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1

Mathematical modeling and stochastic stability analysis of viscoelastic nanobeams using higher-order nonlocal strain gradient theory

Pavlović I. R., Pavlović R., Janevski G.

Archives of Mechanics

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2019

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Vol. 71, nr 2

137--153

EN

This paper analyzes stochastic vibrations of a viscoelastic nanobeam under axial loadings. Based on the higher-order nonlocal strain gradient theory and the Liapunov functional method, bounds of the almost sure asymptotic stability of a nanobeam are obtained as a function of retardation time, variance of the stochastic force, higher-order and lower-order scale coefficients, strain gradient length scale, and intensity of the deterministic component of axial loading. Analytical results from this study are first compared with those obtained from the Monte Carlo simulation. Numerical calculations are performed for the Gaussian and harmonic non-white processes as models of axial forces.

2

Dynamic Stability of Activated Composite Beam-Like Plates

Tylikowski A., Dziedzic K.

Machine Dynamics Research

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2012

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Vol. 36, No. 2

96-105

EN

This paper presents an analytical modelling technique which uses the direct Liapunov method and classical laminated plate theory to predict the dynamic stability of activated shape memory alloy hybrid composite beams loaded by time-dependent in-plane forces. By solving the plane expansional strain state in the beam-like laminated plate, using the dynamic stability results of conventional laminated plates the dynamic stability conditions for both harmonic forces and Gaussian ergodic forces. Analytical results are presented for angle-ply symmetric beam-like plates to demonstrate how the temperature activation affects the stability regions.

3

Influence of a Distributed Delay on Stabilization of Structure Vibration

Tylikowski A.

Mechanics and Mechanical Engineering

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2011

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Vol. 15, nr 4

147--153

EN

The purpose of this theoretical work is to present a stabilization problem of beam with a distributed model of feedback delay. A displacement feedback and particular polarization profiles of piezoelectric sensors and actuators are introduced. The structure is described by integro–partial differential equations with time–dependent coefficient. The uniform stochastic stability criteria of the beam equilibrium are derived using the Liapunov direct method. As the axial force is described by the wide–band gaussian process the dynamic equation has to be written as Itˆo evolution equation with white–noise coefficient and the Itˆo differential rule is applied in order to calculate the differential of Liapunov functional. The influence of the time–deley parameter, stiffness and intensity of axial force on dynamic stability regions is shown.

4

Nonlocal Analysis of Dynamic Instability of Micro-and Nano-Rods

Tylikowski A.

Machine Dynamics Problems

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2009

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Vol. 33, No. 1

104-113

EN

The dynamic stability problem is solved for onedimensional structures subjected to time-dependent deterministic or stochastic axial forces. The stability analysis of structures under time-dependent forces strongly depends on dissipation energy. The simplest model of viscous damping with constant coefficient was commonly assumed in previous papers despite the fact that there are other more sophisticated theories of energy dissipation according to which different engineering constant have different dissipative properties. The paper is concerned with the stochastic parametric vibrations of micro- and nano-rods based on the Eringen's nonlocal elasticity theory and Euler-Bernoulli beam theory. The asymptotic instability, and almost sure asymptotic instability criteria involving a damping coefficient, structure and loading parameters are derived using Liapunov's direct method. Using the appropriate energy-like Liapunov functional sufficient conditions for the asymptotic instability, and the almost sure asymptotic instability of undeflected form of beam are derived. The nonlocal Euler-Bernoulli beam accounts for the scale effect, which becomes significant when dealing with short micro- and nano- rods. From obtained analytical formulas it is clearly seen that the small scale effect decreases the dynamic instability region. Instability regions are functions of the axial force variance, the constant component of axial force and the damping coefficient.

5

Dynamic stability of weak equations of rectangular plates

Tylikowski A.

Journal of Theoretical and Applied Mechanics

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2008

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Vol. 46 nr 3

679-692

EN

The stability analysis method is developed for distributed dynamic problems with relaxed ssumptions imposed on solutions. The problem is motivated by structural vibrations with external time-dependent parametric excitations which are controlled using surfacemounted or embedded actuators and sensors. The strong form of equations involves irregulari- ties which lead to computational difficulties for estimation and control problems. In order to avoid irregular terms resulting from differentiation of force and moment terms, dynamical equations are written in a weak form. The weak form of dynamical equations of linear mechanical struc- tures is obtained using Hamilton’s principle. The study of stability of a stochastic weak system is based on examining properties of the Liapunov functional along a weak solution. Solving the problem is not dependent on assumed boundary conditions.

PL

W pracy rozszerzono możliwości analizy stabilności układów ciągłych na układy z osłabionymi warunkami nakładanymi na rozwiązania. Układy aktywnego tłumienia drgań cienkościennych elementów płytowych mogą zawierać elementy piezoelektryczne oddziaływujące na konstrukcję.W uproszczonym modelu oddziaływanie to sprowadza się do działania momentów gnących lub sił rozłożonych na krawędziach elementu piezoelektrycznego. Wprowadzenie dystrybucji -Diraca i jej pochodnej prowadzi do analitycznego zapisu obciążenia i wprowadza nieregularności do rozwiązania zadania drgań wymuszonych układu ciągłego. Słabą postać równań płyty otrzymano za pomocą zasady Hamiltona. Badanie stateczności stochastycznych układów w formie słabej jest oparte na analizie funkcjonału Lapunowa wzdłuż słabego rozwiązania. Rozwiązanie zadania jest niezależne od przyjętych warunków brzegowych.

6

Semiactive Control of a Shape Memory Alloy Hybrid Composite Shaft under a Torsional Load

Tylikowski A.

Machine Dynamics Problems

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2007

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Vol. 31, No. 2

128-139

EN

The dynamic stability analysis is applied to activated shape memory alloy hybrid structures rotating with the nonconstant angular velocity. The rotating circular cylindrical shell is treated as a beam-like structure subjected to a constant torque. The time-dependent component of angular velocity is assumed in the form of the wide-band Gaussian processes modelled as a Wiener process. In this dynamics study the hybrid elements is treated as a thin angle-ply laminated beam containing both the conventional fibers arbitrary oriented and the activated shape memory alloy fibers parallel to the shaft axis. Using the appropriate energy-like Liapunov functional and the standard stability technique of partial differential equations leads to the effective sufficient criterion for the dynamic stability and the uniform stochastic stability of the shaft equilibrium. The boundaries of stability regions as functions of angular velocity, loading characteristics, damping coefficients, and properties of shaft material are analytically defined. The thermal activation substantially increases stability regions.

7

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.

8

Stateczność dynamiczna konstrukcji pod wpływem czasowo-przestrzennych obciążeń przypadkowych

Tylikowski A.

Zeszyty Naukowe. Mechanika / Akademia Techniczno-Rolnicza w Bydgoszczy

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2004

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Z. 54(243)

23-32

PL

Obciążenia losowe dużych konstrukcji pochodzące zarówno od wiatru, ruchów skorupy ziemskiej lub fal morskich charakteryzują się przestrzenną korelacją. W pracy przedstawiona jest analiza stateczności dynamicznej typowych konstrukcji jednowymiarowych, takich jak maszty, wieże lub mosty wiszące, poddanych obciążeniu zmiennemu w czasie i przestrzeni prowadzącemu do drgań parametrycznych konstrukcji. Wyprowadzone bezpośrednią metodą Lapunowa dostateczne warunki stateczności wyrażone są przez podstawowe parametry konstrukcji i obciążenia, takie jak współczynnik tłumienia wiskotycznego, wymiary, sztywności oraz intensywności i długości korelacji obciążenia.

EN

Random loadings of wind turbulences, sea waves and the earth's crust acting on large structures are characterized by the spatial correlation. In the present paper dynamic stability of one dimensional structures such as masts, towers, chimneys and suspension bridges subject to space and time dependent loading is investigated. Via Liapunov direct method sufficient stability conditions are derived. The stability conditions are expressed by geometrical and stiffness data, the viscous damping coefficient as well as intensities and correlation lengths of loading.

9

Stability of functionally graded plate under in-plane time-dependent compression

Tylikowski A.

Mechanics and Mechanical Engineering

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2004

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Vol. 7, nr 2

5-12

EN

Functionally graded materials have gained considerable attention in the high temperature applications. Linear dynamics equations taking into account a coupling of in-plane and transverse motions are used. Material properties are graded in the thickness direction of the plate according to volume fraction power law distribution. An oscillating temperature causes generation of in-plane time-dependent forces destabilizing the plane state of the plate equilibrium. The asymptotic stability and almost-sure asymptotic stability criteria involving a damping coefficient and loading parameters are derived using Liapunov's direct methods. Effects of power law exponent of the stability domains are studied.

10

Stability of a shaft rotating with fluctuating angular velocity

Tylikowski A.

Mechanics and Mechanical Engineering

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1999

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Vol. 3, nr 2

93--102

EN

In this paper the technique of the dynamic stability analysis proposed for the conven¬tional laminated structures is extended to the structures rotating with the time-dependent angular velocity. The rotating angle-ply symmetrically laminated circular cylindrical shell is treated as a beam-like structure. The shaft is subjected to a constant torque. The ve¬locity stochastic component is assumed in the form of the wide - band Gaussian processes modelled as a Wiener process. The fluctuating component of angular velocity implies a stochastic parametric excitation of shaft motion. The structure buckles dynamically when the axial parametric excitation becomes so large that the structure does not os¬cillate about the unperturbed state, and a new increasing mode of oscillations occurs. The uniform stochastic stability criteria involving a damping coefficient, a rotation speed and geometrical and material parameters are derived using Liapunov's direct method. Formulas determining dynamic stability regions are written explicitly.