MORE ON THE SO-CALLED DYNAMIC STIFFENING EFFECT

The geometric or dynamic stiffening effect, a long known phenomenon, a long with other related phenomena are re-examined in detail. It is sho wn that, in general, for any system with first-order degrees of freedo m (DOFs), e.g., elastic vibrations, which also undergoes zero-order mo tion, i.e., rigid body motion, certain terms might be missed in the eq uations of motion if the first-order DOFs are expressed as linear comb inations of the generalized coordinates. It is also shown that if the rigid body motions (zero-order) are prescribed, the stiffness matrix i s the only part of the equation that might suffer. However, in the cas e of a system for which the zero-order motion is not a prescribed one, but involves degrees of freedom, certain blocks of the generalized ma ss matrix and force vector might also lose some terms. Kane's method i s used to generate the equations necessary in the analysis, which prec isely identifies which terms will be missed in a general case. Three o f the most popular approaches for circumventing this problem of missin g terms are discussed, and retaining the second order terms in the exp ressions for elastic displacements is recommended as the best approach . A novel method based on the use of nonlinear strain-displacement kin ematics is presented which can be employed to generate the correct (up to the first order) form of the equations of motion for a general ela stic system undergoing arbitrary large rigid-body motion. In this meth od the strains, not the displacements, are expressed as linear combina tions of the generalized coordinates, Specializations of the method fo r beams and plates with some examples are presented to shed some light on the theoretical discussions.