The uncertainty quantification in the dynamic response of structures under uncertainties in the geometric and material properties as well as the randomness in the external loading is discussed. The dynamics analysis of the wind turbine blades can be performed using the proposed formalism when the finite element discretization of the corresponding cantilever beam model is further reduced based on the associated spectral behavior. The impact of the uncertainties in the geometry and material properties on the spectral property of the beam is quantified using a computationally efficient solution algorithm for the stochastic eigenvalue problem. The algorithm extends the ideas of power iteration and subspace iteration to the stochastic operators, based on which the a lower dimensional dominant subspace is constructed based on the random eigenvalues and eigenvectors are calculated which subsequently facilitates the computation of the random response. Numerical results are then shown to verify the efficiency of the proposed framework.