The Statistical Energy Analysis (SEA)

1
The Statistical Energy Analysis (SEA)
S E A
by Michael Fischer JASS 2006 in St.
Petersburg
2
The Statistical Energy Analysis (SEA)

1. Methods used for vibration problems

by Michael Fischer JASS 2006 in St.
Petersburg
3
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
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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
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The Statistical Energy Analysis (SEA)
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
Hz
frequency Hz
Wheel of a bike
by Michael Fischer JASS 2006 in St.
Petersburg
6
The Statistical Energy Analysis (SEA)
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
Hz
frequency Hz
by Michael Fischer JASS 2006 in St.
Petersburg
7
The Statistical Energy Analysis (SEA)
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
Hz
frequency Hz
by Michael Fischer JASS 2006 in St.
Petersburg
8
The Statistical Energy Analysis (SEA)
2. Introductory example
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

• The air inside the room also shows
• modes (starting at about 50 Hz)

by Michael Fischer JASS 2006 in St.
Petersburg
9
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/mm2)
• masses of the materials (e.g. concrete 25
  kN/m2)
• 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
10
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
11
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
12
The Statistical Energy Analysis (SEA)
4. Motivation for SEA
by Michael Fischer JASS 2006 in St.
Petersburg
13
The Statistical Energy Analysis (SEA)
5. Deterministic approach modal superposition
mode shape (point of observation)
system response
contribution of the i.th mode
by Michael Fischer JASS 2006 in St.
Petersburg
14
The Statistical Energy Analysis (SEA)
5. Deterministic approach modal superposition
influence of the geometry of excitation
amplification functioninfluence of the frequency
of excitation
by Michael Fischer JASS 2006 in St.
Petersburg
15
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)
contribution of the i.th mode
by Michael Fischer JASS 2006 in St.
Petersburg
16
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
17
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
18
The Statistical Energy Analysis (SEA)
6. Energetic approach 6.3 Averaging over the
points of observation ( Step 1)
Orthogonality of modeshapes
(Summing up the modal energy)
by Michael Fischer JASS 2006 in St.
Petersburg
19
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 dont have to be
  calculated!

by Michael Fischer JASS 2006 in St.
Petersburg
20
The Statistical Energy Analysis (SEA)
6. Energetic approach 6.4 Averaging over the
points of excitation ( Step 2)
mean modal force
modal mass
by Michael Fischer JASS 2006 in St.
Petersburg
21
The Statistical Energy Analysis (SEA)
6. Energetic approach 6.4 Averaging over the
points of excitation ( Step 2)
force
amplification function
total mass
? no information about the modes necessary!
by Michael Fischer JASS 2006 in St.
Petersburg
22
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
amplification
by Michael Fischer JASS 2006 in St.
Petersburg
23
The Statistical Energy Analysis (SEA)
6. Energetic approach 6.5 Averaging over the
frequencies of excitation ( Step 3)
force
total mass
damping
frequency band
by Michael Fischer JASS 2006 in St.
Petersburg
24
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
frequency band
total mass
damping
by Michael Fischer JASS 2006 in St.
Petersburg
25
The Statistical Energy Analysis (SEA)
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). -mean input power in a frequency band
force
frequency band
total mass
? input power is independent from damping
by Michael Fischer JASS 2006 in St.
Petersburg
26
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
27
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
damping
by Michael Fischer JASS 2006 in St.
Petersburg
28
The Statistical Energy Analysis (SEA)
7. Balance of power- hydrodynamic analogy
Expansion to coupled systems -For every
sub-system holds
by Michael Fischer JASS 2006 in St.
Petersburg
29
The Statistical Energy Analysis (SEA)
7. Balance of power- hydrodynamic analogy
Expansion to coupled systems -Energy flow
between two sub-systems
modal energy
coupling loss factor
by Michael Fischer JASS 2006 in St.
Petersburg
30
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)
by Michael Fischer JASS 2006 in St.
Petersburg
31
The Statistical Energy Analysis (SEA)
8. Equations of the SEA
coupling
by Michael Fischer JASS 2006 in St.
Petersburg
32
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
33
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
34
The Statistical Energy Analysis (SEA)
8. Conclusions and look into the future
-example
by Michael Fischer JASS 2006 in St.
Petersburg
35
The Statistical Energy Analysis (SEA)
8. Conclusions and look into the future
by Michael Fischer JASS 2006 in St.
Petersburg
36
The Statistical Energy Analysis (SEA)
Thank you for your attention!
by Michael Fischer JASS 2006 in St.
Petersburg