On the Behavior of Linear Undamped Elastic Systems Perturbed by Follower Forces,
BROWN UNIV PROVIDENCE R I LEFSCHETZ CENTER FOR DYNAMICAL SYSTEMS
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This problem, known as Becks Problem is one of many non-conservative problems encountered in the theory of elastic structures the problem is called nonconservative in that the total energy is not necessarily constant given the follower nature of the load. In the more familiar Euler problem in which the load p is always in the vertical direction energy is conserved as is well known, a bifurcation of static solutions occurs at a critical value of p, the so-called Euler static buckling load. No such phenomenon arises in the case of Becks problem bifurcation of static solutions does not occur. Rather, as the load p is increased, a critical value is surpassed and certain oscillatory solutions no longer remain bounded i.e., dynamic instability occurs. This phenomenon has been investigated in the engineering literature through the use of separation of variables and subsequent analysis of the eigenvalues corresponding to the first few eigenvectors. In this manner, a critical value of the load p is determined below which the motions described by these eigenvectors are bounded, whereas for p above this value at least one such motion is unbounded. For Becks problem, the critical value of p thus obtained is approximately 20.05. Since, in the stable case, all eigenvalues are purely imaginary, such an analysis in not necessarily conclusive.
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