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In this study,Stainless steel coil tube the design of the torsion and compression springs of the wing folding mechanism used in the rocket is considered as an optimization problem. After the rocket leaves the launch tube, the closed wings must be opened and secured for a certain amount of time. The aim of the study was to maximize the energy stored in the springs so that the wings could deploy in the shortest possible time. In this case, the energy equation in both publications was defined as the objective function in the optimization process. The wire diameter, coil diameter, number of coils, and deflection parameters required for the spring design were defined as optimization variables. There are geometric limits on the variables due to the size of the mechanism, as well as limits on the safety factor due to the load carried by the springs. The honey bee (BA) algorithm was used to solve this optimization problem and perform the spring design. The energy values obtained with BA are superior to those obtained from previous Design of Experiments (DOE) studies. Springs and mechanisms designed using the parameters obtained from the optimization were first analyzed in the ADAMS program. After that, experimental tests were carried out by integrating the manufactured springs into real mechanisms. As a result of the test, it was observed that the wings opened after about 90 milliseconds. This value is well below the project’s target of 200 milliseconds. In addition, the difference between the analytical and experimental results is only 16 ms.
In aircraft and marine vehicles,Stainless steel coil tube folding mechanisms are critical. These systems are used in aircraft modifications and conversions to improve flight performance and control. Depending on the flight mode, the wings fold and unfold differently to reduce aerodynamic impact1. This situation can be compared to the movements of the wings of some birds and insects during everyday flight and diving. Similarly, gliders fold and unfold in submersibles to reduce hydrodynamic effects and maximize handling3. Yet another purpose of these mechanisms is to provide volumetric advantages to systems such as the folding of a helicopter propeller 4 for storage and transport. The wings of the rocket also fold down to reduce storage space. Thus, more missiles can be placed on a smaller area of the launcher 5. The components that are used effectively in folding and unfolding are usually springs. At the moment of folding, energy is stored in it and released at the moment of unfolding. Due to its flexible structure, stored and released energy are equalized. The spring is mainly designed for the system, and this design presents an optimization problem6. Because while it includes various variables such as wire diameter, coil diameter, number of turns, helix angle and type of material, there are also criteria such as mass, volume, minimum stress distribution or maximum energy availability7.
This study sheds light on the design and optimization of springs for wing folding mechanisms used in rocket systems. Being inside the launch tube before the flight, the wings remain folded on the surface of the rocket, and after exiting the launch tube, they unfold for a certain time and remain pressed to the surface. This process is critical to the proper functioning of the rocket. In the developed folding mechanism, the opening of the wings is carried out by torsion springs, and the locking is carried out by compression springs. To design a suitable spring, an optimization process must be performed. Within spring optimization, there are various applications in the literature.
Paredes et al.8 defined the maximum fatigue life factor as an objective function for the design of helical springs and used the quasi-Newtonian method as an optimization method. Variables in optimization were identified as wire diameter, coil diameter, number of turns, and spring length. Another parameter of the spring structure is the material from which it is made. Therefore, this was taken into account in the design and optimization studies. Zebdi et al. 9 set goals of maximum stiffness and minimum weight in the objective function in their study, where the weight factor was significant. In this case, they defined the spring material and geometric properties as variables. They use a genetic algorithm as an optimization method. In the automotive industry, the weight of materials is useful in many ways, from vehicle performance to fuel consumption. Weight minimization while optimizing coil springs for suspension is a well-known study10. Bahshesh and Bahshesh11 identified materials such as E-glass, carbon and Kevlar as variables in their work in the ANSYS environment with the goal of achieving minimum weight and maximum tensile strength in various suspension spring composite designs. The manufacturing process is critical in the development of composite springs. Thus, various variables come into play in an optimization problem, such as the production method, the steps taken in the process, and the sequence of those steps12,13. When designing springs for dynamic systems, the natural frequencies of the system must be taken into account. It is recommended that the first natural frequency of the spring be at least 5-10 times the natural frequency of the system to avoid resonance14. Taktak et al. 7 decided to minimize the mass of the spring and maximize the first natural frequency as objective functions in the coil spring design. They used pattern search, interior point, active set, and genetic algorithm methods in the Matlab optimization tool. Analytical research is part of spring design research, and the Finite Element Method is popular in this area15. Patil et al.16 developed an optimization method for reducing the weight of a compression helical spring using an analytical procedure and tested the analytical equations using the finite element method. Another criterion for increasing the usefulness of a spring is the increase in the energy it can store. This case also ensures that the spring retains its usefulness for a long period of time. Rahul and Rameshkumar17 Seek to reduce spring volume and increase strain energy in car coil spring designs. They have also used genetic algorithms in optimization research.
As can be seen, the parameters in the optimization study vary from system to system. In general, stiffness and shear stress parameters are important in a system where the load it carries is the determining factor. Material selection is included in the weight limit system with these two parameters. On the other hand, natural frequencies are checked to avoid resonances in highly dynamic systems. In systems where utility matters, energy is maximized. In optimization studies, although the FEM is used for analytical studies, it can be seen that metaheuristic algorithms such as the genetic algorithm14,18 and the gray wolf algorithm19 are used together with the classical Newton method within a range of certain parameters. Metaheuristic algorithms have been developed based on natural adaptation methods that approach the optimal state in a short period of time, especially under the influence of the population20,21. With a random distribution of the population in the search area, they avoid local optima and move towards global optima22. Thus, in recent years it has often been used in the context of real industrial problems23,24.
The critical case for the folding mechanism developed in this study is that the wings, which were in the closed position before flight, open a certain time after leaving the tube. After that, the locking element blocks the wing. Therefore, the springs do not directly affect the flight dynamics. In this case, the goal of the optimization was to maximize the stored energy to accelerate the movement of the spring. Roll diameter, wire diameter, number of rolls and deflection were defined as optimization parameters. Due to the small size of the spring, weight was not considered a goal. Therefore, the material type is defined as fixed. The margin of safety for mechanical deformations is determined as a critical limitation. In addition, variable size constraints are involved in the scope of the mechanism. The BA metaheuristic method was chosen as the optimization method. BA was favored for its flexible and simple structure, and for its advances in mechanical optimization research25. In the second part of the study, detailed mathematical expressions are included in the framework of the basic design and spring design of the folding mechanism. The third part contains the optimization algorithm and optimization results. Chapter 4 conducts analysis in the ADAMS program. The suitability of the springs is analyzed before production. The last section contains experimental results and test images. The results obtained in the study were also compared with the previous work of the authors using the DOE approach.
The wings developed in this study should fold towards the surface of the rocket. Wings rotate from folded to unfolded position. For this, a special mechanism was developed. On fig. 1 shows the folded and unfolded configuration5 in the rocket coordinate system.
On fig. 2 shows a sectional view of the mechanism. The mechanism consists of several mechanical parts: (1) main body, (2) wing shaft, (3) bearing, (4) lock body, (5) lock bush, (6) stop pin, (7) torsion spring and (8 ) compression springs. The wing shaft (2) is connected to the torsion spring (7) through the locking sleeve (4). All three parts rotate simultaneously after the rocket takes off. With this rotational movement, the wings turn to their final position. After that, the pin (6) is actuated by the compression spring (8), thereby blocking the entire mechanism of the locking body (4)5.
Elastic modulus (E) and shear modulus (G) are key design parameters of the spring. In this study, high carbon spring steel wire (Music wire ASTM A228) was chosen as the spring material. Other parameters are wire diameter (d), average coil diameter (Dm), number of coils (N) and spring deflection (xd for compression springs and θ for torsion springs)26. The stored energy for compression springs \({(SE}_{x})\) and torsion (\({SE}_{\theta}\)) springs can be calculated from the equation. (1) and (2)26. (The shear modulus (G) value for the compression spring is 83.7E9 Pa, and the elastic modulus (E) value for the torsion spring is 203.4E9 Pa.)
The mechanical dimensions of the system directly determine the geometric constraints of the spring. In addition, the conditions in which the rocket will be located should also be taken into account. These factors determine the limits of the spring parameters. Another important limitation is the safety factor. The definition of a safety factor is described in detail by Shigley et al.26. The compression spring safety factor (SFC) is defined as the maximum allowable stress divided by the stress over the continuous length. SFC can be calculated using equations. (3), (4), (5) and (6)26. (For the spring material used in this study, \({S}_{sy}=980 MPa\)). F represents the force in the equation and KB represents the Bergstrasser factor of 26.
The torsion safety factor of a spring (SFT) is defined as M divided by k. SFT can be calculated from the equation. (7), (8), (9) and (10)26. (For the material used in this study, \({S}_{y}=1600 \mathrm{MPa}\)). In the equation, M is used for torque, \({k}^{^{\prime}}\) is used for spring constant (torque/rotation), and Ki is used for stress correction factor.
The main optimization goal in this study is to maximize the energy of the spring. The objective function is formulated to find \(\overrightarrow{\{X\}}\) that maximizes \(f(X)\). \({f}_{1}(X)\) and \({f}_{2}(X)\) are the energy functions of the compression and torsion spring, respectively. The calculated variables and functions used for optimization are shown in the following equations.
The various constraints placed on the design of the spring are given in the following equations. Equations (15) and (16) represent the safety factors for compression and torsion springs, respectively. In this study, SFC must be greater than or equal to 1.2 and SFT must be greater than or equal to θ26.
BA was inspired by bees’ pollen-seeking strategies27. Bees seek by sending more foragers to fertile pollen fields and fewer foragers to less fertile pollen fields. Thus, the greatest efficiency from the bee population is achieved. On the other hand, scout bees continue to look for new areas of pollen, and if there are more productive areas than before, many foragers will be directed to this new area28. BA consists of two parts: local search and global search. Local search searches for more communities near the minimum (elite sites), like bees, and searches less for other sites (optimum or select sites). An arbitrary search is performed in the global search part, and if good values are found, the stations are moved to the local search part in the next iteration. The algorithm contains some parameters: the number of scout bees (n), the number of local search sites (m), the number of elite sites (e), the number of foragers in elite sites (nep), the number of foragers in optimal areas. Site (nsp), neighborhood size (ngh), and number of iterations (I)29. The BA pseudocode is shown in Figure 3.
The algorithm tries to work between \({g}_{1}(X)\) and \({g}_{2}(X)\). As a result of each iteration, optimal values are determined and a population is gathered around these values in an attempt to obtain the best values. Restrictions are checked in the local and global search sections. In a local search, if these factors are appropriate, the energy value is calculated. If the new energy value is greater than the optimal value, assign the new value to the optimal value. If the best value found in the search result is greater than the current element, the new element will be included in the collection. The block diagram of the local search is shown in Figure 4.
Population is one of the key parameters in BA. It can be seen from previous studies that expanding the population reduces the number of iterations required and increases the likelihood of success. However, the number of functional assessments is also increasing. The presence of a large number of elite sites does not significantly affect performance. The number of elite sites can be low if it is not zero30. The size of the scout bee population (n) is usually chosen between 30 and 100. In this study, both 30 and 50 scenarios were run to determine the appropriate number (Table 2). Other parameters are determined depending on the population. The number of selected sites (m) is (approximately) 25% of the population size, and the number of elite sites (e) among the selected sites is 25% of m. The number of feeding bees (number of searches) was chosen to be 100 for elite plots and 30 for other local plots. Neighborhood search is the basic concept of all evolutionary algorithms. In this study, the tapering neighbors method was used. This method reduces the size of the neighborhood at a certain rate during each iteration. In future iterations, smaller neighborhood values30 can be used for a more accurate search.
For each scenario, ten consecutive tests were performed to check the reproducibility of the optimization algorithm. On fig. 5 shows the results of optimization of the torsion spring for scheme 1, and in fig. 6 – for scheme 2. Test data are also given in tables 3 and 4 (a table containing the results obtained for the compression spring is in Supplementary Information S1). The bee population intensifies the search for good values in the first iteration. In scenario 1, the results of some tests were below the maximum. In Scenario 2, it can be seen that all optimization results are approaching the maximum due to the increase in population and other relevant parameters. It can be seen that the values in Scenario 2 are sufficient for the algorithm.
When obtaining the maximum value of energy in iterations, a safety factor is also provided as a constraint for the study. See table for safety factor. The energy values obtained using BA are compared with those obtained using the 5 DOE method in Table 5. (For ease of manufacture, the number of turns (N) of the torsion spring is 4.9 instead of 4.88, and the deflection (xd) is 8 mm instead of 7.99 mm in the compression spring.) It can be seen that BA is better Result. BA evaluates all values through local and global lookups. This way he can try more alternatives faster.
In this study, Adams was used to analyze the movement of the wing mechanism. Adams is first given a 3D model of the mechanism. Then define a spring with the parameters selected in the previous section. In addition, some other parameters need to be defined for the actual analysis. These are physical parameters such as connections, material properties, contact, friction, and gravity. There is a swivel joint between the blade shaft and the bearing. There are 5-6 cylindrical joints. There are 5-1 fixed joints. The main body is made of aluminum material and fixed. The material of the rest of the parts is steel. Choose the coefficient of friction, contact stiffness and depth of penetration of the friction surface depending on the type of material. (stainless steel AISI 304) In this study, the critical parameter is the opening time of the wing mechanism, which must be less than 200 ms. Therefore, keep an eye on the wing opening time during the analysis.
As a result of Adams’ analysis, the opening time of the wing mechanism is 74 milliseconds. The results of dynamic simulation from 1 to 4 are shown in Figure 7. The first picture in Figure. 5 is the simulation start time and the wings are in the waiting position for folding. (2) Displays the position of the wing after 40ms when the wing has rotated 43 degrees. (3) shows the position of the wing after 71 milliseconds. Also in the last picture (4) shows the end of the turn of the wing and the open position. As a result of dynamic analysis, it was observed that the wing opening mechanism is significantly shorter than the target value of 200 ms. In addition, when sizing the springs, the safety limits were selected from the highest values recommended in the literature.
After completion of all design, optimization and simulation studies, a prototype of the mechanism was manufactured and integrated. The prototype was then tested to verify the simulation results. First secure the main shell and fold the wings. Then the wings were released from the folded position and a video was made of the rotation of the wings from the folded position to the deployed one. The timer was also used to analyze time during video recording.
On fig. 8 shows video frames numbered 1-4. Frame number 1 in the figure shows the moment of release of the folded wings. This moment is considered the initial moment of time t0. Frames 2 and 3 show the positions of the wings 40 ms and 70 ms after the initial moment. When analyzing frames 3 and 4, it can be seen that the movement of the wing stabilizes 90 ms after t0, and the opening of the wing is completed between 70 and 90 ms. This situation means that both simulation and prototype testing give approximately the same wing deployment time, and the design meets the performance requirements of the mechanism.
In this article, the torsion and compression springs used in the wing folding mechanism are optimized using BA. The parameters can be reached quickly with few iterations. The torsion spring is rated at 1075 mJ and the compression spring is rated at 37.24 mJ. These values are 40-50% better than previous DOE studies. The spring is integrated into the mechanism and analyzed in the ADAMS program. When analyzed, it was found that the wings opened within 74 milliseconds. This value is well below the project’s target of 200 milliseconds. In a subsequent experimental study, the turn-on time was measured to be about 90 ms. This 16 millisecond difference between analyzes may be due to environmental factors not modeled in the software. It is believed that the optimization algorithm obtained as a result of the study can be used for various spring designs.
The spring material was predefined and was not used as a variable in the optimization. Since many different types of springs are used in aircraft and rockets, BA will be applied to design other types of springs using different materials to achieve optimal spring design in future research.
We declare that this manuscript is original, has not been previously published, and is not currently being considered for publication elsewhere.
All data generated or analyzed in this study is included in this published article [and additional information file].
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