Wyniki wyszukiwania - BazTech

1

Comments on existing analytical solutions to the wave-induced cyclic response of a porous seabed of infinite thickness

Magda Waldemar

Polish Maritime Research

|

2023

|

nr 2

197--206

EN

This paper deals with the wave-induced cyclic response of a porous seabed (by means of oscillating parameters: pore-fluid pressure, soil displacement components, effective normal stress and shear stress components) due to a surface sinusoidal water-wave propagating over a seabed of infinite thickness. The main existing analytical solutions to the governing problem are critically discussed, pointing out their meaningful errors and doubtful items. A phase-lag phenomena is particularly studied as an immanent part of any complex-valued analytical solutions having a cyclic nature.

2

Refined energy method for the elastic flexural-torsional buckling of steel H-section beam-columns. Part II: Comparison and verification for elements LTU and LTR

Giżejowski Marian, Barszcz Anna, Wiedro Paweł

Archives of Civil Engineering

|

2023

|

Vol. 69, nr 2

265--289

EN

In investigations constituting Part I of this paper, the effect of approximations in the flexural-torsional buckling analysis of beam-columns was studied. The starting point was the formulation of displacement field relationships built straightforward in the deflected configuration. It was shown that the second-order rotation matrix obtained with keeping the trigonometric functions of the mean twist rotation was sufficiently accurate for the flexural-torsional stability analysis. Furthermore, Part I was devoted to the formulation of a general energy equation for FTB being expressed in terms of prebuckling stress resultants and in-plane deflections through the factor k1. The energy equation developed there was presented in several variants dependent upon simplified assumptions one may adopt for the buckling analysis, i.e. the classical form of linear eigenproblem analysis (LEA), the form of quadratic eigenproblem analysis (QEA) and refined (non-classical) forms of nonlinear eigenproblem analysis (NEA), all of them used for solving the flexural-torsional buckling problems of elastic beam-columns. The accuracy of obtained analytical solutions based on different approximations in the elastic flexural-torsional stability analysis of thin-walled beam-columns is examined and discussed in reference to those of earlier studies. The comparison is made for closed form solutions obtained in a companion paper, with a scatter of results evaluated for k1 = 1 in the solutions of LEA and QEA, as well as for all the options corresponding to NEA. The most reliable analytical solution is recommended for further investigations. The solutions for selected asymmetric loading cases of the left support moment and the half-length uniformly distributed span load of a slender unrestrained beam-column are discussed in detail in Part II. Moreover, the paper constituting Part II investigates how the buckling criterion obtained for the beam-column laterally and torsionally unrestrained between the end sections might be applied for the member with discrete restraints. The recommended analytical solutions are verified with use of numerical finite element method results, considering beam-columns with a mid-section restraint. A variant of the analytical form of solutions recommended in these investigations may be used in practical application in the Eurocode’s General Method of modern design procedures for steelwork.

PL

W I części niniejszej pracy zastosowano różne rodzaje aproksymacji w analizie wyboczenia giętno-skrętnego elementów ściskanych i zginanych w płaszczyźnie większej bezwładności przekroju. Punktem wyjścia było sformułowanie zależności na pole przemieszczeń w konfiguracji odkształconej. Pokazano, że macierz rotacji, otrzymana przy zachowaniu funkcji trygonometrycznych średniego kąta skręcenia, jest wystarczająco dokładna do analizy stateczności giętno-skrętnej. Szczególną uwagę zwrócono w Części I na sformułowanie ogólnego równania energetycznego dla FTB, wyrażonego w funkcji sił przekrojowych na podstawowej ścieżce równowagi, przed utratą płaskiej postaci zginania II rzędu, a także wpływu efektu ugięć w płaszczyźnie większej bezwładności przekroju, wyrażonego za pomocą współczynnika k1. Otrzymane równanie energetyczne zostało przedstawione w kilku wariantach zależnych od założeń upraszczających, jakie można przyjąć do rozwiązywania problemów wyboczenia giętno-skrętnego, tj. w postaci klasycznej analizy liniowego problemu własnego (LEA), w postaci kwadratowego problemu własnego (QEA) oraz w postaci udoskonalonej (nieklasycznej) analizy nieliniowego problemu własnego (NEA). W części II, w pierwszej kolejności, została zbadana i dyskutowana dokładność otrzymanych rozwiązań analitycznych w odniesieniu do propozycji przedstawionych we wcześniejszych opracowaniach. Przeprowadzone są też porównania dla rozwiązań w postaci zamkniętej uzyskanych w Części I, z oceną rozrzutu wyników, po przyjęciu k1 = 1 w rozwiązaniach odpowiadających LEA i QEA, a także wszystkich opcji w rozwiązaniach odpowiadających NEA. Do dalszych badań rekomendowano najbardziej wiarygodne rozwiązanie analityczne. Szczegółowej weryfikacji poddano rozwiązania uzyskane dla wybranych asymetrycznych przypadków obciążenia: momentem na lewej podporze i równomiernie rozłożonym obciążeniem w połowie długości nieusztywnionego, smukłego elementu ściskanego i zginanego. Ponadto w Części II zbadano, w jaki sposób kryterium wyboczeniowe, uzyskane dla elementu ściskanego i zginanego bez usztywnień poprzecznych i przeciwskrętnych między przekrojami końcowymi, może być zastosowane dla elementu z dyskretnymi stężeniami poprzecznymi. Zalecane rozwiązania analityczne zweryfikowano z wykorzystaniem wyników numerycznych metody elementów skończonych dla elementów stężonych w przekroju środkowym. Wariant analitycznej postaci rozwiązania zalecanego w zaprezentowanych badaniach może być wykorzystany w praktyce w eurokodowej Metodzie Ogólnej (GM).

3

Calculation of the heat power of a tube heat exchanger

Rup Kazimierz

Archives of Thermodynamics

|

2022

|

Vol. 43, no 1

127--140

EN

In this paper, the logarithmic mean temperature difference method is used to determine the heat power of a tube-in-tube exchanger. Analytical solutions of the heat balance equations for the exchanger are presented. The considerations are illustrated by an example solution of the problem. In particular, the heat power of the tube-in-tube heat exchanger is determined taking into account the variants of work in the co-current and counter-current mode. Apart from the analytical solutions, appropriate numerical calculations in Matlab environment have been carried out.

4

Analytical and numerical analysis of injection pump (stepped) shaft vibrations using timoshenko theory

Noga Stanisław, Rejman Edward, Bałon Paweł, Kiełbasa Bartłomiej, Smusz Robert, Szostak Janusz

Acta Mechanica et Automatica

|

2022

|

Vol. 16, no. 3

215--224

EN

The free transverse vibrations of shafts with complex geometry are studied using analytical methods and numerical simulations. A methodology is proposed for evaluating the results of a natural transverse vibration analysis as generated by finite element (FE) models of a shaft with compound geometry. The effectiveness of the suggested approach is tested using an arbitrarily chosen model of the injection pump shaft. The required analytical models of the transverse vibrations of stepped shafts are derived based on the Timoshenko thick beam theory. The separation of variables method is used to find the needed solutions to the free vibrations. The eigenvalue problem is formulated and solved by using the FE representation for the shaft and for each shaft-simplified model. The results for these models are discussed and compared. Additionally, the usefulness of the Myklestad–Prohl (MP) method in the field of preliminary analysis of transverse vibration of complex shaft systems is indicated. It is important to note that the solutions proposed in this paper could be useful for engineers dealing with the dynamics of various types of machine shafts with low values of operating speeds.

5

Analytical solutions for a general mixed boundary value problem associated with motions of fluids with linear dependence of viscosity on the pressure

Vieru D., Fetecau C., Bridges C.

International Journal of Applied Mechanics and Engineering

|

2020

|

Vol. 25, no. 3

181--197

EN

An unsteady flow of incompressible Newtonian fluids with linear dependence of viscosity on the pressure between two infinite horizontal parallel plates is analytically studied. The fluid motion is induced by the upper plate that applies an arbitrary time-dependent shear stress to the fluid. General expressions for the dimensionless velocity and shear stress fields are established using a suitable change of independent variable and the finie Hankel transform. These expressions, that satisfy all imposed initial and boundary conditions, can generate exact solutions for any motion of this type of the respective fluids. For illustration, three special cases with technical relevance are considered and some important observations and graphical representations are provided. An interesting relationship is found between the solutions corresponding to motions induced by constant or ramptype shear stresses on the boundary. Furthermore, for validation of the results, the steady-state solutions corresponding to oscillatory motions are presented in different forms whose equivalence is graphically proved.

6

The application of cumulants to flow routing

Romanowicz Renata J., Doroszkiewicz Joanna

Meteorology Hydrology and Water Management. Research and Operational Applications

|

2019

|

Vol. 7, Iss. 1

15--21

EN

This paper aims to fill a gap between present and past research approaches to modelling flow in open channels. In particular, a history of the analytical solutions of a linearized St. Venant equation is presented. A solution of the linearized St. Venant equation, describing the response of a river channel to a single impulse forcing, the so called Instantaneous Unit Hydrograph (IUH), can be described using cumulants, defined as the moments of a logarithm of a variable. A comparison of analytical and numerical solutions of flood wave propagation under various flow conditions is given. The river reach of Biała Tarnowska is used as an illustration of both approaches. A practical application of simplified solutions to the emulator of a flood wave propagation is suggested showing a link between theory and practice.

7

Estimation of random friction forces on the microbearing cooperating surfaces

Wierzcholski Krzysztof, Miszczak Andrzej

Journal of KONES

|

2019

|

Vol. 26, No. 2

167--174

EN

Presented paper concerns a new mathematical form of stochastic theory of hydrodynamic friction forces occurring on the real cooperating surfaces in computer micro bearing fan and computer microbearing in hard disc driver HDD. This paper presents particularly a new-review of stochastic analytical considerations realized by the authors for friction forces estimation during hydrodynamic lubrication performed on the ground of the measurements of the gap height between two roughness surfaces. After numerous experimental measurements directly follows that the random unsymmetrical increments and decrements of the gap height of computer microbearings have important influence on the load carrying capacities and finally on the friction forces and wear of cooperating surfaces. The main topic demonstrates the influence of the variations of expectancy values and standard deviation of the computer microbearing gap height on the friction forces occurring in the HDD and microbearing fan. Moreover, it is observed the evident connection between the apparent dynamic viscosity and the features of the microbearing superficial surface. Hence after abovementioned remarks follows the corollary that the influence of the microbearing gap height stochastic variations connected with the surface roughness tend moreover indirect from the apparent viscosity into the friction forces variations. The synthetic, complex elaboration of obtained results indicates finally the influence of the random roughness in micro and nano level of microbearing surfaces on the friction forces distribution. A new results contained in this paper are obtained taking into account 3D variations of dynamic viscosity of the non-Newtonian lubricant, particularly variations crosswise the film thickness and influences of surface superficial layer features on the lubricant apparent viscosity.

8

MHD natural convection in saturated porous media with heat generation/absorption and thermal radiation: closed-form solutions

Turkyilmazoglu M.

Archives of Mechanics

|

2019

|

Vol. 71, nr 1

49--64

EN

The present work is devoted to the free convection flow occurring about a heated vertically stretching permeable surface placed in a porous medium under the influence of a temperature dependent internal heat generation or absorption. There are volume radiative heat sources in the fluid and the system is permeated by a uniform magnetic field. It is shown that the governing equations are reducible to a self-similar nonlinear ordinary differential equation of third order whose solutions are constructed analytically in the purely exponential series form. Under special circumstances, closed-form solutions are available which clearly indicate the existence of dual natural convection solutions. Otherwise, analytical solutions are still possible which are shown to be computed from an elegant algorithm without a need to invoke any numerical means. Exact solutions demonstrate, in physical insight that, in the presence of a heat sink absorbing the temperature from the porous medium increases the rate of heat transfer from the wall, whereas a heat source mechanism will surely overheat the system during the wall heating process, resulting in poor heat transfer rates. The presented exact solutions are beneficial for investigation of free convection phenomena in different geometries taking into account more complex physical features in higher dimensions.

9

Detection and Tracking Damage in Composite Structures Elements

Markuszewski D.

Machine Dynamics Research

|

2017

|

Vol. 41, No. 4

183--191

EN

Use of next generation materials in construction of racing yachts pushes the limits of performance under extreme conditions. Light yet durable carbon masts can be built higher without a need to increase diameter which minimizes the effect of the mast on the change of the speed of flow around and distortion. At cost of these benefits, inherent features of the carbon material generate problems during production and mast operation. Our work aims at developing reliable diagnostics tools for carbon yacht poles, which are be capable of providing information about construction defects and damages occurring in the course of operation.

10

The constituent equations of piezoelectric cantilevered three-layer actuators with various external loads and geometry

Mieczkowski G.

Journal of Theoretical and Applied Mechanics

|

2017

|

Vol. 55 nr 1

69--86

EN

This paper presents test results for deformation conditions of three-layer, piezoelectric cantilever converters subjected to various electrical and mechanical boundary conditions. A general solution has been developed based on implementation of piezoelectric triple segments (PTS) to the beam. A working mechanism and conditions for strain of the PTS segment have been determined. Basing on the general solution, for the cantilever actuator subjected to an external force (of single and dual PTS segments) and a uniform load (of single PTS segment), particular solutions have also been developed. Moreover, dimensionless frequencies of the oscillating motion for the analyzed converters have been determined. In the next step, the influence of such factors as length, quantity and position of PTS segments, their relative stiffness and unit weight on values of the obtained frequencies of vibration have been defined. The resulting analytical solutions have been compared with the developed FEM solution.

11

Analytical determination of the order of stress field singularity in some configurations of multiwedge systems for the case of antiplane deformation

Makhorkin M., Makhorkina T.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

|

2017

|

Vol. 6, No 3

45--51

EN

In this paper, the possibility of constructing the analytical expressions to determine the order of the stress singularities in multi-wedge composites of the most prevalent geometric configurations for the case of antiplane deformation is considered. Particularly, the analytical solutions of the corresponding characteristic equations are constructed for three-wedge systems whose components have such geometric characteristics [wzór] is а half-plane and attached to it wedges with the such apical angles: (in the presence and absence of a slit) [wzór] is а half-plane and attached to it wedges with such apical angles [wzór] (in the presence and absence of the slit with outlet angle to the linear materials interface) [wzór] is а half-plane and attached to it wedges with such apical angles [wzór]. The analytical solutions of characteristic equations for composite wedges composed of [wzór] elements with identical apical angles are constructed as well. Additional studies, the results of which have not been included in the materials of the article due to their inconvenience, indicate to that there are analytical solutions of the characteristic equation for a composite of this type with more elements. The obtained results make it possible to study the stress-strain state in multi-wedge systems of the considered configurations not restricting ourselves only to the vicinity of the wedges convergence point. In addition, the use of analytical solutions of characteristic equations in systems with a large number of wedges having the same apical angles gives the additional possibilities for analysis the angularly functionally graded materials.

12

Influence of torsional-bending coupling on transverse vibration of piston engine

Dąbrowski Z., Chiliński B.

Vibrations in Physical Systems

|

2016

|

Vol. 27

76--82

EN

The article presents the analysis of the influence of bending-torsional coupling of vibrations in the crankshaft on transverse vibrations of the engine body. In practice, there is used a simplified model, wherein transverse and torsional oscillations are analyzed independently. With the use of the model of deformable crankshaft, the authors show the influence of bending-torsional coupling on the frequency structure of transverse vibrations. The Introduction presents the problem of vibrations in combustion engines and their modelling. Further, there is presented the elastic model of the crankshaft, together with the applied assumptions and equations of motion describing vibrations in one cylinder combustion engine . Next chapter shows numerical simulation results with their initial analysis. The whole paper is summarized with conclusions about calculations and the possibility to use the results in practice.

13

Coupled lateral-torsional vibrations of a symmetric rotor

Chiliński B., Pakowski R., Stanik Z.

Journal of KONES

|

2016

|

Vol. 23, No. 4

41--47

EN

The paper presents a cross-sectional analysis of dynamics of rotating systems with different characteristics, starting from the search for solutions in an analytical way, and ending with the numerical attempt to verify the results. There is presented a problem of bending and torsional vibrations in rotational systems, using possibly the simplest model of the phenomenon. It is a heavy disk embedded on the deformable massless shaft. The rotor is symmetric and it is assumed that stiffness can be nonlinear. The analysis of vibrations occurring in rotors is made with the use of this model. This is followed by a preliminary discussion on the problem and the conclusions of the pilot studies. There are presented basic problems related to the phenomenon of lateral-torsional coupling in rotational systems with an emphasis on critical states. Formulation of the problem is the conclusion of preliminary part of the paper. Next, there are discussed the selected methods currently used to solve such problems, together with a discussion of their application in this case. In addition, there is discussed a problem of numerical or analytical character of obtained results and the range of using methods in modelling, designing and experimental studies. The next part of the paper focuses on the solution of the model presented in the introduction, using the following analytical or numerical methods: Krylov-Bogolubov, Adams and Runge-Kutta methods. There are presented assumptions, simplifications of the model, technical details of searching for solutions and the results together with a discussion of their correctness. The results, which were obtained from analysis, are polyharmonic. Due to this fact, the spectra of each solution were presented. These aspects are discussed in the context of the solving methods and motion analysis for rotor systems. The whole paper is summarized with a discussion of the results and their confrontation with reality.

14

Edinyj podhod k rešeniû zadač teplogazoobmena pri požarah na različnyh obʺektah

Pashkovskiy P. S., Grekov S. P., Zinchenko I. N.

Bezpieczeństwo i Technika Pożarnicza

|

2016

|

Vol. 44, nr 4

83--88

EN

Aim: The purpose of this study is to expose existing mathematical models dealing with heat and gas exchange, and reduce them to a single transfer equation, for any substance, during fire incidents inmines and other locations, by taking account of heated gas thermal expansion. Introduction: In order to perform an assessment of a fire hazard it is necessary to know the duration of a fire and period in which the fire will reach dangerous levels such as: concentration of toxic gases and saturation by smoke. A fire, manifested by an uncontrolled burning activity is a diffucult, unstable and ultimately not fully understood process, which occurs within a three dimensional expanse. This process is described, in the main, by mathematical equations dealing with behaviour principles involving conservation and conversion of energy, convection and diffused transfer of various gas components as well as smoke. Methods: In order to obtain anequation describing the transfer process for any substance, use is made of the central (finite) differences numerical method, for explicit and implicit conditions, using specific weighting appropriate to each. Results: The study examined the flow of air at velocities below the speed of sound – at less than 10m/s. According to the Bernoulli equation pressure changes insignificantly in relation to atmospheric pressure at such speed. Therefore, an equation describing this condition can be satisfactorily interchanged with an equation for normal conditions. This made it possible to depict a system of equations, in a dimensionless form, to address the behaviour and transformation of energy as well as convective and diffused transfer of components for different gasses and smoke using one universal equation. A numeric approach was developed to address issues. The designed calctulation diagram, incorporating explicit and implicit conditions is realised with the use of a “Miedwiedzcia” template. Specific examples illustrate comparisons between proposed calculations and graph analysis of variations in time and range of heat, and gas release in a ventilated channel. Conclusions: The article advocates a universal solution in a dimensionless form, which with the use of one equation describes a system of equations for conserving and transformation of energy, and transfer of different gas and smoke components by convective and diffused methods. The “Miedwiedzcia” template facilitates the capture of data dealing with the dynamics of heat and gas transfer processes in channels, corridors and mines.

PL

Cel: Celem badań jest generalizacja istniejących modeli matematycznych wymiany masy i gazów oraz stworzenie na ich podstawie jednego równania transferu dowolnej substancji podczas pożarów w kopalniach i w innych obiektach z uwzględnieniem rozszerzenia cieplnego gazów. Wprowadzenie: Aby przeprowadzić ocenę zagrożenia pożarowego, należy znać czas trwania pożaru oraz czas, w jakim osiągnie on niebezpieczne wartości np. stężenia toksycznego gazu i zadymienia. Prawdziwy pożar, będący niekontrolowanym procesem spalania, jest złożonym i nie do końca poznanym procesem niestacjonarnym, który odbywa się w przestrzeni trójwymiarowej. W celu opisania tego procesu wykorzystywane są głównie: równania zasady zachowania i przemiany energii, równanie konwekcyjno-dyfuzyjnego transferu składników różnych gazów i równanie konwekcyjno-dyfuzyjnego transferu dymu. Metody: W celu otrzymania rozwiązania równania transportu dowolnej substancji wykorzystano metodę liczbową różnic skończonych schematów jawnych i niejawnych z określonym ciężarem właściwym jednej z nich. Wyniki: Rozpatrywano przepływ powietrza z prędkością mniejszą niż prędkość dźwięku, poniżej 10 m/s. Przy takich prędkościach, zgodnie z równaniem Bernoulliego, ciśnienie zmienia się nieznacznie w stosunku do ciśnienia atmosferycznego. Dlatego też równanie stanu z dostatecznym stopniem dokładności można zamienić na równanie dla normalnych warunków. Pozwoliło to przedstawić w postaci bezwymiarowej system równań zachowania i przemiany energii oraz konwekcyjno-dyfuzyjnego transferu składników różnych gazów i dymu w jednym uniwersalnym równaniu. Opracowano metodę liczbową rozwiązywania zagadnienia. Przedstawiony łączony schemat obliczeniowy – jawny i niejawny jest realizowany na sześciopunktowym szablonie „Miedwiedzica”. Na konkretnych przykładach przedstawiono porównanie zaproponowanej metody liczbowej z analitycznymi krzywymi zmiany w czasie i w różnych odległościach funkcji wydzielenia ciepła i gazu w wentylowanym kanale. Wnioski: Zaproponowano uniwersalne rozwiązanie w formie bezwymiarowej, które opisuje jednym równaniem system równań zachowania i przemiany energii, konwekcyjno-dyfuzyjny transfer składników różnych gazów i dymu. Przedstawiono schemat obliczeniowy „Miedwiedzica”, który dzięki zastosowaniu metody liczbowej pozwala otrzymać wiarygodne dane o dynamice procesów wymiany masy i gazów w kanałach, korytarzach i kopalniach.

15

Dirac-Kähler particle in Riemann spherical space : analytical and numerical study, visualization

Chichurin A. V., Ovsiyuk E. M., Red’kov V. M.

Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie

|

2015

|

Nr 1(9)

41--54

EN

Exact analytical solutions of the Dirac-Kähler equation for the case of the spherical Riemann space of constant positive curvature are constructed. For non-zero values of the total angular momentum, the radial equations are reduced to a pair of complicated fourth-order differential equations. Employing the factorization approach, we have found four independent fundamental solutions written in terms of combinations of the hypergeometric functions. The energy spectrum differs substantially from the energy spectrum of the Dirac particle in spherical Riemann space. The visualization of the constricted solution is performed. The involved 4-th order differential equations are solved numerically. Results of the numerical analysis are in good agreement with analytical study.

PL

Znaleziono rozwiązania analityczne równania Diraca-Kählera w przypadku przestrzeni sferycznej Riemanna o stałej dodatniej krzywiźnie. W przypadku niezerowych wartości kretu równania różniczkowe promieniowe przekształcają się do pary równan różniczkowych czwartego rzędu. Stosując metodę faktoryzacji, znaleziono jego cztery podstawowe niezależne rozwiazania, które zapisano jako kombinacje funkcji hipergeometrycznych. Wyznaczone widmo energii różni się znacznie od widma energii cząstki Diraca w sferycznej przestrzeni Riemanna. Wykonano wizualizację znalezionych rozwiazań. Równania różniczkowe czwartego rzędu są rozwiązywane numerycznie. Wyniki analizy numerycznej są zgodne z badaniami analitycznymi.

16

Natural vibrations of thick circular plate based on the modified Mindlin theory

Senjanovic I., Hadzic N., Vladimir N., Cho D.-S.

Archives of Mechanics

|

2014

|

Vol. 66, nr 6

389--409

EN

Outline of the modified Mindlin theory is presented in which the Mindlin mathematical model with three differential equations of motion for total deflection and rotations is decomposed into a single equation for pure bending vibrations with transverse shear and rotary inertia effects and two differential equations for in-plane shear vibrations. The governing equations are transformed from orthogonal to polar coordinate system for the purpose of circular plate vibration analysis. The fourth order differential equation of flexural vibrations is split further into two second order equations of Bessel type. Also, the in-plane shear differential equations are transformed to Bessel equation by introducing displacement potential functions. The exact values of natural frequencies are listed and compared with FEM results.

17

Analysis of disturbance torque influence on critical state in rotational systems

Chiliński B.

Transport Problems

|

2013

|

T. 8, z. 4

137--146

EN

Currently most of existing means of transport contains different types of rotational systems. In many cases the dynamics of such rotors substantially can influence exploitation of the whole vehicle. Moreover, in order to minimize mass of the whole object modern construction materials are applied. This causes that the dynamic phenomena may be fundamental of exploitation. The paper presents preliminary analysis of disturbance torque influence on critical state in rotational system. The consideration assumed simple physical object in the form of heavy disk embedded on weightless, elastic shaft. The shaft was supported on two bearings. In particular chapters of paper, path leading from proposition of physical model, by solution of it, to qualitative conclusions about considered object and torque disturbances influence of motion of this system, was presented. In introduction, outline of considered problem and potential opportunities of it, were demonstrated. In the next chapter, physical and mathematical model of the analysed object, was described. Next and also the last but one chapter gives a detailed discussion of mathematical model in the form of nonlinear ordinary differential equations proposed earlier. The first part of the chapter presents the possibility to solve such a problem, then it shows the simplifications which are used. Furthermore, the influence of used simplifications on the shape of analysed problem was demonstrated. Additionally, the possibility of equations solution presented in the paper was discussed. Moreover, the series of interesting properties of analysed system of equations has been shown based on founded approximate solutions. The whole paper was summarized with plans for future work and synthetic conclusions concerning the innovative control method of critical states.

PL

Aktualnie większość istniejących środków transportu zawiera różnego typu układy wirujące. W wielu przypadkach dynamika takich wirników w istotny sposób wpływa na eksploatacje całego pojazdu. Ponadto w celu zminimalizowania masy całego obiektu stosuje się nowoczesne materiały konstrukcyjne. To powoduje, że zjawiska dynamiczne, mogą mieć podstawowe znaczenie eksploatacyjne. Artykuł przedstawia wstępną analizę wpływu zaburzenia momentu skręcającego na stany krytyczne układu wirującego. Do rozważań przyjęto prosty obiekt fizyczny w postaci ciężkiego krążka osadzonego na nieważkim podatnym wale podpartym w dwóch łożyskach. W poszczególnych rozdziałach artykułu przedstawiono drogę prowadzącą od zaproponowania modelu fizycznego, przez jego rozwiązanie do jakościowych wniosków o rozpatrywanym obiekcie i wpływie zaburzenia momentu skręcającego na jego ruch. We wstępie przedstawiono zarys rozpatrywanego problemu oraz potencjalne możliwości wykorzystania opracowanego zagadnienia. W kolejnym rozdziale zaprezentowano model fizyczny oraz matematyczny dla obiektu będącego podstawą niniejszego artykułu. Następny i zarazem przedostatni rozdział w sposób szczegółowy przedstawia dyskusję zaproponowanego wcześniej modelu matematycznego w postaci układu nieliniowych równań różniczkowych zwyczajnych. Na początku rozdziału przedstawiono rozwiązywalność takiego zagadnienia, następnie opisano zastosowane uproszczenia. W dalszym ciągu zademonstrowano wpływ zastosowanych uproszczeń na kształt analizowanych równań. Dodatkowo podjęto dyskusję o możliwości rozwiązania równań przedstawionych w pracy. Ponadto na podstawie znalezionych przybliżonych rozwiązań wykazano szereg ciekawych własności analizowanego układu równań. Całość została podsumowana celami dalszej pracy oraz syntetycznymi wnioskami, dotyczącymi propozycji innowacyjnej metody sterowania stanami krytycznymi.

18

Quantum mechanical scalar particle with polarisability in the Coulomb field, analytical and numerical consideration

Chychuryn A., Red’kov V.

Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie

|

2013

|

Nr 2(6)

73--89

EN

In the article the quantum-mechanical problem of a charged particle with additional characteristic, polarisability, in presence of external Coulomb field is investigated. After separation of the variables the problem is reduced to the Heun type differential equation with four singular points. There are found parameter restrictions when a general solution can be given in a simple analytical form. In the case of zero polarisability, exact analytical solutions can be constructed in hypergeometric functions, their visualization is given. In the case of non-zero polarisability, with the help of numerical methods and functional object DifferentialRoot there are constructed approximate solutions describing quantum-mechanical bound states, their visualization is given. In the paper, applying numerical and analytical methods, and performing visualization of the results, we use the Mathematica 9.

PL

W artykule zbadano problem kwantowo-mechaniczny z naładowaną cząstką i dodatkową charakterystyką polaryzowalności w obecności zewnętrznego pola Coulomba. Po rozdzieleniu zmiennych problem sprowadza się do równania różniczkowego typu Heunego, które ma cztery punkty osobliwe. Znaleziono ograniczenia parametrów, gdy rozwiązanie ogólne zostaje podane w prostej postaci analitycznej. W przypadku zerowej polaryzowalności dokładne rozwiązanie analityczne może być przedstawione w funkcjach hipergeometrycznych (przedstawiono ich wizualizację). W przypadku niezerowej polaryzowalności za pomocą metod numerycznych i funkcjonalnego obiektu DifferentialRoot sporządzono rozwiązania przybliżone, które opisują stany związane z mechaniką kwantową (przedstawiono ich wizualizację). Przy stosowaniu metod numerycznych i analitycznych oraz przy wykonaniu wizualizacji rezultatów użyto programu Mathematica 9.

19

Application of higher order Hamiltonian approach to nonlinear vibrating systems

Askari H., Nia Z. S., Yildirim A., Yazdi M. K., Khan Y.

Journal of Theoretical and Applied Mechanics

|

2013

|

Vol. 51 nr 2

287--296

EN

The higher order Hamiltonian approach is utilized to elicit approximate solutions for two nonlinear oscillation systems. Frequency-amplitude relationships and the model of buc kling of a column and mass-spring system are scrutinized in this paper. First, second and third approximate solutions of examples are achieved, and the frequency responses of the systems are verified by exact numerical solutions. According to the numerical results, we can conclude that the Hamiltonian approach is an applicable method for solving the nonlinear equations, and the accuracy of this method in the second and third approximates is very high and reliable. The achieved results of this paper demonstrate that this method is powerful and uncomplicated for solving of sophisticated nonlinear problems.

PL

W pracy przedstawiono zastosowanie metody Hamiltona wyższego rzędu do wyznaczania przybliżonych rozwiązań analitycznych dla dwóch nieliniowych układów drgających. Szczegółowej analizie poddano charakterystyki amplitudowo-częstościowe modelu ściskanej belki oraz dyskretnego układu sprężysto-inercyjnego. Otrzymano przybliżone rozwiązania pierwszego, drugiego i trzeciego rzędu, a odpowiedzi częstościowe układów porównano z dokładnymi rezultatami symulacji numerycznych. Na ich podstawie oceniono, że metoda Hamiltona jest stosowalna dla układów nieliniowych, a przybliżenia drugiego i trzeciego rzędu stanowią rozwiązania analityczne o wysokiej dokładności. Uzyskane w pracy wyniki przekonują, że zaproponowana metoda jest prostym i jednocześnie bardzo skutecznym narzędziem rozwiązywania nieliniowych problemów układów mechanicznych o dużym stopniu złożoności.

20

Nonlinear flexural analysis of laminated composite plates

Pandey R., Shukla K. K., Jain A.

International Journal of Applied Mechanics and Engineering

|

2008

|

Vol. 13, no 3

707-733

EN

The paper presents an analytical-numerical solution of the nonlinear flexural response of the moderately thick laminated co-rectangular plates subjected to mechanical and thermo-mechanical loadings. It also investigates the effects of non-linear elastic supports on flexural response of laminated composite plates. The problem is formulated using higher order shear deformation theory (HSDT) and von Karman's nonlinearity. The governing nonlinear coupled partial differential equations are linearized utilizing quadratic extrapolation technique. The spatial descretisation of the linear differential equations is carried out using the fast converging finite double Chebyshev series. Numerical results depicting the central displacements and moments of laminated composite plates for different boundary conditions are presented. Effects of plate geometry, elastic foundation parameters, and temperature dependent mechanical and thermal properties of material on the flexural response are presented.