Nonlinear vibrations of beams and plates with fractional derivative elements subject to combined harmonic and random excitations

This paper proposes an efficient approach for estimating reliably the second order statistics of the response of continua excited by combinations of harmonic and random loads. The problem is relevant in several engineering applications, where, for instance, the harmonic load is influenced by significant noise that cannot be neglected when computing the response statistics. The considered problems pertain to the vibration of beams and of plates endowed with fractional derivative elements. In both cases, it is shown that by representing the system response by the linear modes of vibration, systems of nonlinear fractional ordinary differential equations describing the time-dependent variation of the modes amplitudes are obtained. These equations are coupled and are treated by combining the harmonic balance and statistical linearization techniques, leading to the determination of the second-order statistics of the response. Relevant Monte Carlo data demonstrate the reliability of the proposed solution approach. The specific numerical examples considered pertain to simply supported beams, and plates with simply supported stress-free edges conditions.