Transcript The Statistical Energy Analysis (SEA)

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
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 differ considerably.
Level difference
sound pressure - harmonic force
The Statistical Energy Analysis (SEA)

Wheel of a bike
by Michael Fischer
JASS 2006 in St. Petersburg
frequency
HzHz
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.
Level difference
sound pressure - harmonic force
The Statistical Energy Analysis (SEA)
frequency
HzHz
by Michael Fischer
JASS 2006 in St. Petersburg
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
Level difference
sound pressure - harmonic force
The Statistical Energy Analysis (SEA)
frequency
HzHz
by Michael Fischer
JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)
2. Introductory example:
-The air inside the room also shows
modes (starting at about 50 Hz)
by Michael Fischer
mode
empty room
Frequency [Hz]
room with disturbing objects
Frequency [Hz]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
…
56,8
70,2
85,4
90,3
102,6
110,6
114,3
124,3
134,1
141,8
142,7
152,8
159,0
165,6
173,2
173,5
…
49,0
68,6
79,5
85,3
96,2
104,9
107,9
118,7
127,5
134,6
139,8
149,0
149,9
153,9
162,5
171,2
…
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
by Michael Fischer
contribution of the i.th mode
JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)
5. Deterministic approach: modal superposition
influence of the geometry
of excitation
V  
D0
D1  0
D2  D1
 p   dV
j ( 2f )
i 

2
f2
mi * ( 2fi )
(1  2 )  j
fi
i
V
1

1
amplification function
influence of the frequency of excitation
by Michael Fischer
JASS 2006 in St. Petersburg
2
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)

v  

2

i

i  i 

2
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  


i

i  i 

2
Orthogonality of modeshapes

1
2
v 
 
2V 
V

1


2V
m
i
i
2

1
i  i  dV 

2V

F 2  i2
*2
i
 (2fi ) 4
2
    dV
2
i
i
2
i
V

  i  i2 dV
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
 i2 
1
 F 2  i2 dV
V
V

    i 2 dV

V

2

  ( 2  fi )4


  i2
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
F  i
2
2
vi

2
m (2fi )
2
amplification function
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
by Michael Fischer
JASS 2006 in St. Petersburg
fu
The Statistical Energy Analysis (SEA)
6. Energetic approach:
6.5 Averaging over the frequencies of excitation („ Step 3“)
force
F2  
1
vi  2

m  (2fi )  2 2(fo  fu )
2
total mass
by Michael Fischer
damping
frequency band
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:
force
centre frequency
2
E f
m  vi
F2
N



2
8  m  (2fm )   (fo  fu )
total mass
by Michael Fischer
damping
JASS 2006 in St. Petersburg
frequency band
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:
„the mean input power, which is introduced during
one cycle of vibration equals to the dissipated power
due to damping“ (compare SDOF system).
c
-mean input power in a frequency band:
force
F2
N
P

4  m (fo  fu )
_
frequency band
total mass
 input power is independent from damping
by Michael Fischer
JASS 2006 in St. Petersburg
k
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
-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
Pin  Pout
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
i, j
j, ji
by Michael Fischer
JASS 2006 in St. Petersburg
The Statistical Energy Analysis (SEA)
8. Equations of the SEA
k


( 12N1)
 1   1i  N1


i 1

k


 (  21N2 )
  2    2 i  N2



i 2

...
...

 (  N )
...
k1 k


damping
by Michael Fischer
...
...
...
...

( 1k N1)
  E1 
  N1 
 E 
(  2k N2 )    2  
  N2 
  ... 
...
  Ek 
k

   
  k    ik  Nk  Nk 

 
i k
coupling
JASS 2006 in St. Petersburg
 P1 
P 
 2
 ... 
 
Pk 
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