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1
Content available Dynamic stability of a three-layer beam – generalisation of the sandwich structure theory
Magnucki Krzysztof, Magnucka-Blandzi Ewa
Acta Mechanica et Automatica
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2024
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Vol. 18, no. 1
1--7
EN
The work focuses on the dynamic stability problem of a simply supported three-layer beam subjected to a pulsating axial force. Two analytical models of this beam are developed: one model takes into account the non-linear hypothesis of cross-section deformation, and the other takes into account the standard "broken line" hypothesis. Displacements, strains and stresses for each model are formulated in detail. Based on the Hamilton principle, equations of motion are determined for each of these models. These systems of two differential equations for each model are approximately solved with the consideration of the axial pulsating force, and the fundamental natural frequencies, critical forces and the Mathieu equation are determined. Detailed studies are performed for an exemplary family of beams. The stable and unstable regions are calculated for the three pulsating load cases. The values of fundamental natural frequencies and critical forces of exemplary beams calculated from two models are compared.
2
Content available Strength of a bent sandwich beam with clamped ends
Kustosz Joanna, Magnucki Krzysztof, Goliwąs Damian
Pojazdy Szynowe
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2023
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Nr 1-2
3--7
EN
The subject of the work is a symmetrical sandwich beam with clamped ends under uniformly distributed load. The system of two equilibrium equations, formulated taking into account the literature, was solved analytically. The function of the shear effect and the maximum deflection of the beam were determined. The stress state at the clamped end of the beam is described in detail. The significant influence of the shear effect on the normal stresses in the outer layers of the beam near the clamped end was indicated. Exemplary calculations were made for the adopted family of beams. Moreover, the numerical FEM model of the beam was developed and calculations were made for this adopted family of beams. A comparative analysis of the obtained results was carried out.
3
Content available Bending of a stepped sandwich beam: the shear effect
Kustosz Joanna, Magnucki Krzysztof, Goliwąs Damian
Engineering Transactions
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2022
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Vol. 70, iss. 4
373--390
EN
This paper is devoted to the stepped sandwich beam with clamped ends subjected to a uniformly distributed load. The bending problem of the beam is formulated and solved with consideration of the classical sandwich beam of constant face thickness. Two differential equations of equilibrium based on the principle of the stationary potential energy of the classical beam are obtained and analytically solved. Moreover, numerical-FEM models of the beams are developed. Deflections for an exemplary beam family with the use of two methods are calculated. The results of the study are presented in figures and tables.
4
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.
5
Content available remote A shear deformation theory of beams with bisymmetrical Cross-sections based on the Zhuravsky shear stress formula
Magnucki Krzysztof, Lewinski Jerzy, Magnucka-Blandzi Ewa
Engineering Transactions
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2020
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Vol. 68, iss. 4
353--370
EN
This paper is devoted to simply supported beams with bisymmetrical cross-sections under a generalized load. Based on the Zhuravsky shear stress formula, the shear deformation theory of a planar beam cross-section is formulated. The deflections and the shear stresses of exemplary beams are determined. Moreover, the numerical-FEM computations of these beams are carried out. The results of the research are shown in figures and tables.
6
Content available remote Bending of a seven layer beam with foam cores
Magnucki K., Milecki S., Lewiński J., Kędzia P.
Engineering Transactions
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2018
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Vol. 66, iss. 3
249--262
EN
The subject of the paper is a seven layer beam with foam cores. The structure of the beam is symmetrical. The beam is composed of the main core, two inner sheets, two second cores and two outer sheets. The main core and two face cores are metal and polyurethane foams, while the sheets are metal. The analytical model of the beam is developed. The displacement and strain fields are formulated with consideration of the Zig-Zag hypothesis of deformation of a flat cross-section of the beam. The governing differential equations for the seven layer beam are obtained based on the stationary total potential energy. The detailed studies are devoted to deflections and stresses of the beams under a uniformly distributed load. The influence of the foam type of cores on the deflections and stresses of the beam is analysed. Moreover, the numerical FEM-model of the beam is developed. The analytical solution is compared to numerical calculations – FEM studies (ABAQUS System and SolidWorks Simulation). The results of the analysis are presented in Tables and Figures.
7
Content available On application of piezoelectronic shear effect to active damping of transverse vibration in beams
Przybyłowicz P. M.
Journal of Theoretical and Applied Mechanics
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2000
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Vol. 38 nr 3
573-589
EN
In the paper the problem of active damping of transverse vibration of a beam by making use of piezoelectric elements with shear piezoelectric effect employed is taken up. The analysis was motivated by the fact that popular piezoelectric materials made of lead zirconate titanate ceramics (PZT) exhibit very strong shear effect, being roughly three times greater than the longitudinal effect. The shear effect can be applied to beam-like systems by gluing piezoelectric sensors and actuators not to external surfaces of a given structure but placing them e.g. between two parallel strips as it is realised in three-layer sandwich structures, in which the middle layer is to dissipate vibration energy. In the presented model the middle viscoelastic core is replaced with a purely elastic piezoelectric element. Some preliminary results obtained for the proposed model based on a cantilever Timoshenko's beam made of PZT are presented. The method of control, generalised with respect to the piezoelectric shear effect boundary conditions, and finally, the resonance characteristics corresponding to excitation by external harmonicforce are discussed. Particulary advantageous effect of the active damping is confirmed for lower resonances, where the natural damping by internal friction in the material is weak.
PL
O zastosowaniu postaciowego efektu piezoelektrycznego do aktywnego tłumienia drgań poprzecznych belek. W pracy podjęto problem aktywnego tłumienia drgań poprzecznych belki za pomocą elementów piezoelektrycznych przy wykorzystaniu postaciowego efektu piezoelektrycznego. Motywację podjętych badań stanowi fakt, że popularne piezoelektryki ceramiczne wykonane na bazie spieków tlenku cyrkonu i tytanu (PZT) wykazują bardzo silny efekt postaciowy, około trzykrotnie większy od efektu wzdłużnego. Efekt postaciowy może zostać wykorzystany poprzez zastosowanie elementów nie przez naklejenie akumulatorów na zewnętrznych powierzchniach belki, lecz umieszczenie w strukturze samej belki, np. między dwiema równoległymi listwami, podobnie jak w konstrukcjach typu sandwich, gdzie warstwą silnie rozpraszającą energię drgań jest warstwa środkowa - tu zastąpiona piezoelektrykiem. W pracy przedstawiono wstępne wyniki badań bazujące na modelu wspornikowej belki Timoszenki wykonanej z ceramiki PZT. Omówiono metodę sterowania, uogulnione na piezoelektryczny efekt postaciowy warunki brzegowe oraz, jako rezultat końcowy, charakterystki rezonansowe przy wymuszeniu harmoniczną siłą zewnętrzną. Stwierdzono szczególnie korzystne działanie aktywnego tłumienia wykorzystującego efekt postaciowy dla niskich częstości, rezonansowych, słabo tłumionych przez naturalne właściwości dyssypatywne materiału belki.
8
Content available The effects of constrained cross-specional warping on bending of beams
Janecki S.
Journal of Theoretical and Applied Mechanics
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1998
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nr 1
121-144
EN
The equations of motion for a bent beam of compact cross-section, are presented in this work. They are derived by using of the general methodapplicable to one-dimensional models of continuum. The warping constraint of cross-sections, caused by shearing, has been taken into account. An additional parameter characterizing the form of a cross-section warping function is introduced. Two dimensionless shearing coefficients appear in the given equations. One of them characterizes the constrained shearing, and the other one the free shearing. It has been shown that these coefficients do not depend upon the form of the warping function. In a particular case, if the constraint cross-sectional warping does not appear, then the equation of motion and the constitutive relations are the same as in Timoshenko's theory. A series of important examples illustrating the bending theory of beams is presented in the paper. The form of the warping function and the warping constraint of an arbitrary, compact cross-section has been taken into account in these examples. In a particular case, an equation defining the parameter magnitudes of the thickness shear mode, for a simply supported beam of an arbitrary cross-section, is given. A critical analysis the works of Bickford (1982), Ewing (1990), Leung (1990) and Levinson (1981) has been made.
PL
W pracy przedstawiono równania ruchu belki zginanej o dowolnych jednospójnych przekrojach poprzecznych. Uwzględniono skrępowanie deplanacji przekrojów poprzecznych, spowodowanej nierównomiernym ścinamiem. Wprowadzono dodatkowy parametr charakteryzujący kształy funkcji deplanacji przekrojów poprzecznych belki. W podanych równaniach występują dwa niezależne, bezwymiarowe współćzynniki ścinania. Jeden z nich charakteryzuje ścinanie skrępowane, a drugi ścinanie swobodne. Pokazano, że współczynniki te są niezależne od kształtu funkcji deplanacji. Równania ruchu i związki konstytutywne w szczególnym przypadku deplanacji swobodnej sprowadzają się do równań i związków występujących w teorii Timoshenko. W pracy przedstawiono szereg ważnych przykładów ilustrujących teorie zgięcia belek. Wzięto w nich pod uwagę kształt funkcji deplanacji i skrępowanie deplanacji dowolnych, jednospójnych przekrojów poprzecznych. W szczególnym przypadku podano równanie określające wielkość parametrów postaci drgań ścinania poprzecznego belki (thickness shear mode) swobodnie podpartej, posiadającej dowolny przekrój poprzeczny. Przeprowadzono krytyczną analizę wyników prac Levisona (1981), Bickforda (1982), Ewinga (1990), Leunga (1990).
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