The overall response of a hoisting cable system to an inertial load due to transport motion acceleration and deceleration and to external harmonic excitation is studied. A mathematical model of the system is proposed, and the response is simulated numerically. Due to the time-varying length of the cable, the natural frequencies of the system vary slowly, and a transient resonance occurs when one of the frequencies coincides with the frequency of the excitation at some critical time. Near the resonance region, the resonant mode is dominant, and a single-mode approximation of the system response in this region is formulated. A combined perturbation and numerical technique is applied to predict the response during passage through the main resonance.