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  1. 4 Archives of Civil and Mechanical Engineering
  2. 3 Engineering Transactions
  3. 3 Journal of Theoretical and Applied Mechanics
  4. 2 Archives of Civil Engineering
  5. 2 Vibrations in Physical Systems
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  1. 3 Magnucki Krzysztof
  2. 3 Zenkour A. M.
  3. 2 Kędzia Piotr
  4. 2 Magnucki K.
  5. 2 Obara P.
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  1. 1 2023
  2. 1 2022
  3. 3 2020
  4. 1 2019
  5. 4 2018
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1
Content available Free flexural vibration of a sandwich beam on an elastic foundation with variable properties
Magnucki K., Wstawska I., Kedzia P.
Archives of Mechanics
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2023
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Vol. 75, nr 6
619--646
EN
Free flexural vibration of a simply supported sandwich beam on an elastic foundation is the main purpose of the presented investigation. An analytical model of multi-layered beam on elastic foundation has been prepared. The authors submitted an original beam-foundation interaction model which based on variable parameters of the foundation and their influence on the beam response. This explanation leads to the possibility of continuous characterization of the beam-foundation interplay. A nonlinear mathematical function for symmetrical properties of the foundation has been adopted. The frequency equation as a function of geometric and mechanical properties of the beam and the parameters of the elastic foundation was derived using the Galerkin method. The analytical investigation has been divided into two parts: the analysis of elastic foundation with constant and variable properties. The unconventional shape function and the function of deflection have been introduced and employed. Moreover, the finite element analysis has been performed. Sample analytical and numerical calculations have been performed, demonstrating a good concurrence between both models. The difference between analytical and numerical values of the fundamental natural frequency did not exceed 0:5%.
2
Content available Analytical investigation of a beam on elastic foundation with nonsymmetrical properties
Wstawska Iwona, Magnucki Krzysztof, Kędzia Piotr
Engineering Transactions
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2022
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Vol. 70, iss. 1
77--93
EN
The subject of presented analytical and numerical investigation is the stability of an axially compressed beam on an elastic foundation. The shape function of the foundation was assumed. The formula was supplemented with the offset parameter. The critical values of loads were calculated and presented as a function of geometric and mechanical properties of the beam and nonsymmetrical properties of the elastic foundation. The highest values of critical loads can be obtained for the highest values of shape parameter and the lowest values of amplitudes of shape function. The values of critical loads increase with the increase of the value of the offset parameter.
3
Content available Free axisymmetric flexural vibrations of circular plate with symmetrically varying mechanical properties supported on elastic foundation
Magnucki Krzysztof
Vibrations in Physical Systems
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2020
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Vol. 31, nr 2
art. no. 2020217
EN
The subject of the paper is a circular plate with clamped edge supported on elastic foundation. Mechanical properties of the plate symmetrically vary in its thickness direction. Free axisymmetric flexural vibration problem of the plate with consideration of the shear effect is analytically studied. Two partial differential equations of motion based on the Hamilton principle are obtained. The system of equations is analytically solved and the fundamental natural frequency of axisymmetric vibration for example plates is derived.
4
Content available Vibration of a composite elastic beam on an inhomogeneous elastic foundation
Sahin Onur
Journal of Applied Mathematics and Computational Mechanics
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2020
|
Vol. 19, nr 2
107--119
EN
The lowest vibration modes of a composite beam resting on an inhomogeneous elastic foundation in the case of a clamped-free boundary condition are investigated. It is observed that the relation between the frequency and foundation constant has an effect on the valid solution of the problem. An asymptotic procedure is employed to derive the eigenfrequencies as well as the eigenforms revealing that only under certain conditions on the ratios of material parameters. Numerical illustrations are presented to confirm that the obtained asymptotic frequencies agree well with the exact frequencies in the lowest frequency range. Comparisons of asymptotic and exact displacements are also presented and a remarkable agreement is observed for high-contrast beam components.
5
Content available remote Bending and buckling analysis of FGM plates resting on elastic foundations in hygrothermal environment
Zenkour A. M., Radwan A. F.
Archives of Civil and Mechanical Engineering
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2020
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Vol. 20, no. 4
198--220
EN
The present paper deals with the effect of exponential temperature and moisture concentration on the bending and buckling analysis of functionally graded plates resting on two-parameter elastic foundations via a four-variable exponential shear deformation theory. The mechanical properties of the plates are assumed to vary through the thickness. The equations of equilibrium are derived using Hamilton’s principle. The present solutions are derived using Navier’s method. Using Navier’s solution the numerical results are presented and compared well with those available in the literature. Discussions are made to show how the foundation stiffness, hygrothermal loading and other parameters have a significant influence on the bending and buckling analysis of FG plates under hygrothermal and mechanical loading.
6
Content available remote Approximate estimation of stability of homogeneous beam on elastic foundation
Wstawska Iwona, Kędzia Piotr, Magnucki Krzysztof
Engineering Transactions
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2019
|
Vol. 67, iss. 3
429--440
EN
The paper deals with a proposition of obtaining an analytical solution for a beam on elastic foundation. The main objective of presented work was stability analysis of the axially compressed beam. The analytical model was proposed. Shape function for inhomogeneous properties of the foundation was assumed. The Galerkin method was used to calculate the values of critical forces. Main conditions have been defined. The critical loads as a function of geometric and mechanical properties of the beam as well as inhomogeneous properties of the elastic foundation have been calculated.
7
Content available The use of modelling of impacts exerted by means of transport on the environment for ensuring safety
Bril J., Rydygier E.
Scientific Journal of the Military University of Land Forces
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2018
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Vol. 50, No. 2(188)
162--175
EN
The article presents the model research on impacts exerted by means of transport on the structures. In modelling the dynamics of transport systems the dynamic properties of the ground forming the foundation soil for tracks or roadways have been taken into account. The ground has been modelled as an elastic half-space. The dynamics of an infinite mass band being in contact with an elastic half-space has been investigated. As part of the research on impacts exerted by means of transport on structures a model of a problem has been examined where an automotive vehicle, representing a concentrated force in motion, is in contact with a roadway described as a rigid body coupled with an elastic half-space. It has been demonstrated that a surface (Rayleigh) wave propagates in the ground, being a continuous (elastic) medium, and acts on a structure modelled as a rigid body. The research results have been presented in the form of vertical and horizontal transmittances of the ground for different frequencies of loading with different unit forces.
8
Content available Układ dwóch nieskończenie długich belek z wypełnieniem sprężystym pod obciążeniem ruchomym
Ataman M., Szcześniak W.
Autobusy : technika, eksploatacja, systemy transportowe
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2018
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R. 19, nr 12
285--288
PL
W artykule omówiono analityczne rozwiązanie zadania z dynamiki układu dwóch nieskończenie długich belek z wypełnieniem sprężystym. Układ spoczywa na podłożu Winklera i jest obciążony ruchomą siłą skupioną. Ponieważ zagadnienie jest stacjonarne dla obserwatora poruszającego się z obciążeniem, cząstkowe równania różniczkowe opisujące drgania układu transformowano na równania różniczkowe zwyczajne w układzie współrzędnych związanym z ruchomą siłą. Układ równań doprowadzono do jednego równania różniczkowego, zawyżonego rzędu, na ugięcie dolnej belki. Rozwiązanie zadania doprowadzono do prostej całki nieskończonej Fouriera. W pracy przedstawiono obszerny wykaz publikacji z literatury przedmiotu [1-45].
EN
The paper discussed the analytical solution of a dynamic problem of a system of two infinite beams separated by an elastic core. The beams’ system rests on the Winkler foundation and is loaded with a moving concentrated force. Because the problem is stationary for an observer moving with the load, partial differential equations, describing the vibrations of the system, were transformed into ordinary differential equations in the coordinate system related to the moving force. The system of equations was transformed to one differential equation of an eighth order. The equation defines deflection of the lower beam. The solution of the problem was resulted to the simple infinite Fourier integral. An extensive list of publications on the related literature is presented in the paper [1-45].
9
Content available remote Compressive study of functionally graded plates resting on Winkler–Pasternak foundations under various boundary conditions using hyperbolic shear deformation theory
Zenkour A. M., Radwan A. F.
Archives of Civil and Mechanical Engineering
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2018
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Vol. 18, no. 2
645--658
EN
Equilibrium equations of a functionally graded plate resting on two-parameter elastic foundations are derived using hyperbolic shear deformation theory. This theory takes into account the hyperbolic distribution of transverse shear deformation and satisfies that the corresponding shear stresses equal to zero on upper and lower surfaces of the plate without requiring any shear correction factors. Eight different types of boundary conditions are considered. Governing equations are obtained including the plate-foundation interaction. The present results are compared well with the corresponding available in the literature. Effects of boundary conditions, linear (Winkler) modulus and shear foundation (Pasternak) modulus, gradient index, plate aspect ratio, side-to-thickness ratio on the stresses and deflections are all discussed. It is established that the present model is more accurate than some theories developed previously.
10
Content available Trigonometric solution for an exponentially graded thick plate resting on elastic foundations
Zenkour A. M.
Archive of Mechanical Engineering
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2018
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Vol. LXV, nr 2
193--208
EN
This article investigates the solution of exponentially graded (EG) thick rectangular plates resting on two-parameter elastic foundations according to a trigonometric plate theory (TPT). This theory includes the effect of both shear and normal strains thickness without needing to any shear correction factor. The displacement fields contains initial terms of a power series across plate thickness as well as additional trigonometric terms. The material properties of plate is graded such that Lamé’s coefficients convert exponentially in a given constant orientation.Equilibrium equations according to the EG plate resting on Pasternak’s foundations are derived. The solution is obtained by using Navier’s technique. Numerical results for the EG thick plate on elastic foundations are presented, and compared with those available in the literature. The influences of Winkler’s and Pasternak’s parameters, side-to-thickness ratio, inhomogeneity parameter and aspect ratio on the bending responses of EG plates are investigated.
11
Content available remote Analytical solution for beams with multipoint boundary conditions on two-parameter elastic foundations
Aslami M., Akimov P. A.
Archives of Civil and Mechanical Engineering
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2016
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Vol. 16, no. 4
668--677
EN
An efficient analytical method is presented for the closed form solution of continuous beams on two-parameter elastic foundations. The general form of the governing equation is reduced to a system of first-order differential equations with constant coefficients. The system is then solved using Jordan form decomposition for the coefficient matrix and construction of the fundamental solution. Common types of boundary conditions (pinned and roller support, hinge connection, fixed and free end) can be applied to an arbitrary point on the beam. The method has a completely computer-oriented algorithm, computational stability, and optimal conditionality of the resultant system and is a powerful alternative to the analytical solution of beams with multipoint boundary conditions on one- or two-parameter elastic foundations. Examples with different types of loading, boundary conditions, and foundation are presented to verify the method.
12
Content available Effect of damages on crack development in coating on elastic foundation
Mirsalimov V. M., Hasanov S. G.
Journal of Theoretical and Applied Mechanics
|
2015
|
Vol. 53 nr 4
823--836
EN
A fracture mechanics problem for a coating linked to the basis made of another elastic material is considered. It is assumed that in the vicinity of technological defect (crack) in the process of loading in the coating, there will arise prefracture zones (damages) that are modeled as the areas of weakened interparticle bonds of the material. In the prefracture zones (interlayers of the overstressed material), the coating material is deformed beyond the limit of elasticity. It is considered that during loading in the cross section of the coating in the vicinity of the crack, there is an arbitrary number of rectilinear prefracture zones. The condition determining the limiting value of the external load at which the crack growth happens is obtained.
13
Content available Research of work of a rotor crush machine on elastic foundation with the use of graphs
Zinko R.
Vibrations in Physical Systems
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2014
|
Vol. 26
343--350
EN
The use of graphs for the research of a rotor crush machine work on elastic foundation is offered. For machines the structure graph of its design is being written. The graph of levels structure of the generalized coordinates of mathematical model is being formed on the basis of structure graph of machine design. It allows building of the mathematical model, which describe the physical processes of the machine work quickly and qualitatively.
14
Probabilistic assessment of beams rested on an elastic foundation
Michenkova S., Theisz G., Frydrysek K.
Modelowanie Inżynierskie
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2014
|
T. 21, nr 52
118-125
EN
This article deals with the application of the Method of Initial Parameters (MIP) for a straight beam rested along its entire length on an elastic foundation. It presents the derivation of relationships and analytical solutions for one example according to the 1st-order theory (i.e. bending and thermal stress). A deterministic and probabilistic approach is implemented – i.e. the application of the Simulation-Based Reliability Assessment (SBRA) Method (Monte Carlo simulations and probabilistic reliability assessment).
15
Charakterystyki dynamiczne średnio-gabarytowych urządzeń okrętowych z sztywnym i elastycznym fundamentowaniem
Murawski L.
Logistyka
|
2014
|
nr 6
7678--7686
PL
Średniej wielkości urządzenia okrętowe są relatywnie duże w stosunku do urządzeń wykorzystywanych w innych obszarach inżynieryjnych. Pompa balastowa lub wody chłodzącej, w siłowni średniej wielkości statku handlowego, może mieć masę od 400-1500 kg. Dno podwójne statku (miejsce fundamentowania) nie może być uznawane, jako nieskończenie sztywne podczas analiz dynamicznych urządzeń okrętowych. Co więcej, pomiędzy dnem podwójnym a urządzeniem stosuje się dodatkowo sztywne (np. stalowe) lub elastyczne (np. gumowe) podkładki fundamentowe. Różne systemy fundamentowania są przyczyną różnych charakterystyk dynamicznych urządzeń a co za tym idzie różnych ich drgań. Prosta zmiana typu podkładki fundamentowej może zasadniczo zmienić krzywą rezonansową urządzenia. W pracy przedstawiono przykładową analizę pompy balastowej kontenerowca. Podkładki elastyczne (metalowo-gumowe) mają silnie nieliniowe charakterystyki dynamiczne. Autor proponuje uproszczoną metodę obliczeń tego typu konstrukcji. Analizowana pompa może być traktowana, jako ciało nieodkształcalne posadowione na nieliniowych elementach sprężystych (podkładkach). Statyczna i dynamiczna sztywność kadłuba statku (w szczególności dna podwójnego) została uwzględniona podczas obliczeń.
EN
Marine middle-sized devices are quite big in comparison to other engineering area. Ballast or culling water pump can weight from 400 kg up to 1500 kg in the engine room of the medium sized merchant ship. Ships' double bottom (devices' foundation) cannot be treated as an infinitely stiff during dynamic analysis of the device. What is more, stiff (e.g. steel) or elastic (e.g. rubber) foundation pads are used between analysed devices and double bottom. Different foundation system may lead to different dynamic characteristic and different vibration level of the analysed device. Simple change in the type of foundation pads can change resonance curve dramatically. An example of ballast pump of container ship is analysed in the paper. Elastic (rubber) pads have strongly nonlinear dynamic characteristics. The author proposed simplified calculations methodology of that kind of structure. Analysed pump can be treated as an ideally stiff body placed on nonlinear spring elements (pads). Additionally, static and dynamic stiffness of the ship hull (in particular double bottom) is taken into account during analysis.
16
Content available remote Vibration analysis of a thin ring interacting with the annular plate treated as an elastic foundation
Noga S.
Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture
|
2014
|
z. 61, nr 2
131--138
EN
In this paper the free in–plane flexural vibration of a thin ring with elastic foundation of a Winkler type is analysed on the basis of the analytical and numerical methods. Mathematical model of the three–parameter elastic foundation is proposed. The separation of variables method is used to obtain the general solution of the considered system free vibration. Then the eigenvalue problem is solved by using the finite element method. Natural frequencies and normal modes of the system vibration are determined. The achieved results of calculation are discussed and verified by experimental data. FE model is formulated by using ANSYS software. It is important to note that the problems discussed in the paper bring practical advice for engineers dealing with dynamics of circular ring systems like aviation gears and railway wheels, respectively.
PL
W pracy analizowane są drgania giętne w płaszczyźnie cienkiego pierścienia współpracującego z podłożem sprężystym typu Winklera, w oparciu o metody analityczne i numeryczne. Zaproponowano trzyparametryczny model matematyczny podłoża sprężystego. Zagadnienie drgań własnych układu rozwiązano metodą rozdzielenia zmiennych. Następnie wykorzystując metodę elementów skończonych, rozwiązano zagadnienie własne omawianego układu. Otrzymane rozwiązania zweryfikowano badaniami doświadczalnymi. Wymagane modele MES opracowano w środowisku obliczeniowym ANSYS. Warto podkreślić, że omawiane w pracy zagadnienia mogą być pomocne inżynierom, zajmującym się obliczeniami dynamiki układów kołowych (kół zębatych, kół kolejowych).
17
Content available Vibrations and stability of Bernoulli-Euler and Timoshenko beams on two-parameter elastic foundation
Obara P.
Archives of Civil Engineering
|
2014
|
Vol. 60, nr 4
421--440
EN
The vibration and stability analysis of uniform beams supported on two-parameter elastic foundation are performed. The second foundation parameter is a function of the total rotation of the beam. The effects of axial force, foundation stiffness parameters, transverse shear deformation and rotatory inertia are incorporated into the accurate vibration analysis. The work shows very important question of relationships between the parameters describing the beam vibration, the compressive force and the foundation parameters. For the free supported beam, the exact formulas for the natural vibration frequencies, the critical forces and the formula defining the relationship between the vibration frequency and the compressive forces are derived. For other conditions of the beam support conditional equations were received. These equations determine the dependence of the frequency of vibration of the compressive force for the assumed parameters of elastic foundation and the slenderness of the beam.
18
Content available Face wrinkling of sandwich beams under pure bending
Jasion P., Magnucki K.
Journal of Theoretical and Applied Mechanics
|
2012
|
Vol. 50 nr 4
933-941
EN
The paper is devoted to sandwich beams under pure bending. The local buckling problem is analysed. The analytical description of the upper face wrinkling is proposed. From the principle of stationary total potential energy, formulae describing critical stresses in the faces of the beam are derived. The algorithm for determining the critical stresses is shown. Two particular cases of the solution following the core properties are mentioned. The finite element model of the sandwich beam is formulated. The comparison of the results obtained from the proposed analytical model and from FEM analysis is shown for a family of sandwich beams with different thicknesses and core properties.
PL
W pracy omówiono zagadnienie lokalnej stateczności belki trójwarstwowej poddanej czystemu zginaniu. Zaproponowano analityczny model marszczenia górnej, ściskanej okładziny. Z zasady stacjonarności energii potencjalnej wyprowadzono równanie opisujące naprężenia krytyczne. Zaproponowano algorytm pozwalający określić wartość naprężeń krytycznych w zależności od własności rdzenia. Opracowano model numeryczny MES belki trójwarstwowej. Dla rodziny belek przeprowadzono analizę numeryczną na wartości własne, a wyniki porównano z otrzymanymi z zaproponowanego modelu analitycznego.
19
Content available Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the Variational Iteration Method
Younesian D., Saadatnia Z., Askari H.
Journal of Theoretical and Applied Mechanics
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2012
|
Vol. 50 nr 2
639-652
EN
Closed form expressions are obtained for the dynamic response of an elastic beam rested on a nonlinear foundation in this paper. The nonlinear governing equation is solved using the Variational Iteration Method (VIM). An iteration formulation is constructed based on the VIM and the dynamic responses are then obtained. Frequency responses are presented in a closed form and their sensitivity with respect to the initial amplitudes are investigated. A number of numerical simulations are then carried out and performance and validity of the solution procedure is evaluated in the time domain. It is proved that the VIM is quite a reliable and straightforward technique to solve the corresponding set of coupled nonlinear differential equations.
PL
W pracy przedstawiono zamknięte formy rozwiązań opisujących odpowiedź dynamiczną elastycznej belki spoczywającej na nieliniowo sprężystym podłożu. Równanie ruchu rozwiązano przy pomocy iteracyjnej metody wariacyjnej (VIM). Sformułowanie iteracyjne skonstruowane w oparciu o tę metodę pozwoliło na wyznaczenie odpowiedzi dynamicznej belki. Charakterystyki częstościowe przedstawiono w zamkniętej formie i podkreślono ich wrażliwość na wartość amplitud początkowych. Następnie zaprezentowano wyniki kilku symulacji numerycznych w celu oszacowania wydajności i dokładności zastosowanej procedury rozwiązywania równań ruchu w dziedzinie czasu. Wykazano, że metoda VIM jest wystarczająco prosta i wiarygodna w rozwiązywaniu układu sprzężonych, nieliniowych równań różniczkowych.
20
Pomiary energii pękania złączy klejonych z elastyczną warstwą kleju
Budzik M. K., Jumel J., Imielińska K., Shanahan M. E. R.
Inżynieria Materiałowa
|
2010
|
Vol. 31, nr 5
1366-1371
PL
Praca dotyczy opracowywania nowych metod badań adhezji elastycznych klejów, na przykładzie akrylowego kleju "mastic" i porównawczo kleju epoksydowego, na podstawie mechaniki pękania. Przedstawiono wyniki ba- dań przeprowadzonych dla złącza asymetrycznego, czyli np. składającego się z dwóch rożnych materiałów lub tych samych materiałów o rożnej grubości. Metody wyznaczania podstawowych parametrów w próbach mechaniki pękania przeprowadzanych na złączach adhezyjnych wykorzystują najczęściej model belki utwierdzonej, w której front pęknięcia zastępuje się przez sztywne utwierdzenie. Gdy zastosowany klej jest elastyczny, np. ze względu na niski moduł Younga, degradację właściwości itp., obciążenie złącza prowadzi do jego zauważalnych odkształceń, zmieniając wszystkie określane parametry. Używając asymetrycznego testu klina, w którym jeden z klejonych substratów uzbrojono w serię tensometrów oraz modelu Emila Winklera, w pracy zbadano stan odkształcenia kleju i jego wpływ na badane kryteria mechaniki pękania. Stwierdzono, że długość pęknięcia wyznaczona dotychczasowymi metodami jest zawyżana. Jednakże podstawiając długość pęknięcia z jej własnym błędem uzyskuje się prawidłową wielkość energii pękania. Dodatkowym aspektem pracy jest fakt, że przeprowadzone próby dotyczą deformacji kleju odbywającej się przed frontem pęknięcia, nie dotyczą pękania per se, są więc badaniami nieniszczącymi.
EN
Fracture of the adhesive joints was studied in the aluminum plates Bondem with two adhesives: rigid-epoxy or the flexible-acrylic. Since the conventional methods of adhesion assessment, e.g. shear strength are inadequate for the flexible adhesive due to possible adhesive compliance and time /temperature effects the recently developed, instrumented asymmetric wedge test was used. In this contribution we have analyzed the effect of the adhesive layer on the crack length and fracture energy of the bonded joint in mode I test. Within the work two models were compared - standard simple beam theory (SBT) model and elastic foundation Winkler model (Fig. 3). It was found that treating the strain measurements results with SBT model leads to the overestimation of the crack position compared to the energy estimated using the real crack position (Fig. 7 left). However, when the crack length is left with it own overestimation error, the estimated fracture energy is found equal to the one calculated using more sophisticated Winkler model (Fig. 7 right).
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