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1

Vibration analysis of variable thickness functionally graded toroidal shell segments

Vuong Pham Minh, Duc Nguyen Dinh

Archives of Civil and Mechanical Engineering

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2023

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Vol. 23, no. 3

art. no. e207, 2023

EN

In this paper, for the first time, the nonlinear vibration response of toroidal shell segments with varying thickness subjected to external pressure is investigated analytically using Reddy’s third-order shear deformation shell theory. The variable thickness shells are made of functionally graded material (FGM) that is created from ceramic and metal constituents. The material properties of FGM shells are assumed to be gradually graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of constituents. Equations of motion of variable thickness FGM toroidal shell segments are established based on Reddy’s third-order shear deformation shell theory with von Kármán nonlinearity. The Galerkin method and the Runge–Kutta method are used to solve the governing system of partial differential equations of motion, and then the nonlinear vibration response of variable thickness FGM toroidal shell segment is analyzed. A numerical analysis is also performed to show the effects of material and geometrical parameters on the nonlinear vibration response of variable thickness FGM toroidal shell segments.

2

Nonlinear vibration analysis of sandwich plates with honeycomb core and graphene nanoplatelet‑reinforced face‑sheets

Mohammad‑Rezaei Bidgoli E., Arefi Mohammad

Archives of Civil and Mechanical Engineering

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2023

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Vol. 23, no. 1

art. no. e56, 2023

EN

This paper studies nonlinear vibration analysis of a graphene nanoplatelets’ composite sandwich. The core and two face-sheets of composite sandwich plate are fabricated from a honeycomb material and graphene nanoplatelet (GNP) reinforcements, respectively. Displacement field of sandwich plate is developed based on first-order shear deformation theory. Geometric nonlinearity is accounted in the constitutive relations based on von-Karman assumptions. After derivation of the governing partial differential motion equations through Hamilton’s principle, Galerkin’s approach is used to reduce them into a nonlinear equation of motion in terms of transverse defection. The nonlinear frequency is found based on linear frequency and initial conditions, analytically. The nonlinear-to-linear frequency ratio is computed based on significant input parameters of honeycomb structure and graphene nanoplatelets such as thickness-to-length and thickness-to-height ratios, angle of honeycomb, various distribution, weigh fraction and geometric characteristics of graphene nanoplatelets. Before presentation of full numerical results, the comprehensive comparative study is presented for verifcation of the derivation and solution method.

3

Gravity-induced bistable 2DOF piezoelectric vibration energy harvester for broadband low-frequency operation

Liu Chaoran, Zhang Wei, Yu Kaiping, Liao Baopeng, Zhao Rui, Liu Tao

Archives of Civil and Mechanical Engineering

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2023

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Vol. 23, no. 3

art. no. e208, 2023

EN

Bistability has been proven beneficial for vibration energy harvesting. However, previous bistable harvesters are usually cumbersome in structure and are not necessarily capable of low-frequency operation. To resolve this issue, this paper proposes a compact two-degree-of-freedom (2DOF) bistable piezoelectric energy harvester with simple structure by using an inverted piezoelectric cantilever beam elastically coupled with a swinging mass-bar. The swinging mass-bar possesses bistable property due to the combined effect of the gravity and the elastic joint. It is revealed that, under the inter-well periodic motion pattern which has large swinging amplitude, the swinging mass-bar can exert large force and moment on the piezoelectric cantilever beam, thereby generating large electrical output in this process. Moreover, the inter-well periodic swinging motion can occur in a very broad low-frequency region, enabling broadband low-frequency energy harvesting. An experimental prototype is tested under harmonic excitation and sine frequency sweeping excitation; high electrical output is gained in the frequency range of 2 Hz to 12.6 Hz with a peak power of 3.558mW and a normalized power density of 19.52mW/(g2·cm3), which validates the broadband low-frequency energy harvesting capability.

4

Effect of porosity and characteristics of carbon nanotube on the nonlinear characteristics of a simply-supported sandwich plate

Bidgoli E. Mohammad-Rezae, Arefi Mohammad

Archives of Civil and Mechanical Engineering

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2023

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Vol. 23, no. 3

art. no. e214, 2023

EN

This paper presents geometric-based nonlinear formulation of a composite sandwich plate on the elastic foundation based on first-order shear deformation theory. The composite sandwich plate is fabricated from a porous core integrated with two carbon-nanotubes-reinforced face sheets. After developing the kinematic relations based on first-order shear deformation theory, the geometric nonlinearity is accounted based on von-Karman-type nonlinearity. Porosity of the core is modeled based on two known models in terms of porosity coefficient. After presentation of the effective material properties of the core and the carbon nanotube reinforcement in terms of porosity coefficient, volume fraction of carbon nanotube, and basic material properties, the nonlinear governing equations are derived using Hamilton’s principle. Galerkin’s approach is applied to reduce nonlinear governing equations of motion to an ordinary time-dependent differential equation. The nonlinear frequency is analytically found based on linear frequency and initial boundary conditions. Before presentation of full numerical results, a comprehensive comparative study is presented for verification of the derivation and solution procedure. The nonlinear to linear frequency ratio is computed based on significant input parameters of porous core and carbon-nanotube-reinforced face sheets such as type of porosity, porosity coefficient, volume fraction, and type of reinforcement’s distribution.

5

Analysis and control of nonlinear vibration of autonomous vehicle passing through hybrid consecutive speed control humps

Yang Zhiyong, Wang Long, Yu Yanjun, Mou Zhenping, Ou Minghui

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2022

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Vol. 70, nr 6

art. no. e143643

EN

In order to improve the safety and comfort of autonomous vehicles passing through the expressway, relevant departments of expressway construction often design and lay consecutive speed control humps (SCHs) with cross-sections of different shapes according to different road conditions, such as the combination of trapezoidal and sinusoidal SCHs. In this paper, we conduct a study about the nonlinear dynamic characteristics of the autonomous vehicle passing through hybrid SCHs. Firstly, a four-degree-of-freedom (4-DOF) nonlinear model of the vehicle suspension and the speed coupling excitation model under hybrid SCHs are established. Then the fourth-fifth order Runge–Kutta method is used to simulate the nonlinear system, and its nonlinear dynamic characteristics are analyzed. The results show that chaotic motion occurs when the vehicle passes through hybrid SCHs, and the speed range of chaotic motion is obtained. Then, a direct variable feedback control method is used to suppress the chaotic vibration of semi-active suspension vehicles, and the effect is verified by simulation experiments. Finally, this paper presents a multi-objective optimization model based on a genetic algorithm (GA) for active suspension vehicles. The optimization model selects the vertical displacement and pitching angle of the vehicle body as the objective function. The research results of this paper can provide information on the ride comfort’s optimization for autonomous vehicles passing through hybrid SCHs and on the design of vehicle suspension system.

6

Analysis of the associated stress distributions to the nonlinear forced vibrations of functionally graded multi-cracked beams

Chajdi Mohcine, Adri Ahmed, El Bikri Khalid, Benamar Rhali

Diagnostyka

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2021

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Vol. 22, No. 1

101--112

EN

Geometrically non-linear vibrations of functionally graded Euler-Bernoulli beams with multi-cracks, subjected to a harmonic distributed force, are examined in this paper using a theoretical model based on Hamilton's principle and spectral analysis. The homogenisation procedure is performed, based on the neutral surface approach, and reduces the FG beams analysis to that of an equivalent homogeneous multi-cracked beam. The so-called multidimensional Duffing equation obtained and solved using a simplified method (second formulation) previously applied to various non-linear structural vibration problems. The curvature distributions associated to the multi-cracked beam forced deflection shapes are obtained for each value of the excitation level and frequency. The parametric study performed in the case of a beam and the detailed numerical results are given in hand to demonstrate the effectiveness of the proposed procedure, and in the other hand conducted to analyse many effects such as the beam material property, the presence of crack, the vibration amplitudes and the applied harmonic force on the non-linear dynamic behaviour of FG beams.

7

Frequency of free vibration in systems with power-law restoring force

Grega Ivan, Grega Robert, Homisin Jaroslav

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2021

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Vol. 69, nr 2

art. no. e136723

EN

To model the nonlinear behaviour of vibrating systems, Taylor expansion with integer powers is often used. Some systems, however, are inherently nonlinear. In the case of a non-integer real exponent, the power-law system cannot be linearised around the equilibrium position using Taylor expansion. The approach presented here provides a simple estimate of the principal frequency of free vibration in systems with power-law restoring force. Without seeking the precise mathematical form of the output waveform, we only focus on the principal frequency. The first step is the use of dimensional analysis to reduce the number of parameters. Two independent non-dimensional groups are formed and functional dependence between them is sought using numerical simulations. Once this dependence is known, the principal frequency of free vibration can be readily determined for any system properties and any initial conditions. Finally, we compare the numerical results to analytical expressions for a few restoring force exponents.

8

Free vibration of a hyper-elastic microbeam using a new “augmented Biderman model”

Karami Mohammadi Ardeshir, Danaee Barforooshi Saeed

Journal of Theoretical and Applied Mechanics

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2019

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Vol. 57 nr 3

739--750

EN

A new augmented Biderman model inspired by the modified couple stress theory has been introduced to investigate the size effect in addition to nonlinear material effects. Then, this model is used to investigate free vibration of a hyper-elastic microbeam. Classical Biderman strain energy does not include the effect of small size in hyper-elastic materials. In order to consider the effect of small size, terms inspired by the modified couple stress theory are added to the classical Biderman strain energy function. In order to provide the possibility of calculating these terms, a relation between the material constants in the hyper-elastic Biderman model and the linear elastic constants is obtained. The equations of motion of the microbeam is obtained based on the extended Hamilton principle, and then is solved using Galerkin discretization and perturbation methods. The effect of thickness to length scale ratio on the normalized frequency is studied for different modes. It is shown that when thickness gets larger in comparison with the length scale parameter, the normalized frequency tends to classical Biderman results. The results obtained are validated by results of the Runge-Kutta numerical method and indicate an excellent agreement. Mode shapes of the microbeam based on the classical and the augmented models are depicted, where the augmented model anticipates stiffer behavior for hyperelastic microbeams.

9

Comparison between time- and frequency methods for determination of response characteristics of a nonlinear vibration energy harvester

Kulik M.

Zeszyty Naukowe. Elektryka / Politechnika Opolska

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2017

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z. 75

45--46

EN

The paper presents the frequency characteristics of a nonlinear vibration energy harvester determined via the time- and frequency-domain analysis methods. The results of computations using the models compare better for systems with nonlinearity being compensated by the spring force. For highly nonlinear systems the results predicted by the frequency method are too inaccurate.

10

Analiza drgań wieszara cięgnowego jako modelu kolejowej sieci trakcyjnej obciążonej ruchem pantografów

Bryja D., Popiołek A.

Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture

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2017

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z. 64, nr 2/I

177--190

PL

Zwiększenie prędkości jazdy pociągów i rozwój kolei dużych prędkości spowodowały wyraźny wzrost zainteresowania problemami dynamiki kolejowych sieci trakcyjnych. W ostatnich latach pojawiło się w literaturze zagranicznej wiele nowych publikacji przedstawiających zaawansowane numerycznie metody symulacji drgań górnej sieci jezdnej i pantografów. W Polsce temat ten jest stosunkowo mało znany, stąd jednym z celów pracy jest przegląd literatury na temat metod modelowania sieci trakcyjnych. Celem zasadniczym jest przedstawienie oryginalnej metody symulacji drgań sprzężonego układu sieć trakcyjna – pantograf oraz zastosowanie metody do analizy drgań przykładowej sieci trakcyjnej. Metoda bazuje na modelu obliczeniowym przedstawionym przez autorów w odrębnej pracy, sformułowanym na podstawie teorii drgań wiotkiego cięgna z ciągłym rozkładem masy. Górna sieć jezdna jest traktowana jako wstępnie napięty, wieloprzęsłowy wieszar cięgnowy złożony z liny nośnej o niepomijalnym zwisie w przęsłach i przewodu jezdnego podwieszonego za pomocą wiotkich wieszaków nie przenoszących ściskania. Sieć trakcyjna jest obciążona dwoma pantografami poruszającymi się ze stałą prędkością. Pantografy są układami dynamicznymi o dwóch stopniach swobody. Równania ruchu wyprowadzone metodą Lagrange’a – Ritza zostały w tej pracy przekształcone poprzez wyodrębnienie nieliniowych sił, które kompensują wpływ wieszaków ściskanych. Opisano metodę rozwiązania nieliniowych równań ruchu i wyjaśniono sens fizyczny stowarzyszonych z nimi równań liniowych. Przedstawiono przykład symulacji drgań sieci złożonej z dziesięciu przęseł, świadczący o efektywności i możliwościach obliczeniowych prezentowanej metody oraz zbadano wpływ tłumienia w materiale liny nośnej i przewodu jezdnego na charakterystyki dynamiczne badanej sieci.

EN

Increasing train speeds and rapid development of high speed railway systems give rice to growing interest in dynamics of railway overhead wire systems. In recent years, many new publications on advanced numerical methods for computer simulation of vibration of pantographcatenary systems appeared in foreign literature. In Poland, this topic is relatively unknown, hence one of the objectives of this paper is to review the literature on methods for modeling overhead contact lines and pantographs. The main goal is to present an original method for simulation of pantograph and catenary coupled system vibration and the use of method in dynamic analysis of a sample system. The method is based on the computational model which have been presented in a separate article. This model is formulated on the basis of vibration theory of a continuous cable. Catenary is treated as initially tensioned, multi-span cable structure which consists of a carrying cable characterized by non-negligible static sag and a contact wire suspended by means of droppers. The slackening of droppers under compressive forces is taken into account. Catenary is subjected to a passage of two pantographs moving with constant sped, each idealized as twodegree-of-freedom dynamic system. Equations of motion of the system, derived by the use of Lagrange equations and Ritz approximation of catenary displacements, are reexpressed in this paper to extract nonlinear forces which compensate the effects of compressed droppers. The method for solving nonlinear equations of motion is described. It is also explained what is a physical meaning of linear equations associated with these nonlinear. Exemplary simulations are presented for the catenary consisting of ten spans in order to demonstrate efficiency and computing capabilities of the simulation method. An influence of the material damping in carrying rope and this in contact wire, on the dynamic response of analyzed catenary is examined.

11

Dynamic response of rectangular prestressed membrane subjected to uniform impact load

Li D., Zheng Z. L., Liu C. Y., Zhang G. X., Lian Y. S., Tian Y., Xiao Y., Xie X. M.

Archives of Civil and Mechanical Engineering

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2017

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Vol. 17, no. 3

586--598

EN

This paper investigated the dynamic response of rectangular prestressed membrane subjected to uniform impact load theoretically and experimentally. The dynamic response proceeds in two stages, namely, forced vibration and free vibration. Firstly, the maximal displacement for forced vibration is obtained by means of the principle of minimum potential energy based on the theoretical model proposed. Then, equations of motion for the transverse free vibration are derived based on thin-plate theory, and simplified by using Galerkin method. Consequently, the analytic solutions of dynamic parameters, such as frequency, displacement, amplitude, velocity, and acceleration, for free vibration are obtained by means of the multiple-scale perturbation method. In order to identify the reliability of theoretical model, the corresponding experimental study is carried out based on the developed experimental system. Furthermore, the effects of pretension force and load on the dynamic response of membrane are discussed, respectively. The present work provides a theoretical model to calculate the dynamic response of prestressed membrane subjected to uniform impact load, and a set of experimental system to study this problem.

12

Vibrations of a Rotating System Containing a Textile Element

Zajączkowski J.

Fibres & Textiles in Eastern Europe

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2015

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Vol. 23, no. 1 (109)

68--70

EN

The study examined the behaviour of a rotating system containing a textile element. A mathematical model is formulated and a numerical investigation is performed. It has been found that the layer of fibres limits the mass motion and stabilises the rotary motion, thereby preventing a decrease in the speed of the system.

PL

W pracy badane jest zachowanie układu wirującego zawierającego element tekstylny. Sformułowano matematyczny model i przeprowadzono badania numeryczne. Stwierdzono, że warstwa włókien ogranicza ruch masy i stabilizuje ruch obrotowy, nie pozwalając na zmniejszenie prędkości obrotowej układu.

13

Nonlinear vibration analysis of prebuckling and postbuckling in laminated composite beams

Abdollahzadeh G., Ahmadi M.

Archives of Civil Engineering

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2015

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Vol. 61, nr 2

161--178

EN

In this study, we attempt to analyse free nonlinear vibrations of buckling in laminated composite beams. Two new methods are applied to obtain the analytical solution of the nonlinear governing equation of the problem. The effects of different parameters on the ratio of nonlinear to linear natural frequencies of the beams are studied. These methods give us an agreement with numerical results for the whole range of the oscillation amplitude.

PL

Niniejsze opracowanie podejmuje temat analizy swobodnych drgań nieliniowych wyboczenia laminowanych belek kompozytowych. Zastosowano dwie nowe metody w celu uzyskania rozwiązania w postaci kluczowego równania nieliniowego, opisującego ten problem. Przestudiowano wpływ różnych parametrów na stosunek częstotliwości drgań nieliniowych do drgań liniowych w odniesieniu do badanych belek. Metody te umożliwiły nam weryfikację otrzymanych wyników dla całego zakresu amplitudy oscylacji.

14

Resonance Vibrations of an Elastic System Supported on a Textile Layer

Zajączkowski J.

Fibres & Textiles in Eastern Europe

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2014

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Vol. 22, no. 2 (104)

76--79

EN

The vibrations of a flexible system resting on a layer of fibres are investigated in this paper. A mathematical model of the system excited by an electromagnet supplied with an alternating current is formulated. Two resonance regions are found. In one region the frequency of the varying contact force is twice the supply electric current frequency. In the other region the frequency of vibration is equal to that of the current. At the resonances the system loses contact with the layer at the moment when the reaction force becomes equal to zero.

PL

W pracy badane są drgania układu sprężystego spoczywającego na warstwie włókien. Sformułowano model matematyczny układu wzbudzanego elektromagnesem zasilanym prądem zmiennym. Znaleziono dwa obszary rezonansowe. W jednym z nich częstotliwość siły nacisku jest dwukrotnie większa od częstotliwości prądu zasilania. W drugim obszarze częstotliwość drgań jest równa częstotliwości prądu. W rezonansie system traci kontakt z warstwą w chwili gdy siła reakcji staje się równa zeru.

15

The effect of quintic nonlinearity on the investigation of transversely vibrating buckled Euler-Bernoulli beams

Sedighi H. M., Reza A., Zare J.

Journal of Theoretical and Applied Mechanics

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2013

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Vol. 51 nr 4

959--968

EN

A new formulation of vibrations of the axially loaded Euler-Bernoulli beam with quintic nonlinearity is investigated in the present study. The beam nonlinear natural frequency as a function of the initial amplitude is obtained. In this direction, modern powerful analytical methods namely He’s Max-Min Approach (MMA) and Amplitude-Frequency Formulation (AFF) are employed to approximate the frequency-amplitude relationship of the beam vibrations. Afterwards, it is clearly shown that the first term in the series expansions is sufficient to produce a highly accurate approximation of the nonlinear system. Finally, preciseness of the present analytical procedures is evaluated in contrast with numerical calculation methods.

16

Study on the Dynamic Characteristics of Warp in the Process of Weaving

Shen D. F., Ye G. M.

Fibres & Textiles in Eastern Europe

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2013

|

Vol. 21, no. 2 (98)

68--73

EN

The fluctuation process of warp movement is analysed by the nonlinear vibration method. Based on Newton’s Law, a differential equation of warp axial/cross direction movement is established. This paper separates the time variable from the space variable by the variable- separating method, then it gives a numerical solution of the motion equation by the fourth-order Runge-Kutta method. Also this paper discusses influencing factors and variable trends for warp vibration. Finally a method for the control of vibration is introduced.

PL

Proces drgań przy ruchu osnowy jest analizowany przez nieliniową metodę określania drgań. Równanie różniczkowe poprzecznego i osiowego kierunku ruchu osnowy oceniane jest na podstawie prawa Newtona. W pracy tej zmienna czasowa i zmienna przestrzenna rozdzielone są metodą rozdziału zmiennych, co daje rozwiązanie numeryczne równania ruchu przy pomocy metody Runge-Kutty 4-tego rzędu. W artykule omówiono również wpływ czynników i tendencji zmien na drgania osnowy, jak również przedstawiono metodę kontroli drgań.

17

Application of Textile Items in Supports of Rotating Bodies

Zajączkowski J.

Fibres & Textiles in Eastern Europe

|

2013

|

Vol. 21, no. 1 (97)

97--99

EN

The study analysed the possibility of using an element with a textile structure for the design of supports for a spinning body that is not subject to external transverse forces. A numerical simulation was carried out and the phenomena that occur in the system were explained with the use of the energy function.

PL

W pracy poddano analizie możliwość zastosowania elementu o strukturze włókienniczej do konstrukcji podpór ciała wirującego nieobciążonego zewnętrznymi siłami poprzecznymi. Przeprowadzono symulację numeryczną oraz wyjaśniono zachodzące zjawiska z wykorzystaniem funkcji energii.

18

Effectiveness of vibration reduction of a nonlinear dynamic vibration eliminator

Golec Z.

Vibrations in Physical Systems

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2012

|

Vol. 25

153--158

EN

The paper presents the research aimed at effectiveness of elimination of the mechanical system vibration with the use of a nonlinear dynamic vibration eliminator. The considered object has been modeled as a discrete system with two degrees of freedom subjected to force excitation. Nonlinearity of elastic and dissipative properties of the protected object and the eliminator is defined by 3rd order polynomials. Solutions of the equations of motion have been obtained by simulation research with the use of MATLAB® software. Effectiveness of the vibration reduction has been assessed based on the function of the elimination effectiveness defined as the ratio of rms values of the vibration amplitudes of the protected object without the eliminator to the ones occurring with it, in the case of steady motion.

19

Nonlinear vibration analysis of a fixed-fixed beam under oscillating axial load and vibrating magnetic field

Chang T.-P.

Journal of Theoretical and Applied Mechanics

|

2012

|

Vol. 50 nr 2

441-453

EN

The nonlinear vibration behavior of a fixed-fixed beam under oscillating axial load and vibrating magnetic field is investigated in the present study. The transverse magnetic force, transverse magnetic couple, axial force, uniform translation spring force, transverse surface force and the damper are considered in the system. Hamilton’s principle is adopted to derive the equation of motion of the beam system under certain hypotheses, and then Galerkin’s method is utilized to obtain the solution of the system. It can be concluded from the present study that under stable situations, the more the transverse magnetic field increases, the more the displacement and natural frequency of the beam system decrease.

PL

Praca przedstawia analizę nieliniowej dynamiki belki obustronnie zamurowanej i poddanej zewnętrznemu obciążeniu siłą harmoniczną w obecności oscylacyjnie zmiennego pola magnetycznego. W badanym układzie uwzględniono poprzeczną siłę magnetyczną, poprzeczny moment magnetyczny, siłę mechaniczną w kierunku osiowym, jednorodnie rozłożoną poprzeczną siłę sprężystości, poprzeczne obciążenie powierzchniowe oraz tłumienie. Do wyznaczenia równań ruchu zastosowano zasadę Hamiltona przy założeniu pewnych hipotez, a następnie użyto metody Galerkina w celu rozwiązania tych równań. W wyniku przeprowadzonej analizy zaobserwowano, że w stabilnych warunkach wzrost indukcji przecznego pola magnetycznego powoduje ograniczenie drgań belki oraz spadek częstości własnych układu.

20

Chaotic Vibrations of an Object Supported on a Layer of Fibres

Zajączkowski J.

Fibres & Textiles in Eastern Europe

|

2012

|

Vol. 20, no. 6A (95)

75--77

EN

The dynamic properties of a mathematical model of a layer of fibres are studied in this paper. It is shown that the layer significantly restricts the amplitude of vibration of a supported mass, which is subject to a compressive oscillating force. For certain frequencies of sufficiently large oscillatory force a chaotic motion may take place.

PL

W pracy badane są właściwości dynamiczne matematycznego modelu układu warstw włókien. Wykazano, że układ warstw włókien w bardzo istotny sposób ogranicza amplitudę drgań obiektu na nim spoczywającego, poddanego oscylującej sile ściskającej. W pewnym zakresie częstotliwości, dla wystarczająco dużej siły wymuszającej drgania, może pojawić się ruch chaotyczny.