# Global and Local Dynamics of a Bistable Asymmetric Composite Laminated Shell

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equation of Motion for Global Dynamics

_{N}–90

_{N}]

_{T}subjected to the base excitation is considered, as shown in Figure 1. If the number of layers is too large, the bistable characteristic will disappear. The bistable shell with (0/0/0/90/90/90) is a suitable choice, while (0/90) is too thin to bear large loads. In the experimental environment, the exciter acts the base excitation on the center of the shell through a support bar.

_{x}and 2L

_{y}, respectively.

_{N}–90

_{N}]

_{T}, as shown in Figure 1b. The bistability is led by asymmetric residual thermal stress that can be expressed by the thermal expansion coefficient α and temperature difference ΔT between the manufacturing temperature and room temperature.

- (1)
- The bistable plate model takes the zero plane before curing as the datum plane, while the bistable shell model takes the static surface that represents a stable equilibrium configuration after curing as the datum plane.
- (2)
- The bistable plate and shell models are converted to each other by the static displacement generated after curing.
- (3)
- The middle plane is assumed to be a neutral surface.

_{s}, v

_{s}and w

_{s}are the initial displacements and u

_{0}, v

_{0}and w

_{0}are the displacements of any point in the neutral plane along the x, y and z directions.

_{xx}, α

_{yy}and α

_{xy}are thermal expansion coefficients and ΔT is temperature difference between manufacturing temperature and room temperature.

_{ij}and E

_{11}, E

_{22}, G

_{12}, G

_{13}and G

_{23}can be expressed as

_{ij}are defined as extensional stiffnesses, D

_{ij}are defined as the bending stiffnesses and B

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

_{ij}as

_{s}, v

_{s}and w

_{s}can be set as

_{n,m−n}, v

_{n,m−n}and w

_{n,m−n}are coefficients related to curvatures.

_{0}and v

_{0}are transformed into functions of w

_{0}. Substituting Equations (24)–(26) into Equation(20) and integrating the obtained equations in the in-plane domain (x ϵ [–L

_{x}, L

_{x}] and [–L

_{y}, L

_{y}]), a two-degrees-of-freedom nonlinear ordinary differential equation concerning the global dynamics can be obtained

_{1}and α

_{2}, length L

_{x}, width L

_{y}and thickness h are the main factors of the static bifurcation, which is the super-critical pitchfork bifurcation. In order to obtain two stable equilibrium configurations with ideal initial curvatures, appropriate parameters α

_{1}, α

_{2}, L

_{x}, L

_{y}and h should be selected. The material properties collected in Table 1 are selected based on the above principles.

## 3. Three Equilibrium Configurations

_{0}= v

_{0}= w

_{0}= 0, the static equations can be derived as follows:

_{1}which represents x-curvature, (b) donates the static bifurcation curve of w

_{2}which represents y-curvature.

## 4. Equation of Motion for Local Dynamics

_{1}and R

_{2}are radii of curvatures of one of the two stable equilibrium configurations corresponding to the initial curvatures $\frac{{\partial}^{2}{w}_{s}}{\partial {x}^{2}}$ and $\frac{{\partial}^{2}{w}_{s}}{\partial {y}^{2}}$ determined above.

_{0}, v

_{0}and w

_{0}are expanded as follows:

^{u}(x,y), R

^{v}(x,y) and R

^{w}(x,y) are boundary functions, U(x,y), V(x,y) and W(x,y) are spatial functions and r(t) is a temporal function.

^{α}(x,y) = 0. The shape functions are expressed as:

_{m}and T

_{n}are the m-th and n-th order Chebyshev polynomial of the first kind, respectively.

_{b}is the additional stiffness caused by the supporting bar.

**K**and

**M**represent stiffness matrix and mass matrix respectively, p is n-dimensional displacement vector and n = 3MN.

_{1}and R

_{2}of the cylindrical shell. The Rayleigh–Ritz method is used to determine the modal shapes for the boundary conditions of central elastic support, as shown in Figure 3.

_{1}and R

_{2}of the cylindrical shell. The Rayleigh–Ritz method is used to determine the modal shapes for the boundary conditions of central elastic support, as shown in Figure 3.

## 5. Numerical Simulation

#### 5.1. Global Dynamics

_{1}represents the vibration for curvature in the x direction while w

_{2}represents the vibration for curvature in the y direction. Through the comparative study of w

_{1}and w

_{2}, the vibrations of the bistable shell can be determined. When f = 0.2, w

_{1}and w

_{2}remain almost zero around the equilibrium position (0, 0), that is to say, the bistable shell vibrates slightly around the first stable equilibrium configuration, which is the periodic vibration according to Poincaré map shown in Figure 4. When f = 0.35, w

_{1}increases rapidly while w

_{2}remain almost zero around the equilibrium position (0, 0),that is to say, the bistable shell vibrates violently around the first stable equilibrium configuration, which is the chaotic vibration according to Poincaré map shown in Figure 5. When f = 0.425, in a phase after the start, w

_{1}increases rapidly while w

_{2}remain almost zero around the equilibrium position (0, 0), at a certain moment, w

_{1}increases from 0 to 0.2 and remains almost constant while w

_{2}increases from 0 to 0.2 and vibrates violently around the equilibrium position (0.2, 0.2), that is to say, dynamic snap-through occurs, which is the chaotic vibration according to the Poincaré map shown in Figure 6. When f = 0.43, w

_{1}and w

_{2}vibrate violently around the equilibrium position (0.2, 0.2), that is to say, the bistable shell vibrates violently around the second stable equilibrium configuration, which is the chaotic vibration according to the Poincaré map shown in Figure 7. When f = 0.5, w

_{1}and w

_{2}change repeatedly between 0 and 0.2 simultaneously, namely, the constant dynamic snap-through occurs between the two stable equilibrium configurations, which is the chaotic vibration according to the Poincaré map shown in Figure 8. When f = 0.8, w

_{1}vibrates slightly while w

_{2}remain almost zero around the equilibrium position (0, 0), that is to say, the bistable shell vibrates slightly around the first stable equilibrium configuration, which is the quasi-periodic vibration according to the Poincaré map shown in Figure 9. When f = 0.9, w

_{1}and w

_{2}remain almost 0.2 around the equilibrium position (0.2, 0.2), that is to say, the bistable shell vibrates slightly around the second stable equilibrium configuration, which is the periodic vibration according to the Poincaré map shown in Figure 10.

_{1}and w

_{2}vibrate slightly around the equilibrium position (0, 0), when f is located in the interval 0.25~0.43, w

_{1}and w

_{2}vibrate violently around the equilibrium position (0, 0), when f is located in the interval 0.43~0.72, w

_{1}and w

_{2}vibrate violently between the equilibrium positions (0, 0)and (0.2, 0.2) and when f is located in the interval 0.72~1.2, w

_{1}and w

_{2}vibrate slightly around the equilibrium position (0, 0) or (0.2, 0.2).

#### 5.2. Local Dynamics

_{1}and σ

_{2}are two detuning parameters.

_{1}and a

_{2}represent the amplitude of w

_{1}and w

_{2}respectively and φ

_{1}and φ

_{2}and ${\mathsf{\varphi}}_{2}$ represent the phase angle of w

_{1}and w

_{2}respectively.

_{2}= 0. In Figure 12, solid lines AK, FE and dotted line KF represent the amplitude ${\overline{a}}_{1}$ of the first mode and solid lines AB, GC and dotted line KF represent the amplitude ${\overline{a}}_{2}$ of the second mode. When the base excitation amplitude f

_{2}increases gradually from zero, ${\overline{a}}_{1}$ changes along AK and ${\overline{a}}_{2}$ changes along AB. When f

_{2}= 5.8, ${\overline{a}}_{1}$ transfers from AK to DE by snap-through and ${\overline{a}}_{2}$ transfers from AB to BC. When the base excitation amplitude f

_{2}decreases gradually from 8, ${\overline{a}}_{1}$ changes along EF and ${\overline{a}}_{2}$ changes along CG. When f

_{2}= 2, ${\overline{a}}_{1}$ transfers from EF to JA by snap-through and ${\overline{a}}_{2}$ transfers from CG to HA by snap-through. When ${\overline{a}}_{2}$ goes along CG, no matter how f

_{2}changes, ${\overline{a}}_{2}$ remains constant, that is to say, the response of the second mode enters saturation state. This is because the energy applied to the second mode is transferred to the first mode, which means that permeation takes place. Saturation and permeation are the peculiar phenomena of forced vibration of nonlinear multi-degree of freedom system related to 1:2 internal resonance.

_{1}= σ

_{2}= 0. Similar to Figure 12, with the change of base excitation amplitude f

_{2}, saturation and permeation occur.

_{1}. It can be seen from Figure 14 that when σ

_{1}changes from negative to positive, the system shows the softening and hardening nonlinearity successively.

_{1}. Different from Figure 14, with the change of σ

_{1}, the system shows only linear characteristics.

_{2}. It can be seen from Figure 16 that with the change of σ

_{2}, the system shows the hardening nonlinearity.

_{2}. It can be seen from Figure 17 that when σ

_{2}changes from negative to zero, the system shows the linear characteristics with negative slope, while when σ

_{2}changes from zero to positive, the system shows the linear characteristics with positive slope.

_{1}or σ

_{2}is changed, the first mode of the system shows nonlinear characteristics (softening and hardening nonlinearity) while the second mode shows linear characteristics.

## 6. Conclusions

- (1)
- Choosing difference temperature ΔT as the controlling parameter, the super-critical pitchfork bifurcation can be obtained. When ΔT is set to a specific value, three equilibrium configurations corresponding to two stable equilibrium configurations and one unstable equilibrium configuration are determined.
- (2)
- The global dynamics behave as the snap-through between the two stable equilibrium configurations and the vibrations around the two stable equilibrium configurations respectively.
- (3)
- The dynamic snap-through of the bistable system often occurs in chaos. In other words, the bistable system is often accompanied by the chaotic vibration in the process of the dynamic snap-through.
- (4)
- In the global dynamics, the vibrations behave as the periodic vibration, the quasi-periodic vibration and the chaotic vibration.
- (5)
- In the local dynamics, saturation and permeation occur in the process of the 1:2 internal resonance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The bistable asymmetric composite laminated shell model. (

**a**) The bistable asymmetric composite laminated shell. (

**b**) The thickness of the bistable asymmetric composite laminated shell.

**Figure 2.**The static bifurcation diagrams for under actual conditions via the temperature difference ΔΤ, (

**a**) donates the static bifurcation curve of w

_{1}which represents x-curvature, (

**b**) donates the static bifurcation curve of w

_{2}which represents y-curvature.

**Figure 3.**The first four mode shapes of the bistable asymmetric composite laminated shell, (

**a**) the translational mode, (

**b**) the rotational mode, (

**c**) the flexible torsional mode, (

**d**) the flexible bending mode.

**Figure 4.**The periodic motion around the first stable equilibrium configuration when f = 0.2, (

**a**) donates the time-history on the plane $\left(t,{w}_{1}\right)$, (

**b**) donates the phase portrait on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$, (

**c**) donates the time-history on the plane $\left(t,{w}_{2}\right)$, (

**d**) donates the phase portrait on the plane $\left({w}_{2},{\dot{w}}_{2}\right)$, (

**e**) donates three-dimensional phase portrait in space $\left({w}_{1},{\dot{w}}_{1},{w}_{2}\right)$, (

**f**) donates Poincaré map on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$.

**Figure 5.**The chaotic motion around the first stable equilibrium configuration when f = 0.35, (

**a**) donates the time-history on the plane $\left(t,{w}_{1}\right)$, (

**b**) donates the phase portrait on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$, (

**c**) donates the time-history on the plane $\left(t,{w}_{2}\right)$, (

**d**) donates the phase portrait on the plane $\left({w}_{2},{\dot{w}}_{2}\right)$, (

**e**) donates three-dimensional phase portrait in space $\left({w}_{1},{\dot{w}}_{1},{w}_{2}\right)$, (

**f**) donates Poincaré map on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$.

**Figure 6.**The snap-through and chaotic motion between the two stable equilibrium configurations when f = 0.425, (

**a**) donates the time-history on the plane $\left(t,{w}_{1}\right)$, (

**b**) donates the phase portrait on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$, (

**c**) donates the time-history on the plane $\left(t,{w}_{2}\right)$, (

**d**) donates the phase portrait on the plane $\left({w}_{2},{\dot{w}}_{2}\right)$, (

**e**) donates three-dimensional phase portrait in space $\left({w}_{1},{\dot{w}}_{1},{w}_{2}\right)$, (

**f**) donates Poincaré map on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$.

**Figure 7.**The chaotic motion around the second stable equilibrium configuration when f = 0.43, (

**a**) donates the time-history on the plane $\left(t,{w}_{1}\right)$, (

**b**) donates the phase portrait on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$, (

**c**) donates the time-history on the plane $\left(t,{w}_{2}\right)$, (

**d**) donates the phase portrait on the plane $\left({w}_{2},{\dot{w}}_{2}\right)$, (

**e**) donates three-dimensional phase portrait in space $\left({w}_{1},{\dot{w}}_{1},{w}_{2}\right)$, (

**f**) donates Poincaré map on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$.

**Figure 8.**The constant snap-through and chaotic motion between the two stable equilibrium configurations when f = 0.5, (

**a**) donates the time-history on the plane $\left(t,{w}_{1}\right)$, (

**b**) donates the phase portrait on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$, (

**c**) donates the time-history on the plane $\left(t,{w}_{2}\right)$, (

**d**) donates the phase portrait on the plane $\left({w}_{2},{\dot{w}}_{2}\right)$, (

**e**) donates three-dimensional phase portrait in space $\left({w}_{1},{\dot{w}}_{1},{w}_{2}\right)$, (

**f**) donates Poincaré map on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$.

**Figure 9.**The quasi-periodic motion around the first stable equilibrium configuration when f = 0.8, (

**a**) donates the time-history on the plane $\left(t,{w}_{1}\right)$, (

**b**) donates the phase portrait on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$, (

**c**) donates the time-history on the plane $\left(t,{w}_{2}\right)$, (

**d**) donates the phase portrait on the plane $\left({w}_{2},{\dot{w}}_{2}\right)$, (

**e**) donates three-dimensional phase portrait in space $\left({w}_{1},{\dot{w}}_{1},{w}_{2}\right)$, (

**f**) donates Poincaré map on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$.

**Figure 10.**The periodic motion around the second stable equilibrium configuration when f = 0.9, (

**a**) donates the time-history on the plane $\left(t,{w}_{1}\right)$, (

**b**) donates the phase portrait on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$, (

**c**) donates the time-history on the plane $\left(t,{w}_{2}\right)$, (

**d**) donates the phase portrait on the plane $\left({w}_{2},{\dot{w}}_{2}\right)$, (

**e**) donates three-dimensional phase portrait in space $\left({w}_{1},{\dot{w}}_{1},{w}_{2}\right)$, (

**f**) donates Poincaré map on the plane $\left({w}_{1},{\dot{w}}_{1}\right)$.

**Figure 11.**The bifurcation diagrams for w

_{1}and w

_{2}via the base excitation amplitude f. (

**a**) The bifurcation diagram for w

_{1}via the base excitation amplitude f. (

**b**) The bifurcation diagram for w

_{2}via the base excitation amplitude f.

Properties | Data |
---|---|

E_{11}[GPa] | 146.95 |

E_{22}[GPa] | 10.702 |

G_{12}[GPa] | 6.977 |

G_{13}[GPa] | 6.977 |

G_{23}[GPa] | 6.977 |

ν_{12} | 0.3 |

α_{1}[°C]^{−1} | 5.028 × 10^{−7} |

α_{2}[°C]^{−1} | 2.65 × 10^{−5} |

h[mm] | 0.122 |

L_{x}[mm] | 300 |

L_{y}[mm] | 300 |

R^{u}(x) | R^{v}(x) | R^{w}(x) | R^{u}(y) | R^{v}(y) | R^{w}(y) | |
---|---|---|---|---|---|---|

FFFF | 1 | 1 | 1 | 1 | 1 | 1 |

FSFF | 1 | 1 − x | 1 − x | 1 − y | 1 | 1 − y |

SFFF | 1 | 1 + x | 1 + x | 1 + y | 1 | 1 + y |

SSFF | 1 | 1 − x^{2} | 1 − x^{2} | 1 − y^{2} | 1 | 1 − y^{2} |

FCFF | 1 − x | 1 − x | 1 − x | 1 − y | 1 − y | 1 − y |

CFFF | 1 + x | 1 + x | 1 + x | 1 + y | 1 + y | 1 + y |

SCFF | 1 − x | 1 − x^{2} | 1 − x^{2} | 1 − y^{2} | 1 − y | 1 − y^{2} |

CSFF | 1 + x | 1 − x^{2} | 1 − x^{2} | 1 − y^{2} | 1 + y | 1 − y^{2} |

CCFF | 1 − x^{2} | 1 − x^{2} | 1 − x^{2} | 1 − y^{2} | 1 − y^{2} | 1 − y^{2} |

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## Share and Cite

**MDPI and ACS Style**

Dong, T.; Guo, Z.; Jiang, G.
Global and Local Dynamics of a Bistable Asymmetric Composite Laminated Shell. *Symmetry* **2021**, *13*, 1690.
https://doi.org/10.3390/sym13091690

**AMA Style**

Dong T, Guo Z, Jiang G.
Global and Local Dynamics of a Bistable Asymmetric Composite Laminated Shell. *Symmetry*. 2021; 13(9):1690.
https://doi.org/10.3390/sym13091690

**Chicago/Turabian Style**

Dong, Ting, Zhenkun Guo, and Guoqing Jiang.
2021. "Global and Local Dynamics of a Bistable Asymmetric Composite Laminated Shell" *Symmetry* 13, no. 9: 1690.
https://doi.org/10.3390/sym13091690