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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 recent years, a new approach called transmissibility based operational modal analysis has been propounded.The new approach is able to identify the modal parameters of structural systems based on the transmissibility functions, where, unlike conventional methods of operational modal analysis, there is no limiting assumption about the input excitations. In this paper, an effective form of transmissibility called power spectral density transmissibility is used in order to identify the dynamic characteristics of a 5DOF system. The dynamic system that is modeled using MATLAB/Simulink is excited by the different earthquakes such as El Centro, Northridge and Loma Prieta, and white Gaussian noise is also added to its responses with different signal to noise ratios. The modal parameters (natural frequencies and mode shapes) of the numerical model are calculated and extracted based on the singular values and vectors obtained from singular value decomposition of the power spectral density transmissibility matrix. This matrix, unlike the Fourier spectral transmissibility matrix, can be created based on the transferring outputs obtained from just one loading condition; therefore, there is no need to use the outputs of multiple loading conditions, so that it is possible to identify the dynamic characteristics with only one dynamic test. In this research, the modal identification results are evaluated through comparison with the values obtained from exact solution of the system. The comparison shows that the modal parameters extracted from the system responses with different noise levels have a good agreement with the exact values.