| Summary: | <p><strong>I. Introduction </strong></p><p>The subject of research in this work is a process of thread unrolling from a bobbin. The mathematical model of this process considering motion of thread peace on a bobbin and unrolled peace is proposed. The dimension of system of differential equations for this model is constant during deploying.</p><p>The relevance to simulate this process for design of Heliogyro-like solar sails (Heliogyro [1], BMSTU-Sail [2]) is proved. The paper briefly characterizes a blade for such solar sail as a simulation object. It proves the possibility for using a flexible thread model for a long blade because of very small blade thickness (less than 10 μm [3]) relative to blade width and the phenomena of Koriolis forces [4] that lead to buckling failure of blade flatness.</p><p>The major features of the proposed model are:</p><p>-- simulated as a motion of the thread piece both being on a bobbin and its unrolled peace;</p><p>-- splitting a thread length into nodes does not depend on the demand to ensure a sufficient number of nodes on a single thread turn on the coil;</p><p>-- because of avoiding a problem of contact between the thread and bobbin a stable integration of motion equations is provided by the conventional Runge-Kutta method of fourth order with a constant step [5];</p><p>-- in the course of solution the number of freedom degrees (number of motion equation) is constant, thereby simplifying a calculation algorithm.</p><p>The closest mathematical model is proposed in [6].</p><p>The scientific novelty of this research is the approach to solving the problem of unrolling thread from a bobbin using a constant number of motion equations while preserving real kinematics coiling process.</p><p><strong>II. Problem formulation</strong></p><p>In this section the problem of unrolling thread with length L from a bobbin of radius r is posed while any kind of forces are acting on the unrolled peace of thread. Moreover, the law of bobbin rotation φ(t) assumed to be known with the proviso that the model can be modified if φ(t) is the result of solving of ordinary differential equation.</p><p><strong>III. Mathematical model</strong></p><p>In this section the detailed description of proposed mathematical model based on conventional spring-mass model is given.</p><p>The major feature of this model is that on the every integration step all nodes divided into three groups:</p><p> nodes on the bobbin (linked) – their motion is completely determined by the bobbin rotation;</p><p> free nodes – descended from the bobbin and not neighboring nodes on the bobbin;</p><p> boundary nodes - descended from the bobbin neighboring nodes on the bobbin.</p><p>For each nodes group the right parts of equations of motions are different. In these equations the additional corrections are included that result in reducing the impact of discretization on simulation results. These corrections are:</p><p> equations of motion for boundary nodes eliminate unphysical reaction components, resulting from discretization;</p><p> when a node descends from the bobbin its velocity vector turns following motion of already descended nodes, thus simulating the thread bending of very small radius at the point of thread descending.</p><p><strong>IV. Numerical experiment</strong></p><p>The developed mathematical model is tested using the simplified problem of two-blades rotary solar sail deploying from the «BMSTU-Sail» project [2].</p><p>The calculations demonstrated that the proposed mathematical model satisfies the law of conservation of angular momentum. It is also shown that calculation results are almost independent on the utility parameters of mathematical model (e.g. damping coefficient of nodes relative oscillations).</p><p><strong>V. Conclusion</strong></p><p>Major conclusions in this paper are:</p><p> proposed mathematical model satisfies the law of conservation of angular momentum that indirectly confirms its correctness;</p><p> the transverse thread motion is almost independent on the damping and stiffness coefficients, therefore they can be adjusted for calculation stability without accuracy loss;</p><p> based on proposed mathematical model of unrolling a thread from a bobbin the improved model of unrolling from a bobbin ought to be developed for more accurate calculations of solar sail deploying.</p>
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