An iterative successive linearization approach based on energy optimization is presented to estimate the response statistics of a moderately largely vibrating beam endowed with a fractional derivative, subject to combined harmonic and random excitations. The presented method is developed by constructing two sets of optimized linear systems, so that the responses of the new system are equivalent to those of the original system in the sense of average and statistical characteristics. Specifically, the iterative deterministic and statistical linearization methods are utilized successively, based on optimizations of the average and mean square difference between the potential energies associated with the original systems and their linear counterparts. Numerical examples are given with Monte Carlo data to demonstrate the reliability and accuracy of the presented method.
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