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1

The dynamics of a parametrically driven damped pendulum

Das A., Kumar K.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 2

257--266

EN

Ordered and chaotic states of a parametrically driven planar pendulum with viscous damping are numerically investigated. The damping makes the number of chaotic windows fewer but with larger width. Stroboscopic maps of the chaotic motion of the pendulum, driven either subharmonically or harmonically, show strange attractors with inversion symmetry in the phase plane.

2

Kinematically excited parametric vibration of a tall building model with a TMD - Part 2: Experimental analyses

Majcher K., Wójcicki Z., Grosel J., Pakos W., Sawicki W.

Archives of Civil and Mechanical Engineering

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2014

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Vol. 14, no. 1

218--229

EN

This paper constitutes the second part of the article Kinematically excited parametric vibration of a tall building model with a TMD. Part 1: numerical analyses (ACME, in press) by K. Majcher and Z. Wójcicki, which presents the results of theoretical research. This paper presents the experimental verification of those results. The experimental studies were carried out with the use of an especially designed physical model of a tall building, which rested on an earthquake simulator – a shaking table – created for this project. The simulator was used to generate several types of kinematic excitations: harmonic ones, superpositions of harmonic ones and, finally, ones generated on the basis of real seismograms. Vibrations were kinematically excited in the horizontal and vertical directions independently and simultaneously. The vertical component of the earthquake causes the pendulum suspension point to vibrate, thus exciting the pendulum parametrically. The theoretical study indicated a significant influence of this parametric excitation (parametric resonance) on the effectiveness of the Pendulum Tuned Mass Damper (PTMD). Therefore, the experimental analyses were especially focused on the parametric effects' impact on: the PTMD's ability to reduce the building's vibration, and the possibility of parametric resonance of the building due to parametric resonance of the PTMD.

3

Kinematically excited parametric vibration of a tall building model with a TMD - Part 1: Numerical analyses

Majcher K., Wójcicki Z.

Archives of Civil and Mechanical Engineering

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2014

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Vol. 14, no. 1

204--217

EN

This paper undertakes to analyze the research problem of vibration of a tall building with a Pendulum Tuned Mass Damper (PTMD). The vibration of the building-damper system is due to kinematic excitation representing seismic load. It was assumed that during an earthquake the ground can move horizontally and vertically. An analysis of various earthquakes reveals that, sometimes, the vibration has comparable amplitudes in both these directions. It is usually the horizontal vibration that is catastrophic to structures. Vertical vibration is therefore often omitted. As this paper will show, in cases where the TMD model is a pendulum, the vertical ground motion can be transmitted through the building structure to the pendulum suspension point. In such cases, parametric resonance may occur in the system, which is especially dangerous as it amplifies vibration despite the presence of damping. Taking this phenomenon into consideration will make it possible to better secure the structure against earthquakes. As the teams carrying out theoretical and experimental analyses differed, the paper was purposely divided into two parts. In the first part, the idea was formulated and the MES model of the building-TMD system was created. The second part contains an experimental verification of the theoretical analyses.

4

Position paper - fundamental issues in self-excited chatter in turning. Appendix – research into process damping by Altintas et al

Ito Y.

Journal of Machine Engineering

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2013

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Vol. 13, No. 3

7--25

EN

Although observing a considerable number of new interesting behavior in dynamic states of machining, nearly all researches into the chatter vibration have been based on those conducted by Tobias, Tlusty and Merritt. In fact, we have established myriad remedies for the chatter suppression on the strength of their achievements; however, some crucial issues remain still in uncertainties, e.g., validity of penetration effects, definition duly what are the essential features of the chatter vibration. This position paper describes the facing problems of the chatter vibration in detail, aiming at the establishment of a generalized chatter loop. By this chatter loop, we may unveil synthetically the deterioration and improvement of the stability chart by all the influencing attributes, such as forced vibration and non-linearity within the machine-tool-work system.

5

Application of the polynomial chaos approximation to a stochastic parametric vibrations problem

Brząkała A.

Archives of Civil and Mechanical Engineering

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2013

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Vol. 13, no. 2

220--228

EN

In the paper the application of the polynomial chaos expansion in case of parametric vibrations problem is presented. Hitherto this innovative approach has not been applied to such a stochastic problem. The phenomenon is described by a nonlinear ordinary differential equation with periodic coefficients. It can be observed among others in cable-stayed bridges due to periodic excitation caused by a deck or a pylon. The analysis is focused on a real situation for which the problem of parametric resonance was observed (a cable of the Ben–Ahin bridge). The characteristic of the viscous damper is considered as a log-normal random variable. The results obtained by the use of the polynomial chaos approximations are compared with the ones based on the Monte Carlo simulation. The convergence of both methods is discussed. It is found that the polynomial chaos yields a better convergence then the Monte Carlo simulation, if resonant vibrations appear.

6

Stability Analysis of Continuous Systems with Imperfect Boundary Conditions in a Weak Formulation

Tylikowski A.

Machine Dynamics Problems

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2008

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Vol. 32, No. 3

107-116

EN

Dynamics of continuous systems have been considered in a weak (variational) form. Dynamics equation of beam subjected to the axial stochastic force in the weak formulation has been derived. The weak form of dynamics equations of linear mechanical structures is obtained using variational calculus. The almost sure stochastic stability of beam equilibrium, without the previous discretization, has been analysed by means of direct Lyapunov method. The stability analysis method is developed for distributed dynamic problems with relaxed assumptions imposed on solutions. Sufficient stability conditions have been established for the imperfect boundary conditions and two limit cases: the simply supported beam and the clamped beam are obtained.

7

Dynamiczna stateczność słabych równań układów ciągłych

Tylikowski A.

Czasopismo Techniczne. Mechanika

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2008

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R. 105, z. 1-M

241-247

PL

Niniejszy artykuł poświęcony jest analizie dynamiki układów ciągłych w słabym sformułowaniu. Wyprowadzono słabą postać równania dynamiki belki poddanej działaniu osiowej losowo zależnej od czasu siły. Posługując się bezpośrednią metodą Lapunowa, zbadano prawie pewną stabilność stochastyczną prostoliniowej postaci belki bez uprzedniej dyskretyzacji zadania. Wyznaczono warunki dostateczne stabilności belki swobodnie podpartej i sztywno utwierdzonej na obu końcach.

EN

Dynamics of continuous systems have been considered in a weak (variational) form. Dynamics equation of beam subjected to the axial stochastic force in the weak formulation has been derived. The almost sure stochastic stability of beam equilibrium, without the previous discretization, has been analysed by means of direct Lyapunov method. Sufficient stability conditions have been established for the simply supported beam and the clamped beam.

8

Dynamic stability of carbon nanotubes

Tylikowski A.

Mechanics and Mechanical Engineering

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2006

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Vol. 10, nr 1

160-166

EN

The dynamical stability of carbon nanotubes embedded in an elastic matrix under time-dependent axial loading is studied in this paper. Effects of van der Waals interaction forces between the inner and outer walls of nanotubes are taken into account. Using continuum mechanics an elastic beam model is applied to solve the transverse parametric vibrations of two co-axial carbon nanotubes. The physically realizable forces with known probability distributions and uniformly distributed on the both beam edges are assumed as the tube axial loadings. The energy-like functionals are used in the stability analysis. The emphasis is placed on a qualitative analysis of dynamic stability problem. Influence of constant component of axial forces on stability regions is shown. Boundaries of dynamic stability regions are determined using the three models and techniques with different degree of accuracy.

9

The dynamic of a coupled three degree of freedom mechanical system

Sado D.

Mechanics and Mechanical Engineering

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2004

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Vol. 7, nr 1

20-39

EN

In this paper, a nonlinear coupled three degree-of-freedom autoparametric vibration system with elastic pendulum attached to the main mass is investigated numerically. Solutions for the system response are presented for specific values of the uncoupled normal frequency ratios and the energy transfer between modes of vibrations is observed. Curves of internal resonances for free vibrations and external resonances for vertical exciting force are shown. In this type system one mode of vibration may excite or damp another one, and except different kinds of periodic vibration there may also appear chaotic vibration. Various techniques, including chaos techniques such as bifurcation diagrams and: time histories, phase plane portraits, power spectral densities, Poincare maps and exponents of Lyapunov, are used in the identification of the responses. These bifurcation diagrams show many sudden qualitative changes, that is, many bifurcations in the chaotic attractor as well as in the periodic orbits. The results show that the system can exhibit various types of motion, from periodic to quasi-periodic and to chaotic, and is sensitive to small changes of the system parameters.

10

Badania dynamiczne przekładni pasowej z pasami zębatymi

Krasiński M., Stachoń S.

Czasopismo Techniczne. Mechanika

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2004

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R. 101, z. 13-M

63--74

PL

Niniejsza praca dotyczy opisu i analizy dynamicznej przekładni pasowej z pasami zębatymi. Celem jej jest wyznaczenie pierwszego (głównego) i drugiego parametrycznego obszaru rezonansowego obciążonego cięgna pasowego dla dwóch postaci drgań. W pracy wykorzystano analizę dynamiczną przekładni z pasami klinowymi, zawartą w pracach [2] i [4]. W opracowaniu pominięto wpływ zginania pasów na granice obszarów rezonansowych i dlatego skoncentrowano się głównie na obliczeniu zastępczej masy jednostkowej pasa ρz oraz odpowiednim przystosowaniu wzorów zamieszczonych w pracy [4], tak aby skutecznie przeprowadzić całą analizę dynamiczną wg tej samej metody. Dodatkowo przeprowadzono weryfikację eksperymentalną wyników na stanowisku badawczym, opisanym w rozdziale 3, otrzymując prawie identyczne wartości częstotliwości wymuszeń parametrycznych z uzyskanymi dla tych samych danych badanej przekładni obliczonych wg zaproponowanej metody teoretycznej.

EN

The paper deals with a dynamic analysis of a toothed belt transmission. The objective of the work is to determine the first (main) and the second parametric resonant regions of a loaded toothed belt for two forms of parametric vibration. Using results of the papers [2] and [4], the present paper mainly deals with an estimation of the equivalent unit mass of the toothed belt ρz according to V-belt. Thence the formulas from the paper [4] may be used to analyse parametric resonant region limits of a toothed belt transmission neglecting belt flexural rigidity. Furthermore experimental values of parametric frequencies were determined using the investigative stand described in Chap.3. Those values were in agreement with theoretical ones obtained by the method showed in the paper.

11

Niestacjonarne drgania samowzbudno-parametryczne pewnych układów ciągłych

Bar A.

Zeszyty Naukowe Politechniki Rzeszowskiej. Mechanika

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2002

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z. 60 [197]

11-18

EN

The paper presents an approximate analysis based on a two models of the vibration of a mast supported by stay cables. The vibrations of the mast are excited by the separation of the Karman vortex street. It has been demonstrated that the vibration in the discussed structure has the self excited-parametric character. The paper includes tlie frequency and phase characteristics as well as the time plots obtained by way of numerical analysis. The non-autonomous model confirms the presents of vibration of the beat type.

12

Free and parametric vibration of the system: telescopic boom - hydraulic cylinder (changing the crane radius).

Sochacki Wojciech, Tomski Lech

Archive of Mechanical Engineering

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1999

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Vol. XLVI, nr 3

257-271

EN

In this work, authors present the development of a model of the system: telescopic boom - hydraulic cylinder (that one which controls the crane radius of the track crane). The problem is formulated by means of the Hamilton's pronciple and is solved as a geometrically non-linear problem by the small parameter methd. Within the range of free vibration, the vibration frequencies of the system have been determined. The effect of cetrain parameters of the system on its vibration frequency has also been studied. Solution of the Mathieu equation (describing parametric vibration of the system) enables one to determine the dynamic stability regions of the system. Possibility of existence of the parametric resonance in the system has been investigated. It has been fund that, for each of the studied examples, there exists such a wire length for witch the critical value of the coefficient "a" (in the Mathieu equation) is obtained. This means that for specified geometrical and load conditions, the system may loose its dynamic stability (unless vibration damping is considered).

PL

W niniejszej pracy zbudowano model układu wysięgnik teleskopowy-siłownik zmiany wysięgu żurawia samochodowego. Zagadnienie formułuje się z wykorzystaniem zasady Hamiltona i rozwiązuje jako geometrycznie nieliniowe metodą małego parametru. W zakresie drgań swobodnych wyznaczono częstości drgań układu i zbadano wpływ niektórych parametrów układu na jego częstości drgań. Rozwiązanie równania Mathieu opisującego drgania parametryczne układu prowadzi do wyznaczenia obszarów stateczności dynamicznej rozpatrywanego układu. Zbadano możliwość istnienia rezonansów parametrycznych w badanym układzie i stwierdzono, ze w każdym z badanych przypadków istnieje taka długość liny, dla której otrzymuje się krytyczną wartość współczynnika "a" w równaniu Mathieu. Oznacza to, że układ dla pewnych warunków geometrycznych i obciążenia (bez uwzględniania tłumienia) może stracicć stateczność dynamiczną nawet dla małych wartości współczynnika "b" w równaniu Mathieu.