Direct and inverse dynamics of a very flexible beam

The problem of direct and inverse dynamics of a very flexible plane beam was discussed and analyzed. Direct dynamics is referred to as an ordinary dynamics problem, where initial conditions are known, and where three conditions (either forces or motion) are prescribed at each end of the beam as boundary conditions. Partially or fully inverse dynamics are defined to be the case when some or all of the conditions are moved from one end to the other, so that one end is overprescribed, including both forces and motion, and the constraints at the other end of the beam are partially or fully unspecified. A new model for direct and inverse dynamics of a straight or curved planar beam at small strains and large deflections has been derived. Lagrange equations in a local frame of reference and a finite element method were utilized for the formulation. The direct dynamics problems were solved by a fast and relatively simple linear iteration method, and verified by results of existing programs for specific cases. This formulation is particularly useful for more complicated constitutive relations. On the other hand, inverse dynamics solutions were stable for only short periods of time. This result demonstrates the need for further research in the area of inverse dynamics. Practical application of the study may be considered in the field of robotics, where very flexible beams may be used in order to reduce the number of arms and actuators in a robotics system.