# Numerical and Experimental Extraction of Dynamic Parameters for Pyramidal Truss Core Sandwich Beams with Laminated Face Sheets

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Samples

#### 2.2. Numerical Model of Sandwich Beam

#### 2.2.1. Modeling of Laminated Face Sheets

#### 2.2.2. Modeling of an Aluminum Pyramidal Core

#### 2.2.3. Modeling of an Adhesive Layer

#### 2.2.4. Model Assembly and Solution Method

#### 2.3. Damping Model

#### 2.3.1. Strain Energy

#### 2.3.2. Dissipated Energy

#### 2.3.3. Damping Model Validation

#### 2.4. Experimental Set-Up

#### Modal Loss Factors Extraction

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Feng, L.J.; Xiong, J.; Yang, L.H.; Yu, G.C.; Yang, W.; Wu, L.Z. Shear and bending performance of new type enhanced lattice truss structures. J. Mech. Sci.
**2017**, 134, 589–598. [Google Scholar] [CrossRef] - Wang, J.; Liu, W.; Kang, S.; Ma, T.; Wang, Z.; Yuan, W.; Song, H.; Huang, C. Compression performances and failure maps of sandwich cylinders with pyramidal truss cores obtained through geometric mapping and snap-fit method. Compos. Struct.
**2019**, 226, 1–10. [Google Scholar] [CrossRef] - Xiong, J.; Ma, L.; Pan, S.; Wu, L.; Papadopoulos, J.; Vaziri, A. Shear and bending performance of carbon fiber composite sandwich panels with pyramidal truss cores. Acta Mater.
**2012**, 60, 1455–1466. [Google Scholar] [CrossRef] - Xu, G.D.; Yang, F.; Zeng, T.; Cheng, S.; Wang, Z.H. Bending behavior of graded corrugated truss core composite sandwich beams. Compos. Struct.
**2016**, 138, 342–351. [Google Scholar] [CrossRef] - Xu, G.D.; Zeng, T.; Cheng, S.; Wang, X.H.; Zhang, K. Free vibration of composite sandwich beam with graded corrugated lattice core. Compos. Struct.
**2019**, 229, 1–7. [Google Scholar] [CrossRef] - Ruzzene, M. Vibration and sound radiation of sandwich beams with honeycomb truss core. J. Sound Vib.
**2004**, 277, 741–763. [Google Scholar] [CrossRef] - Li, W.; Sun, F.; Wang, P.; Fan, H.; Fang, D. A novel carbon fiber reinforced lattice truss sandwich cylinder: Fabrication and experiments. Compos. Part A Appl. Sci. Manuf.
**2016**, 81, 313–322. [Google Scholar] [CrossRef] - Deshpande, V.S.; Fleck, N.A. Collapse of truss core sandwich beams in 3-point bending. Int. J. Solids Struct.
**2001**, 38, 6275–6305. [Google Scholar] [CrossRef] - O’Masta, M.R.; Dong, L.; St-Pierre, L.; Wadley, H.N.G.; Deshpande, V.S. The fracture toughness of octet-truss lattices. J. Mech. Phys. Solids.
**2017**, 98, 271–289. [Google Scholar] [CrossRef] - George, T.; Deshpande, V.S.; Wadley, H.N.G. Hybrid carbon fiber composite lattice truss structures. Compos. Part A Appl. Sci. Manuf.
**2014**, 65, 135–147. [Google Scholar] [CrossRef] - Zhang, X.; Jin, X.; Xie, G.; Yan, H. Thermo-Fluidic Comparison between Sandwich Panels with Tetrahedral Lattice Cores Fabricated by Casting and Metal Sheet Folding. Energies
**2017**, 10, 906. [Google Scholar] [CrossRef][Green Version] - Yin, S.; Chen, H.; Wu, Y.; Li, Y.; Xu, J. Introducing composite lattice core sandwich structure as an alternative proposal for engine hood. Compos. Struct.
**2018**, 201, 131–140. [Google Scholar] [CrossRef] - Yang, J.S.; Ma, L.; Chaves-Vargas, M.; Huang, T.X.; Schröder, K.U.; Schmidt, R.; Wu, L.Z. Influence of manufacturing defects on modal properties of composite pyramidal truss-like core sandwich cylindrical panels. Compos. Sci. Technol.
**2017**, 147, 89–99. [Google Scholar] [CrossRef] - Xu, M.; Qiu, Z. Free vibration analysis and optimization of composite lattice truss core sandwich beams with interval parameters. Compos. Struct.
**2013**, 106, 85–95. [Google Scholar] [CrossRef] - Spadoni, A.; Ruzzene, M. Structural and Acoustic Behavior of Chiral Truss-Core Beams. J. Vib. Acoust.
**2006**, 128, 616–626. [Google Scholar] [CrossRef] - Arvin, H.; Sadighi, M.; Ohadi, A. A numerical study of free and forced vibration of composite sandwich beam with viscoelastic core. Compos. Struct.
**2010**, 92, 996–1008. [Google Scholar] [CrossRef] - Barkanov, E.; Skukis, E.; Petitjean, B. Characterisation of viscoelastic layers in sandwich panels via an inverse technique. J. Sound. Vib.
**2009**, 327, 402–412. [Google Scholar] [CrossRef] - Yang, J.; Xiong, J.; Ma, L.; Wang, B.; Zhang, G.; Wu, L. Vibration and damping characteristics of hybrid carbon fiber composite pyramidal truss sandwich panels with viscoelastic layers. Compos. Struct.
**2013**, 106, 570–580. [Google Scholar] [CrossRef] - Barkanov, E.; Wesolowski, M.; Hufenbach, W.; Dannemann, M. An effectiveness improvement of the inverse technique based on vibration tests. Comput. Struct.
**2015**, 146, 152–162. [Google Scholar] [CrossRef] - Wesolowski, M.; Barkanov, E. Improving material damping characterization of a laminated plate. J. Sound Vib.
**2019**, 462, 1–12. [Google Scholar] [CrossRef] - Rozylo, P.; Ferdynus, M.; Debski, H.; Samborski, S. Progressive Failure Analysis of Thin-Walled Composite Structures Verified Experimentally. Materials
**2020**, 13, 1138. [Google Scholar] [CrossRef] [PubMed][Green Version] - Huang, Y.; Zhang, Z.; Li, C.; Mao, K.; Huang, Q. Modal Performance of Two-Fiber Orthogonal Gradient Composite Laminates Embedded with SMA. Materials
**2020**, 13, 1102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wang, Y.J.; Zhang, Z.J.; Xue, X.M.; Zhang, L. Free vibration analysis of composite sandwich panels with hierarchical honeycomb sandwich core. Thin Walled Struct.
**2019**, 145, 1–10. [Google Scholar] [CrossRef] - Kłosowski, P.; Zagubień, A. Analysis of material properties of technical fabric for hanging roofs and pneumatic shells. Arch. Civ. Eng.
**2003**, 49, 277–294. [Google Scholar] - Adams, R.; Bacon, D. Measurement of the flexural damping capacity and dynamic Young’s modulus of metals and reinforced plastics. J. Phys. D Appl. Phys.
**1973**, 6, 27–41. [Google Scholar] [CrossRef] - Lin, D.; Ni, R.; Adams, R. Prediction and Measurement of the Vibrational Damping Parameters of Carbon and Glass Fibre-Reinforced Plastics Plates. J. Compos. Mater.
**1984**, 18, 132–152. [Google Scholar] [CrossRef] - Maheri, M.; Adams, R. Finite-element prediction of modal response of damped layered composite panels. Compos. Sci. Technol.
**1995**, 55, 13–23. [Google Scholar] [CrossRef] - Ewins, D.J. Modal Testing: Theory, Practice and Application, 2nd ed.; Research Studies Press LTD: Baldock, UK, 2000. [Google Scholar]

**Figure 1.**Fabrication process of a sandwich beam: (

**a**,

**b**) aluminum pyramidal truss core assembly; (

**c**,

**d**) a sandwich beam assembly with laminated face sheets and pyramidal core. CFRP: carbon fiber reinforced plastic.

**Figure 3.**Development procedure of the numerical model: (

**a**) parent materials assembly; (

**b**) associated mesh; (

**c**) joint modeling with the adhesive layer; (

**d**) joint modeling without the adhesive layer.

**Figure 4.**Experimental assessment of a modal damping: (

**a**) three-dimensional (3D) laser vibrometer set-up; (

**b**) half-bandwidth method for modal loss factor extraction.

**Figure 5.**Experimental Frequency Response Functions (FRFs) of the sandwich beams: (

**a**) Beam 1; (

**b**) Beam 2; (

**c**) Beam 3; (

**d**) Beam 4; (

**e**) Beam 5.

**Figure 6.**Numerical results: (

**a**) Example of the deformed mode shape (mode 1); (

**b**) undeformed state of the adhesive layer; (

**c**) deformed state of the adhesive layer.

**Figure 7.**Sensitivity study of the eigenfrequency change due to the adhesive E modulus variation: (

**a**) Mode 1; (

**b**) Mode 2 and 3; (

**c**) Mode 4 and 5.

Sample | l [mm] | a [mm] | h [mm] | t [mm] | $\Phi $ [mm] |
---|---|---|---|---|---|

Beam 1 | 386.3 | 50.0 | 27.75 | 1.4 | 0 |

Beam 2 | 385.9 | 50.0 | 27.80 | 1.4 | 0 |

Beam 3 | 386.2 | 50.0 | 27.75 | 1.4 | 0 |

Beam 4 | 386.5 | 50.0 | 27.80 | 1.4 | 0 |

Beam 5 | 386.6 | 50.0 | 27.75 | 1.4 | 0 |

Parent Component | Material | E [GPa] | E${}_{1}$ [GPa] | E${}_{2}$ [GPa] | G${}_{12}$ [GPa] | G${}_{13}$ [GPa] | G${}_{23}$ [GPa] | ${\mathit{\nu}}_{12}$ | $\mathit{\nu}$ |
---|---|---|---|---|---|---|---|---|---|

Face sheet | CFRP | - | 142.2 | 10.5 | 7.8 | 7.8 | 3.8 | 0.3 | - |

Core | PA6 | 66.2 | - | - | - | - | - | - | 0.25 |

Adhesive | Epoxy | 1.46 | - | - | - | - | - | - | 0.25 |

Parent Component | Material | ${\mathit{\psi}}_{11}[\%]$ | ${\mathit{\psi}}_{22}[\%]$ | ${\mathit{\psi}}_{12}[\%]$ | ${\mathit{\psi}}_{\mathit{c}}[\%]$ | ${\mathit{\psi}}_{\mathit{a}}[\%]$ | $\mathit{\rho}$ [kg/m${}^{3}$] |
---|---|---|---|---|---|---|---|

Face sheet | CFRP | 0.63 | 8.30 | 5.45 | - | - | 1590.0 |

Core | PA6 | - | - | - | 3.85 | - | 2700.0 |

Adhesive | Epoxy | - | - | - | - | 3.00 | 1200.0 |

Parent Component | Element Type | Number of Elements | Material Model | |
---|---|---|---|---|

Face sheets | S4R | 38,600 | Orthotropic, elastic | |

Core | SC8R | 154,832 | Isotropic, elastic | |

Adhesive | C3D8R | 5376 | Isotropic, elastic | |

Total | 198,808 |

**Table 5.**The predicted values of the specific damping capacity (SDC) coefficients for square laminated plate vs. [26].

Mode | FEM Present | FEM [26] | Difference, % |
---|---|---|---|

1 | 0.0690 | 0.0676 | 2.03 |

2 | 0.0422 | 0.0428 | 1.42 |

3 | 0.0607 | 0.0589 | 2.97 |

4 | 0.0422 | 0.0413 | 2.13 |

5 | 0.0526 | 0.0511 | 2.85 |

6 | 0.0046 | 0.0047 | 2.17 |

Average | 2.26 |

Mode (n) | Experiment (Beam 1) | FEM (I) | FEM (II) |
---|---|---|---|

1 | |||

2 | |||

3 | |||

4 | |||

5 |

$\begin{array}{c}\mathbf{Mode}\hfill \\ \left(\mathbf{n}\right)\hfill \end{array}$ | Mode Type | ${\mathit{f}}_{\mathit{n}}^{\mathit{EXP}},\mathbf{Hz}$ | Beam 1 | Beam 2 | Beam 3 | Beam 4 | Beam 5 | Average * |
---|---|---|---|---|---|---|---|---|

${}^{\mathit{s}}{\mathit{\eta}}_{\mathit{n}}^{\mathit{EXP}},\%$ | ||||||||

1 | Out-of-plane, | ${f}_{1}^{EXP}$ | 382.60 | 381.20 | 381.10 | 377.70 | 381.80 | 381.65 |

(twisting) | ${}^{s}{\eta}_{1}^{EXP}$ | 0.661 | 0.668 | 0.703 | 0.806 | 0.688 | 0.677 | |

2 | Out-of-plane, | ${f}_{2}^{EXP}$ | 1256.60 | 1253.50 | 1254.60 | 1288.60 | 1243.60 | 1254.90 |

(bending) | ${}^{s}{\eta}_{2}^{EXP}$ | 0.365 | 0.363 | 0.367 | 0.454 | 0.533 | 0.365 | |

3 | Out-of-plane, | ${f}_{3}^{EXP}$ | 1482.40 | 1495.10 | 1492.00 | 1506.00 | 1489.00 | 1489.85 |

(twisting) | ${}^{s}{\eta}_{3}^{EXP}$ | 0.593 | 0.576 | 0.591 | 0.767 | 0.803 | 0.587 | |

4 | Out-of-plane, | ${f}_{4}^{EXP}$ | 2081.50 | 2072.50 | 2082.60 | 2085.00 | 1866.00 | 2078.85 |

(bending) | ${}^{s}{\eta}_{4}^{EXP}$ | 0.757 | 0.745 | 0.669 | 0.992 | 0.459 | 0.724 | |

5 | In-plane, | ${f}_{5}^{EXP}$ | 2128.60 | 2127.60 | 2117.80 | 2232.10 | 2122.80 | 2124.65 |

(bending) | ${}^{s}{\eta}_{5}^{EXP}$ | 0.383 | 0.363 | 0.336 | 0.836 | 0.354 | 0.361 |

Mode (n) | ${\mathit{f}}_{\mathit{n}}^{\mathit{EXP}}$, Hz * | ${\mathit{f}}_{\mathit{n}}^{\mathit{FEM}\left(\mathit{I}\right)}$, Hz | ${\mathit{f}}_{\mathit{n}}^{\mathit{FEM}\left(\mathit{II}\right)}$, Hz | $\Delta $ FEM(I), % | $\Delta $ FEM(II), % |
---|---|---|---|---|---|

1 | 381.65 | 383.65 | 419.65 | 0.52 | 9.96 |

2 | 1254.9 | 1281.20 | 1328.80 | 2.10 | 5.89 |

3 | 1489.85 | 1570.50 | 1595.20 | 5.41 | 7.07 |

4 | 2078.85 | 2150.70 | 2280.60 | 3.46 | 9.69 |

5 | 2124.65 | 2159.50 | 2169.20 | 1.64 | 2.13 |

Average | 2.63 | 6.83 |

Mode (n) | ${}^{\mathit{s}}{\mathit{\eta}}_{\mathit{n}}^{\mathit{EXP}}$, % * | ${}^{\mathit{s}}{\mathit{\eta}}_{\mathit{n}}^{\mathit{FEM}\left(\mathit{I}\right)}$, % | $\Delta $, % |
---|---|---|---|

1 | 0.677 | 0.673 | 0.59 |

2 | 0.365 | 0.404 | 10.68 |

3 | 0.587 | 0.525 | 10.56 |

4 | 0.724 | 0.545 | 24.72 |

5 | 0.361 | 0.402 | 11.36 |

Average | 11.58 |

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**MDPI and ACS Style**

Wesolowski, M.; Ruchwa, M.; Skukis, E.; Kovalovs, A. Numerical and Experimental Extraction of Dynamic Parameters for Pyramidal Truss Core Sandwich Beams with Laminated Face Sheets. *Materials* **2020**, *13*, 5199.
https://doi.org/10.3390/ma13225199

**AMA Style**

Wesolowski M, Ruchwa M, Skukis E, Kovalovs A. Numerical and Experimental Extraction of Dynamic Parameters for Pyramidal Truss Core Sandwich Beams with Laminated Face Sheets. *Materials*. 2020; 13(22):5199.
https://doi.org/10.3390/ma13225199

**Chicago/Turabian Style**

Wesolowski, Miroslaw, Mariusz Ruchwa, Eduards Skukis, and Andrejs Kovalovs. 2020. "Numerical and Experimental Extraction of Dynamic Parameters for Pyramidal Truss Core Sandwich Beams with Laminated Face Sheets" *Materials* 13, no. 22: 5199.
https://doi.org/10.3390/ma13225199