Damped Free Response
Physically, there is no such a vibrating system that vibrates forever, that means there is always some kind of damping in the system that dissipates energy. For mathematical simplicity, the damping is modelled as viscous damping. Depending on the magnitude of damping, a damped system can be underdamped, critical damped or overdamped. Critical damping coefficient is determined by the system's mass and spring constant. Under critical damping, damping ratio is unity. critical damping separates nonoscillatory motion from oscillatory motion. When damping ratio is greater than 1, which is called overdamping, the system does not oscillate. for damping ration less than 1, which is called underdamping, the system oscillates with decaying magnitude,as shown in the figure below. For most physical system, damping ratios are less than 1. Actually, most physical systems have damping ratio less than 0.1. With damping in the free vibration system, the mass always restores its equilibrium position even it is disturbed. The greater the damping, the less time it takes to restore its equilibrium position. So in most cases, adequate damping is desireable.
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