In this study, the wings must be folded onto the missile surface. For this reason, a special mechanism has been designed that can rotate on a certain single axis. Figure 1 shows both the folded and deployed configurations in the missile coordinate system (Kulunk and Sahin, 2019). From the folded position to the deployed position, the wing rotates around an axis. This axis has a special orientation with respect to the missile coordinate frame.

Figure 2 shows the section view of the mechanism in both compact (a) and expanded (b) configurations (Kulunk and Sahin, 2019). The mechanism consists of several mechanical components: (1) Main Casing, (2) Wing Shaft, (3) Bearings, (4) Locking Casing, (5) Locking Bushing, (6) Locking Pin, (7) Torsion Spring, and (8) Compression Spring. The wing shaft (2) is connected to a torsion spring (7) through a locking casing (4). These three parts rotate simultaneously after the missile leaves. With this rotational movement, the wing is deployed to its final position. After that, the pin (6) is activated by a compression spring (8) which locks the whole mechanism of the locking casing (4) (Kulunk and Sahin, 2019).

After the missile leaves the tube, the wings must be opened and fixed within a certain time. In this study, this time is expected to be less than 200 *ms*. The maximum energy of the springs (especially the torsion spring) that can be stored will directly affect the opening time of the wing. For this reason, the energies of the springs were used as the optimization target. Elasticity Modulus (E) and Shear Modulus (G) are the most important parameters for the spring design. In this study, high carbon spring wire has been chosen as a spring material. The other parameters were the wire diameter (d), mean coiling diameter (Dm), coiling number (N), and deflection of springs (xd for compression spring and θ for torsion spring) (Shigley et al., 2011). The stored energies for compression\({ (SE}_{x})\) and torsion (\({SE}_{\theta }\)) springs can be calculated using equations (1) and (2) (Shigley et al., 2011).

$${SE}_{x}=\left({ d}^{4} G{ xd}^{2}\right)/\left(16{ Dm}^{3} N\right)$$

1

$${SE}_{\theta }=({d}^{4} E {\theta }^{2} \pi /180)/\left(7776 Dm N\right)$$

2

The mechanical dimensions of the system directly determine the geometric constraints of the springs. In addition, the conditions to which the missile will be exposed should be considered. These factors determine the limits of the spring parameters. Another important constraint is the safety factor. The determination of safety factors is detailed in (Shigley et al., 2011). The safety factor for compression (SFC) springs is defined as the division of maximum allowable stress by stress at solid length. SFC can be calculated using equations (3), (4), (5), and (6) (Shigley et al., 2011). (For the spring material used in this study, \({S}_{sy}=980 MPa\)). In the equations, F indicates the force, and Kb indicates the Bergstrasser factor. In this study, SFC was chosen to be greater than or equal to 1.2 (Shigley et al., 2011).

$$\eta \left(safety\right)=\frac{{S}_{sy}(Max.allowable stress)}{{T}_{s}\left(Stress at solid length\right)}$$

3

$${T}_{s}={K}_{B}\frac{8*F*Dm}{\pi *{d}^{3}}$$

4

$${K}_{B}=\frac{\left(4*\frac{Dm}{d}\right)+2}{\left(4*\frac{Dm}{d}\right)-3}$$

5

$$F=xd*\frac{{d}^{4}*G}{8*{Dm}^{3}*N}$$

6

The safety factor for torsion (SFT) spring is defined as the division of M by k. SFT can be calculated from equations (7), (8), (9), and (10) (Shigley et al., 2011). (For the material used in this study, \({S}_{y}=1600 MPa\)). In the equations, M indicates the moment, k’, the spring rate and Ki, the stress correction factor. The safety angle must be greater than or equal to θ. Tables 1 and 2 show the geometric and safety constraints of the springs.

$$Safety angle=\frac{M*360}{{k}^{{\prime }}}$$

7

$$M=\frac{\pi *{d}^{3}*{S}_{y}}{32*{K}_{i}}$$

8

$${K}_{i}=\frac{4*{\left(\frac{Dm}{d}\right)}^{2}-\frac{Dm}{d}-1}{\left(4*\frac{Dm}{d}\right)*(\frac{Dm}{d}-1)}$$

9

$${k}^{{\prime }}=\frac{{d}^{4}*E}{\text{10,8}*Dm*N}$$

10

Table 1

Constraints of the compression spring design

Parameters & Constraints | Minimum | Maximum | Constraint Type |

Shear Modulus (G) | 83.7E9 Pa | Fixed |

Wire diameter (d) | 0.3 mm | 0.5 mm | Geometric |

Coiling diameter (Dm) | 3.1 mm | 3.6 mm | Geometric |

Coiling Number (N) | 8 | 12 | Geometric |

Deflection (xd) | 6 mm | 9 mm | Geometric |

Factor of Safety (SFC) | 1.2 | Safety |

Table 2

Constraints of the torsion spring design

Parameters & Constraints | Minimum | Maximum | Constraint Type |

Elasticity modulus (E) | 203.4E9 | Fixed |

Wire diameter (d) | 1 mm | 1.7 mm | Geometric |

Coiling diameter (Dm) | 15 mm | 21 mm | Geometric |

Coiling Number (N) | 3 | 7 | Geometric |

Deflection (\(\theta\)) | 130 deg. | 170 deg. | Geometric |

Safety Angle (SFT) | must be larger or equal than\(\theta\) | Safety |

Depending on the mechanism, the spring geometry being within certain limits narrows the solutions of the optimization problem. In this context, a solution can be produced with classical optimization methods, but for this, the entire range for each variable must be searched. Metaheuristic algorithms, on the other hand, can find min-max values much faster by performing multiple searches on the solution set thanks to the population. For this reason, the optimization process was done with BA.