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Damped Free Response

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Physically, there is no 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 modeled as viscous damping.
Depending on the magnitude of damping, a damped system can be underdamped,
critically damped or overdamped. The critical damping coefficient is determined
by the system's mass and spring constant. Under critical damping, the damping
ratio is unity. Critical damping separates nonoscillatory motion from oscillatory
motion. When the damping ratio is greater than 1, which is called overdamping,
the system does not oscillate. For a damping ratio 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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DANIEL J. INMAN, Engineering Vibration, Prentice Hall, Englewood Cliffs,
New Jersey,1994