# Resonant Actuation Based on Dynamic Characteristics of Bistable Laminates

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## Abstract

**:**

## 1. Introduction

## 2. Modeling

#### 2.1. The Finite Element Model

#### 2.2. The Static Behaviour of the Analytical Model

#### 2.3. The Dynamic Behaviour of the Analytical Model

#### 2.4. Prediction of Stable Configuration

_{T}laminate was selected. The accuracy of the analytical model was verified by comparing the out-of-plane displacement of the rectangular laminates. The comparison between the finite element results and the analytical prediction results of the stable configuration under different external forces is shown in Figure 4. It was found that the analytical model can provide satisfactory results compared with the finite element model, and slight error is reasonable.

#### 2.5. Geometric Parameter Analysis of Bistable Unsymmetric Laminates

_{T}and [0

_{2}/90

_{2}]

_{T}layups in an ideal environment were analyzed and the results are shown in Figure 6 and Figure 7.

_{T}and [0

_{2}/90

_{2}]

_{T}bifurcate with the increase of the side lengths of the laminates. Compared with the FEA results, the two methods were similarly successful in predicting the vertex displacement of the rectangular bistable laminates. Figure 6 shows that the FEA model predicted the bifurcation point of a [0/90]

_{T}laminate at around 30 mm. Figure 7 shows that the bifurcation point of the [0

_{2}/90

_{2}]

_{T}laminate predicted by FEA was 60 mm. The analytical model accurately predicted the bifurcation behavior of laminate with the increase of laminate size. The bistable critical length of laminates increased with the thickness. The FEA model is more accurate for predicting the bifurcation point, and the analytical model can not easily converge to a stable configuration when the width of the plate is near the bifurcation point.

_{T}layup were studied in the following experiments. Considering the avoidance of technical difficulties, to compare the theoretical and experimental results, three specimens with different dimensions were prepared. The vertex displacement is indicated by the discrete blue points in Figure 6, showing that the vertex displacements predicted by the theoretical model for the laminate in stable state B were in good agreement with the experimental results, while the errors for the stable state A were relatively large. The source of errors may be due to the use of inaccurate material parameters, the layering cooling process, defects in the test parts, and so on.

## 3. Kinetic Analysis

#### 3.1. Fundamental Frequency Analysis

#### 3.2. Frequency Sweep Experiment

_{T}laminates under external excitation with different frequencies.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The configurations of a bi-stable cross-ply laminate. (

**a**) Saddle-shaped configuration; (

**b**) Cylindrical stable state A; (

**c**) Cylindrical stable state B.

**Figure 3.**The stable configuration of the bistable composite laminate: (

**a**) stable state A; (

**b**) stable state B.

**Figure 4.**Predicted stable configurations of bistable laminates under different applied force, curved surface–theoretical results, discrete points–FEA. (

**a**) 0 N; (

**b**) 0.1 N; (

**c**) 0.2 N.

**Figure 5.**Deformation of the laminate, curved surface–3D scanning, discrete points–Analytical modal.

**Figure 10.**Vibration amplitudes of the bistable plates under low-level sinusoidal excitation at different frequencies.

**Figure 11.**Vibration amplitudes of the bistable plates under higher-level sinusoidal excitation at different frequencies.

Parameter | Value |
---|---|

Axial tensile modulus E_{1}/GPa | 124.9 |

Transversal tensile modulus E_{2}/GPa | 7.9 |

Shear modulus G_{12}/GPa | 5.6 |

Shear modulus G_{23}/GPa | 5.6 |

Poisson’s ratio ν12 | 0.3 |

Longitudinal thermal expansion coefficient α_{1}/°C^{−1} | 4 × 10^{−7} |

Transverse thermal expansion coefficient α_{2}/°C^{−1} | 1.8 × 10^{−5} |

The thickness of layer t/mm | 0.15 |

Applied Force (N) | Analytical Model (mm) | Finite Element Model (mm) | Error |
---|---|---|---|

0 | 38.5675 | 37.3187 | 3.99% |

0.1 | 45.3683 | 46.6199 | 2.68% |

0.2 | 52.2781 | 53.3966 | 2.09% |

Width (mm) | Analytical Model (Hz) | Finite Element Model (Hz) | Test Specimen (Hz) |
---|---|---|---|

70 | 26.77 | 27.6 | 23 |

80 | 20.649 | 21.5 | 15.5 |

90 | 16.4 | 16.5 | 14 |

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**MDPI and ACS Style**

Liu, Y.; Zhang, J.; Pan, D.; Wu, Z.; Wang, Q.
Resonant Actuation Based on Dynamic Characteristics of Bistable Laminates. *Machines* **2023**, *11*, 318.
https://doi.org/10.3390/machines11030318

**AMA Style**

Liu Y, Zhang J, Pan D, Wu Z, Wang Q.
Resonant Actuation Based on Dynamic Characteristics of Bistable Laminates. *Machines*. 2023; 11(3):318.
https://doi.org/10.3390/machines11030318

**Chicago/Turabian Style**

Liu, Yuting, Jiaying Zhang, Diankun Pan, Zhangming Wu, and Qingyun Wang.
2023. "Resonant Actuation Based on Dynamic Characteristics of Bistable Laminates" *Machines* 11, no. 3: 318.
https://doi.org/10.3390/machines11030318