We consider a problem of optimal placement of concentrated masses along a weightless elastic column (rod) 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. When the number of masses is two we find the optimal load analytically. For the case of three or more masses, we obtain conjectures for the globally optimal load that are strongly supported by extensive computational results . To obtain these, we use techniques from nonsmooth optimization, employing the recently developed open-source software package GRANSO (GRadient-based Algorithm for Non-Smooth Optimization) to maximize the load subject to appropriate nonsmooth stability constraints .
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 .
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weightless elastic column (rod)
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When the number of masses is two we find the optimal load analytically. For
The stability constraint is nonconvex and nonsmooth, making the optimization problem quite challenging. We give a detailed analytical treatment for
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three or more masses, we obtain conjectures for the globally optimal load that are
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
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. To obtain these, we use techniques from nonsmooth optimization, employing
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subject to appropriate nonsmooth stability constraints
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