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3 Tylikowski A.

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

100%

Tylikowski A.

tom Vol. 33, No. 1

104-113

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.

Active damping of geometrically nonlinear transverse beam vibrations

100%

Tylikowski A.

tom Vol. 5, nr 2

127-135

The paper is concerned with the stabilization of an elastic beam nonlinear geometrically subjected to a time-dependent axial forcing. The direct Liapunov method is proposed to establish criteria for the almost sure stochastic stability of the unperturbed (trivial) solution of the structure with closed-loop control. We construct the Liapunov functional as a sum of the modified kinetic energy and the elastic energy of the structure The distributed control is realized by the piezoelectric sensor and actuator, with the changing widths, glued to the upper and lower beam surface. The paper is devoted to the stability analysis of the closed-loop system described by the stochastic partial differential equation without a finite-dimensional approach. The fluctuating axial force is modelled by the physically realizable ergodic process. The rate velocity feedback is applied to stabilize the panel parametric vibrations. Calculations are performed for the Gaussian process with given mean value and variance as well as for the harmonic process with an amplitude A.

Dynamic stability of carbon nanotubes

100%

Tylikowski A.

tom Vol. 10, nr 1

160-166

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.