Issue 
Acta Acust.
Volume 8, 2024
Special Issue: Vibroacoustics



Article Number  32  
Number of page(s)  14  
DOI  https://doi.org/10.1051/aacus/2024040  
Published online  30 August 2024 
Scientific Article
Quality criterion and errors of corrected negative SEA loss factors
^{1}
KFB Acoustics sp. z o. o., Acoustic Research and Innovation Center, Oławska 8, 55040 Domasław, Poland
^{2}
Wroclaw University of Science and Technology, Department of Acoustics, Multimedia and Signal Processing, Wybrzeże Stanisława Wyspiańskiego 27, 50370 Wrocław, Poland
^{*} Corresponding author: p.nieradka@kfbacoustics.com
Received:
26
February
2024
Accepted:
19
July
2024
Statistical Energy Analysis (SEA) is a wellknown numerical method for predicting vibroacoustic phenomena in complex systems. The accuracy of SEA models relies on the precise determination of coupling loss factors and damping loss factors. Experimental SEA (ESEA) methods, such as the Power Injection Method are commonly employed to measure these parameters. However, these techniques may yield negative loss factors, which are considered measurement errors. Monte Carlo Filtering (MCF) is one of the procedures, that allows the correction of negative loss factors, but the quality of the results remains unknown. The knowledge of the loss factors’ quality is directly related to the practical applications of SEA, where good quality of the input model parameters (coupling and damping loss factors) correspond to good quality and precise simulations of complex vibroacoustic systems (like trains, vehicle, airplanes, buildings) responses. In a previous study, a total loss factor (TLF) criterion was proposed as a quality indicator for the corrected loss factors. The current paper validates the TLF criterion through a comprehensive analysis of various numerical examples. By expanding the Monte Carlo sample’s value range (search area) and using different probability density functions, we intentionally introduced errors in the loss factors. The TLF criterion demonstrated resilience to increasing errors in certain scenarios, raising concerns about its sensitivity. Nevertheless, it seems, that the TLF criterion remains a good indicator of population stability and large error occurrence.
Key words: Statistical energy analysis / Coupling Loss Factor / Damping Loss Factor / Monte Carlo Filtering / Vibroacoustics
© The Author(s), Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The SEA method has been widely used to perform vibroacoustic energy flow simulations in complex systems [1]. SEA (Statistical Energy Analysis) can address issues where acoustic and vibrational fields interact. It finds application in the automotive sector [2], crafting silent ships [3], designing trains [4], airplanes [5], and architectural structures [6]. SEA was also employed in predicting soundinsulating enclosures’ insertion loss [7], mitigating machinery noise [8], estimating partitions’ sound reduction index [9], and predicting the behavior of other systems, where vibroacoustic energy flow is of concern [10–13]. In order to perform SEA calculations, one needs to identify coupling and damping loss factors (LFs), which can be determined by using one of the ESEA (Experimental SEA) methods. Bies and Hamid pioneered the implementation of a technique called PIM (Power Injection Method), directly derived from the SEA energy balance [14]. PIM is therefore the most straightforward way of calculating LFs and is a reference method for researchers introducing new ESEA methods. PIM enables the experimental determination of both CLF and DLF factors for a structure without the need to disassemble individual system components. Fahy et al. devised a technique, known as IPMT (Input Power Modulation Technique) for determining LF coefficients without the need for input power measurement [15]. The main advantage of the IPMT method lies in the requirement to measure only one receiving point per subsystem. It was specifically developed for the case of two coupled subsystems and relies on specially modulated excitation signals. Another category of methods, termed ERM (Energy Ratio Methods), also eliminates the need for input power measurement. In the case of ERM, knowledge of the energy ratios of individual subsystems during the calculation process is essential [16–19]. Fahy introduced alternative coefficients, namely the power transfer coefficient (PTC) and power dissipation coefficient (PDC), alongside a proposed measurement method [20]. The method’s advantage lies again in its ability to omit input power measurement. His approach provides a more physically meaningful interpretation of the defined coefficients compared to classical loss factors. Ming conducted a comparative analysis between Fahy’s method and PIM measurements, demonstrating convergence of results under modal overlap conditions greater than unity [21]. Ming also presented a method to approximate loss factor values by measuring intensity in the structures under investigation [22]. He observed that the method’s error decreases with increasing frequency and becomes negligible when the modal overlap factor of the receiving subsystem is smaller than that of the transmitting subsystem. Cacciolati et al. developed a methodology for determining loss factors based on mechanical admittance measurements [23]. This method is specifically applicable to pointtopoint connections but necessitates disassembling the system into its components to access the connection points between subsystems.
In the process of PIM measurements, an energy matrix is formulated, and the inverse of this matrix is used to extract loss factors. The sensitivity of the energy matrix to measurement errors can lead to the determination of negative loss factors during its inversion. This effect can occur for every experimentally obtained SEA matrix. When the physical system is properly represented by SEA model, then negative loss factor values lack a physical interpretation and can be regarded as measurement errors [24]. Some researchers focused only on this numerical problem of the ESEA [25, 26], and this approach is also continued in the current article (it is assumed that SEA assumptions are fulfilled). In this paper, we investigate one of the recently proposed methods of correcting negative loss factors, namely the Monte Carlo Filtering (MCF) method [27]. This approach is inspired by a wellknown Monte Carlo simulation scheme already used in many other disciplines of science like statistical physics [28], radiation materials science [29], computational biology [30] and others [31]. In MCF, a statistical ensemble of energy matrices is generated based on the mean value and variance of the experimental data. The population is then filtered to remove all matrices that will produce negative LFs. The final step is to compute the mean value of the obtained LFs. The MCF proved to be successful in correcting negative loss factors [32], but strong dependence of loss factors value on the socalled search area has been pointed out [33]. This observation raises the question of whether the quality of obtained results is acceptable and physically meaningful. Even though some methods of minimizing MCF errors were proposed in the previous paper [33], the appropriate criterion of loss factors quality is still missing.
In this paper, we will show how to utilize total loss factor (TLF) properties to derive the criterion in question. The total loss factor concept was introduced to capture the phenomena of apparent damping increase of the subsystem when connected to other structures. For example, in building acoustics the total losses of the wall can significantly increase when connected to the rest of the building [34–36]. The new method is simple to implement and takes advantage of the principal dependencies between the exact and approximate value of the total loss factor. Experiments performed on nine simple systems proved the usefulness of the proposed approach. In MCF, one can control the search area by increasing the Γ parameter (expansion of the search area) or decreasing the Γ parameter (narrowing the search area). The search area determines the range of values used in energy matrices during Monte Carlo sample generation. If the TLF criterion is unmet, one can state that the permissible value of Γ has been exceeded. The numerical validation have shown though, that the derived TLF criterion is not sensitive to LFs errors caused by using low values of Γ. The LFs quality deterioration caused by other factors (like lack of compliance with the SEA assumptions) will not be detected by the TLF criterion and must be dealt with separately.
2 Monte Carlo filtering
Statistical Energy Analysis (SEA), Power Injection Method (PIM), and Monte Carlo Filtering (MCF) are wellknown and established methods. The reader is referred to the previous publications to learn the details of those methods [1, 14, 27]. In this section, only a brief introduction is provided, which is necessary to easily grasp the main paper’s contributions. SEA system is fully described if the loss matrix [L] is known. From L_{ij} entries of [L] one can easily extract coupling and damping loss factors (CLFs and DLFs) in the following manner: 1) CLF η_{ij} is L_{ij} element multiplied by −1; 2) DLF η_{ii} is equal to sum over the ith column of [L]. In PIM one seeks to obtain [L] through an experimental procedure. It is possible to obtain experimentally the energy matrix [E], where each element E_{ij} stands for energy of subsystem i, when the subsystem j is excited. Energy matrix can be normalized to input power and angular frequency to obtain normalized energy matrix [G]:
where [P] is a diagonal matrix, which P_{ii} entry stands for the power injected into the subsystem i. The loss factor matrix [L] can be obtained by inversion of [G]
This variant of PIM is sometimes called NEMI (Normalized Energy Matrix Inversion) [27]. Operation 2 is sensitive to measurement uncertainties and can produce negative loss factors, which are considered measurement errors. One of the methods to correct negative LF is Monte Carlo Filtering (MCF). MCF is based on generating a large matrix population {[G_{s}]} built around the measured matrix [G]. Each sample matrix [G_{s}] drawn from the normally distributed population is calculated as
The normalized energy increment [ΔG_{s}] is calculated as
where [a] is a matrix of normally distributed random variables with 0 mean values and standard deviations equal to 1, [σ] is a standard deviation matrix and ° stands for elementwise matrix multiplication (Hadamard product). Each entry σ_{ij} of [σ] is the measurement standard deviation of corresponding entry G_{ij} of [G]. Then each matrix entry G_{s,ij} of individual sample matrix [G_{s}] obtained from equation (3) is a normally distributed random variable with mean value μ_{X} = G_{i,j} and standard deviation σ_{X} = σ_{i,j} with probability density function (pdf) f(x) equal to
Different probability density functions can be used to obtain {[G_{s}]}, for example lognormal distribution with parameters μ_{log}, σ_{log} and pdf equal to
To retain the same standard deviation σ_{X} = σ_{i,j} and mean value μ_{X} = G_{i,j} as in the normal distribution case, one can substitute
Irrespectively of the chosen random variable distribution, the obtained set {[G_{s}]} is then filtered to exclude matrices [G_{s}] associated with [L_{s}] that produce negative loss factors. Then, for each remaining correct [G_{s}] the corresponding [L_{s}] and loss factors are calculated. The final result is the mean value of all obtained positive loss factors.
Sometimes the energy increments obtained from (4) are not enough to find the set of correct positive loss factors. Then, the socalled expansion of the search area (ESA) can be utilized by introducing an additional scaling factor Γ [33]
It was shown, that increasing the value of Γ can lead to considerable errors in loss factors [33]. This property of ESA will be utilized in this paper to intentionally introduce errors in LFs to test the TLF criterion behavior. Subscript s stands for “sample” and will be omitted in the next sections, as it is not introducing ambiguity.
3 TLF criterion derivation
In the classical approach, the total loss factor η_{TOT} is determined using the method of structural reverberation time (according to relevant standards). The reverberation time measured in this way takes into account all mechanisms in which energy is dissipated, including losses associated with energy flow to neighboring subsystems. Such a situation can also be reproduced during PIM measurements if one ignores all connections to the other subsystems (treats the selected subsystem as a complete system consisting of only one subsystem i). Then the determined damping loss factor of the selected subsystem i will correspond to the total losses making it equal to η_{TOT,i}. Thus, the following equation applies
On the other hand, the formula for η_{TOT,i} can be determined using the complete energy balance, where the other subsystems are not omitted. If, for the purpose of the example, we assume for now a system consisting of only two subsystems and choose i = 1, then the η_{TOT,1} of subsystem “1” connected to subsystem “2” can be derived using SEA energy balance equation
From the formula (11), a relationship for the average energy of the subsystem “1” can be derived:
Then by substituting (12) into (10), the following expression is obtained
From the equation (13), it can be seen that when the term ωη_{21}E_{2} of the sum P_{1} + ωη_{21}E_{2} is negligible, the following approximation can be used
The term ωη_{21}E_{2} does not affect the value of the expression (13) when the receiving system is heavily damped (the energy E_{2} is very small). Analyzing the form of the formula (13), it can be seen that the approximation equation (14) determines the upper limit of the exact value, η_{TOT,1}
or after simple rearrangement
The formula (14) can be generalized to the case where M receiving subsystems are connected to any ith subsystem. The exact formula for the TLF of a subsystem i then has the form
As in equation (13), when the energies of all receiving subsystems are negligible, the generalized approximation can be used
Having the values determined from equations (18) and (10) for all subsystems (i = 1, 2, …, N), the TLF ratio inequality can be stated independently for each i:
The value of is much easier to determine at the simulation stage, and it should be taken into account that it represents the upper limit of the exact value of η_{TOT}, which is, in turn, easier to determine experimentally (e.g., the total loss factor measurements of a partition in a reverberation chamber during sound insulation measurements).
Note that the inequality 19 can be used as a simple criterion to check the consistency and the stability of the MCF result for systems with the number of subsystems N > 1(for N = 1 we have and the criterion is always satisfied). If η_{TOT} and are calculated from the same Monte Carlo run, then the condition (19) will be always satisfied. Also, if two consecutive MCF runs (two MCF populations) are consistent with each other, then from the first run and η_{TOT,run2} from the second run should also satisfy (19). Conversely, if two consecutive MCF runs are not stable (because of the inherent randomness of the process), then equation (19) will not be satisfied. Suppose the negative loss factor has been successfully corrected in a given band, but the inequality (19) is not satisfied. In that case, the result associated with that band is subject to error, and such a band can be marked as a “poorquality” band. The usefulness of the proposed criterion (inequality (19)) will be demonstrated in the next section, where we show that it is necessary to apply (in some cases) a narrowing of the search area during MCF (setting Γ < 1) to minimize the number of corrected bands that initially failed to meet the TLF criterion.
4 Initial experiments
Initial experiments were first described in the shortened, conference paper version of this article [37]. PIM measurements were performed on nine subsystems in order to validate the proposed approach. Those systems are the same as those used in the previous study [32]. Each system consisted of two steel plates connected by different technical junctions at right angles. Each plate was treated as a single bending wave subsystem. Therefore, responses were measured only along the z axis (direction normal to the plate surface). The geometrical and mechanical properties of the plates are shown in Table 1. The following technical junctions were used to connect plates: welding, rubber junction, point welding, rivets, and bolts. Some systems consisted of the same junction but differed in the damping level. Rubber magnetic tape was used to introduce damping to the plates (Fig. 1). The level of damping was controlled by adding or removing individual rubber stripes.
The geometrical and material properties of the plates.
The plates were suspended loosely on flexible strings. The PCB 086C03 modal hammer was employed to excite the constructions, the PCB 356A03 accelerometer was utilized to measure the response of the excited subsystem at the driving point, and PCB T356A32 accelerometers were employed to measure the responses of both the source and receiver subsystems away from the driving point. Data collection was carried out using HEAD acoustics SQuadriga II. The sensors were fixed to the plates using wax. For averaging purposes, three excitation and six response points were randomly chosen on each subsystem.
Postprocessing of the results was done in two stages. In the first stage, full PIM analysis (2 × 2 matrix inversion) was performed in order to determine all CLFs and DLFs. This allowed to determine by simply summing up DLF and CLFs. At the second stage, one of the plates was treated as if it was an isolated system disconnected from the second plate. This allowed us to determine η_{TOT}, which was equal to DLF, as explained in the previous section. At both stages, MCF was utilized. Finally, it was possible to implement inequality (19) and evaluate the quality of all measured loss factors. The described postprocessing procedure was performed separately for Γ = 1 and Γ < 1 cases. Figure 2 shows the ratio for all tested systems. Each subplot in Figure 2 applies to one system and contains two curves. One curve relates to the effective MCF correction using Γ = 1, while the other relates to the effective MCF correction with Γ < 1. It can be seen from the figure that the results obtained by the MCF method in the basic version (Γ = 1) do not meet the TLF criterion in many frequency bands, despite the full correction of the negative loss factors in all cases. In contrast, narrowing the search area allowed the TLF criterion to be met in all analyzed bands. In order to narrow the search area Γ was set to 0.25 in the considered cases.
Figure 2 TLF criterion utilized in measured systems. Black marker corresponds to Γ = 1 case, while white marker corresponds to Γ < 1 case; (a) Line welding, low damping; (b) Rubber, low damping; (c) Line welding, medium damping; (d) Rubber, medium damping; (e) Line welding, high damping; (f) Rubber, high damping; (g) Point welding, high damping; (h) Bolt junction, high damping; (i) Rivet junction, high damping. 
Results show that during MCF, a proper Γ must be chosen, and the TLF criterion can be helpful to confirm if Γ is not too big. In the considered case, using Γ < 1 can be considered as a reaction (intervention) to poorquality results obtained for the Γ = 1 case. Without utilizing the TLF criterion, badquality results could have been accepted as correct.
5 Numerical validation
Section 4 applies the TLF criterion only to measured data and shows that narrowing the search area can enable the criterion to be met. However, the TLF criterion was not directly compared with the errors in loss factors resulting from the MCF procedure. The TLF criterion can be considered a good quality indicator only if it is sufficiently sensitive to errors appearing in the loss factors. To assess the sensitivity of the TLF criterion to MCF errors, a series of additional analyses were conducted on energy matrices derived from a virtual experiment. The energy matrices were obtained by running SEA simulations in VaOne 2021 software. Afterward, Monte Carlo populations of 5000 samples were produced based on the matrices obtained in this way (the rationale for adopting such a population size is provided later in the text). Expansion of the search area (ESA) was used to intentionally introduce increasing errors into the loss factors. When conducting analyses based on simulated data, the value of the standard deviation σ_{i} from the measurement is undefined. This fact makes the use of the scaling factor Γ problematic when expanding the search area (ESA) since Γ is multiplied by σ_{i} in the energy increment formula. For this reason, when analyzing ESA errors based on simulations, it seems more correct to take the fraction Γ of the total normalized energy of the system [G] as the independent variable. Then the matrix of energy increments [ΔG] for the ESA with constant Γ is determined from the relation
instead of from the formula used previously
where [a] is a matrix of random variables with a standard deviation equal to 1 and a mean equal to 0. For individual elements of the matrix [ΔG], the corresponding substitution is G_{i}Γ = σ_{iΓ}, which allows to analyze the results in the function of Γ. The corresponding value of the Γ parameter can be explicitly reconstructed by recalculating Γ = Γ · G_{i}/σ_{i} when σ_{i} derived from the measurement is available.
5.1 SEA models
The numerical SEA models were prepared in such a way, that they fulfill SEA requirements. The basic SEA assumption of a large value of the modal overlap factor (Table 3) was met by the appropriate selection of the frequency of analysis (a single 1/3octave band with a center frequency of 8 kHz) and damping of the subsystems (the exception is subsystem 2 of system 3 with a deliberately introduced small damping and a modal overlap factor of 1.6). Light coupling requirement was also satisfied (CLFs at least 10 times smaller than DLFs with a deliberate exception in system 3, as seen in Tab. 2). Four different SEA systems were analyzed, each consisting of two subsystems. Systems 1 and 3 (Fig. 3a) consisted of two 1mmthick steel plates connected to each other at right angles. The plates were in the form of squares with side lengths of 2 m. The following steel parameters were assumed. Young’s modulus of 2.1 · 10^{11} Pa, Kirchhoff’s modulus of 8.0 · 10^{10} Pa, Poisson’s number of 0.3125, density of 7800 kg/m^{3}. Systems 2 and 4 (Fig. 3b) consisted of a steel plate (the same as in systems 1 and 3) coupled to an acoustic cavity. The acoustic cavity was in the form of a cube with a side length of 2 m. Systems 1 and 2 differ from systems 3 and 4, respectively, in the degree of damping of subsystem 2.
Figure 3 SEA systems; (a) systems 1 and 3, (b) systems 2 and 4. 
CLF and DLF parameters of the analyzed SEA systems.
Each of the analyzed models is described by a nonsymmetric SEA matrix
It was also important for the overall validation to determine the normalized energy matrices for MCF simulations
The components of the [L] matrix were taken directly from VaOne 2021 software, while the [G] components were obtained by conducting a simple virtual experiment. First, an input power of 1 W was introduced into subsystem 1, and the average energy E_{11} and E_{21} of subsystems 1 and 2 were retrieved. Both energies were normalized with respect to input power and frequency to obtain the components G_{11} and G_{21}. Then, a power of 1 W was similarly introduced into subsystem 2 and, after energy normalization, the elements G_{22} and G_{12} were obtained. The CLF and DLF parameters derived directly from the [L] matrix are summarized in Table 2. The parameters of the systems are summarized in Table 3. MCF simulations for systems 1, 2, 3 and 4 were carried out separately using normal (Sect. 5.2) and lognormal (Sect. 5.3) distributions of the population of elements of the [G] matrix.
Parameters of the analyzed SEA systems.
5.2 Normal distribution
The highly damped systems (systems 1 and 2) and the systems with a weakly damped one of the subsystems (systems 3 and 4) were analyzed separately. The results for system 1 are shown in Figure 4. The top graph shows how the relative error of the loss factors δη changed after the application of ESA (Γ∈< 0.1; 3 >), while the bottom graph shows how the value of changed to assess the fulfillment of the TLF criterion. In the graphs, the error scale was limited to the range of < −100%; 100%> in order to maintain readability for small values of Γ.
Figure 4 Relative error of loss factors and TLF criterion for system 1 as a function of the fraction of normalized energy, MCF population with normal distribution. 
The shape and sign of the error in a function of Γ depends on the employed probability density function and the parameters of the SEA systems. In the case of system 1 and normal distribution, the error of loss factors increases as Γ increases. This is the expected behavior, since small values of Γ can be associated with small measurement uncertainty. Next, the positive error decreases and reaches 0 for a certain Γ_{0}. For Γ > Γ_{0} the error turns negative. This form of error has been observed before for experimental data and the mechanism of forming of the negative part of the error has been tentatively described [33]. A more complete description that includes also positive errors will be the subject of further work. However, using the example of scalars, it can be preliminarily concluded that the positive error has to do with drawing samples with values close to 0. These samples with small values after inversion can take unlimitedly large values and are not properly balanced by the inversion of equally distant from the expected value large samples, which take finitely small values.
Irrespective of the mechanisms of errors, it can be seen that the TLF criterion is not sensitive enough to errors for small Γ and begins to take values markedly different from unity only for large Γ and large error values. It can be seen that for large errors, the ratio of takes alternating values greater and less than unity. Therefore, the TLF criterion defined in the section “TLF criterion derivation” as should be accompanied by the condition . This, in turn, suggests that some experimental results preliminarily identified as meeting the criterion (Sect. 4) may have a large measurement error.
Figure 5 shows the error of system 1 loss factors for Monte Carlo populations of different sizes. It can be seen that adopting a population size larger than 5000 does not introduce significant changes in the magnitude of the errors, so in order to optimize computation time, it was decided to stay with the assumed size of 5000. Figures 5c and 5d show that sporadic, isolated increases in error values of up to 700% can occur, especially for populations with smaller sizes. For all analyzed systems, populations of size 5000 did not include these unrepresentative cases.
Figure 5 Relative error of loss factors in the function of normalized energy fraction and Monte Carlo population size; (a) DLF_{11}, (b) DLF_{22}, (c) CLF_{12}, (d) CLF_{21}. 
The results for system 2 are shown in Figure 6. It can be seen that replacing one of the panels in the SEA system with an acoustic cavity caused a significant increase in the errors of the loss factors, especially CLF. To some extent, this was reflected in the TLF criterion, as the ratio began to take values farther from unity. However, when the area labeled ZOOM in Figure 6 is considered, it can be seen that the TLF criterion is not sensitive to errors appearing for Γ close to 0 (Fig. 7). For example, for Γ = 2 · 10^{−1} the error in determining the CLF is equal to 10%, although the TLF criterion is still satisfied.
Figure 6 Relative error of loss factors and TLF criterion for system 2 as a function of the fraction of normalized energy, MCF population with normal distribution. 
An interesting effect was observed for systems having a weakly damped one of the subsystems. For both system 3 (Fig. 8) and system 4 (Fig. 9), the relative errors of some of the loss factors were much larger than zero already for small values of Γ. For example, for the DLF_{11} coefficient of system 3, a δη < −10% was recorded at Γ = 0.1. This is directly related to the fact that the weakly damped subsystems had higher energy when input power was introduced at a constant value of 1 W. The higher value of energy resulted in an increase in ΔG and consequently caused an increase in the error associated with drawing more distant samples. In future studies, it may be interesting to check the behavior of the MCF method with ΔG independent of σ_{i} and G_{i}. For both system 3 and system 4, the errors associated with the weakly damped subsystem 2 were dominant (DLF_{22} and DLF_{21} form the approximation for subsystem 2). Nevertheless, the ratio of associated with subsystem 2 was not significantly different from the TLF criterion associated with subsystem 1. The sensitivity of the TLF criterion around Γ = 0 was also not satisfactory.
Figure 8 Relative error of loss factors and TLF criterion for system 3 as a function of the fraction of normalized energy, MCF population with normal distribution. 
Figure 9 Relative error of loss factors and TLF criterion for system 4 as a function of the fraction of normalized energy, MCF population with normal distribution. 
5.3 Lognormal distribution
Figures 10–13 show that the loss factors determined from populations with a lognormal distribution have a much lower tendency to take on relative errors with a negative sign (this situation occurred only for system 3). The errors for system 1 (Fig. 10) and 2 (Fig. 11) have a similar form and increase monotonically as Γ increases. For system 3 (Fig. 12) and 4 (Fig. 13), as for the normal distribution, errors related to the loss factors of the weakly damped subsystem 2 dominate, and errors can be far from zero even for small values of Γ. The TLF criterion shows a slight upward trend with increasing errors for all analyzed systems. The values of , however, are very close to 1, which can make it difficult to draw conclusions based on this parameter.
Figure 10 Relative error of loss factors and TLF criterion for system 1 as a function of the fraction of normalized energy, MCF population with lognormal distribution. 
Figure 11 Relative error of loss factors and TLF criterion for system 2 as a function of the fraction of normalized energy, MCF population with lognormal distribution. 
Figure 12 Relative error of loss factors and TLF criterion for system 3 as a function of the fraction of normalized energy, MCF population with lognormal distribution. 
Figure 13 Relative error of loss factors and TLF criterion for system 4 as a function of the fraction of normalized energy, MCF population with lognormal distribution. 
6 Conclusions
The TLF criterion proposed in this work can be considered a useful complement to the process of correcting negative loss factors with the MCF method. Evaluation of the TLF criterion makes it possible to determine whether changing the search area during Monte Carlo sample generation is necessary to obtain a result that makes sense from a physical point of view. This directly relates to the practical applications, where experimentally obtained, highquality loss factors are required as input to numerical SEA simulations of the particular product.The effectiveness of the technique has been demonstrated in measurements of nine different SEA systems consisting of two subsystems. Tests on larger systems are also planned to validate the generalized inequality of the TLF criterion. Based on the performed numerical validation, it can be concluded that the TLF criterion in many cases is sensitive to errors in loss factors only after applying a significant expansion of the search area (thus for large values of Γ or γ). For small expansions of the search area, the TLF criterion is not sensitive enough. Therefore, the TLF criterion can be treated as a necessary, but not sufficient condition for obtaining low errors in MCF. However, it is worth noting that error information is not available during actual measurements, while the TLF ratio can be easily computed. For this reason, the TLF ratio can be treated as an additional diagnostic tool in the MCF process. Then, when the TLF criterion is calculated and found not to be satisfied, a given measurement can be considered a lowquality measurement (two consecutive Monte Carlo runs are not consistent with each other). On the other hand, when the TLF criterion is met, it is not possible to determine whether the measurement is of high or low quality. The results suggest that in addition to condition , it is also important to meet condition .
Conflicts of interest
Authors declared no conflict of interests.
Data availability statement
The research data associated with this article are included within the article.
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All Tables
All Figures
Figure 1 Plate with (a) low damping; (b) medium damping; (c) high damping. Source: [32]. 

In the text 
Figure 2 TLF criterion utilized in measured systems. Black marker corresponds to Γ = 1 case, while white marker corresponds to Γ < 1 case; (a) Line welding, low damping; (b) Rubber, low damping; (c) Line welding, medium damping; (d) Rubber, medium damping; (e) Line welding, high damping; (f) Rubber, high damping; (g) Point welding, high damping; (h) Bolt junction, high damping; (i) Rivet junction, high damping. 

In the text 
Figure 3 SEA systems; (a) systems 1 and 3, (b) systems 2 and 4. 

In the text 
Figure 4 Relative error of loss factors and TLF criterion for system 1 as a function of the fraction of normalized energy, MCF population with normal distribution. 

In the text 
Figure 5 Relative error of loss factors in the function of normalized energy fraction and Monte Carlo population size; (a) DLF_{11}, (b) DLF_{22}, (c) CLF_{12}, (d) CLF_{21}. 

In the text 
Figure 6 Relative error of loss factors and TLF criterion for system 2 as a function of the fraction of normalized energy, MCF population with normal distribution. 

In the text 
Figure 7 Zoom in on the data area from Figure 5. 

In the text 
Figure 8 Relative error of loss factors and TLF criterion for system 3 as a function of the fraction of normalized energy, MCF population with normal distribution. 

In the text 
Figure 9 Relative error of loss factors and TLF criterion for system 4 as a function of the fraction of normalized energy, MCF population with normal distribution. 

In the text 
Figure 10 Relative error of loss factors and TLF criterion for system 1 as a function of the fraction of normalized energy, MCF population with lognormal distribution. 

In the text 
Figure 11 Relative error of loss factors and TLF criterion for system 2 as a function of the fraction of normalized energy, MCF population with lognormal distribution. 

In the text 
Figure 12 Relative error of loss factors and TLF criterion for system 3 as a function of the fraction of normalized energy, MCF population with lognormal distribution. 

In the text 
Figure 13 Relative error of loss factors and TLF criterion for system 4 as a function of the fraction of normalized energy, MCF population with lognormal distribution. 

In the text 
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