# Prediction and Measurement of the Damping Ratios of Laminated Polymer Composite Plates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- The viscoelastic behaviour of the polymer matrix materials;
- The inelastic behaviour of the fibres;
- The inelastic behaviour of the interphase between fibre and matrix;
- The slip at the fibre/matrix interface in case of non-perfect adhesion;
- Thermo-elastic behaviour of fibres and matrix;
- The volume fractions of matrix and fibre.

## 2. Identification of the Complex Engineering Constants of Single Layers

#### 2.1. Linear Viscoelasticty

11->1 | 22->2 | 33->3 |

12->6 | 13->5 | 23->4 |

_{1}is often fibre dominated while E

_{2}is matrix dominated), Poisson’s ratios ${\upsilon}_{12}^{*}$ and ${\upsilon}_{21}^{*}$ cannot simultaneously have a zero tangents delta value.

#### 2.2. Identification of the Complex Engineering Constants

#### 2.2.1. Elastic Part of the Complex Engineering Constants

_{1}and E

_{2}can be found with the ASTM IET test on the two test beams. With these values of the Young’s moduli, a special aspect ratio of the test plate can be computed:

_{i}is the resonance frequency associated to the mode shapes.

^{(i)}, b

^{(i)}, c

^{(i)}and e

^{(i)}can be defined:

_{1}and E

_{2}can be found with IET on the beams. By replacing the plate rigidity values D

_{ij}in (25), using the relationship (14) between the plate rigidities and the engineering constants, (25) can be solved for v

_{12}and G

_{12}. Hence the virtual field method provides starting values for all the engineering constants, without the necessity to actually measure the mode shapes. After obtaining the starting values, further identification with the mixed numerical experimental method shown in Figure 2 can be performed using a sensitivity based gradient method as shown by Sol et al. [23].

#### 2.2.2. Identification of the Imaginary Part of the Complex Engineering Constants

_{ij}in (36) can be found using (34). The simultaneous solution of these two sets of Equations (35) and (36) requires the relation between the tangents $\delta $ of the engineering constants and the tangents $\delta $ of the stiffness matrix ${C}_{ij}^{*}$. The relations can be evaluated based on the expression in the complex stiffness matrix (10) and observing that for relatively small damping values $\mathrm{tan}\delta \cong \delta $. For the complex Poisson’s ratios, it can be seen that:

## 3. Identification of Complex Stiffness Values of Laminates

## 4. Computation of the Modal Damping Ratio of Laminated Plates

_{ij}is partial portions related to the anisotropic plate rigidities D

_{ij}. The modal damping ratio of a mode shape of a laminated plate can be computed similar as (34):

_{i}

_{j}related to the plate rigidities can be calculated by the FE model of the plate.

## 5. Experiments and Validations

#### 5.1. Experiment 1: Laminated Carbon/Epoxy Plates

- E
_{1}_Start = 1.125E + 11 [Pa] E_{1}_Final = 1.092E + 11 [Pa] - E
_{2}_Start = 7.540E + 09 [Pa] E_{2}_Final = 7.303E + 09 [Pa] - v
_{12}_Start = 4.907E - 01 [-] v_{12}_Final = 0.476 [-] - G
_{12}_Start = 3.188E + 09 [Pa] G_{12}_Final = 3.660E + 09 [Pa]

_{S}was prepared in an autoclave. The layers were cut out of the 30 cm width bobbin CMP 200/300 CP0031 carbon/epoxy prepreg material, the same as used for the single layer test. The measured average thickness of this laminate was 2.1 mm and the computed complex anisotropic plate rigidities of this laminated plate are shown in Table 7:

#### 5.2. Experiment 2: Polyester Reinforced with UD Glass Fabric

^{2}oriented in the 0° direction was prepared with hand layup, cured and compressed between two thick aluminium plates. The cured plate had a final thickness of 4.54 mm. The same measurement procedure for obtaining the single layer properties described in Experiment 1 is applied. The size and mass of the two test beams and the resulting Poisson plate are given in Table 10.

_{1}and E

_{2}are not as extreme as for the UD carbon epoxy. The tangents delta values are higher.

#### 5.3. Experiment 3: Epoxy Reinforced with UD Glass Fabric

_{1}and E

_{2}are not as large as for the UD carbon epoxy. The tangents delta values are higher than for the UD carbon epoxy and the glass polyester plates in Experiments 1 and 2.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Elmarakbi, A. Advanced Composite Materials for Automotive Applications: Structural Integrity and Crashworthiness; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2014. [Google Scholar]
- Baker, A.A.; Murray, L. Composite Materials for Aircraft Structures; AIAA/American Institute of Aeronautics, Inc.: Lake Success, NY, USA, 2016. [Google Scholar]
- Jones, R. Mechanics of Composite Materials; McGraw–Hill: New York City, NY, USA, 1975; ISBN 0-07-032790-4-0. [Google Scholar]
- Lazan, B.J. Damping of Materials and Members in Structural Mechanics; Pergamon Press: Oxford, UK, 1968. [Google Scholar]
- Bert, C.W. Material damping: An introductory review of mathematical models, measures and experimental techniques. J. Sound Vib.
**1973**, 29, 129–153. [Google Scholar] [CrossRef] - Adams, R.D. Damping properties analysis of composites. Engineered materials handbook. Compos. ASM
**1973**, 1, 206–217. [Google Scholar] - Treviso, A.; Van Genechten, B.; Mundo, D.; Tournour, M. Damping in composite materials: Properties and models. Compos. Part B Eng.
**2015**, 78, 144–152. [Google Scholar] [CrossRef] - Hashin, Z. Complex moduli of viscoelastic composites: General theory and application to particulate composites. Int. J. Solids Struct.
**1970**, 6, 539–552. [Google Scholar] [CrossRef] - Berthelot, J.M. Damping analysis of laminated beams and plates using the Ritz method. Compos. Struct.
**2006**, 74, 186–201. [Google Scholar] [CrossRef] - Sun, C.T.; Wu, J.K.; Gibson, R.F. Prediction of material damping of laminated polymer matrix composites. J. Mater. Sci.
**1987**, 22, 1006–1012. [Google Scholar] [CrossRef] - Ni, R.G.; Adams, R.D. A rational method for obtaining the dynamic mechanical properties of laminae for predicting the stiffness and damping of laminated plates and beams. Composites
**1984**, 15. [Google Scholar] [CrossRef] - Jong, H.Y. A damping analysis of composite laminates using the closed form expression for the basic damping of Poisson’s ratio. Compos. Struct.
**1999**, 46, 405–411. [Google Scholar] - De Wilde, P.W.; Sol, H. Determination of the material constants using experimental free vibration analysis on anisotropic plates. Exp. Stress Anal.
**1986**, 207–214. [Google Scholar] - Sol, H. Identification of Anisotropic Plate Rigidities Using Free Vibration Data. Ph.D. Thesis, Vrije Universiteit, Brussel, Belgium, October 1986. [Google Scholar]
- McIntyre, M.E.; Woodhouse, J. On measuring the elastic and damping constants of orthotropic sheet materials. Acta Metall.
**1988**, 36, 1397–1416. [Google Scholar] [CrossRef] - Deobald, L.R.; Gibson, R.F. Determination of elastic constants of orthotropic plates by a modal analysis/Rayleigh-Ritz technique. J. Sound Vib.
**1988**, 24, 269–283. [Google Scholar] [CrossRef] - Talbot, J.P.; Woodhouse, J. The vibration damping of laminated plates. Compos. Part A
**1997**, 28, 1007–1012. [Google Scholar] [CrossRef] - El Mahi, A.; Assarar, M.; Sefrani, Y.; Berthelot, J.M. Damping analysis of orthotropic composite materials and laminates. Compos. Part B
**2008**, 39, 1069–1076. [Google Scholar] [CrossRef] - Wesolowski, M.; Barkanov, E. Improving material damping characterization of a laminated plate. J. Sound Vib.
**2019**, 462, 114928. [Google Scholar] [CrossRef] - Marchetti, F.; Ege, K.; Leclère, Q.; Roozen, N.B. On the structural dynamics of laminated composite plates and sandwich structures; a new perspective on damping identification. J. Sound Vib.
**2020**, 474, 115256. [Google Scholar] [CrossRef] [Green Version] - He, J.; Fu, Y. Modal Analysis; Butterworth–Heineman: Oxford, UK, 2001; ISBN 0-7506-5079-6. [Google Scholar]
- Sol, H.; De Visscher, J.; De Wilde, W.P. The Resonalyser method: A nondestructive method for stiffness identification of fiber reinforced composite materials. Non Destr. Test.
**1996**, 237–242. [Google Scholar] - Sol, H.; De Visscher, J.; De Wilde, W.P. A mixed numerical/experimental technique for the nondestructive identification of the stiffness properties of fibre reinforced composite materials. NDT E Int.
**1997**, 30, 85–91. [Google Scholar] [CrossRef] - Lauwagie, T.; Sol, H.; Roebben, G.; Heylen, W.; Shi, Y. Validation of the Resonalyser method: An inverse method for material identification. In Proceedings of the ISMA 2002, International Conference on Noise and Vibration Engineering, Leuven, Belgium, 16–18 September 2002; pp. 687–694. [Google Scholar]
- De Visscher, J.; Sol, H.; De Wilde, W.P.; Vantomme, J. Identification of the damping properties of orthotropic composite materials using a mixed numerical experimental method. Appl. Compos. Mater.
**1997**, 4, 13–33. [Google Scholar] [CrossRef] - Pierron, F.; Grediac, M. The Virtual Fields Method; Springer: London, UK, 2012; ISBN 978-1-7614-1825-8. [Google Scholar]
- Heritage, K.; Frisby, C.; Wolfenden, A. Impulse excitation technique for dynamic flexural measurements at moderate temperature. Rev. Sci. Instrum.
**1988**, 59, 973. [Google Scholar] [CrossRef] - ASTM E1976, Standard Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio by Impulse Excitation of Vibration. Available online: https://www.semanticscholar.org/paper/Standard-Test-Method-for-Dynamic-Young%27s-Modulus%2C-1-Resonance/c5ec8ed15f00f0f7e9be1b1dbb6f93ba64f8611c (accessed on 29 July 2020).
- Frederiksen, P.S. Experimental Procedure and Results for the Identification of Elastic Constants of Thick Orthotropic Plates. J. Compos. Mater.
**1997**, 31, 360–382. [Google Scholar] [CrossRef]

**Figure 1.**Support, excitation position and measurement positions for impulse excitation technique (IET) tests on beams.

**Figure 2.**Mixed numerical experimental method: iteratively updating the engineering constants in the numerical finite element (FE) model of a test plate.

**Figure 3.**Two test beams cut along the two orthotropic directions, and a test plate with edges parallel to the orthotropic material directions.

**Figure 6.**Nodal lines of three mode shapes. (

**a**) The nodal lines of the torsion mode shape, (

**b**) the nodal lines of the saddle mode shape, (

**c**) the nodal line of the breathing mode shape.

**Figure 7.**(

**a**) Suspension wires fixed in the intersection of the nodal lines of torsion and breathing mode shapes, (

**b**) suspension wires fixed in the corners of the Poisson plate.

**Figure 9.**Nodal lines of the mode shapes of the Poisson plate, computed with the FE model. (

**a**) Torsion mode shape, (

**b**) saddle mode shape and (

**c**) breathing mode shape.

**Figure 10.**Test setup for measuring the damping ratios with IET: An aluminium suspension frame and the DJB micro accelerometer connected through a signal conditioning box with a PC; in (

**a**) the accelerometer is fixed with beeswax on the intersection of the nodal lines of the saddle and breathing mode shapes; in (

**b**) the accelerometer is fixed on the intersection of torsion and breathing mode shape.

**Figure 11.**Curve fitted decaying sinusoidal signal. The white zone is the (compressed) recorded sinusoidal signal; the red envelope line is the curve fitted damped exponential function of the signal. The example shows a signal with resonance frequency = 241.2 Hz and a damping ratio value of 0.00237.

**Figure 12.**Computed resonance frequencies, mode shapes and damping ratios of a carbon/epoxy (0.2365 m × 0.1385 m × 0.0021 m) laminated plate with stacking sequence (0° −45° 45° −45° 0°)

_{S.}

Test Specimen | Length [m] | Width [m] | Thickness [m] | Mass [kg] |
---|---|---|---|---|

Beam 1 (0°) | 3.550E-01 | 1.840E-02 | 2.080E-03 | 1.995E-02 |

Beam 2 (90°) | 2.320E-01 | 2.400E-02 | 2.060E-03 | 1.701E-02 |

IET Test Specimen | Frequency [Hz] | Damping Ratio [%] |
---|---|---|

Beam 1 (0°) | 146.5 | 0.037 |

Beam 2 (90°) | 87.5 | 0.372 |

Test Specimen | Length [m] | Width [m] | Thickness [m] | Mass [kg] |
---|---|---|---|---|

Poisson plate | 2.710E-01 | 1.390E-01 | 2.120E-03 | 1.174E-01 |

Mode Shape Type | Frequency [Hz] |
---|---|

Torsion | 94.4 |

Saddle | 241.2 |

Breathing | 264.4 |

Mode Shape Type | Frequency [Hz] | Damping Ratio [%] |
---|---|---|

Torsion | 94.4 | 0.500 |

Saddle | 241.2 | 0.237 |

Breathing | 264.4 | 0.174 |

Engineering Constant | Real Part [GPa] | Imaginary Part [GPa] | Tangents Delta [-] |
---|---|---|---|

Young’s Modulus E_{1} | 109.2 | 8.1E-02 | 0.00074 |

Young’s Modulus E_{2} | 7.303 | 5.4E-02 | 0.00744 |

Major Poisson’s ratio v_{12} | 0.476 | −1.4E-03 | −0.00294 |

Minor Poisson’s ratio v_{21} | 0.032 | 1.2E-04 | 0.00376 |

In-plane Shear Modulus G_{12} | 3.660 | 3.7E-02 | 0.01021 |

Plate Rigidity | Real Part [Nm] | Imaginary Part [Nm] |
---|---|---|

D_{XX} | 56.06 | 0.0631 |

D_{YY} | 16.45 | 0.0523 |

D_{XY} | 12.12 | 0.0081 |

D_{ZZ} | 12.22 | 0.0247 |

D_{XZ} | −3.99 | −0.001 |

D_{YZ} | −3.99 | −0.001 |

Mode | Predicted Frequency [Hz] | Measured Frequency [Hz] | Predicted Damping Ratio [%] | Measured Damping Ratio [%] | Difference Predict-Measured [%] |
---|---|---|---|---|---|

1 | 208.2 | 205.9 | 0.123 | 0.147 | 0.024 |

2 | 246.8 | 244.3 | 0.094 | 0.103 | 0.009 |

3 | 431.1 | 431.0 | 0.148 | 0.177 | 0.029 |

4 | 488.7 | 481.0 | 0.114 | 0.142 | 0.028 |

Mode | Predicted Frequency [Hz] | Measured Frequency [Hz] | Predicted Damping Ratio [%] | Measured Damping Ratio [%] |
---|---|---|---|---|

Beam-X | 394 | 394.5 | 0.082 | 0.082 |

Test Specimen | Length [m] | Width [m] | Thickness [m] | Mass [kg] |
---|---|---|---|---|

Beam 1 (0°) | 2.91E-01 | 2.44E-02 | 4.5E-03 | 5.874E-02 |

Beam 2 (90°) | 2.11E-01 | 2.33E-02 | 4.5E-03 | 4.059E-02 |

Poisson plate | 2.4E-01 | 1.95E-01 | 4.54E-03 | 3.9057E-01 |

Engineering Constant | Real Part [GPa] | Imaginary Part [GPa] | Tangents Delta [-] |
---|---|---|---|

Young’s Modulus E_{1} | 32.7 | 8.5E-02 | 0.0026 |

Young’s Modulus E_{2} | 13.7 | 1.36E-01 | 0.0099 |

Major Poisson’s ratio v_{12} | 0.27 | −1.4E-03 | −0.00506 |

Minor Poisson’s ratio v_{21} | 0.11 | 2.46E-04 | 0.00224 |

In-plane Shear Modulus G_{12} | 4.27 | 5.91E-02 | 0.01384 |

Plate Rigidity | Real Part [Nm] | Imaginary Part [Nm] |
---|---|---|

D_{XX} | 226.5 | 0.75 |

D_{YY} | 109.6 | 1.07 |

D_{XY} | 48.48 | 0.10 |

D_{ZZ} | 51.96 | 0.42 |

D_{XZ} | 46.76 | −0.12 |

D_{YZ} | 2.28 | −0.021 |

Mode | Predicted Frequency [Hz] | Measured Frequency [Hz] | Predicted Damping Ratio [%] | Measured Damping Ratio [%] | Difference Predict-Measured [%] |
---|---|---|---|---|---|

1 | 327.3 | 327 | 0.638 | 0.680 | 0.042 |

2 | 573.2 | 574 | 0.256 | 0.288 | 0.032 |

3 | 738.6 | 736 | 0.418 | 0.502 | 0.086 |

Mode | Predicted Frequency [Hz] | Measured Frequency [Hz] | Predicted Damping Ratio [%] | Measured Damping Ratio [%] | Difference Predict-Measured [%] |
---|---|---|---|---|---|

Beam | 944.8 | 946 | 0.424 | 0.527 | 0.103 |

Test Specimen | Length [m] | Width [m] | Thickness [m] | Mass [kg] |
---|---|---|---|---|

Beam 1 (0°) | 1.62E-01 | 1.885E-02 | 3.83E-03 | 1.615E-02 |

Beam 2 (90°) | 1.26E-01 | 2.84E-02 | 4.22E-03 | 2.073E-02 |

Poisson plate | 0.15E-01 | 1.2E-01 | 3.81E-03 | 9.436E-02 |

Engineering Constant | Real Part [GPa] | Imaginary Part [GPa] | Tangents Delta [-] |
---|---|---|---|

Young’s Modulus E_{1} | 15.1 | 1.30E-01 | 0.00864 |

Young’s Modulus E_{2} | 5.7 | 1.76E-01 | 0.03094 |

Major Poisson’s ratio v_{12} | 0.33 | −1.1E-03 | −0.00335 |

Minor Poisson’s ratio v_{21} | 0.13 | 5.5E-03 | 0.0429 |

In-plane Shear Modulus G_{12} | 1.9 | 6.93E-02 | 0.03648 |

Plate Rigidity | Real Part [Nm] | Imaginary Part [Nm] |
---|---|---|

D_{XX} | 79.94 | 0.92 |

D_{YY} | 34.17 | 1.07 |

D_{XY} | 16.60 | 0.26 |

D_{ZZ} | 17.71 | 0.39 |

D_{XZ} | 16.73 | −0.08 |

D_{YZ} | 2.47 | 0.02 |

Mode | Predicted Frequency [Hz] | Measured Frequency [Hz] | Predicted Damping Ratio [%] | Measured Damping Ratio [%] | Difference Predict-Measured |
---|---|---|---|---|---|

1 | 85.4 | 83 | 1.695 | 1.741 | 0.046 |

2 | 143.1 | 143 | 1.396 | 1.418 | 0.022 |

3 | 185.2 | 182 | 0.757 | 0.801 | 0.044 |

4 | 219.5 | 221 | 1.581 | 1.641 | 0.060 |

5 | 264.6 | 257 | 1.077 | 1.144 | 0.063 |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sol, H.; Rahier, H.; Gu, J.
Prediction and Measurement of the Damping Ratios of Laminated Polymer Composite Plates. *Materials* **2020**, *13*, 3370.
https://doi.org/10.3390/ma13153370

**AMA Style**

Sol H, Rahier H, Gu J.
Prediction and Measurement of the Damping Ratios of Laminated Polymer Composite Plates. *Materials*. 2020; 13(15):3370.
https://doi.org/10.3390/ma13153370

**Chicago/Turabian Style**

Sol, Hugo, Hubert Rahier, and Jun Gu.
2020. "Prediction and Measurement of the Damping Ratios of Laminated Polymer Composite Plates" *Materials* 13, no. 15: 3370.
https://doi.org/10.3390/ma13153370