Lecture 31 : Basics of soil dynamics [ Section 31.3 :
Different modes of vibration; Damped ]
Damped free vibration system
Refer fig.
The governing equation is,
where,
m is the mass of the system,
k is the stiffness of the system,
x is the displacement of the system , is the velocity and, is the acceleration.
Fig. 8.8 Damped free vibration system
Solving as a initial value problem, the boundary conditions are,
The initial displacement and velocity is given as,
Where, is the displacement at time (t) =0 andis the velocity at time (t)=0.
Substituting in the above equation, , i.e. undamped natural frequency.
The solution is in the form of x(t) =
, is the velocity and, 
is the acceleration. 
is the displacement at time (t) =0 and
is the velocity at time (t)=0. 
, i.e. undamped natural frequency. 
