# Comparison of Mode Shapes of Carbon-Fiber-Reinforced Plastic Material Considering Carbon Fiber Direction

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background for Modal Analysis

^{th}FRFs can be obtained; therefore, the final equation regarding FRFs should be written in the matrix formula. The modal vector can be calculated only in multiple degree-of-freedom conditions, and it plays a critical role in the determination of the dynamic characteristics of the target system. The mode shape of the system is representative of the modal vector. If the i-th modal vector is defined as ${\psi}_{i}$, the similarities between modal vectors can be evaluated using the MAC, as formulated below:

## 3. Modal Test of CFRP Specimens

^{3}, elastic modulus of 198 GPa, and Poisson’s ratio of 0.28. Tetra elements were used to generate meshes in the FE model using HyperWorks software (Altair, MI, USA), and the clamping area illustrated in Figure 2 was represented by assigning constraints with six degrees-of-freedom on the same area in the FE model. Modal analysis from the FE model was performed using Virtual.Lab software (Siemens, Munich, Germany) and four flexible modes were determined for a frequency limit of up to 2000 Hz. The first four modes were sufficient to represent the dynamics of interesting simple specimens, and it was difficult to measure higher modes under a limited number of sensors. The derived mode shapes are presented in Figure 3. The FE model was validated by comparing the modal parameters and the corresponding mode shapes by adjusting its material properties. The experimental modal analysis of the SCS13A specimen is explained later.

## 4. Comparison of Mode Shapes

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Configuration of simple carbon-fiber-reinforced plastic (CFRP) specimen with $\theta $ denoting the carbon fiber direction, and it consists of 12 layers of pre-implemented composite fibers (USN 250A).

**Figure 2.**Clamping area and sensor attachment locations of simple CFRP specimen. A: 3 mm; B: 10 mm; C: 40 mm; D: 30 mm; E: 20 mm.

**Figure 3.**Mode shape of finite element (FE) model of simple specimens (gray color) overlapped with original model (orange color): (

**a**) first mode (1st bending); (

**b**) second mode (1st twisting); (

**c**) third mode (2nd bending); (

**d**) fourth mode (2nd twisting).

**Figure 4.**Configuration of simple specimens with attachment of accelerometers: (

**a**) CFRP specimen with zero carbon fiber direction; (

**b**) SCS13A specimen.

**Figure 5.**Experimental mode shape of simple specimens (yellow line) overlapped with the original model (white line): (

**a**) first mode (1st bending); (

**b**) second mode (1st twisting); (

**c**) third mode (2nd bending); (

**d**) fourth mode (2nd twisting).

**Figure 6.**Variations in resonance frequencies according to the carbon fiber direction: (

**a**) : first bending mode; : first twisting mode; (

**b**) : second bending mode; : second twisting mode.

**Figure 7.**Variations in damping coefficients according to the carbon fiber direction: (

**a**) : first bending mode; : first twisting mode; (

**b**) : second bending mode; : second twisting mode.

Specimen | Resonance Frequency (Hz) | Modal Damping (%) | Mode Shape |
---|---|---|---|

CFRP specimen ($\theta ={0}^{\xb0})$ | 218.4 | 1.7 | T |

254.7 | 2.4 | B | |

356.7 | 7.0 | T | |

1103.6 | 2.4 | B | |

1539.6 | 2.3 | B | |

1708.8 | 0.4 | T | |

CFRP specimen ($\theta ={30}^{\xb0})$ | 146.7 | 3.5 | B |

404.6 | 8.0 | T | |

1009.5 | 1.4 | B | |

1143.8 | 2.0 | T | |

1635.0 | 1.9 | T | |

CFRP specimen ($\theta ={45}^{\xb0})$ | 109.2 | 4.9 | B/T |

206.8 | 4.9 | B | |

411.6 | 1.4 | T | |

701.1 | 3.1 | B | |

1103.2 | 0.7 | T | |

CFRP specimen ($\theta ={60}^{\xb0})$ | 82.8 | 4.3 | B/T |

186.1 | 4.9 | B/T | |

326.1 | 3.4 | T | |

535.8 | 1.5 | B | |

1029.4 | 1.9 | T | |

1501.5 | 2.1 | B | |

1905.0 | 3.1 | T | |

CFRP specimen ($\theta ={90}^{\xb0})$ | 75.5 | 5.7 | B |

231.3 | 11.9 | T | |

499.8 | 0.9 | B | |

880.3 | 1.3 | T | |

1364.4 | 1.6 | B | |

1689.7 | 3.3 | T | |

SCS13A specimen | 175.1 | 0.2 | B |

578.3 | 1.1 | T | |

1109.5 | 0.6 | B | |

1927.8 | 0.5 | T |

Mode | Resonance Frequency (Hz) | MAC | ||
---|---|---|---|---|

Experiment | FE Model | Error (%) | ||

#1 | 175.1 | 175.8 | 0.4 | 0.99 |

#2 | 578.3 | 584.1 | 1.0 | 0.99 |

#3 | 1109.5 | 1084.8 | 2.2 | 0.95 |

#4 | 1927.8 | 1871.0 | 2.9 | 0.96 |

Specimen | Resonance Frequency (Hz) | Modal Damping (%) | MAC | Mode Shape |
---|---|---|---|---|

CFRP Specimen #1 ($\theta ={0}^{\xb0})$ | 254.7 | 2.4 | 0.99 | B |

356.7 | 7.0 | 0.99 | T | |

1539.6 | 2.3 | 0.76 | B | |

1708.8 | 0.4 | 0.88 | T | |

CFRP Specimen #2 ($\theta ={30}^{\xb0})$ | 146.7 | 3.5 | 0.90 | B |

404.6 | 8.0 | 0.83 | T | |

1009.5 | 1.4 | 0.60 | B | |

1143.8 | 2.0 | 0.53 | T | |

1635.0 | 1.9 | 0.45 | T | |

CFRP Specimen #3 ($\theta ={45}^{\xb0})$ | 109.2 | 4.9 | 0.75 | B/T |

206.8 | 4.9 | 0.78 | B | |

411.6 | 1.4 | 0.53 | T | |

701.1 | 3.1 | 0.86 | B | |

1103.2 | 0.7 | 0.82 | T | |

CFRP Specimen #4 ($\theta ={60}^{\xb0})$ | 186.1 | 4.9 | 0.96 | B |

326.1 | 3.4 | 0.93 | T | |

535.8 | 1.5 | 0.93 | B | |

1029.4 | 1.9 | 0.90 | T | |

CFRP Specimen #5 ($\theta ={90}^{\xb0})$ | 75.5 | 5.7 | 0.90 | B |

231.3 | 11.9 | 0.91 | T | |

499.8 | 0.9 | 0.97 | B | |

880.3 | 1.3 | 0.99 | T |

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**MDPI and ACS Style**

Kim, C.-J.
Comparison of Mode Shapes of Carbon-Fiber-Reinforced Plastic Material Considering Carbon Fiber Direction. *Crystals* **2021**, *11*, 311.
https://doi.org/10.3390/cryst11030311

**AMA Style**

Kim C-J.
Comparison of Mode Shapes of Carbon-Fiber-Reinforced Plastic Material Considering Carbon Fiber Direction. *Crystals*. 2021; 11(3):311.
https://doi.org/10.3390/cryst11030311

**Chicago/Turabian Style**

Kim, Chan-Jung.
2021. "Comparison of Mode Shapes of Carbon-Fiber-Reinforced Plastic Material Considering Carbon Fiber Direction" *Crystals* 11, no. 3: 311.
https://doi.org/10.3390/cryst11030311