Free Vibration with Viscous Damping

Lecture 3
Contents
1. Types of damping
2. Free Vibration with Viscous Damping
3. Critical Damping Constant and the Damping Ratio
1. Types of damping
a) Viscus damping
b) Coulomb (dry friction) damping
c) Structural damping
2. Free Vibration with Viscous Damping
The viscous damping force F is proportional to the velocity ??? or v and can be expressed as ??=??????
where c is the damping coefficient and ??? is the velocity and F is the damping force.
3. Critical Damping Constant and the Damping Ratio
The critical damping constant ???? can be calculated as follow; ????=2???????=2?????? ??=??????
??2??= ?????? ??= 2 ?? ??????
The solution to the equation of motion as a function of ?? is as follows;
??1,2= (???±???2?1) ???? ??(??)=??1??(? ?? + ???2?1)??????+??2??(? ?? ? ???2?1)??????

The nature of the roots ??1 and ??2 and the behaviour of the solution for the equation of motion for damped free vibration, depends upon the magnitude of damping. It can be seen that the case ??=0 leads to the undamped vibrations discussed in previous lecture. Hence we assume that ???0 and consider the following three cases.
Case 1) Underdamped vibration
when ?? < 1
??24??2c = 2 ?? m????
????=????? ????2=???? ??=????