16 research outputs found
Aimed control of the frequency spectrum of eigenvibrations of elastic plates with a finite number of degrees of freedom of masses by superimposing additional constraints
As is known, for some elastic systems with a finite number of degrees of freedom of masses, for which the directions of motion of the masses are parallel and lie in the same plane, methods have been developed for creating additional constraints that purposefully change the spectrum of natural frequencies. In particular, theory and algorithm for the formation of aimed additional constraints have been developed for the rods, the introduction of each of which does not change any of the modes of natural vibrations, but only increases the value of only one frequency, without changing the values of the remaining frequencies. The distinctive paper is devoted to the method of forming a matrix of additional stiffness coefficients corresponding to such aimed constraint in the problem of natural vibrations of rods. This method can also be applied to solving a similar problem for elastic systems with a finite number of degrees of freedom, in which the directions of motion of the masses are parallel, but not lie in the same plane. In particular, such systems include plates. However, the algorithms for the formation of aimed additional constraints, developed for rods and based on the properties of rope polygons, cannot be used without significant changes in a similar problem for plates. The method for the formation of design constraint schemes that purposefully change the spectrum of frequencies of natural vibrations of elastic plates with a finite number of degrees of freedom of masses, will be considered in the next work. Β© 2021, ASV Publishing House. All rights reserved
ΠΡΠΈΡΠ΅ΡΠΈΠΉ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΠΏΠΎΠ»ΠΊΠΈ ΡΡΠ΅ΡΠΆΠ½Ρ Π΄Π²ΡΡΠ°Π²ΡΠΎΠ²ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΏΡΠΈ Π²Π°ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ Π΅Π΅ ΡΠΎΠ»ΡΠΈΠ½Ρ ΠΈ ΠΎΡΠ΅ΡΡΠ°Π½ΠΈΡ ΡΠΈΡΠΈΠ½Ρ ΠΏΡΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ Π½Π° Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΈΠ»Ρ ΠΈΠ»ΠΈ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΡΡ ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ Π² Π΄Π²ΡΡ Π³Π»Π°Π²Π½ΡΡ ΠΏΠ»ΠΎΡΠΊΠΎΡΡΡΡ ΠΈΠ½Π΅ΡΡΠΈΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ
We have already presented original criterion of minimum material consumption within the design of the outline of the width of the I-shaped bar and the stability constraints or restriction to the value of the first natural frequency in one principal plane of inertia of the cross-section. This paper is devoted in its turn to a criterion for the minimum material capacity of the I-shaped bar with a variation in its thickness and outline of the width, with restrictions to the value of the critical force or restriction to the value of the first natural frequency in two principal planes of inertia of the section.Π Π°Π½Π΅Π΅ Π±ΡΠ» ΡΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²Π°Π½ ΠΊΡΠΈΡΠ΅ΡΠΈΠΉ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΠΏΡΠΈ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΠΎΡΠ΅ΡΡΠ°Π½ΠΈΡ ΡΠΈΡΠΈΠ½Ρ ΠΏΠΎΠ»ΠΎΠΊ ΡΡΠ΅ΡΠΆΠ½Π΅ΠΉ Π΄Π²ΡΡΠ°Π²ΡΠΎΠ²ΠΎΠ³ΠΎ ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ
ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈΠ»ΠΈ Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ Π² ΠΎΠ΄Π½ΠΎΠΉ Π³Π»Π°Π²Π½ΠΎΠΉ ΠΏΠ»ΠΎΡΠΊΠΎΡΡΠΈ ΠΈΠ½Π΅ΡΡΠΈΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ. Π Π΄Π°Π½Π½ΠΎΠΉ ΡΠ°Π±ΠΎΡΠ΅ ΡΠΎΡΠΌΡΠ»ΠΈΡΡΠ΅ΡΡΡ ΠΊΡΠΈΡΠ΅ΡΠΈΠΉ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΠΏΠΎΠ»ΠΊΠΈ ΡΡΠ΅ΡΠΆΠ½Ρ Π΄Π²ΡΡΠ°Π²ΡΠΎΠ²ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΏΡΠΈ Π²Π°ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ Π΅Ρ ΡΠΎΠ»ΡΠΈΠ½Ρ ΠΈ ΠΎΡΠ΅ΡΡΠ°Π½ΠΈΡ ΡΠΈΡΠΈΠ½Ρ ΠΏΡΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ
Π½Π° Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΈΠ»Ρ ΠΈΠ»ΠΈ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ Π² Π΄Π²ΡΡ
Π³Π»Π°Π²Π½ΡΡ
ΠΏΠ»ΠΎΡΠΊΠΎΡΡΡΡ
ΠΈΠ½Π΅ΡΡΠΈΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ
Assessment of the proximity of design to minimum material capacity solution of problem of optimization of the flange width of i-shaped cross-section rods with allowance for stability constraints or constraints for the value of the first national frequency and strength requirements
There are known methods for optimizing the flange width of I-shaped cross-section rods with stability constraints or the constraints for the value of the first natural frequency. Corresponding objective function has the form of the volume of the flange material for the case when only the flange width varies and the cross-section height, wall thickness and flange thickness are specified. Special criterion for assessment of proximit y of corresponding an optimal solution to the design of minimal material capacity was formulated for the considering problem. In this case, the resulting solution may not meet some other unaccounted constraints, for example, strength requirements. Modification of solution in order to meet previously unaccounted constraints does not al-low researcher to consider such design as optimal. In the distinctive paper allowance for strength requirements, stability constraints or constraints for the value of the first natural frequency are proposed within considering problem of optimization. Special approach is formulated, which proposes to assess proximity to the design of minimum of material capacity obtained as a result of optimization. Increment of the objective function and criteria corresponding to constrains and restrictions are under consideration within computational process. Β© 2020, ASV Publishing House. All rights reserved
ΠΡΠΈΡΠ΅ΡΠΈΠΉ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΠΏΠΎΠ»ΠΊΠΈ ΡΡΠ΅ΡΠΆΠ½Ρ Π΄Π²ΡΡΠ°Π²ΡΠΎΠ²ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΏΡΠΈ Π²Π°ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ Π΅Π΅ ΡΠΎΠ»ΡΠΈΠ½Ρ ΠΈ ΠΎΡΠ΅ΡΡΠ°Π½ΠΈΡ ΡΠΈΡΠΈΠ½Ρ ΠΏΡΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ Π½Π° Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΈΠ»Ρ ΠΈΠ»ΠΈ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΡΡ ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ Π² Π΄Π²ΡΡ Π³Π»Π°Π²Π½ΡΡ ΠΏΠ»ΠΎΡΠΊΠΎΡΡΡΡ ΠΈΠ½Π΅ΡΡΠΈΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ
We have already presented original criterion of minimum material consumption within the design of the outline of the width of the I-shaped bar and the stability constraints or restriction to the value of the first natural frequency in one principal plane of inertia of the cross-section. This paper is devoted in its turn to a criterion for the minimum material capacity of the I-shaped bar with a variation in its thickness and outline of the width, with restrictions to the value of the critical force or restriction to the value of the first natural frequency in two principal planes of inertia of the section.Π Π°Π½Π΅Π΅ Π±ΡΠ» ΡΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²Π°Π½ ΠΊΡΠΈΡΠ΅ΡΠΈΠΉ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΠΏΡΠΈ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΠΎΡΠ΅ΡΡΠ°Π½ΠΈΡ ΡΠΈΡΠΈΠ½Ρ ΠΏΠΎΠ»ΠΎΠΊ ΡΡΠ΅ΡΠΆΠ½Π΅ΠΉ Π΄Π²ΡΡΠ°Π²ΡΠΎΠ²ΠΎΠ³ΠΎ ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ
ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈΠ»ΠΈ Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ Π² ΠΎΠ΄Π½ΠΎΠΉ Π³Π»Π°Π²Π½ΠΎΠΉ ΠΏΠ»ΠΎΡΠΊΠΎΡΡΠΈ ΠΈΠ½Π΅ΡΡΠΈΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ. Π Π΄Π°Π½Π½ΠΎΠΉ ΡΠ°Π±ΠΎΡΠ΅ ΡΠΎΡΠΌΡΠ»ΠΈΡΡΠ΅ΡΡΡ ΠΊΡΠΈΡΠ΅ΡΠΈΠΉ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΠΏΠΎΠ»ΠΊΠΈ ΡΡΠ΅ΡΠΆΠ½Ρ Π΄Π²ΡΡΠ°Π²ΡΠΎΠ²ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΏΡΠΈ Π²Π°ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ Π΅Ρ ΡΠΎΠ»ΡΠΈΠ½Ρ ΠΈ ΠΎΡΠ΅ΡΡΠ°Π½ΠΈΡ ΡΠΈΡΠΈΠ½Ρ ΠΏΡΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ
Π½Π° Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΈΠ»Ρ ΠΈΠ»ΠΈ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ Π² Π΄Π²ΡΡ
Π³Π»Π°Π²Π½ΡΡ
ΠΏΠ»ΠΎΡΠΊΠΎΡΡΡΡ
ΠΈΠ½Π΅ΡΡΠΈΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ
ΠΡΠΈΡΠ΅ΡΠΈΠΈ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΡ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΏΡΠΈ ΡΠΎΡΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΡΡΠ΅ΡΠΆΠ½Π΅ΠΉ Ρ ΠΊΡΡΠΎΡΠ½ΠΎ-ΠΏΠΎΡΡΠΎΡΠ½Π½ΡΠΌ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ΠΌ ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΡ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΏΡΠΈ ΠΎΡΠ³Π°Π½ΠΈΡΠ΅Π½ΠΈΡΡ ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈΠ»ΠΈ Π½Π° Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ. Π§Π°ΡΡΡ 1: Π’Π΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΎΡΠ½ΠΎΠ²Ρ
The special properties of optimal systems have been already identified. Besides, criteria has been formulated to assess the proximity of optimal solutions to the minimal material consumption. In particular, the criteria were created for rods with rectangular and I-beam cross-section with stability constraints or constraints for the value of the first natural frequency. These criteria can be used for optimization when the cross sections of a bar change continuously along its length. The resulting optimal solutions can be considered as an idealized object in the sense of the limit. This function of optimal design allows researcher to assess the actual design solution by the criterion of its proximity to the corresponding limit (for example, regarding material consumption). Such optimal project can also be used as a reference point in real design, for example, implementing a step-by-step process of moving away from the ideal object to the real one. At each stage, it is possible to assess the changes in the optimality index of the object in comparison with both the initial and the idealized solution. One of the variants of such a process is replacing the continuous change in the size of the cross sections of the rod along its length with piecewise constant sections. Boundaries of corresponding intervals can be selected based on an ideal feature, and cross-section dimensions can be determined by one of the optimization methods. The distinctive paper is devoted to criteria that allow researcher providing reliable assessment of the endpoint of the optimization process.Π Π°Π½Π΅Π΅ Π±ΡΠ»ΠΈ Π²ΡΡΠ²Π»Π΅Π½Ρ ΠΎΡΠΎΠ±ΡΠ΅ ΡΠ²ΠΎΠΉΡΡΠ²Π° ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ ΠΈ ΡΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²Π°Π½Ρ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ, ΠΎΡΠ΅Π½ΠΈΠ²Π°ΡΡΠΈΠ΅ Π±Π»ΠΈΠ·ΠΎΡΡΡ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΊ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΠΌΡ. Π ΡΠ°ΡΡΠ½ΠΎΡΡΠΈ, Π±ΡΠ»ΠΈ ΡΠΎ Π·Π΄Π°Π½Ρ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ, Π΄Π»Ρ ΡΡΠ΅ΡΠΆΠ½Π΅ΠΉ Ρ ΠΏΡΡΠΌΠΎΡΠ³ΠΎΠ»ΡΠ½ΡΠΌ ΠΈ Π΄Π²ΡΡΠ°Π²ΡΠΎΠ²ΡΠΌ ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΠΌ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ΠΌ ΠΏΡΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ
ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈΠ»ΠΈ Π½Π° Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ. ΠΡΠΈ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ ΠΏΡΠΈΠΌΠ΅Π½ΠΈΠΌΡ ΠΏΡΠΈ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ Π² ΡΠ»ΡΡΠ°ΡΡ
, ΠΊΠΎΠ³Π΄Π° ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΡΡΠ΅ΡΠΆΠ½Ρ Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΠΎ ΠΈΠ·ΠΌΠ΅Π½ΡΡΡΡΡ ΠΏΠΎ Π΅Π³ΠΎ Π΄Π»ΠΈΠ½Π΅. ΠΠΎΠ»ΡΡΠ΅Π½Π½ΡΠ΅ ΠΏΡΠΈ ΡΡΠΎΠΌ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΌΠΎΠ³ΡΡ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°ΡΡΡΡ ΠΊΠ°ΠΊ ΠΈΠ΄Π΅Π°Π»ΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΠΉ ΠΎΠ±ΡΠ΅ΠΊΡ Π² ΡΠΌΡΡΠ»Π΅ ΠΏΡΠ΅Π΄Π΅Π»ΡΠ½ΠΎΠ³ΠΎ. ΠΠ°Π½Π½Π°Ρ ΡΡΠ½ΠΊΡΠΈΡ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠΎΠ΅ΠΊΡΠ° ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΠΎΡΠ΅Π½ΠΈΠ²Π°ΡΡ ΡΠ΅Π°Π»ΡΠ½ΠΎΠ΅ ΠΊΠΎΠ½ΡΡΡΡΠΊΡΠΎΡΡΠΊΠΎΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ ΠΏΠΎ ΠΊΡΠΈΡΠ΅ΡΠΈΡ Π΅Π³ΠΎ Π±Π»ΠΈΠ·ΠΎΡΡΠΈ ΠΊ ΠΏΡΠ΅Π΄Π΅Π»ΡΠ½ΠΎΠΌΡ (Π½Π°ΠΏΡΠΈΠΌΠ΅Ρ, ΠΏΠΎ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ). Π’Π°ΠΊΠΎΠΉ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΠΉ ΠΏΡΠΎΠ΅ΠΊΡ ΡΠ°ΠΊΠΆΠ΅ ΠΌΠΎΠΆΠ΅Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡΡΡ ΠΈ ΠΊΠ°ΠΊ ΠΎΡΠΈΠ΅Π½ΡΠΈΡ ΠΏΡΠΈ ΡΠ΅Π°Π»ΡΠ½ΠΎΠΌ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ, Π½Π°ΠΏΡΠΈΠΌΠ΅Ρ, ΡΠ΅Π°Π»ΠΈΠ·ΡΡ ΠΏΠΎΡΡΠ°ΠΏΠ½ΡΠΉ ΠΏΡΠΎΡΠ΅ΡΡ ΠΎΡΡ
ΠΎΠ΄Π° ΠΎΡ ΠΈΠ΄Π΅Π°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ΅ΠΊΡΠ° ΠΊ ΡΠ΅Π°Π»ΡΠ½ΠΎΠΌΡ. ΠΡΠΈ ΡΡΠΎΠΌ Π½Π° ΠΊΠ°ΠΆΠ΄ΠΎΠΌ ΡΡΠ°ΠΏΠ΅ ΠΏΠΎΡΠ²Π»ΡΠ΅ΡΡΡ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΡ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΡ ΠΏΠΎΠΊΠ°Π·Π°ΡΠ΅Π»Ρ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΡΡΠΈ ΠΎΠ±ΡΠ΅ΠΊΡΠ° ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ, ΠΊΠ°ΠΊ Ρ Π½Π°ΡΠ°Π»ΡΠ½ΡΠΌ, ΡΠ°ΠΊ ΠΈ Ρ ΠΈΠ΄Π΅Π°Π»ΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΠΌ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ΠΌ. ΠΠ΄Π½ΠΈ ΠΈΠ· Π²Π°ΡΠΈΠ°Π½ΡΠΎΠ² ΡΠ°ΠΊΠΎΠ³ΠΎ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠΎΡΡΠΎΠΈΡ Π² Π·Π°ΠΌΠ΅Π½Π΅ Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΠΎΠ³ΠΎ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΡ ΡΠ°Π·ΠΌΠ΅ΡΠΎΠ² ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΡΡΠ΅ΡΠΆΠ½Ρ ΠΏΠΎ Π΅Π³ΠΎ Π΄Π»ΠΈΠ½Π΅ ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠΈΠΌΠΈ ΠΊΡΡΠΎΡΠ½ΠΎ-ΠΏΠΎΡΡΠΎΡΠ½Π½ΡΠΌΠΈ ΡΡΠ°ΡΡΠΊΠ°ΠΌΠΈ. ΠΡΠ°Π½ΠΈΡΡ ΡΡΠ°ΡΡΠΊΠΎΠ² ΠΌΠΎΠ³ΡΡ Π²ΡΠ±ΠΈΡΠ°ΡΡΡΡ Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΈΠ΄Π΅Π°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ΅ΠΊΡΠ°, Π° ΡΠ°Π·ΠΌΠ΅ΡΡ ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΡΡΡΡ ΠΎΠ΄Π½ΠΈΠΌ ΠΈΠ· ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ. Π Π½Π°ΡΡΠΎΡΡΠ΅ΠΉ ΡΡΠ°ΡΡΠ΅ ΠΏΡΠ΅Π΄Π»Π°Π³Π°ΡΡΡΡ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠΈΠ΅ Π½Π°Π΄Π΅ΠΆΠ½ΠΎ ΠΎΡΠ΅Π½ΠΈΠ²Π°ΡΡ ΠΌΠΎΠΌΠ΅Π½Ρ ΠΎΠΊΠΎΠ½ΡΠ°Π½ΠΈΡ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠ°ΠΊΠΎΠΉ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ
ΠΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠ΅ ΠΊΡΠΈΡΠ΅ΡΠΈΡ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΡΡΠ΅ΡΠΆΠ½Π΅ΠΉ ΠΏΡΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ Π΄Π»Ρ ΡΠ»ΡΡΠ°Ρ ΠΊΡΠ°ΡΠ½ΠΎΠΉ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠΉ Π½Π°Π³ΡΡΠ·ΠΊΠΈ
As it is known, special criteria are formulated to evaluate the obtained solution of some optimization problems. In particular, we formulate a criterion that allows us to estimate the proximity of the decision on the rod of the lowest weight and the restrictions on the resistance to the minimum material-intensive for rectilinear rods for certain types of cross sections. The criterion is based on the analysis of stresses from bending moments arising from the loss of stability. If the least critical force is not a multiple, then the form of loss of stability and the corresponding diagram of moments are the only ones. At multiplicity of the least critical load there are multiple forms of loss of stability, and any of their linear combination is also its own form. To estimate the obtained solution, it is necessary to form a combination of multiple forms of buckling and the corresponding diagram of bending moments, which will serve as the basis for the use of the criterion. This paper proposes an approach that allows to determine such a combination of multiple forms, which will be the basis for the application of the criterion of proximity of the obtained solution to the minimum material-intensive.ΠΠ°ΠΊ ΠΈΠ·Π²Π΅ΡΡΠ½ΠΎ, Π΄Π»Ρ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ Π½Π΅ΠΊΠΎΡΠΎΡΡΡ
Π·Π°Π΄Π°Ρ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΡΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²Π°Π½Ρ ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΡΠ΅ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ. Π ΡΠ°ΡΡΠ½ΠΎΡΡΠΈ, ΡΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²Π°Π½ ΠΊΡΠΈΡΠ΅ΡΠΈΠΉ, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠΈΠΉ ΠΎΡΠ΅Π½ΠΈΡΡ Π΄Π»Ρ ΠΏΡΡΠΌΠΎΠ»ΠΈΠ½Π΅ΠΉΠ½ΡΡ
ΡΡΠ΅ΡΠΆΠ½Π΅ΠΉ ΠΏΡΠΈ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
ΡΠΈΠΏΠ°Ρ
ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Π±Π»ΠΈΠ·ΠΎΡΡΡ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΎ ΡΡΠ΅ΡΠΆΠ½Π΅ Π½Π°ΠΈΠΌΠ΅Π½ΡΡΠ΅Π³ΠΎ Π²Π΅ΡΠ° ΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ
ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΊ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΠΌΡ. ΠΡΠΈΡΠ΅ΡΠΈΠΉ ΠΎΡΠ½ΠΎΠ²Π°Π½ Π½Π° Π°Π½Π°Π»ΠΈΠ·Π΅ Π½Π°ΠΏΡΡΠΆΠ΅Π½ΠΈΠΉ ΠΎΡ ΠΈΠ·Π³ΠΈΠ±Π°ΡΡΠΈΡ
ΠΌΠΎΠΌΠ΅Π½ΡΠΎΠ², Π²ΠΎΠ·Π½ΠΈΠΊΠ°ΡΡΠΈΡ
ΠΏΡΠΈ ΠΏΠΎΡΠ΅ΡΠ΅ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ. ΠΡΠ»ΠΈ Π½Π°ΠΈΠΌΠ΅Π½ΡΡΠ°Ρ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΠΈΠ»Π° Π½Π΅ ΠΊΡΠ°ΡΠ½Π°Ρ, ΡΠΎ ΡΠΎΡΠΌΠ° ΠΏΠΎΡΠ΅ΡΠΈ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈ ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠ°Ρ Π΅ΠΉ ΡΠΏΡΡΠ° ΠΌΠΎΠΌΠ΅Π½ΡΠΎΠ² Π΅Π΄ΠΈΠ½ΡΡΠ²Π΅Π½Π½ΡΠ΅. ΠΡΠΈ ΠΊΡΠ°ΡΠ½ΠΎΡΡΠΈ Π½Π°ΠΈΠΌΠ΅Π½ΡΡΠ΅ΠΉ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠΉ Π½Π°Π³ΡΡΠ·ΠΊΠΈ Π²ΠΎΠ·Π½ΠΈΠΊΠ°ΡΡ ΠΊΡΠ°ΡΠ½ΡΠ΅ ΡΠΎΡΠΌΡ ΠΏΠΎΡΠ΅ΡΠΈ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ, ΠΈ Π»ΡΠ±Π°Ρ ΠΈΡ
Π»ΠΈΠ½Π΅ΠΉΠ½Π°Ρ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΈΡ ΡΠ°ΠΊΠΆΠ΅ ΡΠ²Π»ΡΠ΅ΡΡΡ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΠΎΠΉ ΡΠΎΡΠΌΠΎΠΉ. ΠΠ»Ρ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ Π½Π΅ΠΎΠ±Ρ
ΠΎΠ΄ΠΈΠΌΠΎ ΡΡΠΎΡΠΌΠΈΡΠΎΠ²Π°ΡΡ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΈΡ ΠΊΡΠ°ΡΠ½ΡΡ
ΡΠΎΡΠΌ ΠΏΠΎΡΠ΅ΡΠΈ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈ ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΡΡ Π΅ΠΉ ΡΠΏΡΡΡ ΠΈΠ·Π³ΠΈΠ±Π°ΡΡΠΈΡ
ΠΌΠΎΠΌΠ΅Π½ΡΠΎΠ², ΠΊΠΎΡΠΎΡΠ°Ρ ΠΈ Π±ΡΠ΄Π΅Ρ ΡΠ»ΡΠΆΠΈΡΡ ΠΎΡΠ½ΠΎΠ²ΠΎΠΉ Π΄Π»Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΡ ΠΊΡΠΈΡΠ΅ΡΠΈΡ. Π Π΄Π°Π½Π½ΠΎΠΉ ΡΡΠ°ΡΡΠ΅ ΠΏΡΠ΅Π΄Π»Π°Π³Π°Π΅ΡΡΡ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠΈΠΉ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΡΡ ΡΠ°ΠΊΡΡ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΈΡ ΠΊΡΠ°ΡΠ½ΡΡ
ΡΠΎΡΠΌ, ΠΊΠΎΡΠΎΡΠ°Ρ ΡΡΠ°Π½Π΅Ρ ΠΎΡΠ½ΠΎΠ²ΠΎΠΉ Π΄Π»Ρ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΡ ΠΊΡΠΈΡΠ΅ΡΠΈΡ Π±Π»ΠΈΠ·ΠΎΡΡΠΈ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΊ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΠΌΡ
Assessment of the Proximity of Design to Minimum Material Capacity Solution of Problem of Optimization of the Flange Width of I-Shaped Cross-Section Rods with Allowance for Stability Constraints or Constraints for the Value of the First Natural Frequency and Strength Requirements
There are known methods for optimizing the flange width of I-shaped cross-section rods with stability constraints or the constraints for the value of the first natural frequency. Corresponding objective function has the form of the volume of the flange material for the case when only the flange width varies and the crosssection height, wall thickness and flange thickness are specified. Special criterion for assessment of proximity of corresponding an optimal solution to the design of minimal material capacity was formulated for the considering problem. In this case, the resulting solution may not meet some other unaccounted constraints, for example, strength requirements. Modification of solution in order to meet previously unaccounted constraints does not allow researcher to consider such design as optimal. In the distinctive paper allowance for strength requirements, stability constraints or constraints for the value of the first natural frequency are proposed within considering problem of optimization. Special approach is formulated, which proposes to assess proximity to the design of minimum of material capacity obtained as a result of optimization. Increment of the objective function and criteria corresponding to constrains and restrictions are under consideration within computational process.ΠΠ·Π²Π΅ΡΡΠ½Ρ ΠΌΠ΅ΡΠΎΠ΄Ρ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΡΠΈΡΠΈΠ½Ρ ΠΏΠΎΠ»ΠΎΠΊ ΡΡΠ΅ΡΠΆΠ½Π΅ΠΉ Π΄Π²ΡΡΠ°Π²ΡΠΎΠ²ΠΎΠ³ΠΎ ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΏΡΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ
ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈΠ»ΠΈ Π²Π΅Π»ΠΈΡΠΈΠ½Π΅ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ, ΡΡΠ½ΠΊΡΠΈΠΈ ΡΠ΅Π»ΠΈ Π² Π²ΠΈΠ΄Π΅ ΠΎΠ±ΡΠ΅ΠΌΠ° ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»Π° ΠΏΠΎΠ»ΠΎΠΊ, Π΄Π»Ρ ΡΠ»ΡΡΠ°Ρ, ΠΊΠΎΠ³Π΄Π° Π²Π°ΡΡΠΈΡΡΠ΅ΡΡΡ ΡΠΎΠ»ΡΠΊΠΎ ΡΠΈΡΠΈΠ½Π° ΠΏΠΎΠ»ΠΎΠΊ, Π° Π²ΡΡΠΎΡΠ° ΡΠ΅ΡΠ΅Π½ΠΈΡ, ΡΠΎΠ»ΡΠΈΠ½Π° ΡΡΠ΅Π½ΠΊΠΈ ΠΈ ΡΠΎΠ»ΡΠΈΠ½Π° ΠΏΠΎΠ»ΠΊΠΈ Π·Π°Π΄Π°Π½Ρ. ΠΠ»Ρ ΡΡΠΎΠ³ΠΎ Π²Π°ΡΠΈΠ°Π½ΡΠ° ΠΏΠΎΡΡΠ°Π½ΠΎΠ²ΠΊΠΈ Π·Π°Π΄Π°ΡΠΈ Π±ΡΠ» ΡΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²Π°Π½ ΠΊΡΠΈΡΠ΅ΡΠΈΠΉ ΠΎΡΠ΅Π½ΠΊΠΈ Π±Π»ΠΈΠ·ΠΎΡΡΠΈ ΡΠ°ΠΊΠΎΠ³ΠΎ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΊ ΠΏΡΠΎΠ΅ΠΊΡΡ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ. ΠΡΠΈ ΡΡΠΎΠΌ Π² ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΠΎΠΌ ΡΠ΅ΡΠ΅Π½ΠΈΠΈ ΠΌΠΎΠ³ΡΡ Π½Π΅ Π²ΡΠΏΠΎΠ»Π½ΡΡΡΡΡ Π½Π΅ΠΊΠΎΡΠΎΡΡΠ΅ Π΄ΡΡΠ³ΠΈΠ΅ Π½Π΅ΡΡΡΡΠ½Π½ΡΠ΅ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡ, Π½Π°ΠΏΡΠΈΠΌΠ΅Ρ, ΠΏΠΎ ΠΏΡΠΎΡΠ½ΠΎΡΡΠΈ. ΠΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ Ρ ΡΠ΅Π»ΡΡ ΡΠ΄ΠΎΠ²Π»Π΅ΡΠ²ΠΎΡΠ΅Π½ΠΈΡ Π½Π΅ΡΡΡΡΠ½Π½ΡΠΌ ΡΠ°Π½Π΅Π΅ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΠΌ Π½Π΅ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΡΡΠΈΡΠ°ΡΡ ΡΠ°ΠΊΠΎΠΉ ΠΏΡΠΎΠ΅ΠΊΡ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΠΌ. Π Π΄Π°Π½Π½ΠΎΠΉ ΡΡΠ°ΡΡΠ΅ ΠΏΡΠ΅Π΄Π»Π°Π³Π°Π΅ΡΡΡ Π² ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌΠΎΠΉ Π·Π°Π΄Π°ΡΠ΅ ΡΡΠΈΡΡΠ²Π°ΡΡ Π² ΠΏΡΠΎΡΠ΅ΡΡΠ΅ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΏΡΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ
ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈΠ»ΠΈ Π²Π΅Π»ΠΈΡΠΈΠ½Π΅ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ Π΅ΡΡ ΠΈ ΡΡΠ»ΠΎΠ²ΠΈΠΈ ΠΏΡΠΎΡΠ½ΠΎΡΡΠΈ. Π€ΠΎΡΠΌΡΠ»ΠΈΡΡΠ΅ΡΡΡ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄, Π² ΠΊΠΎΡΠΎΡΠΎΠΌ ΠΏΡΠ΅Π΄Π»Π°Π³Π°Π΅ΡΡΡ Π΄Π»Ρ ΠΎΡΠ΅Π½ΠΊΠΈ Π±Π»ΠΈΠ·ΠΎΡΡΠΈ ΠΊ ΠΏΡΠΎΠ΅ΠΊΡΡ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ, ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΠΎΠ³ΠΎ Π² ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠ΅ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ, Π½Π°ΡΡΠ΄Ρ Ρ Π°Π½Π°Π»ΠΈΠ·ΠΎΠΌ Π² ΠΏΡΠΎΡΠ΅ΡΡΠ΅ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΠΉ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΠΉ Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΏΡΠΈΡΠ°ΡΠ΅Π½ΠΈΡ ΡΡΠ½ΠΊΡΠΈΠΈ ΡΠ΅Π»ΠΈ, ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡ Π΅ΡΡ ΠΈ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ, Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΠ·ΡΡΡΠΈΠ΅ ΠΊΠ°ΠΆΠ΄ΠΎΠ΅ ΠΈΠ· ΠΏΡΠΈΠ½ΡΡΡΡ
ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΠΉ
About the solution of a structural class optimization problems. Part 1: Formulation of theoretical foundations problems of the solution procedure
Earlier, the criterion of minimum material consumption was formulated within the outline design of the I-shaped bar width and the stability constraints or restriction to the value of the first natural frequency in one principal plane of the cross-section inertia. In the distinctive paper, we formulate a criterion for the minimum material capacity of the I-shaped bar with a variation in its thickness and outline of the width, with restrictions on the value of the critical force or restriction to the value of the first natural frequency in two principal planes of the section inertia. Β© Published under licence by IOP Publishing Ltd
The solution of structural class optimization problems. Part 2: Numerical examples
Earlier, the criterion of minimum material consumption was formulated within the design of the I-shaped bar width outline and the stability constraints or restriction to the value of the first natural frequency in one principal plane of the cross-section inertia. In the distinctive paper, we formulate a criterion for the minimum material capacity of the I-shaped bar with a variation in its thickness and outline of the width, with restrictions on the critical force value or restriction to the value of the first natural frequency in two principal planes of the section inertia. Numerical examples are presented. Β© Published under licence by IOP Publishing Ltd
ΠΡΠΈΡΠ΅ΡΠΈΠΈ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΡ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΏΡΠΈ ΡΠΎΡΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΡΡΠ΅ΡΠΆΠ½Π΅ΠΉ Ρ ΠΊΡΡΠΎΡΠ½ΠΎ-ΠΏΠΎΡΡΠΎΡΠ½Π½ΡΠΌ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ΠΌ ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΡ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΏΡΠΈ ΠΎΡΠ³Π°Π½ΠΈΡΠ΅Π½ΠΈΡΡ ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈΠ»ΠΈ Π½Π° Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ. Π§Π°ΡΡΡ 2: ΠΡΠΈΠΌΠ΅ΡΡ ΡΠ°ΡΡΠ΅ΡΠ°
The special properties of optimal systems have been already identified. Besides, criteria has been formulated to assess the proximity of optimal solutions to the minimal material consumption. In particular, the criteria were created for rods with rectangular and I-beam cross-section with stability constraints or constraints for the value of the first natural frequency. These criteria can be used for optimization when the cross sections of a bar change continuously along its length. The resulting optimal solutions can be considered as an idealized object in the sense of the limit. This function of optimal design allows researcher to assess the actual design solution by the criterion of its proximity to the corresponding limit (for example, regarding material consumption). Such optimal project can also be used as a reference point in real design, for example, implementing a step-by-step process of moving away from the ideal object to the real one. At each stage, it is possible to assess the changes in the optimality index of the object in comparison with both the initial and the idealized solution. One of the variants of such a process is replacing the continuous change in the size of the cross sections of the rod along its length with piecewise constant sections. Boundaries of corresponding intervals can be selected based on an ideal feature, and cross-section dimensions can be determined by one of the optimization methods. The distinctive paper is devoted to criteria that allow researcher providing reliable assessment of the endpoint of the optimization process, and the second part of the material presented contains corresponding numerical examples, prepared in accordance with the theoretical foundations given in the first part.Π Π°Π½Π΅Π΅ Π±ΡΠ»ΠΈ Π²ΡΡΠ²Π»Π΅Π½Ρ ΠΎΡΠΎΠ±ΡΠ΅ ΡΠ²ΠΎΠΉΡΡΠ²Π° ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ ΠΈ ΡΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²Π°Π½Ρ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ, ΠΎΡΠ΅Π½ΠΈΠ²Π°ΡΡΠΈΠ΅ Π±Π»ΠΈΠ·ΠΎΡΡΡ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΊ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΠΌΡ. Π ΡΠ°ΡΡΠ½ΠΎΡΡΠΈ Π±ΡΠ»ΠΈ ΡΠΎ Π·Π΄Π°Π½Ρ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ, Π΄Π»Ρ ΡΡΠ΅ΡΠΆΠ½Π΅ΠΉ Ρ ΠΏΡΡΠΌΠΎΡΠ³ΠΎΠ»ΡΠ½ΡΠΌ ΠΈ Π΄Π²ΡΡΠ°Π²ΡΠΎΠ²ΡΠΌ ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΠΌ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ΠΌ ΠΏΡΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡΡ
ΠΏΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΡΡΠΈ ΠΈΠ»ΠΈ Π½Π° Π²Π΅Π»ΠΈΡΠΈΠ½Ρ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΎΡΡ ΡΠΎΠ±ΡΡΠ²Π΅Π½Π½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ. ΠΡΠΈ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ ΠΏΡΠΈΠΌΠ΅Π½ΠΈΠΌΡ ΠΏΡΠΈ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ, ΠΊΠΎΠ³Π΄Π° ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΡΡΠ΅ΡΠΆΠ½Ρ Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΠΎ ΠΈΠ·ΠΌΠ΅Π½ΡΡΡΡΡ ΠΏΠΎ Π΅Π³ΠΎ Π΄Π»ΠΈΠ½Π΅. ΠΠΎΠ»ΡΡΠ΅Π½Π½ΡΠ΅ ΠΏΡΠΈ ΡΡΠΎΠΌ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΌΠΎΠ³ΡΡ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°ΡΡΡΡ ΠΊΠ°ΠΊ ΠΈΠ΄Π΅Π°Π»ΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΠΉ ΠΎΠ±ΡΠ΅ΠΊΡ Π² ΡΠΌΡΡΠ»Π΅ ΠΏΡΠ΅Π΄Π΅Π»ΡΠ½ΠΎΠ³ΠΎ. ΠΡΠ° ΡΡΠ½ΠΊΡΠΈΡ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠΎΠ΅ΠΊΡΠ° ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΠΎΡΠ΅Π½ΠΈΠ²Π°ΡΡ ΡΠ΅Π°Π»ΡΠ½ΠΎΠ΅ ΠΊΠΎΠ½ΡΡΡΡΠΊΡΠΎΡΡΠΊΠΎΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ ΠΏΠΎ ΠΊΡΠΈΡΠ΅ΡΠΈΡ Π΅Π³ΠΎ Π±Π»ΠΈΠ·ΠΎΡΡΠΈ ΠΊ ΠΏΡΠ΅Π΄Π΅Π»ΡΠ½ΠΎΠΌΡ (Π½Π°ΠΏΡΠΈΠΌΠ΅Ρ, ΠΏΠΎ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ). Π’Π°ΠΊΠΎΠΉ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΠΉ ΠΏΡΠΎΠ΅ΠΊΡ ΡΠ°ΠΊΠΆΠ΅ ΠΌΠΎΠΆΠ΅Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡΡΡ ΠΈ ΠΊΠ°ΠΊ ΠΎΡΠΈΠ΅Π½ΡΠΈΡ ΠΏΡΠΈ ΡΠ΅Π°Π»ΡΠ½ΠΎΠΌ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ, Π½Π°ΠΏΡΠΈΠΌΠ΅Ρ, ΡΠ΅Π°Π»ΠΈΠ·ΡΡ ΠΏΠΎΡΡΠ°ΠΏΠ½ΡΠΉ ΠΏΡΠΎΡΠ΅ΡΡ ΠΎΡΡ
ΠΎΠ΄Π° ΠΎΡ ΠΈΠ΄Π΅Π°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ΅ΠΊΡΠ° ΠΊ ΡΠ΅Π°Π»ΡΠ½ΠΎΠΌΡ. ΠΡΠΈ ΡΡΠΎΠΌ Π½Π° ΠΊΠ°ΠΆΠ΄ΠΎΠΌ ΡΡΠ°ΠΏΠ΅ ΠΏΠΎΡΠ²Π»ΡΠ΅ΡΡΡ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΡ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΡ ΠΏΠΎΠΊΠ°Π·Π°ΡΠ΅Π»Ρ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΡΡΠΈ ΠΎΠ±ΡΠ΅ΠΊΡΠ° ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ, ΠΊΠ°ΠΊ Ρ Π½Π°ΡΠ°Π»ΡΠ½ΡΠΌ, ΡΠ°ΠΊ ΠΈ Ρ ΠΈΠ΄Π΅Π°Π»ΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΠΌ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ΠΌ. ΠΠ΄Π½ΠΈ ΠΈΠ· Π²Π°ΡΠΈΠ°Π½ΡΠΎΠ² ΡΠ°ΠΊΠΎΠ³ΠΎ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠΎΡΡΠΎΠΈΡ Π² Π·Π°ΠΌΠ΅Π½Π΅ Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΠΎΠ³ΠΎ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΡ ΡΠ°Π·ΠΌΠ΅ΡΠΎΠ² ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΡΡΠ΅ΡΠΆΠ½Ρ ΠΏΠΎ Π΅Π³ΠΎ Π΄Π»ΠΈΠ½Π΅ ΠΊΡΡΠΎΡΠ½ΠΎ-ΠΏΠΎΡΡΠΎΡΠ½Π½ΡΠΌΠΈ ΡΡΠ°ΡΡΠΊΠ°ΠΌΠΈ. ΠΡΠ°Π½ΠΈΡΡ ΡΡΠ°ΡΡΠΊΠΎΠ² ΠΌΠΎΠ³ΡΡ Π²ΡΠ±ΠΈΡΠ°ΡΡΡΡ Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΈΠ΄Π΅Π°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ΅ΠΊΡΠ°, Π° ΡΠ°Π·ΠΌΠ΅ΡΡ ΠΏΠΎΠΏΠ΅ΡΠ΅ΡΠ½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΡΡΡΡ ΠΎΠ΄Π½ΠΈΠΌ ΠΈΠ· ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ. Π Π΄Π°Π½Π½ΠΎΠΉ ΡΡΠ°ΡΡΠ΅ ΠΏΡΠ΅Π΄Π»Π°Π³Π°ΡΡΡΡ ΠΊΡΠΈΡΠ΅ΡΠΈΠΈ, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠΈΠ΅ Π½Π°Π΄Π΅ΠΆΠ½ΠΎ ΠΎΡΠ΅Π½ΠΈΠ²Π°ΡΡ ΠΌΠΎΠΌΠ΅Π½Ρ ΠΎΠΊΠΎΠ½ΡΠ°Π½ΠΈΡ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠ°ΠΊΠΎΠΉ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ, ΠΏΡΠΈΡΠ΅ΠΌ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΠ΅ΠΌΠ°Ρ Π²ΡΠΎΡΠ°Ρ ΡΠ°ΡΡΡ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»Π° ΠΏΡΠ±Π»ΠΈΠΊΠ°ΡΠΈΠΈ ΡΠΎΠ΄Π΅ΡΠΆΠΈΡ ΠΏΡΠΈΠΌΠ΅Ρ ΡΠ°ΡΡΠ΅ΡΠ° Π² ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΠΈΠΈ Ρ ΠΈΠ·Π»ΠΎΠΆΠ΅Π½Π½ΡΠΌΠΈ Π² ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΈ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΠΎΡΠ½ΠΎΠ²Π°ΠΌΠΈ