske Sung and Bouthier Bernhard : (, ) ηω e( ) W dis, the local average of the total energ densit is the injected or dissipated poer b eternal elements (forces, joint, ) 9

20 USE ENERGY CONSERVATION LAW EQUATION OF -D ENERGY CONSERVATION Integration over direction leads to a sstem ith one dimension in L 0 I ( ) d+ I (, ) L L [ ] + W (, ) d W ( ) 0 dis = δ, 0 ( ) E the densit of the total energ integrated in direction (in J/m). The differential form of the conservation la is then ritten b dφ d ( ) ( ) = W ( ) + ηω E δ L 0 (, ) d ηω E ( ) η ω e = For a point force in ( ), 0

21 USE ENERGY CONSERVATION LAW EXEMPLE WITH ONE FORCE dφ d ( ) ( ) = W ( ) + ηω E δ Wδ ( ) φ ( ) W δ ( ) dηωe = 0 φ ( ) 0 L Injected poer in the plate W 0 L The evolution of the averaged poer flo along the dimension For an isolated plate sstem, the boundar conditions are φ 0 = 0 φ ( L ) W ηωe L = 0 ( ), = W The loss factor can be estimated η = ω E b: L = ω φ ma L L 0 0 e φ (, ) min dd

22 USE ENERGY CONSERVATION LAW ONE FORCE AND JOINT Use the folloing differential equation Wδ ( ) dφ d ( ) ( ) = Wδ ( ) ξ ω L E ( ) ( ) + ηω E δ 0 0 plate plate force Plate 0 ξ L ωe ( ) δ ( ) 0 L joint Plate W 0 0 ξ LωE L ( ) 0 W Dissipated poer b joint joint = ωξ L + ( ) + E ( ) E 0 ξ c t g d ωl 0 = ξ ωl E ( ) 0

23 USE ENERGY CONSERVATION LAW CHARACTERIZATION OF JOINT 0 0 uppe r part loe r part joint The average densit of energ in each of the to plates: L 0 ε < e >= e( ) dd L, < e >= 0 ( L ) L L 0 L 0 e + ε (, ) dd Position of the joint identified b the measured forces The loss factor φ η = ma ω E φ L min The linear loss factor of densit of dissipation ξ = ( φ φ ) ωη E + E ( L ) ma min L [ 0 0 ] ( E + E )/ At 668 Hz for the plate loss factor and for the linear loss factor of the joint are respectivel 0.0 % and 5%

24 EFFECTIVE PARAMETER IDENTICAFATION OF D STRUCTURES FROM MEASUREMENTS USING A SCANNING LASER VIBROMETER Introduction Methodsfor for evaluating parameters of of structures Energ Energ methods b b using using measuring data data b b a Scanning Laser Laser Vibrometer Estimation of of fleural avebumbers and loss factorin in -D structures Energ Energ methods to to obtain obtain dispersion curve curve General techniques for for computation Results of of measurements from the the Scanning Laser Vibrometer 4

25 Introduction (con t( con t) Methods to compute avenumbers The The finite-difference-approimation method Use Use three three accelerometers to to estimate the the fleural fleural avenumbersin in one-dimensional structures such such as as beams beams It Itis is directl directl based based on on the the ave ave equation associated ith iththe the far-field far-field approimation Disadvantage Disadvantage Too Toohigh sensitivit to to phase phase differences beteen sensors due due to to the the use use of of the the finite finite difference technique 5

26 Introduction (con t( con t) Methods to compute avenumbers Use Use of of Fourier Transform (SFT) (SFT) Determinethe the maimum of of avenumber spectrum in in beams, beams, hich hich as as then then used used to to identif identif the the value value of of natural natural fleural fleural avenumber To To reduce reduce the the distortions brought b b Spatial Spatial Fourier Fourier Transform (SFT) (SFT) a regressive methodas proposed Disadvantage Disadvantage The The use use of of the the direct direct Fourier Fourier Transform results results significant errors errors in in the the computations because of of truncated signals. 6

27 Introduction (con t( con t) Methods to compute avenumbers Spatial correlation approach Correlationof of the the measurements ith ith the the avefield e jk t e jk t The The choice choice of of k that the gives the t, k t that maimises the correlation gives the best best estimate of of the the fleural fleural avenumbers It It is is used used for for estimation of of avenumbersin in D D structures 7

28 ESTIMATION OF FLEXURAL WAVENUMBER IN TWO-DIMENSIONAL STRUCTURES First step Use non dissipative energ equation of plate to derive the effective fleural avenumbers 8

29 ESTIMATION OF FLEXURAL WAVENUMBER (con t) Derive fleural avenumbers from energ concept A thin isotropic plate ecited b one or or more mechanical forces F i neglecte the structural dissipation and the losses b radiation The equation of of Kirchhoffis is epressed b D jω 4 4 ( v k v) = δ ( r r ) B F i i i D k B v is is the bending stiffness of of plate, natural fleural avenumberin in vacuum,, fleural velocit 9

30 ESTIMATION OF FLEXURAL WAVENUMBER (con t) Derive fleural avenumbers from energ concept Multipling the above equation b the comple conjugate of the velocit ields D j ω ( ) 4 4 vv k v = F v ( r ) δ ( r r ) B = i ( Wi + jqi ) δ ( r ri ) Consider a non-dossipation plate. In the zone here there are nonecitation forces, no damping, no absorptions, e can obtain to equations: i i i i D 4 Im{ vv } = 0 ω I s = 0 Leading the divergence of the structural intensit to be zero. 0

31 ESTIMATION OF FLEXURAL WAVENUMBER (con t) Derive fleural avenumbers from energ concept Re { 4 } v v k v = 0 B Estimators of effective avenumber of fleural aves: γ a = Re v { 4 vv } / 4 γ b = Re 4 { vv } v / 4 The brackets < > denote the spatial average over the points, that is, outside the mechanical ecitation zones.

32 CONSIDERATION OF DISSIPATIVE TERMS EFFECTIVE LOSS FACTOR Second step Introduice dissipative terms in plate equation to obtain an estimator of loss factor

33 CONSIDERATION OF DISSIPATIVE TERMS EFFECTIVE LOSS FACTOR If the dissipations, losses due to structural dissipation and losses b radiations, are taken into consideration, equation of Kirchhoff are epressed b jω ( 4 D v ω ρh v) = + δ ( r r ) p a F i i i D = D + ( jη) is the comple bending stiffness η the structural loss factor p a is the acoustic radiation pressure on the to sides of the plate

34 CONSIDERATION OF DISSIPATIVE TERMS EFFECTIVE LOSS FACTOR Assumptions non eternal mechanical forces no local damping or absorptions Estimator of the total loss factor 4 * { v v } = η η 4 * a { v v } Im η T = + Re η the structural loss factor η a the loss factor due to acoustic radiations I n 0 ηa = ωρh v ρ cσ = ωρh σ is radiation efficienc coefficient Maimum Magnitude order at critical frequenc η a 7 for brass plate < < T 4 η < 5 0 4

35 THREE EFFECTIVE ESTIMATORS FOR D STRUCTURES γ a γ b η T = = = Re Re Re 4 { v v } / 4 v 4 { v v } Im v 4 * { v v } { 4 v v * } / 4 Local WavenumberW Estimator Averaged Wavenumber Estimator Loss Factor Estimator The are derived from energetic conception : the are independent of the resolution in avenumber domain 4 The are function of v and v The are based on the assumption : there are no eternal mechanical forces and no local damping. Develop computation methods 5

36 METHOD OF COMPUTATION OF ESTIMATORS Third step Find solutions to compute the double Laplacian of vibrating velocit and to eclude the points in local ecitation or absorbing zones 6

37 METHOD OF COMPUTATION OF ESTIMATORS Pre-processing and Spatial Fourier Transform (SFT) γ b = Re 4 { v v } v / 4 To compute the double 4 Laplacian v of the vibrating velocit, the technique of avenumber processing associated ith the Spatial Fourier Transform (SFT) is emploed. To reduce the distorsions caused b truncated signal, Pre-processing such as mirror method is applied before performing SFT. 4 SFT v a + ( K K ) V ( K, K ) 7

38 METHOD OF COMPUTATION OF ESTIMATORS (con t) Method to remove ecitation or damping zones from computations An eperimental eample is used to sho ho to eclude the data in ecitation or damping zones A brass plate ith dimension mm The plate is ecited b a shaker Normal vibrating velocit as measured b using Scanning Laser vibrometer { 4 } Map of Im vv proportional to eteral poer flo due to forces acting on the brass plate ( f = 500 Hz) Hotpots Damping zone Ecitation zone 8

39 METHOD OF COMPUTATION OF ESTIMATORS (con t) Method to remove ecitation or damping zones from computations Use of Histogram of 4 γ a The histogram shos the distributions of the values of 4 estimator over the plate γ a The unanted values are negative ones and ver large ones Zones of unanted values The points corresponding to those values are the zones of energ transfer due to eternal forces 9

40 METHOD OF COMPUTATION OF ESTIMATORS (con t) Method to remove ecitation or damping zones from computations Map of estimator 4 γ a Trace map of estimator Trace the points in the ecitation or damping zones (circles in can color) It is shon that the ecluding points in ecitation or damping zones can be determined b the methods proposed here. 40

41 METHOD OF COMPUTATION OF ESTIMATORS (con t) Method to remove ecitation or damping zones from computations Ho to determine values in the ecitation zones Construct a vector in the a of sorting the values Im{ 4 vv } in ascending order. Select a threshold value ith the help of the curve of the vector The values greater than the threshold value are unanted values and are removed from the computation, resulting in ecluding the ecitation or damping zones. 4

42 METHOD OF COMPUTATION OF ESTIMATORS (con t) Eamples Histogram of at frequenc indicated γ a The values γ b are averaged over the points selected using the histogram of of γ a 4 The D / ρ h = ω / γ is is b computed and shonb ello band Eperimental values have good agreement ith Table values 4

43 EXPERIMENTAL RESULTS OF EFFECTIVE PARAMETERS Divergence of the structural intensit free edge 0 mm 90 mm clamped edge 850 mm Scanning Laser Vibrometer A scanning vibrometer use a OFV 00 optical head To galvo-driven mirrors direct the laser beam horizontall and verticall measurement points Structure for testing Test assembl consists of to steel plates of thickness mm The to opposite edges are clamped. The to other edges are free A normal point force is acting on the plate 4

44 EXPERIMENTAL RESULTS OF EFFECTIVE PARAMETERS The curve in dashed line is the avenumber computed b ith k = ω ρh D ( ) 4 B D e e = ω ω ω ω ω ρh γ 4 b ( ω) dω 44