Dynamics of multilink rod system with constraints: a plane problem in finite element formulation

In this paper, the dynamic of a structure composed of flexible rod elements connected via hinges is modeled. It is assumed that the hinges have constraints - rigid and non-rigid, controlled and uncontrolled ones. Mathematically, they are considered as differential ones in integrable or non-integrable forms. Mathematical model is formulated based on the finite element method taking into account finite deformations and the nonlinearity of the inertial forces. The rod element ends are considered to be connected with rigid bodies whose dimensions are small relative to the element length. Each finite element is associated with a local coordinate system for which the displacements, angles of rotation, the translational and rotational speed are strictly considered. Shape functions are taken as quasi-static approximations of local displacement and rotation angles of element cross-sections. Absolute displacements and rotation angles of element boundary cross-sections are taken as generalized coordinates of the problem. The dynamic equations are obtained using d'Alembert-Lagrange principle. It is considered that the generalized coordinates are subjected to the linear relations relative to the generalized velocities. Variation of the problem functional for which to look for the steady-state value is transformed by the addition of the constraint equations multiplied by the undefined Lagrange multipliers. The variational problem for the transformed functional is solved as a free. The stationarity conditions together with the differential equations of constraints determine the desired values of the generalized coordinates. This paper proposes an approach that allows to avoid cumbersome calculations of the nonlinear inertial members without simplification of the physical model and (or) changing the original structure of equations. An example of deploying rod system consisting of three flexible rods connected in series via hinges is considered. The solution of nonlinear dynamic equations is obtained numerically using the integral curve length parameter as a problem argument. This transformation gives a system of resolving equations the best conditioning of the numerical solution process.