Damped System
Vibration systems may encounter damping of following types:
(i) Internal molecular friction.
(ii) Sliding friction
(iii) Fluid resistance
Generally mathematical model of such damping is quite complicated and not suitable for vibration analysis.
Simplified mathematical model (such as viscous damping or dash-pot) have been developed which leads to simplified formulation.
A mathematical model of damping in which force is proportional to displacement i.e., Fd = cx is not possible because with cyclic motion this model will encounter an area of magnitude equal to zero as shown in figure. So dissipation of energy is not possible with this model.
The damping force (non-linearly related with displacement) versus displacement curve will enclose an area, it is referred as the hysteresis loop, that is proportional to the energy lost per cycle.
Viscously damped free vibration :
Viscous damping force is expressed as,
c is the constant of proportionality and it is called damping co-efficient.
From free body diagram, we have
Let us assume a solution of the following form
where s is a constant (can be a complex number) and t is time.
So that 
and 
, on substituting in the equation of motion, we get,
From the condition that x is a solution for all values of t, above equation gives a characteristic equation (Frequency equation) as
The above equation has the following form
solution of which is given as 
Hence, solution of can be written as
Hence the general solution is given by the equation
where A and B are integration constants to be determined from initial conditions
Substituting.
The term outside the bracket in RHS is an exponentially decaying function. The term
can have three cases.
(i) then exponents will be real numbers.
a. No oscillation is possible
b. This is an overdamped system
Hence the equation takes the following form
(iii) Critical case between oscillatory and non-oscillatory motion :
Damping corresponding to this case is called critical damping, cc
Any damping can be expressed in terms of the critical damping by a non-dimensional number S called the damping ratio
S = c/cc
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