Statistical Energy Analysis

(1)

1

Statistical Energy Analysis

1 2 3 1 Long. 1 Bend. 2 Long. 3 Long. 2 Bend. 3 Bend.

[

E

]

=

[

C

]

-1

[

W

]

Statistical Energy Analysis

• Lectures

– Time: Wednesday 10 - 12 – Place: TA 201

– Lecturer: Prof. Björn Petersson

• Tutorials

– Time: Wednesday 12 - 14 – Place: TA K001

– Tutor: Dipl. Ing. Wolfgang Weith

(2)

2

SEA

• 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

SEA

• 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 Statistical Energy Analysis. Butterworth-Heinemann, Boston. • W. Wöhle, 1984. Statistische Energieanalyse der Schalltransmission,

Kap. 1.10, Taschenbuch Akustik (Eds. Fasold, Kraak, Schirmer), VEB Verlag Technik, Berlin.

• R.J.M. Craik, 1996. Sound Transmission through Buildings Using Statistical Energy Analysis. Ashgate, Aldershot.

(3)

3

SEA

• Module is defined by: The compendium, the lectures and tutorials.

• Tutorials comprise: Problems and reports.

Statistical Energy Analysis

1 2 3 1 Long. 1 Bend. 2 Long. 3 Long. 2 Bend. 3 Bend.

[

E

]

=

[

C

]

-1

[

W

]

(4)

4

SEA

• Why analyse and predict vibro-acoustic behaviour

SEA

– Structural strength

• Displacement 1~ Strain

– Sound, noise, reliability

• Velocity ~1 Power

– Human reliability/stress, “comfort”

• Acceleration 1~ Physiologics ©

(5)

5

SEA

• Vibration hazard

• Built-up

structures

– Antenna

SEA

(6)

6

• Built-up structures

– Heat pump • Structure-borne waves • Liquid-borne waves • Airborne waves

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• Vibro-Acoustics

Generation Transmission Propagation Radiation

The physical process

(7)

7

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• Tool boxes and Tools

Generation Transmission Propagation Radiation

Mechanics Thermo-dyn. Fluid-dyn. Hydraulics Dynamics Acoustics Dynamics Acoustics Acoustics T ool boxe s T ool s No general ! Impedance/ Mobility theory Phase analysis Wave theory SEA Wave theory SEA Hybrides © Prof. B.A.T. Petersson

SEA

– Structural strength

• Displacement 1~ Strain

– Sound, noise, reliability

• Velocity ~1 Power

– Human reliability/stress, “comfort”

• Acceleration 1~ Physiologics ©

(8)

8

SEA

• Single-degree-of-freedom-system, SDOF

M R K=_1/C x ©

Prof. B.A.T. Petersson

SEA

• Forced vibration

M R K x F © Prof. B.A.T. Petersson

(9)

9

SEA

• Transient and

stationary

vibrations

SEA

• Two coupled SDOF Systems

(10)

1

SEA

• Black-box approach

F₍1₎ Y₍1₎ v₍1₎

SEA

(11)

2

SEA

• Noise signal

x₍t₎ t ©

SEA

• Narrow band spectrum

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 Gx x ( w) 100 80 60 40 20 0 w Gxx ( 1 ) 1

(12)

3

SEA

• Narrow band correlation

-15 -10 -5 0 5 10 15 Cx x ( t) -1.0 -0.5 0.0 0.5 1.0 t ©

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• Broad band spectrum

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 Gx x ( w) 100 80 60 40 20 0 w 1 Gxx ( 1 )

(13)

4

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• Broad band correlation

200 150 100 50 0 -50 -100 Cx x ( t) -1.0 -0.5 0.0 0.5 1.0 t ©

SEA

• Evaluation in frequency bands

1

+ =

(14)

5

SEA

• Evaluation in frequency bands

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• Two-sided spectrum

3 2 1 0 -1 Sx x ( w), [( EU ) 2 / ra d / s] -100 -50 0 50 100 w, [rad/s] Sxx ( 1 ), 1_,

(15)

6

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• Single-sided spectrum

5 4 3 2 1 0 -1 Gx x ( w), [( EU ) 2 / ra d / s] 140 120 100 80 60 40 20 0 w, [rad/s] 1_, Gxx ( 1 ),

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• Single-sided spectrum

25 20 15 10 5 0 Gx x (f ), [( EU ) 2 / H z ] 20 15 10 5 0 f, [Hz] © Prof. B.A.T. Petersson

(16)

1

SEA

• Magnitude

and phase

of mobility

– point or input mobility © Prof. B.A.T. Petersson

SEA

• Exhaust system and car floor panel

– Two contact points

(17)

2

SEA

• Substructured exhaust system

FF 1 FF2 FE 2 FE 1 vE 1 vE2 vF 2 vF 1 © Prof. B.A.T. Petersson

SEA

• Impedance and mobility representations

[

Z

₁

]

[

Z

₂

]

[

Z

₁

]

+ [

Z

₂

]

[

Z

_tot

]

[

Y

_tot

]

= [

Z

_tot

]

-1

[

Y

₁

]

[

Y

₂

]

[

Y

₁

]

-1

+ [

Y

2

]

-1

[

Z

_tot

]

[

Y

_tot

]

= [

Z

_tot

]

-1 © Prof. B.A.T. Petersson

(18)

3

SEA

• Transfer mobility

F_j v_i © Prof. B.A.T. Petersson

SEA

• Coupled subsystems

(19)

4

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• Coupled subsystems

– One subsystem blocked

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• Two spring-coupled subsystems

M₁ R₁ K₁=1/C₁ v₁ _M 2 R₂ K₂=1/C₂ v₂ K_c © Prof. B.A.T. Petersson

(20)

1

SEA

• Energy flow between two coupled oscillators

* W₂₁ = B(E₁ - E₂) /= - W₁₂ /

* Directly proportional to difference in decoupled energies * v₂= 0

* Proportionality factor positive definite so that energy flows from the more energetic oscillator

* Proportionality factor symmetric so that the flow is reciprocal

SEA

• Energy flow between two coupled oscillators

* W₂₁ = B(E₁ - E₂) /= - W₁₂ /

* Directly proportional to difference in actual energies * v1 2 0 , v2 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

(21)

2

SEA

• Axially, freely vibrating rod

L x S Density: 1 Young’s modulus: E ©

SEA

• First four

eigenfunctions

of an axially

free rod

n=0 n=1 n=2 n=3 v₍x₎ x ©

(22)

3

SEA

• Two-mode system

(23)

Institut für Technische Akustik

SEA

• Modal synthetis

p

(

x

)

v

(

A

)

Ω

(24)

Institut für Technische

Akustik

SEA

• Modal synthetis - a sum of SDOFs

v

(

A

)

=

F

₀

1

i

ω

M

n

eff

(

A

)

+

1

i

ω

C

n

eff

(

A

)

+

R

n

eff

(

A

)

∑

M

_neff

(

A

)

=

M

n

(

A

)

κ

_n

(

A

)

C

n eff

(

A

)

=

κ

n

(

A

)

ω

_n2

M

_n

(

A

)

R

n eff

(

A

)

=

ω

n 2

ω

η

M

_n

(

A

)

κ

_n

(

A

)

(25)

Akustik

SEA

• Arctan (arg) / tan

-1

(arg)

-2 -1 0 1 2

ar

ct

an

(a

rg

)

-20 -10 0 10 20

arg

(26)

Akustik

SEA

• Complex Young’s modulus

σ

(

t

)

=

Re

[

E

ε

ˆ

e

j

ω

t

] =

E

0

ε

ˆ

{

cos(

ω

t

)

−

η

sin(

ω

t

)

}

η

=

W

diss

2

π

W

rev

(27)

SEA

• Half-power bandwidth

(28)

Akustik

SEA

• Averaged mean square velocity

Over all possible force positions

Over all possible response positions

Over time

Over a frequency band

v

2

∆

ω

=

F

∆

ω

2

M

tot

2

π

2

ωη

∆

N

∆

ω

(29)

SEA

• Modal synthetis

p

(

x

)

v

(

A

)

Ω

©

(30)

Akustik

SEA

• Averaged mean square velocity

Over all possible force positions

Over all possible response positions

Over time

Over a frequency band

©

v

2

∆

ω

=

F

∆

ω

2

M

tot

2

π

2

ωη

∆

N

∆

ω

(31)

Akustik

SEA

• Coupled energy reservoirs

Subsystem 1

Subsystem 2

©

(32)

Akustik

SEA

• Simply supported beam

L

S, I

E,

ρ

v

(

x

)

©

(33)

SEA

• Occurrence of eigen-frequencies

©

(34)

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• Cumulated

number of

modes

©

(35)

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• Modal density

»

n = dN/df

©

(36)

1

SEA

• SDOF coupled to a plate

SEA

• Free-body-diagram

Fm v_m F₁ F₂ v₁ v₂ M R C © Prof. B.A.T. Petersson

(37)

2

SEA

• Point force excited, infinite thin plate

F = F^ej1t

x y

z

SEA

• Oscillator - plate interface

F₁ F₂ v₂ v₁ v_osc © Prof. B.A.T. Petersson

(38)

3

SEA

• Autospectrum of plate velocity

(39)

Akustik

SEA

• SDOF coupled to a plate

©

v

osc

∆

2

ω

=

v

0

(

ω

)

2

1

Ω

Y

osc

(

ω

)

2

Y

pl

(

ω

)

+

Y

1

(

ω

)

2

d

Ω

d

ω

Ω

∫

∆

ω

∫

(40)

Akustik

SEA

• Autospectrum of average plate velocity

©

1.00 |v 0 | 2 w

ω

(41)

Akustik

SEA

• Input mobility of oscillator

©

|Y 1 | w

ω

|

Y

1

|

(42)

Akustik

SEA

• Input mobility of plate at a point

©

|Y p l | w

ω

|

Y

p

l

|

(43)

Akustik

SEA

• Magnitude of summed mobilities

©

|Y p l + Y 1 | w

ω

|

Y

p

l

+

Y

1

|

(44)

Akustik

SEA

• Inverted magnitude of summed mobilities

©

1 /| Y p l + Y 1 | 2 w

ω

1

/|

Y

p

l

+

Y

1

|

2

(45)

Akustik

SEA

• Narrow band coupling

©

|Y o sc | 2 /| Y p l + Y 1 | 2 w

ω

|

Y

o

sc

|

2

/

|

Y

p

l

+

Y

1

|

2

(46)

Akustik

SEA

• Autospectrum of average plate velocity

1.00 |v 0 | 2 w ©

(47)

Akustik

SEA

• Arbitrary “force spectrum”

©

|Y p l | w

1

/|

Y

|

2

ω

1

/|

Y

|

2

(48)

Akustik

SEA

• Narrow band coupling

©

|Y tr f | 2 w

ω

|

Y

tr

f

|

2

(49)

Akustik

SEA

• Transfer mobility of “specimen 1” for a band

∆ω

©

|Y tr f | 2 D w/M ax [| Y tr f | 2 ] w Dw

ω

|

Y

tr

f

|

2

∆

ω

/

M

ax

[|

Y

tr

f

|

2

]

∆ω

(50)

Akustik

SEA

• Transfer mobility of “specimen 2” for a band

∆ω

©

|

Y

tr

f

|

2

∆

ω

/

M

ax

[|

Y

tr

f

|

2

]

ω

∆ω

(51)

Akustik

SEA

• Integrals and band averages for the two different

realisations

MS1=0.035 MS2=0.38

ω

B

an

d

e

n

er

g

ie

s

MS1=0.35 ©

(52)

Akustik

SEA

• Uncertainty in eigenfrequencies of a plate

0.30 0.25 0.20 0.15 0.10 0.05 0.00

P

S

D

140 120 100 80 60 40 20 0

Natural Frequency, [Hz]

©

(53)

SEA

• Pickup

transfer

function

©

H

=

p

ear

(54)

SEA

Transfer

mobilities

of ordinary

beer cans

©

(55)

1

SEA

• A system consisting of two subsystems

System boundary

SEA

System boundary

(56)

2

SEA

• Reciprocity; point to point quantities

F 1 v 2 F 2 v 1 ©

SEA

• First hypothetical experiment

F _v d Radiating structure Vibrating rigid disc © Prof. B.A.T. Petersson

(57)

3

SEA

• First hypothetical experiment

F’ v’ Radiating structure Vibrating rigid disc © Prof. B.A.T. Petersson

Spatial coupling Temporal coupling

strong

weak

strong

weak

(58)

4

Statistical Energy Analysis

2 ₃

1

2 ₃

1

(59)

5

2 ₃

1

(60)

1

SEA

System boundary

SEA

System boundary

(61)

2

SEA

• Two spring-coupled subsystems

M 1 R 1 K₁=_1/C₁ v 1 M 2 R 2 K₂=_1/C₂ v 2 K c © Prof. B.A.T. Petersson

SEA

• Two sets of mode groups coupled one-to-one

11 12 1n 1N 2N 2n 22 21 © Prof. B.A.T. Petersson

(62)

3

SEA

• Modal gap and bandwidth

3 |Y _|2 b |3 1- 32| Bandwidth r=₂1 b ©

SEA

• Uniform distribution of a mode in a band

3 2₍3₎ 13 3 1n 1/13 © Prof. B.A.T. Petersson

(63)

4

SEA

• Basic assumptions in SEA

– 1. 3

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

2(3)₌const.; _{3 4 13}

(64)

Akustik

SEA

• Basic assumptions in SEA

– 1.

ω

_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

©

f

(

ω

)

=

1

∆

ω

;

ω

n

∈∆

ω

E

m

=

E

∆

ω

N

(65)

Akustik

• Mode to mode coupling

©

(66)

Akustik

Statistical Energy Analysis

Spatial coupling

Temporal coupling

strong

weak

strong

weak

©

v

(

x

,

t

)

=

v

n

Φ

n

(

x

)cos(

ω

n

t

)

n

(67)

Akustik

• Effective mobility

F

1

F

2

F

3

©

Y

11

Σ

=

Y

11

+

Y

12

F

2

F

1

+

Y

13

F

3

(68)

Akustik

• Coupling lossfactor

“E

₀

”

“E

₀

”

“E

₀

”

©

η

ij

=

W

ij

(69)

Akustik

Statistical Energy Analysis

Loss factor:

Coupling loss factor:

Reciprocity:

Power transfer

coefficient:

Dissipation

coefficient

or modal

overlap

factor:

©

η

qq

=

W

diss

ω

E

rev

η

qp

=

W

p

→

q

ω

E

p

η

pq

ω

n

q

=

η

qp

ω

n

p

µ

qq

=

η

qq

ω

n

q

µ

pq

=

η

pq

ω

n

q

(70)

Akustik

• Power balance

1

2

11,

in

11,

diss

22,

diss

12

21

22,

in

©

(71)

Akustik

Statistical Energy Analysis

“E

₀

”

“E

₀

”

“E

₀

”

©

η

ij

=

W

ij

(72)

Akustik

Loss factor:

Coupling loss factor:

Reciprocity:

Power transfer

coefficient:

Dissipation

coefficient

or modal

overlap

factor:

©

η

qq

=

W

diss

ω

E

rev

η

qp

=

W

p

→

q

ω

E

p

η

pq

ω

n

q

=

η

qp

ω

n

p

µ

qq

=

η

qq

ω

n

q

µ

pq

=

η

pq

ω

n

q

(73)

Akustik

Statistical Energy Analysis

• Point coupling

1

2

F

₀

F

1

F

2

v

1

v

2

;

v

10

=

Y

10

F

₀

©

(74)

Akustik

• Two obliquely propagating waves

+

-+

+

-+

-©

(75)

Akustik

• Group speed

©

(76)

Akustik

• Diffuse incidence

d

θ

l

cos

θ

©

(77)

Akustik

Statistical Energy Analysis

• Structure-fluid coupling

V

<|

p

R

|

2

>

S

<|

v

S

|

2

>

©

(78)

Akustik

• Radiation efficiency

S

©

v

p

2

W

p

=

ρ

cSv

p 2

σ

=

W

ρ

cSv

2_p

W

=

σρ

cS v

(

x

,

y

)

2

v

(

x

,

y

)

2

(79)

Akustik

R

ad

ia

ti

o

n

i

n

d

ex

1

0

lo

g

σ

5log(P/2

λ

_c

)

0

Area

S

Perimeter

P

10log(

λ

_c2

/S)

λ

_c

=c/f

_c

2

f

_c

f

_c

f

_c

/4

3c/P

c

2

/(2Sf

c

)(1/4S-1)

100

λ

c/P

2

• Radiation efficiency

©

(80)

Akustik

Statistical Energy Analysis

“E

₀

”

“E

₀

”

“E

₀

”

©

η

ij

=

W

ij

(81)

Akustik

• Point coupling

1

2

F

₀

F

1

F

2

v

1

v

2

;

v

10

=

Y

10

F

₀

©

(82)

Akustik

• Diffuse incidence

d

θ

l

cos

θ

©

(83)

Akustik

• Structure-fluid coupling

V

<|

p

R

|

2

>

S

<|

v

S

|

2

>

©

(84)

Akustik

Statistical Energy Analysis

Physical system

1

F

2

1

2

Π

11,

in

Π

11,

diss

Π

22,

diss

Π

12

Π

21

SEA model

(flexural waves only)

©

(85)

Akustik

Power balance

⇔

©

Π

_1,_in

{

}

=

ω

η

11

n

1

+

η

21

n

1

−η

12

n

2

−η

₂₁

n

₁

η

₂₂

n

₂

+

η

₁₂

n

₂













E

₁m

E

₂m

















∆ω

Π

_1,_in

{

}

=

ω

µ

11

+

µ

21

−

µ

12

−

µ

₂₁

µ

₂₂

+

µ

₁₂













E

₁m

E

₂m

















∆ω

(86)

Akustik

Physical system

F

1

2

3

Bending modes

Long. modes

4

Bending modes

Long. modes

3

4

Bending modes

Long. modes

©

(87)

Akustik

SEA model

1

3

Π

11,

in

Π

21

Π

13

Π

31

5

Π

35

Π

53

2

4

6

Π

12

Π

43

Π

34

Π

65

Π

56

Π

24

Π

42

Π

46

Π

64

Π

51

©

(88)

Akustik

Transmission efficiencies - 10log

τ

B2B3

τ

B3B3

τ

L2B3

τ

L1B3

t

1

=

t

3

=15 mm

t

2

=7 mm

©

(89)

Akustik

Transmission efficiencies - 10log

τ

L1L3

τ

B2L3

t

1

=

t

3

=15 mm

t

2

=7 mm

©

(90)

Akustik

Statistical Energy Analysis

• Model of ships section

made of cardboard

and plastic film

©

(91)

Akustik

• Comparison measurements and calculations

» Velocity level difference (bulkheads 241 and 264)

(92)

Statistical

Energy

Analysis

• Cross-section of a ship

indicating deck and

hull plates

©

(93)

Akustik

-20 -10 0 10 20 ²L v , [ dB ] 102 103 Frequency, [Hz]

» Velocity level difference (bulkheads 264 and 277)

©

(94)

Akustik

• Double-bottom structure

©

(95)

Akustik

Statistical Energy Analysis

• Double-bottom structure — SEA model

©

(96)

Akustik

©

(97)

Akustik

• Coupled SDOFs

1

2

F

(

ω

)

ω ∈

∆

ω

E

1

=

E

1,Tot

E

2

=

E

2,Tot

;

ψ

2

(

x

2

)

©

(98)

Akustik

Statistical Energy Analysis

• Coupled multi-modal subsystems

1

2

F

(

ω

) ;

ω ∈

∆

ω

E

1

,

N

1

E

2

,

N

2

ψ

n

x

1

)

ψ

2n

x

2

)

©

(99)

Akustik

• 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

l

x

l

y

l

x

+

∆

l

x

l

y

+

∆

l

y

i

j

Variations in the boundary conditions

Variations in the position

Power input changes

©

p

(

x

)

ψ

_n

(

x

)

d

Ω

(100)

Akustik

Statistical Energy Analysis

1

2

F

(

ω

) ;

ω ∈

∆

ω

E

1

,

N

1

E

2

,

N

2

E

α

E

σ

©

(101)

Akustik

• Partial variances

•

1) Äquipartition

Var[

E

α

]

•

2) Modale Dichte

Var[

N

1

] ~ Σ

α

•

3) Randbedingung

Var[

B

σα

]

•

4) Position

Var[

x

]

©

v

(

x

,

t

)

2

=

2

ψ

σ

2

M

₂

B

σα

E

α

∑

ωη

σ

+

B

σα

α

∑









∑

ψ

(

x

)

2

(102)

Akustik

• 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

_α2

−

ω

_σ2

)

2

+

(

η

_α

ω

_α

+

η

_σ

ω

_σ

)(

η

_α

ω

_α

ω

_σ2

+

η

_σ

ω

_σ

ω

_α2

)



1

∆

ω

∫

B

σα

(

ω

α

)

d

ω

α

B

σα ω_α

=

2

K

c 2

M

_α

M

_σ

π

2

(

ηω

+

η

_σ

ω

_σ

)

ω

∆

ωω

_σ2

(103)

1

• Five factors associated with spatial variance

1. E_n2 _E m 2. N_i2_N j 3. Y_i(0,y) ₃ Y_j(0,y) 4. ₂ n(xi) 32n(xj) 5. W_in,i 2_W in,j l_x l_y l_x+₂l_x l_y+₂l_y i j

Variations in the boundary conditions

Variations in the position

Power input changes

(104)

2 • Point-excited finite plate

Statistical Energy Analysis

v(x,t) F(x_s,t)

• Eigenvalue distribution

(105)

3

• Effect of

eigenvalue

distribution

• Confidence

coefficients

(106)

4 Institute

of Technical Acoustics

Statistic Energy Analysis

• Uncertainty assessment • Complete variance

Institute of Technical Acoustics

Statistical Energy Analysis

• Power balance equation — Matrix notation

1C E = P

(107)

5 Institute

of Technical Acoustics

Statistical Energy Analysis

• Normalised variance of subsystem energy

G_pi= P_i/1

Institute of Technical Acoustics

Statistical Energy Analysis

• Example of uncertainty propagation for three-plate system

Uncertain parameters: l and h

(108)

6 Institute

of Technical Acoustics

Statistical Energy Analysis

• Confidence band for energy in subsystem 3

--- 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

(109)

7

Statistical Energy Analysis

• 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)