Method for a Fast and Simple Dynamic Analysis of 2D and 3D Mechanisms

Abstract

Here, an approach to describe the dynamics of 2D and 3D mechanisms that leads to a simpleand fast analysis is presented. Any position of an elastic mechanism may be defined usingonly rotations. Because in space the finite angles are not vectors, the Euler–Rodriguesparameters were adopted to describe the 3D rotations. The assembling process of the links intothe whole mechanism is natural and follows the standard finite element scheme. Lagrange multipliersare used only to impose the closed loop conditions. The approach developed in this work can be applied to 3D mechanisms composed by beams and having internal hinges as kinematics pairs, but itcan be generalized to other link shapes. The method is used either for mechanisms with rigid links,or with elastic ones, even for very deformable links that completely change the initial configuration.Both the floating and the absolute reference frame approach may be used, depending on the problem;passing from one formulation to the other is quite natural.

If the links may be considered beams, the method starts from the exact equations written forthe deformed shape of each link and this provides a good accuracy. In this paper, a special finiteelement is presented: the unknowns of the problems being the nodal rotations or nodal Euler–Rodriguesparameters. Few nodes are requested for good accuracy. In general, as the number degrees of freedomper node is smaller than in the classical finite element approach, an important reducing of the totalnumber of nodal unknowns is obtained leading to an important reducing of the computer time. TheEuler–Bernoulli beam model was adopted, but the implementation of the Timoshenko beam model thattakes the shear efforts into account, is not difficult.