Statistical Energy Analysis

Transcription

1 1 2 1 Long. 2 Long. 3 Long. 3 1 Bend. 2 Bend. 3 Bend. [E] = [C] -1 [W] Lectures Time: Wednesday Place: TA 201 Lecturer: Prof. Björn Petersson Tutorials Time: Wednesday Place: TA K001 Tutor: Dipl. Ing. Wolfgang Weith 1

2 Lecture plan Introduction, Background Classical Dynamics Signal Analysis Energy Flow in Built-up Systems Modal Analysis Energy Considerations Reciprocity Considerations Multi-modal Systems Multi-modal Coupling Coupling Lossfactor Variance and Confidence Computational Uncertainty Module Notes: Compendium on Structural Acoustics Can be purchased at the ITA secretariat Additional reading: R.H. Lyon and R.G. DeJong,1995. Theory and Applications of. Butterworth-Heinemann, Boston. W. Wöhle, Statistische Energieanalyse der Schalltransmission, Kap. 1.10, Taschenbuch (Eds. Fasold, Kraak, Schirmer), VEB Verlag Technik, Berlin. R.J.M. Craik, Sound Transmission through Buildings Using. Ashgate, Aldershot. 2

3 Module is defined by: The compendium, the lectures and tutorials. Tutorials comprise: Problems and reports Long. 1 Bend. 2 Long. 3 Long. 2 Bend. 3 Bend. [E] = [C] -1 [W] 3

4 Why analyse and predict vibro-acoustic behaviour Why analyse and predict vibro-acoustic behaviour Structural strength Displacement ~ Strain Sound, noise, reliability Velocity ~ Power Human reliability/stress, comfort Acceleration ~ Physiologics 4

5 Vibration hazard Built-up structures Antenna 5

6 Built-up structures Heat pump Structure-borne waves Liquid-borne waves Airborne waves Vibro-Acoustics The physical process Generation Transmission Propagation Radiation 6

7 Tool boxes and Tools Generation Transmission Propagation Radiation Tool boxes Mechanics Thermo-dyn. Fluid-dyn. Hydraulics Dynamics Acoustics Dynamics Acoustics Acoustics Tools No general! Impedance/ Wave theory Mobility theory Phase analysis Wave theory Hybrides Why analyse and predict vibro-acoustic behaviour Structural strength Displacement ~ Strain Sound, noise, reliability Velocity ~ Power Human reliability/stress, comfort Acceleration ~ Physiologics 7

8 Single-degree-of-freedom-system, SDOF M x K=1/C R Forced vibration F M x K R 8

9 Transient and stationary vibrations Two coupled SDOF Systems 9

10 Black-box approach F( ) Y( ) v( ) 1

11 Noise signal x(t) t Narrow band spectrum G xx xx (w) ( ) w

12 Narrow band correlation C xx (t) t Broad band spectrum G xx ( ) (w) w

13 Broad band correlation C xx (t) t Evaluation in frequency bands 1 + = 4

14 Evaluation in frequency bands Two-sided spectrum S xx (w), [(EU) 2 /rad/s] S xx ( ), , w, [rad/s] 5

15 Single-sided spectrum G xx (w), [(EU) 2 /rad/s] G xx ( ), , w, [rad/s] Single-sided spectrum 25 G xx (f), [(EU) 2 /Hz] f, [Hz] 6

16 Magnitude and phase of mobility point or input mobility Exhaust system and car floor panel Two contact points 1

17 Substructured exhaust system F F 1 F F 2 v F 1 v F 2 F E 1 F E 2 v E 1 v E 2 Impedance and mobility representations [Z 1 ] [Z 2 ] [Y 1 ] [Y 2 ] [Z 1 ] + [Z 2 ] [Z tot ] [Y tot ] = [Z tot ] -1 [Y 1 ] -1 + [Y 2 ] -1 [Z tot ] [Y tot ] = [Z tot ] -1 2

18 Transfer mobility F j v i Coupled subsystems 3

19 Coupled subsystems One subsystem blocked Two spring-coupled subsystems v 1 M 1 K c M 2 v 2 K 1 =1/C 1 R 1 K 2 =1/C 2 R 2 4

20 Energy flow between two coupled oscillators * W 21 = B(E 1 - E 2 ) /= - W 12 / * Directly proportional to difference in decoupled energies * v 2 = 0 * Proportionality factor positive definite so that energy flows from the more energetic oscillator * Proportionality factor symmetric so that the flow is reciprocal Energy flow between two coupled oscillators * W 21 = B(E 1 - E 2 ) /= - W 12 / * Directly proportional to difference in actual energies * v 1 0, v 2 0 * Proportionality factor positive definite and symmetric so that energy flows from the more energetic oscillator and the flow is reciprocal * With only one oscillator externally excited, the maximum of the other is that of the excited one 1

21 Axially, freely vibrating rod x Density: Young s modulus: E L S First four eigenfunctions of an axially free rod v(x) n=0 n=1 n=2 n=3 x 2

22 Two-mode system 3

23 Modal synthetis p(x) Ω v(a)

24 Modal synthetis - a sum of SDOFs 1 v(a) = F 0 1 iωm eff n (A) + iωc eff n (A) + R n eff (A) M n eff (A) = M n (A) κ n (A) C n eff (A) = κ n (A) ω n 2 M n (A) R n eff (A) = ω n 2 ω η M n (A) κ n (A)

25 Arctan (arg) / tan -1 (arg) 2 1 arctan(arg) arg

26 Complex Young s modulus E = E 0 ( 1 + jη) η = W diss 2πW rev σ(t) = Re[ Eˆε e jωt ] = E 0 { ˆε cos(ω t) η sin(ω t) }

27 Half-power bandwidth

28 Averaged mean square velocity Over all possible force positions Over all possible response positions v 2 ω = F 2 ω M 2 tot π N 2ωη ω Over time Over a frequency band

29 Modal synthetis p(x) Ω v(a)

30 Averaged mean square velocity Over all possible force positions Over all possible response positions v 2 ω = F 2 ω M 2 tot π N 2ωη ω Over time Over a frequency band

31 Coupled energy reservoirs Subsystem 1 Subsystem 2

32 Simply supported beam L E, ρ v(x) S, I

33 Occurrence of eigen-frequencies

34 Cumulated number of modes

35 Modal density» n = dn/df

36 SDOF coupled to a plate Free-body-diagram v 2 F 2 R M v m C F m v 1 F 1 1

37 Point force excited, infinite thin plate ^ F = Fe j t z x y Oscillator - plate interface v osc F 1 v 1 F 2 v 2 2

38 Autospectrum of plate velocity 3

39 SDOF coupled to a plate 2 v osc ω = v 0 (ω) 2 1 Ω ω Ω Y osc (ω) 2 Y pl (ω) + Y 1 (ω) 2 dωdω

40 Autospectrum of average plate velocity v ωw

41 Input mobility of oscillator Y 1 ωw

42 Input mobility of plate at a point Y pl ωw

43 Magnitude of summed mobilities pl 1 Y pl +Y 1 ωw

44 Inverted magnitude of summed mobilities 1/ Y 1 2 pl +Y 1 2 wω

45 Narrow band coupling Y / Y pl +Y 1 2 osc 2 / Y pl +Y 1 2 wω

46 Autospectrum of average plate velocity v ω w

47 Arbitrary force spectrum 1/ Y 2 pl 1/ Y 2 w ω

48 Narrow band coupling Y trf 2 ω w

49 Transfer mobility of specimen 1 for a band ω Y Y trf 2 Dw/Max[ Y trf 2 ] trf 2 ω trf 2 ] Dw ω w ω

50 Transfer mobility of specimen 2 for a band ω Y trf 2 ω /Max[ Y trf 2 ] ω ω

51 Integrals and band averages for the two different realisations Band energies MS1=0.35 MS1=0.035 MS2=0.38 ω

52 Uncertainty in eigenfrequencies of a plate PSD Natural Frequency, [Hz]

53 Pickup transfer function H = p ear F input

54 Transfer mobilities of ordinary beer cans

55 A system consisting of two subsystems System boundary A system consisting of two subsystems System boundary 1

56 Reciprocity; point to point quantities F 1 v 2 F 2 v 1 First hypothetical experiment Radiating structure Vibrating rigid disc F v d 2

57 First hypothetical experiment Radiating structure Vibrating rigid disc v F Spatial coupling Temporal coupling strong strong weak weak 3

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60 A system consisting of two subsystems System boundary A system consisting of two subsystems System boundary 1

61 Two spring-coupled subsystems v 1 M 1 K c M 2 v 2 K 1 =1/C 1 R 1 K 2 =1/C 2 R 2 Two sets of mode groups coupled one-to-one n n 1N 2N 2

62 Modal gap and bandwidth Y 2 b Bandwidth r=2 b 1-2 Uniform distribution of a mode in a band ( ) 1/ 1n 3

63 Basic assumptions in 1. n is a stochastic variable with 2. The energy within the band is equally distributed amongst the modes 3. The lossfactor is constant within the band ( ) =const.; 4

64 Basic assumptions in 1. ω n is a stochastic variable with f (ω) = 1 ω ; ω n ω 2. The energy within the band is equally distributed amongst the modes E m = E ω N 3. The lossfactor is constant within the band η(ω) = const. ; ω ω

65 Mode to mode coupling

66 n v(x,t) = v n Φ n (x)cos(ω n t) Spatial coupling Temporal coupling strong strong weak weak

67 Effective mobility F 1 F 2 F 3 Y 11 Σ = Y 11 + Y 12 F 2 F 1 + Y 13 F 3 F 1

68 Coupling lossfactor E 0 E 0 η ij = W ij ωe 0 E 0

69 Loss factor: η qq = W diss ωe rev Coupling loss factor: η qp = W p q ωe p Reciprocity: η pq ωn q = η qp ωn p Dissipation coefficient or modal overlap factor: Power transfer coefficient: µ qq = η qq ωn q µ pq = η pq ωn q

70 Power balance 11,in 22,in ,diss 22,diss

72 Loss factor: η qq = W diss ωe rev Coupling loss factor: η qp = W p q ωe p Reciprocity: η pq ωn q = η qp ωn p Dissipation coefficient or modal overlap factor: Power transfer coefficient: µ qq = η qq ωn q µ pq = η pq ωn q

73 Point coupling 1 F 0 F 1 F 2 v 1 ; v 10 = Y 10 F 0 2 v 2

74 Two obliquely propagating waves

75 Group speed

76 Diffuse incidence lcos θ l θ dθ

77 Structure-fluid coupling S V < p R 2 > < v S 2 >

78 Radiation efficiency v p 2 S 2 W p = ρcsv p σ = W 2 ρcsv p W = σρcs v(x, y) 2 v(x, y) 2 S

79 Radiation efficiency 5log(P/2λ c ) 0 Radiation index 10logσ 10log(λ c 2 /S) λ c =c/f c Area S Perimeter P c 2 /(2Sf c )(1/4S-1) 3c/P 100λc/P 2 f c /4 f c 2f c

81 Point coupling 1 F 0 F 1 F 2 v 1 ; v 10 = Y 10 F 0 2 v 2

82 Diffuse incidence lcos θ l θ dθ

83 Structure-fluid coupling S V < p R 2 > < v S 2 >

84 Physical system 1 F 2 Π 11,in model (flexural waves only) Π Π 12 Π 11,diss Π 22,diss

85 Power balance { Π 1,in }= ω η n + η n η 12 n 2 η 21 n 1 η 22 n 2 + η 12 n 2 E 1 m E 2 m ω { Π 1,in }= ω µ + µ µ 12 µ 21 µ 22 + µ 12 E 1 m E 2 m ω

86 Physical system F Bending modes Long. modes Bending modes Long. modes 3 4 Bending modes Long. modes

87 model Π 11,in Π 51 Π 31 Π Π 35 Π 13 Π 21 Π 12 Π 43 Π 34 Π 65 Π 56 Π Π 24 Π 64 Π 46

88 Transmission efficiencies - 10logτ τ B2B3 τ B3B3 τ L2B3 τ L1B3 t 1 =t 3 =15 mm t 2 =7 mm

89 Transmission efficiencies - 10logτ τ L1L3 τ B2L3 t 1 =t 3 =15 mm t 2 =7 mm

90 Model of ships section made of cardboard and plastic film

91 Comparison measurements and calculations» Velocity level difference (bulkheads 241 and 264) ²L v, [db] Frequency, [Hz]

92 Statistical Energy Analysis Cross-section of a ship indicating deck and hull plates

93 Comparison measurements and calculations» Velocity level difference (bulkheads 264 and 277) ²L v, [db] Frequency, [Hz]

94 Double-bottom structure

95 Double-bottom structure model

96 Comparison measurements and calculations

97 Coupled SDOFs F(ω) ω ω 1 2 E 1 =E 1,Tot E 2 =E 2,Tot ; ψ 2 (x 2 )

98 Coupled multi-modal subsystems F(ω) ; ω ω ψ n x 1 ) ψ 2n x 2 ) 1 2 E 1, N 1 E 2, N 2

99 Five factors associated with spatial variance 1. E n E m 2. N i N j 3. Y i (0,y) Y j (0,y) 4. ψ n (x i ) ψ n (x j ) 5. W in,i W in,j p(x)ψ n (x)dω = p(x)ψ m (x)dω l x l x + l x i l y j l y + l y Variations in the boundary conditions Variations in the position Power input changes

100 Coupled multi-modal subsystems F(ω) ; ω ω E α E σ 1 2 E 1, N 1 E 2, N 2

101 Partial variances v(x,t) 2 2 = ψ σ 2M 2 α B σα E α ψ (x) 2 ωη σ + B σα α 1) Äquipartition Var[E α ] 2) Modale Dichte Var[N 1 ] ~ Σ α 3) Randbedingung Var[B σα ] 4) Position Var[x]

102 Average coupling M α K c M σ B σα = 2(η α ω α + η σ ω σ )K c 2 M α M σ (ω 2 α ω 2 σ ) 2 + (η α ω α + η σ ω σ )(η α ω α ω 2 σ + η σ ω σ ω 2 α ) 2 1 ω B ω α 2K σα (ω α )dω α B σα = c π (ηω + η σ ω σ ) 2 M α M σ 2 ω ωω σ

103 Coupled multi-modal subsystems F( ) ; 1n (x 1 ) 2n (x 2 ) 1 2 E 1, N 1 E 2, N 2 Five factors associated with spatial variance 1. E n E m 2. N i N j 3. Y i (0,y) Y j (0,y) 4. n (x i ) n (x j ) 5. W in,i W in,j l x l x + l x i l y j l y + l y Variations in the boundary conditions Variations in the position Power input changes 1

104 Point-excited finite plate F(x s,t) v(x,t) Eigenvalue distribution 2

105 Effect of eigenvalue distribution Confidence coefficients 3

106 e of Technical Acoustics Statistic Energy Analysis Uncertainty assessment Complete variance e of Technical Acoustics Power balance equation Matrix notation C E = P 4

107 e of Technical Acoustics Normalised variance of subsystem energy G pi = P i / e of Technical Acoustics Example of uncertainty propagation for three-plate system Uncertain parameters: l and h 5

108 e of Technical Acoustics Confidence band for energy in subsystem coupling length Strengths of Power Balance» Involves a coarse idealization with few parameters» Limited number of degrees of freedom» Gives a genuine feel for the physics» Cheap and quick computations» Framework for design and modelling» Provides upper limit for the reponse» Basis for energy flow analysis 6

109 Weaknesses of Power Balance» No information of confidence limits» Expert modelling» Not suitable for narrow band or tonal excitation» Not suitable for periodic/repeated systems» Unreliable (but not invalid) for low modal overlap» (Cannot handle indirect transmission)» (Cannot handle non-resonant, free field subsystems) 7