Free Vibration of
damped SDOFS
16MECH80H
DR. MOHAMED LOTFY
Free Vibration of damped SDOFS
The equation of motion of a single degree of freedom system was
derived to be:
It can be written in the form
where
Types of motions
The solution of the second order differential equation with damping
term can be classified into three types of motion according to the
damping ratio:
1- Critically damped response
2- Under damped response
3- Overdamped response
Solution of EOM
Underdamped Systems
The characteristic equation of underdamped free vibrating single
degree of freedom systems is:
Solving the quadratic equation gives:
The solution of the D.E. was assumed to be:
Substituting with roots of C.E. in the E.O.M gives:
Which can be rewritten as:
Applying Initial conditions, the constants are
calculated to be.
Underdamped Systems (ctd.)
Substituting with constants into the response equation gives:
where is the damped frequency
and
Effect of damping on natural
frequency
For light damped structures, the damped frequency can be
approximately considered as same as natural frequency of the
undamped system
Logarithmic Decrement
Decay of motion
The coefficient of damping in systems can be estimated using the
amplitude of two successive peaks as follows:
Logarithmic Decrement
Decay of motion (ctd.)
In practical applications, it is very hard to measure an amplitude
difference between two successive peaks.
An alternative expression relating peaks sebarated by j cycles can
be derived as follows:
As a rule of
thumb, if the
damping ratio
of a system is
0.1, The
response of the
system will
reach 50% of its
max.
amplitude in
only one cycle
Overdamped Systems
When the damping ration of the system is greater than one, the
characteristic equation roots are:
Where
Recalling the general assumed response:
The dynamic response of an overdamped system is as follows:
