Кафедра "Прикладна математика"

Постійне посилання колекціїhttps://repository.kpi.kharkov.ua/handle/KhPI-Press/4610

Офіційний сайт кафедри http://web.kpi.kharkov.ua/apm

Від 1981 року кафедра має назву "Прикладна математика", первісна назва – кафедра теоретичної й математичної фізики.

Кафедра теоретичної й математичної фізики була заснована в 1947 році. Організатором і першим завідувачем цієї кафедри був відомий вчений-математик, фахівець із конструктивної теорії функцій, член-кореспондент Української Академії наук Наум Ілліч Ахієзер. У 1970 році кафедра цілком чітко взяла курс на дослідження прикладних питань математики, і ще тоді припускалося перейменування кафедри в кафедру "Прикладна математика".

Кафедра входить до складу Навчально-наукового інституту комп'ютерного моделювання, прикладної фізики та математики Національного технічного університету "Харківський політехнічний інститут".

У складі науково-педагогічного колективу кафедри працюють: 1 доктор технічних наук, 1 доктор фізико-математичних наук, 5 кандидатів технічних наук, 4 кандидата фізико-математичних наук; 2 співробітника мають звання професора, 8 – доцента, 1 – старшого наукового співробітника.

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Analysis of free vibration of porous power-law and sigmoid functionally graded sandwich plates by the R-functions method
(Shahid Chamran University of Ahvaz, 2023) Kurpa, Lidiya; Shmatko, Tetyana; Awrejcewicz, Jan; Timchenko, Galina; Morachkovska, Iryna
Investigation of free vibration of porous power and sigmoid-law sandwich functionally graded (FG) plates with different boundary conditions is presented in this paper. The FG sandwich plate includes three layers. The face layers are fabricated of functionally graded material (FGM) and middle layer (core) is isotropic (ceramic). Imperfect sigmoid FG sandwich plates with even and linear-uneven porosities and nonporous core layer are studied. Developed approach has been realized in the framework of a refined theory of the first-order shear deformation theory (FSDT) using variational methods and the R-functions theory. The analytical expressions are obtained for calculating the elastic characteristics with the assumption that the values of Poisson's ratio are the same for constituent FGM materials. For rectangular plates, the obtained results are compared with known results and a good agreement is obtained. Vibration analysis of a complex-shaped porous sandwich plate made of FGM has been performed. The effect of various parameters on the dynamic behavior of the plate, such as the type and values of porosity coefficients, power index, lay-up scheme, types of FGM, has been studied.
Free vibration analysis of FGM shell with complex planform in thermal environments
(Wydawnictwo Politechniki Łódzkiej, 2019) Awrejcewicz, Jan; Kurpa, Lidiya; Shmatko, Tetyana
Summary. In the present study free vibrations of FGM shallow shells of an arbitrary planform in thermal environment are investigated via R-functions method (RFM). First-order shear deformation theory of shallow shells is employed. Material properties are assumed to be temperature-dependent and expressed as nonlinear functions of temperature. The generic material properties are not only functions of temperature, but also functions of thickness direction. It is supposed that material properties vary through thickness according to a power-law distribution of the constituent’s volume fraction. The developed method is based on the combined applications of the R-functions theory, variational Ritz’s method. A comparison of the obtained results with available ones is carried out for rectangular plates and shallow shells. Vibration of shell panels with complex planform and different boundary conditions including mixed ones are studied. Solution structures and related admissible functions for shells with complex planform have been constructed by the R-functions theory. The effect of the temperature rise, geometry of the shell, material properties and constituent volume fraction index is examined.
Vibration analysis of laminated functionally graded shallow shells with clamped cutout of the complex form by the Ritz method and the R-functions theory
(Brazilian Association of Computational Mechanics, 2019) Kurpa, Lidiya; Shmatko, Tetyana; Awrejcewicz, Jan
The R-functions theory and Ritz approach are applied for analysis of free vibrations of laminated functionally graded shallow shells with different types of curvatures and complex planforms. Shallow shells are considered as sandwich shells of different types: a) face sheets of the shallow shells are made of a functionally graded material (FGM) and their cores are made of an isotropic material; b) face sheets of the shallow shells are isotropic, but the core is made of FGM. It is assumed that FGM layers are made of a mixture of metal and ceramics and effective material properties of layers are varied accordingly to Voigt’s rule. Formulation of the problem is carried out using the first-order (Timoshenko’s type) refined theory of shallow shells. Different types of boundary conditions, including clamped, simply supported, free edge and their combinations, are studied. The proposed method and the created computer code have been examined on test problems for shallow shells with rectangular planforms. In order to demonstrate the possibility of the developed approach, novel results for laminated FGM shallow shells with cut of the complex form are presented. Effects of different material distributions, mechanical properties of the constituent materials, lamination scheme, boundary conditions and geometrical parameters on natural frequencies are shown and analyzed.
Nonlinear vibrations of functionally graded shallow shells of a complex planform in thermal environments
(CongressLine Ltd., Hungary, 2017) Awrejcewicz, Jan; Kurpa, Lidiya; Shmatko, Tatiana
Geometrically nonlinear vibrations of FGM shallow shells of an arbitrary planform subjected to thermal environment are investigated with the use of R-functions theory and variational methods. Nonlinear firstorder shear deformation shallow shells are employed. Material properties are assumed to be temperaturedependent and varying along the thickness direction accordingly to Voigt’s law. The developed method is based on the combined applications of R-functions theory, variational Ritz’s method, procedure by Bubnov-Galerkin, and Runge-Kutta’s approach. The effect of the temperature rise, geometry of the shell, and constituent volume fraction index is examined. A comparison of the obtained results with available ones is also carried out for rectangular plates and shallow shells.
Stability investigation of nonlinear vibrations of plates by R-functions method
(Wydawnictwo Politechniki Łódzkiej, 2007) Awrejcewicz, Jan; Kurpa, Lidiya; Mazur, Olga
The parametric vibrations of plates with cutouts subjected to in-plane periodic and compressive loads, are studied. The proposed approach is based on R-functions method and the classical variational approach. The influence of cutouts parameters, as well as static factors of load on stability regions and nonlinear vibrations are investigated.
Research of Stability and Nonlinear Vibrations by R-Functions Method
(Springer Science+Business Media B. V., 2009) Awrejcewicz, Jan; Kurpa, Lidiya; Mazur, Olga
Investigation of the parametric vibration of the orthotropic plates subjected to periodic in plane forces by multi-modal approximation and R-functions method
(NTU "KhPI", 2010) Awrejcewicz, Jan; Kurpa, Lidiya; Mazur, Olga
The original method of studying parametric vibrations of orthotropic plate with complex shape is proposed. Suggested approach is based on combined application of variational methods and the R-functions theory. Using the proposed method and developed software the regular and chaotic regimes of T-shaped plate are analyzed.
On the parametric vibrations and meshless discretization of orthotropic plates with complex shape
(De Gruyter, 2010) Awrejcewicz, Jan; Kurpa, Lidiya; Mazur, Olga
Investigation of the stress-strain state of the laminated shallow shells by R-functions method combined with spline-approximation
(WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011) Awrejcewicz, Jan; Kurpa, Lidiya; Osetrov, Andrey
The bending behavior of the laminated shallow shells under static loading has been studied using the R functions theory together with the spline-approximation. Formulation is based on the first order shear deformation theory. Due to usage of the R-functions theory the laminated shallow shells with complex shape and different types of the boundary conditions can be investigated. Application of the spline-approximation allows getting reliable and validated results for non concave domains and domains with holes. The proposed method is implemented in the appropriate software in framework of the mathematical package MAPLE. The analysis of influence of certain factors (curvature, packing of layers, geometrical parameters, boundary conditions) on a stress-strain state is carried out for shallow shells with cut-outs. The comparison of obtained results with those already known from literature and results obtained by using ANSYS are also presented.
Nonlinear vibration of orthotropic shallow shells of the complex shape with variable thickness
(Wydawnictwo Politechniki Łódzkiej, 2011) Awrejcewicz, Jan; Kurpa, Lidiya; Shmatko, T.
Early R-functions theory [1] combined with variational methods have been applied to linear [2] and nonlinear vibration problems [3,4] of the shallow shells theory of the constant thickness. In the present study, we first apply R-functions theory in order to investigate the geometrically nonlinear vibrations of orthotropic shallow shells of complex shape with variable thickness. Mathematical formulation is made in the framework of classical geometrically nonlinear theory of thin shallow shells. For a discretization of the original system in time, approximation of unknown functions is carried out by using a single mode approach. In order to construct a system of basic functions, the proposed algorithm includes sequence of the linear problems such as finding eigen functions of the linear vibrations of shallow shells with variable thickness and auxiliary tasks of the elasticity theory. The linear problems are solved by the R-functions method. The developed approach allows reducing the original problem to the corresponding problem of solving nonlinear ordinary differential equations (ODEs), whose coefficients are presented in analytical form. In order to solve the obtained system of ODEs the Bubnov-Galerkin method is applied. The proposed algorithm is implemented within an automated system POLE-RL [1]. Numerical examples of large-amplitude flexible vibrations of shallow orthotropic shells with complex shape and variable thickness are introduced demonstrating merits and advantages of the R-functions method. Comparison of the obtained results regarding shells with rectangular plans with the other methods confirms the reliability of the proposed method.