Study of the natural frequencies for two- and three-dimensional curved beams under rotational movement

Abstract

Dynamic analysis of highly elastically deformed bodies subjected to rotational movement is very important from a practical point of view: the centrifugal forces may considerably modify the body configuration. The classical large displacement theory is complicated and it leads to very time-consuming computer codes. In this work, a simple numerical method to study the dynamics of 3D beam systems is presented. The computing of the eigenfrequencies, taking into consideration the centrifugal stiffening effect even for very large displacements of elastic beams, is also developed. The elastic deformed configuration of one bent beam may be uniquely expressed only in rotations, or alternative parameters such as the Euler-Rodrigues quaternion, as functions of a curvilinear coordinate. The equations for the Euler-Bernoulli 3D beam model, written for the deformed beam, are solved. Consequently the equations are exact even for really large displacements. This approach leads to a non-incremental method and to a fast and accurate approach easy to program for computer applications. Passing from the absolute nodal coordinate formulation to the floating reference frame formulation becomes very easy, even natural.