Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc. A98-39914 AIAA-98-4969 OPTIMIZATION OF STABILITY OF A FLEXIBLE MISSILE UNDER FOLLOWER THRUST Oleg N. Kirillov* and Alexander P. Seyranian" The Lomonosov Moscow State University, Moscow, Russia ABSTRACT This paper addresses two formulations of a dy- namic problem of structural optimization. We con- sider a beam moving in space under a tangential end force as an idealization of a flexible missile. This nonconservative system can lose stability under a certain critical end force either by flutter or by diver- gence. That depends on the mass and/or stiffness dis- tributions of the beam. We first consider a non-uniform beam supposing its cross-sections are similar geometric figures. We are searching for an optimal mass distribution of the beam with the constant volume constraint in the sense of maximization of the critical end force. In the sec- ond formulation of the problem we study a uniform beam carrying a nonstructural mass. In this case our goal is to find an optimal distribution of the non- structural mass with constant volume constraint. For the first problem the mass distribution of the beam with the critical flutter load p* « 290 is ob- tained. In the second case it is shown with the use of Pontryagin's maximum principle that optimal solu- tions belong to the «bang-bang» type. Optimal distri- butions of nonstructural mass with two and four switching points are presented. I. INTRODUCTION The stability of a uniform beam moving in space under a tangential end force, as an idealization of a flexible missile, has been investigated first by Gopak [1], Feodosiev [2], Beal [3], and Goroshko [4]. The separated dimensionless differential equation of this problem describing transverse vibrations of the beam s .IV (1.1) with the boundary conditions \x=0 * Postgraduate-student, Department of Applied Mechanics, The Lomonosov Moscow State University. ** Professor, Institute of Mechanics, The Lomonosov Moscow State University. Copyright © 1998 by O. N. Kirillov and A. P. Seyranian. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. =1 =0, M '"| ;t=1 =0, (1.2) as shown in detail in [1-4]. Primes denote differentia- tion with respect to x. It has been found that this system loses stability by flutter under a critical end force p* = 109.69 [2]. Sundararajan [5] considered the problem of op- timal arrangement of nonstructural mass along such a beam. He plotted graphs showing dependence of the critical load on the displacement of the concentrated mass along the beam and found the optimal point. But this result seems to be not accurate since he used only two modes in Bubnov-Galerkin approximation. Sundararajan [5] also attempted to find an optimal continuous nonstructural mass distribution. In this paper we use and develop further the ideas and methods presented in Seyranian and Sha- ranyuk [8], Pedersen and Seyranian [9], and Seyra- nian [10]. II. BASIC RELATIONS Consider a flexible beam moving in space under a tangential end force with non-uniform cross- sections. It is assumed that the beam carries a non- structural mass. This system is described by the fol- lowing equations and boundary conditions [4]: mU-(Q(s)U') +(£/£/")" =0 where: M= \m\ o * w jl/j-w»*» (21) E/C/"U=0,(£/t/")' s=0 =0, EJU"\ s=l =0,(EJU")'\ s=l =0, I is the length of the beam, ", - the total mass of the beam, P - the follower force, 2063 American Institute of Aeronautics and Astronautics Downloaded by Helmholtz-Zentrum Dresden-Rossendorf, Bibliothek on September 27, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.1998-4969