Modeling of Closed Kinematic Chains with Flexible Links

A way is presented of modeling complex mechanical systems, and particularly closed kinematic chains, that are comprised of elastic components. The motion of each link is viewed using a moving reference frame and the zero tip deformation constraint is introduced to define the orientation of the reference frames. That analysis is combined with the differential-algebraic formulation and the equation of motion is given in constrained form. We investigate the effect of the shortening of the projection and show that ignoring it can lead to gross inaccuracies. Introduction Modeling of manipulators and closed kinematic chains that have flexible links has been considered by many researchers. In order to simplify partial differential equations associated with flexible links some of them used assumed modes technic to derive equations of motion [1][4]. Others used discretisation of the links like finite element method [5]-[8], or finite segment modeling combined with Kane’s equation formulation [9]. Some of the earliest work was done by Neubauer at. al. [10] where they derived equations of motion by using both assumed modes method and finite element method. They considered slider-crank mechanism and only transverse vibrations of the connecting rod. Since modeling of flexible dynamical systems is very complex most of the early papers considered only transverse vibrations [1],[4],[6], and/or only one link was flexible [2],[3],[4],[6],[10]. Introduction of powerful computers enables researchers to consider both axial and transverse vibrations, and allow all links to be flexible [5],[7],[8],[9]. In this paper, we consider dynamical systems that are comprised of elastic linkages. We write the equations of motion using constrained generalized coordinates and we describe the elastic motion of the linkages in terms of moving coordinate systems. Most of the papers [1], [3]-[10] neglect coupling effect between axial and transverse vibrations ( i.e,. shortening of the projection ). Therefore, we analyze the characteristics of the response as a function of the shortening of the projection and variety of other factors. Using constrained generalized coordinates to derive the equations of motion is more suitable than using a set of independent generalized coordinates in a variety of circumstances. For example, in a closed kinematic chain it is difficult to obtain a set of independent generalized coordinates and to express the remaining coordinates in terms of the independent coordinates [10]. Barring a few exceptions, the same situation is encountered even when a method that can handle nonholonomic constraints by means of generalized speeds is used. 1 Associate Professor 2 Graduate Assistant