Dynamic behavior of a translating flexible beam with a prismatic joint

The dynamic modeling of a single-link flexible beam having both rotational and translation motions and the effects of higher-order dynamics on the response of the beam are investigated. In the literature, modeling of flexible link has often been carried out with the assumed modes technique. In this approximation, only two or three modes are retained and the rest are truncated, which in many cases is inadequate for high performance control system development. In this study, an analytical model of the beam, characterized by an infinite number of modes, is developed using Euler-Bernoulli beam equation and modal expansion method. The infinite-dimensional transcendental transfer function for the manipulator is formulated without modal approximation that is conveniently transformed into state-space form. A preliminary study of the beam acceleration is conducted through computer simulations to demonstrate the effects of higher-order dynamics. In this approach, adding modes or increasing the order of transfer function is straightforward and provides a straightforward check of the effect of ignoring higher-order terms. A numerical example is presented to illustrate the results.