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1

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.

2

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.

3

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.

4

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.