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```									    The Statistical Energy Analysis (SEA)

SEA

by Michael Fischer   JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

1. Methods used for vibration problems:

by Michael Fischer   JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

1. Methods used for vibration problems:
-Usually we are dealing with models like FEM, (BEM) and analytical models
which enable us to calculate for deterministic 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.

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

2. Introductory example :

sound pressure - harmonic force
-The figure shows for the 18 points
in the room all measured transfer

Level difference
functions between the harmonic

-lt can clearly be seen, that at
higher frequencies the transfer
functions differ considerably.

                frequency Hz
Wheel of a bike           Hz

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

2. Introductory example:

sound pressure - harmonic force
- reason for the high differences:

Level difference
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.

frequency Hz
Hz

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

2. Introductory example:

sound pressure - harmonic force
- Even slight temperatur differences in
the room, which practically cannot be

Level difference
eliminated, influence the positions of the
Eigenfrequencies so that a detailed
prediction cannot be given

frequency Hz
Hz

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

2. Introductory example:                mode   empty room       room with disturbing objects
Frequency [Hz]   Frequency [Hz]
-The air inside the room also shows    1      56,8             49,0
modes (starting at about 50 Hz)       2      70,2             68,6
3      85,4             79,5
4      90,3             85,3
5      102,6            96,2
6      110,6            104,9
7      114,3            107,9
8      124,3            118,7
9      134,1            127,5
10     141,8            134,6
11     142,7            139,8
12     152,8            149,0
13     159,0            149,9
14     165,6            153,9
15     173,2            162,5
16     173,5            171,2
…      …                …

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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)
•   ...

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

3. Historical example:
In the early 1960s:
-prediction of the vibrational response to
rocket noise of satellite launch vecicles and
-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

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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 which
are 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.

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

4. Motivation for SEA:

by Michael Fischer   JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

5. Deterministic approach: modal superposition

mode shape (point of observation)

velocity, pressure    i  i
i

system response             contribution of the i.th mode

by Michael Fischer      JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

5. Deterministic approach: modal superposition

influence of the geometry                         V  
of excitation
D0
D1  0

 p   dV  i
j ( 2f )
D2  D1

i      V
                                1

mi * ( 2fi ) 2
f2                                               
(1  2 )  j                        1   2

fi

amplification function
influence of the frequency of excitation

by Michael Fischer       JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

6. Energetic approach:
6.1 Shift to energy
-In the first step a shift from velocities to energy is carried out.
-the mean kinetic energy is proportional to the mean square velocity

mode shape (point of observation)

2
                  
v 
2


 i
i  i 



contribution of the i.th mode

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

6. Energetic approach:
6.2 Averaging in the SEA
- Now we increase the prediction accuracy by appropriate averaging
in several steps

by Michael Fischer       JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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

- by this step the phase information gets lost

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

6. Energetic approach:
6.3 Averaging over the points of observation („ Step 1“)
2
                   
v 
2


i
i  i 


Orthogonality of modeshapes

2
1                                 1
               i  i  dV                     
2                                                            2        2
v                                                                   i dV
2V                                                     i
2V
V          i                                    i       V

1           F 2  i2

2V
   m
i
*2
 (2fi ) 4
2

  i  i2 dV
i                                V

(„Summing up the modal energy“)

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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 don‘t have to be calculated!

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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

mean modal force

1
 F 2  i2 dV
V
 i2                  V
2
  i2
               
    i 2 dV   ( 2  fi )4
               
     V         

modal
mass

by Michael Fischer            JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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

2
F  i
2
amplification function
vi2   
m2 (2fi )4

total mass

 no information about the modes necessary!

by Michael Fischer          JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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

-To simplify the mean square velocity
once again, we assume several similar
modes N in a frequency band

fl   fu

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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

2         F2            1
vi        2               
m  (2fi )  2 2(fo  fu )

total mass       damping     frequency band

by Michael Fischer             JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

6. Energetic approach:
6.5 Averaging over the frequencies of excitation („ Step 3“)
Energy within a certain frequency band:

centre frequency
force

2
m  vi            F2            N
E f                               
2      8  m  (2fm )   (fo  fu )

frequency band
total mass       damping

by Michael Fischer          JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)
F t 
6. Mean input power
-We are looking at one „sub-system“ (frequency band)
-We assume a steady state vibration:                       c       k
„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:

force
_
F2       N
P                                      frequency band
4  m ( fo  fu )
total mass

 input power is independent from damping

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

7. Balance of power- hydrodynamic analogy

Mean input power P

Energy E in the sub-system

Dissipated energy

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

7. Balance of power- hydrodynamic analogy
-every sub-system is considered as a
energy reservoir
Pin  Pout
-The dissipated energy
is proportional to the absolute dynamic
energy E of the sub-system:

Pdiss  2fm  E  

damping

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

7. Balance of power- hydrodynamic analogy
Expansion to coupled systems:
-For every sub-system holds:
Pi,in  Pi,out

Pi,diss  2fm  Ei  i

by Michael Fischer           JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

7. Balance of power- hydrodynamic analogy
Expansion to coupled systems:
-Energy flow between two sub-systems:

 Ei E j 
Pij  2fm  ij  Ni    
 Ni N j 
        

modal energy
coupling loss factor

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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)

Pi,in  Pi,diss         P
j, ji
i, j

by Michael Fischer       JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

8. Equations of the SEA
        k                                                      
 1   1i  N1      ( 12N1)        ...        ( 1k N1)      E1 
      i 1                                                       N1   P 
                           k                                    E   1
 (  21N2 )        2    2i  N2   ...        (  2k N2 )    2   P2 
                         i 2                                    N2   ... 
         ...                ...         ...             ...        ...   
                                                                   Ek  Pk 
 (  N )                                              k        
...         ...     k    ik  Nk  Nk 
                                                                 
k1 k
                                                     i k       

damping                                                 coupling

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

8. Equations of the SEA
-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

by Michael Fischer         JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

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

by Michael Fischer        JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

8. Conclusions and look into the future
-example:

by Michael Fischer   JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)

8. Conclusions and look into the future

Rail-Impedance-Model RIM

by Michael Fischer   JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)