A new formulation for the direct dynamic simulation of flexible mechanisms based on the Newton–Euler inverse method

This paper is concerned with the direct dynamic analysis of mechanisms comprising flexible members. It is first recalled how the time integration of the dynamic equations governing the motion of a flexible mechanism is conditioned by the availability of the so-called iteration matrix. The latter is obtained not only from the mass matrix, which is sufficient in the rigid case, but also from the stiffness and damping tangent matrices. Hence, in a context based upon relative co-ordinates where the deformation of the flexible bodies is described by component modes, two formulations, a classical one and a new one, which permit the determination of this iteration matrix are presented. The classical formulation is based upon the Lagrange's theorem. The new one consists in a direct calculation of the residues of the dynamic equations by using the inverse method of Newton–Euler. In this case, the iteration matrix is obtained by a numerical derivation technique which proves well-conditioned. A first simple example is treated in order to compare the two formulations. A more complex one illustrates the real capabilities of the new formulation.