Equivalent linearization technique in non-linear stochastic dynamics of a cable-mass system with time-varying length

In this paper the transverse vibrations of a vertical cable carrying at its lower end a concentrated mass and moving slowly vertically within the host structure are considered. It is assumed that longitudinal inertia of a cable can be neglected, with the longitudinal motion of the concentrated mass coupled with the lateral motion of the cable. An expansion of the lateral displacement of a cable in terms of approximating functions is used. The excitation of vibrations of a cable-mass system is a base-motion excitation due to the sway motion of a host tall structure. Such a motion of a structure is assumed to result from action of the wind, hence it may be adequately idealized as a narrow-band random process. The narrow-band process is represented as the output of a system of two linear filters to the input in form of a Gaussian white noise process. The non-linear problem is dealt with by an equivalent linearization technique, where the original non-linear system is replaced with an equivalent linear one, whose coefficients are determined from the condition of minimization of a mean-square error between the non-linear and the linear system. The mean value and variance of the transverse displacement of the cable as well as those of a longitudinal motion of the lumped mass are determined with the aid of an equivalent linear system and compared with the response of the original non-linear system subjected to the deterministic harmonic excitation.