BazTech - Yadda

1

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.

2

Numerical investigation of geometrically nonlinear clamped uniform rods and rods with sections varying exponentially free vibration

Abdeddine E., Majid A., Bouzida N., Bouzida Z., Zarbane Kh.

Journal of Achievements in Materials and Manufacturing Engineering

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2022

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Vol. 112, nr 2

49--63

EN

Purpose: The present paper is intended to investigate the problem of linear and non-linear longitudinal free vibration of uniform rods and rods whose cross-sections vary exponentially at large vibration amplitudes. Design/methodology/approach: The method adopted consists in discretizing the energy term on linear kij and non-linear rigidity tensor bijkl, as well as the mass tensor mij. Therefore, the formulation of this structure is based on Lagrange equations and the harmonic balance method so as to obtain the nonlinear algebraic equations. These latter are solved numerically and analytically through the explicit and linearized method. Findings: The response of Clamped-Clamped uniform and non-uniform rods on our structure are highlighted in the amplitude frequency and associated first three mode shapes. Moreover, this research leads to study the influence of the exponential slope on the maximum displacement, thus emphasizing the non-uniform bars usefulness. The obtained results are then compared with the available literature with a view to validating this theory. Research limitations/implications: As a perspective, the method used in this paper would be pushed to study the FDM material, taking into account other parameters related to additive manufacturing, and later to be validated experimentally. Practical implications: Longitudinal vibrations are important in mechanical structures; therefore, the determination of their dynamic behaviour needs to be understood. In the present study, the effect of the displacement amplitude on the exponential slope of the structure was analysed, which led to the determination of the reduction range of the vibration amplitude under resonance. However, this should be taken into account in the design process. Besides, the usefulness of the non-linearity geometric effects was demonstrated to examine these structures by considering all the parameters involved. Originality/value: A linearized procedure is used to solve a nonlinear algebra equation. The use of this method leads to reduce calculation time contrary to iterative methods.

3

Finite element model for analysis of characteristics of shrouded rotor blade vibrations

Zinkovskii Anatoliy, Savchenko Kyrylo, Onyshchenko Yevheniia, Polishchuk Leonid, Nazerke Abilkaiyr, Zhumazhanov Bagashar

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2022

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T. 12, nr 4

11--16

EN

The paper presents the approaches to FE modelling of blade airfoil, contact between the shrouds and operational damage. The regularities are established concerning the influence of the finite element type, finite element mesh and model of contact interaction on the spectrum of natural frequencies of blade assemblies. The use of the developed computational models is substantiated to determine the forced vibration characteristics of the selected objects of investigation. Based on the performed numerical experiments it was substantiated of finite element model selection for analysis of characteristics of shrouded rotor blade vibrations.

PL

W artykule przedstawiono podejścia do modelowania elementów skończonych płata łopaty, styku osłon oraz uszkodzeń eksploatacyjnych. Ustalono prawidłowości dotyczące wpływu typu elementu skończonego, siatki elementów skończonych oraz modelu interakcji stykowej na widmo częstotliwości drgań własnych zespołów łopatek. Uzasadnione jest wykorzystanie opracowanych modeli obliczeniowych do wyznaczania charakterystyk drgań wymuszonych wybranych obiektów badań. Na podstawie przeprowadzonych eksperymentów numerycznych uzasadniono wybór modelu elementów skończonych do analizy charakterystyk drgań osłoniętych łopat wirnika.

4

A discrete model for geometrically nonlinear free and forced vibrations of stepped and continuously segmented Euler-Bernoulli AFG beams (SAFGB) carrying point masses

Moukhliss Anass, Rahmouni Abdellatif, Bouksour Othman, Benamar Rhali

Diagnostyka

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2022

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Vol. 23, No. 4

art. no. 2022401

EN

A discrete model is applied to handle the geometrically nonlinear free and forced vibrations of beams consisting of several different segments whose mechanical characteristics vary in the length direction and contain multiple point masses located at different positions. The beam is presented by an N degree of freedom system (N-Dof). An approach based on Hamilton's principle and spectral analysis is applied, leading to a nonlinear algebraic system. A change of basis from the displacement basis to the modal basis has been performed. The mechanical behavior of the N-Dof system is described in terms of the mass tensor mij, the linear stiffness tensor kij, and the nonlinear stiffness tensor bijkl. The nonlinear vibration frequencies as functions of the amplitude of the associated vibrations in the free and forced cases are predicted using the single mode approach. Once the formulation is established, several applications are considered in this study. Different parameters control the frequency-amplitude dependence curve: the laws that describe the variation of the mechanical properties along the beam length, the number of added masses, the magnitude of excitation force, and so on. Comparisons are made to show the reliability and applicability of this model to non-uniform and non-homogeneous beams in free and forced cases.

5

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.

6

Criterion for transient behaviour in a nonlinear Duffing oscillator

Szczebiot Ryszard, Jordan Andrzej

Przegląd Elektrotechniczny

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2019

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R. 95, nr 4

196--199

EN

The paper proposes a criterion for determining transient behaviour in a nonlinear Duffing oscillator. For this purpose studies of specific attractors typical of the system have been conducted. Exactly defined deviation value of ∆ with respect to the mean value of the surface areas bounded by the successive trajectory cycles has been assumed as the termination of the transient behaviour.

PL

W pracy zaproponowano kryterium wyznaczania czasu trwania procesu przejściowego w nieliniowym oscylatorze Duffinga. W tym celu badano specyficzne atraktory charakteryzujące ten układ. Za kryterium końca procesu przejściowego przyjęto ściśle zdefiniowaną wartość odchyłki ∆ od wartości średniej pól powierzchni ograniczonych kolejnymi cyklami trajektorii.

7

Linear and geometrically nonlinear free and forced vibrations of multi-cracked beams

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

Diagnostyka

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2019

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Vol. 20, No. 1

111--125

EN

The linear and geometrically nonlinear free and forced vibrations of Euler-Bernoulli beams with multicracks are investigated using the crack equivalent rotational spring model and the beam transfer matrix method. The Newton Raphson solution of the transcendental frequency equation corresponding to the linear case leads to the cracked beam linear frequencies and mode shapes. Considering the nonlinear case, the beam transverse displacement is expanded as a series of the linear modes calculated before. Using the discretised expressions for the total strain and kinetic energies and Hamilton’s principle, the nonlinear amplitude equation is obtained and solved using the so-called second formulation, developed previously for similar nonlinear structural dynamic problems, to obtain the multi-cracked beam backbone curves and the corresponding amplitude dependent nonlinear mode shapes. Considering the forced vibration case, the nonlinear frequency response functions obtained numerically near to the fundamental nonlinear mode using a single mode approach show the effects of the number of cracks, their locations and depths, and the level of the concentric harmonic force. The inverse problem is explored using the frequency contour plot method to identify crack parameters, such as the crack locations and depths. Satisfactory comparisons are made with previous analytical results.

8

Tolerance modelling of nonstationary problems of microheterogeneous media and structures

Jędrysiak J., Domagalski Ł., Marczak J., Pazera E.

Vibrations in Physical Systems

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2018

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Vol. 29

art. no. 2018001

EN

In this note a certain review of applications of a non-asymptotic modelling approach, called the tolerance modelling, is presented. Some objects and thermomechanical problems are shown, with a general outline of this method and an example of application for nonlinear vibrations of periodic beams.

9

Nonlinear dynamic phenomena in fixed and rotating mechanical structures

Warmiński J.

Vibrations in Physical Systems

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2018

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Vol. 29

art. no. 2018002

EN

A review of selected nonlinear phenomena which may occur in fixed or rotating structures has been presented in the paper. At first, a self, parametrically and externally exited oscillator with added time delay control has been studied. It has been shown that the interaction between different vibration types may produce an untypical resonance curve, with five solutions occurring, observed by an internal resonance loop. The existence of the loop may be controlled by adding a time delay input signal. A proper selection of the time delay may reduce the loop or eliminate it totally. In the second problem a rotating hub-beam structure has been studied. The blade, apart from passive layers, has been composed of two active PZT layers which enabled active vibration control. A nonlinear coupling of the structure (plant) and the controller resulted in the so called saturation phenomenon which has been effectively used for the vibration reduction.

10

Numerical modelling of toothed gear dynamics

Margielewicz J., Gąska D., Wojnar G.

Zeszyty Naukowe. Transport / Politechnika Śląska

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2017

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z. 97

105--115

EN

This paper presents the results of computer simulations of a gear model, where the variable stiffness of the meshing and backlash are considered. The outcome of such assumptions is a non-linear mathematical model in which chaotic phenomena can occur. During model studies, attention was paid to the identification of areas limited by the physical parameters, for which the analysed system behaved chaotically. To determine the ranges of irregular gear behaviour, numerical procedures were used to plot the bifurcation diagram, the Lyapunov exponent, the amplitude-frequency distribution and the Poincaré cross section.

11

Linear and geometrically non-linear frequencies and mode shapes of beams carrying a point mass at various locations. An analytical approch and a parametric study

Adri A., Benamar R.

Diagnostyka

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2017

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Vol. 18, No. 2

13--21

EN

In the present paper, the frequencies and mode shapes of a clamped beam carrying a point mass, located at different positions, are investigated analytically and a parametric study is performed. The dynamic equation is written at two intervals of the beam span with the appropriate end and continuity conditions. After the necessary algebraic transformations, the generalised transcendental frequency equation is solved iteratively using the Newton Raphson method. Once the corresponding program is implemented, investigations are made of the changes in the beam frequencies and mode shapes for many values of the mass and mass location. Numerical results and plots are given for the clamped beam first and second frequencies and mode shapes corresponding to various added mass positions. The effect of the geometrical non-linearity is then examined using a single mode approach in order to obtain the corresponding backbone curves giving the amplitude dependent non-linear frequencies.

12

A discrete model for nonlinear vibrations of a simply supported cracked beams resting on elastic foundations

Khnaijar A., Benamar R.

Diagnostyka

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2017

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Vol. 18, No. 3

39--46

EN

The present paper introduces a discrete physical model to approach the problem of nonlinear vibrations of cracked beams resting on elastic foundations. It consists of a beam made of several small bars, evenly spaced, connected by spiral springs, presenting the beam bending stiffness. The crack is modeled by a spiral spring with a reduced stiffness and the Winkler soil stiffness is modeled using linear vertical springs. Concentrated masses, presenting the inertia of the beam, are located at the bar ends. The nonlinear effect, due to the axial forces in the bars resulting from the change in their length, is presented by longitudinal springs. This model has the advantage of simplifying parametric studies, because of its discrete nature, allowing any modification in the mass and the stiffness matrices, and in the nonlinearity tensor, to be made separately. After establishing the model, various practical applications are performed without the need of going through all the formulation again. Numerical linear and nonlinear results are given, corresponding to a cracked simply supported beam.

13

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.

14

Regular and chaotic dynamics of a 4-DOF mechanical system with dry friction

Kosińska A., Awrejcewicz J., Grzelczyk D.

Vibrations in Physical Systems

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2016

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Vol. 27

195--202

EN

In this paper the model of four degree-of-freedom mechanical sliding system with dry friction is considered. One of the components of the mentioned system rides on driving belt, which is driven at constant velocity. This model corresponds to a row of carriage laying on a guideway, which moves at constant velocity with respect to the guideway as a foundation. From a mathematical point of view the analyzed problem is governed by four second order differential equations of motion, and numerical analysis is performed in Mathematica software. Some interesting behaviors are detected and reported using Phase Portraits, Poincaré Maps and Lyapunov Exponents. Moreover, Power Spectral Densities obtained by the Fast Fourier Transform technique are reported. The presented results show different behaviors of the system, including periodic, quasi-periodic and chaotic orbits.

15

Non-linear vibrations of a non-uniform beam with symmetrically located piezoelectric patches

Kuliński K., Przybylski J.

Vibrations in Physical Systems

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2016

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Vol. 27

203--210

EN

In this paper the non-linear vibration behaviour and its modification due to the piezoelectric actuation of a beam with varying cross section and resting on an elastic foundation has been discussed. Due to assumed end conditions the stretching force emerges during the system vibrations. That force can be modified by an axial residual force to enhance or reduce the value of vibrations frequency of the beam. The system is divided onto three segments with the central segment consisted of the core beam and two colocally and perfectly bonded piezo patches. In order to obtain the approximate solutions of the non-linear frequency of the systems the Lindstedt-Poincare method has been utilized. Vast number of numerical results shows that not only the structural parameters of the system have significant effect on its non-linear vibration behaviour at a given amplitude but also the residual force and the elastic foundation modulus.

16

Natural frequencies of layered elongated cylindrical panels for geometrically nonlinear deformation at discrete consideration of components

Marchuk M., Goriachko T., Pakosh V.

Vibrations in Physical Systems

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2016

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Vol. 27

255--264

EN

The proposed and verified the technique of finding a finite number of first natural frequencies for geometrically nonlinear vibrations of layered elongated cylindrical panels at discrete consideration of components. The influence of the radius of curvature on the natural frequencies of three- and five-layered panels is investigated. The dependence between the volume of filler three-layer panels and the lowest natural frequency has been established.

17

An analytical-numerical approach to analysis of large amplitude vibrations of slender periodic beams

Domagalski Ł.

Vibrations in Physical Systems

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2016

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Vol. 27

99--106

EN

The paper is devoted to analysis of geometrically nonlinear vibrations of beams with geometric and material properties periodically varying along the axis. The 1-D Euler-Bernoulli theory of beams with von Kármán nonlinearity is applied. An analytical-numerical model based on non-asymptotic tolerance modelling approach and Galerkin method is applied. The linear natural frequencies and mode shapes are determined and the results are confirmed by comparison with a finite element model. Forced damped vibrations analysis in the large deflection range is performed to illustrate complex behaviour of the system.

18

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.

19

Modeling and parameter identification of vibrations of a double torsion pendulum with friction

Czerwiński E., Olejnik P., Awrejcewicz J.

Acta Mechanica et Automatica

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2015

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Vol. 9, no. 4

204--212

EN

The purpose of this paper is to investigate a double torsion pendulum with planar frictional contact. The single torsion pendulum with one-degree-of-freedom is an angular equivalent of the linear harmonic oscillator. The second degree of freedom has been obtained by adding a free body to the inverted single torsion pendulum. The free body’s angular displacement is caused by frictional forces appearing in the interface (contact zone) between the free body and the pendulum column’s head kinematically excited at its base by a mechanism with torsion spiral spring. An experimental station has been set up and run to find most unknown parameters of the pendulum from the time series of state variables taken as inputs to the Nelder-Mead method of identification. The obtained results proved significant usability of the identification method in the case of numerical simulation of the pendulum’s dynamical model. It has not been satisfactorily proved in the case of time characteristics coming from a real system that exhibits also some unrecognized physical effects.

20

Nonlinear vibrations of rotating system near resonance

Starosta R., Sypniewska-Kamińska G.

Vibrations in Physical Systems

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2014

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Vol. 26

265--272

EN

The paper concerns analysis of nonlinear vibration of the rotating system consisted of two disks and shaft. The analytical multiple time scale method is applied to the analysis dynamics of the system near main resonance. The transition phenomenon depending on the value of the nonlinearity parameter is discussed. All the analytical results have been confirmed numerically.