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Several recent product developments for vibration and motion control have needed passive viscous damping, in addition to traditional elastomer-based hysteretic damping, to be successful in their respective applications. In addition to attenuating steady-state vibration, an important function of these recent product developments is to control motion from impulsive or mechanical shock input. Examples are the cab mounts of off-highway vehicles that need damping in the vertical direction to control cab motion from ground input through the vehicle and some torsionally flexible couplings that need damping to control torque spikes from shift shocks or other transient events. In this work, the theoretical damped impulse response quantities of displacement, velocity, acceleration, force, jerk, yank, and jounce are investigated. This work shows that, for certain response quantities, there is a specific magnitude of damping that minimizes response from impulsive or mechanical shock input.