Abstract
Structural optimization of non-conservative systems with respect to stability criteria is a research area with important applications in fluid-structure interactions, friction-induced instabilities, and civil engineering. In contrast to optimization of conservative systems where rigorously proven optimal solutions in buckling problems have been found, for nonconservative optimization problems only numerically optimized designs have been reported. The proof of optimality in non-conservative optimization problems is a mathematical challenge related to multiple eigenvalues, singularities in the stability domain, and non-convexity of the merit functional. We present here a study of optimal mass distribution in a classical Ziegler pendulum where local and global extrema can be found explicitly. In particular, for the undamped case, the two maxima of the critical flutter load correspond to a vanishing mass either in a joint or at the free end of the pendulum; in the minimum, the ratio of the masses is equal to the ratio of the stiffness coefficients. The role of the singularities on the stability boundary in the optimization is highlighted, and an extension to the damped case as well as to the case of higher degrees of freedom is discussed.
Highlights
Structural optimization of conservative and non-conservative systems with respect to stability criteria is a rapidly growing research area with important applications in industry [1,2,3].Optimization of conservative elastic systems such as the problem of the optimal shape of a column against buckling is already non-trivial, because some optimal solutions could be multi-modal and correspond to a multiple semi-simple eigenvalue which creates a conical singularity of the merit functional [3]
It is known that mass and stiffness modification can increase the critical flutter load by hundreds percent, which is an order of magnitude higher than typical gains achieved in optimization of conservative systems [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]
Temis and Fedorov [17] found for a free-free beam moving under the follower thrust an optimal design with a critical flutter load that exceeds the load for a uniform beam by 823 %
Summary
Structural optimization of conservative and non-conservative systems with respect to stability criteria is a rapidly growing research area with important applications in industry [1,2,3]. We note that the very notion of the follower forces was debated in [20,21,22], the Beck column [19] as well as its discrete analogues [23,24,25] including the Ziegler pendulum [26], remain popular models for investigating mode-coupling instabilities in non-conservative systems and related optimization problems. The known optimized shapes of the Beck column or of a free-free rod moving under follower thrust have an almost vanishing cross-section, e.g., at the free end, which means vanishing mass of a finite element in the corresponding discretization [9,10,11,12, 17, 18] Another intriguing feature of optimizing non-conservative systems is the ‘wandering’ critical frequency at the optimal critical load. We formulate an optimization problem for an m-link Ziegler pendulum and discuss some hypotheses on plausible optimal solutions and their expected properties
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