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Citation: Neubauer, M.; Schwaericke, F.; Radmann, V.; Sarradj, E.; Modler, N.; Dannemann, M. Material Selection Process for Acoustic and Vibration Applications Using the Example of a Plate Resonator. Materials 2022,15, 2935. https:// doi.org/10.3390/ma15082935

Academic Editor: Leif Kari

Received: 20 March 2022 Accepted: 13 April 2022 Published: 18 April 2022

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Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/ 4.0/).

Article Material selection process for acoustic and vibration applications using the example of a plate resonator

Moritz Neubauer 1,* , Felix Schwaericke 2, Vincent Radmann 2, Ennes Sarradj 2, Niels Modler 1 and Martin Dannemann 3

1Institute of Lightweight Engineering and Polymer Technology (ILK), Technische Universität Dresden, Holbeinstraße 3, 01307 Dresden, Germany; [email protected] 2Institute of Fluid Dynamics and Engineering Acoustics, Technische Universität Berlin, Einsteinufer 25, 10587 Berlin, Germany; [email protected] (F.S.); [email protected] (V.R.); [email protected] (E.S.) 3Faculty of Automotive Engineering, Institute of Energy and Transport Engineering, Westsächsische Hochschule Zwickau, Scheffelstraße 39, 08012 Zwickau, Germany; [email protected] *Correspondence: [email protected]; Tel: +49-463-4026

Abstract: In this work, a new method for selecting suitable materials is presented. This method has a high potential for a variety of engineering applications, such as the design of sound-absorbing and vibration-loaded structures, where a large number of different requirements have to be met. The method is based on the derivation of functional dependencies of selected material parameters. These dependencies can be used in parameter studies to consider parameter combinations that lie in the range of real existing and targeted material groups. This allows the parameter space to be reduced, the calculation to be accelerated, and suitable materials to be (pre-selected) for the application, which contributes to a more target-oriented design. The method is applied to the example of a resonator. For this purpose, a semi-analytical model is implemented to calculate the plate performance as well as the loss of the reflected plate and the performance of the transmitted fluid flows, taking into account the relative silencer speed and silencing, taking into account the influence of the transmitted fluid flow velocity.

Keywords: method of material selection; plate resonator; thermoplastic; model; acoustic liner

1. Introduction

Numerous methodologies have been introduced that aim to provide a guiding structure in the process of selecting the optimal material for a particular application [ 1 ]. The frameworks described by Ashby et al. [

2 ] and Van Kesteren et al. [

3 ] coincide in the concept that material selection includes the basic steps of defining requirements and certain criteria, the screening of materials to generate a set of potential candidates, the comparison and subsequent ranking of suitable candidates, and the final step of selecting the optimal material. In this context, Ashby charts are often used for screening because they allow easy and convenient comparison of property ratios of groups of materials. The weighted property methods (WPM) as well as the multi-criteria decision (MCDM) are suitable methods to narrow down the number of potential materials [ 4 8 ]. In this context, MCDM techniques apply material selection decision matrices and criteria sensitivity analysis to enable a logical ranking of considered materials [

9 11 ].An alternative approach to supporting the decision-making process in material selection lies in the application of artificial intelligence [

12 14 ]. The so-called expert systems consist of modules for knowledge acquisition, inference, as well as a user interface module that enables the simulation of human experts reasoning and decision making [ 15 17 ]. For the present work, the material property charts introduced by Ashby and Cebon [ 18 ] are used to identify functional dependencies of the relevant material properties in order to reduce the parameter space and obtain a coherent representation of the relationships that influence the target variable.

The Commission shall adopt delegated acts in accordance with the opinion of the Standing Committee on Plants, Animals, Food and Rural Affairs and the Committee on Plants, Animals, Food and Rural Affairs and the Committee on Plants, Food and Rural Affairs and the Committee on Plants, Food and Rural Affairs and the Committee on Plants, Animals, Food and Rural Affairs and the Committee on Plants, Food and Rural Affairs and the Committee on Plants, Food and Rural Affairs and the Committee on Plants, Agriculture, Fisheries and Rural Affairs.

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Materials 2022,15, 2935 2 of 15

For materials used in complex applications, analytical models are often derived and subsequently used to relate the relevant parameters. However, depending on the complexity of the application, the target parameters may depend on a large number of material parameters whose actual influence on the target variable is unknown. This makes it rather difficult to identify the material parameters with significant influence on the target variable as well as appropriate value ranges that can be used as a basis for material selection. Additionally, some material parameters show dependencies that cannot be resolved. A typical example is the observation that a material generally cannot have both high stiffness and high material damping. This is a typical conflict of targets for many applications in the field of vibration and acoustic material. Due to the vast number of relevant parameters, the modum of a plate spectrometer (PR) provides a rather complex method of resource-intensive analysis within the context of the proposed functional parameters, allowing the selection of a more coherent, time-based, and in a manner that supports the selection of material parameters.

Plate silencers, consisting of expansion chambers fully covered with a plate, are especially used for attenuating low frequencies [

Although plate resonators are already used in practice, it turned out that the experimental results obtained and the practical experience with plate resonators agree with the established computational models only under restrictions [

The design of this type of resonance absorber is therefore considered complex and requires additional supporting measurements [22]. The most advanced model currently used for describing plate resonators is the one developed by Huang and Wang [

It is a semi-analytical model that describes the interaction of the plate with the underlying cavity and the duct above. Excited by a plane wave, the system of plate and cavity resonates, and the plate radiates sound back into the duct. This causes the radiated sound to overlap with the incoming sound, resulting in partial cancellation. In addition, part of the sound energy in this process is dissipated by internal losses in the material of the plate.

Plate silencers have many advantages over silencers made of porous material or perforated surfaces. They can be exposed to contaminated air, heat, frost, and humidity without being damaged. Due to their smooth surfaces, their flow resistance is relatively low [25 ]. There are numerous applications where the advantageous properties of the smooth and robust sheet surface can be effectively utilized, e.g., as suspended transparent ceiling installations made of plastic films to meet high air quality requirements in clean rooms [22 , 26 ].

In the automotive industry, they are used as engine compartment linings and in mufflers to attenuate the broadband and low frequency sound emitted by the engine, while saving costs and installation space [24 , 28 ].

In this context, the performance of the plate silencer is significantly influenced by the material properties of the plate, so it is important to find a suitable material for the specific application.

The aim of the present work is to introduce a novel model-assisted methodology for material selection using the example of the plate silencer. After describing the general steps of the method for model-based material selection, the underlying model of a plate resonator as well as the relevant properties are introduced. Subsequently, the task of material selection is described taking into account functional dependencies between properties and the interaction between the selection process and the model.

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Materials 2022,15, 2935 3 of 15 allows two material parameters to be reduced to one and subsequently visualize its effect on the target variable in a frequency spectrum. This enables the coherent analysis of the effect of the relevant material parameters on a target variable over a wide range of values and the identification of practice-relevant trends. After deriving and presenting the general and applied method of model-based material selection, the derived results are discussed.

2. Materials and Methods

The generalized approach to model-based material selection for complex applications is described below.The methodology shown in Figure 1 is subsequently applied to the selection of the plate material for a plate resonator.

Materials 2022, 15, x FOR PEER REVIEW 3 of 15 resonator as well as the relevant properties are introduced. Subsequently, the task of material selection is described taking into account functional dependencies between properties and the interaction between the selection process and the model. The functional dependence allows two material parameters to be reduced to one and subsequently visualize its effect on the target variable in a frequency spectrum. This enables the coherent analysis of the effect of the relevant material parameters on a target variable over a wide range of values and the identification of practice-relevant trends. After deriving and presenting the general and applied method of model-based material selection, the derived results are discussed.

2. Materials and Methods

Figure 1. Methodology of model-based material selection for complex applications.

2.1 Screening of materials

Figure 1. Methodology of model-based material selection for complex applications.

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Materials 2022,15, 2935 4 of 15

2.1 Screening of materials

The starting point is the definition of requirements resulting from the intended application and the subsequent development of the model to describe the relationships and phenomena of the application. In the course of the modelling process, the relevant material parameters are identified. Together with the defined requirements, they enable pre-selection by excluding groups of materials due to certain specifications exceeding the limits of the associated properties [

In addition to model-specific parameters, this may include requirements arising from the product lifecycle, such as restrictions in the fields of manufacturing technologies or recycling.

2.2 Preliminary Parameter Study (independent variation)

The objective of the preliminary parameter study is a first exploration of the influence of particular material parameters. This enables the identification of parameters with a significant effect on the target variable in order to select them for a subsequent step of determining the functional dependencies. Here, a list (e.g., see Table 1) of parameters required to model the problem can be used. In order to narrow down the parameter space, a design point as well as a maximum range are defined. This is done by examining the distribution of property values of the target material group. For properties that are scattered over a large range, a logarithmic step size is beneficial for analyzing its effect on the target variable. Starting from the design point, the parameter is then varied towards the defined limits. However, in the case that the material parameter and the variable parameter have a comprehensible direct parameter, this is not considered to have an effect on the target variable due to the fact that the target variable has a more conflicting effect on the target variable, but if this needs to be investigated, it needs to be considered to have a wider impact on the target variable.

Table 1.Values of geometry and material parameters for modelling a 2D plate silencer.

Parameter Symbol Unit Value/Value Range Design Point

Parameters of geometry

Length of cavity lCmm 60 60

Height of the cavity hCmm 30 30

Cavity width wCmm ∞ ∞ Duct height hDmm 60 60 Duct width wDmm ∞ ∞ Plate thickness hPmm 0.3 0.3

Parameter of the material

Youngs module EN/m2E ∈{107; 108; 109; 1010} 109

Poisson ratio ν- 0.4 0.4 Loss modulus η-η∈{0.0015; 0.015; 0.15} 0.015

Density ρkg/m3ρ∈{850; 950; 1050; 1150} 1050

2.3. Determination of functional dependence

The independent variation of properties in parameter studies leads to large areas in the parameter space not being represented by real materials. This unnecessary increase in parameter space increases computational resources time, and complicates the selection of appropriate materials. In order to perform the parameter studies in the range of existing materials, the identification of functional dependencies between the previously defined

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Materials 2022,15, 2935 5 of 15 relevant material parameters using material parameter charts is a useful tool. For the representation and assessment, the relevant material parameter can therefore be plotted on a respective axis in charts [

18 ]. The visualization can support the process of identifying functional dependencies of the selected material properties [

Depending on the size of the selected group of materials and the range of properties in terms of magnitude, this can be performed in a linear single or double logarithmic representation. Subsequently, the resulting plots are checked for existing functional dependencies. After a relationship has been identified, a corresponding base function is selected. Then, using appropriate mathematical regression methods, the function that best describes the given dependencies is determined.

2.4. Parameter study

Based on the functional relationship, derived in Section 2.3, value pairs can now be calculated to serve as input variables for the model-based parameter study. The step size between the pairs should be based on the step size determined in Section 2.3, to allow an analysis of the effect on the target variable. If unfavorable step sizes are chosen, it may be difficult to assess the effect of the parameter variation of the value pairs on the target variable. In this case, it is advisable to reduce the step size to make the effects on the target variable clear. Based on the observed trends on the target variable, a material could be selected. However, it may be required to apply a finer estimate over the entire parameter range if more sections are of particular resolution interest.

2.5. Selection

In the last step, a parameter combination is selected, which provides the desired result with respect to the target variable. Depending on the accuracy of the functional dependence, the material can be selected directly according to the identified parameter combination. In case the determined parameter configuration differs slightly from existing materials, the parameter combination of one or more materials that are in the target range can be analyzed and the desired material ((s) can be selected.

3. Application of the Methodology presented for the selection of a plate material for a plate resonator

In this section, a brief overview of the plate resonator model is provided before applying the presented material selection method to determine the plate material of the silencer.

You know what ?

3.1. ApplicationPlate silencer

The methodology introduced in the present work is validated during the design process of an acoustic plate silencer, more specifically through the selection of its plate material. The requirements for the plate derive from the intended applications as a silencer as well as from the general environmental aspects of the product lifecycle. In general, the plate silencer can be used in ducts such as exhaust systems. A simple model of a plate silencer is shown in Figure 2. Here, a plate capable of oscillating (Figure 2, green) replaces the upper wall facing the duct. Behind this plate is a volume, called a cavity. As sound propagates along the duct, the sound is excited by the pressure difference above and below it.

In this context, the validity of this model was verified through the comparison of experimental results, which showed a satisfactory level of conformity [

In addition, the validated model has been used in further parameter studies in the context of acoustic investigations, including the analysis of sandwich and composite plates [

Therefore, this model was chosen to

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Materials 2022,15, 2935 6 of 15 demonstrate the material selection process of a plate silencer. To evaluate the performance of the plate silencer, the transmission loss can be used, which was set as the target variable for the current material selection process.

Materials 2022, 15, x FOR PEER REVIEW 6 of 15 of the plate silencer, the transmission loss can be used, which was set as the target variable for the current material selection process.

Figure 2. (a) Duct with defined height hD, width wD and integrated resonator liner, (b) design of a plate resonator with 1 its film layer 2 the enclosed cavity, comprising the cell length lC, the cell height hc, the cell width wc and 3 the bottom layer. (c) Detailed view of the film with the relevant material properties, being density ρ, Youngs material module E, Poisson ratio and the loss factor η as well as the film thickness hp.

Therefore, the bending differential equation of the plate must be solved for its velocity. In this context, the right side of Equation (1) corresponds to the excitation of the plate. The sound fields above and below the plate can be represented by a kind of impedance for the duct ZD, and the cavity ZC (see Figure 2). Additionally, such an impedance matrix can also be formulated for the plate (L), which contains the properties of the plate material.

In order to solve the differential equation, a Galerkin method was used. Thus, with the impedance matrices of the duct cavity plate, and the incident sound I, the following system of equations can be solved for the modal amplitude of the plate velocity, v. (L+ZC+ZD) v =−I

(a) The following:

To solve this equation, a modular implementation of the plate silencer model was coded in python. With the determined velocity, the radiated sound power of the plate can be calculated, allowing the transmitted and reflected sound power to be calculated as well. The transmission loss, representing the target variable, can be computed from the transmitted sound power Ptrans, and the incident sound power, as follows: TL =−10log PinPtrans

♬ Pin.

(b)

Figure 2. (a) Duct with defined height hD, width wD and integrated resonator liner, (b) design of a plate resonator with 1 its film layer

2 the enclosed cavity, comprising the cell length lC, the cell height hc, the cell width wc and

3 the bottom layer. (c) Detailed view of the film with the relevant material properties, being density

ρ , Youngs modulus E , Poisson ratio

h and the loss factor η as well as the film thickness hp.

ZC (see Figure 2). In addition, such an impedance matrix can also be formulated for the plate (

L ), which contains the properties of the plate material.

In order to solve the differential equation, a Galerkin method was used. Thus, with the impedance matrices of the duct cavity plate, and the incident sound

I, the following system of equations can be solved for the modal amplitude of the plate velocity, v.

L+ZC+ZDv=−I(1) To solve this equation, a modular implementation of the plate silencer model was coded in python. With the determined velocity, the radiated sound power of the plate can be calculated, allowing the transmitted and reflected sound power to be calculated as

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Materials 2022,15, 2935 7 of 15 well. The transmission loss, representing the target variable, can be computed from the transmitted sound power Ptrans, and the incident sound power Pin, as follows: TL =−10 log

Ptrans Pin . (2)

The sound power dissipated due to the intrinsic damping behavior of the plate is derived from the difference between the incident sound power on one side and the reflected and transmitted sound power on the other side.

The Python implementation of the plate silencer model allows to calculate single-sided and double-sided configurations. In the procedure described here, the single-sided configuration was applied (see Figure 2). The basis of the calculation of the transmission loss is the input parameters contained in the impedance matrices. As shown in Table 1, they are divided into geometry and material parameters.

Since the calculation was performed in two dimensions, the displayed width of the duct wD , and the cavity wC , is considered infinite. The dimensions of the resonator were chosen to create a silencer small enough to be used in a machine or vehicle. The purpose here was to demonstrate that a plate resonator can be used as a low-frequency silencer requiring very little space. At the same time, the small dimensions limit the number of modes to be dampened in the system, which reduces the computation time and allows the simulation of more material configurations.

3.2. Process of selection of materials

In the following section, the above method is applied to the material selection of the plate of a plate silencer.

3.2.1. Screening of materials

The identification of a suitable material group is primarily based on the field of application. Depending on this, the goal in noise control is often to effectively attenuate low frequencies, since they are more difficult to suppress than high frequencies. Secondly, the material identification process is influenced by the requirements of the design and manufacturing process. For the presented example of a plate silencer, the material group of thermoplastics and thermoplastic elastomers was chosen. Compared to, e.g., metals, these materials have a higher internal damping characteristic, which can lead to a higher overall damping of the silencer. In addition, they are easy to process, can be joined using a variety of joining techniques, and are relatively easy to manufacture, even in the desired low plate thicknesses.

3.2.2.Preliminary Parameter Study

After the process of selecting the material group is completed, the second step is to conduct a preliminary parameter study for the first exploration in order to assess the effect of the individual material parameters on the target variable. Therefore, the effect of the Youngs module, loss factor, and density on the transmission loss was analyzed. For this purpose, the value range, step size, and design point were defined first. As described in Section 2.2 of the presented method, a logarithmic and linear step size was determined based on the value distribution of the material parameters for the selected material group. The parameter densities and design points that were finally applied are listed in Table 1. Starting from the design point of the preliminary parameter study, the simulation leads to a reference spectrum of the transmission loss (see Figure 3a).

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Materials 2022,15, 2935 8 of 15 thickness, the influence of this parameter need not be further investigated, leaving the Youngs module and the loss factor for further investigations.

Materials 2022, 15, x FOR PEER REVIEW 8 of 15

Figure 3. Results of the preliminary parameter study, (a) reference result of the design point, (b) variation of Youngs module E, (c) variation of density ρ, (d) variation of loss factor η.

The loss factor widens and reduces the transmission loss peaks in the spectrum. The effect is of different intensity for the main peak compared to the smaller peaks above and below it (see Figure 3d). Compared to the density, there is a significantly larger range of value for the loss factor within the material group of thermoplastics and thermoplastic elastomers. Therefore, and due to the fact that the loss factor has a higher qualitative as well as a more diverse effect on the transmission loss, further analysis of the parameter is required.

Based on the objective of low-frequency attenuation and the analysis of the results shown in Figure 3b, it appears that high values for the Youngs module are preferable. Nevertheless, it seems that a more detailed study on the influence of the Youngs module, taking into account smaller step sizes, is necessary.

3.2.3. Determination of functional dependencies

In order to reduce and maintain the parameter space in the range of existing thermoplastic polymers, the identification of a functional dependence between the relevant material parameters is targeted, as described in Section 2.3 of the introduced method. Since the preliminary parameter study implied the relevance of the Young's modulus and the mechanical loss factor with respect to the target variable, these two material parameters were considered for the determination of a functional dependence. For the purpose of identifying a dependence, the corresponding pairs of values of the material parameters were analyzed in the form of a material property chart introduced by Ashby [34], varying the modules of both axes. In Figure 4, the logarithmic mean values of the Young's modulus and the mechanical loss factor taking into account the material-envelopes of the sub-classes of the Young's axes and the logarithmic loss factor representative of the logarithmic parameters (see Figure 4).

3.2.3. Determination of functional dependencies

In Figure 4, the logarithmic mean values of Young's modulus and mechanical loss factor taking into account the material-envelopes of 211 representative sub-classes of the material class of thermoplastics and thermoplastic elastomers were analyzed on the basis of

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Materials 2022,15, 2935 9 of 15 a logarithmic scale for both axes [

The graphic display of the parameters reveals the linear dependence in a double logarithmic representation, leading to a polynomic interrelation in the linear space (see Figure 4).

Materials 2022, 15, x FOR PEER REVIEW 9 out of 15

Mechanical loss factor over Youngs module of thermoplastic and thermoplastic elastomer polymers: the fitting curve represents the functional dependence between the two material parameters as well as the two sets of parameter pairs with different step sizes for the subsequent effect analysis (material data from [35]).

Applying a polynomial base function, the numerical relationship was established by linear regression in the form of the following equation: η =0.0335∙E−0.633.

(iii) The following:

3.2.4. Parameter study

In order to perform the parameter study, corresponding pairs of material parameters with a logarithmic step size were selected, covering the entire value range of the target material group (Figure 4, orange diamonds).The key advantage of this approach is that the number of parameters to be included in the optimization process is reduced to one, since the corresponding value of the other parameter results from the determined functional relationship.

0,001 0,01 0,1 1 10

0.001 0.01 0.1 1 10 Mechanical loss coefficient tan δ[-]

Young's Modulus [GPa] Logarithmic mean values

Low resolution step width

High-resolution step width

Fitting curve

Applying a polynomial base function, the numerical relationship was established by linear regression in the form of the following equation: η=0.0335·E−0.633. (3)

3.2.4. Parameter study

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Materials 2022,15, 2935 10 out of 15 Materials 2022, 15, x FOR PEER REVIEW 10 out of 15

Figure 5. Results of the parameter study with functional dependence between Youngs modulus, E, and the corresponding loss factor, η.

Therefore, the range and the step size between the parameter pairs was further reduced (Figure 4, red diamonds). In the corresponding chart, three effects become visible now (see Figure 6). On the one hand, the two main peaks were shifted to higher frequencies for higher values of the Youngs module. On the other hand, the amplitude of the first main peak increased with the increasing Youngs module, while the amplitude of the second peak decreased. Furthermore, the third effect showed that the frequency shift was not equidistant. Thus, the frequency shift steps of the first main peak became smaller as the Youngs module increased, while those of the second main peak module gained. At this point, it would already be possible to proceed to the next and gain a larger material, since the influence of the small step size of the Youngs plate step step step method with the loss of the Youngs behavior is closely related to the small size of the data, however, since the reduction of the size of the sample size of the sample size of the sample size is not necessarily related to a greater loss of the size of the sample size of the sample.

Therefore, the range and step size between the parameter pairs was further reduced (Figure 4, red diamonds). In the corresponding chart, three effects become visible now (see Figure 6). On the one hand, the two main peaks were shifted to higher frequencies for higher values of the Youngs module. On the other hand, the amplitude of the first main peak increased with increasing Youngs module, while the amplitude of the second peak decreased. Furthermore, the third effect showed that the frequency shift was not equidistant. Thus, the frequency shift steps of the first main peak became smaller as the Youngs module increased, while those of the second main peak became larger. FOR Materials 2022, 15, x PEER REVIEW 11 of 15

Figure 6. Results of the parameter study with reduced range and step size.

However, in order to support the formulation of an optimization goal and subsequently draw an elaborated conclusion on the material selection, a continuous spectral analysis showing the effect of the relevant parameters on the target variable over the entire parameter range of the material group could be beneficial. Since in the present case the computation time is reasonable using average computational resources, a more detailed calculation can be performed over the entire target value range. The results were presented in the form of a color map (see Figure 7), which is comparable to a topographic map. In this context, the yellow areas are the peaks of the transmission losses, as shown in the figures above. A cut through the color loss module provides the values over the frequency spectrum of a single value pair of Young's display value and loss factor. This parameter corresponds to the frequency spectrum (see Figure 6), which corresponds to the two yellow and yellow lines in the chart, and can be seen in Figure 7, illustrated as a consistent red line loss.

In addition, s-shaped patterns became apparent in the color map (see Figure 7). These patterns are caused by the eigenmodes of the plate. Following the colored area starting from the lowest Youngs module at approximately 600 Hz to higher values of the Youngs module, the s-shaped form of the transmission loss over the frequency spectrum becomes more apparent. Between these s-patterns, discontinuities occurred, becoming more distinctive with increasing Youngs module. These discontinuities correspond to a shift of the eigenfrequency from a higher to a lower mode of the module. This explains why no clear trend could be observed within the preliminary parameter study applying a larger size step (see Figure 5).

Figure 6. Results of the parameter study with reduced range and step size.

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Materials 2022,15, 2935 11 of 15

At this point, it would already be possible to proceed to the next step and select a material, since the influence of the Youngs module coupled with the loss factor on the transmission loss is readily apparent in the narrower parameter range. However, in the present case, there was a significant difference between the behavior of the transmission loss with a large step size of the Youngs module and a small step size. Thus, it seems worth investigating the entire target parameter range of the Youngs module linked to the loss factor with a finer resolution to gain a better understanding of the behavior of the plate silencer. Since this step requires a greater computational effort than decreasing the range with the step size (see Figure 6), it is not necessarily recommended within the framework of the material selection method.

Materials 2022, 15, x FOR PEER REVIEW 12 out of 15 peaks of the transmission loss become higher and narrower, whereas for higher loss factors, the peak widens and the transmission loss decreases.

Figure 7. Results of the parameter study with a fine parameter resolution.

3.2.5. Selection

Based on the observed trends, the specific Youngs module combined with the loss factor can now be selected to achieve the desired characteristics for the plate silencer. Thus, to realize the objective of low-frequency attenuation mentioned above, a high Youngs module is preferable, because higher transmission losses appear at lower frequencies for higher Youngs modules. In contrast, lower Youngs modules decrease the transmission losses and shift them to higher frequencies.

In order to select an optimum parameter range, it seems reasonable to move along curve A, due to the high transmission losses below 600 Hz. However, curve A simultaneously shows sharply decreasing transmission losses in the frequency range up to 600 Hz. Due to this and the fact that in the present case two transmission loss peaks in a red-frequency range are considered superior to one, the optimal range of the Youngs frequency range was determined between 1 and 3 MPa (Figure 7, Young and Young respectively). This method provides an apparent difference in performance between the performance of the low-frequency range and the performance of the low-frequency range.

Figure 7. Results of the parameter study with a fine parameter resolution.

In addition, s-shaped patterns became apparent in the color map (see Figure 7). These patterns are caused by the eigenmodes of the plate.

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Materials 2022,15, 2935 12 of 15 from the lowest Youngs module at approximately 600 Hz to higher values of the Youngs module, the s-shaped form of the transmission loss over the frequency spectrum becomes more apparent. Between these s-patterns, discontinuities occurred, becoming more distinctive with increasing Youngs module. These discontinuities correspond to a shift of the eigenfrequency from a higher to a lower odd mode of the plate. This explains why no clear trend could be observed within the preliminary parameter study applying a wider step size (see Figure In addition, Figure 7 shows that the s-shaped patterns became more distinct and brighter with Youngs module, and faded and somewhat darker with increasing loss factor.

3.2.5. Selection

In order to select an optimal parameter range, it seems reasonable to move along curve A, due to the high transmission losses below 600 Hz. However, curve A simultaneously shows sharply decreasing transmission losses in the existing frequency range up to 600 Hz. At a Youngs modulus of 1 MPa, a second curve B appeared at around 400 Hz. Thus, the mode shift between E=1 and 3 MPa revealed a region with two peaks. Above a Youngs modulus of 3 MPa, curve A turned to higher frequencies outside the target range. In this range, curve B shows high transmission losses. However, curve A simultaneously shows sharply decreasing transmission losses in the existing frequency range up to 600 Hz. Due to this and the fact that in the present case two transmission loss peaks in a red-frequency range are considered superior to one, the optimal range of the Youngs is determined closely depending on the range of 1 and 3 MPa (Fig 7), whereas determined at the end of this method, all three materials in the scope of the MPCT can be used (see Figure 3).

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Materials 2022,15, 2935 13 of 15 Materials 2022, 15, x FOR PEER REVIEW 13 of 15 of the material selection process. As in this case, where two peaks at a lower frequency may be superior to one with a higher transmission loss, depending on the application. Consequently, the range of 1 to 3 MPa provides the optimal parameter pairs of Youngs modulus and loss factor that are used to select real existing materials. Therefore, the materials that match the parameter pairs closely are identified in the material chart (see Figure 4). Since the actual existing materials correspond only to a limited extent to the functional dependence, the transmission loss curves of the three materials: styrene methyl methacrylate (SMMA), acrylonitone butylene styrene (ABS), and polyethylene terephthalate (PCT) must be determined separately (see Figure 8).

Figure 8. Transmission loss for specific plate materials [35].

4. Conclusions

A new method of model-based material selection has been developed and successfully transferred to the application of the plate resonator. The method allows the visualization and coherent analysis of the influences on the target parameter over a certain value range by determining a functional dependency between relevant material parameters. By adjusting the step sizes and displaying the results, a method has been established that can be applied to the process of material selection of similarly complex systems. In the presented example, the functional relationship represents the target parameter range well and allows a reduction of the parameter space. However, the accuracy of the functional relationship depends on the parameter range considered, which leads to a reduction of the material parameter range if the accuracy of the material range is low. Therefore, the method greatly relies on the functional dependency between the relevant parameters of the target material.

4. Conclusions

A new method of model-based material selection has been developed and successfully transferred to the application of the plate resonator. The method allows the visualization and coherent analysis of the influences on the target parameter over a certain value range by determining a functional dependence between relevant material parameters. By adjusting the step sizes and displaying the results, a method has been established that can be applied to the process of material selection of similarly complex systems. In the presented example, the functional relationship represents the target parameter range well and allows a reduction of the parameter space. However, the accuracy of the functional relationship depends on the parameter range considered, which leads to a reduction of the material parameter range if the accuracy of the target material range is low. Therefore, the method greatly relies on the functional dependence between the relevant parameters of the target material range.

Author Contributions:

Conceptualization, M.N., F.S., V.R. and M.D.; methodology, M.N., M.D. and

F.S.; investigation, M.N., F.S. and V.R.; writing original draft preparation, M.N., V.R. and F.S.;

writing review and editing, M.N., V.R., M.D. and E.S.; visualization, M.N., V.R. and F.S.; supervision,

N.M., E.S. and M.N.; project administration, N.M., E.S. and M.N.; acquisition of funding, N.M. and E.S.

All authors have read and agreed with the published version of the manuscript.

Funding: Funded by the German Research Foundation (DFG),GermanResearch

The Federal Ministry for Economic Affairs and Climate Action (contract numbers: 20E1915B and 20E1915C), based on a decision of the German Bundestag, is gratefully grateful for the financial support of the work in the framework of the LuFo VI-1 project FLIER (Flexible wall structures for acoustic LINERs).

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Data Availability Statement: The data presented in the current study are available upon request from the corresponding author.

Conflicts of Interest: The authors declare that there is no conflict of interest.

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The Commission shall adopt implementing acts in accordance with the procedure referred to in paragraph 1.

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