# Developing Equations for Free Vibration Parameters of Bistable Composite Plates Using Multi-Objective Genetic Programming

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Formulation of a Bistable Laminate

## 3. Numerical and Experimental Methods

^{−7}for automatic stabilization [31,44,45].

## 4. Genetic Programming Methodology

_{i}is the weight of ith gene and w

_{0}is the bias term. What makes MOGP a reliable tool for developing formulations for a set of data is its relatively higher efficiency and capability in modelling nonlinear complex problems [52]. MOGP can optimize the complexity development and the goodness of fit to restrict the model from over-expansion and over complexity.

## 5. Results

#### 5.1. Determining the Natural Frequencies

#### 5.2. Mathematical Expressions

_{1}= a/b and x

_{2}= t, contribute to the generation of the GP population and training-validation process. Several runs are conducted, and according to the explanations of Section 4, four GP models with the highest performance are achieved for the natural frequencies of the composite laminates that are presented in Table 4.

^{2}and MRE are as follows:

^{2}) and Mean Relative Error (MRE) are herein calculated for the Training and Validation, and the Test, subsets. Table 5 presents the calculated R

^{2}and MRE for the proposed formulae. As per this table, it is seen that the R

^{2}values are higher than 97 percent for the Train and Validation and over 96 percent for the Test data. Since the Test data had not been seen by the GP algorithm, and because the R

^{2}and MRE values of the Train and Validation set are very close to the Test data, the models are able to predict the natural frequencies accurately and they do not overfit in the training process.

#### 5.3. Parametric Study and Sensitivity Analysis

_{max}(x

_{i}) and NF

_{min}(x

_{i}) are the maximum and minimum output values when the ith parameter is substituted in the GP formulas, while the average value of the other parameter is used. The results of the sensitivity analysis are presented in Table 7. According to the table, the natural frequencies of bistable laminates are more sensitive to the thickness rather than the length ratio. This is an advantage that makes it possible to design bistable structures with a desired vibration response by setting the thickness. It is seen that the contributions of the parameters in NF1 and NF4 are approximately similar, wherein the sensitivity of these two models to a/b is about one-fourth of t, which indicates the significantly higher contribution of the thickness in these modes. Nevertheless, different results can be seen for the sensitivity analysis of the second and third natural frequencies. Although these natural frequencies are more sensitive to the a/b compared to NF1 and NF4, thickness still plays a more vital role.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Two stable equilibria shapes of a cross-ply bistable composite plate, (

**a**) the first stable (

**b**) the second stable configuration.

**Figure 4.**The random points chosen to apply load (

**a**) and to extract the dynamic response of the bistable plate (

**b**).

**Figure 5.**Stable shapes of a cross-ply (0/90) bistable composite, (

**a**) the first stable (

**b**) the second stable shape.

**Figure 6.**Experimental method to collect the desired signal from the vibrational response of a bistable composite plate: (

**a**) the used set up, (

**b**) the two stable shapes of the bistable laminate, and (

**c**) an appropriate sensor in different points of the laminate.

**Figure 7.**Associated mode shapes with the first four natural frequencies around the first stable configuration.

**Figure 8.**Comparison of the FE outputs with the predictions GP models for the natural frequencies: NF1 train and validation (

**a**) and test (

**b**) data; NF2 train and validation (

**c**) and test (

**d**) data; NF3 train and validation (

**e**) and test (

**f**) data; and NF4 train and validation (

**g**) and test (

**h**) data.

**Figure 9.**Results of the parametric studies on the inputs and output of the GP models. (

**a**) the effect of a/b variation on NF1 (

**b**) the effect of t variation on NF1 (

**c**) the effect of a/b variation on NF2 (

**d**) the effect of t variation on NF2 (

**e**) the effect of a/b variation on NF3 (

**f**) the effect of t variation on NF3 (

**g**) the effect of a/b variation on NF4 (

**h**) the effect of t variation on NF4.

**Table 1.**The properties of (0/90) laminate [41].

${\mathit{E}}_{\mathbf{11}}\left[GPa\right]$ | ${\mathit{E}}_{\mathbf{22}}\left[GPa\right]$ | ${\mathit{G}}_{\mathbf{12}}\left[GPa\right]$ | ${\mathit{\nu}}_{\mathbf{12}}$ | ${\mathit{\alpha}}_{\mathbf{11}}\left[\mathbf{1}\mathbf{/}\mathbf{\mathbb{C}}\right]$ | ${\mathit{\alpha}}_{\mathbf{22}}\left[\mathbf{1}\mathbf{/}\mathbf{\mathbb{C}}\right]$ | ${\mathit{t}}_{\mathit{p}\mathit{l}\mathit{y}}\left[mm\right]$ | $\Delta \mathit{T}\left[\mathbf{\mathbb{C}}\right]$ |
---|---|---|---|---|---|---|---|

147 | 10.71 | 6.98 | 0.3 | $5.03\times {10}^{-7}$ | $2.65\times {10}^{-5}$ | 0.225 | 160 |

Parameter | Setting |
---|---|

Functions | +, −, ×, /, exp, Ln, power, add3, mult3 ^{1} |

Population size | 10,000 |

Number of generations | 15,000 |

Maximum number of genes | 2 |

Maximum tree depth | 2 |

Training set | 0.75 |

Validation set | 0.15 |

Crossover events | 0.85 |

High-level crossover | 0.20 |

Low-level crossover | 0.80 |

Sub-tree mutation | 0.90 |

^{1}add3(a,b,c) = a + b + c, mult3(a,b,c) = a × b × c.

Natural Frequency (Hz) | 1st | 2nd | 3rd | 4th |
---|---|---|---|---|

FEA | 33.81 | 63.91 | 130.22 | 143.19 |

NF | Formula |
---|---|

NF1 | $187.5{x}_{2}{}^{{x}_{2}{}^{{x}_{2}{}^{3.469}}}-3510{x}_{1}{}^{2}{x}_{2}{}^{1.351}\times {\left(1.911{x}_{1}{}^{2}\right)}^{{x}_{2}{}^{{x}_{2}}}\times exp\left({x}_{2}-5.51{x}_{1}\right)-3.302$ |

NF2 | $516.8-503.7\times {\left(0.5091{x}_{1}{}^{2.707}{x}_{2}\right)}^{0.5487{x}_{1}{}^{{x}_{1}}\times {x}_{2}{}^{{x}_{1}+1}}$ |

NF3 | $704.2{x}_{2}{}^{\left(1.45\times {0.5467}^{{x}_{1}}\right)}\times {1.469}^{-1.809{x}_{1}{}^{2}}+27.13$ |

NF4 | $450.6{x}_{2}\times {\left({x}_{1}{x}_{2}{}^{2}\right)}^{-0.2237{x}_{1}}\times {1.039}^{2{x}_{2}}\times {4.331}^{{\left(0.0413\right)}^{{x}_{1}}}-40.43$ |

^{x}

_{1}

^{= a/b, x}

_{2}

^{= t}

**Table 5.**The statistical indices for the Training and Validation, and the Test, subsets in the GP formulas.

Model | Train and Validation (75%, 15% Data) | Test (10% Data) | ||
---|---|---|---|---|

R^{2} (%) | MRE (%) | R^{2} (%) | MRE (%) | |

NF1 | 97.31 | 2.23 | 96.79 | 2.86 |

NF2 | 99.07 | 1.82 | 97.12 | 2.10 |

NF3 | 98.80 | 1.65 | 98.24 | 2.04 |

NF4 | 98.95 | 1.41 | 98.42 | 1.89 |

Model | k | k′ |
---|---|---|

NF1 | 0.993 | 1.002 |

NF2 | 0.997 | 1.002 |

NF3 | 0.999 | 0.998 |

NF4 | 1.000 | 0.999 |

Model | S(a/b)% | S(t)% |
---|---|---|

NF1 | 25.7 | 74.3 |

NF2 | 41.6 | 58.4 |

NF3 | 45.1 | 54.9 |

NF4 | 22.8 | 77.2 |

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**MDPI and ACS Style**

Saberi, S.; Hosseini, A.S.; Yazdanifar, F.; Castro, S.G.P.
Developing Equations for Free Vibration Parameters of Bistable Composite Plates Using Multi-Objective Genetic Programming. *Polymers* **2022**, *14*, 1559.
https://doi.org/10.3390/polym14081559

**AMA Style**

Saberi S, Hosseini AS, Yazdanifar F, Castro SGP.
Developing Equations for Free Vibration Parameters of Bistable Composite Plates Using Multi-Objective Genetic Programming. *Polymers*. 2022; 14(8):1559.
https://doi.org/10.3390/polym14081559

**Chicago/Turabian Style**

Saberi, Saeid, Alireza Sadat Hosseini, Fatemeh Yazdanifar, and Saullo G. P. Castro.
2022. "Developing Equations for Free Vibration Parameters of Bistable Composite Plates Using Multi-Objective Genetic Programming" *Polymers* 14, no. 8: 1559.
https://doi.org/10.3390/polym14081559