Single Degree of Freedom, Free Damped Vibration
From ControlTheoryPro.com

Contents
 1 Introduction to Single Degree of Freedom, Free Damped Vibration
 2 Viscous Damping^{[1]} for Free Damped Vibration
 3 Coulomb (Dry Friction) Damping^{[2]} for Free Damped Vibration
 4 Hysteretic Damping^{[3]} for Free Damped Vibration
 5 Equations for Single Degree of Freedom, Free Undamped Vibration^{[4]}
 6 Notes
1 Introduction to Single Degree of Freedom, Free Damped Vibration
Undamped systems make for good educational examples. Real systems always dissipate energy as they move. This dissipation of energy dampens the system motion. If the damping is very small then an undamped system may be accurately approximated by an undamped system.
Energy dissipation occurs through friction. Often multiple kinds of friction are present but in each operation mode for a system a certain form of friction is dominant. The forms of friction are viscous, Coulomb, and hysteretic.
2 Viscous Damping^{[1]} for Free Damped Vibration
3 Coulomb (Dry Friction) Damping^{[2]} for Free Damped Vibration
4 Hysteretic Damping^{[3]} for Free Damped Vibration
5 Equations for Single Degree of Freedom, Free Undamped Vibration^{[4]}
Givens 

Constant Mass 
Spring stiffness is Constant 
Spring mass is negligible 
6 Notes
Beards, C. F. 1995 Engineering Vibration Analysis with Applications to Control Systems. ISBN 034063183X