PROCEEDINGS of the23rd 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., PolandABSTRACT
Statistical energy analysis (SEA) predicts the average response of a population. This population consists ofsimilar vibroacoustic structures subjected to white noise forces at high frequencies. However, usually inengineering applications, the SEA method is used to determine the response of a particular structure. It resultsfrom the assumption, that in the case of complex structures the response of a single member from thepopulation 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 wasdetermined. 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 averagecovering 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, theability 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 weldedplates. 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 beconsidered 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 theyare associated with a specific structure (1). Six structures (SEA systems) were tested in the work. Eachsystem 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 welding,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".

1p.nieradka@kfb-acoustics.com4265

2. THEORY
2.1 Coupling Loss Factor of Line Junctions
The Coupling Loss Factor of line-coupled 2D subsystems is determined from (1):ߟଵଶ ൌ‫ܿܮ‬௚ଵ߬ߨ߱ܵଵ ௕(1)where ܿ௚ଵ is wave group speed of subsystem 1, ‫ ܮ‬is length of connection, ߱ is angular frequency, ܵଵis 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 coefficient assuming rigid
plate connection (3) and models allowing to take into account the finite stiffness and resistance of thejoints (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, bysolving a system of linear equations, where the column of free words are average input powers, and themain 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:ߟ ൅ߟʹߨ݂ ቀ ଵെߟ ଵଶଵଶെߟଶଵ‫ܧ‬ଵܲଵߟଶ ൅ ߟଶଵ ቁ ൬‫ܧ‬ଶ ൰ ൌ ൬ܲଶ ൰(2)where ݂ is frequency in Hz, ߟଵ and ߟଶ are damping loss factors (DLF), ߟଶଵ and ߟଵଶ are couplingloss factors (CLF) computed as shown in 2.1. After solving the system of equations (2), the averagesquare of vibration velocity of the subsystem with the mass ‫ ܯ‬can be determined from the equation:‫ ݒ‬ଶ ൌ ‫ܧ‬Ȁ‫ܯ‬.

(3)

Statistical Energy Analysis works best if modal overlap ‫ܯ‬௢ is greater than unity (6):‫ܯ‬௢ ൌ ݊ߟ߱ ൐ ͳ(4)݊ ൌ ܵ߱Ȁሺʹߨܿ௚ ܿ௙ ሻ.

(5)where ݊ is asymptotic modal density:

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 estimatedbased on the parameter ߛ௜௝ :ߛ௜௝ ൌ ߟ௜௝ Ȁߟ௜ .
Coupling is assumed to be weak, if ߛ௜௝ ‫ ͳ ا‬condition holds.

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(6)

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 thestrings (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

ValueLength, L

0.49 mWidth, W

0.49 mThickness, h0.002 mDensity, ߩ7884 kg/m3

Young Modulus, E200 GPa

Poisson Number, ߪ0.3

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

Weld A

12-1, 2-3, 4-5

Weld B

6-7, 8-9, 10-11

Geometry

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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଺଴ ‫݂ ڄ‬ሻ(7)where ܴܶ଺଴ is structural reverberation time
The plates were then joined and a further series of measurements was carried out. Mechanicalpower 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 mechanicalvelocity was determined on the source and receiver subsystem (on each plate the signal was picked up
using accelerometers in six randomly selected places). Excitat ion of the plate in many places andaveraging of individual responses approximates the "rain-on-the-roof" excitation, which is one of theassumptions 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 freewords contained the measured TLF (ߟଵ௧ ǡ ߟଶ௧ )ͳ൬݁ଵଶߟଵ௧݁ଶଵ ߟଵ൰ ቀߟ ቁ ൌ ቀߟ ቁ.
ͳଶଶ௧

(7)

Using the results from (7) the second system of equations was solved, allowing to determine the CLF
coefficients (ߟଵଶ ǡ ߟଶଵ ሻͳ൬െ݁ଵଶ4268݁ଶଵ ߟଶെ݁ଶଵ ߟଵଶ൰ ቀߟ ቁ ൌ ቀ݁ ߟ ቁ.
ͳଶଵଵଶ ଵ(8)

4. RESULTSVelocity Level Difference, D [dB]

4.1 Comparison of Welds
Figure 2 presents the averaged Velocity Level Difference ‫ ܦ‬for welds A and B. Energy ratios fromwhich the results can be determined are listed in table 3.

1/3-octave band center frequency [Hz]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 ofthe measurements were divided into two groups. The first group (the representation of the populationof plates with weld A) has been assigned six parameters determined from the relationship:݁஺ǡ௜ ൌ‫ܧ‬௦௢௨௥௖௘ǡ஺ǡ௜Ǣ ݅ ൌ ͳǡ ʹǡ ǥ ǡ ͸‫ܧ‬௥௘௖௘௜௩௘௥ǡ஺ǡ௜

(9)Similarly, six parameters were assigned to the second group (representation of the plate population
with weld B):‫ܧ‬௦௢௨௥௖௘ǡ஻ǡ௜݁஻ǡ௜ ൌǢ ݅ ൌ ͳǡ ʹǡ ǥ ǡ ͸(10)‫ܧ‬௥௘௖௘௜௩௘௥ǡ஻ǡ௜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 inequations (7) and (8). Then, the Welch's t-test at 0.05 significance level was conducted to check thefollowing null hypothesis ‫ܪ‬଴ ǣ The average mechanical energy ratio of the source and receiver plate isindependent of the type of weld used. Table 3 shows the determined average energy ratios for eachweld type, standard deviations ߪ and test results.

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Table 3 – Results of two-tailed Welch’s t-test݂݁ഥ஺ ሺߪ௘ಲ ሻ݁஻ ௘ಳ ሻതതതሺߪ

‫ݐ‬‫ݐ‬଴Ǥ଴ହ

100

1.60 (0.85)

1.02 (0.17)

1.63

2.51

125

1.09 (0.20)

1.08 (0.06)

0.13

160

1.01 (0.12)

1.29 (0.30)

200

1.08 (0.25)250݂݁ഥ஺ ሺߪ௘ಲ ሻ݁஻ ௘ಳ ሻതതതሺߪ‫ݐ‬‫ݐ‬଴Ǥ଴ହ

10001.30 (0.23)

1.52 (0.36)

1.24

2.28

2.461250

1.29 (0.25)

1.44 (0.69)

0.50

2.42

2.13

2.40

1600

1.15 (0.17)

1.22 (0.13)

0.79

2.25

0.97 (0.11)

0.98

2.38

2000

1.27 (0.15)

1.15 (0.11)

1.65

2.25

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

2.423150

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ሾ ሿሾ ሿ

For each 1/3-octave band center frequency ݂ (except for 400 Hz), the calculated statistic ‫ ݐ‬is lessthan the critical statistics ‫ݐ‬଴Ǥ଴ହ at 0.05 significance level (two tailed test). Therefore, there is no reason
to reject the null hypothesis. Only for the 400 Hz, the null hypothesis can be rejected. However, for 400
Hz, the observed difference between ‫ ܦ‬for welds A and B is only 0.6 dB and in practice, thisdifference can be neglected.

Value (CLF or DLF)

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), CL F
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 omit ted in the course of thecalculations because they are contradictory in the classic SEA approach.

1/3-octave band center frequency [Hz]Figure 3 – Coupling Loss Factors and Damping Loss Factors

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The DLF and CLF parameters determined by the ERM method were compared with each otherusing the relationship (6) to check the coupling strength. Additionally, a modal overlap was designatedusing (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 consideredfrequency 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 ‫ ܦ‬was observed in relation to the theory. In this situation, not all modes are excited evenlyand only a few vibration modes are responsible for energy transmission, and the share of the othermodes is insignificant (8).

Velocity Level Difference, D [dB]

4.3 SEA Simulation
Figure 4 shows the simulated (SEA) and measured (averaged for all structures) velocity level
difference ‫ ܦ‬between 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 theform 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 isunfounded. This means that indicators such as modal overlap and ߛ௜௝ fulfilled their role.

1/3-octave band center frequency [Hz]Figure 4 – SEA simulation results vs averaged measurement results

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5. CONCLUSIONS
When we consider ensemble average, we may allow subsystems with different weld types to beincluded 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 andB. 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 thereverberation time method (disconnected plates) were observed. Differences may result from theinaccuracy 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 areresponsible for the energy transmission between subsystems (the cont ribution of the other modes isnegligible). This phenomenon causes that the SEA method (which assumes the even excitation of allmodes) 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 fortechnical support during the project.

REFERENCES
1. Le Bot A. Foundations of Statistical Energy Analysis in Vibroacoustics. 198 Madison Avenue, NewYork, USA: Oxford University Press; 2015.
2. Lyon RH, DeJong RG. Theory and Application of Statistical Energy Analysis. Newton, Massachusetts,
USA: Butterworth-Heinemann; 1995.
3. Cremer L, Heckl M. Structure-Borne Sound (2nd edition). Berlin, Germany: Springer-Verlag; 1988.
4. Patil VH, Manik DN. Determination of coupling loss factors (CLF) for right angled plates. Proc
ICSV22; 12-16 July 2015; Florence, Italy 2015.
5. Leung R, Pinnington R. Wave propagations through right angle joint with compliance-flexural incident
wave. Journal of Sound and Vibration. 1990;142:31-46.
6. Le Bot A, Cotoni V. Validity diagrams of statistical energy analysis. Journal of Sound and Vibration.
2010;329:221-235.
7. Cuschieri JM, Sun JC. Use of Statistical Energy Analysis For Rotating Machinery, Part I: Determinationof Dissipation and Coupling Loss Factors Using Energy Ratios. Journal of Sound and Vibration.
1994;170:181-190.
8. Maxit L, Guyader JL. Extension of SEA model to subsystems with non-uniform modal energy
distribution. Journal of Sound and Vibration. 2003;265:337-358.

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