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The dynamic stability problem of a Timoshenko beam supported by a generalized Pasternak-type viscoelastic foundation subjected to compressive axial loading, where rotary inertia is neglected, is investigated. Each axial force consists of a constant part and a time-dependent stochastic function. By using the direct Liapunov method, bounds of the almost sure asymptotic stability of a beam as a function of viscous damping coefficient, variance of the stochastic force, shear correction factor, parameters of Pasternak foundation, and intensity of the deterministic component of axial loading are obtained. With the aim of justifying the use of the direct Liapunov method analytical results are firstly compared with numerically obtained results using Monte Carlo simulation method. Numerical calculations are further performed for the Gaussian process with a zero mean as well as a harmonic process with random phase. The main purpose of the paper is to point at significance damping parameter of foundation on dynamic stability of the structure. [Project of the Serbian Ministry of Education, Science and Technological Development, Grant no. 174011]

The stochastic stability problem of a viscoelastic Voigt-Kelvin balanced rotating shaft subjected to action of axial forces at the ends is studied. The shaft is of circular cross-section, it rotates at a constant rate about its longitudinal axis of symmetry. The effect of rotatory inertia of the shaft cross-section and external viscous damping are included into account. The force consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, external damping coefficient, retardation time, angular velocity, and geometric and physical parameters of the shaft. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ2 as well as for harmonic process with amplitude H.

The dynamic stability problem of the thin-walled beams subjected to end moments is studied. Each moment consists of constant part and time-dependent stochastic non-white function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability condition is obtained as function of stochastic process variance, damping coefficient, geometric and physical parameters of the beam. The stability regions for I-cross section and narrow rectangular cross section are shown in variance - damping coefficient plane when stochastic part of moment is Gaussian zero-mean process with variance σ2 and harmonic process with amplitude A.

Abstract: The dynamic stability problem of the symmetrically laminated cross-ply plates made of the viscoelastic Voigt–Kelvin material, compressed by time-dependent stochastic membrane forces, is investigated. The effect of rotatory inertia is included in the present formulation. It is assumed that all elastic moduli have the same retardation times. By using the direct Liapunov method, bounds of the almost sure stability of cross-ply plates as a function of viscous damping coefficient, retardation time, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, cross-ply ratio and intensity of the deterministic components of axial loading are obtained. Numerical calculations are performed for the Gaussian process with a zero mean, as well as an harmonic process with random phase. [Copyright &y& Elsevier]

Abstract: The dynamic instability problem of symmetrically laminated cross-ply plates, compressed by time-dependent stochastic membrane forces, is investigated. The effect of shear deformation is included in the formulation. By using the direct Liapunov method, bounds of the almost sure instability of cross-ply plates are obtained. Furthermore, it is shown how the viscous damping coefficient, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, and cross-ply ratio influence the instability regions. A special attention is given to the proper choice of two shear correction factors of the laminated plate. Numerical calculations are performed for the Gaussian process with a zero mean and variance as well as for the harmonic process with an amplitude A. [Copyright &y& Elsevier]

In this paper the stochastic stability problem of a moving elastic band subjected to action in-plane acting forces is investigated. Each force consists of a constant part and a time-dependent zero mean stochastic function. By using the direct Liapunov functional method, almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, damping coefficient, and geometric and physical parameters of the band. Numerical calculations are performed for infinite mode and compared with known results. Almost sure stability regions are shown for infinite and first mode the two-dimensional density probability function, and for higher modes when the, edge load Gaussian or harmonic process is known. [ABSTRACT FROM AUTHOR]

Abstract: The dynamic stability problem of a viscoelastic Voigt–Kelvin rotating shaft subjected to action of axial forces at the ends is studied. The shaft is of circular cross-section, it rotates at a constant rate about its longitudinal axis of symmetry. The effect of rotatory inertia of the shaft cross-section is included in the present formulation. Each force consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, retardation time, angular velocity, and geometric and physical parameters of the shaft. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ 2 as well as for harmonic process with amplitude H. [Copyright &y& Elsevier]

Abstract: The stochastic instability problem associated with an axially loaded cross-ply laminated beam is formulated. The effects of shear deformation and rotatory inertia are included in the present formulations. The beam is subjected to time-dependent deterministic and stochastic forces. By using the direct Liapunov method, bounds for the almost sure instability of beams as a function of viscous damping coefficient, variance of the stochastic force, ratio of principal lamina stiffnesses, shear correction factor, number of layers, mode numbers and geometrical ratio, are obtained. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ 2 as well as for harmonic process with an amplitude A. [Copyright &y& Elsevier]