Formation of a family of trajectories of free spherical motion of a spacecraft as a rigid body, providing reorientation of its axis of dynamic symmetry to a given position

DOI: 10.34759/trd-2021-121-03


Korovaytseva E. A.

Institute of Mechanics Lomonosov Moscow State University, 1, Michurinsky prospect, Moscow, 119192, Russia



The presented work studies opportunities of automatic segmentation method implementation for solving problems of axisymmetric static deforming of soft shells of revolution at large displacements and strains.

Mathematical statement of the problem includes four quasilinear differential equations and nine nonlinear algebraic equations. Geometrical relations of thin shells nonlinear theory are employed herewith, and elastic potentials of various views are being engaged for nonlinear physical relations formulation. The opportunity of composite shell analysis is supposed, which corresponds to the statement of multipoint boundary problem.

Algorithm of the parametric differentiation method is used for the problem solution. The initial relations of the nonlinear boundary problem are being differentiated herewith by the preselected parameter of the problem solving continuation. As the result, a system of interconnected quasilinear boundary and nonlinear initial problems is being formed. The result of the obtained system solving while studying shell deforming under conditions of large displacements and strains depends on simultaneous successful setting of a whole range of computational algorithm parameters. As far as the method of arbitrary search of parameters leads to irrational increasing the time of computer work with the program, the article suggests a reasonable setting of one of the algorithm parameters, namely, the number of segments into which the integration interval of quasi-linear boundary value problem is divided. The said number of segments is being determined at the stage of preprocessing with the help of authored automatic segmentation method, which was successfully applied earlier when solving the linear boundary value problems.

The article presents results of the three problems solutions on soft shell inflation by the pressure, uniformly distributed along meridian, using parameter differentiation method, one of the steps of which is the automatic segmentation method.

Selection of inflation problem of a cylinder from Mooney-Rivlin material fixed at its ends by a roller is stipulated by the existence of this problem analytical solution. Three cases of arbitrary selection of a number of segments, into which the shell is divided, and a case of automatic segmentation are considered. The article shows that it is possible in the last case to obtain an optimal combination of solution accuracy and iteration processes convergence rate. The choice of a hemisphere inflation from neohookean material fixed along the equator by a roller problem is dictated not only by the existence of analytical solution of this problem, but also by the presence of singular coefficients in resolving equation system Jacobi matrix at the problem integration interval. The said feature leads to the fact that in a fairly large vicinity of the shell pole the condition, used as a criterion of shell division into segments, is violated in each point of the meridian. As the result, the number of segments determined by the automatic segmentation method appears to be quite large. However, the problem solution result turned out to be one of the best among the shell segmentation options considered from the viewpoint of both solution accuracy and iteration processes convergence rate. It is worth emphasizing that in both examples under consideration, one of the options of arbitrary setting of segments number lead to a principally unsatisfactory solution result. Thus, the automatic segmentation method application is necessary to obtain correct solution. The problem of inflation of a hinged hemisphere of neohookean material is selected as it does not have analytical solution. As the result of calculations, it was established that automatic segmentation leads to the lowest iteration processes convergence rate among all the shell meridian division considered options. Along with this, with insufficient number of segments the solution obtaining turned out to be impossible. The performed studies revealed that automatic segmentation method application was necessary for the computer groundless actions minimizing concerning the calculation algorithm parameters setting, as well as for the problem solution results accuracy and iteration processes convergence rate increasing. Along with this, when solving problems, which resolving equation system has singular coefficients at some point of integration interval, the said method application requires additional studies, and apparently, application of some auxiliary techniques.


nonlinear deforming, nonlinear boundary-value problem, soft shell, hyperelastic material, segmentation method, parameter differentiation method


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