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The dynamic instability problem of coupled nanobeam system subjected to compressive axial loading is investigated. The paper is concerned with the stochastic parametric vibrations of nanobeams based on Eringen's nonlocal elasticity theory of Helmholtz and bi-Helmholtz type of kernel and Euler-Bernoulli beam theory. Each pair of axial forces consists of a constant part and a time-dependent stochastic function. By using the direct Liapunov method, bounds of the almost sure asymptotic instability of a coupled nanobeam system as a function of viscous damping coefficient, stiffness of the coupling elastic medium, variances of the stochastic forces, scale coefficients, and intensity of the deterministic components of axial loading are obtained. Analytical results are verified with the numerical results obtained from the Monte Carlo simulation. Numerical calculations are performed for the Gaussian process with a zero mean as well as a harmonic process with random phase. [ABSTRACT FROM AUTHOR]

The moment Lyapunov exponents and Lyapunov exponent of a two-dimensional system under stochastic parametric excitation are studied. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises. Approximate analytical results for the pth moment Lyapunov exponents are compared with the numerical values obtained by the Monte Carlo simulation approach. The results are applied to the study of the almost-sure and the moment stability of the stationary solutions of the elastic beam subjected to the stochastic axial load. The boundaries of the almost-sure and the moment stability of the elastic beam as the function of the damping coefficient, spectral density of the stochastic force and mode number are obtained. [ABSTRACT FROM AUTHOR]