# Study on Dynamic Snap-Through and Nonlinear Vibrations of an Energy Harvester Based on an Asymmetric Bistable Composite Laminated Shell

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Model

_{Lam}denotes the volume of the asymmetric composite laminated plate.

_{0}, v

_{0}and w

_{0}in Equations (1)–(5) are replaced by static displacements ${u}_{s}^{\mathrm{I}}$, ${v}_{s}^{\mathrm{I}}$ and ${w}_{s}^{\mathrm{I}}$.

_{Lam1}denotes the potential energy of the asymmetric composite laminated plate, and ${X}_{1}={u}_{s}^{\mathrm{I}}$, ${X}_{2}={v}_{s}^{\mathrm{I}}$ and ${X}_{3}={w}_{s}^{\mathrm{I}}$.

_{1}, f

_{2}and f

_{3}by Equations (9)–(13), the static bifurcation diagram Figure 2 can be graphically presented as below.

_{z}is homogenized in the thickness direction

_{x}, L

_{x}] and y∈[−L

_{y}, L

_{y}]), selecting the first three degrees of freedom and introducing dimensionless parameters,

## 3. Numerical Simulation

#### 3.1. Dynamic Snap-Through and Nonlinear Vibrations

#### 3.2. Frequency Sweeping

#### 3.3. Amplitude Sweeping

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Emam, S.A.; Nayfeh, A.H. On the nonlinear bynamics of a buckled beam subjected to a primary-resonance excitation. Nonlinear Dyn.
**2004**, 35, 1–17. [Google Scholar] [CrossRef] - Abou-Rayan, A.M.; Nayfeh, A.H.; Mook, D.T. Nonlinear response of a parametrically excited buckled beam. Nonlinear Dyn.
**1993**, 4, 499–525. [Google Scholar] [CrossRef] - Lacarbonara, W.; Nayfeh, A.H.; Kreider, W. Experimental validation of reduction methods for nonlinear vibrations of distributed-parameter systems: Analysis of a buckled beam. Nonlinear Dyn.
**1998**, 17, 95–117. [Google Scholar] [CrossRef] - Daynes, S.; Potter, K.D.; Weaver, P.M. Bistable Prestressed Buckled Laminates. Compos. Sci. Technol.
**2008**, 68, 3431–3437. [Google Scholar] [CrossRef] - Li, H.; Dai, F.; Weaver, P.M.; Du, S. Bistable hybrid symmetric laminates. Compos. Struct.
**2014**, 116, 782–792. [Google Scholar] [CrossRef] - Daynes, S.; Nall, S.J.; Weaver, P.M.; Potter, K.D.; Margaris, P.; Mellor, P.H. Bistable composite flap for an airfoil. J. Aircr.
**2010**, 47, 334–338. [Google Scholar] [CrossRef] - Daynes, S.; Weaver, P.M.; Trevarthen, J.A. A morphing composite air inlet with multiple stable shape. J. Intell. Mater. Syst. Struct.
**2011**, 22, 961–973. [Google Scholar] [CrossRef] - Guest, S.D.; Pellegrino, S. Analytical models for bistable cylindrical shells. Proc. R. Soc. A
**2006**, 462, 839–854. [Google Scholar] [CrossRef] - Seffen, K.A. ‘Morphing’ bistable orthotropic elliptical shallow shells. Proc. R. Soc. A
**2007**, 463, 67–83. [Google Scholar] [CrossRef] - Brinkmeyer, A.; Santer, M.; Pirrera, A.; Weaver, P.M. Pseudo bistable self-actuated domes for morphing applications. Int. J. Solids Struct.
**2012**, 49, 1077–1087. [Google Scholar] [CrossRef] [Green Version] - Brinkmeyer, A.; Pirrera, A.; Santer, M.; Weaver, P.M. Pseudo bistable pre-stressed morphing composite panels. Int. J. Solids Struct.
**2013**, 50, 1033–1043. [Google Scholar] [CrossRef] [Green Version] - Coburn, B.H.; Pirrera, A.; Weaver, P.M.; Vidoli, S. Tristability of an orthotropic doubly curved shell. Compos. Struct.
**2013**, 96, 446–454. [Google Scholar] [CrossRef] - Eckstein, E.; Pirrera, A.; Weaver, P.M. Multi-mode morphing using initially curved composite plates. Compos. Struct.
**2014**, 109, 240–245. [Google Scholar] [CrossRef] - Hyer, M.W. Some observations on the cured shape of thin unsymmetric laminates. J. Compos. Mater.
**1981**, 15, 175–194. [Google Scholar] [CrossRef] - Hyer, M.W. Calculations of the room-temperature shapes of unsymmetric laminates. J. Compos. Mater.
**1981**, 15, 296–310. [Google Scholar] [CrossRef] - Hyer, M.W. The room-temperature shapes of four-layer unsymmetric cross-ply laminates. J. Compos. Mater.
**1982**, 16, 318–340. [Google Scholar] [CrossRef] - Peeters, L.J.B.; Powell, P.C.; Warnet, L. Thermally induced shapes of unsymmetric laminates. J. Compos. Mater.
**1996**, 30, 603–626. [Google Scholar] [CrossRef] - Schlecht, M.; Schulte, K.; Hyer, M.W. Advanced calculation of the room-temperature shapes of thin unsymmetric composite laminates. Compos. Struct.
**1995**, 32, 627–633. [Google Scholar] [CrossRef] - Schlecht, M.; Schulte, K. Advanced calculations of the room-temperature shapes of unsymmetric laminates. J. Compos. Mater.
**1999**, 33, 1472–1490. [Google Scholar] [CrossRef] - Cho, M.; Choi, M.H.; Chung, H.C.; Ahn, K.J.; Eom, Y.S. A study on the room-temperature curvature shapes of unsymmetric laminates including slippage effects. J. Compos. Mater.
**1998**, 32, 460–482. [Google Scholar] [CrossRef] - Gigliotti, M.; Wisnom, M.R.; Potter, K.D. Development of curvature during the cure of AS4/8552 [0/90] unsymmetric composite plates. Compos. Sci. Technol.
**2003**, 63, 187–197. [Google Scholar] [CrossRef] - Dai, F.; Li, H.; Du, S. Design and analysis of a tri-stable structure based on bi-stable laminates. Compos. Part A
**2012**, 43, 1497–1504. [Google Scholar] [CrossRef] - Dai, F.; Li, H.; Du, S. A multi-stable wavy skin based on bi-Stable laminates. Compos. Part A
**2013**, 45, 102–108. [Google Scholar] [CrossRef] - Dai, F.; Li, H.; Du, S. A multi-stable lattice structure and its snapthrough behavior Among Multiple States. Compos. Struct.
**2013**, 97, 56–63. [Google Scholar] [CrossRef] - Wang, G.Q.; Liao, W.H. A bistable piezoelectric oscillator with an elastic magnifier for energy harvesting enhancement. J. Intell. Mater. Syst. Struct.
**2016**, 28, 392–407. [Google Scholar] [CrossRef] - Cottone, F.; Basset, P.; Vocca, H.; Gammaitoni, L.; Bourouina, T. Bistable electromagnetic generator based on buckled beams for vibration energy harvesting. J. Intell. Mater. Syst. Struct.
**2013**, 25, 1484–1495. [Google Scholar] [CrossRef] - Harne, R.L.; Wang, K.W. A review of the recent research on vibration energy harvesting via bistable systems. Smart Mater. Struct.
**2013**, 22, 023001. [Google Scholar] [CrossRef] - Stanton, S.C.; Mcgehee, C.C.; Mann, B.P. Nonlinear dynamics for broadband energy harvesting: Investigation of a bistable piezoelectric inertial generator. Phys. D Nonlinear Phenom.
**2010**, 239, 640–653. [Google Scholar] [CrossRef] - Mann, B.P.; Owens, B.A. Investigations of a nonlinear energy harvester with a bistable potential well. J. Sound Vib.
**2010**, 329, 1215–1226. [Google Scholar] [CrossRef] - Arrieta, A.F.; Hagedorn, R.; Erturk, R.; Inman, R.J. A piezoelectric bistable plate for nonlinear broadband energy harvesting. Appl. Phys. Lett.
**2010**, 97, 174103. [Google Scholar] [CrossRef] [Green Version] - Lee, A.J.; Inman, D.J. A multifunctional bistable laminate: Snap-through morphing enabled by broadband energy harvesting. J. Intell. Mater. Syst. Struct.
**2018**, 29, 2528–2543. [Google Scholar] [CrossRef] [Green Version] - Syta, A.; Bowen, C.R.; Kim, H.A.; Rysak, A.; Litak, G. Experimental analysis of the dynamical response of energy harvesting devices based on bistable laminated plates. Meccanica
**2015**, 50, 1961–1970. [Google Scholar] [CrossRef] [Green Version] - Emam, S.A.; Hobeck, J.; Inman, D.J. Experimental investigation into the nonlinear dynamics of a bistable laminate. Nonlinear Dyn.
**2019**, 95, 3019–3039. [Google Scholar] [CrossRef] - Emam, S.A.; Inman, D.J. A review on bistable composite laminates for morphing and energy harvesting. Appl. Mech. Rev.
**2015**, 67, 060803. [Google Scholar] [CrossRef] - Pellegrini, S.P.; Tolou, N.; Schenk, M.; Herder, J.L. Bistable vibration energy harvesters: A review. J. Intell. Mater. Syst. Struct.
**2013**, 24, 1303–1312. [Google Scholar] [CrossRef] - Lu, Z.Q.; Shao, D.; Fang, Z.W.; Ding, H.; Chen, L.Q. Integrated vibration isolation and energy harvesting via a bistable piezo-composite plate. J. Vib. Control
**2020**, 26, 779–789. [Google Scholar] [CrossRef] - Borowiec, M.; Rysak, A.; Betts, D.N.; Bowen, C.R.; Kim, H.A.; Litak, G. Complex response of a bistable laminated plate: Multiscale entropy analysis. Eur. Phys. J. Plus
**2014**, 129, 1–7. [Google Scholar] [CrossRef] [Green Version] - Reddy, A.N. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis; CRC Press: Boca Raton, FL, USA, 2004; pp. 200–216. [Google Scholar]
- Pirrera, A.; Avitabile, D.; Weaver, P.M. Bistable plates for morphing structures: A refined analytical approach with high-order polynomials. Int. J. Solids Struct.
**2010**, 47, 3412–3425. [Google Scholar] [CrossRef] [Green Version] - Sembiring, L.; van Ormondt, M.; van Dongeren, A.; Roelvink, D. A validation of an operational wave and surge prediction system for the Dutch coast. Nat. Hazards Earth Syst. Sci.
**2015**, 15, 1231–1242. [Google Scholar] [CrossRef] [Green Version] - Gude, M.; Hufenbach, W.; Kirvel, C. Piezoelectrically driven morphing structures based on bistable unsymmetric laminates. Compos. Struct.
**2011**, 93, 377–382. [Google Scholar] [CrossRef]

**Figure 3.**The two stable cylindrical shells after curing, (

**a**) the cylindrical Shell I, (

**b**) the cylindrical Shell II.

**Figure 4.**The two new stable cylindrical shells with a piezoelectric patch on the surface, (

**a**) the cylindrical Shell I, (

**b**) the cylindrical Shell II.

**Figure 5.**The dynamic model for the bistable energy harvester based on the asymmetric bistable composite laminated shell.

**Figure 6.**The voltage outputs around the two stable states respectively, (

**a**) the time-history graphs around the two stable states respectively, (

**b**) the spectrum diagram for the lower stable state, (

**c**) the spectrum diagram for the upper stable state.

**Figure 7.**The dynamic snap-through for the voltage output, (

**a**) the time-history graph, (

**b**) the spectrum diagram.

**Figure 8.**The vibrations of the asymmetric bistable composite laminated shell, (

**a**) the vibrations around the two stable states respectively, (

**b**) the dynamic snap-through between the two stable states.

**Figure 11.**The dynamic snap-through and chaotic vibration, (

**a**) the time-history graph, (

**b**) the Poincaré map.

**Figure 12.**The bifurcation diagrams for displacement w and voltage V via the base excitation frequency Ω when amplitude f = 0.2, (

**a**) the dynamic displacement, (

**b**) the voltage output of harvester.

**Figure 13.**The bifurcation diagram for displacement w and voltage V via the base excitation frequency Ω when amplitude f = 0.3, (

**a**) the dynamic displacement, (

**b**) the voltage output of harvester.

**Figure 14.**The bifurcation diagram for displacement w and voltage V via the base excitation frequency Ω when amplitude f = 0.6, (

**a**) the dynamic displacement, (

**b**) the voltage output of harvester.

**Figure 16.**The bifurcation diagram for displacement w and voltage V via the base excitation amplitude f when frequency Ω = 8, (

**a**) the dynamic displacement, (

**b**) the voltage output of harvester.

**Figure 17.**The bifurcation diagram for displacement w and voltage V via the base excitation amplitude f when frequency Ω = 10, (

**a**) the dynamic displacement, (

**b**) the voltage output of harvester.

**Figure 18.**The bifurcation diagram for displacement w and voltage V via the base excitation amplitude f when frequency Ω = 13, (

**a**) the dynamic displacement, (

**b**) the voltage output of harvester.

**Figure 19.**The bifurcation diagram for displacement w and voltage V via the base excitation amplitude f when frequency Ω = 15, (

**a**) the dynamic displacement, (

**b**) the voltage output of harvester.

**Figure 20.**The bifurcation diagram for displacement w and voltage V via the base excitation amplitude f when frequency Ω = 16, (

**a**) the dynamic displacement, (

**b**) the voltage output of harvester.

Properties | Description | Data | Unit |
---|---|---|---|

E_{11} | The longitudinal modulus | 147 | GPa |

E_{22} | The transverse modulus | 10.7 | GPa |

G_{12} | The shear modulus | 7 | GPa |

G_{13} | The shear modulus | 7 | GPa |

G_{23} | The shear modulus | 7 | GPa |

ν_{12} | The major Poisson’s ratio | 0.3 | – |

α_{1} | The longitudinal coefficient of thermal expansion | 5 × 10^{−7} | [°C]^{−1} |

α_{2} | The transversal coefficient of thermal expansion | 2.649 × 10^{−5} | [°C]^{−1} |

h | The thickness of the fiber | 0.122 | mm |

L_{x} | The length of the fiber | 300 | mm |

L_{y} | The width of the fiber | 300 | mm |

Properties | Description | Data | Unit |
---|---|---|---|

E_{11} | The longitudinal modulus | 30.336 | GPa |

E_{22} | The transverse modulus | 15.857 | GPa |

G_{12} | The shear modulus | 5.515 | GPa |

G_{13} | The shear modulus | 5.515 | GPa |

G_{23} | The shear modulus | 6.823 | GPa |

ν_{12} | The major Poisson’s ratio | 0.31 | – |

e_{31} [10^{−9} mm/V] | The piezoelectric constant | −210 | 10^{−9} mm/V |

e_{32} [10^{−9} mm/V] | The piezoelectric constant | −210 | 10^{−9} mm/V |

e_{33} [10^{−9} mm/V] | The piezoelectric constant | 460 | 10^{−9} mm/V |

h | The thickness of the MFC | 0.3 | mm |

L_{x} | The length of the MFC | 85 | mm |

L_{y} | The width of the MFC | 57 | mm |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dong, T.; Chen, X.; Zhang, J.
Study on Dynamic Snap-Through and Nonlinear Vibrations of an Energy Harvester Based on an Asymmetric Bistable Composite Laminated Shell. *Symmetry* **2021**, *13*, 2405.
https://doi.org/10.3390/sym13122405

**AMA Style**

Dong T, Chen X, Zhang J.
Study on Dynamic Snap-Through and Nonlinear Vibrations of an Energy Harvester Based on an Asymmetric Bistable Composite Laminated Shell. *Symmetry*. 2021; 13(12):2405.
https://doi.org/10.3390/sym13122405

**Chicago/Turabian Style**

Dong, Ting, Xinhua Chen, and Jun Zhang.
2021. "Study on Dynamic Snap-Through and Nonlinear Vibrations of an Energy Harvester Based on an Asymmetric Bistable Composite Laminated Shell" *Symmetry* 13, no. 12: 2405.
https://doi.org/10.3390/sym13122405