Stochastic responses of nonlinear inclined cables with an attached damper and random excitations

Flexible and lightweight cables are extensively used in engineering structures, which are prone to produce nonlinear deformation and nonlinear vibrations under random excitations. Usually, dampers are installed at certain positions of the cables to reduce the vibration response. The present paper investigates the stochastic responses of the nonlinear inclined cables with an attached damper under Gaussian white noise and wide-band noise excitations. First, the dynamical model of an inclined cable is established and the differential equations for each mode of vibration are derived by using Galerkin’s discretization method. Then, the stochastic linearization method is applied to derive the stochastic responses of the generalized displacements. The effectiveness of the truncated order, the effects of the excitation amplitude, damper installation position and damping coefficient are studied by investigating the stochastic responses. Since stochastic linearization is not applicable to systems with strong nonlinearity, stochastic averaging of energy envelope and quasi-Hamiltonian systems are adopted to study the main modal vibration of the inclined cables. The probability density functions of energy and generalized displacement are calculated. The comparisons between the results derived from the theoretical method and those derived from the numerical simulation showed the accuracy of the analytical results.