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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
  load and the sound pressure.

 -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
  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




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)




          Thank you for your attention!




by Michael Fischer   JASS 2006 in St. Petersburg

				
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