Geometrically Nonlinear Formulations of Beams in Flexible Multibody Dynamics

[+] Author and Article Information
J. Mayo, J. Dominguez

Department of Mechanical Engineering, University of Seville, Av. Reina Mercedes s/n, 41012 Seville, Spain

A. A. Shabana

Department of Mechanical Engineering, University of Illinois at Chicago, 842 West Taylor St., Chicago, IL 60607-7022

J. Vib. Acoust 117(4), 501-509 (Oct 01, 1995) (9 pages) doi:10.1115/1.2874490 History: Received October 01, 1993; Revised April 01, 1994; Online February 26, 2008


In this paper, the equations of motion of flexible multibody systems are derived using a nonlinear formulation which retains the second-order terms in the strain-displacement relationship. The strain energy function used in this investigation leads to the definition of three stiffness matrices and a vector of nonlinear elastic forces. The first matrix is the constant conventional stiffness matrix; the second one is the first-order geometric stiffness matrix; and the third is a second-order stiffness matrix. It is demonstrated in this investigation that accurate representation of the axial displacement due to the foreshortening effect requires the use of large number or special axial shape functions if the nonlinear stiffness matrices are used. An alternative solution to this problem, however, is to write the equations of motion in terms of the axial coordinate along the deformed (instead of undeformed) axis. The use of this representation yields a constant stiffness matrix even if higher order terms are retained in the strain energy expression. The numerical results presented in this paper demonstrate that the proposed new approach is nearly as computationally efficient as the linear formulation. Furthermore, the proposed formulation takes into consideration the effect of all the geometric elastic nonlinearities on the bending displacement without the need to include high frequency axial modes of vibration.

Copyright © 1995 by The American Society of Mechanical Engineers
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