By George Herrmann
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Extra resources for Dynamic Stability of Structures. Proceedings of an International Conference Held at Northwestern University, Evanston, Illinois
20) Another quantity of considerable interest is the value of the axial force, P, acting along the column at time, i. It is given by the equation P/PE = ξ-UP-e^ (21) Dynamic stability of structures 31 Here Ρ Ε is the Euler l o a d : P^ = EASE = n^EI/L^ (22) where Λ is the cross-sectional area and / the moment of inertia of the cross section of the column. As it was assumed in the derivation of this equation that the curvature of the column could be represented by d^y/dx^, the equation is vahd only for moderately large lateral displacements; they should not exceed, say, one-twentieth of the length of the column.
N A R D O , S H A R A D A . PATEL and N . J. H O F F , A n Experimental Investigation of the         Maximum Loads Supported by Elastic Columns in Rapid Compression Tests, Polytechnic Institute of Brooklyn Department of Aeronautical Engineering PIBAL Report N o . 296 (1955). EUGENE SEVIN, O n the elastic bending of columns due to dynamic axial forces including effects of axial inertia, J. , 27, p. 125 (1960). A. S. VOL'MIR, O b ustoychivosti cilindricheskikh obolochek pri dinamicheskom nagruzhenii, Dokladi Akademii Nauk.
This is, however, not true since what is proved is not that the assumption of small perturbations in unjustified, but that neglect of higher order terms compared with linear terms is unjustified. Sommerfeld treats in detail a special case of this kind—the sleeping top—in which it is possible to solve explicitly the full nonlinear p r o b l e m ; the result is the conclusion that the motion is stable, but the small oscillation criterion leads to large perturbations and hence to the wrong conclusion.