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In this study random vibration of a cantilever tapered beam under distributed stationary stochastic excitation with Gaussian probability density function is investigated. early free vibration analysis is performed to obtain the mode shapes of beam in form of Bessel functions, then the response is described in summation of mode shapes, and auto correlation of response is shaped by considering the mode shapes of tapered beam, also spectral density matrix of excitation is derived with cooperation of mode shapes and two dummy variables. in next step by means of frequency response and taking Fourier integral of autocorrelation of response, spectral density of displacement is computed and by using spectral density of displacement, variance of random displacements for various positions along the beam are achieved. Finally elasticity equation is applied to derive random strain and stress of beam. Comparing the variance of random stress with yield stress of beam leads to obtain probability of beam failure.

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In this study a new approach for investigating the flutter speed of nonlinear aeroelastic systems is proposed. In this approach, the compatibility of nonlinear random vibration analysis based on the statistical properties of response is used and extended to the nonlinear aeroelastic systems to analyze the instability of these systems with using neither time domain analysis nor limit cycle oscillations. To this aim a 2-degree nonlinear airfoil with cubic torsional spring under quasi steady flow is considered as an aeroelastic system. At first, one random Gaussian white noise is added to the aerodynamic lift force then the statistical linearization and the random vibration analysis of the nonlinear systems are used to obtain a nonlinear map of response-variance with flow velocity as the control parameter. This nonlinear map leads to a nonlinear algebraic equation which consists of two parameters as the flow velocity and variance of the response. By solving this nonlinear equation for various flow velocities, the flutter speed is considered as the maximum of response-variance. Finally the jump phenomenon is also investigated where tangent bifurcation occurs.