Piecewise Constant Sections Research Articles

Structural and thermodynamic properties of fluids whose molecules interact via one-, two-, and three-step potentials

The structural and thermodynamic properties of fluids whose molecules interact via potentials with a hard-core plus a square well, a square shoulder, and a second square well, are considered. Those properties are derived by using a (semi-analytical) rational-function approximation method as a particular case of the more general formulation provided earlier involving potentials with a hard-core plus n piecewise constant sections. Comparison of the results with recent simulation data confirms the usefulness of the approach.

Estimation of the Accuracy of a Control System with a Fuzzy PID Controller Based on the Approximation of the Static Characteristic of the Controller

The problem of estimating the accuracy of an automatic control system with a fuzzy PID controller is solved. To describe a fuzzy controller, its static characteristic is used, which is approximated by two piecewise-linear and one piecewise-constant sections. This approach makes it possible to study the system as a linear one at each section of the approximated characteristic, and accordingly develop the calculation methods known in control engineering, taking into account the features of the system under consideration. In the article, to calculate the error in the steady state, the theorem on the final value of the original is used. For two different types of second-order control objects — static and astatic — on the basis of this theorem, analytical expressions are obtained that relate the accuracy of the control system with the values of the target and disturbance with a different structure of the controller (P-, PI-, PD-). When conducting experimental studies, the fuzzy PID controller was compared with a linear one tuned by the method of the maximum stability. Research results show that a fuzzy controller ensures the accuracy of the control system is not worse than a linear one, while increasing the dynamics of the system. The analytical expressions presented in the article make it possible to assess the accuracy of a control system with a fuzzy controller and can be used as a technique for adjusting the controller based on the accuracy requirements.

Об одной задаче оптимизации конструкций с учетом требований устойчивости, прочности, при ограничениях первой частоты собственных колебаний

В одной из предыдущих работ авторов данной статьи рассматривалась задача оценки близости решения к проекту минимальной материалоёмкости при оптимизации ширины полок стержней двутаврового поперечного сечения при ограничениях по устойчивости или величины первой частоты собственных колебаний с учётом требований прочности в случае непрерывного изменения варьируемых параметров по длине стержня. При этом известно, что в строительной практике в основном стержни проектируются с кусочно-по­стоянным изменением параметров сечения. В другой статье авторов был сформулирован критерий оценки оптимальных решений при формирования кусочно-постоянных участков полки стержней двутаврового сечения при ограничениях по устойчивости или величине первой частоты собственных колебаний, но без учёта требований прочности. В данной статье рассматривается задача оценки оптимальных решений при формировании кусочно-постоянных участков ширины полок стержней двутаврового сечения при ограничениях по устойчивости или величины первой частоты собственных колебаний, но с учётом требований прочности и некоторых конструктивных ограничений.

Estimated closeness of optimum piecewise constant section width in I-rods with stability or first eigen-frequency limits to the predicted minimum material consumption with regard to strength requirements

The previous research described estimated closeness of optimum piecewise constant section width in I-rods with stability or first eigen-frequency limits to the predicted minimum material consumption with regard to strength requirements in continuous change in variable rod parameters. It is however known that in construction, rods are generally designed with piecewise constant change in the section parameters. Besides, in another work, the criterion was formulated for assessment of optimum solutions of piecewise constant sections of I-rods with stability or the first eigen-frequency limits, without considering the strength requirements. This paper focuses on a more general problem of estimated closeness of optimum piecewise constant section width in I-rods with stability or first eigen-frequency limits to the predicted minimum material consumption with regard to strength requirements.

Assessment criterion for optimum design solutions of piecewise constant sections in I-rods with stability or first eigen-frequency limits

The criterion for the minimum material consumption of strips strengthening the I-rod with stability or first eigen-frequency limits is formulated in previous studies for the case of continuous change of the variable parameter. It is known that this solution allows evaluating a real design project not only by the criterion of its proximity to the minimum material consumption, but also by the reference point in the real design. In many cases, it is used to replace the continuous change in the variable size of the rod-strengthening piecewise constant sections. The boundaries of these sections are based on the minimum material consumption. The width of the strengthening strips is determined by the optimization methods. The paper proposes the criterion allowing to correctly assess the termination of the optimization processes.

ASSESSMENT CRITERION FOR OPTIMUM DESIGN SOLUTIONS OF PIECEWISE CONSTANT SECTIONS IN RODS OF RECTANGULAR CROSS-SECTION WITH STABILITY OR FIRST EIGEN-FREQUENCY LIMITS

Specific properties of optimum systems have been already considered in previous research. Moreover, the criteria were proposed for a correct assessment of proximity of optimum to minimum material consumption. In particular, the criteria are proposed for rods of rectangular crosssection with stability or first eigen-frequency limits. These criteria can be used for problem optimization, when the rod cross-sections continuously change longitudinally. The obtained optimum solutions can be considered as a perfect limited object. This optimum project function allows researcher to assess the real design solution using the proximity limit criterion (for example, material consumption limit). This kind of optimum design can also be used as a guideline for real design in terms of a stage-by-stage process of transition from a perfect to real object. In this case, it is possible to assess changes in the object optimality at each stage as compared to the initial and idealized solutions. In particular, one of the variants of the process includes replacing the rod with continuous longitudinally varying cross-sections by a rod with piecewise constant sections. The section boundaries can be based on a perfect object, and cross-sections can be determined by one of the optimization methods. This paper presents criteria, which ensure the reliable definition of the time of completion of the optimization process.

ASSESSMENT CRITERION FOR OPTIMUM DESIGN SOLUTIONS OF PIECEWISE CONSTANT SECTIONS IN RODS OF I-SHAPED CROSS-SECTION WITH STABILITY OR FIRST EIGEN-FREQUENCY LIMITS

Specific properties of optimum systems have been already considered in previous research. Moreover, the criteria were proposed for a correct assessment of proximity of optimum to minimum material consumption. In particular, the criteria are proposed for rods of rectangular crosssection with stability or first eigen-frequency limits. These criteria can be used for problem optimization, when the rod cross-sections continuously change longitudinally. The obtained optimum solutions can be considered as a perfect limited object. This optimum project function allows researcher to assess the real design solution using the proximity limit criterion (for example, material consumption limit). This kind of optimum design can also be used as a guideline for real design in terms of a stage-by-stage process of transition from a perfect to real object. In this case, it is possible to assess changes in the object optimality at each stage as compared to the initial and idealized solutions. In particular, one of the variants of the process includes replacing the rod with continuous longitudinally varying cross-sections by a rod with piecewise constant sections. The section boundaries can be based on a perfect object, and cross-sections can be determined by one of the optimization methods. This paper presents criteria, which ensure the reliable definition of the time of completion of the optimization process.

ASSESSMENT CRITERIA OF OPTIMAL SOLUTIONS FOR CREATION OF RODS WITH PIECEWISE CONSTANT CROSS-SECTIONS WITH STABILITY CONSTRAINTS OR CONSTRAINTS FOR VALUE OF THE FIRST NATURAL FREQUENCY. PART 2: NUMERICAL EXAMPLES

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-by­step 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.

ASSESSMENT CRITERIA OF OPTIMAL SOLUTIONS FOR CREATION OF RODS WITH PIECEWISE CONSTANT CROSS-SECTIONS WITH STABILITY CONSTRAINTS OR CONSTRAINTS FOR VALUE OF THE FIRST NATURAL FREQUENCY. PART 1: THEORETICAL FOUNDATIONS

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-by­step 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.

Схема дослідження поздовжніх коливань стрижня кусково-сталого перерізу

Схема дослідження поздовжніх коливань стрижня кусково-сталого перерізу

Structure of discrete-potential fluids interacting via two piece-wise constant potentials with a hard-core

The structure of fluids whose molecules interact via potentials with a hard-core plus two piece-wise constant sections of different widths and heights has been studied by the perturbation theory based on the power series of an inverse temperature. The perturbation theory based on the reference system which incorporates both the repulsive and attractive potentials predicts accurate radial distribution function and direct correlation function of the discrete-potential fluids for the whole density regions. In particular, it is the most successful for the square-shoulder plus square-shoulder fluid and shifted square-shoulder fluid with a purely repulsive potential rather than the square-well plus square-well fluids with an attractive potential. It is found that the present theory predicts a more accurate structure than the rational-function approximation and conventional integral equations such as Percus–Yevick and hypernetted-chain theories in most of the cases.

Structural properties of fluids interacting via piece-wise constant potentials with a hard core

The structural properties of fluids whose molecules interact via potentials with a hard core plus two piece-wise constant sections of different widths and heights are presented. These follow from the more general development previously introduced for potentials with a hard core plus n piece-wise constant sections [A. Santos, S. B. Yuste, and M. Lopez de Haro, Condens. Matter Phys. 15, 23602 (2012)] in which use was made of a semi-analytic rational-function approximation method. The results of illustrative cases comprising eight different combinations of wells and shoulders are compared both with simulation data and with those that follow from the numerical solution of the Percus-Yevick and hypernetted-chain integral equations. It is found that the rational-function approximation generally predicts a more accurate radial distribution function than the Percus-Yevick theory and is comparable or even superior to the hypernetted-chain theory. This superiority over both integral equation theories is lost, however, at high densities, especially as the widths of the wells and/or the barriers increase.

Rational-function approximation for fluids interacting via piece-wise constant potentials

The structural properties of fluids whose molecules interact via potentials with a hard-core plus n piece-wise constant sections of different widths and heights are derived using a (semi-analytical) rational-function approximation method. The results are illustrated for the cases of a square-shoulder plus square-well potential and a shifted square-well potential and compared both with simulation data and with those that follow from the (numerical) solutions of the Percus-Yevick integral equation.