Summary We consider the problem of optimal placement of concentrated masses along a massless elastic column that is clamped at one end and loaded by a nonconservative follower force at the free end. The goal is to find the largest possible interval such that the variation in the loading parameter within this interval preserves stability of the structure. The stability constraint is nonconvex and nonsmooth, making the optimization problem quite challenging. We give a detailed analytical treatment for the case of two masses, arguing that the optimal parameter configuration approaches the flutter and divergence boundaries of the stability region simultaneously. Furthermore, we conjecture that this property holds for any number of masses, which in turn suggests a simple formula for the maximal load interval for $n$ masses. This conjecture is strongly supported by extensive computational results, obtained using the recently developed open-source software package granso (GRadient-based Algorithm for Non-Smooth Optimization) to maximize the load interval subject to an appropriate formulation of the nonsmooth stability constraint. We hope that our work will provide a foundation for new approaches to classical long-standing problems of stability optimization for nonconservative elastic systems arising in civil and mechanical engineering.

中文翻译：

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寻找具有随动载荷和可重新定位的集中质量的最强稳定无质量柱

总结 我们考虑集中质量沿着一端夹紧并在自由端由非保守从动力加载的无质量弹性柱的最佳放置问题。目标是找到最大可能的区间，以使该区间内加载参数的变化保持结构的稳定性。稳定性约束是非凸的和非光滑的，使得优化问题非常具有挑战性。我们对两个质量的情况进行了详细的分析处理，认为最佳参数配置同时接近稳定区域的颤振和发散边界。此外，我们推测该属性适用于任意数量的质量，这反过来又提出了一个简单的公式来计算 $n$ 质量的最大负载间隔。这个猜想得到了广泛的计算结果的强烈支持，该结果是使用最近开发的开源软件包 granso（基于梯度的非平滑优化算法）在适当制定非平滑稳定性约束条件下最大化加载间隔。我们希望我们的工作将为解决土木和机械工程中出现的非保守弹性系统的经典长期稳定性优化问题提供新方法的基础。