Question

For a single-degree-of-freedom system, if the damping ratio is critically damped, what is the specific characteristic of its free vibration response when subjected to an initial displacement?

Accepted Answer

For a single-degree-of-freedom system, which is a mechanical system whose motion can be completely described by a single coordinate (for example, a simple mass-spring-damper system moving along one direction), the damping ratio (represented by the Greek letter zeta, ζ) is a dimensionless parameter that describes how oscillations in a system decay after a disturbance. When the damping ratio is critically damped, it specifically means that ζ = 1. This condition represents the precise amount of damping required to bring the system back to its equilibrium position as quickly as possible without oscillating. The equilibrium position is the stable resting position where the net force on the system is zero. When subjected to an initial displacement, meaning the system is moved from its equilibrium position and then released, the specific characteristic of its free vibration response is that the system does not oscillate. Instead, it moves smoothly and directly back towards the equilibrium position without crossing it or undergoing any repeated back-and-forth motion. The system achieves the fastest possible return to equilibrium without any overshoot or oscillation, asymptotically approaching the equilibrium position over time. This response is a non-oscillatory, exponential decay of the displacement.

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For a single-degree-of-freedom system, if the damping ratio is critically damped, what is the specific characteristic of its free vibration response when subjected to an initial displacement?