Fischer SEA

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The Statistical Energy Analysis (SEA)

by Michael Fischer JASS 2006 in St. Petersburg

1.Methods used for vibration problems:


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1.Methods used for vibration problems:-Usually we are dealing with models like FEM, (BEM) and analytical models

which enable us to calculate fordeterministic loads and defined model

parameters deterministic responses.

-Typically the calculated value is given in detail with respect to frequency,time and location.

-However, the level of discretization of time/frequency and the geometric

data has to be defined at the basis of theoretical considerations regarding

wave-lengths, eigenmodes etc.

-The following introductory example shows, that at higher frequencies the

reliability of the result of calculation might be considerably reduced.


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2. Introductory example:- room (25 m3)

- limited by a steel plate

- one of the boundary surfaces is excited by a harmonic load

- 18 points in the room are considered


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2. Introductory example:-The figure shows for the 18 points

in the room all measured transfer

functions between the harmonic

load and the sound pressure.

-lt can clearly be seen, that at

higher frequencies the transfer

functions differconsiderably.

Hz

f requency Hz

Leve

ldifference

soun

dpressure

-harmon

icforce

Wheel of a bike



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2. Introductory example:- reason for the high differences:

different contributions of single modes

which are close together regarding

their eigenfrequency.

So e.g. in the centre of the room and a

tonal excitation at 250 Hz, a difference

of about 20 dB (factor 10) between the

individual functions is observed.

Hz

f requency Hz

Leve

ldifference

soun

dpressure

-harmon

icforce


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2. Introductory example:- Even slight temperatur differences in

the room, which practically cannot be

eliminated, influence the positions of the

Eigenfrequencies so that a detailed

prediction cannot be given

Hzf requency Hz

Leve

ldifference

soun

dpressure

-harmon

icforce


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2. Introductory example:-The air inside the room also shows

modes (starting at about 50 Hz)

mode empty roomFrequency [Hz]

room with disturbing objectsFrequency [Hz]

1

2

3

4

5

67

8

9

10

11

12

1314

15

16

56,8

70,2

85,4

90,3

102,6

110,6114,3

124,3

134,1

141,8

142,7

152,8

159,0165,6

173,2

173,5

49,0

68,6

79,5

85,3

96,2

104,9107,9

118,7

127,5

134,6

139,8

149,0

149,9153,9

162,5

171,2


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2. Introductory example:

Possible Uncertainties of

boundary conditions (e.g. clamped/free edge) dynamic material properties (e.g. concrete: E ~ 30kN/mm^2)

masses of the materials (e.g. concrete: 25 kN/m^2)

damping

load distribution (e.g. position of the machine)

frequency of excitation (e.g. velocity of train)

...


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3. Historical example:In the early 1960s:

-prediction of the vibrational response to

rocket noise of satellite launch vecicles and

their payloads

-problem: the frequency range of significant

response contained the natural frequencies of

a multitude of higher order modes:

-the Saturn launch vehicle possessed about

500.000 natural frequencies

in the range 0 to 2000 Hz


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4. Motivation for SEA:-The both examples above are leading to the insight that

at higher frequencies a method with less detailing has to be accepted.

-A detailed analysis at the basis of FEM approach (input at a point of

excitation, output at a point of observation) would lead to results whichare very sensitive to slight changes in the input parameters

(factor 10!).

-In order to obtain acceptable sensitivities of the results, but to describe

nevertheless the system response, we will give the results in an averaged

sense.


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4. Motivation for SEA:


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5. Deterministic approach: modal superposition

i iipressurevelocity,

contribution of the i.th mode

mode shape (point of observation)

system response


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5. Deterministic approach: modal superposition

jf

f

fj

fm

dVp

i

ii

V

i

i

)1(

)2(

)2(*2

22

influence of the geometry

of excitation

amplification function

influence of the frequency of excitation

1

1 2

V

0D0

1D

12DD


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6. Energetic approach:6.1 Shift to energy

2

i ii

2v

-In the first step a shift from velocities to energy is carried out.

-the mean kinetic energy is proportional to the mean square velocity

contribution of the i.th mode

mode shape (point of observation)


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6. Energetic approach:6.2 Averaging in the SEA


- Now we increase the prediction accuracy by appropriate averaging

in several steps


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6. Energetic approach:6.3 Averaging over the points of observation ( Step 1)


- by this step the phase information gets lost


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2

i

ii2v


6. Energetic approach:6.3 Averaging over the points of observation ( Step 1)

i V

2i

2

i4

i

2*i

2i

2

i V

2i

2i

2

i

ii2

dV)f2(m

F

V2

1

dVV2

1dV

VV2

1v

(Summing up the modal energy)


Orthogonality of modeshapes


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6. Energetic approach:6.4 Averaging over the points of excitation ( Step 2)


-By this averaging, the information about the shape of the individual

eigenmodes is eliminated and has no longer to be considered

This means: the modes dont have to be calculated!


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2

4

2

2

22

2

2

1

i

i

V

i

V

i

i

fdV

dVFV

)(

modal

mass

mean modal force



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v F

m fi

i

i

22 2

2 42

( )

no information about the modes necessary!

total mass

amplification function

force


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6. Energetic approach:6.5 Averaging over the frequencies of excitation ( Step 3)


uflf

-To simplify the mean square velocity

once again, we assume several similar

modes N in a frequency band


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)ff(2

1

2)f2(m

Fv

uoi2

22

i

force

total mass damping frequency band


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)ff(

N

)f2(m8

F

2

vmE

uom

22

if

force

total mass dampingfrequency band

Energy within a certain frequency band:

centre frequency


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6. Mean input power


force

total mass

frequency band

input power is independent from damping

)ff(

N

m4

FP

uo

2_

tF

kc

-We are looking at one sub-system (frequency band)

-We assume a steady state vibration:

the mean input power, which is introduced during

one cycle of vibration equals to the dissipated power

due to damping (compare SDOF system).

-mean input power in a frequency band:


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7. Balance of power- hydrodynamic analogy


Mean input power P

Energy E in the sub-system

Dissipated energy


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outin PP


Ef2P mdiss

-every sub-system is considered as a

energy reservoir

-The dissipated energy

is proportional to the absolute dynamic

energy E of the sub-system:

damping


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iimdiss,i Ef2P


Expansion to coupled systems:

-For every sub-system holds:

out,iin,i PP


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Expansion to coupled systems:

-Energy flow between two sub-systems:

j

j

i

iiijmij

N

E

N

ENf2P

modal energy

coupling loss factor


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8. Equations of the SEA


The governing equations can be derived by considering:

the loss of energy by damping

the energy flow between every pair of sub-systems (coupling)

j,iij,j

diss,iin,i PPP


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1 11

1 12 1 1 1

21 2 2 22

2 2 2

1

1

1

2

2

1

2

i

i

k

k

ii

k

k

k k k ik

i k

k

k

k

k

N N N

N N N

N N

E

N

E

N

E

N

P

P

( ) ... ( )

( ) ... ( )

... ... ... ...

( ) ... ...

......

Pk

damping coupling


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-Related to the different possible deflection patterns

(e.g. bending, shear, torsional waves):

each part of the structure might appear as various energy reservoirs

and thus described by various governing equations.

-FE: usually a high dicretization of the structure is necessary

-SEA: based on calculation of global values

computational costs are much smaller

interactive planning by the engineer is possible


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8. Conclusions and look into the future


-Energy methods have a huge impact on the methodology of noise

and vibration prediction

-especially hybrid methods can carry out vibroacoustic investigations

with a good confidence


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-example:


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Rail-Impedance-Model RIM


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by Michael Fischer JASS 2006 in St Petersburg

Thank you for your attention!


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Fischer SEA