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1

Steady-state vibrations of a beam structure under moving horizontal force-analytical approach

Ataman Magdalena, Szcześniak Wacław

Archives of Civil Engineering

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2025

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Vol. 71, nr 1

543--558

EN

The following paper presents a dynamic analysis of a railway superstructure subjected to horizontal moving loads. The superstructure including the substructure was modelled with two infinitely long beams connected by an elastic layer. The structure rests on a Winkler elastic bed carrying loads in the horizontal direction. The analysed system is loaded by a moving horizontal force that moves at a constant speed tangentially to the upper beam. Similar solution and calculations can be provided for the loaded lower beam. The problem was solved analytically by bringing the two equations of motion into a single differential equation with an higher order. The solution is illustrated with a computational example and the results are analysed in detail. The relationships between the horizontal displacements and the characteristics of the structure and the subsoil, and between the strains and the characteristics of the structure and the subsoil, were shown in the corresponding diagrams obtained using the analytical method. The effect of load velocity on the horizontal displacements and strains of the system under study was also investigated and the results are shown in the figures. The problem solved and the results obtained can be applied to the dynamic analysis of the railway superstructure and the substructure. The results can be used as a benchmark for FEM analysis of more complex engineering structures under moving loads caused by railroad vehicles. The issue is particularly relevant for high-speed railways.

PL

W artykule przedstawiono analizę dynamiczną nawierzchni kolejowej poddanej działaniu poziomych obciążeń ruchomych. Konstrukcja nawierzchni wraz z podtorzem zamodelowana została dwiema nieskończenie długimi belkami połączonymi warstwą sprężystą. Struktura spoczywa na podłożu sprężystym Winklera przenoszącym obciążenia w kierunku poziomym. Analizowany układ obciążony jest ruchomą siłą poziomą, która przesuwa się ze stałą prędkością stycznie do górnej lub dolnej belki. Zadanie rozwiązano analitycznie doprowadzając dwa równania ruchu do jednego równania różniczkowego o zawyżonym rzędzie. Zaproponowane modele nawierzchni kolejowej mają zastosowanie do analizy dynamicznej zarówno nawierzchni podsypkowych, jak i bezpodsypkowych. Zastosowanie analitycznych metod rozwiązania zadań pozwala na analizę wpływu poszczególnych parametrów w pełnym zakresie prędkości oraz danych charakteryzujących nawierzchnię i podłoże. Przedstawione w pracy rozwiązania mogą mieć zastosowanie do badania stateczności toru kolejowego zarówno w planie jak i w profilu, przy uwzględnieniu odpowiednich sztywności belki oraz parametrów opisujących podłoże gruntowe. Uzyskane rozwiązania analityczne mogą być wykorzystane do weryfikacji wyników otrzymanych metodą elementów skończonych. Zjawisko fali podłużnej jest także istotne przy projektowaniu łożysk i przyczółków w mostach kolejowych i drogowych. W dalszej perspektywie należałoby dodatkowo przebadać wpływ sił wywołanych wężykowaniem pojazdów szynowych na odpowiedź dynamiczną nawierzchni i podtorza. Przeanalizowane w artykule zagadnienie jest szczególnie istotne w przypadku kolei dużych prędkości.

2

Modeling the impact of elastic bodies taking into account dry positional friction and the coefficient of velocity restoration

Bohomolov Olexiy, Zavhorodnii Oleksiy, Hurskyi Petro, Bredykhin Vadym, Bohomolov Oleksandr, Shchur Taras, Walichnowska Patrycja, Kruszelnicka Weronika, Tomporowski Andrzej

Advances in Science and Technology. Research Journal

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2025

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Vol. 19, no. 7

25--35

EN

Considering dry positional friction in the model of collinear elastic impact, the stiffness of the system during compression is greater than its stiffness during decompression. The coefficients of the relative motion equation of the bodies involved in the impact have different values at these stages. They depend not only on the geometry of the bodies in the interaction region and the mechanical properties of their materials, but also on the velocity restitution coefficient, which is one of the input parameters of the model, determined a priori through experiments. In the case of power-law nonlinearities in the stiffness of the dynamic system, which corresponds to the solutions of the contact problem in the theory of elasticity, the equation of relative motion of the bodies has closed analytical solutions. During the compression phase, the solution is expressed with an Ateb-sine and its powers, while during the decompression phase, it is expressed with an Ateb-cosine and its powers. To simplify the numerical implementation of the solutions, an approximation of these special functions by trigonometric functions is proposed. This approximation ensures a calculation accuracy of three significant digits after the decimal point. Examples of calculations are provided, demonstrating that accounting for positional friction leads to a decrease in the maximum compression of the bodies, an increase in the peak impact force, and an extension of its duration over time. Variations of impacts between bodies with surfaces of second- and fourth-order in the dynamic interaction region are calculated. The model has much in common with the classical model, but additionally considers the actual value of the velocity restitution coefficient, which is less than one, in collinear impacts. The presented theory applies only to elastic impacts with low initial collision velocities, where no plastic deformations occur.

3

Analytical solution for one-dimensional consolidation of unsaturated soils under dynamic load

Liu Zhi-Yi, Song Yu, Zhou Feng-Xi, Wang Li-Ye

Journal of Theoretical and Applied Mechanics

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2023

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Vol. 61 nr 1

11--22

EN

Firstly, in this paper, based on the theory of the porous elastic medium and combined with the effective stress principle of unsaturated soil, a set of governing equations is established to describe consolidation of the unsaturated soil. Secondly, an analytical expression under any dynamic loads is obtained with the help of Laplace integral transformation. Finally, analysis of numerical examples under specific boundary conditions is made to discuss one-dimensional consolidation characteristics under harmonic loads and the influence of factors on the consolidation characteristics of unsaturated soil, such as excitation frequency and initial saturation.

4

Influence of Nonlinear Shear Modulus Change of Elastomeric Shell of a Composite Tractive Element with a Damaged Structure on its Stress State

Belmas Ivan, Kolosov Dmytro, Onyshchenko Serhii, Bilous Olena, Tantsura Hanna

Inżynieria Mineralna

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2023

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no. 1

147--154

EN

Purpose. Determination of a dependency of a stress state for composite elastomer-cable tractive element with a broken structure on a nonlinear dependency of shear modulus on deformations in the elastomeric shell. Methods. Analytical solution of a model of a composite tractive element with disturbed structure and a deformation-dependent shear modulus of an elastomeric shell. Findings. Algorithm for determining a stress state of a composite tractive element with broken structure and a deformation-dependent shear modulus. Scientific novelty. Character of dependency for a stress state of a composite tractive element on a nonlinear dependency of shear modulus on deformations. Practical significance. A possibility to determine the dependency of a stress state of a composite elastomer-cable tractive element on a nonlinear shear modulus. Scientific novelty. Character of dependency for a stress state of a composite tractive element on a nonlinear dependency of shear modulus on deformations. Practical significance. A possibility to determine the dependency of a stress state of a composite elastomer-cable tractive element on a nonlinear shear modulus allows considering the effect of this phenomenon on the tractive element strength and ensures an increase of its operational safety.

PL

Artykuł przedstawia studium i analizę funkcjonalną wymagań światowego przemysłu metalurgicznego co do jakości rud żelaza w podziemnych kopalniach Ukrainy. Stwierdzono zależności wpływu kształtu i parametrów przestrzeni kompensacyjnych na ich stateczność i wskaźniki jakości rudy. Udowodniono, że komora wyrównawcza w kształcie trapezu pionowego charakteryzuje się największą stabilnością i jest stabilna w zakresie wszystkich rozważanych głębokości, nawet w rudach o twardości 3–5 punktów. Mniejszą stateczność wykazuje komora kompensacji pionowej o kształcie sklepionym z niewielkimi spadkami w przyczółku sklepienia komory w rudach o twardości 3–5 punktów na głębokości 2000 m. Komora z opadami o rożnym natężeniu występuje w dolnej części nachylonych odsłonięć namiotu w rudach o twardości 3–5 punktów na głębokości 1750 m lub większej. Pomieszczenie kompensacji poziomej ma najmniejszą stateczność; spadki występują w rudach o twardości 3–5 punktów na głębokości 1400 m, a na głębokościach 1750–2000 m pozostają stabilne tylko w rudach twardszych. Stwierdzono, że zastosowanie komór kompensacyjnych o dużej stabilności umożliwia osiągnięcie ich maksymalnej objętości, zwiększenie ilości wydobywanej czystej rudy, zmniejszenie jej rozrzedzenia, poprawę jakości wydobywanej masy rudy, a co za tym idzie, wzrost jej ceny i konkurencyjności rynkowej.

5

Free vibration analysis of composite face sandwich plate strengthens by Al2O3 and SiO2 nanoparticles materials

Al-Shablle Marwan, Njim Emad Kadum, Jweeg Mushin J., Al-Waily Muhannad

Diagnostyka

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2023

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Vol. 24, No. 2

art. no. 2023205

EN

The main novelty of the paper is that analytical, experimental, and numerical analyses are used to investigate the free vibration problem of a sandwich structure in which Nanocomposites skins (SiO2/epoxy and Al2O3/epoxy) at different densities are used as the face sheet. The volume fraction's of nanoparticle addition varies (0% to 2.5%). The present free vibration was derived based on Kirchhoff's theory and aspiration to obtain the natural frequency. The results show that in structures with SiO2 nanoparticles with a density of 1180 kg⁄m3, the optimum increase (VF = 2.5%) is 50% in Young's modulus and 22% in natural frequency, while at a density of 1210 kg⁄m3 is 56 % in Young's modulus and 24.5% in natural frequency. Furthermore, the same structures reinforced with Al2O3 Nano-particles show that at the density of 1180 kg⁄m3 , the optimum (VF=2.5%) parentage increase in Young's modulus is 41% and 19% in natural frequency, while at the density of 1210 kg⁄m3 is 46% in Young's modulus and 21% in natural frequency. A numerical investigation was used to validate the obtained results of the analytical solution. The findings also show an acceptable convergence between analytical and numerical techniques with a maximum discrepancy not exceeding 3%.

6

Refined energy method for the elastic flexural-torsional buckling of steel H-section beam-columns. Part I: Formulation and solution

Giżejowski Marian, Barszcz Anna, Wiedro Paweł

Archives of Civil Engineering

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2023

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Vol. 69, nr 1

513--537

EN

Closed form solutions for the flexural-torsional buckling of elastic beam-columns may only be obtained for simple end boundary conditions, and the case of uniform bending and compression. Moment gradient cases need approximate analytical or numerical methods to be used. Investigations presented in this paper deal with the analytical energy method applied for any asymmetric transverse loading case that produces a moment gradient. Part I of this paper is devoted entirely to the theoretical investigations into the energy based out-of-plane stability formulation and its general solution. For the convenience of calculations, the load and the resulting moment diagram are presented as a superposition of two components, namely the symmetric and antisymmetric ones. The basic form of a non-classical energy equation is developed. It appears to be a function dependent upon the products of the prebuckling displacements (know from the prebuckling analysis) and the postbuckling deformation state components (unknowns enabling the formulation of the stability eigenproblem according to the linear buckling analysis). Firstly, the buckling state solution is sought by presenting the basic form of the non-classical energy equation in several variants being dependent upon the approximation of the major axis stress resultant M𝑦 and the buckling minor axis stress resultant Mz. The following are considered: the classical energy equation leading to the linear eigenproblem analysis (LEA), its variant leading to the quadratic eigenproblem analysis (QEA) and the other non-classical energy equation forms leading to nonlinear eigenproblem analyses (NEA). The novel forms are those for which the stability equation becomes dependent only upon the twist rotation and its derivatives. Such a refinement is allowed for by using the second order out-of-plane bending differential equation through which the minor axis curvature shape is directly related to the twist rotation shape. Secondly, the effect of coupling of the in-plane and out-of-plane buckling forms is taken into consideration by introducing approximate second order bending relationships. The accuracy of the classical energy method of solving FTB problems is expected to be improved for both H- and I-section beam-columns. The outcomes of research presented in this part are utilized in Part II.

PL

Rozwiązania w postaci zamkniętej dla wyboczenia giętno-skrętnego (FTB) sprężystych belek-słupów można uzyskać tylko dla prostych warunków brzegowych oraz przypadku równomiernego zginania i ściskania. Przypadki zmiennego momentu zginającego wymagają zastosowania przybliżonych metod analitycznych lub numerycznych. Badania przedstawione w niniejszym artykule dotyczą analitycznej metody energetycznej, stosowanej dla dowolnych przypadków asymetrycznych obciążeń poprzecznych, wywołujących nierównomierny moment zginający. Cześć I prezentowanego artykułu jest w całości poświęcona badaniom teoretycznym nad energetyczną formułą utraty stateczności z płaszczyzny zginania i jej ogólnemu rozwiązaniu. Dla wygody obliczeń obciążenie i wykres momentów zginających przedstawiono jako superpozycje dwóch składowych: symetrycznej i antysymetrycznej. Opracowano podstawową postać nieklasycznego (udoskonalonego) równania energetycznego. Jest ono funkcjonałem zależnym od iloczynów odkształceń stanu przedwyboczeniowego, przemieszczeń osi pręta i ich pochodnych, odpowiednio - 𝑢0 i 𝑤0, oraz składowych stanu odkształcenia pokrytycznego, przemieszczenia z płaszczyzny zginania przedkrytycznego i kąta skręcenia, odpowiednio - 𝑣0 i 𝜙𝑥. Przemieszczenia przedwyboczeniowe 𝑢0 osi pręta i 𝑤0 w płaszczyźnie zginania są znane i mogą być powiązane z siłą osiową 𝑁 i momentem zginającym względem osi głównej 𝑀𝑦 otrzymanymi z analizy pierwszego rzędu (LA). Składowe stanu deformacji 𝑣0 i 𝜙𝑥 z płaszczyzny płaskiego stanu zginania oraz ich pochodne są niewiadomymi umożliwiającymi sformułowanie problemu stateczności jako problemu wartości własnych (LBA). W artykule, po pierwsze, poszukiwane jest rozwiązanie stanu wyboczenia poprzez przedstawienie podstawowej postaci nieklasycznego równania energetycznego w kilku wariantach, zależnych od aproksymacji momentu 𝑀𝑧 , a mianowicie klasycznego, prowadzącego do analizy liniowego problemu własnego (LEA) i kwadratowego problemu własnego (QEA) oraz innych form prowadzących do nieliniowych analiz problemów własnych (NEA). Nowe formy to te, dla których równanie stateczności zależy tylko od kąta skręcenia i jego pochodnych. Takie udoskonalenie jest możliwe, gdy do zginania z płaszczyzny zastosowane zostanie równanie różniczkowe drugiego rzędu, za pomocą którego krzywizna osi słabszej jest bezpośrednio powiązana z kątem skręcenia. Po drugie, uwzględniono efekt sprzężenia form wyboczenia w płaszczyźnie i z płaszczyzny zginania przedwyboczeniowego przez wprowadzenie przybliżonych zależności zginania drugiego rzędu. Dzięki uwzględnieniu tych efektów znacznie poprawiono dokładność klasycznej metody energetycznej rozwiazywania problemów FTB elementów ściskanych i zginanych w płaszczyźnie większej bezwładności przekroju, zarówno w wypadku przekroju dwuteowego H, jak i I. Wyniki tej części są wykorzystywane w Części II, dotyczącej porównania i weryfikacji rozwiązań uzyskanych w formie zamkniętej w Części I artykułu.

7

A unified design model for estimating tunnel performance considering multiple excavation stoppages

Wu Kui, Xing Chenzhe, Yang Yuezong, Shao Zhushan, Zhao Nannan, Chu Zhaofei

Archives of Civil and Mechanical Engineering

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2023

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Vol. 23, no. 4

art. e260, 1--14

EN

In tunnel construction, excavation stoppages are often encountered due to many irresistible factors. This study reveals the influence mechanism of excavation stoppages on tunnel mechanical response by using mathematical analytical method. Firstly, the mathematical expressions for stress release coefficient considering multiple excavation stoppages are redefined and provided, based on the previous stress release coefficient in the negative exponential function form. Secondly, the mechanical model of a deep circular lined tunnel considering excavation stoppages is established, and the unified analytical solutions for tunnel displacement and liner pressure during the unlined and lined (excavation and excavation stoppage) stages are provided, respectively. Furthermore, the proposed solutions are able to capture the average tunnel deformation and liner pressure in Rongjiawan tunnel (an excavation stoppage took place) and agree well with the numerical results. Finally, a parametric investigation is performed, including the influences of starting time and duration time of excavation stoppage, excavation rate and liner installation time. Results show that both the final tunnel displacement and liner pressure are unaffected by starting time and duration time of excavation stoppage. However, their convergence rates are significantly affected by these two excavation stoppage factors. A higher excavation rate before stoppage causes a larger tunnel displacement and a smaller liner pressure. The influence of re-excavation rate is too low to be paid attention. A higher liner pressure can be induced by an earlier liner installation. The findings in this study are helpful for the evaluation of excavation stoppage-caused influence in tunnels.

8

A new and simple model for predicting soil erosion based on hole erosion tests

Cai Weiling, Bora Manash Jyoti, Pekkat Sreeja, Bordoloi Sanandam, Garg Ankit, Sekharan Sreedeep

Acta Geophysica

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2023

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Vol. 71, no. 2

823--836

EN

Determination of erosion characteristics is of great significance to assess the erodibility of geomaterials that are subjected to seepage force. The erosion characteristics indicate soil particle removal in term of internal erosion that might occur in earthen structures. Hole erosion test (HET) is a simple and effective approach to determine erosion characteristics. It is noted that there are not many studies that focus on the development of a theoretical model describing the erosion characteristics and the associated process of soil particle detachment in HETs. The aim of this study is to propose a simple model based on Bernoulli’s principle to interpret erosion characteristics of geomaterials in HETs. An analytical equation was deduced from a physically based model incorporating Bernoulli’s principle and erosion constitutive law for internal erosion within a soil pipe driven by pressure gradient. The analytical equation could be applied to determine soil particle removal, radial erosion propagation, erosion coefficient, and critical shear stress. A series of HETs were performed under different flow rate to verify the proposed model. The obtained results demonstrated that the proposed model allowed for reasonably predicting the amount of soil particle removal and understanding erosion characteristics of soils through the HET.

9

Analysis, evaluation, and optimization of bio-medical thermo-resistive micro-calorimetric flow sensor using an analytical approach

Babaelahi Mojtaba, Sadri Somayyeh

Metrology and Measurement Systems

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2022

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Vol. 29, nr 1

109--125

EN

Sensitive MEMS-based thermal flow sensors are the best choice for monitoring the patient’s respiration prompt diagnosis of breath disturbances. In this paper, open space micro-calorimetric flow sensors are investigated as precise monitoring tools. The differential energy balance equation, including convection and conduction terms, is derived for thermal analysis of the considered sensor. The temperature-dependent thermal conductivity of the thin silicon-oxide membrane layer is considered in the energy balance equation. The derived thermal non-linear differential equation is solved using a well-known analytical method, and a finite-element numerical solution is used for the confirmation. Results show that the presented analytical model offers a precise tool for evaluating these sensors. The effects of flow and thin membrane film parameters on thermo-resistive micro-calorimetric flow sensors’ performance and sensitivity are evaluated. The optimization has been performed at different flow velocities using a genetic algorithm method to determine the optimum configuration of the considered flow sensor. The geometrical parameters are selected as a decision variable in the optimization procedure. In the final step, using optimization results and curve-fitting, the expressions for the optimum decision variables have been derived. The sensor’s optimum configuration is achieved analytically based on flow velocity with the analytical terms for optimum decision variables.

10

Analytical evaluation of the influence of adding rubber layers on free vibration of sandwich structure with presence of nano-reinforced composite skins

Al-Shablle M., Al-Waily M., Njim E.K.

Archives of Materials Science and Engineering

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2022

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Vol. 116, nr 2

57--70

EN

Purpose: Developing structural designs that offer superior vibration properties is still a major challenge, but they stay solid and lightweight simultaneously. Composite faces are frequently used in insulating constructions as an alternative to sheet metal roofs. Rubber overlays have been added to reduce waves' natural frequency and fade time. Design/methodology/approach: The mechanical properties and the natural frequency calculation of the materials that make up the composite structural panels designed for structural applications with the addition of rubber layers were studied in this study. Findings: The results showed the addition of rubber layers with SiO2 nanoparticles with a density of 1180 kg m3, and the optimal decrease (VF = 2.5%) is 38.5% in the natural frequency while at a density of 1210 kg/m3, it is 40.2% in the natural frequency. While the addition of rubber layers with Al2O3 nanoparticles shows a density of 1180 kg/m3, the optimum reduction (VF = 2.5%) is 41% in HF while at a density of 1210 kg/m3 36.8% in an NF 41% during a density of 1210 kg/m3 38.4%. Research limitations/implications: Certain hypotheses were used to apply Kirchhoff's theory to solve the mathematical model of the structure. Practical implications: The work was carried out on the faces of nanocomposites made of SiO2/epoxy and Al2O3/epoxy with different densities and polylactic acid core. The inclusion of nanoparticles as a percentage of the fraction size ranges from 0% to 2.50%. Originality/value: This study's results shed light on the fundamental behaviour of the components that make up the sandwich in the presence of rubber layers.

11

Analytical solution of the electro-mechanical flexural coupling between piezoelectric actuators and flexible-spring boundary structure in smart composite plates

Gohari Soheil, Mozafari F., Moslemi N., Mouloodi Saeed, Sharifi S., Rahmanpanah Hadi, Burvill Colin

Archives of Civil and Mechanical Engineering

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2021

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Vol. 21, no. 1

546--570

EN

An analytical solution has been developed developed in this research for electro-mechanical flexural response of smart laminated piezoelectric composite rectangular plates encompassing flexible-spring boundary conditions at two opposite edges. Flexible-spring boundary structure is introduced to the system by inclusion of rotational springs of adjustable stiffness which can vary depending on changes in the rotational fixity factor of the springs. To add to the case study complexity, the two other edges are kept free. Three advantages of employing the proposed analytical method include: (1) the electro-mechanical flexural coupling between the piezoelectric actuators and the plate’s rotational springs of adjustable stiffness is addressed; (2) there is no need for trial deformation and characteristic function—therefore, it has higher accuracy than conventional semi-inverse methods; (3) there is no restriction imposed to the position, type, and number of applied loads. The Linear Theory of Piezoelectricity and Classical Plate Theory are adopted to derive the exact elasticity equation. The higher-order Fourier integral and higher-order unit step function differential equations are combined to derive the analytical equations. The analytical results are validated against those obtained from Abaqus Finite Element (FE) package. The results comparison showed good agreement. The proposed smart plates can potentially be applied to real-life structural systems such as smart floors and bridges and the proposed analytical solution can be used to analyze the flexural deformation response.

12

Analysis of the transient flow of non Newtonian power law fluids in homogeneous reservoirs with the elastic outer boundary

Zhao Chaochao, Min Chao

Acta Geophysica

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2021

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Vol. 69, no. 5

1865--1875

EN

Since some oil fields have entered the late stage of high water-cut development, the use of polymer flooding and other oil recovery technologies makes the fluid be of non-Newtonian characteristics, but the existing elastic outer boundary percolation model ignores this point. Firstly, this paper establishes a non-Newtonian power-law fluid percolation model with the elastic outer boundary condition. Secondly, the analytical solution of the percolation model is obtained in the Laplace space by the structural method. Thirdly, the double logarithmic characteristic curves are drawn. Finally, the sensitivity analysis of the parameters is carried out. The results show that the wellbore storage coefficient and skin factor play important roles in the early stage of the double logarithmic curves. For the radial flow stage and the later flow stage, the main influence factors are the power-law index, the elastic coefficient and outer boundary radius, respectively. It is concluded that the percolation model established in this paper provides a theoretical basis for reasonably solving the problem that the previous data deviation could not explain the reservoir parameters well. Meanwhile, the application of the structural method can provide a scientific calculation method for the well test analysis of non-Newtonian power-law fluids.

13

The fourth-order ordinary differential equation with the fractional initial/boundary conditions

Siedlecki Jarosław

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

79--87

EN

The initial/boundary value problem for the fourth-order homogeneous ordinary differential equation with constant coefficients is considered. In this paper, the particular solutions an ordinary differential equation with respect to the set of boundary conditions are studied. At least one of the boundary conditions is described by a fractional derivative. Finally, a few illustrative examples of particular solutions to the considered problem are shown.

14

A new mathematical model to calculate the equilibrium scour depth around a pier

Gazi Ainal Hoque, Afzal Mohammad Saud

Acta Geophysica

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2020

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Vol. 68, no. 1

181--187

EN

This paper sheds light on the formulation of a new equilibrium local scour depth equation around a pier. The total bed materials removed from the scour hole due to the force exerted by the fowing fuid after colliding with the pier in the fow feld are estimated. At the equilibrium condition, the shape of the scour hole around the pier may take any form, viz. linear, circular, parabolic, triangular, or combination of diferent shapes. To consider that, two functions are assumed at the stoss and the lee sides of the pier. The total volume of bed materials removed from the scour hole of an arbitrary shape at the stoss and the lee sides of the pier is obtained by integrating the two functions. The equilibrium scour depth is formed by applying the energy balance theorem. An example problem is illustrated and the results are compared with the equations presented by Melville and Coleman (Bridge scour. Water Resources Publication, Colorado, 2000) and HEC-18 (Richardson and Davis in Evaluating scour at bridges, HEC-18. Technical report no. FHWA NHI, 2001).

15

Differential equation of y = f(y') and interpretation of the solution in Mathematica program

Czajkowski Andrzej Antoni, Oleszak Wojciech Kazimierz, Frączak Piotr Stanisław

Problemy Nauk Stosowanych

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2019

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T. 10

25--34

EN

Introduction and aims: The paper presents a method of solving y=f(y') equations. The main aim of the work is to show how to solve this type of differential equations. In addition, the purpose of the discussion is to present the appropriate algorithms in Mathematica program, which are used to present the geometric interpretation of the obtained solutions. Material and methods: The sources contain material on the subject of differential equations. The method of mathematical analysis has been used. Results: In the analysis of selected examples, the method of substitution of new variable t has been used and the solution of the studied differential equation has been obtained in the form of the system of equations x=x(t) and y=y(t). Conclusion The solution of the differential equation of the type y=f(y') in the form of a system of equations x=x(t) and y=y(t) can be interpreted graphically using an appropriately used algorithm in Mathematica numerical program.

PL

Wstęp i cele: W pracy przedstawiono metodę rozwiązywania równań typu y=f(y'). Głównym celem pracy jest pokazanie sposobu rozwiązywania tego typu równań różniczkowych. Ponadto celem rozważań jest przestawienie odpowiednich algorytmów w programie Mathematica, które służą do przedstawienia interpretacji geometrycznej otrzymanych rozwiązań. Materiały i metody: Źródła zawierają materiał dotyczący tematyki równań różniczkowych. Zastosowano metodę analizy matematycznej. Wyniki: W analizie wybranych przykładów zastosowano metodę podstawienia nowej zmiennej t i otrzymano rozwiązanie badanego równania różniczkowego w postaci układu równań x=x(t) i y=y(t). Wniosek: Rozwiązanie równania różniczkowego typu y=f(y') w postaci układu równań x=x(t) i y=y(t) można zinterpretować graficznie stosując odpowiednio zastosowany algorytm w programie numerycznym Mathematica.

16

Differential equation of x = f(y') and interpretation of the solution in Mathematica program

Czajkowski Andrzej Antoni, Oleszak Wojciech Kazimierz, Frączak Piotr Stanisław

Problemy Nauk Stosowanych

|

2019

|

T. 10

15--24

EN

Introduction and aims: The paper presents a method of solving x=f(y') equations. The main aim of the work is to show how to solve this type of differential equations. In addition, the purpose of the discussion is to present the appropriate algorithms in Mathematica program, which are used to present the geometric interpretation of the obtained solutions. Material and methods: The sources contain material on the subject of differential equations. The method of mathematical analysis has been used. Results: In the analysis of selected examples, the method of substitution of new variable t has been used and the solution of the studied differential equation has been obtained in the form of the system of equations x=x(t) and y=y(t). Conclusion: The solution of the differential equation of the type x=f(y') in the form of a system of equations x=x(t) and y=y(t) can be interpreted graphically using an appropriately used algorithm in Mathematica numerical program.

PL

Wstęp i cele: W pracy przedstawiono metodę rozwiązywania równań typu y=f(y'). Głównym celem pracy jest pokazanie sposobu rozwiązywania tego typu równań różniczkowych. Ponadto celem rozważań jest przestawienie odpowiednich algorytmów w programie Mathematica, które służą do przedstawienia interpretacji geometrycznej otrzymanych rozwiązań. Materiały i metody: Źródła zawierają materiał dotyczący tematyki równań różniczkowych. Zastosowano metodę analizy matematycznej. Wyniki: W analizie wybranych przykładów zastosowano metodę podstawienia nowej zmiennej t i otrzymano rozwiązanie badanego równania różniczkowego w postaci układu równań x=x(t) i y=y(t). Wniosek: Rozwiązanie równania różniczkowego typu y=f(y') w postaci układu równań x=x(t) i y=y(t) można zinterpretować graficznie stosując odpowiednio zastosowany algorytm w programie numerycznym Mathematica.

17

Analytical design of selected geotechnical solutions which protect civil structures from the effects of underground mining

Misa Rafał, Sroka Anton, Tajduś Krzysztof, Dudek Mateusz

Journal of Sustainable Mining

|

2019

|

Vol. 18, iss. 1

1--7

EN

This paper presents the authors' computational methods based on Knothe's theory. The methods enable the estimation of the reduction coefficient for effects which originate from mining operations performed via the application of a longitudinal structure which is sunk in to the ground. It could be, for example, a partition, which as a structural gap fulfils the function of an expansion grout, or via breaking the subsoil continuity (e.g. because of creating a peat-filled ditch or using a natural gap). Demonstrative calculations have been carried out in a few cases, i.a. to protect a structure situated in the vicinity of a planned tunnel. Additionally, some examples of the discontinuity zone which impact the obtained deformation values have been presented. The calculation method has been tested in case studies. The results of the calculations clearly show the positive influence of the applied geotechnical solutions on the minimisation of mining damage.

18

Screw dislocations in piezoelectric laminates with four or more phases

Wang X., Chen L., Schiavone P.

Archives of Mechanics

|

2019

|

Vol. 71, nr 3

239--262

EN

We present an analytical solution to the problem of a screw dislocation in a four-phase piezoelectric laminate composed of two piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media. A new version of the complex variable formulation is proposed such that the 2 × 2 real symmetric matrix appearing in the formulation becomes dimensionless. Using analytic continuation, the original boundary value problem is reduced to the identification of a single 2D analytic vector function which is completely determined following rigorous solution of the resulting linear recurrence relations in matrix form. An explicit expression for the image force acting on the piezoelectric screw dislocation is obtained once the single 2 × 2 real matrix function is identified. We also discuss the solution for a screw dislocation in an N-phase piezoelectric laminate composed of N − 2 piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media.

19

Analytical and numerical solving of linear non-homogeneous differential equations of the first-order with changeable coefficients by using constant variation method and application of Mathematica program

Czajkowski A. A., Oleszak W. K., Dyrdał J.

Problemy Nauk Stosowanych

|

2018

|

T. 8

5--20

EN

Introduction and aim: The paper presents the analytical and numerical algorithm of solving linear nonhomogeneous equations of the first order with changeable coefficients. The aim of the work is to show the algorithms for solving equations both analytically and numerically. The additional aim is to show numerical algorithms and graphical interpretation of solutions. Material and methods: Some selected equations have been chosen from the subject literature. In the solutions the constant variation method has been presented. Results: The paper presents the selected linear non-homogeneous equations of the first order with changeable coefficients containing exponential, logarithmic, trigonometric and cyclometric functions. Conclusion: Taking into account the constant variation method it is possible to solve the first order linear nonhomogeneous differential equations with changeable coefficients. Using the Mathematica program it is possible quickly get a solution and create its graphical interpretation.

PL

Wstęp i cel: W pracy pokazano algorytmy analityczny i numeryczny rozwiązywania równań różniczkowych liniowych niejednorodnych pierwszego rzędu o zmiennych współczynnikach. Celem pracy jest pokazanie algorytmu rozwiązywania równań zarówno sposobem analitycznym jak i numerycznym. Ponadto również dodatkowym celem jest pokazanie algorytmów numerycznych oraz interpretacji graficznej rozwiązań. Materiał i metody: Wybrane równania zaczerpnięto z literatury przedmiotu. W rozwiązaniach równań zastosowano metodę wariacji stałej. Wyniki: W pracy opracowano wybrane równania różniczkowe liniowe niejednorodne pierwszego rzędu o zmiennych współczynnikach zawierających funkcje wykładnicze, logarytmiczne, trygonometryczne i arcus. Wniosek: Stosując metodę uzmienniania stałej jest możliwe rozwiązywanie równań różniczkowych liniowych niejednorodnych pierwszego rzędu o zmiennych współczynnikach. Wykorzystując program Mathematica można szybko uzyskać rozwiązanie oraz sporządzić jego interpretację graficzną.

20

Analytical and numerical solving of linear non-homogeneous differential equations of the second-order with changeable coefficients by using constant variation method and application of Mathematica program

Czajkowski A. A., Oleszak W. K., Skorny G. P., Udała R.

Problemy Nauk Stosowanych

|

2018

|

T. 8

21--38

EN

Introduction and aim: The paper presents the analytical and numerical algorithm of solving linear nonhomogeneous equations of the second order with changeable coefficients. The aim of the work is to show the algorithms for solving equations both analytically and numerically. The additional aim is to make some graphical interpretation of solutions. Material and methods: Some selected equations have been chosen from the subject literature. In the solutions the constant variation method has been presented. Results: The paper presents the selected linear non-homogeneous equations of the second order with constant coefficients containing linear, homographic, logarithmic and trigonometric functions. Conclusion: Taking into account the constant variation method it is possible to solve the second order linear non-homogeneous differential equations with changeable coefficients. Using the Mathematica program it is possible quickly get a solution and create its graphical interpretation.

PL

Wstęp i cel: W pracy pokazano algorytm analityczny i numeryczny rozwiązywania równań różniczkowych liniowych niejednorodnych drugiego rzędu o zmiennych współczynnikach. Celem pracy jest pokazanie algorytmu rozwiązywania równań zarówno sposobem analitycznym jak i numerycznym. Ponadto dodatkowym celem jest interpretacji graficznej rozwiązań. Materiał i metody: Wybrane równania zaczerpnięto z literatury przedmiotu. W rozwiażanich równań zastosowano metodę wariacji stałej. Wyniki: W pracy opracowano wybrane równania różniczkowe liniowe niejednorodne drugiego rzędu o zmiennych współczynnikach zawierających funkcje liniowe, homograficzne, logarytmiczne i trygonometryczne. Wniosek: Stosując metodę uzmienniania stałej jest możliwe rozwiązywanie równań różniczkowych liniowych niejednorodnych drugiego rzędu o zmiennych współczynnikach. Wykorzystując program Mathematica można szybko uzyskać rozwiązanie oraz sporządzić jego interpretację graficzną.