The method for analyzing mechanics of thin elastic rod systems using the differential approach

This paper proposes a method for analyzing geometrically nonlinear dynamic deformation of rod system (several interconnected rods) using differential model of a thin elastic rod. The deformation of each rod is defined by 12 differential equations with boundary conditions. Boundary conditions for each rod are formed by kinematics, loads and interconnections between rods. Appropriate nonlinear boundary value problem is solved numerically. It needs many times to solve a large system of linear equations with matrix formed by a Jacobi matrix of differential equations, boundary conditions and conditions of interconnections between the rods. On the main diagonal of this matrix the blocks are formed by differential equations and boundary conditions for each rod. These blocks are linked by elements formed by the equations of connection between the rods. The paper proposes the method for solving sparse linear systems with blocks on the main diagonal, while these blocks are related with each other by a small number of equations. The solution of original sparse system is splitted to few solutions of smaller systems (for each block on the main diagonal) and to the solution of SLAE size of equal to number of nonzero lines linking the matrix blocks. Therefore, the proposed method is effective with a comparatively small (compared with size of the original matrix) count of nonzero lines linked with isolated matrix blocks on the main diagonal. The paper presents the solution of the test problems of large displacements loaded frames. The numerical results of the proposed method are compared with the results of the calculation in the finite element program ANSYS. The results of calculations are practically the same; the accuracy of matches depends on models discretization.