The response of a secondary system mounted on a torsionally coupled yielding primary system under white-noise excitation is presented. The main objective is to examine the effect of rotary coupling and non-linearity of the primary structural system on the response characteristics of the linear secondary system in a stochastic environment. The responses are obtained by the frequency domain spectral analysis using a linearisation technique. The behaviour of the S-system is examined under a set of parametric variations. These parameters include the uncoupled lateral frequency, the eccentricity ratio, the ratio of uncoupled lateral to rotational frequencies, the damping ratio, and the hysteretic parameters for the primary structure. Also, they include the frequency, the damping ratio, the normalised eccentricity for the secondary system, and the mass ratio of the two systems. The results of the study indicate that the yielding of the primary system may result in an increased response of the secondary system under certain conditions, and that the rotary coupling of the primary system can have a significant influence on the responses of the secondary system.