Nonlinear Effects in the Static and Dynamic Behavior of Beams and Rotor Blades

摘要

Recent conclusions concerning alleged 'effects of transformation sequence' on the static and dynamic behavior of an end-loaded cantilevered, Euler/Bernoulli beam are analyzed and found to be false. The analysis contained in that paper illustrates several erroneous ideas that must be avoided in dynamics analysis in general. Such errors include a violation of the law of impenetrability (i.e., a violation of continuity of the deformation), a failure to obtain equations of motion to the same order of approximation as desired in the solution, a failure to distinguish among rotational kinematical variables of different types, and a failure to properly relate that which is measured in an experiment to that which is calculated with theory. Correct equations containing all nonlinear terms through third degree in the unknowns are developed in the present paper, and the resulting deflections and rotations are shown to be independent of the choice of rotational variables as expected. Another outcome of the present investigation concerns partial differential equations of motion, for flexural-flexural-torsional deformation of a cantilever beam with a tip mass, that contain all nonlinearities through second degree only. When these equations are specialized for static behavior, perturbation analysis reveals that the torsional variable is of higher order than the flexural variables; then, for consistency, all nonlinearities disappear from the flexural equations. It follows that nonlinear equations governing flexural deformation cannot be consistent unless they include all nonlinear terms up through third order. Reprints. (edc)