# Material Parameter Identification for Acoustic Simulation of Additively Manufactured Structures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Additive Manufacturing

^{®}(thermoplastic polyurethane, TPU) from NinjaTek).

## 3. Experimental Investigation

#### 3.1. Setup and Specimens

#### 3.2. Material Parameter Identification for Static Case

^{3}and the span is selected to be 64 mm. In Figure 5 the test setup with the beam specimen between the supports and the compression fin is schematically illustrated.

#### 3.3. Results

## 4. Parameter Study by Numerical Investigation

#### 4.1. Mechanical Model and Numerical Solution

_{0}= 3 × 10

^{9}N/m

^{2}in order to receive a conservative mesh size by the convergence study. Applying ${E}_{0}$ to the three thickness setups $t=\{1,3,6\}$ mm, the mesh size is reduced systematically, until $\Delta {h}^{2}$ is smaller than 0.1 dB. In Figure 9, the maximum error is plotted in dependency on the mesh size. The chosen mesh size for each beam is further marked in the figure. Finally, this results in FE models with 12 k, 24 k and 48 k dof for the 6, 3 and 1 mm beam, respectively. The mesh is shown above Figure 9.

#### 4.2. Parameter Identification

#### 4.2.1. Frequency-Independent Parameters

#### 4.2.2. Frequency-Dependent Parameters

^{9}N/m

^{2}from one sample to the next. The motivation is to avoid non-physical jumps. Finally, the contour plots in Figure 12 are created which show the FRAC distribution over frequency and flexural modulus. The identified values are marked by the +. On the first view, the procedure works quite well with the exception of Beam_1a and Beam_3a. For these specimens, the identified values are not laying on a recognizable curve as FRAC is indicating several best-fitting flexural moduli for one frequency sample.

^{9}N/m

^{2}and 3.7 × 10

^{9}N/m

^{2}which is about twice the value due to a 6 times thicker beam. The sensitivity reduces comparing the 3 mm beam and the 6 mm beam which indicates a non-linear behavior of the flexural modulus in dependency on the thickness. For ABH structures, besides the usual stiffness reduction by a lowered thickness, an additional effect can be expected by the manufacturing process itself. Considering this effect in mechanical models, a more precise and reliable prediction may be the result.

#### 4.3. Final Results

## 5. Conclusions

- dependence of homogenized material parameter (Young’s modulus, loss factor) on frequency and thickness
- Young’s modulus decreases significantly with decreasing thickness

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

3D | Three-dimensional |

ABH | Acoustic Black Hole |

AM | Additive manufacturing |

${\delta}_{1}$ | First error criterion |

dof | degrees of freedom |

E | Young’s modulus |

elPaSo | Elementary parallel solver (in-house code) |

$\eta $ | Loss factor |

f | Frequency |

F | Exciting Force |

FOM | Full order model |

FRAC | Frequency Response Assurance Criterion |

${h}^{2}$ | Mean squared admittance |

H | Frequency response function |

MEX | Material extrusion |

MOR | Model order reduction |

MSA | Mean squared admittance |

${N}_{f}$ | Number of frequency points |

${N}_{p}$ | Number of surface points |

$\nu $ | Poisson’s ratio |

PLA | Polyactic acid |

$\varrho $ | Density |

ROM | Reduced order model |

t | Thickness of beam specimens |

TPU | Thermoplastic polyurethane |

v | Surface velocity |

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**Figure 1.**Tip of Acoustic Black Holes (ABH) beam with (

**right**) and without (

**left**) additional damping material.

**Figure 2.**Overview of beam specimens with different system complexity: simple beam structure made of PLA (

**front**), beam structure with an ABH made of PLA (

**center**), and beam structure with an ABH made of PLA with an additional damping treatment made of an thermoplastic polyurethane (

**back**).

**Figure 9.**Maximum error in dependency on mesh sizes for each beam specimen with two elements over the beam’s thickness (chosen mesh size marked with circle).

**Figure 10.**Comparison of ${h}^{2}$ for two elements (dashed line) and three elements (marks) in thickness direction for each beam.

**Figure 11.**${\delta}_{1}$ contour plot for each specimen with identified optima under variation of E and $\eta $ (frequency-independent, constant values).

**Figure 12.**FRAC contour plot for each specimen with identified optima under variation of E (frequency-dependent, constant value for $\eta $).

**Figure 13.**Identified flexural moduli by Figure 12 for Beam_6a/b with fitted linear curve.

**Figure 14.**Identified flexural moduli by Figure 12 for Beam_3a/b with fitted linear curve.

**Figure 15.**Identified flexural moduli by Figure 12 for Beam_1a/b with fitted linear curve.

**Figure 16.**Overall fitted (

**a**) linear curves for the three different manufactured thicknesses and (

**b**) flexural modulus in dependency on frequency and thickness.

**Figure 18.**Experimental and numerical response of each specimen with frequency-dependent material parameters E and $\eta $.

Material | PLA |
---|---|

Temperature Build Platform | 60 °C |

Temperature Nozzle | 215 °C |

Layer Thickness | 0.0002 m |

Raster Angle | ±45° |

Perimeter Shells | 2 |

Flow Rate | 105% |

Infill Percentage | 100% |

Extrusion Width | 0.0004 m |

Extrusion Speed | 0.05 m/s |

Specimen | Length (m) | Width (m) | Thickness (m) | Mass (kg) | Effective Density (kg/m^{3}) |
---|---|---|---|---|---|

Beam_6a | 0.200 | 0.020 | 0.0063 | 0.03081 | 1222.6 |

Beam_6b | 0.200 | 0.020 | 0.0062 | 0.03067 | 1236.7 |

Beam_3a | 0.200 | 0.020 | 0.0032 | 0.01613 | 1260.2 |

Beam_3b | 0.200 | 0.020 | 0.0033 | 0.01643 | 1244.7 |

Beam_1a | 0.200 | 0.020 | 0.0012 | 0.00588 | 1225.0 |

Beam_1b | 0.200 | 0.020 | 0.0012 | 0.00579 | 1206.3 |

**Table 3.**Overview of frequency boundaries and signal type for separately excited and investigated frequency ranges.

Frequency Range (Hz) | Signal Type |
---|---|

0–1000 | Pseudo Random |

1000–2000 | Sweep |

2000–3000 | Sweep |

3000–4000 | Sweep |

4000–6000 | Sweep |

Material | PLA |
---|---|

Flexural modulus | 3196.75 × 10^{6} N/m^{2} |

Standard deviation | 68.98 × 10^{6} N/m^{2} |

**Table 5.**Number of matched moments in the reduced-order model (ROM) in order to ensure a maximum $\Delta {h}^{2}$ < 0.1 dB for the parameter identification.

Model | Moments Matched |
---|---|

6 mm beam | 2 |

3 mm beam | 3 |

1 mm beam | 5 |

Parameter | Unit | Range | Delta | |
---|---|---|---|---|

Flexural modulus | E | N/m^{2} | 1.5 × 10^{9}–6.0 × 10^{9} | 0.06 × 10^{9} |

Loss factor | $\eta $ | 0.005–0.15 | 0.005 | |

Density | $\rho $ | kg/m^{3} | constant (see Table 2) | |

Poisson’s ratio | $\nu $ | constant (0.35) |

Specimen | Thickness (m) | Flexural Modulus (N/m^{2}) | Loss Factor |
---|---|---|---|

Beam_6a | 0.0063 | 4.20 × 10^{9} | 0.090 |

Beam_6b | 0.0062 | 4.44 × 10^{9} | 0.095 |

Beam_3a | 0.0032 | 4.38 × 10^{9} | 0.080 |

Beam_3b | 0.0033 | 3.96 × 10^{9} | 0.060 |

Beam_1a | 0.0012 | 3.42 × 10^{9} | 0.055 |

Beam_1b | 0.0012 | 3.24 × 10^{9} | 0.065 |

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**MDPI and ACS Style**

Rothe, S.; Blech, C.; Watschke, H.; Vietor, T.; Langer, S.C. Material Parameter Identification for Acoustic Simulation of Additively Manufactured Structures. *Materials* **2021**, *14*, 168.
https://doi.org/10.3390/ma14010168

**AMA Style**

Rothe S, Blech C, Watschke H, Vietor T, Langer SC. Material Parameter Identification for Acoustic Simulation of Additively Manufactured Structures. *Materials*. 2021; 14(1):168.
https://doi.org/10.3390/ma14010168

**Chicago/Turabian Style**

Rothe, Sebastian, Christopher Blech, Hagen Watschke, Thomas Vietor, and Sabine C. Langer. 2021. "Material Parameter Identification for Acoustic Simulation of Additively Manufactured Structures" *Materials* 14, no. 1: 168.
https://doi.org/10.3390/ma14010168