Stochastic Response of an Inclined Shallow Cable with Linear Viscous Dampers under Stochastic Excitation

Considering the coupling between the in-plane and out-of-plane vibration, the stochastic response of an inclined shallow cable with linear viscous dampers subjected to Gaussian white noise excitation is investigated in this paper. Selecting the static deflection shape due to a concentrated force at the dampers location and the first sine term as shape functions, a reduced four-degree-of-freedom system of nonlinear stochastic ordinary differential equations are derived to describe dynamic response of the cable. Since only polynomial-type terms are contained, the fourth-order cumulant-neglect closure together with the C-type Gram-Charlier expansion with a fourth-order closure are applied to obtain statistical moments, power spectral density and probabilistic density function of the cable response, whose availability is verified by Monte Carlo method. Taking a typical cable as an example, the influence of several factors, which include excitation level and direction as well as damper size, on the dynamic response of the cable is extensively investigated. It is found that the sum of mean square in-plane and out-of-plane displacement is primarily independent of the load direction when the excitation level and viscous coefficient of the damper are fixed. Moreover, the peak frequency and half-band width of the spectra of both the in-plane and the out-of-plane displacements are increasing with excitation level when the damper size is constant. It is also observed that, even though the actual optimal damper size is slightly greater than the one obtained by the complex modal theory, the difference of statistical moment of the cable caused by these two damper size is negligible, so the vibration reduction effect provided by the theoretical optimal viscous coefficient is satisfactory.