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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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materials

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; niels.molder@tu-dresden.de

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: moritz.neubauer@tu-dresden.de; Tel.: +49-351-463-4026

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 respective application,

which contributes to a more target-oriented design. The method is applied to the example of a plate

resonator. For this purpose, a semi-analytical model is implemented to calculate the transmission

loss as well as the reflected and dissipated sound power of plate silencers, taking into account the

influence of flow velocity and fluid temperature on the performance of plate silencers.

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

1. Introduction

Numerous methodologies have been introduced that aim to provide a guiding struc-

ture in the process of selecting the optimal material for a particular application [

]. The

frameworks described by Ashby et al. [

] and Van Kesteren et al. [

] 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 mate-

rial. In this context, Ashby charts are often used for screening because they allow facile 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 [

–

]. In this context, MCDM techniques

apply material selection decision matrices and criteria sensitivity analysis to enable a logical

ranking of considered materials [

–

]. An alternative approach of supporting the decision

process in material selection lies in the application of artificial intelligence [

–

]. 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 [

–

]. For the present work, the material property charts introduced by

Ashby and Cebon [

] 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.

Materials 2022,15, 2935. https://doi.org/10.3390/ma15082935 https://www.mdpi.com/journal/materials

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 complex-

ity 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 vari-

able as well as suitable value ranges that can be used as a basis for material selection.

In addition, 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 acoustics. Due to the vast number of relevant parameters,

the modeling of a plate resonator (PR) is rather complex, leading to computational and

time-intensive parameter studies in the context of an optimization. Therefore, a method

is required to specify the stepwise model-based material selection for acoustic PR liners.

Consequently, the narrowing of the parameter space and the identification of parameters

with significant influence on the target variable through material group analysis enables a

time- and resource-efficient material selection. In this context, the presented determination

of a functional dependence between the material parameters allows a more decisive and

coherent analysis of their effects on the target variable. The representation of the parameters

of interest within a spectrum provides additional information that can support the choice of

the formulated optimization goal and is a possible complement to the method of applying

performance indices.

Plate silencers, that consist 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 obtained experimental results 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 to silencers made of porous material or per-

forated surfaces. They can be exposed to contaminated air, heat, frost, and humidity

without being damaged. Due to their smooth surfaces, their flow resistance is compara-

tively low [

]. There are numerous applications where the advantageous properties of

the smooth and robust sheet surface can be effectively utilized, e.g., as suspended trans-

parent ceiling installations made of plastic films to meet high air quality requirements

in clean rooms [

]. Furthermore, plate resonators are widely used as low-frequency

tuned silencers in industrial ventilation systems as well as in the pipework of central air

conditioning systems to reduce noise emissions [

]. 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 [

In this context, the performance of the plate silencer is significantly influenced by the

material properties of the plate. Hence, it is important to find a suitable material for the

specific application.

The aim of the presented 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 considering functional dependencies among properties and

the interaction between the selection process and the model. The functional dependency

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 value range 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 1is 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 ma-

terial selection is described considering functional dependencies among properties and

the interaction between the selection process and the model. The functional dependency

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 value range 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 applica-

tions is described below. The methodology shown in Figure 1 is subsequently applied to

the selection of the plate material for a plate resonator.

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

2.1. Material Screening

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

Materials 2022,15, 2935 4 of 15

2.1. Material Screening

The starting point is the definition of requirements resulting from the intended ap-

plication and the subsequent development of the model to describe the relationships and

phenomena of the application. In the course of the modeling process, the relevant ma-

terial 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

can include requirements arising from the product lifecycle, such as restrictions in the fields

of manufacturing technologies or recycling. After the selection of suitable material groups,

the fundamental parameter set(s) is/are established either by experimental characterization

or database query.

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 carried out by examining the

distribution of property values of the targeted material group. For properties that scatter

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. In case the material parameter and the target variable have a direct proportional

relationship, this parameter is not considered further due to its comprehensible effect on the

target variable. However, if the material parameter has a non-facile but contrary effect that

leads to conflicting targets, this parameter has a relevant influence on the target variable

that needs to be investigated in more detail.

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

Parameter Symbol Unit Value/Value Range Design Point

Geometry parameter

Cavity

length lCmm 60 60

Cavity

height hCmm 30 30

Cavity width

wCmm ∞ ∞

Duct height hDmm 60 60

Duct width wDmm ∞ ∞

Plate

thickness hPmm 0.3 0.3

Material parameter

Young’s

modulus 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

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 [

]. The visualization can support the process of identi-

fication of functional dependencies of the selected material properties [

]. Depending

on the size of the selected group of materials and the range of the 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 basis function is selected. Then, us-

ing suitable 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 resolution over the entire parameter

range in case more sections are of particular interest. In this way, the effects of the relevant

parameters can be mapped across the entire value range. This assumes that the required

computational effort remains feasible.

2.5. Selection

In the last step, a parameter combination is selected, that provides the desired result

with respect to the target variable. Depending on the accuracy of the functional dependency,

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 Presented Methodology 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 apply-

ing the presented material selection method to determine the plate material of

the silencer

3.1. Application—Plate 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 closed volume,

called a cavity. As sound propagates along the duct, the plate is excited by the pressure

difference above and below it. As a result, one part of the sound is reflected, and another

part is dissipated by the vibration of the plate and its interaction with the cavity. To

describe this process, a model developed by Huang and Wang is applied, describing a

two-dimensional plate silencer with a simply supported plate [

]. In this context, the

validity of this model was verified through the comparison of experimental results, that

showed a satisfying level of accordance [

]. 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

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 present 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 present material selection process.

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

plate resonators 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, Young’s modulus 𝐸, Poisson ratio 𝜐 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 veloc-

ity. 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 sys-

tem of equations can be solved for the modal amplitude of the plate velocity, 𝑣.

(𝐿+𝑍𝐶+𝑍𝐷)𝑣 =−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 well.

The transmission loss, representing the target variable, can be computed from the trans-

mitted sound power 𝑃trans, and the incident sound power 𝑃𝑖𝑛, as follows:

𝑇𝐿 =−10log|𝑃𝑡𝑟𝑎𝑛𝑠

𝑃𝑖𝑛 |.

(2)

Figure 2.

(

) Duct with defined height

, width

and integrated resonator liner, (

) design of a

plate resonators with



its film layer



the enclosed cavity, comprising the cell length

, the cell

height

, the cell width

and



the bottom layer. (

) Detailed view of the film with the relevant

material properties, being density

, Young’s modulus

, Poisson ratio

and the loss factor

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

, and the cavity

(see Figure 2). Additionally, such an impedance matrix can

also be formulated for the plate (

), 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

, the following system

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

L+ZC+ZDv=−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

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



. (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 configu-

ration was applied (see Figure 2). The foundation of the computation of the transmission

loss is the input parameters contained in the impedance matrices. As shown in Table 1,

they are divided in geometry and material parameters.

Since the calculation was performed in two dimensions, the displayed width of the

duct

, and the cavity

, 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 damped in the system, which reduces the computation time and allows the

simulation of more material configurations.

3.2. Material Selection Process

In the following section, the above-presented method is applied to the material selec-

tion of the plate of a plate silencer.

3.2.1. Material Screening

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. Secondarily,

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 first exploration in order to assess the effect

of the individual material parameters on the target variable. Therefore, the effect of the

Young’s modulus, loss factor, and density on the transmission loss was analyzed. For this

purpose, the value range, step size, and the 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 ranges 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). From there, all three

parameters were varied independently towards the defined limits. For the density and

the loss factor, the effects were distinct and predictable. The density results in a relative

change in frequency, with a higher density shifting the spectrum to lower frequencies

(see Figure 3c). Since the value range of the density within the targeted material group is

comparatively small and the mass of the plate can also be changed by varying the plate

Materials 2022,15, 2935 8 of 15

thickness, the influence of this parameter does not need to be investigated further, leaving

the Young’s modulus 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 Young’s modulus 𝐸, (c) variation of density ρ, (d) variation of loss factor η.

The loss factor broadens 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 exists a significantly larger

value range for the loss factor within the material group of thermoplastics and thermo-

plastic elastomers. Therefore, and due to the fact that the loss factor has a higher qualita-

tive as well as a more diverse effect on the transmission loss, further analysis of the pa-

rameter is required.

The influence of the Young’s modulus is as yet undetermined, since no clear effect

can be observed. Based on the goal of low-frequency attenuation and analyzing the results

shown in Figure 3b, it appears that high values for the Young’s modulus are preferable.

Nevertheless, it seems that a more detailed study on the influence of the Young’s modu-

lus, considering 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 thermo-

plastic polymers, the identification of a functional dependence between the relevant ma-

terial 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 regard to the target variable, these two material parameters

were considered for the determination of a functional dependency. For the purpose of

identifying a dependency, 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 scales of both axes. In Figure 4, the logarithmic mean values of Young´s modulus and

mechanical loss factor considering the material-envelopes of 211 representatives sub-clas-

ses of the material class of thermoplastics and thermoplastic elastomers were analyzed on

a logarithmic scale for both axes [35]. The graphical display of the parameters reveals the

linear dependency in a double logarithmic representation, leading to a polynomic inter-

relationship in linear space (see Figure 4).

Figure 3.

Results of the preliminary parameter study, (

) reference result of the design point,

(b) variation of Young’s modulus E, (c) variation of density ρ, (d) variation of loss factor η.

The loss factor broadens 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 exists a significantly larger value

range 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.

The influence of the Young’s modulus is as yet undetermined, since no clear effect

can be observed. Based on the goal of low-frequency attenuation and analyzing the results

shown in Figure 3b, it appears that high values for the Young’s modulus are preferable.

Nevertheless, it seems that a more detailed study on the influence of the Young’s modulus,

considering 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 ther-

moplastic 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 regard to the target variable, these two material parameters

were considered for the determination of a functional dependency. For the purpose of

identifying a dependency, the corresponding pairs of values of the material parameters

were analyzed in the form of a material property chart introduced by Ashby [

], varying

the scales of both axes. In Figure 4, the logarithmic mean values of Young

s modulus and

mechanical loss factor considering the material-envelopes of 211 representatives sub-classes

of the material class of thermoplastics and thermoplastic elastomers were analyzed on

Materials 2022,15, 2935 9 of 15

a logarithmic scale for both axes [

]. The graphical display of the parameters reveals

the linear dependency in a double logarithmic representation, leading to a polynomic

interrelationship in linear space (see Figure 4).

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

Figure 4. Mechanical loss factor over Young’s modulus of thermoplastic and thermoplastic elasto-

mer polymers: the fitting curve represents the functional dependence between the two material pa-

rameters 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 basis 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

In order to perform the parameter study, corresponding pairs of the material param-

eters with a logarithmic step size were selected, covering the whole value range of the

targeted 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. The resulting chart displays the variation of the transmission loss

for the target value range, where the effect of the parameter combination is not clearly

apparent (see Figure 5).

0.001

0.01

0.1

0.001 0.01 0.1 1 10

Mechnaical loss coefficent tan δ[-]

Young´s Modulus [GPa]

Logarithmized mean values

Low resolution step width

High resolution step width

Fitting curve

Figure 4.

Mechanical loss factor over Young’s modulus 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 basis 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

In order to perform the parameter study, corresponding pairs of the material param-

eters with a logarithmic step size were selected, covering the whole value range of the

targeted 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. The resulting chart displays the variation of the transmission loss

for the target value range, where the effect of the parameter combination is not clearly

apparent (see Figure 5).

Materials 2022,15, 2935 10 of 15

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

Figure 5. Results of the parameter study with functional dependence between Young’s modulus, E,

and the corresponding loss factor, η.

Therefore, the range and the step size between the parameter pairs was further re-

duced (Figure 4, red diamonds). In the corresponding chart, three effects become visible

now (see Figure 6). On one side, the two main peaks were shifted to higher frequencies

for higher values of the Young’s modulus. On the other side, the amplitude of the first

main peak increased with increasing Young’s modulus, 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

Young’s modulus increased, while those of the second main peak became larger.

At this point, it would already be possible to proceed to the next step and select a

material, since the influence of the Young’s modulus 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 Young’s modulus and a small step size. Thus, it seems to

be worthwhile to investigate the whole target parameter range of the Young’s modulus

linked with 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.

Figure 5.

Results of the parameter study with functional dependence between Young’s 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 one side, the two main peaks were shifted to higher frequencies for

higher values of the Young’s modulus. On the other side, the amplitude of the first main

peak increased with increasing Young’s modulus, 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

Young’s modulus increased, while those of the second main peak became larger.

Materials 2022, 15, x FOR 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 subse-

quently 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 pre-

sented 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 chart affords the transmission loss values

over the frequency spectrum of a single value pair of Young’s modulus and loss factor.

This corresponds to the type of plots already known (see Figure 6) and illustrates two

sections inserted in the color map (Figure 7, red and yellow dashed line). It can be seen

that, depending on the combination of Young’s modulus and loss factor, one or two peaks

appear in transmission loss over the frequency spectrum. This observation is consistent

with that in Figure 6, as the parameter range displayed there corresponds to the area be-

tween the yellow and the red dashed line in Figure 7.

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 Young’s modulus at approximately 600 Hz to higher values of the

Young’s modulus, the s-shaped form of the transmission loss over the frequency spectrum

becomes more apparent. In between these s-patterns, discontinuities occurred, becoming

more distinctive with increasing Young’s modulus. 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 5). In Addition, Figure 7 shows that the s-shaped patterns be-

came more distinct and brighter with increasing Young’s modulus, and faded and became

a little darker with increasing loss factor. This implies that for higher Young’s moduli, the

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

Materials 2022,15, 2935 11 of 15