Response of MDOF nonlinear system with fractional derivative damping and driven by fractional Gaussian noise

The integration of fractional calculus into stochastic dynamics has introduced two innovative concepts, i.e., fractional derivative damping (FDD) and fractional Gaussian noise (fGn). To date, most studies on fractional stochastic dynamics have exclusively focused on either FDD or fGn, rather than concurrently exploring both. It is the first attempt for this paper to investigate a multi-degree-of-freedom (MDOF) nonlinear system with FDD and driven by fGn at the same time. Employing the generalized harmonic balance technique, FDD is equivalently decomposed into a quasi-linear restoring force and a quasi-linear damping force. Consequently, the original random Lagrangian system with FDD is transformed into the stochastically excited and dissipated Hamiltonian system. Utilizing the characteristic of the relatively flat power spectral density of fGn, it is approximated as wideband noise. This enables the application of the stochastic averaging method for quasi-integrable Hamiltonian system under wideband noise. Two detailed examples are carried out to provide the technical procedures and numerical verifications.