The Effect of the Weld Type on Ensemble Average in SEA

Abstract

Statistical energy analysis (SEA) predicts the average response of a population. This population consists of similar vibroacoustic structures subjected to white noise forces at high frequencies. However, usually in engineering applications, the SEA method is used to determine the response of a particular structure. It results from the assumption, that in the case of complex structures the response of a single member from the population will not significantly differ from the average. The aim of the research was to check the above assumption on a simple structures. Six similar mechanical structures were tested. Each structure consisted of two plates welded together at right angles. The vibration (velocity) level reduction between the plates was determined. From the point of view of the SEA, each of the six L-shaped structures is represented by the same SEA system composed of two 2D subsystems connected by line junction. However, the resulting line junctions have been achieved by two different welding techniques. This detail is omitted during SEA calculations. Vibration reduction obtained on a single structure was compared with ensemble average covering the entire six-element sample. The influence of the weld type on transmission was considered. The obtained results were compared with the SEA prediction.

Full text

PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany
The Effect of the Weld Type on Ensemble Average in SEA
Paweł Nieradka1; Sebastian Szarapow1
1 KFB Acoustics sp. z o. o., Poland
ABSTRACT
Statistical energy analysis (SEA) predicts the average response of a population. This population consists of
similar vibroacoustic structures subjected to white noise forces at high frequencies. However, usually in
engineering applications, the SEA method is used to determine the response of a particular structure. It results
from the assumption, that in the case of complex structures the response of a single member from the
population will not significantly differ from the average. The aim of the research was to check the above
assumption on a simple structures. Six similar mechanical structures were tested. Each structure consisted of
two plates welded together at right angles. The vibration (velocity) level reduction between the plates was
determined. From the point of view of the SEA, each of the six L-shaped structures is represented by the same
SEA system composed of two 2D subsystems connected by line junction. However, the resulting line
junctions have been achieved by two different welding techniques. This detail is omitted during SEA
calculations. Vibration reduction obtained on a single structure was compared with ensemble average
covering the entire six-element sample. The influence of the weld type on transmission was considered. The
obtained results were compared with the SEA prediction.
Keywords: SEA, Transmission, Junctions
1. INTRODUCTION
The energy of the bending waves is directly related to the radiated acoustic power. Therefore, the
ability to predict bending wave transmission in structures is an important aspect of noise and vibration
control. Structures of great practical importance, which are the object of numerous studies, are steel
constructions. Steel constructions commonly found in mechanical engineering often consist of welded
plates. Hence there is a strong need to predict energy transmission in such systems.
The Statistical Energy Analysis (SEA) is a popular tool for conducting vibroacoustic simulations in
the high frequency range. The basic parameter used in SEA simulations is called Coupling Loss Factor
(CLF), which describes subsystem losses resulting from energy flow to another subsystem. Welded
joints are often approximated by line junctions for which there exists a theoretical CLF formula. The
work will investigate the influence of the geometry of the applied welds on the transmission of bending
waves. This effect is not included in the basic formula for the simple line connection and can be
considered in the context of ensemble average. Using the SEA method, one obtain results averaged
within similar structures (ensembles). This means, that SEA results are only approximations when they
are associated with a specific structure (1). Six structures (SEA sys tems) were tested in the work. Each
system consisted of two steel plates (subsystems) connected at right angles. There were the following
differences between the systems:
a) different arrangement of the plates during w elding,
b) different location of the weld.
In the further part of the work it will be checked whether the introduced changes have a significant
impact on the transmission of vibrations and whether all six systems can be assigned to one set called
"structures similar to one another".
1 p.nieradka@kfb-acoustics.com
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2. THEORY
2.1 Coupling Loss Factor of Line Junctions
The Coupling Loss Factor of line-coupled 2D subsystems is determined from (1):
ߟଵଶ ൌܮܿ௚ଵ
ߨ߱ܵଵ߬௕
(1)
where ܿ௚ଵ is wave group speed of subsystem 1, ܮ is length of connection, ߱ is angular frequency, ܵଵ
is the surface area of the first subsystem, ߬௕ is bending wave transmission coefficient averaged over
the angle of incidence. As one can see, the CLF coefficient is directly proportional to the transmission
coefficient. In the literature, one can find formulas for the transmission co efficient assuming rigid
plate connection (3) and models allowing to take into account the finite stiffness and resistance of the
joints (4, 5). The transmission coefficient was determined in the present study based on the model
shown in (4). Mechanical parameters of the weld were also assumed as in (4): stiffness was set to 100
GN/m and damping was set to 1.8 kNs/m.
2.2 Statistical Energy Analysis of Two Coupled Plates
The SEA method allows to determine the average mechanical energies of subsystems, ܧ, by
solving a system of linear equations, where the column of free words are average input powers, and the
main matrix is formed by DLF and CLF (2). In the case of two coupled plates, the SEA model comes
down to the solution of the following system:
ʹߨ݂ቀߟଵ൅ߟଵଶ െߟଶଵ
െߟଵଶ ߟଶ൅ߟଶଵቁ൬ܧଵ
ܧଶ൰ൌ൬
ܲଵ
ܲଶ൰
(2)
where ݂ is frequency in Hz, ߟଵ and ߟଶ are damping loss factors (DLF), ߟଶଵ and ߟଵଶ are coupling
loss factors (CLF) computed as shown in 2.1. After solving the system of equations (2), the average
square of vibration velocity of the subsystem with the mass ܯ can be determined from the equation:
ݒଶൌܧȀܯ.
(3)
Statistical Energy Analysis works best if modal overlap ܯ௢ is greater than unity (6):
ܯ௢ൌ݊ߟ߱൐ͳ
(4)
where ݊ is asymptotic modal density:
݊ൌܵ߱Ȁሺʹߨܿ
௚ܿ௙ሻ.
(5)
In equation (5) ܿ௚ is subsystem group speed, ܿ௙ is phase speed, ܵ is the area of the plate.
In addition, the subsystems should be weakly coupled. The coupling strength can be estimated
based on the parameter ߛ௜௝:
ߛ௜௝ ൌߟ
௜௝Ȁߟ௜.
(6)
Coupling is assumed to be weak, if ߛ௜௝ اͳ condition holds.
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3. MEASUREMENTS
3.1 Tested Objects
The research objects were six mechanical constructions. Each construction consisted of two steel
plates welded at right angles. The structures during the measurements were freely suspended on the
strings (two anchor points were placed on each plate). Each plate was assigned a number from 1 to 12,
which was used to identify them. The mechanical and geometric parameters of the plates are
summarized in Table 1. Internal losses of plates can be found in section 4.2.
Table 1 – Parameters of plates
Parameter
Val u e
Length, L
0.49 m
Width, W
0.49 m
Thickness, h
0.002 m
Density, ߩ
7884 kg/m
3
Young Modulus, E
200 GPa
Poisson Number, ߪ
0.3
3.2 Types of Junctions
The individual pairs of plates have been coupled together using one of two joints: Weld A and Weld
B. Differences between welds are shown in Table 2. Welds were made using MAG (Metal Active Gas)
technique.
Table 2 – Types of tested welds
Junction
Pairs of plates
Geometry
Photo
Weld A
12-1, 2-3, 4-5
Weld B
6-7, 8-9, 10-11
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3.3 Damping and Coupling Loss Factors
The work concerns the transmission of bending waves, that is why the components of the velocity
vector perpendicular to the plate are considered. Before merging of plates, measurements of internal
losses (ߟ, DLF) by the structural reverberation time were carried out:
ߟ ൌ ʹǤʹȀሺܴܶ଺଴ ڄ݂ሻ
(7)
where ܴܶ଺଴ is structural reverberation time
The plates were then joined and a further series of measurements was carried out. Mechanical
power was injected at individual points selected on the plates using the vibration exciter. Each plate
was excited in three randomly selected places. Then, for each excitation, the average mechanical
velocity was determined on the source and receiver subsystem (on each plate the signal was picked up
using accelerometers in six randomly selected places). Excitation of the plate in many places and
averaging of individual responses approximates the "rain -on-the-roof" excitation, which is one of the
assumptions of the SEA method. The energy ratios of the receiver and source plates have been
determined. Additionally, TLF (Total Loss Factor) measurements were taken based on the structural
reverberation time of the entire structure to determine CLF and DLF using the Energy Ratio Method
(7). The system during measurements is shown in Figure 1.
Figure 1 - Plate 2 excited at point 1. Response measured on Plate 2 at point 3
In order to determine DLF (ߟଵ, ߟଶ) the system of equations was solved, where the main matrix
consisted of the energy ratios ݁௜௝ (receiver plate ݅ to the source plate ݆), while the column of free
words contained the measured TLF (ߟଵ௧ǡߟଶ௧)
൬ͳ݁
ଶଵ
݁ଵଶ ͳ൰ቀߟଵ
ߟଶቁൌቀߟଵ௧
ߟଶ௧ቁ
.
(7)
Using the results from (7) the second system of equations was solved, allowing to determine the CLF
coefficients (ߟଵଶǡߟ
ଶଵሻ
൬ͳെ݁
ଶଵ
െ݁ଵଶ ͳ൰ቀߟଵଶ
ߟଶଵቁൌቀ݁ଶଵߟଶ
݁ଵଶߟଵቁ
.
(8)
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4. RESULTS
4.1 Comparison of Welds
Figure 2 presents the averaged Velocity Level Difference ܦ for welds A and B. Energy ratios from
which the results can be determined are listed in table 3.
Figure 2 - Measurement results for welds A and B
In order to determine if there is a difference in energy transmission using welding A or B, the results of
the measurements were divided into two groups. The first group (the representation of the population
of plates with weld A) has been assigned six parameters determined from the relationship:
݁஺ǡ௜ ൌܧ௦௢௨௥௖௘ǡ஺ǡ௜
ܧ௥௘௖௘௜௩௘௥ǡ஺ǡ௜ Ǣ݅ ൌ ͳǡ ʹǡǥǡ͸
(9)
Similarly, six parameters were assigned to the second group (representation of the plate population
with weld B):
݁஻ǡ௜ ൌܧ௦௢௨௥௖௘ǡ஻ǡ௜
ܧ௥௘௖௘௜௩௘௥ǡ஻ǡ௜ Ǣ݅ ൌ ͳǡ ʹǡǥǡ͸
(10)
where ܧ௦௢௨௥௖௘ǡ௑ǡ௜ – spatially averaged energy of the source system for joint X (sample i), ܧ௥௘௖௘௜௩௘௥ǡ௑ǡ௜
– spatially averaged energy of the receiver system for joint X (sample i). The size of individual groups
is 6, because for a given structure two energy ratios have been determined by exchanging the source
and receiver plates. It is worth noting that definitions (9) and (10) have the energy of the receiving
system in the denominator, that's why they are the inverse of parameters from the main matrix in
equations (7) and (8). Then, the Welch's t-test at 0.05 significance level was conducted to check the
following null hypothesis ܪ଴ǣ The average mechanical energy ratio of the source and receiver plate is
independent of the type of weld used. Table 3 shows the determined average energy ratios for each
weld type, standard deviations ߪ and test results.
1/3-octave band center frequency [Hz]
Velocity Level Difference, D [dB]
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Table 3 – Results of two-tailed Welch’s t-test
݂
ሾሿ
݁஺
ഥሺߪ
௘ಲሻ
݁஻
ത
ത
ത
ሺߪ௘ಳሻ
ݐ
ݐ଴Ǥ଴ହ
݂
ሾሿ
݁஺
ഥሺߪ
௘ಲሻ
݁஻
ത
ത
ത
ሺߪ௘ಳሻ
ݐ
ݐ଴Ǥ଴ହ
100
1.60 (0.85)
1.02 (0.17)
1.63
2.51
1000
1.30 (0.23)
1.52 (0.36)
1.24
2.28
125
1.09 (0.20)
1.08 (0.06)
0.13
2.46
1250
1.29 (0.25)
1.44 (0.69)
0.50
2.42
160
1.01 (0.12)
1.29 (0.30)
2.13
2.40
1600
1.15 (0.17)
1.22 (0.13)
0.79
2.25
200
1.08 (0.25)
0.97 (0.11)
0.98
2.38
2000
1.27 (0.15)
1.15 (0.11)
1.65
2.25
250
1.13 (0.09)
1.07 (0.12)
0.94
2.25
2500
1.20 (0.20)
1.17 (0.27)
0.19
2.25
315
0.99 (0.09)
0.95 (0.03)
1.08
2.42
3150
1.23 (0.19)
1.14 (0.16)
0.86
2.24
400
1.04 (0.13)
1.19 (0.07)
2.38
2.32
4000
1.73 (0.66)
1.60 (0.16)
0.46
2.49
500
1.17 (0.16)
1.18 (0.11)
0.11
2.26
630
1.59 (0.20)
1.46 (0.34)
0.77
2.30
800
1.20 (0.35)
1.28 (0.11)
0.55
2.44
For each 1/3-octave band center frequency ݂ (except for 400 Hz), the calculated statistic ݐ is less
than the critical statistics ݐ଴Ǥ଴ହ at 0.05 significance level (two tailed test). Therefore, there is no reaso n
to reject the null hypothesis. Only for the 400 Hz, the null hypothesis can be rejected. However, for 400
Hz, the observed difference between ܦ for welds A and B is only 0.6 dB and in practice, this
difference can be neglected.
4.2 CLF and DLF measurement
The results averaged for all six structures will be considered in the further part of the work, because
in point 4.1 no differences between welds A and B have been proven. Figure 3 shows the DLF
determined on the disconnected plates, DLF determined by the ERM method (on joined plates), CLF
determined by the ERM method and CLF determined from the theoretical formula (1). During
measurement, few CLF values were negative for some measuring points. Such cases can occur when
the SEA system assumptions are not met. Negative values have been omitted in the course of the
calculations because they are contradictory in the classic SEA approach.
Figure 3 – Coupling Loss Factors and Damping Loss Factors
Value (CLF or DLF)
1/3-octave band center frequency [Hz]
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The DLF and CLF parameters determined by the ERM method were compared with each other
using the relationship (6) to check the coupling strength. Additionally, a modal overlap was designated
using (4). The smallest value of parameter ߛ௜௝ occurred for 4000 Hz and was equal to 1.74. The largest
modal overlap value (0.04) occurred at 3150 Hz. Thus, it can be seen that in the whole considered
frequency range, the tested plates do not meet the assumptions of SEA.
A small value of the modal overlap parameter explains the occurrence of smaller measured CLF
values in relation to the theoretical ones. It also explains the results from point 4.3, where a higher
value of ܦ was observed in relation to the theory. In this situation, not all modes are excited evenly
and only a few vibration modes are responsible for energy transmission, and the share of the other
modes is insignificant (8).
4.3 SEA Simulation
Figure 4 shows the simulated (SEA) and measured (averaged for all structures) velocity level
difference ܦ between plates. The obtained results correspond to the differences observed between
coupling loss factors from figure 3. The greater theoretical value of the CLF factor is reflected in the
form of very small values of the simulated ܦ. The analysis of the chart in Figure 4 clearly shows that,
from the point of view of the SEA, the division of the system in question into two subsystems is
unfounded. This means that indicators such as modal overlap and ߛ௜௝fulfilled their role.
Figure 4 – SEA simulation results vs averaged measurement results
1/3-octave band center frequency [Hz]
Velocity Level Difference, D [dB]
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5. CONCLUSIONS
When we consider ensemble average, we may allow subsystems with different weld types to be
included in averaging process, because there was no difference in vibration transmission when using
different welds. Only for 400 Hz, a statistically significant difference was found between welds A and
B. However, the difference in transmission for 400 Hz is small (0.6 dB) and in practical applications
may be ignored.
Differences between the DLF parameters determined by the ERM method (joined plates) and the
reverberation time method (disconnected plates) were observed. Differences may result from the
inaccuracy of the measurement methods and the influence of the joint on the overall system damping .
The plates in the tested system were not weakly coupled and the modal overlap was less than one.
Under such conditions, not all modes are excited evenly and only a few vibration modes are
responsible for the energy transmission between subsystems (the cont ribution of the other modes is
negligible). This phenomenon causes that the SEA method (which assumes the even excitation of all
modes) predicts higher transmission (smaller velocity level difference, higher coupling loss factor)
than the measurement results indicate. In such situations, other simulation methods like SmEdA (8)
may provide better results.
ACKNOWLEDGEMENTS
Financial support for this research from KFB Acoustics sp. z o.o. is gratefully acknowledged. We
would like to send special thanks to Patryk Kobyłt, Marek Dwornik and Bartosz Chmielewski for
technical support during the project.
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