# Nonlinear Dynamics of an Unsymmetric Cross-Ply Square Composite Laminated Plate for Vibration Energy Harvesting

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equation of Motion

_{x}and 2L

_{y}, respectively, and the thickness of the unsymmetric cross-ply square composite laminated plate is 2H. The edge lengths of the piezoelectric patch in the x and y directions are L

_{1}and L

_{2}, respectively, and the thickness of the piezoelectric patch is h

^{E}. Three variables u, v and w are used to denote the displacements of the plate in the x, y and z directions. Under the experimental conditions, the center of the plate is connected with the exciter through a supporting bar. Therefore, it is assumed that the center of the plate is fixed and the four edges are kept free, and a base excitation is given at the center. The unsymmetric cross-ply square composite laminated plate is subjected to a base excitation Y. Figure 2 shows the two stable equilibrium positions of the structure subjected to thermal stress.

#### 2.1. Analysis of Unsymmetric Composite Laminated Plate

_{0}, v

_{0}and w

_{0}denote the longitudinal and transverse displacements of the mid-plane, and z is the transverse coordinate of the plate.

_{0}} represent the membrane strains, and {k} denotes the bending curvature.

_{xx}, α

_{yy}and α

_{xy}are the coefficients of the linear expansion. ΔT is the temperature change. Q

_{ij}is the material elastic constant, and the expression is shown as follows:

_{11}and E

_{22}are the modulus of elasticity, v

_{12}and v

_{21}are the Poisson’s ratios, G

_{12}is the shear modulus and all these coefficients belong to the unsymmetric cross-ply square composite laminated plate.

_{ij}are the extensional stiffnesses, D

_{ij}are the bending stiffnesses and B

_{ij}are the bending–extensional coupling stiffnesses, which are defined in terms of the lamina stiffnesses Q

_{ij}as

#### 2.2. Analysis of Piezoelectric Patch

_{ij}are the piezoelectric stress constants. ${\in}_{ii}$ is the dielectric constant. ${c}_{ij}^{E}$ are the material elastic constants, and the expression is shown as follows:

#### 2.3. Equation of Motion of the Unsymmetric Composite Laminated Plate with the Piezoelectric Patch

_{total}, U

_{total}and δW

_{total}are the kinetic energy, the strain energy and the virtual work done of the unsymmetric composite laminated plate with a piezoelectric patch.

_{lam}and V

_{lam}are the area and volume of the unsymmetric cross-ply square composite laminated plate, respectively. n

_{lam}is the layer number of the plate.

_{pie}and V

_{pie}are the area and volume of the piezoelectric patch, respectively. n

_{pie}is the number of piezoelectric patches.

_{lam}and ρ

_{pie}are the density of the unsymmetric cross-ply square composite laminated plate and the piezoelectric patch, respectively.

_{z}is the electric field applied in the z direction, and D

_{z}is the electric displacement in the z direction.

_{s}is odd in the x and even in the y directions.

_{s}is even in the x and odd in the y directions.

_{s}is even in the x and y directions, and it vanishes at the origin.

_{n,m}

_{−}

_{n}(t), v

_{n,m}

_{−}

_{n}(t) and w

_{n,m}

_{−}

_{n}(t) to the replace the functions u

_{n,m}

_{−}

_{n}, v

_{n,m}

_{−}

_{n}and w

_{n,m}

_{−}

_{n}that are independent of the time in Equations (31)–(33). Therefore, the dynamic displacements are considered by the following shape functions.

_{n,m}

_{−}

_{n}, v

_{n,m}

_{−}

_{n}and w

_{n,m}

_{−}

_{n}can represent the static curvature of the unsymmetric composite laminated plate, and u

_{n,m}

_{−}

_{n}(t), v

_{n,m}

_{−}

_{n}(t) and w

_{n,m}

_{−}

_{n}(t) can represent the time-varying curvature of the unsymmetric cross-ply square composite laminated plate.

## 3. Analysis of Nonlinear Behaviors and Energy Harvesting

_{11}= 146.95 GPa, E

_{22}= 10.702 GPa, G

_{12}= 6.977 GPa, α

_{1}= 5.028 × 10

^{−7}°C

^{−1}, α

_{2}= 2.65 × 10

^{−5}°C

^{−1}, v

_{12}= 0.3, ρ

_{lam}= 1570 kg/m

^{3}, h = 0.122 mm, L

_{x}=150 mm, L

_{y}=1500 mm, ${E}_{11}^{E}$ = 30.336 GPa, ${E}_{22}^{E}$ = 15.857 GPa, ${G}_{12}^{E}$ = 5.515 GPa, h

^{E}= 0.3 mm, ${v}_{12}^{E}$ = 0.31, L

_{1}= 85 mm, L

_{1}= 57 mm, ρ

_{pie}= 5440 kg/m

^{3}, e

_{31}= −210 × 10

^{−12}C/N and e

_{32}= −210 × 10

^{−12}C/N.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The dynamic model of the unsymmetric composite laminated plate with a piezoelectric patch.

**Figure 2.**The two stable equilibrium positions of the unsymmetric cross-ply square composite laminated plate subjected to thermal stress.

**Figure 5.**The periodic motion around one of the two single-well chaotic attractors when f = 0.01. (

**a**) The time history, (

**b**) the phase portrait, (

**c**) the voltage of the harvester, (

**d**) the Poincare map and (

**e**) the power spectrum.

**Figure 6.**The chaotic motion around one of the two single-well chaotic attractors f = 0.2. (

**a**) The time history, (

**b**) the phase portrait, (

**c**) the voltage of the harvester, (

**d**) the Poincare map and (

**e**) the power spectrum.

**Figure 7.**The chaotic motion and one-way snap-through around the two-well chaos attractor f = 0.27. (

**a**) The time history, (

**b**) the phase portrait, (

**c**) the voltage of the harvester, (

**d**) the Poincare map and (

**e**) the power spectrum.

**Figure 8.**The chaotic motion and multiple-way snap-through around the two-well chaos attractor f = 0.28. (

**a**) The time history, (

**b**) the phase portrait, (

**c**) the voltage of the harvester, (

**d**) the Poincare map and (

**e**) the power spectrum.

**Figure 9.**The chaotic motion and multiple-way snap-through around the two-well chaos attractor f = 0.29. (

**a**) The time history, (

**b**) the phase portrait, (

**c**) the voltage of the harvester, (

**d**) the Poincare map and (

**e**) the power spectrum.

**Figure 10.**The chaotic motion and constant snap-through around the two-well chaos attractor f = 0.35. (

**a**) The time history, (

**b**) the phase portrait, (

**c**) the voltage of the harvester, (

**d**) the Poincare map and (

**e**) the power spectrum.

**Figure 11.**The chaotic motion around another one of the two single-well chaotic attractors f = 0.37. (

**a**) The time history, (

**b**) the phase portrait, (

**c**) the voltage of the harvester, (

**d**) the Poincare map and (

**e**) the power spectrum.

**Figure 12.**The quasi-periodic motion around another one of the two single-well chaotic attractors f = 0.45. (

**a**) The time history, (

**b**) the phase portrait, (

**c**) the voltage of the harvester, (

**d**) the Poincare map and (

**e**) the power spectrum.

**Figure 13.**The periodic motion around another one of the two single-well chaotic attractors f = 0.56. (

**a**) The time history, (

**b**) the phase portrait, (

**c**) the voltage of the harvester, (

**d**) the Poincare map and (

**e**) the power spectrum.

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**MDPI and ACS Style**

Jiang, G.; Dong, T.; Guo, Z.
Nonlinear Dynamics of an Unsymmetric Cross-Ply Square Composite Laminated Plate for Vibration Energy Harvesting. *Symmetry* **2021**, *13*, 1261.
https://doi.org/10.3390/sym13071261

**AMA Style**

Jiang G, Dong T, Guo Z.
Nonlinear Dynamics of an Unsymmetric Cross-Ply Square Composite Laminated Plate for Vibration Energy Harvesting. *Symmetry*. 2021; 13(7):1261.
https://doi.org/10.3390/sym13071261

**Chicago/Turabian Style**

Jiang, Guoqing, Ting Dong, and Zhenkun Guo.
2021. "Nonlinear Dynamics of an Unsymmetric Cross-Ply Square Composite Laminated Plate for Vibration Energy Harvesting" *Symmetry* 13, no. 7: 1261.
https://doi.org/10.3390/sym13071261