In this paper, we develop a vibrating string model to describe the oscillations within a bio-mimetic movable pulley actuator, where transmission point disturbances can induce resonances, and so jeopardise system performance. The dynamics of longitudinal axial vibrations are formulated by a wave equation on a slowly time-varying spatial domain with a moving mass and a small harmonic disturbance at the mass-point. Due to the slow spatial domain variation, a singular perturbation problem arises. Utilizing the semigroup method, we establish the existence and uniqueness of the system’s solution. Through an averaging technique and an interior layer analysis, we construct formal asymptotic approximations for the solution. Our findings reveal that, for specific disturbance frequencies, many oscillation modes jump up from small order ε amplitudes to those of order ε. To suppress the resonances, we introduce viscous damping of varying orders. By employing multiple time-scales perturbation methods, we demonstrate that different orders of the viscous damping produce distinct anti-resonance results. Lastly, numerical simulations validate both the accuracy of our analytical results and the efficacy of the anti-resonance strategies employed.