Optimal design of carbon fiber collapsible hinge for space application

2.1. Finite element model of carbon fiber collapsible hinge

The finite element model of carbon fiber collapsible hinge is built based on ABAQUS composite materials processing module shown in Fig. 1. A single layer composite can be considered as a transversely isotropic homogeneous material. The elastic properties are defined based on five lamina engineering coefficients. $E_1$ is longitudinal elastic modulus of each composite materials layer. $E_2$ is transverse elastic modulus. ${\mu }_{12}$ is longitudinal Poisson’s ratio. $G_{12}$ and $G_{23}$ are shear elastic modulus. The material properties of hinge are shown in Table 1. In order to eliminate the effect of coupling between extension and bending. The hinge is composed of ten carbon fiber composite material layers which stacked orthogonally and symmetrically as ${\left[0/90/0/90/0\right]}_s$. The thickness of each layer is 0.1 mm. The model is meshed as reduced integration shell elements shown in Fig. 1. The value of fundamental frequency extracted through linear perturbation analysis step is 133 HZ under free vibration by using ABAQUS.

Fig. 1Finite element model of carbon fiber collapsible hinge

2.2. Modal experiment

As it shown in Fig. 2, the free modal experiment is carried out to verify the results of finite element model. The hinge is hung up to ensure the free boundary conditions. Hammering modal method is carried out to analyze free vibration of the hinge. There are 11 output points distributed on the axial direction of the hinge. The exciting position is set at the end of the hinge. The vibration signal is acquired through one-direction acceleration sensor fixed on the test output points. The signal collector is VibPilot-8 made by M+P international company. The vibration signal is analyzed by M+P international SO analyzer software. The fundamental frequency obtained based on the modal experiment is 135 HZ. The relative error between finite element model and experiment is small enough. Also, the first mode shape acquired through finite element model is in good agree with result of modal experiment.

Fig. 2Free modal experiment of hinge

4.1. Flow chart of optimization

The size of the hinge is crucial to the fundamental frequency of the structure and the maximum stress when folding. In order to increase the stiffness of the structure and reduce the maximum stress, the optimal design of the size of the slots is carried out in this paper. The calculation based on finite element model cost tremendous amounts of time, especially using dynamic explicit step to analyze quasi-static folding process. So surrogate model is built for optimization iteration instead of finite element model by using co-simulation software ISIGHT. The sample algorithm is Latin hypercube. Radial basis function neural network is chosen as surrogate model to reflect the nonlinear relationships between the optimization objectives and variables. The flow chart of optimization is shown in Fig. 4.

Fig. 4Flow chart of optimization

a)

b)

4.2. Optimization model

The fundamental frequency of the structure $F$ and the maximum stress $S$ during the folding process are optimization objectives. The aim of the optimization design is to increase $F$ while decrease $S$. The total objective $G$ is shown in Eq. (1), while ${\omega }_1$ and ${\omega }_2$ are object weights. This multi-objectives optimization is carried out based on non-dominated sorting genetic algorithm. The design variables of the optimization are the length of slot $L$, the width of slot $W$ and the diameter of the tube hinge $D$. The range of the design variables is shown in Table 2. The mass of the whole hinge $M$ must not be bigger than the mass $m$ before optimization. So, the constrains of optimization is shown in Eq. (2):

4.3. Sensitivity analysis and optimization results

The sensitivity analysis is based on the sample data obtained from finite element model through ABAQUS based on Latin hypercube sample algorithm. As it shown in Fig. 5, the most critical parameter to the fundamental frequency and maximum stress is the diameter of the tube hinge $D$. Then the width of the slots $W$ is the second influential parameter to the $F$ and $S$. While the length of the slots $L$ contributes least to the $F$ and $S$.

The non-dominated sorting genetic algorithm is chosen as optimization algorithm with 50 generation and 30 samples of each generation. The results of this multi-objectives optimization are Pareto solutions shown in Table 3. The optimal result contains three noninferior solutions. The fundamental frequency $F$ is increased by 9 %-16 %, while the maximum stress $S$ is decreased by 12 %-15 %. The bottom solution in Table 3 is chosen to be manufactured in consideration of engineering requirements finally.

Fig. 5Sensitivity of each parameter