Shape Optimization of Nonconservativly Loaded Beamswith a Stability Criterion

Temis, Yu. M.; Fedorov, Ilya

doi:10.32326/1814-9146-2007-69-1-24-37

Cited by 3 publications

(8 citation statements)

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“…Note that in [11] a local ex-tremum of the flutter load for the free-free beam carrying a point mass was found to be at the cuspidal point on the flutter boundary too. The global maximum of the critical flutter load for the undamped Ziegler's pendulum is at infinity when m 2 = 0.The global maximum corresponds to a vanishing mass at the free end of the column which qualitatively is in agreement with the numerically found optimized designs of the Beck's column available in the literature[9,13,14,15,16,17,18]. Indeed, all known optimized designs of the Beck's column are characterized by the vanishing cross-sections at the free end.…”

supporting

confidence: 83%

“…2. The most interesting is the fact that with the increase of the critical flutter load the higher and higher modes were reported to be involved into the coupling that indicates the onset of flutter [9,13,14,15,16,17,18]. Our simple model shows that this phenomenon seems to be natural for the optimal design that causes the degeneracy in the mass matrix that gives rise to the critical frequency that increases without bounds.…”

Section: Divergence: Both Modesmentioning

confidence: 74%

“…Usually, in the structural optimization problems the uniformly distributed stiffness and mass are considered as the initial design that is a starting point in optimization procedures. The critical load of the optimized structure is conventionally compared to that of the same structure with the uniform mass and stiffness distributions [4,5,6,7,8,9,10,13,14,15,16,17,18].…”

Section: Divergence: Both Modesmentioning

confidence: 99%

“…For example, Ringertz [9] reported an 838% increase of the critical flutter load for the Beck's column [19] from 20.05 for a uniform design to 188.1 for an optimized shape. Recently Temis and Fedorov [17] found for a free-free beam moving under the follower thrust an optimal design with the critical flutter load that exceeds that for a uniform beam by 823%. We note that despite the very notion of the follower forces was debated in [20,21,22], the Beck's column [19] as well as its discrete analogues [23,24,25] including the Ziegler's pendulum [26] remain popular models for investigation of mode-coupling instabilities in non-conservative systems and related optimization problems.…”

Section: Introductionmentioning

confidence: 99%

“…Another intrigue of optimization of non-conservative systems is the 'wandering' critical frequency at the optimal critical load. During the optimization the eigenvalue branches experience numerous mutual overlappings and veerings [4,6,7,27,28,29] with the tendency for the critical frequency to increase and to correspond to higher modes [9,14,15,16,17,30]. This puzzling behavior of the critical frequency still waits for its explanation.…”

Section: Introductionmentioning

confidence: 99%

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Re‐visiting structural optimization of the Ziegler pendulum: singularities and exact optimal solutions

Kirillov

2011

Proc Appl Math and Mech

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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 non-conservative optimization problems only numerically optimized designs were reported. The proof of optimality in the non-conservative optimization problems is a mathematical challenge related to multiple eigenvalues, singularities on the stability domain, and non-convexity of the merit functional. We present a study of the optimal mass distribution in a classical Ziegler's 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 extension to the damped case as well as to the case of higher

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supporting

confidence: 83%

Section: Divergence: Both Modesmentioning

confidence: 74%

Section: Divergence: Both Modesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Re‐visiting structural optimization of the Ziegler pendulum: singularities and exact optimal solutions

Kirillov

2011

Proc Appl Math and Mech

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Finding the strongest stable massless column with a follower load and relocatable concentrated masses

Kirillov

Overton

2021

The Quarterly Journal of Mechanics and Applied Mathematics

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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.

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Optimization of compressed wooden bars of variable section according to the criterion of maximum critical load

Shorstov

Yazyev

Чепурненко

2023

Construction and Architecture

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The article presents a numerical and analytical solution to the problem of optimizing compressed rods according to the criterion of the maximum critical load with a possible loss of stability in two planes. We consider a wooden bar of rectangular cross section, the height of which varies according to a linear law, and the width is constant. The calculation is performed in an elastic formulation. A constraint on the constancy of the rod mass is introduced. Variable parameters are the ratio of the minimum section height to the maximum, as well as the ratio of the section width to the maximum height. Numerical search for the optimal solution is performed in the MATLAB environment using the method of sequential quadratic programming. It has been established that the use of rods with a cross-sectional height that varies linearly along the length is effective only if the rod is fixed in at least one of the planes according to the “pinched-free end” scheme. It has been established that the increase in the critical load when using rods with a linearly varying cross-sectional height can be up to 22%.

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Shape Optimization of Nonconservativly Loaded Beamswith a Stability Criterion

Cited by 3 publications

References 8 publications

Re‐visiting structural optimization of the Ziegler pendulum: singularities and exact optimal solutions

Re‐visiting structural optimization of the Ziegler pendulum: singularities and exact optimal solutions

Finding the strongest stable massless column with a follower load and relocatable concentrated masses

Optimization of compressed wooden bars of variable section according to the criterion of maximum critical load

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