In this paper, trajectory control of arbitrary shape mandrel in three-dimensional circular braiding is studied. To obtain accurate trajectory, offset of mandrel is predicted and compensated for trajectory of mandrel. Firstly, the equation of the force of all yarns on three-dimensional mandrel is given. Then offset of mandrel in single layer braiding machine is analyzed via finite element software. Learning these data via back propagation neural network algorithm, offset of mandrel at each moment is derived. The trajectory generation of three-dimensional mandrel based on offset compensation by roll pitch yaw transformation is given. Lastly, braiding angle for the mandrel is analyzed theoretically. In the practical engineering, this method is proven to effectively reduce the error of braiding angle and helpful for the precise control of the trajectory of arbitrary shape mandrel.
Braiding carriers, which are the important parts of a braiding machine, have the functions such as carrying braiding materials, controlling tension of carbon fiber, and driving carbon fiber movement. During the braiding process, two groups of carbon fibers braided in clockwise and counter clockwise direction contact each other and form relative motion, which causes friction and fuzzing. In order to improve this situation, the structural parameters of the carriers need to be optimized. In this paper, the kinematics and dynamics models were established based on the structure of braiding carriers. The micro-element method was used to analyze the relationship between the fiber length released from the yarn barrel, the rotation angle of the lever, and the tension of the carbon fiber. To limit the fluctuant range of carbon fiber tension, and to alleviate the fluffing phenomenon caused by the two groups of carbon fiber in contact with each other, antlion algorithm was used to optimize the structural parameters of braiding carriers. The simulation results showed that the tension of the carbon fiber can meet the processing requirements by adjusting the starting angle of each stage of carrier, the length of lever, the elastic coefficient of springs, and pre-compression of springs. It can be known that the structural parameters of braiding carriers optimized by antlion algorithm could meet the requirement of carbon fiber tension.
It is well known that the notion of ∗-modules plays an important role in the generalization of Morita equivalences. Recently, Wei et al. introduced the concept of ∗n-modules, which induces an equivalence of particular subcategories. In this paper, the dual situation of ∗n-modules, called co -∗n-modules, is considered, and their characterizations and properties are derived. We also present a characterization in terms of category duality, which can be regarded as a generalization of Morita duality. In particular, it is found that the co -selfsmall n-cotilting modules are co -∗n-modules.
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