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Nonlinear dynamics of a plane elastic rod system in a reduced quasi-static bending formulation

Russkikh S.V.

#### Abstract:

The nonlinear dynamics of a flat elastic rod system is considered, which consists of an arbitrary number of elastic inextensible rods connected at the ends by elastic-viscous hinges allowing large relative angles of rotation. The displacements of each rod are described by its final rotation as a rigid body relative to a straight line connecting two adjacent hinge nodes, and a bend with a small transverse displacement. Active control of the system is carried out with the help of horizontal and vertical forces applied in the hinge nodes. The equations of motion of a composite system with an arbitrary number of core elements in a fixed coordinate system are based on the principle of possible displacements and are presented in the form of finite formulas convenient for numerical integration using standard programs and algorithms implemented in computer algebra languages. The reduction of the initial system of equations is carried out by quasi-static bending by neglecting the inertia of the bending forms of the movement of the rods and excluding generalized coordinates representing these forms, which are the angles between the tangent to the curved axis of the rod and its undeformed axis. Thus, “fast variables” are excluded from the equations of motion of the system. An algorithm for converting the initial equations into reduced system equations for an arbitrary number of core elements of the system is presented. An example of a numerical solution of the problem of the reaction of a rod system to an arbitrary perturbing pulse in full and reduced formulations is considered. Comparisons are made and estimates of the accuracy and complexity of numerical integration are given when considering a complete system of nonlinear differential equations and equations of a reduced system.

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