On the stability of a free-free beam under axial thrust subjected to directional control

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Abstract

This paper attempts to elucidate the mechanism of the loss of stability of a free-free beam subjected to a direction-controlled thrust. An adequate description of this basic problem does not appear to be available in the current literature. In contradiction to the claims made by several well-known investigations, it is shown here that (1) for a uniform beam under a constant thrust, the first non-zero eigenvalue represents an unstable bending mode, and (2) under directional control, the critical thrust varies with the sign and the magnitude of the control angle. Numerical data that have helped in understanding the problem are included.

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    Through idealizing the flexible missile as a uniform free-free beam under the action of the end thrust, Beal [19] investigated the dynamic stability of the free-free beam subjected to constant and pulse following force and introduced the direction control mechanism following the force. Wu [20] applied the finite element discretization of the adjoint formula to study the stability of the free-free beam under control with the action of the force. Yang and Xi [21] conducted theoretical and experimental studies on any free-free beams of which the cross-sections along their span were impacted.

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    When the control system was included, it was found that the critical follower force magnitude only corresponded to a reduction of the lowest frequency of zero. Wu [33] studied the stability of a free–free beam under a controlled follower force by using finite element discretization with an adjoint formulation. Park and Mote Jr. [27] studied the maximum controlled follower force on a free–free beam carrying a concentrated mass.

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    These instabilities are severe when the frequency of the thrust variation is in the vicinity of longitudinal natural frequencies. Wu [13,14] examined the stability of a free–free beam under axial thrust subjected to directional control using finite element approach. Propulsive thrust of a launch vehicle/missile is a follower force.

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