An introduction to Statistical Energy Analysis

Mathias Barbagallomathias.barbagallo@wsp.com

LTHGuest lecture

12/12/18

1

Myself

• 2002-2005 BSc in Telecom Eng, Politecnico Milano

• voice activity detector GSM networks, Siemens

• 2005-2008 MSc in Sound Eng and Design, Politecnico

• 2006-2008 MSc in Sound and Vibration, Chalmers

• parametric array loudspeakers

• 2008-2013 PhD in Technical Acoustics, KTH

• SEA, porous materials, variational principles

2

Myself

• 2013-2016, Consultant, Lloyd’s Register Consulting

• N&V; marine sector

• 2016-2017, Consultant, Creo Dynamics

• N&V; R&D, ultrasound, nonlinear acoustics, car industry (Volvo cars)

• 2017-now, Consultant, WSP

• N&V; urban and industrial sectors.

3

This lecture

• Material produced at KTH

• internal courses

• seminars/lectures within Mid-Mod EU project

• together with my supervisor Svante Finnveden (former professor at KTH now at ACAD)

4

Motivation

p

v

• Rocket to the moon (Smith, 1962)

- some 200k modes

- modelling effort

5

Motivation

• Vibroacoustic predictions

- low frequency: measurements, FEM

- high frequency: today’s topic

• Definition of high frequency

6

Motivation

• Broadband excitation of a structure

• Not too high damping ⇒ reverberant field

• Peaks, natural frequencies ⇔ modes

Fahy, Gardonio

7

Motivation

• Clear peaks

• Peaks characteristic of the structure

• Not clear peaks

• Peaks dependent on small geometric variations, discontinuities

8

Motivation

• Magnitudes of the structure-borne frequency response functions of 98 nominally identically cars (Isuzu Rodeo); force applied on the wheel hub, microphone at the driver’s seat. (Kompella, Bernhard)

9

Motivation

• 12 measurements on the same car (Kompella, Bernhard)

10

Motivation

• Vibroacoustic responses of 141 identical beer cans (Fahy)

11

Motivation - What did we see?

• What is changing between various cars from the same production line?

• What is changing between measurements on the same car?

• We may see what high frequency means

• We may have a measure for this

12

Motivation

1.15 m

0.7 m

h = 1 mm

• Input power to a plate with nominal dimensions (left), to 8 plates with nominal dimensional + std(0.02) on h (right)

13

Motivation - What can we say?• Two problems with high frequency models:

- technical: small mesh compared to wavelength (FE)

- profound: chaotic response

- deterministic;

- apparently random;

- unpredictable.

• Useful approach - i.e. not the only but a useful one:

- quick and efficient

- accounting for statistical variation

- employing a measure resistant to variations

14

Answer

• Statistical Energy Analysis

- statistical

- energy

- analysis

- Seems Easy, Ain’t

• Frequency, space, ensemble average response

15

Answer

• Input power to 8 plates with nominal dimension + std(0.02) (black); SEA prediction (red)

16

Formulation

nowadays models may look like CAD drawings...17

Formulation: 0th step

• To start

- a structure

- identify sources

- linear motion

car

wind noise road roughness

engine

vibration fatigue risk

cabine noise

18

Formulation: 1st Step• Subdivide structure/vibroacoustic field into SEA

Elements

- elements may be substructures

- elements may be elements of the response

- elements are quite large

- very difficult task!

Roof bending

Roof longitudinal

Engine

Cabin air volume

19

Formulation: 2nd step

• Power balance for element i

P iin = P i

diss +�

j �=i

P ijcoup

• SEA built upon conservation of energy!

20

Formulation: 3rd step

• Dissipated power

P idiss = �⇥E

- damping loss factor

- angular frequency

- total vibroacoustic energy

- nothing really new (Lord Rayleigh)

�

E�

21

Formulation: 4th step• Coupling power

- number of modes in a band

- introduce:

- modal density [1/rad/s]

- modal energy (power!) [W]

- conductivity [-]

P ijcoup ⇥

�Ei

�Ni� Ej

�Nj

⇥

�Ni

ni =�Ni

��

ei =Ei

ni

Cij = ⇥ni�ij = ⇥nj�ji = Cji

22

Formulation: 5th Step

23

SEA quantities: Energy

• Energy as dependent variable, unknown

• Frequency, space, ensemble averaged energies (ergodic assumption)

• SEA deals with:

- resonant motion

- reverberant wave motion

• Strain energy (density) = Kinetic energy (density)

24

SEA quantities: Energy

• Retrieve vibroacoustic quantities!

E = e · V ; E = e · S; E = e · Le � ⇥p2⇤; e � ⇥v2⇤

<> space, frequency, ensemble average!

power

measure of kinetic energy

measure of strain energy

LW = 10 logW

Wref;

Lv = 10 logv2

v2ref;

Lp = 10 logp2

p2ref

25

SEA quantities: modal density

• Modal density: number of resonance per unit frequency

ni =�Ni

�⇥; n =

1

�⇥

- expected difference between natural frequency [rad/s]

26

SEA quantities: modal density

Craik, Sound transmission through buildings using statistical energy analysisFahy, Noise and vibration

Real part of mobility for statistic ie diffuse conditions:

27

SEA quantities: modal density

• Statistics implies little/no difference between various boundary conditions and/or geometries!

• Infinite structures just as good for some applications.

28

SEA quantities: modal density

Cremer, Heckl, Pettersson, Structure-borne sound

29

SEA quantities: modal overlap factor

• Modal overlap factor

P idiss = ⇥⇤E = Me

M = ⇥⇤n; M =⇥⇤

�⇤

- quantifies dissipation

- quantifies probability of a resonance

30

M = ⇥⇤n; M =⇥⇤

�⇤

SEA quantities: modal overlap factor

(Brian Mace)

31

SEA quantities: modal overlap factor

Finnveden, Energy methods (lecture notes KTH course)

Damping loss factor (structures):

frequency separation where response has decreased of factor 2 (3 dB)

• “Half-power bandwidth” method

approx. dampig loss factor = 7 Hz / 140 Hz = 0,05

32

SEA quantities: modal overlap factor

Lyon, Statistical energy analysis

33

SEA quantities: Conductivity

(Brian Mace)

34

SEA quantities: Conductivity

(Brian Mace)

• A statistical approach based on a low population!

• Thermodynamics: Avogadro’s number, 10^23

• SEA: vibroacoustic natural modes, 0 - 10^4?

35

SEA quantities: Conductivity

(Brian Mace)

36

Potential Flow Model

• Evaluate vibrations in the plate when turbulent flow in the pipe (MWL KTH)

• Believe in SEA

37

Potential Flow Model

Finnveden, JSV 273 (2004)

38

Potential Flow Model

Finnveden, JSV 273 (2004)

• Finite element mesh in the cross-section

• Waves propagating in the long direction

• Tunnel is a long structure

39

Potential Flow Model

Finnveden, JSV 273 (2004)

• Now we know which waves propagate in the structure

• Approximations in respect to frequency ranges

40

Potential Flow Model

Finnveden, JSV 273 (2004)

41

• Compact support

- only direct couplings

SEA quantities: Conductivity

• Travelling wave estimate:

• no transfer of energy back

• regardless of rest of the structure

Conductivity given by:

when calculating C

42

room 1 room 2

M1 M2C12

e1 e2

pin

Room acoustics exercise

43

Room acoustics exercise1. E = 2⇤ep⌅V = 2

1

2

p2

⌅0c2V [J ] ;

2. e =E

n

⇤J

s

⌅= [W ] ;

3. n =k2V

2⇤2c[s] ;

4. M = ⌃⇥n =k2A

8⇤2[�]; A = �S;

5. C =

�Wt

e1

⇥

C12M2

�0

=⇧Wi

e1[�];

6. ⇧ =Wt

Wi; R = 10 log

1

⇧; Wi =

k2S

8⇤2e1;

7. ⇥ C = ⇧k2S

8⇤2

room volume Vspeed of sound cair density rho_0

wavenumber k = ω/c

partition areas Sabsorption areas A

incident power Wi

transmitted power Wt

transmission factor tau

44

�M1 + C �C�C M2 + C

⇥ �e1e2

⇥=

�Pin

0

⇥;

e =eV

n=

2�2⇥p2⇤⇤2⇥0c

;

• Standard texts

1. Lp1 = Lw + 10 log

��

4�r2+

4

A�

⇥;

2. Lp2 = Lp1 � 10 log

�A

⇥S+ 1

⇥typical approximation

direct field is not SEA

Room acoustics exercise

room constant

45

• Symmetric formulation

• Sub-structuring

• Distinction between coupling and damping losses

• (nothing really new but a good framework!)

• (good news: you already know a bit of SEA!)

Room acoustics exercise

46

room 1 room 2

M1 M2

Mw

Cw2Cw1

C12

wall

e1 e2

ew

Room acoustics exercise

47

room 1 room 2wall

Room acoustics exercise

• Coincidence frequency of a plate

mass per unit area

bending stiffness

48

Mi = �iniei

C1w =

�P 1wc

ew

⇥

C1wM1

�0

=�0cS⇥1w⇥v2⇤

ew;

ew =Ew

nw=

St�⇥v2⇤nw

;

nw =S

3.6cLt;

� C1w = �0cS

3.6cLt2�⇥1w;

Room acoustics exerciseRadiated power by a simply-

supported plate placed in an infinite baffle radiating into a semi-infinite

room (Leppington)

Leppington, The acoustic radiation efficiency of rectangular panels, Proceedings of the Royal Society of London A, 382 (1982)

plate thickness tplate density rho

49

Room acoustics exercise

• Radiation efficiency (kf, fluid wavenumber; ks structure wavenumber):

Leppington, The acoustic radiation efficiency of rectangular panels, Proceedings of the Royal Society of London A, 382 (1982)

50

Room acoustics exercise

• Mass-law, forced motion below coincidence frequency:

• insert indirect coupling between two rooms as direct coupling

• Transmitted and incident power in a room, diffuse field

• Coupling loss factor (as per definition):

Craik, Sound transmission through buildings using statistical energy analysis

51

Room acoustics exercise

non-resonant transmission coefficient by Leppington

Craik, Sound transmission through buildings using statistical energy analysis

52

Room acoustics exercise

• Use eq. 4.24 to obtain Rf (non-resonant sound reduction index) and use it to compute tau12 and then eta12 i.e. the coupling loss factor describing the non-resonant transmission between two acoustic volumes (two slides ago).

53

Room acoustics exercise

• Finalise: get sound pressure levels in the two rooms.

• Functions of modal energy, retrieve pressure.

• Check formula in e.g. ISO standard 16283-1:2014 to retrieve R from sound pressure levels in the two rooms, wall area and absorption area of the receiving room.

• Done!

Craik, Sound transmission through buildings using statistical energy analysis

54

Validity of SEA

• N>>1; many modes in a given frequency band

• M>>1; we need disorder, i.e. overlapping modes

• low energy decay within an element == constant energy density within an element

• weak coupling: Cij/Mi << 1, i.e. flow of exchanged vibroacoustic energy small compared to internal losses. Think how Cij is estimated.

Le Bot, Cotoni, JSV 329 (2010)

Careful! If losses are too high, then energy decay will occur! Perhaps M ≈ 1 works better.

55

Why SEA?• Only method for high frequency vibroacoustics

supported by commercial software

• Efficient (numerically cheap)

- less demand on input data

- immediate answers

- impossible problems

• We are ignorant of many things

- why bother with unrealistic precision?

- mean response of “typical” product

• Alternate view point

56

Why SEA?

• Successful with:

• satellites, launch vehicles

• airplane

• ships

• buildings

• vehicles

57

Why not SEA?• No details in results

- poor frequency resolution

- no spatial resolution

• Software

- lack of routines for parameter estimation (not so true actually)

- lack of bench mark examples

• Connection to deterministic methods

• Not universally applicable (narrowband excitation

- how accurate?

- valid for this problem?58

Why not SEA?

• In none of my jobs as consultants I have used SEA for client-work!

• Instead:

• semi-empirical methods based on measured transfer functions

• Stripped down SEA model based on measured reduction indexes and impact sound isolation

• Educated guess work

• Why is it so?

59

References

• Lyon, DeJong “Statistical Energy Analysis” (Buttworth-Heinemann)

• Fahy, Gardonio “Sound and structural vibration” (Elsevier Academic Press)

• Fahy “Statistical energy analysis: a critical overview” (Phil. Trans. R. Soc. Lond. A (1994) 346, 431-447)

• Cremer, Heckl, Pettersson, Structure-Borne Sound (Springer)

• SD2170 course materials by Svante Finnveden (MWL/KTH)

• Wind tunnel example: Finnveden, JSV 273 (2004)

• Le Bot, Cotoni, Validity diagrams for statistical energy analysis, JSV 329 (2010)

60

What to bring home?

• Should you get a job at a place where they use SEA, then you can say you had a lecture on it. :)

• Otherwise there is still something useful:

- interpretation of high-frequency results from deterministic calculations or single measurements should be done carefully

- relevance or irrelevance of structural/geometrical/modelling details in a given problem - sometimes ignorance really is bliss

- think about which waves dominate the response:

- what is the dominating physics?

- think about energy flow model:

- which sources? which receivers? where is the energy flowing to?

61